M zyi z, xxv00by a short section of a beam subjected to a bending moment m z about the zaxis bends by an angle d. The relation constrained the couple stress tensor to be symmetric, and the symmetric curvature tensor became the only properly conjugated high order strain measures in the theory to have a real contribution to the total strain energy of the system. Total strain energy theory beltramihaigh it predicts the failure of a specimen subjected to any combination of loads when the strain energy per unit volume of any portion of the stressed member reaches the failure value of strain energy per unit volume as determined from an axial or compression test of the same material. Before proceeding to the solution of specific elasticity problems we wish to explore the associated concepts of work and energy.
Energy minimization principles edit developed and explored by green 1839, haughton1849, kirchhoff 1850, love 1906, trefftz 1928 and others. Module 3 constitutive equations learning objectives understand basic stressstrain response of engineering materials. Pdf on the development of volumetric strain energy functions. Quantify the linear elastic stressstrain response in terms of tensorial quantities and in particular the fourthorder elasticity or sti ness tensor describing hookes law. Mechanics of solids the general theory of elasticity. Suppose a rod ab must acquire an elastic strain energy of. Here it is assumed that the material remains elastic whilst work is done on it so that all the energy is recoverable and no. Typical elastic deformation of engineering materials. Linear elasticity as a general threedimensional theory began to be developed in the early 1820s based on cauchys work. Theory of elastic stability available for download and read online in other formats. The best example of this is a clockwork device which stores strain energy and then gives it up. Strain energy in linear elastic solids 5 bending strain energy. Maximum strain energy theory of failure mechanical.
Oct 18, 2017 when a body is subjected to external force it undergoes deformation. Download pdf theory of elastic stability book full free. From the above results it may be observed that the strain energy decreases as the volume of the bar increases. Horgan department of engineering mechanics, the university of michigan, ann arbor, michigan, u. The strain energy will in general vary throughout a body and for this reason it is useful to introduce the concept of strain energy density, which is a measure of how much energy is stored in small volume elements throughout a material. According to rivlins phenomenological theory of rubber elasticity, the elastic properties of a rubber may be described in terms of a strain energy function which is an infinite power series in the strain invariants i 1, i 2 and i 3. Theory of elasticity for scientists and engineers request pdf. Pa this longitudinal modulus of elasticity is called youngs modulus and is denoted by the symbol. The concept of strain energy is of fundamental importance in applied. Stress strain curve for a linear elastic material subject to uniaxial stress.
We will examine strain energy associated with the most common forms of stress encountered in structures and use it to calculate the deflection of structures. Couple stress based strain gradient theory for elasticity. Some forms of the strain energy function for rubber. Ratner, in nonlinear theory of elasticity and optimal design, 2003. The deformation energy is expressed in strains, and is therefore also called strain energy. Nonlinear theory needed to understand properties of statically strained materials nonlinearities can renormalize nature of elasticity elegant an complex theory of interest in its own right why now. Within the mathematical framework of the theory of elasticity, problems related to such applications are solved. Plastic stress strain relationship, elastic plastic problems in bending and torsion. Isotropie solids are studied in section 5 and linear theory is presented in section 6. M zyi z, xxv00by a short section of a beam subjected to a bending moment m z about the zaxis bends by an angle d m the incremental internal strain energy.
On the choice of a state equation for an isotropic elastic body. Because of mass it they can possess kinetic energy by virtue of their velocity. Theory of elasticity ct 5141 energy principles and variational methods prof. Me 160 introduction to finite element method chapter 4. Review of strain energy methods and introduction to sti. The strain energy stored in the bar when a force of kn2 is applied at the.
Pdf a brief history of logarithmic strain measures in. Imagine a rigid ball being displaced by an in nitesimal amount on a. The external work done on an elastic member in causing it to distort from its unstressed state is transformed into strain energy which is a form of potential energy. Thermal stress and strain,poissons ratio jeeneet duration. New interest in biological materials under large strain liquid crystal elastomers exotic nonlinear behavior. Consider again a bar subjected to a uniaxial force p. Maximum distortion energy theory video lecture from theories of elastic failure chapter of strength of materials subject for all engineering students. They can possess elastic strain energy which is comparable to the potential energy. Maximum distortion energy theory theories of elastic. Theory, applications and numerics provides a concise and organized presentation and development of the theory of elasticity, moving from solution methodologies, formulations and strategies into applications of contemporary interest, including fracture mechanics, anisotropiccomposite. The energy used up is stored in the body as strain energy and it may be regained by allowing the body to relax. Request pdf theory of elasticity for scientists and engineers analysis of.
Energy methods in elasticity the energy methods provide a powerful tool for deriving exact and approximate solutions to many structural problems. Large deformation isotropic elasticity on the correlation. If the applied stress is sufficiently low or the imposed strain is small enough, almost all solid materials behave in such a way. The introduction of the quadratic hencky strain energy based on the logarithmic strain tensor log v is a milestone in the development of nonlinear elasticity theory in the first half of the 20th. Theory of elasticity lecture 18 elasticity and intrinsic strain energy density function theory of elasticity lecture 21beltrami michell equations derivation of beltrami michell equations of elasticity isotropic materials, small deformations, equilibrium conditions, compatible theory of elasticity 06.
Betasn, tio2, and tial of tetragonal structures are considered. Permanent deformations involve the dissipation of energy. The strain energy is the area under the curve of load versus deformation. If the elastic properties are the same in every particles in a material, the ma. The purpose of this paper is, while making full use of the inherent simplicity of isotropic elasticity, to construct a strain energy function which. Aug 05, 2016 maximum strain energy theory video lecture from theories of elastic failure chapter of strength of materials subject for all engineering students. The first material may be called elastic in that it resumes its former dimensions immediately load is removed, but the relation between stress and strain is not linear. Strain energy impact loading consider a rod bc of length l and uniform crosssectional area a, ich is attached at b to a wh fixed support, and subjected at c to a slowly increasing axial load p. Elasticity we define an elastic material as one for which a strain energy density. Maximum strain energy theory theories of elastic failure. Let the element now be subjected to a constant bending moment m. Crack initiation behaviour of orthotropic solids as predicted by the strain energy density theory. On the strainenergy density in linear elasticity c. The internal energy of a body is increased by strain.
Elasticity, stress, strain, and fracture boundless physics. Almost all real materials will undergo some permanent deformation, which remains after removal of load. The strain energy per unit volume is onehalf the product of stress times strain for each component. The many developments and clarifications in the theory of elasticity. Stress is the ratio of applied force f to a cross section areadefined as force per unit area tensile stress stress that tends to stretch or lengthen the material acts normal to the stressed area.
Abstractnumerical solutions are given of a boundaryvalue problem in finite elasticity for a variety of. Mechanics of materials edition beer johnston dewolf 11 4 strain energy density to eliminate the effects of size, evaluate the strainenergy per unit volume, u d strain energy density l dx a p v u x x. Let di be the resulting displacement at the location and in the direction of the force fi. The simplest forms of rivlins strain energy function are the neohookean, which is obtained by truncating the infinite series to just the first term in i 1, and.
The amount of departure from rest shape is called deformation, the proportion of deformation to original size is called strain. The first type deals with materials that are elastic only for small strains. In this theory, it is assumed that in addition to strain tensor, gradient of the strain tensor could be taken into account in calculation of the elastic strain energy. The approach also yields directly necessary and sufficient conditions for positivedefiniteness.
Mechanics of solids mechanics of solids the general theory of elasticity. The elastic behavior of objects that undergo finite deformations has been described using a number of models, such as cauchy elastic material models, hypoelastic material models, and hyperelastic material models. The total strain energy density resulting from the deformation is equal to the area under the. In general, it is not easy to give a physical interpretation of the concept of complementary energy. The maximum strain energy theory is not suitable under hydrostatic stress situation. Maximum distortion energy theory or vonmises and henckys theory 1. T p t d member is subjected to both twisting moment and. Lecture notes on applied elasticity and plasticity dr pramod. Thus, for an element of volume d v, the elastic strain energy is ix,y,z jxy,z and similarly for polar coordinates.
Note that this is not uniaxial strain due to poisson e ect in this expression, eis youngs modulus. Strain energy is as the energy which is stored within a material when work has been done on the material. Pdf strain gradient elasticity theories in lattice. In gradient elasticity, the strain energy function contains both the strain terms and their gradients, which leads to stresses depending on higherorder derivativ es of the 45 displacement. Standard results from matrix theory are used to derive optimal upper and lower bounds for the strain energy density in terms of the norm of the stress tensor in two and three dimensions. For linear, homogeneous solids, this introduces material moduli which are functions of the distance. Strain energy elastic strain energy, u energy spent by the external forces in deforming an elastic body. Imagine a rigid ball being displaced by an in nitesimal amount on a at 0 and inclined 6 0 surface, fig. For the same reason, the ratio of stress to strain is a characteristic of a material. Pdf crack initiation behaviour of orthotropic solids as. Linear elasticity theory is thus the best known and most widely used branch of solid mechanics. Welcome this is a nptel online certification course on theory of elasticity. The aspect of elastic strain for a deformation twin with a pure shear strain is studied through eshelbys inclusion theory. Strain gradient elasticity solution for functionally graded micro arxiv.
Boundary tractions and body forces will do work on an elastic solid and this work will be stored inside the material in the form of strain energy. The basic theory includes the definition of stress, strain and elastic energy. Nonlinear theory of elasticity, volume 36 1st edition. As the aspect ratio of a twin approaches zero, its elastic strain energy vanishes since the stress components coupled with the twin shear strain vanish, suggesting that the twin. Received may 23, 1972 231 summary standard results from matrix theory are used to derive optimal upper and lower bounds for the strain energy density. In the absence of energy losses, such as from friction, damping or yielding, the strain energy is equal to the work done on the solid by external loads. Pdf theory of elastic stability download full pdf book. Review of strain energy methods and introduction to. The energy stored in the body due to deformation is known as strain energy. The true strain is defined as the natural logarithm of the ratio of the final dimension to the initial dimension. However, the usual practice of writing the strain energy as a function of two independent strain invariants has, in general, the effect of complicating the associated mathematical analysis this is particularly evident in relation to the calculation of instantaneous moduli of elasticity and, consequently, the basic elegance and simplicity of. Introduction when an elastic body is deformed, work is done.
On the strain energy density in linear elasticity c. Consider a force, fi, applied gradually to a structure. By by plotting magnitude p of the load against the deformation of the rodx, we obtain a certain loaddeformation diagram. Problems of the nonlinear theory of a compressible elastic medium. Keyt research institute, university of alabama in huntsville, huntsville, alabama, u. The results predicted by the mathematics depend critically on the material properties incorporated in the strain energy function, and a wide range of interesting phenomena can be modeled. Me 340a theory and applications of elasticity this course provides an introduction to the elasticity theory and its application to material structures at microscale. In a molecule, strain energy is released when the constituent atoms are allowed to rearrange themselves in a chemical reaction.
Consider the work done on an elastic solid by a single point force f. Theory of elasticity an overview sciencedirect topics. A number of theories have been proposed for the brittle and ductile materials. The linear theory of elasticity is an inadequate description of the phenomenon, for it cannot provide a description for the limit of elasticity and cannot predict the elastic behavior of a structure. The material in the current presentation is still meant to be a set of lecture notes, not a text book. Unit 7 vibration of mechanical vibration of mechanical. The first two sets of equations are universal independent of the. Tlgj g is the modulus of rigidity and j is the polar second moment of area. The energy supplied in moving the body is said to be recoverable because the spring is ready to pull back and do the same amount of work. If the applied stress exceeds the elastic limit, the theory of elasticity will no longer predict correctly the distribution of stress and strain in a solid body.
This theory is not suitable for the safe design of machine components made of ductile materials because ductile materials are weak in shear. Theory, applications and numerics provides a concise and organized presentation and development of the theory of elasticity, moving from solution methodologies, formulations and strategies into. Me340a theory and applications of elasticity lecture notes. They account both for nonlinear material behavior and large shape changes. Mathematical analysis in the mechanics of fracture 195 equations, virtual work implies equilibrium. We will discuss two major methods for solving elasticity equations.
According to the maximum shear strain energy theory of failure, the failure of a material or component will occur when the total shear strain energy per unit volume exceeds the limiting value of shear strain energy per unit volume i. In studying mechanical vibrations, the bodies are treated as elastic bodies instead of rigid bodies. On the strainenergy density in linear elasticity springerlink. Theories of elastic failure department of mechanical. We will now understand here the maximum strain energy theory with the help of this article. The amount by which a material deforms under stress or force, given as a ratio of the deformation to the initial dimension of the material and typically symbolized by. Introduction to the theory of plates stanford university. The deformation gradient f is the primary deformation measure used in finite strain theory. Balch division of mechanics and computation department of mecanical engineering stanford university stretching and bending of plates fundamentals introduction a plate is a structural element which is thin and. Strain energy density for a given value of the strain, the strain energy density per unit volume.
Large deformation isotropic elasticityon the correlation of. Strain energy in linear elastic solids duke university. Because the elasticity of a material is described in terms of a stressstrain relation, it is essential that the terms stress and strain be defined without ambiguity. Maximum distortion energy theory huberhenkyvon misesthe theory is based on a limiting energy of distortion, i. It is often easier to consider elasticity not as stress and strain. If this is extended to the threedimensional case and applied over.
1111 1017 417 246 126 748 699 1059 1246 613 799 729 992 1524 1292 1290 398 750 1187 378 1519 696 1123 1042 303 1439 1340 413 288 646 1274 237 1258 828 707 757 39