Foundations of Mechanics of Materials Part 1 [PDF]

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Jeremiah Rushchitsky



Foundations of mechanics of materials: Part 1



JEREMIAH RUSHCHITSKY



FOUNDATIONS OF MECHANICS OF MATERIALS: PART 1



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Foundations of mechanics of materials: Part 1 1st edition © 2021 Jeremiah Rushchitsky & bookboon.com ISBN 978-87-403-3706-8 Peer review by Prof. Surkay Akbarov, Prof. Yaroslav A. Zhuk, Prof. Volodymyr Zozulya



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Contents



CONTENTS Foreword



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1 Basic information on mechanics



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2 Basic information on mechanics of materials. Theory of elasticity. Short description of linear theory of elasticity



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3 Some additional fundamental facts from the linear theory of elasticity



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4 Short description of nonlinear theory of elasticity. Part 1



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5 Short description of nonlinear theory of elasticity. Part 2



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6 Short description of linearized theory of elasticity



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7 Short description of strength of materials



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8 Focus on composite materials. Different models of elastic deformation



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9 Focus on composite materials. Structural model of elastic mixtures



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10 Focus on new materials Nanomechanics of materials



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11 Focus on new materials. Mechanics of auxetic materials



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Foreword



FOREWORD On the auditory. Goals of the book presented. Structure of this book. Three basic parts of the book. The most known classical book (text-book) on mechanics of materials. The most popular books (text-books) on mechanics of materials. Structure of the single chapter. On comments. On bibliography. On questions.



The proposed 20 chapters are created as an answer to the very big request for the courses on the mechanics of materials. This request is displayed in the sufficiently great number of proposed books, their supernormal volume (a very great number of pages), the constant interest of publishers to such books, a great number of republications of the popular in academic community books on mechanics of materials as well as a presence of the course “Mechanics of Materials” in the universities of all continents. The presented short course is intended as the small addendum to the imposing size classical books on the mechanics of materials. The chapters let know of the hearer-reader in brief terms the information on fundamentals of mechanics of materials and have for an object to help him to systemize of his knowledge in the area of modern mechanics of materials, to understand the common back-ground and interdependence of models and theories of the modern mechanics of materials. It seems to be natural that, first of all, the classical book-textbook “Timoshenko S.P. Gere J.M. (1972) Mechanics of Materials. Van Nostrand Reinhold Company, New York” is the world standard for any author. Seemingly, this is not the accidental fact that the author of the proposed chapters is working for the last fifty years just in the area of mechanics of materials at the Institute of Mechanics named after S.P. Timoshenko - the founder and first director of this institute (1918). In this book, the fundamental assumptions, the primary notions, and the basic models and theories of mechanics of materials are stated and commented on. The proposed in the chapters information has differed essentially from the information that is proposed in the classical and modern books on the mechanics of materials and complements them just in the fundamental aspects. First, it is concise and contains 276 pages only. Second, it is based on understanding the mainline of studying the mechanics of materials that consists of conditionally of four parts using the more and more complicated approaches and models – the strength of materials, the linear theory of elasticity, the linearized theory



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Foreword



of elasticity, the nonlinear theory of elas ticity. At that, the focus is made on the nonlinear theory of elasticity as the most adequate theory. Third, it includes the foundations of theories that reflect other basic properties of materials  – thermoelasticity, viscoelasticity, plasticity, elastoplasticity, piezoelasticity, magnetoelasticity, diffusional elasticity. The structure of this book is as follows: PART I Chapter 1. Basic information on mechanics. Chapter 2. B  asic information on mechanics of materials. Theory of elasticity. Short description of linear theory of elasticity. Chapter 3. Some additional fundamental facts from the linear theory of elasticity Chapter 4. Short description of nonlinear theory of elasticity. Part 1. Chapter 5. Short description of nonlinear theory of elasticity. Part 2. Chapter 6. Short description of linearized theory of elasticity. Chapter 7. Short description of strength of materials. Chapter 8. Focus on composite materials. Different models of elastic deformation Chapter 9. Focus on composite materials. Structural model of elastic mixture. Chapter 10. Focus on new materials. Nanomechanics of materials. Chapter 11. Focus on new materials. Mechanics of auxetic materials. Each chapter is finished by comments, bibliography that includes the list of basic and additional books and list of questions.



The most known classical book (text-book) on mechanics of materials



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Foreword



Timoshenko S.P., Gere J.M. (1972) Mechanics of Materials. Van Nostrand Reinhold Company, New York. 670 p.



Gere J.M., Goodno B.J. (2012) Mechanics of Materials. 8th edition. Cangage Learning Custom Publishing, Stanford. 1152 p. The most popular books (text-books) on mechanics of materials



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Foreword



Bedford A.M., Liechti K.M. (2020). Mechanics of Materials. Springer, Berlin. 1039 p.



Handbook of Mechanics of Materials (2019) Editor-in-Chief Hsueh C-H. Springer, Berlin. 2464 p.



Muvdi B.B, Elhouar S. (2016) Mechanics of Materials. With Application in Excel. CRC Press, Boca Raton. 723 p



Vable M. (2015) Advanced Mechanics of Materials. 2nd edition. Michigan Technological University. 594 p.



Ghavami P. (2015) Mechanics of Materials. An Introduction to Engineering Technology. Springer, Berlin. 260 p.



Beer F.P., Johnson Jr E.R., De Wolf J.T., Mazurek D.F. (2014). Mechanics of Materials. McGraw Hill Education, New York. 896 p.



Janco R., Hucko B. (2013). Introduction to Mechanics of Materials. Part I. Part II. Ventus Publishing ApS, Copenhagen (free text-book in BookBooN.com) 160 p. 234 p.



Gupta V. (2013) An Introduction to Mechanics of Materials, Alpha Science Int., 500p.



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Foreword



De Silva W. (2013) Mechanics of Materials (Computational Mechanics and Applied Analysis). CRC Press, Boca Raton. 466p.



Hibbeler R.C. (2011) Mechanics of Materials. 10th edition in 2017. Prentice Hall, New York. 910 p.



Allen J.H. Mechanics of Materials for Dummies (2011) Wiley, New York. 384p.



Kiusalaas J., Pytel A. (2010). Mechanics of Materials. 2nd edition, Cangage Learning Custom Publishing, Stanford. 576 p.



Vable M. (2009) Mechanics of Materials. 2nd edition. Michigan Technological University. 594 p.



Philpot T.A. (2008). Mechanics of Materials: An Integrated Learning System. John Wiley & Sons, New York. 744 p.



Urugal A.C. (2007). Mechanics of Materials: An Integrated Approach. John Wiley & Sons, New York. 736 p.



Bedford A.M., Liechti K.M. (2000). Mechanics of Materials. Prentice Hall, New York. 627 p.



Urugal A.C. (1995). Mechanics of Materials: An Integrated Approach. John Wiley & Sons, New York. 736 p.



Popov E. (1976). Mechanics of Materials. 2nd edition. Prentice Hall, New York. 590 p.



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Foreword



Each chapter-lecture contains at the end the comments to problems considered, the bibliography (the list of books and original articles on the chapter subject for further reading), and the list of question, which will enable the reader to turn to the cited books and to study more deeply some aspect of the chapter. Comments are concentrated mainly on fragments not reflected sufficiently in the chapter-lecture and important for the in-depth study. The bibliography is intended to show the wealth of the problems in hand (mainly, the theoretical models problems, and in a few chapters, only), on the one hand, and to help in the in-depth study, on the other hand. The questions are the main goal to formulate the staring point for in-depth discussion of some aspects of the chapter. The depth of discussion will depend on the reader and his intentions. The genesis of this book can be found in the author’s years of research and teaching while heading of department at S.P. Timoshenko Institute of Mechanics (National Academy of Sciences of Ukraine), a member of the Center for Micro and Nanomechanics at Engineering School of University of Aberdeen (Scotland) and a professor at Physical-Mathematical Faculty of the Igor Sikorsky National Technical University of Ukraine “KPI”.



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Basic information on mechanics



1 BASIC INFORMATION ON MECHANICS Mechanics as science. Definition of mechanics. Antique roots of mechanics. Mecha-nics as a precision science, having the axiomatics. Material body. Material. States of material substance. Classification of mechanics by different attributes. Different classifications of mechanics. The classic division. Continuity. Continuum mechanics. Diffe-rent divisions of mechanics. Classifiers of mechanics.



Definition 1.1. Mechanics is the science on the equilibrium of mechanical motion of material bodies as a change over time in the space of the mutual position of material bodies or their parts under action of forces (in the modern interpretation, of not only the forces of mechanical nature: for example, forces of electromagnetic or diffusion nature, temperature). The word mechanics comes from the Greek word “ μηχανική – mechanics”, which in turn comes from another Greek word “μηχανή – an instrument, a building.” In ancient Greece, the words “μηχανική – mechanics” and “τηχνή – technique” meant “the art of constructing machines”. Mechanics in its original form (ANTIQUE MECHANICS) was initiated by two needs: works in construction (sometimes quite grandiose at that time) and the auxiliary mechanisms necessary for these works and the problem of interaction between the moving solids and water (air) during navigation (flight). Thus, it has been so happened historically that in all stages of the development of mankind, starting with the ancient world, mechanics and engineering business in many cases were considered as a whole. Therefore, the specificity of mechanics as a science is that it is a science of a fundamental nature and, at the same time, the relevance of mechanics is determined by the significance of its problems for engineering. Simultaneously, mechanics is an exact science in the sense that it is extremely active in mathematics in its theoretical constructs and the experiments play a crucial role in the development and testing of theories. A precision of mechanics as the physical science was caused by the interest of the famous mathematician David Gilbert, who formulated his SIXTH PROBLEM in the following way: “Mathematical consideration of the axioms of physics. A study of the foundations of geometry suggests the following problem: Consider by the same way, with the help of axioms, those physi- cal sciences in which mathematics is already playing an important role today; above all, they are the theory of probability and mechanics.“ 11



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Basic information on mechanics



The Gilbert’s problems were voiced in 1900 at the 2nd International Congress of Mathematicians. In the 2nd half of the 20 century, Walter Noll and Clifford Truesdell built the AXIOMATICS OF MECHANICS (6 axioms concerning the material body, 3 axioms concerning mass, 4 axioms concerning force, 3 axioms on reference systems, 2 axioms on inertia, 2 axioms on energy). This allowed C. Trusdell to assert that the analytical mechanics in the sense that it is that part of mechanics, which is not the computational or experimental one, can be called the rational mechanics. The term “rational” is interpreted here in the sense of Lagrange’s concept of rationality. Thus, mechanics belongs to a narrow circle of sciences that have their own axiomatics. The primary concept in mechanics ​​is the notion of the material body. Definition 1.2. The material body (material substance, physical substance, substance) is defined as a collection of discrete formations (atoms, molecules, and more complex formations of them), having the mass of rest and occupying the part of space. The word ”material” has a Latin origin: “material” means a tree (as a substance), material, substance.



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FOUNDATIONS OF MECHANICS OF MATERIALS PART 1



Basic information on mechanics



THE AGGREGATE AND PHASE STATES OF MATTER (SUBSTANCE) ARE DISTINGUISHED. Mechanics describes the mechanical motion of matter for all its states. The aggregate states are divided into four types: GASEOUS, PLASMA, SOLID, LIQUID. The criterion for differing these states is the presence of different types of motion of discrete formation and density of packaging (distances among the formations).



THE GASEOUS AGGREGATE STATE allows the translational, rotary, and oscillatory motions of formations (atoms, molecules). In this state, the distances between the formations are large (the packing density of the formations is small).



THE PLASMA AGGREGATE STATE is formed only by the atomized gas with an equal number of positive and negative charges. This is one of the options for the gaseous state. It stands out separately only because the substance in the universe is composed just of the plasma. At that, the plasma is divided into four types - the plasma itself, Bose- Einstein condensate, fermion condensate, and quark-gluon plasma.



THE SOLID AGGREGATE STATE allows only the oscillatory motion of formations around fixed centers of equilibrium with frequencies 10131014 oscillation per second. There are no translational and rotary motions in it. In this state, the distances between the formations are small, or, in other words, the packing density of the formation is large.



THE LIQUID AGGREGATE STATE by the nature of the motions of formations are close to the gaseous state and by the nature of the packaging to the solid state.



The phase states are divided into three types: GASEOUS, CRYSTALLINE, LIQUID. The criterion for differing these states is the order in the mutual placement of formations.



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THE GASEOUS AGGREGATE AND PHASE STATES almost coincide.



Basic information on mechanics



THE CRYSTALLINE PHASE STATE is characterized by the “far” order in the placement of molecules (formations), when the order is maintained at distances exceeding the size of the molecules (formations) by many hundred times. Here the ordering is observed at the long distances. THE SOLID AGGREGATE STATE corresponds to two phase states - CRYSTALLINE AND GLASSY.



THE LIQUID PHASE STATE is a state of “near” order in the placement of molecules (formations), when the order is saved at distances of several molecules (formations), and ordering is observed only in the immediate neighborhood of the molecule, and further. the placement is unpredictable. For this state, also the term “AMORPHOUS PHASE STATE” is used. It can be liquid or solid (glassy). The solid amorphous phase state is significantly different from the liquid amorphous phase state. Therefore, it is sometimes isolated as the separate phase state, and such substances are called the GLASSES.



Mechanics as science clearly takes into account in their models the differences in the states of matter. The motion of a substance in a liquid state (aggregate and liquid phase) and a gaseous state (aggregate and phase) is studied by mechanics of liquid, gas, and plasma (fluid mechanics). The motion of a substance in the solid aggregate state is studied by mechanics of materials (solid mechanics) since the materials themselves are understood as a substance in the solid aggregate state. The property of a material to have the mass and shape is described by the notion of hardness, which is ability of the material of a certain mass and shape to retain its shape or to prevail over this shape in comparison with other possible shapes. Mechanics distinguishes matter according to the criterion of form change and is divided by this criterion into four groups. Here, two basic notions are first used: the material point and the rigid body.



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Basic information on mechanics



Definition 1.3. The material point is the body having a mass and size of which is neglected in comparison with the distance to which it is displaced. Definition 1.4. The rigid body is the body that does not change its dimensions with the given shape (does not deform). THE GROUP 1 includes mechanics of the material point, THE GROUP 2 includes mechanics of the systems of material points, THE GROUP 3 includes mechanics of the rigid body, and THE GROUP 4 includes the substances that change their form (are deformed). THE GROUP 4 is traditionally divided into two main sections: mechanics of liquid, gas, and plasma (fluid mechanics) and solid mechanics as well as a series of sections intermediate between these two sections that describe the motion of substance with special characteristics: soils, bulk substances, substances with properties, intermediate between the properties of solids and liquids (rheological substances), etc. Mechanics as science has now become a sufficiently branched area of ​​knowledge. In the division of mechanics into the separate sections, the different criteria and attributes are used and in this way, the different divisions are received. The velocity of motion is an important value in the classification of mechanics. The motion of bodies with velocities close to the speed of light, where the classical Newton’s laws are incorrect, is considered in the part of physics called THE THEORY OF RELATIVITY, and the motion of elementary particles at the atomic or subatomic level is considered in THE QUANTUM MECHANICS. These two cases of motion are, of course, not related to the study of both classical mechanics and modern mechanics. The classic division is the division of mechanics by the type of physical models used in the study of phenomena. According to this criterion, mechanics traditionally is divided into five main sections: MECHANICS OF THE MATERIAL POINT, MECHANICS OF THE SYSTEMS OF MATERIAL POINTS, MECHANICS OF THE RIGID BODY, FLUID MECHANICS, SOLID MECHANICS. The construction of models is the main tool of mechanics in the study of mechanical phenomena and all the above sections include a sufficiently large number of different models.



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Basic information on mechanics



It is based on the point of view that, considering mechanics, we consider, in fact, the sequence of physical models. The primary assumption in most of the models is the assumption of CONTINUITY of the material body. Continuity is introduced in mechanics in the following way. The initial fact is that modern phy-sics represents matter as a system of a large number of interconnected and interacted particles, which were previously defined as discrete formations. Since a description of the change in the body shape, while the motion of each particle being taken into account, is a very difficult prob-lem, it was found that solving the problem is inappropriate. The point is that knowledge of the individual motion of the particle (the number of particles in 1 cm3 is of the order 1022) gives a picture of the microscopic motion of all particles of the body. Experience has been shown that a change in the body shape is successfully described as a manifestation of the macroscopic motion of the body as a whole. THE MAIN TOOL in the transition from nanoscale or microscale description to the macro-scale description is THE PRINCIPLE OF CONTINUITY, in which the body in the form of a discrete particle system is replaced by the body with a continuous system of points (CONTINUUM) that occupies the same area of​​ space. At the same time, each point of the continuum attributes a certain set of average physical properties (density and a series of thermodynamic parameters) that are obtained by the procedure of averaging of the parameters of the nano- or microscopic motion. The typical representatives of the mechanics of the rigid body are: the CELESTIAL AND ORBITAL MECHANICS, the SECTION OF FLUID MECHANICS - HYDROMECHANICS, the SECTION OF SOLID MECHANICS – the THEORY OF ELASTICITY and the THEORY OF PLASTICITY If in the study of sufficiently complex phenomena the models from two or three sections mentioned above are commonly used, then such studies are conventionally attributed to the GENERAL MECHANICS or to its separate parts (for example, to MAGNETOHYDRODYNAMICS). Division of mechanics by the methods used in the study of phenomena includes three main sections: ANALYTICAL MECHANICS, COMPUTATIONAL MECHANICS, EXPERIMENTAL MECHANICS.



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Basic information on mechanics



In the past, only the corresponding problems of mechanics of the rigid body were referred to THE ANA-LYTICAL MECHANICS. At present, the analytical mechanics is interpreted in the broader sense as mechanics that is using the analytical approaches, when it refers to the corresponding problems of fluid mechanics and solid mechanics. The general theoretical problems: the formulation of an exact statement of problems, theorems of uniqueness and existence of solutions, variational principles, and other related questions are also referred to the analytical mechanics. In the study of certain problems, the methods and approaches of two or three sections are commonly used. THE COMPUTATIONAL MECHANICS is developing very strongly in our time and it was initially thought that it would push the analytical mechanics because of the great difficulties in the analytic representation of the complex problems of mechanics. The most popular are several methods of computer analysis of the problems of mechanics. The first among them is the method of finite elements, which proposed by R. Courant and further adapted to mechanics by M. Terner, R. Clog, H. Martin, L. Top, J. Argiris, S. Cellsi. Currently, the number of utilized worldwide commercial software products that are developed in computational mechanics. THE EXPERIMENTAL MECHANICS has a very rich history from ancient times to the modern ones. The modern experimental technology is high-tech and expensive. Therefore, experiments in mechanics are carried out mainly in rich countries. The division of mechanics on the base of the characteristic size of the internal structure includes four sections: MACRO-MECHANICS, MESO-MECHANICS, MICROMECHANICS, NANO-MECHANICS. This kind of division of mechanics was introduced into the scientific practice when only the NANO-MECHANICS started its development as the new area of mechanics despite that the notions of MACRO -MECHANICS and MICRO-MECHANICS were used in mechanics many years ago relative to composite materials and the notion of MESO-MECHANICS was used relative the metals. Thus, only the nanomechanics is a product of the twenty-first century, it is developed successfully within the framework of the continuum mechanics after the invention of the atomic force microscope (AFM) and scanning tunneling electron microscope (STEM), that are able to differ the nanostructure of material. The leading in science countries spent big finance on a study of nano-mechanics. The division of mechanics on the base of correspondence to the specific areas of practical human activity includes many sections:



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Basic information on mechanics



BUILDING MECHANICS, CELESTIAL MECHANICS, MECHANICS OF SPACE FLIGHT, BUILDING MECHANICS OF AIRCRAFT AND SHIP, MINING MECHANICS, GEOTECHNICAL MECHANICS, GEOMECHANICS, MECHANICS OF COMPOSITE MATERIALS, BIOMECHANICS AND MECHANICS OF MAN, MECHANICS OF ENVIRONMENT, and others. The division of mechanics by the nature of phenomena contains only two sections: STATICS (studying the body equilibrium) and DYNAMICS (studying the body motion), to which in many cases the STABILITY and FRACTURE are added, which include both static and dynamic phenomena. A more detailed division of mechanics into sections (subsections, subsubsections, etc.) - the more in detail classification of mechanics) is contained in the classifiers of mechanics, which are developed in various international specialized editions on mechanics and references to which are widely used in scientific publications. The three most common classifications of mechanics are as follows:



The classic classifier Uniform Decimal Code (UDC) is least adapted to the contemporary mechanics



The newest classifier Mathematics Subject Classification (MSC) is prepared by two of the world’s largest abstract journals – Mathematical Reviews (USA) and Zentralblatt für Mathematik (Germany) and reflects the current state of mechanics



The latest classifier Applied Mechanics Reviews Subject Classification (AMR), is prepared by the world’s largest abstract journal – Applied Mechanics Reviews (USA) and is most adapted to the contemporary mechanics



Newton’s laws of motion, set forth by I. Newton in 1687 in the famous work “Philosophiae Naturalis Principia Mathematica”, forms the basis of all sections of mechanics. The theoretical mechanics or general mechanics studies general laws and principles relating to the mechanical motion of bodies, and general theorems and equations arising from these laws and principles. It is traditionally divided into STATICS, KINEMATICS, and DYNAMICS. The models of mechanics of the material point, mechanics of the systems of material points, mechanics of the rigid body are used here.



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Basic information on mechanics



The basic concepts of statics are the notions: force, mass, motion, a moment of force, a center of mass, friction. The statics of solids, like statics of liquids, have also been studied in ancient times. Like the Archimedes “law on the equilibrium of floating bodies”, the Archimedes “law on the equilibrium of a lever” Definition 1.5. A lever is called a long rod, which is based on a distance from one end to the fixed support and to which vertically forces are applied on both ends. is formulated as follows - the product of one force at a distance from the point of its application to the point of reference is equal to the product of another force at a distance from the point of its application to the point of support. It is considered the first scientific fact of theoretical mechanics. The basic concepts of kinematics are: trajectory, passed distance, speed, acceleration, rotation, angular velocity. The dynamics in the reduced form is studied in the theoretical mechanics and more fully in the analytical mechanics. THE ANALYTICAL MECHANICS is based on Newton’s laws, concepts of generalized coordinates, the construction of kinematics and kinetics of the rigid body, concepts of work and potential energy. The general equations of the Lagrange motion of a holonomic mechanical system with a finite number of degrees of freedom have made it possible to reduce the solution of any problem of the motion of the system to the mathematical problem of solving the differential equations. This fact underlies the analytic mechanics. It contains the theory of motion of holonomic systems (with positional restraints) and nonholonomic-kinematic systems, methods of constructing the first and second-order Lagrange equations, the Euler-Lagrange equations, Appel and Chaplygin equations of relative motion, canonical equations (including the theory of their integration on the basis of the Hamilton-Jacobi theorem), the theory of perturbations, the variational principles of mechanics (including the introduction of the notion of action by Hamilton, and a number of principles: the Hamilton-Ostrogradsky principle, the stationary action principle Lagrange. The development of analytical mechanics in recent times has led to the creation of new divisions, such as the theory of jet motion, the subsequent transition from the dynamics of discrete systems to the dynamics of continuous systems, etc. Mechanics contains a number of parts, which are simultaneously the parts of other sciences. These parts include the already mentioned relativistic mechanics and quantum mechanics, which are the parts of physics, as well as celestial mechanics, which is also part of the analytical mechanics and the part of astronomy.



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Basic information on mechanics



The history of mechanics is reach and diverse. Many excellent books are devoted to this theme. Therefore, in the following statement, the historical sketches of some parts of mechanics will be presented. Historical sketch. Celestial mechanics studies the motion of celestial bodies as the rigid bodies on the basis of laws relating to the gravitation forces. It was founded by Newton. It is believed that before Newton, the celestial mechanics studied the kinematics of the motion of celestial bodies, and it has its origins since ancient times. Despite the false idea of ​​the motion of celestial bodies around the Earth, the eclipse was predicted fairly accurately and the calendars were synchronized with the motion of the Earth around the Sun. Introduced by Apollonius of Perge, epicycles allow a well-described motion of planets. During the Renaissance, a great step was taken in the development of celestial mechanics, when N.Copernicus placed the Sun in the center of the universe, and J.Kepler formulated three laws of the motion of planets. But only the work of I. Newton marked the formation of celestial mechanics as a science. Newton’s law of universal gravitation “The bodies are mutually attracted to a force that is proportional to their mass and inversely proportional to the square of the distance between them” is the basis of the celestial mechanics. Newton’s theory of gravity has allowed us to construct the differential equations describing the motion of celestial bodies. To solve them, the theory of perturbations was construc-ted in the 18th century, the largest contribution to which made Euler and Lagrange. As a result, the existence of a new planet - Uranus was theoretically foreseen and therefore experimentally observed. In the 20th century, Einstein proposed a new theory of gravitation, known as the general theory of relativity. He refined the results of Kepler and Newton, based on the field equations proposed by Einstein. The initiated by Einstein’s theory of relativity was responded to the proposal of Poincaré: “Perhaps, we must build a completely new mechanics, yet foggy, in which the inertia increases with the speed and the speed of light is the limit one.” In this way, a relativistic mechanics arose, based on a new understanding of gravity. 20th century is also characterized by the appearance in the theoretical concepts of the CONCEPT of CHAOS, which is often understood as a transition from one mode of motion to another one. It was found that the solutions of equations of the celestial mechanics are very sensitive to changes in the initial conditions and small changes in the initial conditions lead to a large evolution of the solution. The features of equations of celestial mechanics are such that evolution due to chaotic instability is very slow. At the most recent stage of development, it is possible to indicate such a section of applied celestial mechanics as astrodynamics, which studies the motion of artificial satellites and the transition of satellites from one orbit to another. The modern celestial mechanics takes into account not only traditional gravity forces but also zero gravity: gas resistance, heat release, the interaction between radiation and matter, comet jet, tidal friction, etc. 20



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Basic information on mechanics



MECHANICS OF FLUID, GAS, AND PLASMA studies, in accordance with the name, the motion of fluid, gas, and plasma. The peculiarity of this type of substance is its fluidity or easy mobility. Models of mechanics of fluid, gas, and plasma necessarily take into account the properties of continuity and fluidity. The main models of fluid, gas and plasma are the model of an ideal (non-viscous) fluid and the model of viscous (Newtonian) fluid. The model of nonviscous fluid is an idealization that does not take into account the existence of shear stresses in the fluid. This simplifies the analysis of motion, while the viscous fluid model takes these stresses into account, considering them proportional to the gradient of the velocity in the direction perpendicular to the shear plane. For example, water is a viscous fluid. The socalled non-Newtonian fluids are studied by rheology. Most fluids with a molecular structure consisting of long molecules exhibit the properties of non-Newtonian fluids.  The study of plasma as an ionized substance required the consideration of the interaction of the plasma with the electromagnetic field. The study of plasma motion was stimulated by the great technical need for the creation of magnetohydrodynamic gene-rators, the study of the behavior of high-temperature plasma in strong magnetic fields (thermonuclear reaction), and others like that. THE MAGNETOHYDRODYNAMICS studies the motion of not only plasma, but also of liquid metals, saline water, and electrolytes. Having the basic properties of continuity and fluidity, gases and fluids differ in the molecular structure. In the fluid, the distance between the molecules is small and therefore molecular forces of adhesion arise between the molecules. These forces act on the outer surface of the fluid and their action is such that the fluid is very compressed. The small changes in pressure observed in the slow-motion cause small changes in the fluid volume. Therefore, the fluids are generally considered to be little compressible and, in most models, the fluid is considered to be incompressible.  In contrast to the fluid, the distance among the molecules in the gas is large and the interaction among the molecules is weak. Therefore, gas is much more compressible than fluid. Just difference in compliance with compression forms the main difference between the fluid and gas. Under certain circumstan- ces, the fluid can be compressed, and the gas is non-compressive. Therefore, the same models of fluid, gas, and plasma are used for fluid and gas.  Note 1.1. The statics of fluid was studied even in ancient times. The mentioned above Archimedes law “the main vector of the pressure of the fluid on the surface of the body immersed in this fluid, is equal to the weight of the fluid in the body volume and directed toward the opposite force of weight” is considered the first scientific fact of mechanics of fluid, gas, and plasma. 21



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Basic information on mechanics



Historical sketch. The foundations of modern mechanics of fluid, gas, and plasma were laid in the Middle Ages. Here such scientists should be listed: Leonardo da Vinci (he described observations and experiments), E.Toricelli (he invented the barometer), I.Newton (he investigated the viscosity, in his honor the viscous fluid is called the Newtonian), B.Pascal (he formulated the law of isotropy of normal stresses in a fluid – Pascal’s law), D.Bernoulli (he proposed the mathematical fluid dynamics). Contribution to the theory of motion of the ideal fluid was made by L.Euler, J.D’Alembert, J.Lagrange, P.S.Laplace, S.D.Poisson, and in the theory of the motion of the viscous fluid – J.L.Puiseuille and G.Hagen. In the 19th century, H.Navier and G.Stokes proposed the basic equations - the Navier-Stokes equation. H.Helmholtz created the doctrine of vortices. Definition 1.6. The vortex is called the form of motion of the fluid, which is described by an antisymmetric part of the velocity gradient; accordingly, the motion of a fluid occurs without a vortex, if this part is absent; the vortex motion is called the vortex motion with the presence of vortices.  The study of the non-vortex motion of fluid forms a separate part of fluid mechanics. Already in the 20th century, L.Prandtl and T. von Karman developed the theory of the boundary layer. O.Reynolds, J.Taylor, A.Kolmogorov made a further contribution to the understanding of the viscosity and turbulence of the motion in the problems of fluid mechanics. The separate study of the statical and dynamical problems in mechanics of fluid, gas, and plasma is a traditional one. THE HYDROSTATICS explains several phenomena of everyday life: the changes in atmospheric pressure when changing the height; ascending trees or oils in water; the horizontal surface of the water and its flat shape in an arbitrary shape of the container, etc. The hydrostatics creates the foundation for hydraulics and engineering, which studies the preservation and transport of fluid, and is partly important for medicine, meteorology, geophysics, astrophysics, and other sciences. The hydrodynamics and aerodynamics exist as separate specialized sections of the dynamics of fluid, gas, and plasma. which have a very wide range of applications.  The following part of mechanics having the great importance in this book is the SOLID MECHANICS (MECHANICS OF MATERIALS). It will be described in the next chapter. 



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Comments Comment 1.1. The tailoring of models of mechanical motion of substance is always a difficult problem. The usual way in modern mechanics is called the phenomenological approach. According to the approach, the parameters of phenomenological model are strongly defined theoretically, further physical experiments for the determination of physical constants are described, and finally, the theory has to predict new phenomena. For fundamental sciences, the necessity of attention to experiments and practice had been formulated as far back by Leibniz in his motto theoria cum praxi, he urged that theory be combi-ned with practical application, and thus Leibniz is claimed as the father of applied science, and the Leibniz’s motto it understood as the necessity for any theory to amplify with experimentations. 200 years later, Boltzman, stated: “nothing is so practical as the theory”. In 1926 in a talk between Werner von Heisenberg and Albert Einstein, Heisenberg stated that each theory, in its building, must cor-respond to only those observed by this time fact. Einstein answered, that it could be wrong to try to build the theory only on observed facts. Really, it happens the vice versa - theory determines, what we can observe. Comment 1.2. The general theory of models is worthy to separate comment. Here we will follow the Stanford Encyclopedia of Philosophy which stated that although the models play an important role in science and are one of the principal instruments of modern science, there remain significant lacunas in the understanding of what models are and of how they work. The philosophical literature studies: probing models, phenomenological models, computational models, developmen-tal models, explanatory models, impoverished models, testing models, idealized models, theoretical models, scale models, heuristic models, caricature models, didactic models, fantasy mo-dels, toy models, imaginary models, mathematical models, substitute models, iconic models, formal models, analog models, instrumental models et cetera. Let us concentrate on the phenomenological models together with empirical and semiempirical models. A traditional definition of phenomenological models in philosophy takes them to be models that only represent the observable properties of their real objects and refrain from postulating hidden mechanisms and the like. But also, the phenomenological models are thought of as models that are independent of theories. This, however, seems to be too strong. Many phenomenological models, while failing to be derivable from a theory, incorporate principles and laws associated with theories. Just this last sentence is close to the used in mechanics understanding the phenomenological model. This model arises in the case when the universal physical laws do not help to study some mechanical object (these laws are not enough to build an adequate model). Then the phenomenological laws are used in mechanics and the constructed theory is called the phenomenological theory.



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Definition 1.7. The phenomenological law is thought in mechanics as the well substantiated empirical law with the restricted and also well substantiated area of application. To the classical examples of such laws can be related: the Hooke’s on the linear relation between the force and extension, the Fourier’s law on the linear relation between the heat flow and temperature gradient. Note 1.2. In essence, every theory is phenomenological, since otherwise we would have reached an absolute penetration into the nature of things, which is impossible in principle. The empirical models have the less universal character as compared with the phenomenological ones. Note 1.3. The term empirical comes from the Greek word for experience - ἐμπειρία. To describe the notions of empirical model and then the empirical theory, it is necessary to introduce the chain of more abstract notions. First, the notion of empirical evidence has to be formulated: Definition 1.8. The empirical evidence is an information received by means of the senses, particularly by observation and experimentation, in the form of recorded data, which may be the subject of analysis. The statements and arguments depending on the empirical evidence are often referred to as a posteriori (following experience) as distinguished from a priori (preceding it). A priori knowledge or justifi-cation is independent of experience, whereas a posteriori knowledge or justification is dependent on experience or empirical evidence.  Note 1.4. In science, the de Groot’s empirical closed cycle is well-known: 1.Observation: The observation of a phenomenon and inquiry concerning its causes. 2. Induction: The formulation of hypotheses - generalized explanations for the phenomenon. 3. Deduction: The formulation of ex-periments that will test the hypotheses (i.e. confirm them if true, refute them if false). 4. Testing: The procedures by which the hypotheses are tested and data are collected. 5. Evaluation: The interpretation of the data and the formulation of a theory - an abductive argument that presents the results of the experiment as the most reasonable explanation for the phenomenon.



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In mechanics, the empirical and semi-empirical models are distinguished by the criterion of the degree of universality and size of substantiated area of application. In general, the phenomenological, semi-empirical, and empirical models are supported by experiment and observation but not necessarily supported by theory.



Further reading I. Oxford Research Encyclopedia of Physics, 2020, Foster, B (Editor-in-Chief ) Oxford University Press & American Institute of Physics, Oxford. II. Encyclopedia of Continuum Mechanics. 2019, Altenbach H & Öchsner A (eds), Springer. Berlin. III. Stanford Encyclopedia of Philosophy. 2018, Stanford University Press, CSLI, Stanford. IV.  Encyclopedia of Physics Research, 2012, Devins, NB & Ramos, JP (eds) Nova Science Publishers, New York. V Encyclopedia of Physics (in 2 vols) 2005, Lerner, RG & Trigg, GL (eds) Wiley-VCH, VI. MacMillan Encyclopedia of Physics (in 4 vols), 1996, Rigden, JS (ed) MacMillan. VII. The Encyclopedia of Physics, 1990, Besancon, RM (ed) Springer, Berlin. VIII. Flugge’s Encyclopedia of Physics. 1988, Springer, Berlin. 1.1. A  scione, L & Grimaldi, A 1993, Elementi di meccanica del continuo (Elements of continuum mechanics). Massimo, Napoli. (In Italian) 1.2. Blekhman, II, Myshkis, AD & Panovko, YH 1976, Applied Mathematics: Subject, Logics, Features of Approaches. Naukova Dumka, Kiev. (In Russian) 1.3. Eringen, AC 1967, Mechanics of Continua. John Wiley, New York. 1.4. Eslami, MR, Hetnarski, RB, Ignaczak, J, Noda, N, Sumi, N & Tanigawa, Y 2013, Theory of Elasticity and Thermal Stresses. Explanations, Problems and Solutions. Series “Solid Mechanics and Its Applications” Springer Verlag Netherlands, Amsterdam. 1.5. Fung, YC 1965, Foundations of Solid Mechanics. Prentice Hall, Englewood Cliffs. 1.6. Germain, P 1973, Cours de mécanique des milieux continus. Tome 1. Théorie générale. Masson et Cie Editeurs, Paris. (in French) 1.7. Gurtin, ME 1981, An Introduction to Continuum Mechanics. Academic Press, New York. 1.8. Iliushin, AA 1990, Mechanics of Continuum, Moscow University Publishing House, Moscow. (in Russian) 1.9.  Lur’e, AI 1990, Nonlinear Theory of Elasticity. North-Holland Series in Applied Mathematics and Mechanics, North-Holland, Amsterdam. 1.10. Maugin, GA 1988, Continuum Mechanics of Electromagnetic Solids. North Holland, Amsterdam. 1.11. Prager, W 1961, Introduction to Mechanics of Continua. Ginn, Boston. 1.12. Sedov, LI 1970, Mechanics of Continuum, in 2 vols. Nauka, Moscow. (in Russian)



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1.13. Spencer, AJM 1980, Continuum Mechanics. Longman, London. 1.14. Truesdell, C 1972, A First Course in Rational Continuum Mechanics. The John Hopkins University, Baltimore.



Questions 1.1. W  hich facts from the ancient history of mechanics you would like to add? Formulate briefly these facts. 1.2. Which parts of modern physics, besides mechanics, are using the notion “continuum” as the basic one? List on these parts and estimate the level of application. 1.3. Write a few areas of natural science (excluding mathematics) that have the axiomatics. 1.4. Try to substantiate the advantage of science to have the axiomatics. If you think that this is not essential fragment in science, you can propose other essential fragments and substantiate this proposal. For example, a correspondence of the basic experiments. 1.5. Add the list of areas of mechanics, which are not mentioned in this chapter. 1.6. Which additional historical sketches you could insert into the text of this chapter ? 1.7. Try to estimate in which a way you could to use the proposed encyclopedias.



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FOUNDATIONS OF MECHANICS OF MATERIALS PART 1



BASIC INFORMATION ON MECHANICS OF MATERIALS. THEORY OF ELASTICITY. SHORT DESCRIPTION OF LINEAR THEORY OF ELASTICITY



2 BASIC INFORMATION ON MECHANICS OF MATERIALS. THEORY OF ELASTICITY. SHORT DESCRIPTION OF LINEAR THEORY OF ELASTICITY Definition of mechanics of materials. Classifications and main properties of materials. Main theories of materials. Theory of elasticity: division on four parts. Material continuum. Equipped continuum. Short description of linear theory of elasticity. Vector of displacement. Cauchy-Green strain tensor. Stress. The Euler-Cauchy cutting principle. Internal stress tensor. Balance equations. Motion equations. Kinetic and potential energy. Constitutive equations. Three classical kinds of symmetry of materials – orthotropy, transversal isotropy, and isotropy. Lame equations. Boundary and initial conditions.



To begin with, let us repeat the definition from Lecture 1. Definition 2.1. SOLID MECHANICS (MECHANICS OF MATERIALS) is the part of mechanics that studies the deformation of materials. The modern mechanics of materials and solid mechanics are not always identical. The difference is observed in the accents when carrying out research. The modern interpretation of research on solid mechanics relates them to three modern sections of science - applied mathematics, mechanical and other engineering, material science. During a few centuries, when the solid mechanics was developed theoretically and experimentally mainly as the theory of elas-ticity, the theoretical part of solid mechanics was very close to the applied mathematics and sometimes the theory of elasticity was considered as a part of mathematical physics. Nowa-days, the prevailing number of both theoretical and experimental studies in mechanics of material is referred to last two sections, which focus on the materials and their models. The materials are classified according to the various features: 



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1. Traditional classification - by the field of use and physical-chemical structure machine-building and construction materials, polymer and composite materials, ceramic and glass ma-terials, etc.  2. Classical classification - on the level of internal structure consideration homogeneous and heterogeneous materials.  3. Modern classification - divides materials into five types: Type 1. Metals and alloys. Type 2. Polymers. Type 3. Ceramics and glass. Type 4. Composites. Type 5. Natural materials (wood, leather, cotton /wool /silk, bone, coal, ice). THERE ARE SEVEN MAIN PROPERTIES OF MATERIALS WHILE THEY BEING DEFORMED.  1. ELASTICITY: the body instantaneously takes on the initial configuration after eliminating the causes of deformation (deformations are reversible).  2.1. PLASTICITY: the body does not take on the initial configuration after eliminating the causes of deformation (deformations are irreversible).  2.2. ELASTOPLASTICITY: the process of deformation is elastic to some value of deformation, and when it exceeds this value becomes plastic.  2.3. HARD-PLASTICITY: to some extent, the intensity of the external action of the body does not change the configuration (the body is not deformed), and when the value exceeds this value becomes plastic.  3.1. THERMOELASTICITY: the temperature changes lead to the elastic deformation and vice versa.  3.2. THERMOPLASTICITY: the temperature changes lead to plastic deformation and vice versa. 4.1. VISCOSITY: the dependence of internal forces that arise during deformation, not on deformations (which is characteristic of elastic and plastic deformation), but on the rate of deformation (which is characteristic of liquids).  4.2. VISCOELASTICITY: at the same time, the body has the properties of elasticity and viscosity, which is manifested in the existence of creep and relaxation phenomena.  Definition 2.2. The creep of deformations consists in increasing the deformations under conditions of stress constancy.  Definition 2.3. The relaxation of stresses consists of decreasing stresses under conditions of deformation constancy.  4.3. VISCOPLASTICITY: the phenomenon of creep is presented and at the same time the body is deformed plastic.  5. DIFFUSIONAL ELASTICITY: diffusion is the cause of elastic deformation and vice versa.



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6. ELECTROELASTICITY: the electric field is the cause of deformation and vice versa. 7. MAGNETOELASTICITY: the magnetic field is the cause of deformation and vice versa.  Based on each of the properties listed above, the corresponding theories are constructed in the mechanics of materials: THE THEORY OF ELASTICITY, THE THEORY OF PLASTICITY, THE THEORY OF THERMOELASTICITY, THE THEORY OF VISCOELASTICITY, THE THEORY OF CREEP, THE THEORY OF DIFFUSIONAL ELASTICITY, THE THEORY OF ELECTROELASTICITY, THE THEORY OF MAGNETOELASTICITY. As it follows from the list above, the property of elasticity is presented practically in all theories (except partially the theory of plasticity). Therefore, the theory of elasticity can be treated as the key and basic theory in the mechanics of materials. From this point of view, this theory is worthy to be commented more in detail. First of all, the theory of elasticity can be divided into four parts: I. The strength of materials. II. The linear theory of elasticity. III. The linearized theory of elasticity. IV. The nonlinear theory of elasticity. At that, the strength of materials can be considered as the separate part very conditionally only despite a big relationship with the theory of elasticity. Note 2.1. There are different criteria for distinguishing the shown above parts of the theory of elasticity. One of the frequently used criteria is that Part I uses the simplest approximate models (mainly, one-dimensional ones) and is considered as a set of standard tools for engineering mechanics. Next three parts are built on the stronger mathematical apparatus and include models of more complicate structure, that are able to describe the richer set of mechanical effects. Part II is based on the one linear model. Part III is based on the different linearized models. Part IV is based on the different nonlinear models.



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Historical sketch. It is believed that the mechanics of elastic materials was initiated by Leonardo da Vinci and Galileo Galilei. Further contributions were made by R.Hooke (Hooke’s law) and I.Newton (Newton’s laws). The following steps were taken in the analysis of one-dimensional problems of mechanics of materials in the 18th century: G.Leibniz, D.Bernoulli, L.Euler, J.D’Alembert, J.Lagrange analyzed the equilibrium and stability of the rods. The beginning of the mechanics of materials as continuum mechanics is associated with H.Navier, A.Cauchy, S.Poisson, P.Clapeiron, G.Lame, B.Saint-Venan. Navier himself derived the equation of the theory of elasticity for isotropic bodies. In the second half of the 19th century, G.Kirchhoff completed the formation of the theory of plates (thin plates), G.Green and Lord Kelvin considered the theory of elasticity of anisotropic bodies, H.Hertz built the theory of the impact of bodies. The theory of thin-walled shells developed at the end of the 19th century by A.Love and H.Lamb, which in the middle of the 20th century was clarified by W.Koiter and V.V. Novozhilov. The study of waves in elastic bodies begins with the work of S. Poisson, A.Cauchy, G.Stokes. Lord Rayleigh made a significant contribution to the theory of elastic waves at the end of the 19th century. The study of the concentration of stre-sses was initiated by G.Kirsch, having solved the so-called Kirsch problem about the concentration of stresses near a circular hole. At the beginning of the 20th century, G.V. Kolosov and C.Inglis generalized this problem for the case of an elliptical hole, which allowed A.Griffith to introduce the concept of a crack in a brittle body and initiate the theory of cracks. The next step in the development of the theory of cracks was made by G.Irwin. In the 20th century, the theory of cracks was developed very actively in many countries within the framework of various models of materials. The transition in the theory of elasticity to the study of nonlinearity is done in the 20th century and is associated with R.Rivlin, M.Mooney, L.Treloar, F.Murnaghan, A.Signorini, V.V.Novozhilov, A.I.Lurie, A.Eringen, R.Ogden, E.Arruda. Being placed between the linear theory of elasticity and nonlinear theory of elasticity, the linearized theory of elasticity is also well developed. This theory is based on the linearization of the basic relationships of the nonlinear theory of elasticity and, in this way, represents some simplification of nonlinear theory. The pioneer works are related to the middle of 20th century and are associated with M.A.Biot, L.S.Leibensohn, A.Yu.Ishlinsky, Yershov L.V., D.D.Ivlev, A.I.Lurie, B.R.Seth, Z.Wesolowski, A.E.Green, R.S.Rivlin, R.T.Shield. This theory obtained strong mathematical formulation and good application to different non-classical and applied areas of mechanics in publications of A.N.Guz and his scientific school.



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The historical sketch for the strength of materials is stated in the brilliant form by S.P.Timo-shenko. Many tens of years ago the strength of materials is formed as the classical approach to the engineering problems of mechanics. It seems to be rational to start with a description of the basic formulations of the theory of elasticity from the linear theory. Just this part (Part II) of the theory of elasticity is studied most deeply and is the most developed. Also, in the statement of foundations of Part I and Parts III and IV, the structure and basic notions of Part II form the initial skeleton, which is simplified for Part I and complicated for Parts III and IV. So, Part II has to be considered first of all. At the beginning, the fundamental idea in the description of deformation of materials– the idea of continuum description - must be repeated once again. It means that the real piece of material is changed on the continuum of the same shape. Accordingly, the material is changed on the area of 3D space, in each point of which the density U  x1 , x2 , x3 is given. In this way, the material continuum is introduced. Further, this continuum is equipped in the linear theory of elasticity by three new functions (from point of view of mathematical formalism), which also are determined in each point of the continuum area. From point of view of mechanics, these functions have mechanical sense and are new mechanical notions. Let us start of the brief information on the mentioned above four parts with the Part II – the linear theory of elasticity. Note here that this theory studied only the linear deformation of the continuum. The displacements are introduced as the distances  u1 , u2 , u3 between the point in the non    deformed state and this point after deformation P  x1 , x2 , x3 , t uk



 xk  xk .(2.1)



The displacements (2.1) form the VECTOR OF DISPLACEMENTS. They must be small in the linear theory of elasticity and depend on the spatial coordinates and time G u  x1 , x2 , x3 , t



 u  x , x , x , t , u  x , x , x , t , u  x , x , x , t .(2.2) 1



1



2



3



2



1



2



3



3



1



2



3



Note 2.2. It is always assumed in the linear theory of elasticity that the deformations, the displacements and other similar basic mechanical parameters (strains and stresses) depend on the location of the point in the initial (non-deformed) state. This is substantiated just by the assumption on the smallness of values of displacements (2.1). To the point, the displacements have the dimension and are measured in meters. For the most of engineering materials, the linear approach admits displacements of order 103 104 m.



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The next after displacement basic notion is the notion of the STRAIN. The strain is introduced into the theory of elasticity on the base of some considerations of a mechanical character. When the piece of material (body, area of a continuum) is deformed, then the distances between points of body are changed, what means that the length of linear element (straight line linked two neighboring points) and angles between coming from the same point two linear elements are changed. Let the linear element joins the points P1* and P2*. After deformation, this element joints the points P1* and P2*. Then the relative extensioncan be defined as follows H



P*1 P2*  P1 P2 P1 P2



'P1 P2 .(2.3) P1 P2



This formula gives the simplest example of deformation. If to assume the smallness of gradients of displacements ui ,k  wui wxk , what is logical in the linear theory of elasticity, then the LINEAR CAUCHY-GREEN STRAIN TENSOR can be introduced H ik  x1 , x2 , x3 , t



1  ui ,k  x1, x2 , x3 , t  uk ,i  x1, x2 , x3 , t .(2.4) 2



According to (2.4), the strains are small and dimensionless. They really form the symmetric tensor of the 2nd rank, because meet the definition of such tensor. Usually, the linear Cauchy-Green strain tensor is supplemented by the rotation tensor Zik  x1 , x2 , x3 , t



1  ui ,k  x1, x2 , x3 , t  uk ,i  x1, x2 , x3 , t .(2.5) 2



THE THIRD AND LAST BASIC NOTION OF THE LINEAR THEORY OF ELASTICITY is the notion of stress. It is based on the fundamental in physics notion of the force. The force must have the point of application, direction of action, and intensity. Therefore, the force is described mathematically as the vector. Usually, the forces are divided on the mass forces and the surface forces (the concentrated force can be considered as the limit case of the surface one). The most used in the theory of elasticity mass force is the inertia force. The surface force must be applied to the points of the body surface. The classical example of such force is the force of pressure when the body is exposed to the action of other bodies. On the base of the notion of the force, the new notion of internal stresses (or stresses) is introduced.



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An introduction of this notion is based on the statement that the action of external forces forms the internal forces in the body. Therefore, the described above forces are called external forces. The classical in the theory of elasticity way of introducing the stresses is called THE EULER-CAUCHY CUTTING PRINCIPLE. According to this principle, the body is virtually divided into two parts by an arbitrary continuous surface. The action of part 2 on part 1 is changed on the action of the surface forces and the body still is in the state of equilibrium. Then on the surface, the infinitesimal element of G surface dS is chosen and the surface force acting on this element is denoted by dP . G G G It is assumed further that dP tN dS , where tN is the stress vector which is applied to G G the surface dS with the normal N . If the vector tN is decomposed on components in three orthogonal directions linked with dS (i.e., on a tangent, normal, and binormal), then the obtained vectors generate three stresses. The values of these stresses can be meant as values of obtained vectors, related to the surface element dS (divided on the area of surface element dS ). Just, therefore, the stresses can be briefly characterized as the force related to the area. They are measured in Pa (Pascal). The INTERNAL STRESS TENSOR is introduced as follows. The infinitesimal coordinate tetrahedron is introduced and it is supposed to be in the balance by the action of forces G dP applied to the four tetrahedron faces. The main conclusion from an analysis of the tetrahedron balance is apparently the conclusion that the quantities of nine stresses on the three coordinate faces form the tensor of the 2nd rank. So, the nine quantities V nm  x1 , x2 , x3 , t , which are called the STRESSES, form the STRESS TENSOR. Now, the basic equations of the linear theory of elasticity can be shown. These equations include three sets of equations. The first one is equations of motion (equilibrium). They are formulated based on the law of balance of momentum in the usual form of three equations of motion U



w 2ui wt 2



V ik ,k  Fi .(2.6)



G



Here F  x1 , x2 , x3 , t



^F  x , x , x , t , F  x , x , x , t , F  x , x , x , t ` is the external volume force. 1



1



2



3



2



1



2



3



3



1



2



3



Note 2.3. An analysis of balance equations (conservation laws) has proceeded from the well-known statement: ”the great laws of classic physics can be considered as one general law of conservation”. Definition 2.4. Definition 2.4.Definition 2.4. Definition 2.4. Definition 2.4.



The balance of2.4. the moment of momentum has the usual corollary that the stress tensor is Definition symmetric. Definition 2.4. Definition 2.4.



on 2.4.



nition 2.4.



Definition 2.4.



Definition 2.4. Definition 2.4.



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BASIC INFORMATION ON MECHANICS OF MATERIALS. THEORY OF ELASTICITY. SHORT DESCRIPTION OF LINEAR THEORY OF ELASTICITY



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Note 2.4. Three equations (2.5) include nine unknown functions – three displacements and six stresses. This means that new six equations are needed to obtain the closed system of equations. The mentioned above six equations follow from an analysis of the balance of energy stored by the body in the process of deformation. At this place, two new functions must be introduced - the kinetic energy K  u1 , u2 , u3 , U as the function of velocities and density and the internal energy U  H11 ,..., H 23 as the function of strain tensor. The energy of a body E is defined as the sum of kinetic energy K of a body and of internal energy of a body U . Definition 2.4.



Definition 2.4. Definition 2.4.



Definition 2.4. The elastic material is defined strongly as a material which can be in the natural (free of stresses) state and in a neighborhood of this state the stresses in present time can be defined one-to-one by values either of deformation gradient or strain tensor at present time V ik



Fik  H lm (2.7)



or for the rectilinear symmetry V ij



Aijkl H kl  AijklmnH kl H mn  AijklmnpqH lmH mnH pq " . (2.8)



Here, the fourth rank tensor Aijkl defines the linear properties of elastic materials, when tensors of higher ranks are absent in (2.7). This means that the linear elastic material is defined as the material, in which the relations between stresses and strains are linear and have the form V ij



Aijkl H kl .(2.9)



In this way, the necessary six equations relative stress and strain tensors components are obtai- ned. But, at that the equations (2.9) introduced six new unknown functions – the components of the strain tensor. And still the number of equations is less of the number of unknown functions. In the linear theory of elasticity, the different procedure of introduction of relations (2.9) is often used. First, the assumption is adopted that the specific internal energy U is an analytical function of the strain tensor components Definition 2.4. U



U  H lm . (2.10)



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From point of view of mathematical formalism, the energy U H nm can be expanded owing its analyticity into the Taylor series by the six independent variables H IK function in the neighborhood of “the point” with zeros coordinates H IK 0 (it is assumed that two first terms are zero because the initial state is zero) e  H IK



M N 1



¦ 1 M ! ª«¬w



M



M 1



ª¬e  H IK º¼ H IK



0



m1 m e  wH ij " wH sr N º» ¼ H IK



  we wH ij



H IK 0



 1 3! ª« w 3e wH ij wH kl wH mn H IK ¬



0



0



H



m1



ij



H ij  1 2! ª« w 2 e wH ij wH kl H ¬



" H sr



IK



0



mN



"



{ Cijkl º» H ij H kl  (2.11) ¼



{ Cijklmn º» H ij H kl H mn  ", m1  "  mN ¼



M.



The linear model of elastic deformation corresponds to saving in (2.11) only the quadratically nonlinear summands e  H IK



1 2! ª¬« w 2e wH ij wH kl H



IK



0



{ Cijkl º» H ij H kl ¼



.(2.12)



The equation (2.12) is obtained in assumption of arbitrary symmetry properties in the material and contains therefore 81 constants Ciklm , which can be formed together as the quadratic matrix 9 u 9. The based on this matrix 4th rank tensor is called the Definitionof 2.5. Definition The 2.5. symmetry Definition 2.5. tensor of elastic constants. strain tensor decreases the number of Definition 2.5. independent elastic constants from 81 to 36. For the next decreasing number of constants, the additional symmetry of a material is needed. The theory of elasticity Definition 2.5. considers mainly three kinds of symmetry Definition 2.5. 2.5. – orthotropy, transversal isotropy, and Definition isotropy. The classical linear theory of elasticity is traditionally concentrated on the case of isotropy. Definition 2.5. The material is called the isotropic one, when its mechanical properties are identical in any direction. Definition 2.5.



The formula for the internal energy U (2.12) generates a few formulas of the general character. Definition 2.4.



Euler’s formula on the uniform functions U



1 2  wU wH lm H lm . (2.13)



Clapeyron’s formula U



1 2 V lmH lm .(2.14)



Castigliano’s formula H lm



 wU wV lm . (2.15) 35



BASIC INFORMATION ON MECHANICS OF MATERIALS. THEORY OF ELASTICITY. SHORT DESCRIPTION OF LINEAR THEORY OF ELASTICITY



FOUNDATIONS OF MECHANICS OF MATERIALS PART 1



Betti’s formula for two stress-strain states  H lm ,V lm ,  H lm



,V lm



in the given point of body V lm H lm



V lm



H lm . (2.16)



To write the missing six equations, the first law of thermodynamics is used which in the considered here case is stated as equality of increments of the energy and the work of external volume and surface forces GK  u1 , u2 , u3 , U  GU  H11 ,..., H 23 G  F1 , F2 , F3  G  S1 , S2 , S3 ,



The formula (2.13) testifies that elasticity decreases the anisotropy level in materials, since it additionally increases the symmetry owing to equalities Cijkl



C jikl , Cijkl



Cijlk , Cijkl



Cklij .



The number of independent constants decreases from 36 to 21. The simplest case of symmetry (the highest symmetry) is the case of isotropy. Then the number of independent elastic cons- tants is 2 and formula (2.13) becomes very simple V ij



OH kk G ij  2PH ij . (2.17)



These equations are called the CONSTITUTIVE EQUATIONS. The constants O , P are called the Lame elastic constants. Six equations (2.17) together with equations (2.4)-(2.6), forms the basic system of 15 linear equations relative to 15 unknown functions uk , H nm ,V il . The most used in the linear theory is the system of Lame equations which follows from (2.4)-(2.6),(2.17) by excluding the strains and stresses and includes 3 coupled partial differential equations of the 2nd order U ui ,tt



P ui ,kk   O  P uk .ki  Fi . (2.18)



THE SYSTEM (2.18) IS THE BASIC ONE IN THE LINEAR THEORY OF ELASTICITY. It permits to solve mathematically the prevailing part of the problem on equilibrium (static problems) and motion (dynamic problems) of elastic bodies. The necessary boundary and initial conditions have the classical form. Three types of BOUNDARY CONDITIONS are mostly used:



36



BASIC INFORMATION ON MECHANICS OF MATERIALS. THEORY OF ELASTICITY. SHORT DESCRIPTION OF LINEAR THEORY OF ELASTICITY



FOUNDATIONS OF MECHANICS OF MATERIALS PART 1



Type 1. Three components of the displacement vector surface S uk  x1 , x2 , x3 , t



S



uko  x1S , x2 S , x3 S , t



G u



are given on the body



 x1S , x2 S , x3S  S .



Type 2. Three conditions of equality of values of the internal stresses on the body surface V ik nk S and the external surface force V ik nk



S



where



Si  x1S , x2 S , x3S , t , G n



^nk  x1S , x2 S , x3S , t ` is the normal to surface S .



Type 3. The surface is divided on two disjoint parts S Su  SV . The condition of Type 1 are given on the part Su and the condition of Type 2 are given on the part SV . These conditions are called the mixed conditions. Note 2.5. Different kinds of mixed conditions can be formed by the procedure, when at the point of surface are given one or two components from Type 1 and two or one components from Type 2. For example, in a case when S is the coordinate plane x1 x1o, the normal to the plane component of displacement u1  x1 , x2 , x3 , t S u1o  x1S , x2 S , x3S , t and two tangential components of surface stresses V1m







Sm x1o , x2 S , x3S , t



S







m



2,3



are given.



The classical INITIAL CONDITIONS in the linear theory of elasticity are formulated as an assignment of the displacements and velocities at the initial moment t o uk  x1 , x2 , x3 , t uko  x1 , x2 , x3 , t o , uk  x1 , x2 , x3 , t uko  x1 , x2 , x3 , t o .



Let us recall that here the brief information on the four parts of the theory of elasticity is stated. The linear theory of elasticity is chosen as the first one, because it forms the basic skeleton with which other three parts are constructed – strength of materials as the part with many simplifying assumptions that are used in the engineering practice and the linearized and nonlinear theories of elasticity as the parts with more exact and complicate models that extend significantly the area of application.



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Comments Comment 2.1. In mechanics, certain methods of substantiating the transition from a model of a discrete structure of a substance to a model of a continuum structure of a substance are known, based on a statistical description of discrete systems. As one of such methods, we indicate the procedure given in the book of Ilyushin quoted below. Here a system of a large number of fixed particles is considered. This system can be simple or complex. For example, a monatomic gas can be considered a simple system in which a particle (molecule) has three translational degrees of freedom. Diatomic gas is a complex system of molecules, where each molecule has three translational and two rotational degrees of freedom. Next, the total energy (kinetic + potential) of the system is studied. For a given function (including potential-internal energy) that describes the system, the average values over time and over the ensemble are determined, which, generally speaking, is characteristic of statistical mechanics, and not mechanics. That is, further the system of particles (molecules) is studied by the methods of statistical mechanics. For such a function, the ergodicity theorem is assumed to be valid - the mean values over time and over the ensemble coincide. Note 2.6. Just this approach was implemented by Kolmogorov in the axiomatization of probability theory and further by Noll and Truesdell in the axiomatization of mechanics. Note 2.7. The ergodicity theorem is physically meaningless if the characteristic time of the system is commensurate (longer) with the time it takes to determine the average values used in the theorem. It is believed that the average over the ensemble of functions characterizing a discrete system are indeed observable quantities. This allows us to move on to the macroscopic characteristics of the system and thereby use the concept of the continuum. In other words, if a macroscopically very small volume of a substance with a large number of microscopically small particles (molecules) is considered and a system with a macroscopically small time exceeding the characteristic time of the system is observed, then the system is equilibrium one and the ergodicity theorem holds for it. It is considered that statistical interpretations help in understanding the physical meaning in the formal transition from a discrete model of matter to a continuum. Comment 2.2. Let us start with the stated in this chapter sentence that the modern interpretation of research on solid mechanics relates them to three modern sections of science - applied mathematics, mechanical and other engineering, material science. In fact, the modern mechanics, as a science and a profession, have disappeared into the three modern



38



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sections of science mentioned above. The vast majority of universities in the world do not train specialists in mechanics. At the same time, about 60 developed countries of the world have the national organizations of mechanicians and are united in the International Union of Theoretical and Applied Mechanics (IUTAM) and European mechanicians are united in the European Mecha-nics Society (EuroMech). Each of three sciences - applied mathematics, mechanical and other enginee-ring, material science - treat mechanics in its way. For example, at one of the International Congress of Industrial and Applied Mathematics (ICIAM), the plenary lecturer formulated the essence of applied mathematics in this way: There is no special science applied mathematics but applied mathematicians nonetheless exist. These are specialists who use the achievements of mathematics for non-mathematical purposes, allowing the use of non-mathematical means to justify their actions. From this point of view, the assignment of mechanics to applied mathematics, perhaps, reduces the prestige of mechanics as fundamental science.



Further reading 2.1. A  scione, L & Grimaldi, A 1993, Elementi di meccanica del continuo (Elements of continuum mechanics). Mas simo, Napoli. (In Italian) 2.2. Ashby, MF 2005, Materials Selection in Mechanical Design, 3rd ed., Elsevier, AmsterdamTokyo. 2.3. Atkin, RJ & Fox, N 1980, An Introduction to the Theory of Elasticity, Longman, London. 2.4. Bell, JF 1973, Experimental foundations of solid mechanics. Flugge’s Handbuch der Physik, Band VIa / 1, pringer Verlag, Berlin. 2.5. Dagdale, DS & Ruiz, C 1971, Elasticity for Engineers. McGraw Hill, London. 2.6. Ericksen, JL 1998, Introduction to the Thermodynamics of Solids, Applied Mathematical Sciences, vol. 131, Springer, Berlin 2.7. Eringen, AC 1967, Mechanics of Continua. John Wiley, New York. 2.8. Eschenauer, H & Schnell, W 1981, Elasticitätstheorie I, Bibl. Inst., Mannheim. (In German) 2.9. Eslami, MR, Hetnarski, RB, Ignaczak, J, Noda, N, Sumi, N & Tanigawa, Y 2013, Theory of Elasticity and Thermal Stresses. Explanations, Problems and Solutions. Springer Series “Solid Mechanics and Its Applications,Springer Verlag Netherlands, Amsterdam. 2.10. Fraeijs de Veubeke, BM 1979, A Course of Elasticity. Springer, New York. 2.11. Fu, YB 2001, Nonlinear Elasticity: Theory and Applications. London Mathematical Society Lecture Note Series,Cambridge University Press, Cambridge. 2.12. Fung, YC 1965, Foundations of Solid Mechanics. Prentice Hall, Englewood Cliffs. 2.13. Germain, P 1973, Cours de mécanique des milieux continus. Tome 1. Théorie Générale. Masson et Cie Editeurs, Paris. (In French) 2.14. Gurtin, ME 1981, An Introduction to Continuum Mechanics. Academic Press, New York.



39



FOUNDATIONS OF MECHANICS OF MATERIALS PART 1



BASIC INFORMATION ON MECHANICS OF MATERIALS. THEORY OF ELASTICITY. SHORT DESCRIPTION OF LINEAR THEORY OF ELASTICITY



2.15. Hahn, HG 1985, Elastizitätstheorie. B.G.Teubner, Stuttgart. (In German) 2.16. Holzapfel, GA 2000, Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Birkhauser, Zurich. 2.17. Iliushin, AA 1990, Mechanics of Continuum, Moscow University Publishing House, Moscow. (In Russian) 2.18. Johns, DJ 1965, Thermal Stress Analysis, Pergamon Press, Oxford. 2.19.  Kobayashi, AS (ed) 1987, Handbook on experimental mechanics. Prentice-Hall, Englewood Cliffs. 2.20. Korotkina, MR 1988, Elektromagnetoelasticity, Moscow University Press, Moscow. (In Russian) 2.21. Love, AEH 1944, The Mathematical Theory of Elasticity. 4th ed. Dover Publications, New York. 2.22. Lur’e AI 1999, Theory of Elasticity. Springer Series in Foundations of Engineering Mechanics. Springer, Berlin. 2.23.  Lur’e, AI 1990, Nonlinear Theory of Elasticity. North-Holland Series in Applied Mathematics and Mechanics, North-Holland, Amsterdam. 2.24. Maugin, GA 1988, Continuum Mechanics of Electromagnetic Solids. North Holland, Amsterdam. 2.25. Müller, W 1959, Theorie der elastischen Verformung. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig. (In German) 2.26. Nowacki, W 1962, Thermoelasticity. Pergamon Press, Oxford. 2.27. Nowacki, W 1970, Theory of Elasticity. PWN, Warszawa. (In Polish, In Russian) 2.28. Podstrigach, JS & Povstenko, YZ 1985, Introduction into mechanics of surface phenomena in solids, Naukova Dumka, Kiev. (In Russian) 2.29. Prager, W 1961, Introduction to Mechanics of Continua. Ginn, Boston. 2.30. Ratner, LW 2003, Non-Linear Theory of Elasticity and Optimal Design. Elsevier, London. 2.31. Rushchitsky, JJ & Tsurpal, SI 1998, Waves in Materials with the Microstructure. S.P. Timoshenko Institute of Mechanics, Kiev. (In Ukrainian) 2.32. Savin, GN & Rushchitsky, JJ 1976, Elements of Mechanics of Hereditary Media. Vyshcha Shkola, Kyiv. (In Ukrainian) 2.33. Sedov, LI 1970, Mechanics of Continuum, in 2 vols. Nauka, Moscow. (In Russian) 2.34. Slaughter, WS 2001, Linearized Theory of Elasticity. Birkhauser, Zurich. 2.35. Sneddon, IN & Berry, DS 1958, The Classical Theory of Elasticity, vol.VI, Flügge Encyclopedia of Physics. Springer Verlag, Berlin. 2.36. Sokolnikoff, IS 1956, Mathematical Theory of Elasticity. McGraw Hill Book Co, New York. 2.37. Sommerfeld, A 1964, Thermodynamics and Statistical Mechanics. Academic Press, New York. 2.38. Spencer, AJM 1980, Continuum Mechanics. Longman, London.



40



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2.39. S  tarovoitov, E & Naghiyev, FBO 2012, Foundations of theTheory of Elasticity, Plasticity, and Viscoelasticity, Apple Academic Press, Palo Alto. 2.40. Storakers, B & Larsson, P-L 1998, Introduktion till finit elasticitetteori. Hallfasthetslara, KTH. (In Swedish) 2.41.  Taber, LA 2004, Nonlinear Theory of Elasticity: Applications in Biomechanics. Birkhauser, Zurich. 2.42. Timoshenko, SP & Goodyear, JN 1970, Theory of Elasticity, 3rd ed. McGraw Hill, Tokyo. 2.43. Truesdell, C 1969, Rational Thermodynamics. McGraw-Hill Book Company, New York. 2.44. Truesdell, C 1972, A First Course in Rational Continuum Mechanics. The John Hopkins University, Baltimore.



Questions 2.1. W  hich more complicated combinations of mechanical properties of materials (for example, the property of gyroelasticity) exist, occurring in the real practice and reflecting in the me chanical theories and do not mention in the chapter? Indicate the degree of development of such theories. 2.2. Is the property of viscoelasticity characteristic for materials only? By other words, it is possible to speak about the viscoelastic materials and the viscoelastic fluids? 2.3. Formulate similarity and distinction between rheology and viscoelasticity. 2.4. Presence of which properties will need the attraction of thermodynamical considerations when the mechanical model being created? 2.5. Which property of material causes energy dissipation when the material being deformed? 2.6. Find the not mentioned here classifications of mechanics and compare them with the basic ones. 2.7. Which text-book on the theory of elasticity you prefer and which book you would add to the proposed list above? Formulate the advantages of preferred book.



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3 SOME ADDITIONAL FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY



Three kinds of basic mathematical problems. Direct and inverse problems. Static and dynamic problems. General theorems. Examples: Saint-Venant’s principle, Kirchhoff’s theorem on uniqueness. Three-dimensional problems. Representations of the general solution: Papkovich-Neuber’s representation, Boussinesq-Galerkin representation, Lame’s representation, Lowe’s representation, Helmholtz’s representation, Boussi- nesq’s representation. TwoThreeand kindsin-plane of basic states. mathematical problems. Direct and inv dimensional problems. Anti-plane dynamic problems. General theorems. Examples: Saint-Ven Airy’s function. Plane and stressDirect states. Three kinds of basicdisplacement mathematical problems. and inverse problems. Static and theorem on uniqueness. Three-dimensional problems. Repr dynamic problems. General theorems. Examples: Saint-Venant’s principle, Kirchhoff’s Generalized plane stress state. Kolosov-Muskhelishvili’s solution: Papkovich-Neuber’s representation, Boussinesqtheorem on uniqueness. Three-dimensional problems. Representations of the general functions. Levi’s theorem. Kirsch’s problem. Method Lowe’s of Lame’s representation, representation, Helmholtz’s solution: Papkovich-Neuber’s representation, Boussinesq-Galerkin representation, nesq’s representation. Two-dimensional problems. Anti-plan conformal mappings. One-dimensional problems. Universal Lame’s representation, Lowe’s representation, Helmholtz’s representation, BoussiAiry’s function. Plane displacement and stress states. Gene nesq’s representation. Two-dimensional problems. Anti-plane and in-plane states. deformations. Rods and beams. Torsion and bending. Kolosov-Muskhelishvili’s functions. Levi’s theorem. Kirsch’s hree kinds of basic mathematical problems. Direct and inverse problems. Static and states. Airy’s function. Plane displacement and stress plane stress state. Plates and shells. Dynamic problems - vibrations and Generalized problems. Universal defo formal mappings. One-dimensional ynamic problems. General theorems. Kolosov-Muskhelishvili’s Examples: Saint-Venant’sfunctions. principle,Levi’s Kirchhoff’s theorem. Kirsch’s problem. Method of conTorsion bending. Plates and shells. Dynamic problems waves. Approximate numerical methods (FDM,FEM,BEM). eorem on uniqueness. Three-dimensional problems. ofand the general problems. Universal deformations. Rods and beams. formal mappings.Representations One-dimensional proximate numerical methods (FDM,FEM,BEM). olution: Papkovich-Neuber’s representation, Boussinesq-Galerkin representation, Torsion and bending. Plates and shells. Dynamic problems - vibrations and waves. Apame’s representation, Lowe’s representation, representation, BoussiproximateHelmholtz’s numerical methods (FDM,FEM,BEM). First, the linear theory of elasticityAnti-plane formulates three kindsstates. of basic mathematical problems: esq’s representation. Two-dimensional problems. and in-plane of basic mathematical problems. Direct plane and inverse Static and ry’s function.Three Planekinds displacement and stress states. Generalized stressproblems. state. dynamic problems. General theorems. Examples: Saint-Venant’s principle, Kirchhoff’s olosov-Muskhelishvili’s functions. Levi’s theorem. Kirsch’s problem. Method of conThe 1stonbasic problem consists in the determination of the stressoftensor in the area V theorem uniqueness. Three-dimensional problems. Rods Representations the general problems. Universal deformations. and beams. rmal mappings. One-dimensional solution: Papkovich-Neuber’s representation, Boussinesq-Galerkin representation, orsion and bending. Plates problems - vibrations and and its boundary S when occupied byand theshells. body Dynamic and the displacement vector in thewaves. area VApLame’smethods representation, Lowe’s representation, Helmholtz’s representation, Boussiroximate numerical (FDM,FEM,BEM). the external volume and surface forcesproblems. are given.Anti-plane and in-plane states. nesq’s representation. Two-dimensional Airy’s function. Plane displacement and stress states. Generalized plane stress state. Kolosov-Muskhelishvili’s functions. Levi’s theorem. Kirsch’s problem. Method of connd The 2mappings. basic problem consists problems. in the determination of the displacement vector and the Universal deformations. Rods and beams. formal One-dimensional Torsion bending. shells. Dynamic - vibrations and waves. by the problems body when the external volumeApforces and stress and tensor in thePlates area Vandoccupied proximate numerical methods (FDM,FEM,BEM).



displacements on the surface



S



are given.



The 3rd basic problem (mixed problem) consists in the determination of the stress tensor and the displacement vector in the area V occupied by the body when the external volume forces and the mixed boundary conditions are given. The stated above basic problems form together the direct problem. It consists in solving one of the basic problems and determination nine functions (six components of the stress tensor and three components of the displacement vector) depending on the external action on the body. The inverse problem consists in that the displacements or stresses are given, and then rest unknown functions including the external forces should be found. 42



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Note 3.1. Solving the inverse problem is essentially simpler. For example, when the displacements are given. They have to be continuously differentiable and then the Saint-Venant’s relations are fulfilled identically. Further, the stresses are determined from the constitutive equations and the external forces are determined from the equilibrium equations and boundary conditions. To the point, the Saint-Venant’s relations are called the compatibility conditions (the continuum after deformation rests the continuum) and they express the fact that displacements are not independent. These relations have in the Cartesian coordinates the form H mq ,np  H np ,mq  H nq ,mp  H mp ,nq



0 . (3.1)



The linear theory of elasticity divides the problems on the static and dynamic ones what means that in the first case the equilibrium equations and boundary conditions should be considered and in the second case – the motion equations and corresponding boundary and initial conditions. The static problems include first the set of general theorems giving the mathematical tools in the study of problems. These theorems include the existence and uniqueness theorems and many other theorems and statements that have a name (Castigliana’s theorem, Clapeyron’s theorem, Green’s theorem, etc). Consider briefly, as an example, such general statement named the Saint-Venant’s principle. The principle, later called the Saint-Venant’s principle, was formulated in the study of the problem of deformations arising in the loaded cylinders and prisms. SaintVenant studied the possibility of an approximate solution to the problem in the case when the load on the end face is statically equivalent, but not identical to the load for which he built the exact solution. As a result, the principle of comparing the strain distributions caused by statically equivalent loads was described. The first general exposition of the Saint-Venant’s principle is given by Boussinesq: A balanced system of external forces applied to an elastic body, all points of application of which lie inside a given sphere causes deformations of a negligible magnitude at distances from the sphere that are quite large compared to its radius. In general, the principle of Saint-Venant’s has a long history. For example, Lowe noted that the Saint-Venant’s principle is “known as the principle of the elastic equivalence of statically equipollent systems of load”. In the modern linear theory of elasticity, the Saint-Venant’s principle justification scheme is such that the rate of attenuation of the elastic energy of the deformation of a cylindrical body is estimated



43



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using the lowest natural frequency of the body part. The body is assumed to have an arbitrary regular cross-section. A self-balanced system of forces is applied to one of the ends of the cylinder; the body does not experience other external actions. The problem is to determine the regularity of attenuation of the elastic energy of deformation (internal energy) of the cylinder with a distance from the loaded end. This principle has great importance for theory and engineering practice. As one more example, the Kirchhoff’s theorem on uniqueness of solutions of the direct problems of the linear theory of elasticity is shown below. This theorem has two basic assumptions – the elastic body has any discontinuities and any initial deformations. Then the basic equations of the linear theory of elasticity have a unique solution. The proof is transparent and accurate on each step. Step 1. It is formulated by contradiction. So, let two non-identical solutions uk , uk



exist, which fulfill the basic equations and boundary conditions in displacements. Step 2. Write for both solutions the equations of equilibrium for an isotropic body P u i ,kk   O  P u k .ki  Fi



P u



i ,kk   O  P u



k .ki  Fi



0 , (3.2) 0 , (3.3)



and boundary condition on S Su  SV V ik nk



V



ik nk



SV



SV



Pi  x1S , x2 S , x3S , t



 x1S , x2 S , x3S  SV , (3.4)



Pi  x1S , x2 S , x3S , t



u k  x1 , x2 , x3 , t



u



k  x1 , x2 , x3 , t



Su



Su



 x1S , x2 S , x3S  SV , (3.5)



uko  x1S , x2 S , x3S , t



 x1S , x2 S , x3S  Su , (3.6)



uko  x1S , x2 S , x3S , t



 x1S , x2 S , x3S  Su (3.7)



are fulfilled. Step 3. Introduce new denotations  uk



 uk  uk



, H k







H k  H k



, V ik



V ik  V ik



(3.8)



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and subtract equation (3.3) from equation (3.2) and boundary conditions (3.5) from (3.4) as well as (3.7) from (3.6). As a result, new formulas are obtaining   P ui ,kk   O  P uk .ki 



V ik nk



0



SV



0 , (3.9)



 x1S , x2 S , x3S  SV ,



 uk  x1 , x2 , x3 , t



Su



0



 x1S , x2 S , x3S  Su (3.10)



Note 3.2. The formulas (3.9),(3.10) testify that the solution (3.8) corresponds to zero external forces and boundary conditions. So, it is necessary to prove that this solution corresponds to the non-deformed state. Step 4. Write the internal energy (work of deformation) of the body for the state of deformation (3.8) U







U  H ij



 







³ ª¬ PH mnH mn  1 2 O H kk



V



2º



¼



dV (3.11)



and recall that the work (3.11) is equal to the work of external and surface forces, which is zero for the state (3.8). Thus  







³ ª¬ PH mnH mn  1 2 O H kk



V



2º



¼



dV



0 (3.12)



Because the Lame constants O , P are non-negative, then the integrand is always positive. This means that 



H mn



0 o H mn



. (3.13) H mn



It follows from the generalized Hooke’s law (link between stresses and strains) that the stresses are also zero 



V mn



0 o V mn



. (3.14) V mn



 Step 5. To prove the identity of displacements uk uk  uk



0 , it is necessary to integrate the strains 



H mn



1 2  um,n  un,m



0 . (3.15)



The equation (3.15) has a solution, from which only one conclusion follows: the displacements are distinguished by only the motion of the body as the rigid body what means that the body is non-deformed.



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Thus, the initial assumption that two solutions of the identical equations of equilibrium and boundary conditions can be different becomes false. It is believed that the proof is finished and the solution of the basic problem of the linear theory of elasticity is unique. Quod erat demostrandum. After the general theorems, the three sections of static problems are considered separately: Section 1. Three-dimensional (spatial) problems. Section 2. Two-dimensional problems (including the plane problems). Section 3. One-dimensional problems. Section 1 includes a lot of solved particular problems and a row of fundamental facts. Let us show here some of these last and consider the case of isotropy of mechanical properties only. To begin with, note that the basic system of equations in this section is as follows P ui ,kk   O  P uk .ki  Fi



0 (3.16)



G



G



G



or P'u   O  P grad div u  F 0 G



In the general analysis of equation (3.16), the case when the external force F is absent and the body can be loaded by only the surface forces is very fruitful. The equation (3.16) is simplified to the form ui ,kk  ª¬1   O P º¼ uk .ki



0 . (3.17)



Differentiation of equation (3.16) by the coordinate xi gives new equation ui ,kki  1   O P uk .kii



0,



from which follows that the dilatation e uk .k is the harmonic function 'uk ,k 'e 0 . Apply further the Laplace operator to equation (3.17). As a result, the very interesting fact is noted – the components of the vector of displacements are the biharmonic functions ''uk 0. Afterward, the harmonic and biharmonic functions became the big place in the mathematical analysis of equations of the linear theory of elasticity. For example, Almansi showed that the functions M  xk\ fulfill the biharmonic equation when the functions M ,\ are the harmonic ones. The next example is referred to Trefftz, who considered the elastic half-space x3 t 0 and chose the displacements in the form uk Mk  x3\ k which includes six harmonic functions.



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So, the main advantage of the analysis of the basic equation by the use of harmonic and biharmonic functions is that the equation to be solved became essentially simp-ler. The linear theory of elasticity contains some class of general representations of solutions or general solutions, the basic concept of which consists in using the harmonic and biharmonic functions. The most known is Papkovich-Neuber’s representation



uk



M,k   xn\ n ,k  4 1  X \ n (3.18)



or



G u



G G G grad M  r ˜\  4 1  X \ ,



where the new unknown functions (one scalar function and one vector function) fulfill the equations G G 4 1  X P '\  F



G G 0, 4 1  X P 'M  r ˜ F



0.(3.19) G



Note 3.3. In the case when the external force F is absent, the equations (3.19) have the solutions in the form of harmonic functions and the representation (3.18) became very fruitful. It can use the big set (catalog) of different harmonic functions. The next variant of representation of the general solution is called Boussinesq-Galerkin representation G u



G



G



1 2P grad div G  2 ª¬1  X P º¼ 'G ,(3.20)



where G G ''G  ª¬1  O  2 P º¼ F



0 .(3.21)



G



Note 3.4. In the case when the external force F is absent, the equation (3.21) is transformed in the biharmonic one and has the solution in the form of biharmonic functions. The simplest general representation uses one only scalar function M  x1 , x2 , x3 . It is called Lame’s representation or the elastic displacement potential.



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In the case of absence of the external force, the function M  x1 , x2 , x3 is a harmonic one, and this representation gives the very simple formula uk



M,k .(3.22)



Consider now the case of axisymmetric state. This means some simplification of the equilibrium equations (3.12) and needs its representation through the stresses and in the cylindrical coordinates ^r ,M , z` V rr ,r  V rz , z  1 r V rr  V MM V rz ,r  V zz , z  1 r V rz



0 , (3.23)



0 . (3.24)



The simple general representation of solutions of equations (3.23),(3.24) is Lowe’s representation through one biharmonic function F  r , z ur



 F ,rz , u z



 F , zz  2 1 Q 'F , (3.25)



V rr



2 P ª¬  F ,rr  Q 'F º¼ , z , (3.26)



V MM



2 P ª¬  1 r F ,r  Q 'F º¼ , z (3.27)



V zz



2 P ª¬  F , zz  2 1 Q 'F º¼ , z ,(3.28)



V rz



2 P ª¬  F , zz  2 1 Q 'F º¼ ,r . (3.29)



Note 3.5. In the proof of relations (3.25)-(3.29), the starting point is using Helmholtz’s representation ur



 1  2Q ) ,r  2 1 Q < , z , (3.30)



uz



 1  2Q ) , z  2 1 Q 1 r  r < ,r , (3.31)



One more general representation for the axisymmetric state is Boussinesq’s representation ur



M  z\ ,r ,



V rr



2 P ª¬M  z\ ,rr Q ' M  z\ º¼ , (3.33)



uz



M  z\ , z  4 1 Q \ , (3.32)



48



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SOME ADDITIONAL FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY



V MM



2 P ª¬1 r M  z\ ,r Q ' M  z\ º¼ , (3.34)



V zz



2 P ª¬M  z\ , zz   2 Q ' M  z\ º¼ , (3.35)



V rz



2 P ª¬M  z\ ,rz  2 1 Q \ ,r º¼ , (3.36)



' M  z\ 2\ , z



Thus, the static basic problems have the power analytical tool for solving diverse three-dimensional problems. At that, mechanics of materials has also the modern experimental base and reach experience in experiments as well as all arsenal of computational mechanics. Nowadays, any problem on the statics of the threedimensional linearly elastic body of the complex shape can be analyzed and solved with some level of exactness. Section 2 traditionally considered plane problems. But also some facts of the general two-dimensional state are presented in this section. Let the stress-strain state depends only on two coordinates x1 , x2. Then the stress-strain state can be described by two different ones: the anti-plane and in-plane states. The ANTI-PLANE STATE is characterized by only the vertical component of the displacement G vector u  x1 , x2 ^0;0; u3  x1 , x2 ` . The equilibrium equation takes on a look V 3E ,E



0 D,E



1;2



u3 E ,E



or



0 .(3.37)



The IN-PLANE STATE is characterized by only two components of the displacement G vector u  x1 , x2 ^u1  x1 , x2 ; u2  x1 , x2 ;0` . But further analysis is brought up on two cases – the plane displacement state and the plane stress state. They will be discussed later. And now one fundamental fact should be shown. Let us write the corresponding to the in-plane state equations of equilibrium V ED ,E



0, V E 3,E



0 D,E



1;2 .(3.38)



Now, the Airy’s function F  x1 , x2 is introduced V DE



 F,DE  GDE F,JJ , D , E , J



1;2 ,(3.39)



which fulfills the first equation in (3.38) and is the two-dimensional biharmonic function



49



FOUNDATIONS OF MECHANICS OF MATERIALS PART 1



''F



0



 'f



SOME ADDITIONAL FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY



f ,11  f ,22 .(3.40)



To fulfill the second equation, it is necessary to introduce one more function \  x1 , x2 , for determination of which the two-dimensional Poisson’s equation should be solved '\



c (c is arbitrary constant).



(3.41)



Return now to the plane displacement and stress states. They correspond to different real mechanical situations. The plane displacement state arises in the analysis of deformation of the long in direction Ox3 prismatic body loaded by the surface forces which do not depend on the coordinate x3 . It is believed then that in any cross-section of the body the plane deformed state is formed. The plane stress state arises in the analysis of the deformation of the small thickness plate loaded by the forces in the plate plane. In some conditions, the plate is undergoing the plane stress state. Plane displacement state. Mathematical description. The equations of equilibrium include two equations and have the form (3.38). The correspond-ding Lame’s equations have the form of the system of two equations



 O  P  u,1  u,2 ,D  P'uD



0 ,(3.42)



The constitutive equations are as follows V 11



¬ª E 1  Q 1  2Q ¼º ¬ª1 Q H11  Q H 22 ¼º , (3.43)



V 22



ª¬ E 1  Q 1  2Q º¼ ª¬1 Q H 22  Q H11 º¼ , (3.44)



V 33



¬ª E 1  Q 1  2Q ¼º  H11  H 22 ,(3.45)



V 12



2 PH12 .(3.46)



The Cauchy relations are also simple and linear H DD



uD ,D , H12



1 2  u1,2  u2,1 .(3.47)



Definition 3.1. Definition Plane 3.1.



stress state. Mathematical description.



Definition 3.1.



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The equations of equilibrium (3.38) and Cauchy relations (3.46) remain and coincide with the prior case. Definition 3.1. The body is in the plane stress state, if Definition 3.1.



V 33 V D 3



Definition 3.1. 0 .(3.48)



Definition 3.1. So, 3.1. the Definition Definition 3.1.



Definition 3.1.



initial assumption about stresses is (3.48), whereas in the case of the plane displacement state the corresponding assumption is another one: V D 3 0.



3.1. Therefore, theDefinition constitutive equations differ from (3.43)-(3.46) Definition 3.1.Definition 3.1.



^



`,(3.49)



^



` ,(3.50)



V 11



2 P H11  ª¬Q 1 Q º¼  H11  H 22



V 22



2 P H 22  ª¬Q 1 Q º¼  H11  H 22



V 12



2 PH12 .(3.51)



But the problem is the three-dimensional and G u  x1 , x2 , x3



^u  x , x , x ; u  x , x , x ; u  x , x , x `. 1



1



2



3



2



1



2



3



3



1



2



3



The equations of equilibrium are identical with the corresponding equations of the plane displacement state (3.42) if the Lame constant O in (3.42) is changed on the constant  O ¬ª 2OP  O  2P ¼º . Thus, in both cases, the same mathematical equation should be solved. The problem on the plane stress state can be generalized and simultaneously reduced to the problem on the plane displacement state. For this, the plate is considered which is loaded only at its contour by the surface forces. These forces have to be the symmetric relative to its middle plane Ox1 x2 . In this statement, the problem is three-dimensional. But in the assumption that the plate is sufficiently thin, the components of stress tensor can be averaged over the thickness. Then the state of the plate does not depend on coordinate x3. This state is called the generalized plane stress state. All basic equations of the generalized plane stress state are identical with the corresponding equations of the plane displacement state. Thus, despite three different from point of view of mechanics statements, the one only kind of equations should be analyzed.



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Now, some words about the tools that are used in the analysis of the plane problems of the linear theory of elasticity. First, the Airy’s function should be mentioned, that leads to the analysis of the biharmonic equation and is historically the pioneer one. Second, the method of complex stress functions should be described as probably most known. At the beginning, the complex variable z x  iy rei- , z x  iy rei- is introduced and then all basic relations are written through this variable.The biharmonic equation in the complex variable is as follows F, zzzz



0 .(3.52)



In the case of the simple-connected body, the function F can be represented by two holomorphic (regular) functions what is expressed by the Goursat’s formula F  z Re ª¬ zM  z  \  z º¼ .(3.53)



In the theory of elasticity, the functions M  z ,\  z are called the KolosovMuskhelishvili’s functions. The next formulas are important in the analysis of the plane problems V11  z  V 22  z 4 Re M c  z 2 ª¬M c  z  M c  z º¼ ,(3.54) V11  z  V 22  z  2V12



2 ª¬ zM cc  z  \ c  z º¼ ,(3.55)



2 P ª¬u1  z  iu2  z º¼ NM  z  zM c  z  \  z .(3.56)



The constant N is different for different plane states: N



3  4Q for the plane displacement state and N



¬ª 3 Q 1  Q ¼º



for the plane stress



state. The general structure of the Kolosov-Muskhelishvili functions depends on the connectedness of the area. In the case of simple-connected area, they can be represented in the power series M z



n f



¦a z n 0



n



n



,\  z



n f



¦ b z .(3.57) n 0



n



n



52



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SOME ADDITIONAL FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY



In the case of two-connected (ring-shaped) areas they have another representation M  z  ^ X  iY ª¬ 2S 1  N º¼` ln  z  z  \ z



^N



n f



¦ a z  z ,



n f



n



n f



n



`  X  iY ln  z  z  ¦ b  z  z , (3.58)



ª¬ 2S 1  N º¼



n f



n



where X  iY characterizes the force flow over the internal contour of the area. Note 3.5. The Levi’s theorem is very useful in the analysis of the multi-connected areas: If the resultant vector of forces applied to each contour separately is zero, then the solution expressed by the Kolosov-Muskhelishvili potentials does not depend on the elastic constants. Basing on this theorem, the stress state in the body made of a certain material can be determined using the body made of another material. This is used in the photoelasticity. The classical example of the application of formulas (3.58) is Kirsch’s problem. This problem is solved by Kirsch in 1898 and is shown in many books on the linear theory of elasticity. It corresponds to the real problem on the plate of essential length and width with the small hole of radius r o placed at the center of the plate. The plate is undergoing the action of the uniform tension stress V xx V o along the longitudinal direction. Usually, the solution in stresses is written in the polar coordinates V rr  r ,-



V 2 ^ª«¬1   r r º»¼  ª«¬1  4  r r



V --  r ,-



V 2 ^ª«¬1   r r º»¼  ª«¬1  3 r r º»¼ cos 2-` ,(3.60)



o



o



o 2



o 2



o 2



4  3  r r o º» cos 2¼



` ,(3.59)



o 4



V r-  r ,-  V o 2 ª«1  2  r r o  3  r r o ¬ 2



4



º sin 2- .(3.61) »¼



The formulas (3.59)-(3.61) are simplified at the contour of the hole and the stresses take here the maximal values V --  r o ,- V o 1  2cos 2- , V rr  r o ,- V r-  r o ,- 0 .(3.62)



Thus, at the point - S 2 , the stress V -- takes the value 3 and attenuated very quickly with increa-sing the distance from the point to the hole. It is believed that this is one of the local effects. The Kirsch problem opened the new direction in the linear theory of elasticity – the stress concentration around holes, necks, cuts et cetera.



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Note 3.6. Often, the notion of stress concentration factor is used. Usually, it is applied to the value of stress at a given point. It shows the ratio of the maximal local stress to the value of stress without the stress concentrator. The progress in solving the plane problems of the linear theory of elasticity is impressive from a different point of view. Let us stop in one direction – the development of new methods of solving the problem. The method of conformal mapping is worthy to be mentioned here above all. It allows not only to solve many important partial problems, but it also was developed essentially from point of view of mathematics. The method of conformal mapping transforms the basic relations of the plane problem from the Cartesian coordinates into the curvilinear ones and permit in that way to simplify the procedure of solving. Note 3.7. The conformal mapping of the certain simple-connected area of the plane ] [  iK into the analogical area of the plane z x  iy is realized by the analytical function z z ] . The word conformal is used here because this kind of mapping preserves the angles between two curves at the point of their intersection. The simplest example of using conformal mapping. The exterior of the circle of radius r o is mapped into the interior of the unit circle. The mapping function has the form z ]



r



] .(3.63)



o



Then for the case, when the infinite plate with the circular hole of the radius r o is stretched in direction Ox by the constant stress V o , a solution in the complex potentials is as follows M z



V 4 ª«¬ z   2  r o



o 2















2 2 z º , \  z  V o 2 ª z   r o 1 z   r o z 3 º .(3.64) »¼ «¬ »¼



The considered problem is, in fact, the Kirsch’s problem. Therefore, the corresponding stresses are shown by formulas (3.59)-(3.61).



Section 3. One-dimensional problems The analysis of these problems is caused by their simplicity and great technical importance. So, if the simplicity of the class of one-dimensional problems is mentioned, then first of all the one more class of problems should be considered here. They are also simple and one-dimensional by the mathema- tical description. At that, they have a general character and unusual practical applications.



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Thus, the class of universal deformations should be briefly considered. The universal deforma- tions (uniform deformations, universal states) occupy a special place in the theory of elasticity just owing to their universality. It consists in that the theoretically and experimentally determined elastic constants of material in samples, in which the universal deformation is created purposely, are valid also for all other deformed states both samples and any different products made of this material. It is considered therefore that the particular importance of universal deformation (their fundamentality) consists of a possibility to use them in the determination of properties of materials from tests. To realize the universal deformation, two conditions have to be fulfilled: 1. Uniformity of deformation must not depend on the choice of material. 2. The deformation of material has to occur by using only the surface loads. In the theory of infinitesimal deformations, the next kinds of universal deformations are studied in detail: simple shear, simple (uniaxial) tension-compression, uniform volume (omniaxial) tension-compression. In the linear theory of elasticity, the experiment with a sample, in which the simple shear is realized, allows determining the elastic shear modulus P . The experiment with a sample, in which the uniaxial tension is realized, allows determining Young’s elastic modulus E and Poisson’s ratio Q . The experiment with a sample, in which the uniform compression is realized, allows determining the elastic bulk modulus k . While being passed from the linear model of very small deformations to the models of nonsmall (moderate or large) ones, that is, from the linear mechanics of materials to nonlinear mechanics of materials, the universal states permit to describe theoretically and experimentally many nonlinear phenomena. The history of mechanics testifies the experimental observation in the XIX century of the nonlinear effects that arose under the simple shear and were named later by names of Poynting and Kelvin. After about a hundred years in the XX century, these effects were described theoretically within the framework of the nonlinear Mooney-Rivlin model. The mechanics of composite materials is one more area of application of universal deformations. The simplest and most used model, in this case, is the model of averaged (effective, reduced) moduli. In the theory of effective moduli, the composite materials of the complex internal structure with internal links are treated usually as the homogeneous elastic media. A possibility to create in such media the states with universal deformations was used in the evaluation of effective moduli by different authors and different methods.



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It was found that it is sufficient for isotropic composites to study the energy stored in the elementary volumes of composites under only two kinds of universal deformations: simple shear and omniaxial compression. Universal deformation of simple shear. The experiments on simple shear are realized on the sufficiently long beam of quadratic cross-section, in which the uniform deformation is created on some distance from the ends. The lower side of the beam is fixed rigidly and the surface tangential constant load T2 is app- lied to the upper side. The deformation of the beam can be described by one component of the deforma-tion gradient u1,2  wu1 wx2 . The component u1,2 and the shear angle J are linked as follows u1,2



tan J



W ! 0. (3.65)



In the linear theory, the shear angle is assumed to be small and then J | tan J W . The Cauchy-Green strain tensor is characterized by only three nonzero components H11



1 2  u1,1  u1,1  u1,k u1,k 1 2  u1,2u1,2  u1,3u1,3 W 2 ;



H12



H 21



1 2  u1,2  u2,1  u1,k u2,k 1 2 W .



The principal extensions are written through the shear angle by formulas O1 1, O2 O3 W . Universal deformation of uniaxial tension. A rod in the form of a straight long cylinder (of circular or quadratic cross-section)with the axis in direction of axis Ox1 is considered when the lateral surface of the rod is free of stresses. The rod is stretched in the axial direction. Then the uniform stress-strain state is formed in the rod except for the area near the ends. It is characterized by only one nonzero component V 11 of the stress tensor and two nonzero components H11 , H 22 H 33 of the strain tensor (or two principal extensions O1 , O2 O3 ). This kind of deformations is used for the introduction of the Young modulus and Poisson’s ratio instead of two classical Lame elastic constants. Perhaps, the oldest and exhausting procedu-res are shown in classical Love’s book. Let us use the adopted at that time notations and write the standard representation of the Hooke law through the Lame moduli O , P Xy



2 PH xy ; Z x



2 PH zx ; Yz



Xx



O'  2 PH xx ; Y y



2 PH yz ,



O'  2 PH yy ; Z z



O'  2 PH zz ; (3.66)



where the notation of dilatation is used ' H xx  H yy  H zz .



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The classical procedure of introducing the Young modulus and the Poisson’s ratio is shown below. Toward this end, the universal deformation of uniaxial tension is considered, when the axis is chosen in direction Ox and the prism is stretched at the ends by a uniform tension T . The stress state of a prism is uniform and is characterized by only the stress X x T (other stresses are zero ones). In this case, the Hooke law becomes simpler T O'  2 PH xx , 0 O'  2 PH yy , 0 O'  2 PH zz .



The expression for dilatation is obtained by adding all three equalities above T



(3O  2 P ) ' o '



T (3O  2 P ).



The substitution of the last expression for dilatation into the first equality (3.66) gives relations T



ª¬O  3O  2 P º¼ T  2 PH xx o T



ª¬ P  3O  2 P  O  P º¼ H xx .



The last expression represents the elementary law T EH xx of link between tension and strain of prism, in which the Young modulus E is used. Comparison of this law with relation (3.66) gives the classical expression for the Young modulus through the Lame moduli E



ª¬ P  3O  2 P  O  P º¼ . (3.67)



The substitution of expression for dilatation into the second and third equalities (2) gives relations H yy



H zz



ª¬ O 2  O  P º¼ H xx ,



which express the classical Poisson’s law on the transverse compression under the longitudinal extension and permit to introduce the Poisson’s ratio V



 H



yy



H xx



 H zz H xx  O 2  O  P . (3.68)



Thus, the Poisson’s ratio is one of the characteristics of linear deformation of elastic material and is considered as the basic notion of linear elasticity. But the ratio of transverse strain to the longitudinal one can be used in any model of nonlinear elasticity (and not only elasticity).



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In this case, this ratio will have its representation in each model and possibly will not be constant quantity for any level of strain. Universal deformation of uniform (omniaxial) compression-tension. A sample has the shape of a cube, to sides of which the uniform surface load (hydrostatic compression) is applied. Then the uniform stress state is formed in the cube. The normal stresses are equal with each other V 11 V 22 V 33 , and the shear stresses V ik  i z k are absent. This type of universal deformation is defined by the following components of displacement gradients u1,1



u2,2



u3,3



H ! 0; u1,1  u2,2  u3,3



3H



e; uk ,m



 wuk



wxm 0  k z m . (3.69)



The Cauchy-Green strain tensor is as follows H11 H 22



H 33 H  1 2 H 2 ; H ik



0



 i z k , (3.70)



and the algebraic invariants are written in the form I1



H11  H 22  H 33



e; I 2



H11



2



  H 22   H 33 ; I 3 2



2



3 3 3 H11  H 22  H 33 . (3.71)



The principal extensions are equal to each other O1 O2 O3 . Definition 3.2.



NoteDefinition 3.8. The3.2. universal deformations will be also discussed in the following chapters. Definition 3.2.



Most often, Definition the linear 3.2.theory of elasticity relates to the one-dimensional problems the Definition 3.2. problems on the deformation of the RODS and BEAMS. Definition 3.2. The rod is an elongated body, the two dimensions of which (height and width) are small compared to the third size (length). In the same sense, the term “beam” is sometimes used. Note 3.9. Usually, the term “rod” refers to bodies of elongated shape, which resist only the forces of tension-compression and torsion. Note 3.10. Usually, the term “beam” refers to bodies of elongated shape, which resist only the forces of bending. Thus, three kinds of deformations are studied for the rods and beams – the tensioncompression, torsion, and bending. The tension-compression causes the elongation strains in the rod and saves the middle line of the rod (no deflection). The torsion causes the shear strains in the rod and no deflection. The bending causes the deflection of the beam. Definition 3.3. Definition 3.3. Definition 3.3. Definition 3.3. Definition 3.3.



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IN FACT, THE RODS AND BEAMS CAN BE ANALYZED IN THE LINEAR THEORY OF ELASTICITY AS THE SPATIAL BODIES WITHIN THE FRAMEWORK OF THE THREE-DIMENSIONAL STATEMENT. BUT THE SOLVING OF THESE PROBLEMS IS DIFFICULT AND MEETS MANY MATHEMATICAL PROBLEMS. THEREFORE, THE LINEAR THEORY OF ELASTICITY DEVELOPED DIFFERENT APPROXIMATE APPROACHES THAT PERMIT TO ANALYZE THE RODS AND BEAMS AS THE ONE-DIMENSIONAL PROBLEMS. THIS IS AN OBJECT OF THE STRENGTH OF MATERIALS, WHICH WILL BE DISCUSSES IN CHAPTER 7. The next after rods and beams objects of analysis of the linear theory of elasticity are PLATES and SHELLS. Most of the textbooks on the linear theory of elasticity do not include the theories of plates and shells. These theories are thinking as some separate part of the general theory of elasticity. Definition 3.3. The plate is defined as the body of finite or infinite sizes bounded by two planes symmetric relative to some plane (middle plane; usually, the coordinate plane Oxy ) one dimension of which (height, thickness) is smaller than two other dimensions (length and width). Thus, the plate is characterized by the length, width, and thickness The plates are divided into thin and thick ones. The THIN PLATE is restricted by the values of the ratio of the smaller side (length or width) to the thickness ! 10. The thin plates are divided on the stiff and flexible ones. The criterion is here the value of the ratio of the smaller side (length or width) to the deflection of the plate. The small values (0.2-0.5) correspond to the stiff plate, and the values ! 0.5 correspond to the flexible plates (membranes). In the analysis of the thin stiff plates, some basic hypotheses are used. The 1st Kirchhoff hypothesis (hypothesis of the straight normal). A segment of the normal to the middle plane rests under bending the straight and normal to the middle surface. The 2nd Kirchhoff’s hypothesis (hypothesis on not pressing the layers of the plate). The stresses vertical to the middle plane are ignored as compared with the stresses in the plate plane.



59



ion 3.4.



FOUNDATIONS OF MECHANICS OF MATERIALS PART 1



SOME ADDITIONAL FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY



The 3rd hypothesis. The deflections of the plate are supposed so small that the membrane forces in the middle plane can be ignored. The main unknown function is the deflection w  x, y and the rest necessary characteristics of deformation of the plate (displacements, strains, stresses) are expressed through the deflection. The fundamental equation in the theory of bending the plates is called the Sophie Germain-La-grange’s equation ''w



q



D ,(3.72)



^



`



where q is the external bending load, D Eh 2 ª¬12 1 Q 2 º¼ is the so-called cylindrical stiffness, E is Young’s modulus, Q is the Poisson’s ratio, h is the thickness of the plate. The THICK PLATE is restricted by the values of the ratio of the smaller side to the thickness  10. Usually, it is analyzed within the framework of the three-dimensional theory. Definition 3.4.



Note here once again that most of the textbooks on the linear theory of elasticity do not include the theories of plates and shells. These theories are thinking as some Definition 3.4. Definition 3.4. Definition separate part 3.4. of the general theory of elasticity. Definition 3.4. The SHELL is defined as the body of finite or infinite sizes bounded by two surfaces symmetric relative to some surface (middle surface of the shell) a distance between which (thickness) is smaller than two other dimensions (length and width). The shells are divided into three types by the criterion of the Gaussian curvature of the surfaces which form the shell: the shell of positive curvature (for example, the spherical or ellipsoidal surface), the 3.4. shell of zeroth curvature (for example, the cylindrical or conical Definition surface), the shell of the negative curvature (for example, the surface of the hyperboloid of one sheet). Note 3.11. Some points on the torus (toroidal surface) have positive, some have negative, and some have zero Gaussian curvature. The shells can be thin and thick ones. A criterion is chosen as the ratio of the shell thickness to the minimal radius of the middle surface. For the THIN SHELL, this ratio has to be 1 20 ,..., 1 30 .



60



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The THICK SHELL is usually analyzed within the framework of the three-dimensional theory. In the analysis of the thin shells, two basic hypotheses are used. The Kirchhoff-Love’s hypothesis (hypothesis of the straight normal). A segment of the normal to the middle surface rests under loading the straight and normal to the middle surface and does not change its length. The 2nd hypothesis (hypothesis on not pressing the shell layers). The stresses normal to the middle surface are ignored as compared with the stresses acting in the surface are-as located parallel to the middle surface. The stress state of thin shells is characterized by two kinds of forces: Kind 1. The stresses arising in the middle surface (it is characterized by coordinates [ ,K ) – the normal forces N[ , NK and shear forces N[K , NK[ . Kind 2. The stresses arising under bending the shell – the transversal forces Q[ , QK , the bending moments M [ , MK , the turning moments H [K , HK[ . The linear theory3.5. of Definition 3.5. Definition 3.5. Definition 3.5. Definition Definition 3.5. Definition 3.5.



elasticity highlights the case of the shallow shells.



Definition 3.5. The SHALLOW SHELL is defined as the shell that has a small clearance over the plane on which it rested. Definition 3.5. 3.5. Definition



Definition 3.5.



Definition 3.5.



3.5. The basicDefinition system of equations consists of two equations relative two unknown functions – the deflection w  x, y and the stress function )  x, y



1 Eh '')  ' k w



0,  ' k )  D''w q, (3.73)



where ' k f k x f, xx  k y f, yy and k x , k y are the principal curvatures of the middle plane of the shallow shell. Note here once again that most of the textbooks on the linear theory of elasticity do not include the theories of plates and shells. These theories are thinking as some separate part of the general theory of elasticity. Finishing the presentation of some additional fundamental facts from the linear theory of elasticity, let us dwell attention to the distinction between the statical and dynamical problems. In contrary to the statical analysis, the dynamical analysis is based on the additional Definition 3.6. Definition 3.6. Definition 3.6. Definition 3.6. Definition 3.6. Definition 3.6.



Definition 3.6. 3.6. Definition Definition 3.6.Definition Definition 3.7. 3.7. Definition 3.7. Definition 3.7. Definition Definition 3.7. 3.7. Definition 3.6. Definition 3.6.



61



efinition 3.6.



efinition 3.6.



efinition 3.7.



efinition 3.7.



FOUNDATIONS OF MECHANICS OF MATERIALS PART 1



SOME ADDITIONAL FUNDAMENTAL FACTS FROM THE LINEAR THEORY OF ELASTICITY



introducing the time as a new independent variable and the inertial forces as a new kind of forces. As a result, the basic equations of the statics (which are the elliptic type) P ui ,kk   O  P uk .ki  Fi



0 , (3.74)



are transformed in the case of dynamics into the hyperbolic type P ui ,kk   O  P uk .ki  Fi



U ui ,tt . (3.75)



This permits to describe the change of the state of deformation in time and study very important stationary and non-stationary mechanical phenomena including the vibrations and waves. Definition 3.6. The vibrations of the elastic body are understood as a change of parameters of deformation of the body (displacement, strain, stress) which occur more or less regularly in time. Sometimes, some alternative definitions can be proposed. As an example, below the quite different definition is written. Definition 3.7. The vibrations of the elastic body are understood as the repeated limited motion relative to some mean value of the body state, which can be often the equilibrium state. Note 3.12. A motion is assumed as oscillatory, when it takes place in the neighbourhood of some fixed state, is restricted in its variation from this state, and is repeated in most cases. Four classical types of vibrations are studied. Type 1. Natural vibrations. They occur in the elastic body after external excitation (jerk). The motion is supplemented after jerk by the internal forces. Type 2. Forced vibrations. They occur under the action of the external periodic forces, which act independently on the vibrations of the body. Type 3. Parametric vibrations. They differ from the forced vibrations by the kind of the external forces. In the case of the forced vibrations, the parameters of the body rest invariable, whereas in the case of the parametric vibrations the vibrations are excited by the periodic change of any parameter of the body. Type 4. Self-vibrations. They occur without the action of the external periodic forces.



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The theory of vibrations of the linearly elastic bodies is the well-developed and structured part of the linear theory of elasticity. Also, it is the part of the general theory of vibrations, which at present includes practically all spectrum of sciences from physics to politics. The theory of waves of the linearly elastic bodies like the mentioned above theory of vibrations is the well-developed and structured part of the linear theory of elasticity. Also, it is the part of the general theory of waves, which at present includes practically all spectrum of sciences from physics to politics. Let us start with two different definitions of the wave. Definition from Encarta® World English Dictionary: an oscillation that travels through a medium by transferring energy from one particle or point to another without causing any permanent displacement of the medium. Definition from the well-known Whitham’s book: a wave is any recognizable signal that is transferred from one part of the medium to another with a recognizable velocity of propagation. Some common attributes of waves can be specified: the observed in certain place of body disturbance must propagate with a finite velocity to some other place of this body; as a rule, the process must be close to oscillatory, if it is observed in time. Waves in elastic bodies are classified by different indications and actually the different classifications exist parallelly. For example, the smoothness of the solution in the form of wave was turned out to be critical in theoretical wave analysis. Knowledge of the solution smoothness is equivalent to knowledge of its continuity or discontinuity, and also their quantitative estimates (types of discontinuities, order of continuity, etc). The situation when waves corresponding to discontinuous and continuous solutions are studied separately was formed long ago. The branch of study associated with discontinuous solutions treats a wave as a singular surface motion relative to some given smooth physical field. That is to say, the wave motion is understood as motion in the space of a field jump on a given surface. The second branch is associated with continuous solutions describing a continuous motion. Two classes of waves are considered here. Hyperbolic waves are obtained as solutions of differential equations of hyperbolic or ultrahyperbolic types and, consequently, are clearly defined by the type of equation.



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Dispersive waves are defined by the form of solution.



Definition 3.8.



Definition 3.8. The body, in which the wave propagates, is dispersive and the wave Definition Definition 3.8.3.8. Definition 3.8. Definition 3.8. themselves is dispersive, if the wave is mathematically represented in the form of given function F of the phase M kx  Z t ( x ( x is the spatial coordinate, k is the wave number, Z is the frequency, and t is time), and if the phase velocity v Z k of the wave depends nonlinearly on frequency. Very often, the dispersion is fixed in the form of nonlinear function Z W  k . Note 3.13. The solutions of type u F (kx  Z t ) are admitted not only to the hyperbolic differential equations, but the parabolic one, and also some integral equations. Note 3.14. The criteria of hyperbolic and dispersive waves are not mutually exclusive; hyperbolic and dispersive waves are therefore encountered simultaneously. Let us write here the classification standard in physics and differing from the mentioned above hyperbolic - dispersive by the kinematic attribute. It consists of four types: Type 1. Solitary waves or pulses – the sufficiently short in time and irregular locally given in a body disturbances. Type 2. Periodic (most often, harmonic) waves, which are characterized by the disturbances in all the body. Type 3. Wave pockets – the regular locally given in a body disturbances. Type 4. Trains of waves – the harmonic wave pockets. Usually, the textbooks on the linear theory of elasticity include the information on some kinds of elastic waves. List of classical linear elastic waves consists of many items: The volume and shear waves. Helmholtz’s theorem and Sommerfeld conditions. Plane waves and Christoffel’s equations. Reflection of the plane harmonic waves. Spherical waves. Cylindrical and torsional waves. Surface waves (Rayleigh’s waves). Cylindrical, spherical and other surface waves. Love’s waves in layer. Lamb’s waves in plate. Waves in rods, plates and shells (elementary and exact theory). Finally, consider the APPROXIMATE METHODS of solving the problems of the linear theory of elasticity. They can be divided into two groups. Group 1. Methods of approximate solving the boundary problems for the differential equations of the linear theory of elasticity.



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First of all, the finite-difference method (FDM) should be related to this group. Also, the Bubnov-Galerkin’s method and Kantorovich-Vlasov’s methods should be mentioned here. Group 2. Direct methods. They are often called the variational methods because they are based on the differential equations but the variational principles of mechanics. The finite element (FEM) and boundary element methods (BEM) should be related to this group. Note 3.15. Two basic variational methods are used here – the Lagrange’s principle based on variations of displacements and the Castigliano’s principle based on variations of stresses. The mentioned approximate numerical methods are actively used and modified. There are many commercial computer packages for solving the big classes of problems.



Comments Comment 3.1. The proofs of theorems and rigorous reasonings have an important goal of removing the doubts. Only a professional mathematician can enjoy the formal justification of each step of a long line of reasoning. As for the removal of doubts, there is a story about D’Alembert: Unsuccessfully explaining the proof of some theorem to one noble pupil, he said: Sir! Honestly, this theorem is right! The reaction of the noble pupil was instant: Oh, sir! That’s enough. You’re a nobleman and I’m a nobleman. And your honest word is the best of proof. Comment 3.2. In this chapter, many facts from the linear theory of elasticity are formulated in some abstract form. This contradicts in some cases to the understanding of the mechanics of materials as a science having also the function of being useful for engineers. Sometimes they joke that the presentation of the solution in the form 7 arc tan 2  2 ln 7 is not acceptable 11



for engineers. It should have to engineer the form 0.338.



65



5



3



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Further reading 3.1. Achenbach, JD 1973, Wave Propagation in Elastic Solids. North-Holland, Amsterdam. 3.2. Ang, W-T 2007, A Beginner’s Course in Boundary Element Methods. Universal Publishers,  Boca Raton, USA. 3.3.  Ascione, L & Grimaldi, A 1993, Elementi di meccanica del continuo (Elements of Continuum Mechanics). Massimo, Napoli. (In Italian) 3.4. Atkin, RJ & Fox, N 1980, An Introduction to the Theory of Elasticity, Longman, London. 3.5. Banerjee, PK 1994, The Boundary Element Methods in Engineering. 2nd ed., McGrawHill,  London. 3.6. Bedford, A & Drumheller, DS 1994, Introduction to Elastic Wave Propagation. John Wiley, Chichester. 3.7. Chaskalovic, J 2008, Finite Elements Methods for Engineering Sciences, Springer Verlag, Berlin. 3.8. Chen, PJ 1972, Wave Motion in Solids. Flügge‘s Handbuch der Physik, Band VIa/3. Springer Verlag, Berlin. 3.9. Dagdale, DS & Ruiz, C 1971, Elasticity for Engineers. McGraw Hill, London. 310. Den Hartog JP 2007, Mechanical Vibrations, 12th ed. Dover Civil and Mechanical Engineering, Mineola. 3.11. Eschenauer, H & Schnell, W 1981, Elasticitätstheorie I, Bibl. Inst., Mannheim. (In German) 3.12. Eslami, MR, Hetnarski, RB, Ignaczak, J, Noda, N, Sumi, N & Tanigawa, Y 2013, Theory of Elasticity and Thermal Stresses. Explanations, Problems and Solutions. Series Solid Mechanics and Its Applications, Springer Verlag Netherlands, Amsterdam. 3.13. Fedorov, FI 1968, Theory of Elastic Waves in Crystals. Plenum Press, New York. 3.14. Fraeijs de Veubeke, BM 1979, A Course of Elasticity. Springer, New York. 3.15. Graff, KF 1991, Wave Motion in Elastic Solids. Dover, London. 3.16. Harris, JG 2001, Linear Elastic Waves. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge. 3.17. Hudson, JA 1980, The Excitation and Propagation of Elastic Waves. Cambridge University Press, Cambridge. 3.18. Hahn, HG 1985, Elastizitätstheorie. B.G.Teubner, Stuttgart. (In German) 3.19. Inman, DJ 2007, Engineering Vibration, 3rd ed. Prentice Hall, New York.  3.20. Katsikadelis, JT 2002, Boundary Elements Theory and Applications. Elsevier, Amsterdam. 3.20. LeVeque, RJ 2007, Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, New York. 3.21. Love, AEH 1944, The Mathematical Theory of Elasticity. 4th ed. Dover Publications, New York. 3.22. Lur’e AI 1999, Theory of Elasticity. Series “Foundations of Engineering Mechanics”. Springer, Berlin.



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3.23. M  agnus K 1976, Schwingungen. Eine Einfuhrung in die theoretische Behandlung von Schwingugsprobleme. Teubner, Stuttgart. (In German) 3.24. Müller, W 1959, Theorie der elastischen Verformung. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig. (In German) 3.25. Nowacki, W 1970, Theory of Elasticity. PWN, Warszawa. (In Polish, In Russian) 3.26. Olver,  P 2013,  Introduction to Partial Differential Equations. Chapter 5: Finite differences. Springer, Berlin. 3.27. Reddy, JN 2006, An Introduction to the Finite Element Method.  3rd ed., McGraw-Hill, New York.  3.28. Reynolds, DD 2016, Engineering Principles of Mechanical Vibration, 4th ed. Trafford On Demand Publishing, Bloomington, USA. 3.29. Royer, D & Dieulesaint, E 2000, Elastic Waves in Solids (I,II). Advanced Texts in Physics. Springer, Berlin. 3.30. Rushchitsky, JJ 2011, Theory of Waves in Materials. Ventus Publishing ApS, Copenhagen. 3.31. Slaughter, WS 2001, Linearized Theory of Elasticity. Birkhäuser, Zurich. 3.32. Sneddon, IN & Berry, DS 1958, The Classical Theory of Elasticity, vol.VI, Flügge Encyclopedia of Physics. Springer Verlag, Berlin. 3.33. Sokolnikoff, IS 1956, Mathematical Theory of Elasticity. McGraw Hill Book Co, New York. 3.34. Starovoitov, E & Naghiyev, FBO 2012, Foundations of the Theory of Elasticity, Plasticity, and Viscoelasticity, Apple Academic Press, Palo Alto. 3.35. Strikwerda, J 2004, Finite Difference Schemes and Partial Differential Equations. 2nd ed. SIAM, New York.  3.36. Timoshenko, SP & Goodyear, JN 1970, Theory of Elasticity, 3rd ed. McGraw Hill, Tokyo. 3,37. Tongue, BH 2001, Principles of Vibration, Oxford University Press, Oxford. 3.38. Wrobel, LC & Aliabadi, MH 2002, The Boundary Element Method. In 2 vols.  John Wiley & Sons, New York. 3.39. Zienkiewicz, OC, Taylor, RL & Zhu, JZ 2005, The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, New York.



Questions 3.1. E  stimate the role of inverse problems in the linear theory of elasticity. Compare with the importance of the inverse problems in geophysics. 3.2. Read something about the existence theorem. Find two Korn’s inequalities and Fichera’s proof of this theorem. 3.3. Show the link among the different general representation of solutions. 3.4. Find and describe the solved simple problem within the framework of the anti-plane state. Comment the simplicity and practicability of this problem.



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3.5. R  epeat the step-by-step procedure of proving that the generalized plane stress state is identical with the plane displacement state. 3.6. Comment an importance of the Airy’s function. 3.7. Find a few handbooks on the stress concentration around holes, necks, cuts and formulate the most important applications of results on the stress concentration. 3.8. Describe shortly the fundamentality of universal deformations in the theory of elasticity. 3.9. Choose the most convenient for you book devoted to rods and beams. 3.10. Choose the most convenient for you book devoted to plates and shells. 3.11. Which modern book on vibrations of the elastic body you prefer? 3.12. Look for the modern books on plates and shells and choose one book on plates and one book on shells, which you could recommend your colleagues.



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SHORT DESCRIPTION OF NONLINEAR THEORY OF ELASTICITY. PART 1



4 SHORT DESCRIPTION OF NONLINEAR THEORY OF ELASTICITY. PART 1 Kinematic notions: motion, configuration, reference systems, deformation, gradient of deformation, fundamental quadratic form, fundamental metric tensor, strain tensors. Kinetic notions: force, moment, stress, stress tensors. Balance equations. Motion equa-tions. Internal energy of deformation and constitutive equations. Generally elastic material, hypoelastic material, hyperelastic material.



Let us recall that here the brief information on the four parts of the theory of elasticity is continued. The linear theory of elasticity was chosen and described in Chapter 2 as the first one because of it forms the basic skeleton with which other three parts are constructed – the nonlinear and linearized theories of elasticity as the parts with more exact and complicated models that extend significantly the area of application and the strength of materials as the part with many simplifying Kinematic notions: motion, configuration, reference systems, deformation, gradient Kinematic notions: motion, configuration, reference systems, deformation, gradient of of assumptions that are used in engineering practice.



deformation, fundamental quadratic fundamental metric tensor, strain tensors. deformation, fundamental quadratic formform fundamental metric tensor, strain tensors. notions: force, moment, stress, stress tensors. Balance equations. Motion equaKinetic notions: force, moment, stress, stress tensors. Balance equations. Motion equaKinetic tions. Internal energy of deformation and constitutive equations. Generally elastic mations. Internal energy of deformation and constitutive equations. Generally elastic mais looking appropriate to consider now the brief description of the nonlinear theory terial, hypoelastic material, hyperelastic material. terial, hypoelastic material, hyperelastic material.



It of elasticity and then pass on to the linearized theory of elasticity. The main difference between the linear and nonlinear theories is that the nonlinear theory abandons the linear description of the deformation process and admits the finite (large) deformations of the body. A description of nonlinear deformation needs more complicated mathematical tools. Despite of this, the nonlinear theory of elasticity is constructed very well and is considered as the phenomenological theory with axiomatic providing the theory structure. First of all, this theory uses the most exact definitions and strong mathematical procedures. This can be demonstrated in the way of introduction of the difference between the nondeformed and deformed states of the material. The basic and primary notion in the theory of elasticity is the notion of a material continuum. It permits to identify the real body with the geometrical domain of 3D space, which this body occupies. In such a way, the real domain is transforming into some physical abstraction, called the body. Suppose the body B in Euclidean space \3 is given. Definition Definition 4.1. 4.1.



Definition Definition 4.2. 4.2.



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FOUNDATIONS OF MECHANICS OF MATERIALS PART 1



Definition 3.8. 4.1. is the mapping of a set B on the domain F B ,t of Definition 4.1.4.1.Definition Definition The motion Definition 4.1. 3 B , t  \1. certain time t x F ( X , t ), X  the spaceDefinition \ at a 4.1. Definition 4.1. Definition 4.1.



Definition 3.8.



The velocity and the acceleration are defined in the standard physical way x F  X , t , x F  X , t . Definition 4.2. Definition 4.2. Definition 4.2. Definition 4.2. image F B ,t Definition 4.2. The



on 4.1.



on 4.2.



on 4.3.



SHORT DESCRIPTION OF NONLINEAR THEORY OF ELASTICITY. PART 1



F at4.2. Definition of the mapping the moment t is called the



configuration.



Definition 4.2.



Definition 4.3. Definition 4.3. Definition 4.3. configuration said sometimes that the Definition 4.3.



It is moment.Definition 4.4.



Definition 4.3. Definition 3.8.



Definition 4.4. Definition 4.4. Definition 4.4.



Definition 4.3.



Definition 4.4.



is like the photograph of the motion at a fixed Definition 4.4.



Definition 4.3. The configuration of a body at time t is called THE ACTUAL CONFIGURATION. Definition 4.4. The configuration of a body at any arbitrarily chosen initial moment is called THE REFERENCE CONFIGURATION.



on 4.4.



Kinematic motion, systems, deformation,isgradient of Definition 4.5. Thenotions: description of configuration, body motion reference by the reference configuration deformation, fundamental quadratic form fundamental metric tensor, strain tensors. called THE REFERENCE DESCRIPTION. notions: force, moment, stress, stress tensors. Balance equations. Motion equaKinetic tions. Internal energy of deformation and constitutive equations. Just this description is mainly used in the nonlinear theory of elasticity.Generally elastic material, hypoelastic material, hyperelastic material. Definition 4.5.



The reference and actual configurations are linked with the Lagrangian and Eulerian reference systems. Definition 4.5.



Definition 4.5.



Definition THE LAGRANGIAN SYSTEM is characterized by that material particles of a body are Definition 4.5. Definition 4.5.4.5. individualized - each particle is associated with Cartesian coordinates xk (or the curvilinear coordinates x k ). This individualization is carried out in the reference configuration. It is further supposed that in the process of mo-ving (transition from the reference configuration to the actual one) the coordinates xk don’t vary, i.e., the particle and its coordinates are interlinked forever.



THE EULERIAN SYSTEM is characterized by that the particle occupies the point in the actual confi-guration given by coordinates X D (or X D ). In this case, the coordinates of the particle are not linked with a motion, because the last has already been taken place and a body is already in the actual configuration. Consider the motion anew and choose in \3 the Lagrangian {x k } and the Eulerian { X D } reference systems as well as assume that transitions x k x k  X D , X D X D  x k Definition 4.1. from one system to Definition another are 4.6.given. Then the motion in the reference description D can be denoted by FN ( X , t ) or x m FNm ( X D , t ) . Definition 4.7. Definition 4.6. Definition 4.7.



Definition 4.6. 4.2. Definition Definition Definition 4.6. Definition 4.6.4.6. Definition 4.7. Definition 4.8 Definition Definition 4.7. Definition 4.7.4.7.



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SHORT DESCRIPTION OF NONLINEAR THEORY OF ELASTICITY. PART 1



It is usually assumed that functions FNm are continuously differentiable up to the necessary order, real, and one-value. Definition 4.6.



Definition 4.6. THE DEFORMATION OF BODY is meant as a change of the Definition 4.7. shape or dimensions of the body. Definition 4.6.



Definition 4.6. 4.6. Definition 4.6. Definition



Definition 4.7. THE GRADIENT OF DEFORMATION is defined as the vector



Definition Definition4.6. 4.7. Definition 4.6. Definition 4.7.



Definition 4.8 Definition 4.7. 4.7. Definition 4.7. Definition G G { ’F k X D4.6. , t , FDm F { Fk X D , t Definition







Definition 4.7. Definition 4.8















x,mD



w F km  X D , t . (4.1) wX D



Definition 4.6. Definition 4.6. Definition 4.7.



Definition 4.8 THE IN MATERIALS is meant as a process in Definition 4.8 DEFORMATION Definition 4.8 Definition 4.8 x B pass from4.7. the reference configuration BR to the actual one B. which the particleDefinition Definition 4.7. Definition 4.8 Definition 4.6. It is described by THE VECTOR OF DISPLACEMENTS OF THE PARTICLE. Definition 4.8 Definition 4.6.4.8 Definition



x B



G u



n 4.8



^u



Definition 4.8 Definition 4.8 Definition 4.7.



m



` { ^u , u 2 , u 3` , u m  X D , t 1



Definition 4.6. Definition 4.7. Definition 4.6. Definition 4.7.



x m  X D , t  X m .(4.2) Definition 4.7. Definition 4.8



THE CONCOMITANT COORDINATE SYSTEM with Lagrangian coordinates is mainly Definition 4.8 Definition 4.8 chosen in the nonlinear theory of elasticity for the analytical description of the configuration. Definition 4.8 In this case, the notion of coordinate transform dT i  wT i w- k d- k a