General Theorems in Elasticity

**3**

**2. Historical development in elasticity**

**Chapter 1**

*Ezgi Günay*

**1. Introduction**

the material.

Introductory Chapter: Analytical

In this section, the historical development of the "elasticity theory" was presented briefly, and recent studies performed about the elasticity concept were categorized and listed according to their basic engineering problem groups. The mentioned literature survey has been performed by searching the keywords "elasticity," "analytic," and "solution" between the years 2014 and 2018. The most important general aspects of the "elasticity theory" were described in four groups as "the unknowns," "the used equations," "the modeling procedures," and "solution methods." In the future, in the consideration of these explained theoretical, numerical, and experimental properties, the researchers can be concentrating on the origin of the problem and new solution methods in deciding the exact nature of

The elasticity concept of solid materials is the deformation with the external force application and recovery to its original shape after the forces removed. In the strength measurement of the material, stress (force per area) and strain (deformation per unit length) criteria have been used. The elasticity theory was presented in order to explain the basic theoretical concepts and their analytical solution methods, the deformations that were assumed to be very small and corresponding stress distributions. The classical elasticity theory was explained by theorems of "uniqueness of solution" and "existence of solution" as they have been declared in the basic mathematical concepts. The "uniqueness of solution" theorem was restricted to a single solution space by satisfying the related boundary or the initial conditions. If there were no any boundary or initial conditions, the solution space would have to be infinity. The "existence of solution" theorem was created by explaining the default displacement functions, checking the equilibrium equations for stress definitions, and satisfying the partial differential equations with the infrastructure of the default solutions. The purpose of the elasticity theory was the determination of this unique and exact solution in elastic region of the material. In linear elastic region, superposition method and combined loading applications are widely used in engineering.

The historical development of the concept of "elasticity" by considering mathematics, physics, and engineering mechanics was summarized in **Figure 1** [1, 2]. The scientific studies performed on engineering problems have been grouped as analytical, numerical, and experimental. The main solution techniques listed below

and Numerical Approaches in

Engineering Elasticity

#### **Chapter 1**

## Introductory Chapter: Analytical and Numerical Approaches in Engineering Elasticity

*Ezgi Günay*

#### **1. Introduction**

In this section, the historical development of the "elasticity theory" was presented briefly, and recent studies performed about the elasticity concept were categorized and listed according to their basic engineering problem groups. The mentioned literature survey has been performed by searching the keywords "elasticity," "analytic," and "solution" between the years 2014 and 2018. The most important general aspects of the "elasticity theory" were described in four groups as "the unknowns," "the used equations," "the modeling procedures," and "solution methods." In the future, in the consideration of these explained theoretical, numerical, and experimental properties, the researchers can be concentrating on the origin of the problem and new solution methods in deciding the exact nature of the material.

The elasticity concept of solid materials is the deformation with the external force application and recovery to its original shape after the forces removed. In the strength measurement of the material, stress (force per area) and strain (deformation per unit length) criteria have been used. The elasticity theory was presented in order to explain the basic theoretical concepts and their analytical solution methods, the deformations that were assumed to be very small and corresponding stress distributions. The classical elasticity theory was explained by theorems of "uniqueness of solution" and "existence of solution" as they have been declared in the basic mathematical concepts. The "uniqueness of solution" theorem was restricted to a single solution space by satisfying the related boundary or the initial conditions. If there were no any boundary or initial conditions, the solution space would have to be infinity. The "existence of solution" theorem was created by explaining the default displacement functions, checking the equilibrium equations for stress definitions, and satisfying the partial differential equations with the infrastructure of the default solutions. The purpose of the elasticity theory was the determination of this unique and exact solution in elastic region of the material. In linear elastic region, superposition method and combined loading applications are widely used in engineering.

#### **2. Historical development in elasticity**

The historical development of the concept of "elasticity" by considering mathematics, physics, and engineering mechanics was summarized in **Figure 1** [1, 2]. The scientific studies performed on engineering problems have been grouped as analytical, numerical, and experimental. The main solution techniques listed below

#### **Figure 1.**

*Development of elasticity between the sixteenth and twentieth centuries.*

form the first step aspects in performing the experiments and obtaining the numerical solutions by considering innovations: (i) characteristics of the solution methods, (ii) learning the mathematical theories, (iii) the physics of the problem, and (iv) learning the problem-solving methodologies. The second step aspects have been listed as "solving problems by mathematical techniques" and "obtaining new formulas." The scientific progress has been continued thanks to the studies done since the sixteenth century. The development in scientific area occurred in the elasticity concept has been summarized and visualized in the consideration of the scientists who have lived between the sixteenth and twentieth centuries and their studies [1, 2]. These famous scientists were Galilei (1564–1642), Mariotte (1620–1684), Hooke (1635–1703), Leibniz (1646–1716), Bernoulli (1700–1782), Baumgarten (1706–1757), Euler (1707–1783), Coulomb (1736–1806), Young (1773–1829), Poisson (1781–1840), Navier (1785–1836), Cauchy (1789–1857), Saint-Venant (1797–1886), Borchardt (1817–1880), Rankine (1820–1872), Kirchhoff (1824–1887), Maxwell (1831–1879), Clebsch (1833–1872), Kohlrausch (1840–1910), Amagat (1841–1915), Voigt (1850–1919), Mallock (1851–1933), Lamme (1864–1924), Röntgen (1872,1919), Synge (1897–1995), and Everett (1930–1982) (**Figure 1**).

#### **3. Classification of engineering problems in the context of elasticity**

In this section, the results of the literature review on elasticity were evaluated by referring to the articles (total number of articles, 157) between 2014 and 2018. Important information has gained from the literature survey about the elasticity theory and its related recent engineering solutions, as well as information about

**5**

**Figure 2.**

*Classification of the basic elasticity problems and their solution techniques.*

*Introductory Chapter: Analytical and Numerical Approaches in Engineering Elasticity*

the theoretical, numerical, and experimental scientific researches and scientific innovations. The brief classification of the main engineering problems was sum-

The studies evaluated in the literature review were listed below in 10 main headings. The distribution of articles corresponding to research concepts is presented in **Figure 3**. These are (1) historical development, (2) analytical and experimental studies related to the finite element method (FEM), (3) experimental studies, (4) analytical studies and finite element analysis (FEA), (5) analytical studies,

*DOI: http://dx.doi.org/10.5772/intechopen.82328*

marized in **Figure 2**.

*Introductory Chapter: Analytical and Numerical Approaches in Engineering Elasticity DOI: http://dx.doi.org/10.5772/intechopen.82328*

the theoretical, numerical, and experimental scientific researches and scientific innovations. The brief classification of the main engineering problems was summarized in **Figure 2**.

The studies evaluated in the literature review were listed below in 10 main headings. The distribution of articles corresponding to research concepts is presented in **Figure 3**. These are (1) historical development, (2) analytical and experimental studies related to the finite element method (FEM), (3) experimental studies, (4) analytical studies and finite element analysis (FEA), (5) analytical studies,

**Figure 2.** *Classification of the basic elasticity problems and their solution techniques.*

*Elasticity of Materials - Basic Principles and Design of Structures*

form the first step aspects in performing the experiments and obtaining the numerical solutions by considering innovations: (i) characteristics of the solution methods, (ii) learning the mathematical theories, (iii) the physics of the problem, and (iv) learning the problem-solving methodologies. The second step aspects have been listed as "solving problems by mathematical techniques" and "obtaining new formulas." The scientific progress has been continued thanks to the studies done since the sixteenth century. The development in scientific area occurred in the elasticity concept has been summarized and visualized in the consideration of the scientists who have lived between the sixteenth and twentieth centuries and their studies [1, 2]. These famous scientists were Galilei (1564–1642), Mariotte (1620–1684), Hooke (1635–1703), Leibniz (1646–1716), Bernoulli (1700–1782), Baumgarten (1706–1757), Euler (1707–1783), Coulomb (1736–1806), Young (1773–1829), Poisson (1781–1840), Navier (1785–1836), Cauchy (1789–1857), Saint-Venant (1797–1886), Borchardt (1817–1880), Rankine (1820–1872), Kirchhoff (1824–1887), Maxwell (1831–1879), Clebsch (1833–1872), Kohlrausch (1840–1910), Amagat (1841–1915), Voigt (1850–1919), Mallock (1851–1933), Lamme (1864–1924), Röntgen (1872,1919),

Synge (1897–1995), and Everett (1930–1982) (**Figure 1**).

*Development of elasticity between the sixteenth and twentieth centuries.*

**3. Classification of engineering problems in the context of elasticity**

In this section, the results of the literature review on elasticity were evaluated by referring to the articles (total number of articles, 157) between 2014 and 2018. Important information has gained from the literature survey about the elasticity theory and its related recent engineering solutions, as well as information about

**4**

**Figure 1.**

**Figure 3.**

*The results of the literature review on elasticity were evaluated by referring to the 157 articles between 2014 and 2018.*

(6) analytical and FEA studies related to the specified boundary conditions, (7) continuum mechanics problems and solutions, (8) analytical and numerical analysis solutions, (9) typical engineering application problems, and (10) solution techniques. The types of elasticity problems have been grouped according to the science innovations and related industrial applications. The numerical problems have been solved in three basic steps. The first step was to check the basic differential equations in terms of satisfaction with the placement of the estimated displacement functions. The second step was to check the "initial values" or the "boundary conditions" of the problem [3–5]. Values were substituted into the differential equations in order to satisfy the conditions at these defined coordinates or at time domains. The boundary conditions have been classified in two groups as "the essential" (displacement) and "the natural" (force) boundary conditions. The initial conditions were the first-stage variations named initiative and time-dependent variables. The third step was the satisfaction of the continuity conditions on the compatibility equations by means of assumed displacement functions. The basic elasticity problems were grouped into 26 subtitles as described in **Figure 2**. In this figure, the number of generally used proposed solution techniques analytically and numerically was equal to eight.

#### **4. General principles in the elasticity theory**

Elasticity concept is explainable by the natural elastic behavior of the materials. In elastic region, material deformed in a nonpermanent form up to the elastic limit was reached. The relationship between stress (*σ*) and strain (*ε*) under loading and unloading cases was explained by the linear and nonlinear equations. The slopes of the linear curves developed in linear elastic region were known as Young's modulus E, and shear modulus G, of the materials under tensile/compression and torsion tests. During these tests, total calculated area under the linear curves was defined as the total potential energy stored in the material. Proportionally, stress development

**7**

*Introductory Chapter: Analytical and Numerical Approaches in Engineering Elasticity*

and strains occurred in the structure according to the applied load. Principally, application of the stress distributions should be very slowly; on the other hand, at each incremental loading step, the equilibrium state and its equilibrium equations of the specimen should be satisfied. This controlled operation and action-reaction principle have worked under the control mechanism of the testing machine. The total work done by incremental external forces "dW" was equal to total potential energy stored incrementally "dU" in the structure of linear elastic region. Using this principle, the governing equations were satisfied by dW − dU = 0. Otherwise, in the case which used high strain rates ε, the material behavior would have been ̇ examined in the material nonlinearity concept. In the nonlinear elastic material experimental tests, the resulting stress-strain curves represented the combination of the behavior of nonlinear continuous or multiple nonlinear continuous forms. In nonlinear curves, the stored potential energy "U" developed in the elastic limit range was calculated in the consideration of two areas: the first area under the σ − ε nonlinear curve described as the stored potential energy by strain increments ε + d and the second area above the curve, known as the complementary potential energy by stress increments σ + d stored in the material. Both linear and nonlinear elasticity equations were derived according to the assumption that during loading and unloading stages of the experiments, the material stores its potential energy within the molecules and there was no loss of energy. As known in the molecular concept, the binding energy keeps the molecules together at any instant of time, and in the lack of energy loss such as heat or light, there will be no loss in the total mass of the molecular system. This phenomenon shows us that the system, which has no energy loss, does not combine (no binding status) with another solid object or with atoms that oscillates at short distances. Otherwise, in the case of the material decreases in amount as losing its mass as energy in the form of heat or light during the binding process, the removed energy corresponding to the removed mass can

mass change in the system, and c is the speed of light, respectively. The elasticity solutions were grouped in terms of a variety of the material, geometry, and loading types. Generally, the used geometries were selected as bar-, beam-, plate-, and shell-type isotropic or composite-type structures. In order to obtain analytical and numerical solutions, the three-dimensional elasticity problems can be reduced into two-dimensional problems in the consideration of the plane stress and plane strain concepts of the elasticity. By these methods the total number of unknowns will be equal to total numbers of equations. Otherwise, some unknown values will stay in unsolvable or undefined forms. Geometrical, material, and loading symmetries reduce problem-solving difficulties in the analytical and numerical models. On the other hand, continuity conditions in geometries automatically satisfies the continuity conditions in the analytical and numerical solutions of elasticity. For example, the existence of the fourth-order partial derivatives of the assumed solution approximation functions is checking the continuity and compatibility equations. Singularity problems may be discarded by omitting the very small holes, empty spaces, gaps in macroscale, or dislocations and beside these the distances between small particles in microscale. In the case of a three-dimensional problem in elasticity, 15 unknowns were defined as mentioned below. These were six stress components, six strain components, and three displacement components. These unknown values were to be calculated by using 15 elasticity equations, three equilibrium equations, six stress-strain relationships, and six strain-displacement relationships. Continuity conditions were satisfied by considering the six compatibility equations which were derived from 15 elasticity equations in three-dimensional problems. Boundary conditions and the initial conditions were both defined on the boundaries and at the starting time domains, respectively, in order to obtain the solutions under the

. Here, E is the binding energy, m is the

*DOI: http://dx.doi.org/10.5772/intechopen.82328*

be explained by Einstein's equation E = mc2

#### *Introductory Chapter: Analytical and Numerical Approaches in Engineering Elasticity DOI: http://dx.doi.org/10.5772/intechopen.82328*

and strains occurred in the structure according to the applied load. Principally, application of the stress distributions should be very slowly; on the other hand, at each incremental loading step, the equilibrium state and its equilibrium equations of the specimen should be satisfied. This controlled operation and action-reaction principle have worked under the control mechanism of the testing machine. The total work done by incremental external forces "dW" was equal to total potential energy stored incrementally "dU" in the structure of linear elastic region. Using this principle, the governing equations were satisfied by dW − dU = 0. Otherwise, in the case which used high strain rates ε, the material behavior would have been ̇ examined in the material nonlinearity concept. In the nonlinear elastic material experimental tests, the resulting stress-strain curves represented the combination of the behavior of nonlinear continuous or multiple nonlinear continuous forms. In nonlinear curves, the stored potential energy "U" developed in the elastic limit range was calculated in the consideration of two areas: the first area under the σ − ε nonlinear curve described as the stored potential energy by strain increments ε + d and the second area above the curve, known as the complementary potential energy by stress increments σ + d stored in the material. Both linear and nonlinear elasticity equations were derived according to the assumption that during loading and unloading stages of the experiments, the material stores its potential energy within the molecules and there was no loss of energy. As known in the molecular concept, the binding energy keeps the molecules together at any instant of time, and in the lack of energy loss such as heat or light, there will be no loss in the total mass of the molecular system. This phenomenon shows us that the system, which has no energy loss, does not combine (no binding status) with another solid object or with atoms that oscillates at short distances. Otherwise, in the case of the material decreases in amount as losing its mass as energy in the form of heat or light during the binding process, the removed energy corresponding to the removed mass can be explained by Einstein's equation E = mc2 . Here, E is the binding energy, m is the mass change in the system, and c is the speed of light, respectively. The elasticity solutions were grouped in terms of a variety of the material, geometry, and loading types. Generally, the used geometries were selected as bar-, beam-, plate-, and shell-type isotropic or composite-type structures. In order to obtain analytical and numerical solutions, the three-dimensional elasticity problems can be reduced into two-dimensional problems in the consideration of the plane stress and plane strain concepts of the elasticity. By these methods the total number of unknowns will be equal to total numbers of equations. Otherwise, some unknown values will stay in unsolvable or undefined forms. Geometrical, material, and loading symmetries reduce problem-solving difficulties in the analytical and numerical models. On the other hand, continuity conditions in geometries automatically satisfies the continuity conditions in the analytical and numerical solutions of elasticity. For example, the existence of the fourth-order partial derivatives of the assumed solution approximation functions is checking the continuity and compatibility equations. Singularity problems may be discarded by omitting the very small holes, empty spaces, gaps in macroscale, or dislocations and beside these the distances between small particles in microscale. In the case of a three-dimensional problem in elasticity, 15 unknowns were defined as mentioned below. These were six stress components, six strain components, and three displacement components. These unknown values were to be calculated by using 15 elasticity equations, three equilibrium equations, six stress-strain relationships, and six strain-displacement relationships. Continuity conditions were satisfied by considering the six compatibility equations which were derived from 15 elasticity equations in three-dimensional problems. Boundary conditions and the initial conditions were both defined on the boundaries and at the starting time domains, respectively, in order to obtain the solutions under the

*Elasticity of Materials - Basic Principles and Design of Structures*

(6) analytical and FEA studies related to the specified boundary conditions, (7) continuum mechanics problems and solutions, (8) analytical and numerical analysis solutions, (9) typical engineering application problems, and (10) solution techniques. The types of elasticity problems have been grouped according to the science innovations and related industrial applications. The numerical problems have been solved in three basic steps. The first step was to check the basic differential equations in terms of satisfaction with the placement of the estimated displacement functions. The second step was to check the "initial values" or the "boundary conditions" of the problem [3–5]. Values were substituted into the differential equations in order to satisfy the conditions at these defined coordinates or at time domains. The boundary conditions have been classified in two groups as "the essential" (displacement) and "the natural" (force) boundary conditions. The initial conditions were the first-stage variations named initiative and time-dependent variables. The third step was the satisfaction of the continuity conditions on the compatibility equations by means of assumed displacement functions. The basic elasticity problems were grouped into 26 subtitles as described in **Figure 2**. In this figure, the number of generally used proposed solution techniques analytically and numeri-

*The results of the literature review on elasticity were evaluated by referring to the 157 articles between 2014 and* 

Elasticity concept is explainable by the natural elastic behavior of the materials. In elastic region, material deformed in a nonpermanent form up to the elastic limit was reached. The relationship between stress (*σ*) and strain (*ε*) under loading and unloading cases was explained by the linear and nonlinear equations. The slopes of the linear curves developed in linear elastic region were known as Young's modulus E, and shear modulus G, of the materials under tensile/compression and torsion tests. During these tests, total calculated area under the linear curves was defined as the total potential energy stored in the material. Proportionally, stress development

**6**

cally was equal to eight.

**Figure 3.**

*2018.*

**4. General principles in the elasticity theory**

limitation of approximate and true percentage minimum error calculations. In the case of three-dimensional elasticity problem, 15 unknown values have to be solved by 15 governing equations (the list of the unknowns were six stress components [σ*<sup>x</sup>* σ*<sup>y</sup>* σ*<sup>z</sup>* τ*xy* τ*yz* τ*xz*] and six strain components [<sup>ε</sup>*<sup>x</sup>* <sup>ε</sup>*<sup>y</sup>* <sup>ε</sup>*<sup>z</sup>* <sup>γ</sup>*xy* <sup>γ</sup>*yz* <sup>γ</sup>*xz*], and additionally the three displacement components [u v w]) [3, 4]. In solid mechanics and elasticity theory, the governing partial differential equations, the constitutive and kinematics equations, and the initial and boundary conditions have been all defined. However, if at least one of the above conditions has remained partially or entirely unknown, then one has a so-called inverse problem (**Figure 2**) [5]. On the other hand, the elasticity "inverse problem" has been defined for the problems in which they consist of recovering the missing displacements to the solution space corresponding to the applied force data by using the iterative calculation steps. Obviously, lost or uncalculated data developing on one part of a whole domain boundary have directly affected the final configuration of the stress-strain and displacement components and their resultant solution spaces at the other part of this boundary. The proposed solutions were both numerical and analytical (**Figure 2**). Inverse problem of elasticity in other words Cauchy problem (Cauchy-Navier equations of elasticity) has been defined on the accessible outer boundary of the structure. The Cauchy stress tensor components were related with the infinitesimal (incremental calculations) strain tensor components which have been identified in deformed configuration with successive iterations.

The stress-strain relationship in terms of indicial notation is given below:

$$
\sigma\_{\vec{\imath}\vec{\jmath}} = \mathcal{Z}\,\mu\epsilon\_{\vec{\imath}\vec{\jmath}} + \lambda\delta\_{\vec{\imath}\vec{\jmath}}\,\varepsilon\_{kk} \tag{1}
$$

**9**

**Author details**

Ankara, Turkey

Ezgi Günay

provided the original work is properly cited.

\*Address all correspondence to: ezgigunay@gazi.edu.tr

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Mechanical Engineering Department, Engineering Faculty, Gazi University,

*Introductory Chapter: Analytical and Numerical Approaches in Engineering Elasticity*

the forces and the potential energy stored has been explained by the material elastic constants. The mechanical response of a homogeneous isotropic linearly elastic material can be explained by two physical constants, Young's modulus and Poisson's ratio. The elastic properties of particle composites, consisting in a dispersion of nonlinear (spherical or cylindrical) nonhomogeneities into a linear solid matrix, were explained by homogenization procedure. The linear elastic constants of fiber composite materials have been defined according to their three principle directions [6]. These principle directions coincided with the fiber orientations located in each layer. By contrast, the physical-mechanical properties of nonlinear elastic materials have generally been described by parameters which have formations as the scalar functions of the deformation, and their material properties have been determined

*DOI: http://dx.doi.org/10.5772/intechopen.82328*

by selecting the suitable solution techniques.

Here, *μ*, λ are the Lamé constants. The Cauchy strain components represent the geometrical nonlinearity of the material according to the deformed configuration.

The inverse problem solution depends on the stepwise calculated and so updated Cauchy stress and strain distributions, over the whole boundary of the geometry. Experimentally, tractions and displacements have been measured by nondestructive tests. In isotropic, fiber, and particulate composite material concepts, the stress-strain distributions σ − ε have been examined according to the defined total number of elastic constants in stiffness [C] matrix. The inverse of the stiffness matrix named as the compliance matrix [*S*] <sup>=</sup> [*C*] −1 includes the elastic constants in ε <sup>−</sup> σ strain versus stress equations. In the generalized Hook's law, anisotropic crystalline materials have been defined with 36 constants. Strain energy function has to be used to show that the number of independent material constants can be reduced from 36 to 21. The solution techniques as iterative methods, inverse method, semi-inverse method, variational formulation, finite element method, finite volume method, and meshless method have been listed in **Figure 2**. The experimental solution techniques have been explained by tensile, compression, torsion, impact, and bending mechanical tests. Nondestructive tests (NDT) have been used to obtain informational data from the surfaces of the materials (nanoindentation-hardness testing).

#### **5. Conclusion**

In this introduction chapter, the historical development of the elasticity concept and its engineering properties were presented briefly. According to Newton's action and reaction principle, the materials behave linear or nonlinear elastically under typical loading. Elasticity theory provides necessarily required equations and solution techniques. The action-response principle defined between the work done by

#### *Introductory Chapter: Analytical and Numerical Approaches in Engineering Elasticity DOI: http://dx.doi.org/10.5772/intechopen.82328*

the forces and the potential energy stored has been explained by the material elastic constants. The mechanical response of a homogeneous isotropic linearly elastic material can be explained by two physical constants, Young's modulus and Poisson's ratio. The elastic properties of particle composites, consisting in a dispersion of nonlinear (spherical or cylindrical) nonhomogeneities into a linear solid matrix, were explained by homogenization procedure. The linear elastic constants of fiber composite materials have been defined according to their three principle directions [6]. These principle directions coincided with the fiber orientations located in each layer. By contrast, the physical-mechanical properties of nonlinear elastic materials have generally been described by parameters which have formations as the scalar functions of the deformation, and their material properties have been determined by selecting the suitable solution techniques.

### **Author details**

*Elasticity of Materials - Basic Principles and Design of Structures*

limitation of approximate and true percentage minimum error calculations. In the case of three-dimensional elasticity problem, 15 unknown values have to be solved by 15 governing equations (the list of the unknowns were six stress components [σ*<sup>x</sup>* σ*<sup>y</sup>* σ*<sup>z</sup>* τ*xy* τ*yz* τ*xz*] and six strain components [<sup>ε</sup>*<sup>x</sup>* <sup>ε</sup>*<sup>y</sup>* <sup>ε</sup>*<sup>z</sup>* <sup>γ</sup>*xy* <sup>γ</sup>*yz* <sup>γ</sup>*xz*], and additionally the three displacement components [u v w]) [3, 4]. In solid mechanics and elasticity theory, the governing partial differential equations, the constitutive and kinematics equations, and the initial and boundary conditions have been all defined. However, if at least one of the above conditions has remained partially or entirely unknown, then one has a so-called inverse problem (**Figure 2**) [5]. On the other hand, the elasticity "inverse problem" has been defined for the problems in which they consist of recovering the missing displacements to the solution space corresponding to the applied force data by using the iterative calculation steps. Obviously, lost or uncalculated data developing on one part of a whole domain boundary have directly affected the final configuration of the stress-strain and displacement components and their resultant solution spaces at the other part of this boundary. The proposed solutions were both numerical and analytical (**Figure 2**). Inverse problem of elasticity in other words Cauchy problem (Cauchy-Navier equations of elasticity) has been defined on the accessible outer boundary of the structure. The Cauchy stress tensor components were related with the infinitesimal (incremental calculations) strain tensor components which have been identified in deformed configuration with successive

The stress-strain relationship in terms of indicial notation is given below:

constants in ε <sup>−</sup> σ strain versus stress equations. In the generalized Hook's law, anisotropic crystalline materials have been defined with 36 constants. Strain energy function has to be used to show that the number of independent material constants can be reduced from 36 to 21. The solution techniques as iterative methods, inverse method, semi-inverse method, variational formulation, finite element method, finite volume method, and meshless method have been listed in **Figure 2**. The experimental solution techniques have been explained by tensile, compression, torsion, impact, and bending mechanical tests. Nondestructive tests (NDT) have been used to obtain informational data from the surfaces of the materials (nanoin-

In this introduction chapter, the historical development of the elasticity concept and its engineering properties were presented briefly. According to Newton's action and reaction principle, the materials behave linear or nonlinear elastically under typical loading. Elasticity theory provides necessarily required equations and solution techniques. The action-response principle defined between the work done by

the stiffness matrix named as the compliance matrix [*S*] <sup>=</sup> [*C*]

*σij = 2ij + ij εkk* (1)

Here, *μ*, λ are the Lamé constants. The Cauchy strain components represent the geometrical nonlinearity of the material according to the deformed configuration. The inverse problem solution depends on the stepwise calculated and so updated Cauchy stress and strain distributions, over the whole boundary of the geometry. Experimentally, tractions and displacements have been measured by nondestructive tests. In isotropic, fiber, and particulate composite material concepts, the stress-strain distributions σ − ε have been examined according to the defined total number of elastic constants in stiffness [C] matrix. The inverse of

−1

includes the elastic

**8**

dentation-hardness testing).

**5. Conclusion**

iterations.

Ezgi Günay Mechanical Engineering Department, Engineering Faculty, Gazi University, Ankara, Turkey

\*Address all correspondence to: ezgigunay@gazi.edu.tr

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An Overview of Stress-Strain

Pulkit Kumar, Moumita Mahanty

and Amares Chattopadhyay

Abstract

are discussed.

1. Introduction

11

Analysis for Elasticity Equations

The present chapter contains the analysis of stress, analysis of strain and stress-strain relationship through particular sections. The theory of elasticity contains equilibrium equations relating to stresses, kinematic equations relating to the strains and displacements and the constitutive equations relating to the stresses and strains. Concept of normal and shear stresses, principal stress, plane stress, Mohr's circle, stress invariants and stress equilibrium relations are discussed in analysis of stress section while strain-displacement relationship for normal and shear strain, compatibility of strains are discussed in analysis of strain section through geometrical representations. Linear elasticity, generalized Hooke's law and stress-strain relations for triclinic, monoclinic, orthotropic, transversely isotropic, fiber-reinforced and isotropic materials with some important relations for elasticity

Keywords: analysis of stress, analysis of strain, Mohr's circle, compatibility of

If the external forces producing deformation do not exceed a certain limit, the deformation disappears with the removal of the forces. Thus the elastic behavior implies the absence of any permanent deformation. Every engineering material/ composite possesses a certain extent of elasticity. The common materials of construction would remain elastic only for very small strains before exhibiting either plastic straining or brittle failure. However, natural polymeric composites show elasticity over a wider range and the widespread use of natural rubber and similar composites motivated the development of finite elasticity. The mathematical theory of elasticity is possessed with an endeavor to decrease the computation for condition of strain, or relative displacement inside a solid body which is liable to the activity of an equilibrating arrangement of forces, or is in a condition of little inward relative motion and with tries to obtain results which might have been basically essential applications to design, building, and all other helpful expressions

The elastic properties of continuous materials are determined by the underlying molecular structure, but the relation between material properties and the molecular structure and arrangement in materials is complicated. There are wide classes of materials that might be portrayed by a couple of material constants which can be

strain, stress-strain relation, generalized Hooke's law

in which the material of development is solid.

### **References** Chapter 2

[1] Todhunter I. A history of the theory of elasticity and of strength of the materials from Galilei to the present time, Vol. II. Saint-Venant to Lord Kelvin. King's College, Cambridge: Pearson; 1893. 574 p. http://www.archive.org/details/ historyoftheoryo02todhuoft

[2] Todhunter I. A history of the theory of elasticity and of strength of the materials from Galilei to the present time. In: Saint-Venant to Lord Kelvin, Part 1. Vol. 2. King's College Cambridge Pearson; 2014. 348 p. DOI: 10.1017/ CB09781107280076

[3] Timoshenko SP, Goodier JN. Theory of Elasticity. 3rd ed. New York: McGraw-Hill; 1970. 580 p

[4] Ugural AC, Fenster S. Advanced Mechanics of Materials and Applied Elasticity. 5th ed. New Jersey: Prentice Hall; 2012. 680 p

[5] Karageorghis A, Lesnic D, Marin L. The method of fundamental solutions for three-dimensional inverse geometric elasticity problems. Computers and Structures. 2016;**166**:51-59. DOI: 10.1016/j.compstruc.2016.01.010

[6] Jones RM. Mechanics of Composite Materials. New York: Hemisphere Publishing Co; 1975. 355 p

#### **References** Chapter 2

## An Overview of Stress-Strain Analysis for Elasticity Equations

Pulkit Kumar, Moumita Mahanty and Amares Chattopadhyay

#### Abstract

The present chapter contains the analysis of stress, analysis of strain and stress-strain relationship through particular sections. The theory of elasticity contains equilibrium equations relating to stresses, kinematic equations relating to the strains and displacements and the constitutive equations relating to the stresses and strains. Concept of normal and shear stresses, principal stress, plane stress, Mohr's circle, stress invariants and stress equilibrium relations are discussed in analysis of stress section while strain-displacement relationship for normal and shear strain, compatibility of strains are discussed in analysis of strain section through geometrical representations. Linear elasticity, generalized Hooke's law and stress-strain relations for triclinic, monoclinic, orthotropic, transversely isotropic, fiber-reinforced and isotropic materials with some important relations for elasticity are discussed.

Keywords: analysis of stress, analysis of strain, Mohr's circle, compatibility of strain, stress-strain relation, generalized Hooke's law

#### 1. Introduction

If the external forces producing deformation do not exceed a certain limit, the deformation disappears with the removal of the forces. Thus the elastic behavior implies the absence of any permanent deformation. Every engineering material/ composite possesses a certain extent of elasticity. The common materials of construction would remain elastic only for very small strains before exhibiting either plastic straining or brittle failure. However, natural polymeric composites show elasticity over a wider range and the widespread use of natural rubber and similar composites motivated the development of finite elasticity. The mathematical theory of elasticity is possessed with an endeavor to decrease the computation for condition of strain, or relative displacement inside a solid body which is liable to the activity of an equilibrating arrangement of forces, or is in a condition of little inward relative motion and with tries to obtain results which might have been basically essential applications to design, building, and all other helpful expressions in which the material of development is solid.

The elastic properties of continuous materials are determined by the underlying molecular structure, but the relation between material properties and the molecular structure and arrangement in materials is complicated. There are wide classes of materials that might be portrayed by a couple of material constants which can be

**10**

*Elasticity of Materials - Basic Principles and Design of Structures*

[1] Todhunter I. A history of the theory of elasticity and of strength of the materials from Galilei to the present time, Vol. II. Saint-Venant to Lord Kelvin. King's College, Cambridge: Pearson; 1893. 574 p. http://www.archive.org/details/ historyoftheoryo02todhuoft

[2] Todhunter I. A history of the theory of elasticity and of strength of the materials from Galilei to the present time. In: Saint-Venant to Lord Kelvin, Part 1. Vol. 2. King's College Cambridge Pearson; 2014. 348 p. DOI: 10.1017/

[3] Timoshenko SP, Goodier JN. Theory

of Elasticity. 3rd ed. New York: McGraw-Hill; 1970. 580 p

[4] Ugural AC, Fenster S. Advanced Mechanics of Materials and Applied Elasticity. 5th ed. New Jersey: Prentice

[5] Karageorghis A, Lesnic D, Marin L. The method of fundamental solutions for three-dimensional inverse geometric elasticity problems. Computers and Structures. 2016;**166**:51-59. DOI: 10.1016/j.compstruc.2016.01.010

[6] Jones RM. Mechanics of Composite Materials. New York: Hemisphere

Publishing Co; 1975. 355 p

CB09781107280076

Hall; 2012. 680 p

determined by macroscopic experiments. The quantity of such constants relies upon the nature of the crystalline structure of the material. In this section, we give a short but then entire composition of the basic highlights of applied elasticity having pertinence to our topics. This praiseworthy theory, likely the most successful and best surely understood theory of elasticity, has been given numerous excellent and comprehensive compositions. Among the textbooks including an ample coverage of the problems, we deal with in this chapter which are discussed earlier by Love [1], Sokolnikoff [2], Malvern [3], Gladwell [4], Gurtin [5], Brillouin [6], Pujol [7], Ewing, Jardetsky and Press [8], Achenbach [9], Eringen and Suhubi [10], Jeffreys and Jeffreys [11], Capriz and Podio-Guidugli [12], Truesdell and Noll [13] whose use of direct notation and we find appropriate to avoid encumbering conceptual developments with component-wise expressions. Meriam and Kraige [14] gave an overview of engineering mechanics in theirs book and Podio-Guidugli [15, 16] discussed the strain and examples of concentrated contact interactions in simple bodies in the primer of elasticity. Interestingly, no matter how early in the history of elasticity the consequences of concentrated loads were studied, some of those went overlooked until recently [17–22]. The problem of the determination of stress and strain fields in the elastic solids are discussed by many researchers [23–33]. Belfield et al. [34] discussed the stresses in elastic plates reinforced by fibers lying in concentric circles. Biot [35–38] gave the theory for the propagation of elastic waves in an initially stressed and fluid saturated transversely isotropic media. Borcherdt and Brekhovskikh [39–41] studied the propagation of surface waves in viscoelastic layered media. The fundamental study of seismic surface waves due to the theory of linear viscoelasticity and stress-strain relationship is elaborated by some notable researchers [42–46]. The stress intensity factor is computed due to diffraction of plane dilatational waves by a finite crack by Chang [47], magnetoelastic shear waves in an infinite self-reinforced plate by Chattopadhyay and Choudhury [48]. The propagation of edge wave under initial stress is discussed by Das and Dey [49] and existence and uniqueness of edge waves in a generally anisotropic laminated elastic plates by Fu and Brookes [50, 51]. The basic and historical literature about the stress-strain relationship for propagation of elastic waves in kinds of medium is given by some eminent researchers [52–57]. Kaplunov, Pichugin and Rogersion [58–60] have discussed the propagation of extensional edge waves in in semi-infinite isotropic plates, shells and incompressible plates under the influence of initial stresses. The theory of boundary layers in highly anisotropic and/or reinforced elasticity is studied by Hool, Kinne and Spencer [61, 62].

equilibrated on the basis of Newton's third law. The internal forces are always

a) b)

Stress ð Þ¼ σ lim

where ΔF is the internal force on the area ΔA surrounding the given point.

Body forces always act on every molecule of a body and are proportional to the volume whereas surface force acts over the surface of the body and is measure in terms of force per unit area. The force acting on a surface may resolve into normal stress and shear stress. Normal stress may be tensile or compressive in nature. Positive side of normal stress is for tensile stress whilst negative side is for compressive.

Figure 2(a) shows the rectangular components of the force vector ΔF referred

to corresponding axes. Taking the ratios ΔFx=ΔAx, ΔFy=ΔAx, ΔFz=ΔAx, three quantities that set up the average intensity of the force on the area ΔAx When the limit ΔA ! 0, the above ratios are characterized as the force intensity acting on X-face at point O. These values associated with three intensities are defined as the "Stress components" related with the <sup>X</sup>‐face at point <sup>O</sup>. The stress component parallel to the surface are called "Shear stress component," is indicated by τ: The

Forces which act on an element of material may be of two types:

ΔA!0

ΔF

<sup>Δ</sup><sup>A</sup> (1)

To examine these internal forces at a point O in Figure 1(a), inside the body, consider a plane MN passing through the point O. If the plane is divided into a number of small areas, as in the Figure 1(b), and the forces acting on each of these are measured, it will be observed that these forces vary from one small area to the next. On the small area ΔA at point O, a force ΔF will be acting as shown in Figure 1(b). From this the concept of stress as the internal force per unit area can be understood. Assuming that the material is continuous, the term "stress" at any point across a small area ΔA can be defined by the limiting equation as below.

present even though the external forces are not active.

An Overview of Stress-Strain Analysis for Elasticity Equations

DOI: http://dx.doi.org/10.5772/intechopen.82066

Forces acting on a (a) body, (b) cross-section of the body.

i. body forces and

Figure 1.

ii. surface forces.

13

2.1 Concept of normal stress and shear stress

This chapter addresses the analysis of stress, analysis of strain and stress-strain relationship through particular sections. Concept of normal and shear stress, principal stress, plane stress, Mohr's circle, stress invariants and stress equilibrium relations are discussed in analysis of stress section while strain-displacement relationship for normal and shear strain, compatibility of strains are discussed in analysis of strain section through geometrical representations too. Linear elasticity generalized Hooke's law and stress-strain relation for triclinic, monoclinic, orthotropic, transversely isotropic and isotropic materials are discussed and some important relations for elasticity are deliberated.

#### 2. Analysis of stress

A body consists of huge number of grains or molecules. The internal forces act within a body, representing the interaction between the grains or molecules of the body. In general, if a body is in statically equilibrium, then the internal forces are

An Overview of Stress-Strain Analysis for Elasticity Equations DOI: http://dx.doi.org/10.5772/intechopen.82066

determined by macroscopic experiments. The quantity of such constants relies upon the nature of the crystalline structure of the material. In this section, we give a short but then entire composition of the basic highlights of applied elasticity having pertinence to our topics. This praiseworthy theory, likely the most successful and best surely understood theory of elasticity, has been given numerous excellent and comprehensive compositions. Among the textbooks including an ample coverage of the problems, we deal with in this chapter which are discussed earlier by Love [1], Sokolnikoff [2], Malvern [3], Gladwell [4], Gurtin [5], Brillouin [6], Pujol [7], Ewing, Jardetsky and Press [8], Achenbach [9], Eringen and Suhubi [10], Jeffreys and Jeffreys [11], Capriz and Podio-Guidugli [12], Truesdell and Noll [13] whose use of direct notation and we find appropriate to avoid encumbering conceptual developments with component-wise expressions. Meriam and Kraige [14] gave an overview of engineering mechanics in theirs book and Podio-Guidugli [15, 16] discussed the strain and examples of concentrated contact interactions in simple bodies in the primer of elasticity. Interestingly, no matter how early in the history of elasticity the consequences of concentrated loads were studied, some of those went overlooked until recently [17–22]. The problem of the determination of stress and strain fields in the elastic solids are discussed by many researchers [23–33]. Belfield et al. [34] discussed the stresses in elastic plates reinforced by fibers lying in concentric circles. Biot [35–38] gave the theory for the propagation of elastic waves in an initially stressed and fluid saturated transversely isotropic media. Borcherdt and Brekhovskikh [39–41] studied the propagation of surface waves in viscoelastic layered media. The fundamental study of seismic surface waves due to the theory of linear viscoelasticity and stress-strain relationship is elaborated by some notable researchers [42–46]. The stress intensity factor is computed due to diffraction of plane dilatational waves by a finite crack by Chang [47], magnetoelastic shear waves in an infinite self-reinforced plate by Chattopadhyay and Choudhury [48]. The propagation of edge wave under initial stress is discussed by Das and Dey [49] and existence and uniqueness of edge waves in a generally anisotropic laminated elastic plates by Fu and Brookes [50, 51]. The basic and historical literature about the stress-strain relationship for propagation of elastic waves in kinds of medium is given by some eminent researchers [52–57]. Kaplunov, Pichugin and Rogersion

Elasticity of Materials ‐ Basic Principles and Design of Structures

[58–60] have discussed the propagation of extensional edge waves in in

reinforced elasticity is studied by Hool, Kinne and Spencer [61, 62].

semi-infinite isotropic plates, shells and incompressible plates under the influence of initial stresses. The theory of boundary layers in highly anisotropic and/or

relationship through particular sections. Concept of normal and shear stress, principal stress, plane stress, Mohr's circle, stress invariants and stress equilibrium relations are discussed in analysis of stress section while strain-displacement relationship for normal and shear strain, compatibility of strains are discussed in analysis of strain section through geometrical representations too. Linear elasticity

generalized Hooke's law and stress-strain relation for triclinic, monoclinic, orthotropic, transversely isotropic and isotropic materials are discussed and some

important relations for elasticity are deliberated.

2. Analysis of stress

12

This chapter addresses the analysis of stress, analysis of strain and stress-strain

A body consists of huge number of grains or molecules. The internal forces act within a body, representing the interaction between the grains or molecules of the body. In general, if a body is in statically equilibrium, then the internal forces are

Figure 1. Forces acting on a (a) body, (b) cross-section of the body.

equilibrated on the basis of Newton's third law. The internal forces are always present even though the external forces are not active.

To examine these internal forces at a point O in Figure 1(a), inside the body, consider a plane MN passing through the point O. If the plane is divided into a number of small areas, as in the Figure 1(b), and the forces acting on each of these are measured, it will be observed that these forces vary from one small area to the next. On the small area ΔA at point O, a force ΔF will be acting as shown in Figure 1(b). From this the concept of stress as the internal force per unit area can be understood. Assuming that the material is continuous, the term "stress" at any point across a small area ΔA can be defined by the limiting equation as below.

$$\text{Stress}\,(\sigma) = \lim\_{\Delta A \to 0} \frac{\Delta F}{\Delta A} \tag{1}$$

where ΔF is the internal force on the area ΔA surrounding the given point. Forces which act on an element of material may be of two types:


Body forces always act on every molecule of a body and are proportional to the volume whereas surface force acts over the surface of the body and is measure in terms of force per unit area. The force acting on a surface may resolve into normal stress and shear stress. Normal stress may be tensile or compressive in nature. Positive side of normal stress is for tensile stress whilst negative side is for compressive.

#### 2.1 Concept of normal stress and shear stress

Figure 2(a) shows the rectangular components of the force vector ΔF referred to corresponding axes. Taking the ratios ΔFx=ΔAx, ΔFy=ΔAx, ΔFz=ΔAx, three quantities that set up the average intensity of the force on the area ΔAx When the limit ΔA ! 0, the above ratios are characterized as the force intensity acting on X-face at point O. These values associated with three intensities are defined as the "Stress components" related with the <sup>X</sup>‐face at point <sup>O</sup>. The stress component parallel to the surface are called "Shear stress component," is indicated by τ: The

Figure 2. (a) Force components of ΔF acting on small area centered at point O and (b) stress components at point O.

shear stress component acting on the <sup>X</sup>‐face in the Y-direction is identified as <sup>τ</sup>xy: The stress component perpendicular to the face is called "Normal Stress" or "Direct stress" component and is denoted by σ.

From the above discussions, the stress components on the <sup>X</sup>‐face at point <sup>O</sup> are defined as follows in terms of force intensity ratios

$$\begin{aligned} \sigma\_{\mathbf{x}} &= \lim\_{\Delta A\_{\mathbf{x}} \to 0} \frac{\Delta F\_{\mathbf{x}}}{\Delta A\_{\mathbf{x}}}\\ \tau\_{\mathbf{xy}} &= \lim\_{\Delta A\_{\mathbf{x}} \to 0} \frac{\Delta F\_{\mathbf{y}}}{\Delta A\_{\mathbf{x}}}\\ \tau\_{\mathbf{xx}} &= \lim\_{\Delta A\_{\mathbf{x}} \to 0} \frac{\Delta F\_{\mathbf{x}}}{\Delta A\_{\mathbf{x}}} \end{aligned} \tag{2}$$

normal stresses are referred to as principal stresses and the plane in which these

Invariants mean those amounts that are unexchangeable and do not differ under various conditions. With regards to stress components, invariants are such quantities that don't change with rotation of axes or which stay unaffected under transformation, from one set of axes to another. Subsequently, the combination of stresses at a point that don't change with the introduction of co-ordinate axis is

Numerous metal shaping procedures include biaxial condition of stress. On the off chance that one of the three normal and shear stresses acting on a body is zero, the state of stress is called plane stress condition. All stresses act parallel to x and y axes. Plane pressure condition is gone over in numerous engineering and forming applications. Regularly, slip can be simple if the shear stress following up on the slip planes is adequately high and acts along favored slip direction. Slip planes may be inclined with respect to the external stress acting on solids. It becomes necessary to transform the stresses acting along the original axes into the inclined planes. Stress

Consider the plane stress condition acting on a plane as shown in Figure 4. Let us investigate the state of stresses onto a transformed plane which is inclined at an

Let by rotating of the x and y axes through the angle θ, a new set of axes X' and Y<sup>0</sup> will be formed. The stresses acting on the plane along the new axes are obtained when the plane has been rotated about the z axis. In order to obtain these transformed stresses, we take equilibrium of forces on the inclined plane both

Thus, the expression for transformed stress using the direction cosines can be

normal stresses act is called principal plane.

An Overview of Stress-Strain Analysis for Elasticity Equations

DOI: http://dx.doi.org/10.5772/intechopen.82066

change ends up essential in such cases.

2.4.1 Stress transformation in plane stress

perpendicular to and parallel to the inclined plane.

angle θ with respect to x, y axes.

written as

15

called stress invariants.

Stress components acting on cube.

2.4 Plane stress

Figure 3.

and the above stress components are illustrated in Figure 2(b).

#### 2.2 Stress components

Three mutually perpendicular coordinate axes x, y, z are taken. We consider the stresses act on the surface of the cubic element of the substance.When a force is applied, as mean that the state of stress is perfectly homogeneous throughout the element and that the body is in equilibrium as shown in Figure 3. There are nine quantities which are acting on the faces of the cubic and are known as the stress components.

In matrix notation, the stress components can be written as

$$
\begin{pmatrix}
\sigma\_{\rm x} & \tau\_{\rm xy} & \tau\_{\rm xx} \\
\tau\_{\rm yx} & \sigma\_{\rm y} & \tau\_{\rm yx} \\
\tau\_{\rm xx} & \tau\_{\rm xy} & \sigma\_{\rm x}
\end{pmatrix}
\tag{3}
$$

which completely define the state of stress in the elemental cube. The first suffix of the shear stress refers to the normal to the plane on which the stress acts and the second suffix refer to the direction of shear stress on this plane. The nine stress components which are derived in matrix form are not all independent quantities.

#### 2.3 Principal stress and stress invariants

Let us consider three mutually perpendicular planes in which shear stress is zero and on these planes the normal stresses have maximum or minimum values. These

An Overview of Stress-Strain Analysis for Elasticity Equations DOI: http://dx.doi.org/10.5772/intechopen.82066

Figure 3. Stress components acting on cube.

shear stress component acting on the <sup>X</sup>‐face in the Y-direction is identified as <sup>τ</sup>xy: The stress component perpendicular to the face is called "Normal Stress" or "Direct

(a) Force components of ΔF acting on small area centered at point O and (b) stress components at point O.

a) b)

σ<sup>x</sup> ¼ lim ΔAx!0

τxy ¼ lim ΔAx!0

τxz ¼ lim ΔAx!0

Three mutually perpendicular coordinate axes x, y, z are taken. We consider the stresses act on the surface of the cubic element of the substance.When a force is applied, as mean that the state of stress is perfectly homogeneous throughout the element and that the body is in equilibrium as shown in Figure 3. There are nine quantities which are

> σ<sup>x</sup> τxy τxz τyx σ<sup>y</sup> τyz τzx τzy σ<sup>z</sup>

which completely define the state of stress in the elemental cube. The first suffix of the shear stress refers to the normal to the plane on which the stress acts and the second suffix refer to the direction of shear stress on this plane. The nine stress components which are derived in matrix form are not all independent quantities.

Let us consider three mutually perpendicular planes in which shear stress is zero and on these planes the normal stresses have maximum or minimum values. These

1

CA (3)

and the above stress components are illustrated in Figure 2(b).

acting on the faces of the cubic and are known as the stress components. In matrix notation, the stress components can be written as

0

B@

2.3 Principal stress and stress invariants

14

From the above discussions, the stress components on the <sup>X</sup>‐face at point <sup>O</sup> are

ΔFx ΔAx 9 >>>>>>=

>>>>>>;

(2)

ΔFy ΔAx

ΔFz ΔAx

stress" component and is denoted by σ.

2.2 Stress components

Figure 2.

defined as follows in terms of force intensity ratios

Elasticity of Materials ‐ Basic Principles and Design of Structures

normal stresses are referred to as principal stresses and the plane in which these normal stresses act is called principal plane.

Invariants mean those amounts that are unexchangeable and do not differ under various conditions. With regards to stress components, invariants are such quantities that don't change with rotation of axes or which stay unaffected under transformation, from one set of axes to another. Subsequently, the combination of stresses at a point that don't change with the introduction of co-ordinate axis is called stress invariants.

#### 2.4 Plane stress

Numerous metal shaping procedures include biaxial condition of stress. On the off chance that one of the three normal and shear stresses acting on a body is zero, the state of stress is called plane stress condition. All stresses act parallel to x and y axes. Plane pressure condition is gone over in numerous engineering and forming applications. Regularly, slip can be simple if the shear stress following up on the slip planes is adequately high and acts along favored slip direction. Slip planes may be inclined with respect to the external stress acting on solids. It becomes necessary to transform the stresses acting along the original axes into the inclined planes. Stress change ends up essential in such cases.

#### 2.4.1 Stress transformation in plane stress

Consider the plane stress condition acting on a plane as shown in Figure 4. Let us investigate the state of stresses onto a transformed plane which is inclined at an angle θ with respect to x, y axes.

Let by rotating of the x and y axes through the angle θ, a new set of axes X' and Y<sup>0</sup> will be formed. The stresses acting on the plane along the new axes are obtained when the plane has been rotated about the z axis. In order to obtain these transformed stresses, we take equilibrium of forces on the inclined plane both perpendicular to and parallel to the inclined plane.

Thus, the expression for transformed stress using the direction cosines can be written as

#### Figure 4. Representation of stresses on inclined plane.

$$\begin{aligned} \sigma\_{\mathbf{x'}} &= l\_{\mathbf{x'}\mathbf{x}}^2 \sigma\_{\mathbf{x}} + l\_{\mathbf{x'}\mathbf{y}}^2 \sigma\_{\mathbf{y}} + 2l\_{\mathbf{x'}\mathbf{x}} l\_{\mathbf{x'y}} \sigma\_{\mathbf{xy}} \\ &= 2 \cos^2 \theta \sigma\_{\mathbf{x}} + 2 \sin^2 \theta \sigma\_{\mathbf{y}} + 2 \cos \theta \sin \theta \sigma\_{\mathbf{xy}} \end{aligned} \tag{4}$$

The plane on which the principal normal stress acts, the shear stress is zero and vice versa. The angle corresponding to the principal planes can be obtained from

The transformation equations of plane stress which are given by Eq. (5) can be represented in a graphical form (Figure 5) by Mohr's circle. The transformation equations are sufficient to get the normal and shear stresses on any plane at a point, with Mohr's circle one can easily visualize their variation with respect to plane

2

cos 2θ þ τxy sin 2θ

sin 2<sup>θ</sup> <sup>þ</sup> <sup>τ</sup>xy cos 2<sup>θ</sup> (9.2)

is for the principal

(9.1)

for the principal normal planes and tan 2<sup>θ</sup> <sup>¼</sup> <sup>τ</sup>xy <sup>σ</sup>x�σ<sup>y</sup>

tan 2<sup>θ</sup> <sup>¼</sup> <sup>τ</sup>xy <sup>σ</sup>x�σ<sup>y</sup>

shear plane.

orientation θ.

Figure 5.

17

Mohr's circle diagram.

2

2.4.2 Mohr's circle for plane stress

2.4.2.1 Equations of Mohr's circle

Rearranging the terms of Eq. (5), we get

and

τx0

<sup>σ</sup><sup>x</sup><sup>0</sup> � <sup>σ</sup><sup>x</sup> <sup>þ</sup> <sup>σ</sup><sup>y</sup>

An Overview of Stress-Strain Analysis for Elasticity Equations

DOI: http://dx.doi.org/10.5772/intechopen.82066

<sup>2</sup> <sup>¼</sup> <sup>σ</sup><sup>x</sup> � <sup>σ</sup><sup>y</sup> 2

<sup>y</sup><sup>0</sup> ¼ � <sup>σ</sup><sup>x</sup> � <sup>σ</sup><sup>y</sup> 2

Similarly, write for the y' normal stress and shear stress. The transformed stresses are given as

$$\begin{aligned} \sigma\_{\mathbf{x'}} &= \frac{\sigma\_{\mathbf{x}} + \sigma\_{\mathbf{y}}}{2} + \frac{\sigma\_{\mathbf{x}} - \sigma\_{\mathbf{y}}}{2} \cos 2\theta + \tau\_{\mathbf{xy}} \sin 2\theta\\ \sigma\_{\mathbf{y'}} &= \frac{\sigma\_{\mathbf{x}} + \sigma\_{\mathbf{y}}}{2} - \frac{\sigma\_{\mathbf{x}} - \sigma\_{\mathbf{y}}}{2} \cos 2\theta - \tau\_{\mathbf{xy}} \sin 2\theta \end{aligned} \tag{5}$$

and

$$
\tau\_{\mathbf{x}'\mathbf{y}'} = \frac{\sigma\_{\mathbf{y}} - \sigma\_{\mathbf{x}}}{2} \sin 2\theta + \tau\_{\mathbf{x}\mathbf{y}} \cos 2\theta
$$

where σ<sup>x</sup><sup>0</sup> and τ<sup>x</sup><sup>0</sup> <sup>y</sup><sup>0</sup> are respectively the normal and shear stress acting on the inclined plane. The above three equations are known as transformation equations for plane stress.

In order to design components against failure the maximum and minimum normal and shear stresses acting on the inclined plane must be derived. The maximum normal stress and shear stress can be found when we differentiate the stress transformation equations with respect to θ and equate to zero. The maximum and minimum stresses are known as principal stresses and the plane of acting is named as principal planes.

Maximum normal stress is given by

$$
\sigma\_1, \sigma\_2 = \frac{\sigma\_\text{x} + \sigma\_\text{y}}{2} \pm \sqrt{\left(\frac{\sigma\_\text{x} - \sigma\_\text{y}}{2}\right)^2 + \sigma\_\text{xy}^2} \tag{6}
$$

and maximum shear stress is

$$
\tau\_{\text{max}} = \sqrt{\left(\frac{\sigma\_{\text{x}} - \sigma\_{\text{y}}}{2}\right)^2 + \tau\_{\text{xy}}^2} \tag{7}
$$

$$\text{with } \tau\_{\text{max}} = \frac{\sigma\_1 - \sigma\_2}{2}. \tag{8}$$

An Overview of Stress-Strain Analysis for Elasticity Equations DOI: http://dx.doi.org/10.5772/intechopen.82066

The plane on which the principal normal stress acts, the shear stress is zero and vice versa. The angle corresponding to the principal planes can be obtained from tan 2<sup>θ</sup> <sup>¼</sup> <sup>τ</sup>xy <sup>σ</sup>x�σ<sup>y</sup> 2 for the principal normal planes and tan 2<sup>θ</sup> <sup>¼</sup> <sup>τ</sup>xy <sup>σ</sup>x�σ<sup>y</sup> 2 is for the principal shear plane.

#### 2.4.2 Mohr's circle for plane stress

The transformation equations of plane stress which are given by Eq. (5) can be represented in a graphical form (Figure 5) by Mohr's circle. The transformation equations are sufficient to get the normal and shear stresses on any plane at a point, with Mohr's circle one can easily visualize their variation with respect to plane orientation θ.

#### 2.4.2.1 Equations of Mohr's circle

σ<sup>x</sup><sup>0</sup> ¼ l 2 x0 <sup>x</sup>σ<sup>x</sup> þ l 2 x0

Representation of stresses on inclined plane.

The transformed stresses are given as

and

τx0

Maximum normal stress is given by

and maximum shear stress is

where σ<sup>x</sup><sup>0</sup> and τ<sup>x</sup><sup>0</sup>

for plane stress.

Figure 4.

as principal planes.

16

<sup>¼</sup> 2 cos <sup>2</sup>

Elasticity of Materials ‐ Basic Principles and Design of Structures

<sup>σ</sup><sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>σ</sup><sup>x</sup> <sup>þ</sup> <sup>σ</sup><sup>y</sup>

<sup>σ</sup><sup>y</sup><sup>0</sup> <sup>¼</sup> <sup>σ</sup><sup>x</sup> <sup>þ</sup> <sup>σ</sup><sup>y</sup>

<sup>y</sup><sup>0</sup> <sup>¼</sup> <sup>σ</sup><sup>y</sup> � <sup>σ</sup><sup>x</sup> 2

<sup>σ</sup>1, <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>σ</sup><sup>x</sup> <sup>þ</sup> <sup>σ</sup><sup>y</sup>

τmax ¼

2

r

�

with <sup>τ</sup>max <sup>¼</sup> <sup>σ</sup><sup>1</sup> � <sup>σ</sup><sup>2</sup>

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ<sup>x</sup> � σ<sup>y</sup> 2 � �<sup>2</sup>

Similarly, write for the y' normal stress and shear stress.

<sup>y</sup>σ<sup>y</sup> þ 2lx<sup>0</sup>

θσ<sup>x</sup> <sup>þ</sup> 2sin<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>σ</sup><sup>x</sup> � <sup>σ</sup><sup>y</sup> 2

<sup>2</sup> � <sup>σ</sup><sup>x</sup> � <sup>σ</sup><sup>y</sup>

xlx<sup>0</sup> <sup>y</sup>τxy

θσ<sup>y</sup> þ 2 cos θ sin θτxy

cos 2θ þ τxy sin 2θ

<sup>2</sup> cos 2<sup>θ</sup> � <sup>τ</sup>xy sin 2<sup>θ</sup>

<sup>y</sup><sup>0</sup> are respectively the normal and shear stress acting on the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ<sup>x</sup> � σ<sup>y</sup> 2 � �<sup>2</sup>

> þ τ<sup>2</sup> xy

þ τ<sup>2</sup> xy

<sup>2</sup> : (8)

sin 2θ þ τxy cos 2θ

inclined plane. The above three equations are known as transformation equations

In order to design components against failure the maximum and minimum normal and shear stresses acting on the inclined plane must be derived. The maximum normal stress and shear stress can be found when we differentiate the stress transformation equations with respect to θ and equate to zero. The maximum and minimum stresses are known as principal stresses and the plane of acting is named

(4)

(5)

(6)

(7)

Rearranging the terms of Eq. (5), we get

$$
\sigma\_{\mathbf{x'}} - \frac{\sigma\_{\mathbf{x}} + \sigma\_{\mathbf{y}}}{2} = \frac{\sigma\_{\mathbf{x}} - \sigma\_{\mathbf{y}}}{2} \cos 2\theta + \tau\_{\mathbf{xy}} \sin 2\theta
$$

(9.1)

and

$$\tau\_{\mathbf{x'}\mathbf{y'}} = -\left(\frac{\sigma\_{\mathbf{x}} - \sigma\_{\mathbf{y}}}{2}\right)\sin 2\theta + \tau\_{\mathbf{xy}}\cos 2\theta \tag{9.2}$$

Figure 5. Mohr's circle diagram.

Squaring and adding the Eqs. (9.1) and (9.2), result in

$$\left(\sigma\_{\mathbf{x'}} - \frac{\sigma\_{\mathbf{x}} + \sigma\_{\mathbf{y}}}{2}\right)^2 + \tau\_{\mathbf{x'}\mathbf{y'}}^2 = \left(\frac{\sigma\_{\mathbf{x}} - \sigma\_{\mathbf{y}}}{2}\right)^2 + \tau\_{\mathbf{xy}}^2\tag{10}$$

y, z direction respectively. Then the stress equilibrium relation or equation of

þ Fx ¼ 0,

9 >>>>>>>=

>>>>>>>;

(15)

þ Fy ¼ 0,

þ Fz ¼ 0:

While defining a stress it was pointed out that stress is an abstract quantity which cannot be seen and is generally measured indirectly. Strain differs in this respect from stress. It is a complete quantity that can be seen and generally measured directly as a relative change of length or shape. In generally, stress is the ratio of change in original dimension and the original dimension. It is the dimensionless constant quantity.

Strain may be classified into three types; normal strain, shear strain and volu-

The normal strain is the relative change in length whether shearing strain relative change in shape. The volumetric strain is defined by the relative change in volume.

Consider a line element of length Δx emanating from position (x, y) and lying in the x-direction, denoted by AB in Figure 6. After deformation the line element

AB <sup>¼</sup> uxð Þ� <sup>x</sup> <sup>þ</sup> <sup>Δ</sup>x; <sup>y</sup> uxð Þ <sup>x</sup>; <sup>y</sup>

<sup>Δ</sup><sup>x</sup> : (16)

<sup>∂</sup><sup>y</sup> : (18)

(17)

, having undergone a translation, extension and rotation. The particle that was originally at x has undergone a displacement uxð Þ x; y and the other end of the line element has undergone a displacement uxð Þ x þ Δx; y : By

> <sup>ε</sup>xx <sup>¼</sup> <sup>∂</sup>ux ∂x

This partial derivative is a displacement gradient, a measure of how rapid the displacement changes through the material, and is the strain at (x, y). Physically, it

Similarly, by considering a line element initially lying in the y-direction, the

<sup>ε</sup>yy <sup>¼</sup> <sup>∂</sup>uy

motion in terms of stress components are given by

An Overview of Stress-Strain Analysis for Elasticity Equations

DOI: http://dx.doi.org/10.5772/intechopen.82066

3. Analysis of strain

3.1 Types of strain

3.2.1 Normal strain

B0

the definition of normal strain

<sup>ε</sup>xx <sup>¼</sup> <sup>A</sup><sup>0</sup>

In the limit Δx ! 0, Eq. (16) becomes

strain in the y-direction can be expressed as

<sup>B</sup><sup>∗</sup> � AB

represents the (approximate) unit change in length of a line element.

occupies A<sup>0</sup>

19

3.2 Strain-displacement relationship

metric strain.

∂σ<sup>x</sup> ∂x þ ∂τyx ∂y þ ∂τzx ∂z

∂τxy ∂x þ ∂σ<sup>y</sup> ∂y þ ∂τzy ∂z

∂τxz ∂x þ ∂τyz ∂y þ ∂σ<sup>z</sup> ∂z

For simple representation of Eq. (10), the following notations are used

$$
\sigma\_{\rm av} = \frac{\sigma\_{\rm x} + \sigma\_{\rm y}}{2}, \; r = \left[ \left( \frac{\sigma\_{\rm x} - \sigma\_{\rm y}}{2} \right)^2 + \sigma\_{\rm xy}^2 \right]^{1/2} \tag{11}
$$

Thus, the simplified form of Eq. (10) can be written as

$$(\sigma\_{\mathbf{x'}} - \sigma\_{\mathbf{av}})^2 + \tau\_{\mathbf{x'y'}}^2 = r^2 \tag{12}$$

Eq. (12) represents the equation of a circle in a standard form. This circle has σ<sup>x</sup><sup>0</sup> as its abscissa and τ<sup>x</sup><sup>0</sup> <sup>y</sup><sup>0</sup> as its ordinate with radius r. The coordinate for the center of the circle is ð Þ σav; 0 .

Mohr's circle is drawn by considering the stress coordinates σ<sup>x</sup> as its abscissa and τxy as its ordinate, and this plane is known as the stress plane. The plane on the element bounded with xy coordinates in the material is named as physical plane. Stresses on the physical plane M is represented by the point M on the stress plane with σ<sup>x</sup> and τxy coordinates.

Stresses on the physical plane which is normal to i.e. N, is given by the point N on the stress plane with σ<sup>y</sup> and τyx: O is the intersecting point of line MN and which is at the center of the circle and radius of the circle is OM. Now, the stresses on a plane, making θ inclination with x axis in physical plane can be determined as follows.

An important point to be noted here is that a plane which has a θ inclination in physical plane will make 2θ inclination in stress plane M. Hence, rotate the line OM in stress plane by 2θ counter clockwise to obtain the plane M<sup>0</sup> . The coordinates of M<sup>0</sup> in stress plane define the stresses acting on plane M<sup>0</sup> in physical plane and it can be easily verified.

$$
\sigma\_{\mathbf{x'}} = \mathbf{PO} + r \cos \left(2\theta\_p - 2\theta\right) \tag{13}
$$

where PO <sup>¼</sup> <sup>σ</sup>xþσ<sup>y</sup> <sup>2</sup> , r <sup>¼</sup> <sup>σ</sup>x�σ<sup>y</sup> 2 � �<sup>2</sup> <sup>þ</sup> <sup>τ</sup><sup>2</sup> xy h i<sup>1</sup>=<sup>2</sup> , cos 2θ<sup>p</sup> <sup>¼</sup> <sup>σ</sup>x�σ<sup>y</sup> <sup>2</sup><sup>r</sup> , sin 2θ<sup>p</sup> <sup>¼</sup> <sup>τ</sup>xy 2r. On simplifying Eq. (13)

$$
\sigma\_{\mathbf{x'}} = \frac{\sigma\_{\mathbf{x}} + \sigma\_{\mathbf{y}}}{2} + \frac{\sigma\_{\mathbf{x}} - \sigma\_{\mathbf{y}}}{2} \cos 2\theta + \tau\_{\mathbf{xy}} \sin 2\theta \tag{14}
$$

Eq. (14) is same as the first equation of Eq. (5). This way it can be proved for shear stress τ<sup>x</sup><sup>0</sup> <sup>y</sup><sup>0</sup> on plane M<sup>0</sup> (do yourself).

#### 2.4.3 Stress equilibrium relation

Let σx, τyx, τzx are the stress components acting along the x-direction, τxy, σy, τzy are the stress components acting along the y-direction and τxz, τyz, σ<sup>z</sup> are the stress components acting along the z-direction. The body forces Fx, Fy, Fz acting along x, y, z direction respectively. Then the stress equilibrium relation or equation of motion in terms of stress components are given by

$$\begin{cases} \frac{\partial \sigma\_{\text{x}}}{\partial \mathbf{x}} + \frac{\partial \tau\_{\text{yx}}}{\partial \mathbf{y}} + \frac{\partial \tau\_{\text{xx}}}{\partial \mathbf{z}} + F\_{\text{x}} = \mathbf{0}, \\\\ \frac{\partial \sigma\_{\text{xy}}}{\partial \mathbf{x}} + \frac{\partial \sigma\_{\text{y}}}{\partial \mathbf{y}} + \frac{\partial \tau\_{\text{xy}}}{\partial \mathbf{z}} + F\_{\text{y}} = \mathbf{0}, \\\\ \frac{\partial \tau\_{\text{xx}}}{\partial \mathbf{x}} + \frac{\partial \tau\_{\text{yx}}}{\partial \mathbf{y}} + \frac{\partial \sigma\_{\text{z}}}{\partial \mathbf{z}} + F\_{\text{z}} = \mathbf{0}. \end{cases} \tag{15}$$

#### 3. Analysis of strain

Squaring and adding the Eqs. (9.1) and (9.2), result in

<sup>þ</sup> <sup>τ</sup><sup>2</sup> x0

For simple representation of Eq. (10), the following notations are used

<sup>σ</sup><sup>x</sup> ð Þ <sup>0</sup> � <sup>σ</sup>av <sup>2</sup> <sup>þ</sup> <sup>τ</sup><sup>2</sup>

τxy as its ordinate, and this plane is known as the stress plane. The plane on the element bounded with xy coordinates in the material is named as physical plane. Stresses on the physical plane M is represented by the point M on the stress plane

<sup>2</sup> , r <sup>¼</sup> <sup>σ</sup><sup>x</sup> � <sup>σ</sup><sup>y</sup>

<sup>y</sup><sup>0</sup> <sup>¼</sup> <sup>σ</sup><sup>x</sup> � <sup>σ</sup><sup>y</sup> 2 � �<sup>2</sup>

2 � �<sup>2</sup>

> x0 <sup>y</sup><sup>0</sup> ¼ r

Eq. (12) represents the equation of a circle in a standard form. This circle has σ<sup>x</sup><sup>0</sup>

Mohr's circle is drawn by considering the stress coordinates σ<sup>x</sup> as its abscissa and

Stresses on the physical plane which is normal to i.e. N, is given by the point N on the stress plane with σ<sup>y</sup> and τyx: O is the intersecting point of line MN and which is at the center of the circle and radius of the circle is OM. Now, the stresses on a plane, making θ inclination with x axis in physical plane can be determined as

An important point to be noted here is that a plane which has a θ inclination in physical plane will make 2θ inclination in stress plane M. Hence, rotate the line OM

in stress plane define the stresses acting on plane M<sup>0</sup> in physical plane and it can be

xy

Let σx, τyx, τzx are the stress components acting along the x-direction, τxy, σy, τzy are the stress components acting along the y-direction and τxz, τyz, σ<sup>z</sup> are the stress components acting along the z-direction. The body forces Fx, Fy, Fz acting along x,

h i<sup>1</sup>=<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>σ</sup><sup>x</sup> � <sup>σ</sup><sup>y</sup> 2

<sup>þ</sup> <sup>τ</sup><sup>2</sup>

<sup>þ</sup> <sup>τ</sup><sup>2</sup> xy

� �1=<sup>2</sup>

<sup>y</sup><sup>0</sup> as its ordinate with radius r. The coordinate for the center of

<sup>σ</sup><sup>x</sup><sup>0</sup> <sup>¼</sup> PO <sup>þ</sup> <sup>r</sup> cos 2θ<sup>p</sup> � <sup>2</sup><sup>θ</sup> � � (13)

, cos 2θ<sup>p</sup> <sup>¼</sup> <sup>σ</sup>x�σ<sup>y</sup>

xy (10)

<sup>2</sup> (12)

. The coordinates of M<sup>0</sup>

2r.

<sup>2</sup><sup>r</sup> , sin 2θ<sup>p</sup> <sup>¼</sup> <sup>τ</sup>xy

cos 2θ þ τxy sin 2θ (14)

<sup>y</sup><sup>0</sup> on plane M<sup>0</sup> (do yourself).

(11)

<sup>σ</sup>x<sup>0</sup> � <sup>σ</sup><sup>x</sup> <sup>þ</sup> <sup>σ</sup><sup>y</sup> 2 � �<sup>2</sup>

Elasticity of Materials ‐ Basic Principles and Design of Structures

<sup>σ</sup>av <sup>¼</sup> <sup>σ</sup><sup>x</sup> <sup>þ</sup> <sup>σ</sup><sup>y</sup>

as its abscissa and τ<sup>x</sup><sup>0</sup>

the circle is ð Þ σav; 0 .

follows.

18

easily verified.

where PO <sup>¼</sup> <sup>σ</sup>xþσ<sup>y</sup>

On simplifying Eq. (13)

2.4.3 Stress equilibrium relation

with σ<sup>x</sup> and τxy coordinates.

Thus, the simplified form of Eq. (10) can be written as

in stress plane by 2θ counter clockwise to obtain the plane M<sup>0</sup>

2 � �<sup>2</sup> <sup>þ</sup> <sup>τ</sup><sup>2</sup>

<sup>2</sup> , r <sup>¼</sup> <sup>σ</sup>x�σ<sup>y</sup>

<sup>σ</sup><sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>σ</sup><sup>x</sup> <sup>þ</sup> <sup>σ</sup><sup>y</sup>

Eq. (14) is same as the first equation of Eq. (5). This way it can be proved for shear stress τ<sup>x</sup><sup>0</sup>

While defining a stress it was pointed out that stress is an abstract quantity which cannot be seen and is generally measured indirectly. Strain differs in this respect from stress. It is a complete quantity that can be seen and generally measured directly as a relative change of length or shape. In generally, stress is the ratio of change in original dimension and the original dimension. It is the dimensionless constant quantity.

#### 3.1 Types of strain

Strain may be classified into three types; normal strain, shear strain and volumetric strain.

The normal strain is the relative change in length whether shearing strain relative change in shape. The volumetric strain is defined by the relative change in volume.

#### 3.2 Strain-displacement relationship

#### 3.2.1 Normal strain

Consider a line element of length Δx emanating from position (x, y) and lying in the x-direction, denoted by AB in Figure 6. After deformation the line element occupies A<sup>0</sup> B0 , having undergone a translation, extension and rotation.

The particle that was originally at x has undergone a displacement uxð Þ x; y and the other end of the line element has undergone a displacement uxð Þ x þ Δx; y : By the definition of normal strain

$$\varepsilon\_{\text{xx}} = \frac{A'B^\* - AB}{AB} = \frac{\mu\_{\text{x}}(\infty + \Delta\mathfrak{x}, y) - \mu\_{\text{x}}(\mathfrak{x}, y)}{\Delta\mathfrak{x}}.\tag{16}$$

In the limit Δx ! 0, Eq. (16) becomes

$$
\varepsilon\_{\infty} = \frac{\partial u\_{\infty}}{\partial \mathfrak{x}}\tag{17}
$$

This partial derivative is a displacement gradient, a measure of how rapid the displacement changes through the material, and is the strain at (x, y). Physically, it represents the (approximate) unit change in length of a line element.

Similarly, by considering a line element initially lying in the y-direction, the strain in the y-direction can be expressed as

$$
\varepsilon\_{\mathcal{Y}} = \frac{\partial u\_{\mathcal{Y}}}{\partial \mathbf{y}}.\tag{18}
$$

#### 3.2.2 Shear strain

The particles A and B in Figure 6 also undergo displacements in the y-direction and this is shown in Figure 7(a). In this case, we have

$$B^\*B' = \frac{\partial u\_\mathcal{y}}{\partial \mathfrak{x}} \Delta \mathfrak{x}.\tag{19}$$

and hence the shear strain can be written in terms of displacement gradients as

∂ux ∂y þ ∂uy ∂x

In similar manner, the strain-displacement relation for three dimensional body

: (22)

∂uy ∂z þ ∂uz ∂y : (23)

2

<sup>ε</sup>xy <sup>¼</sup> <sup>1</sup> 2

∂z ,

∂ux ∂z þ ∂uz ∂x , <sup>ε</sup>yz <sup>¼</sup> <sup>1</sup>

As seen in the previous section, there are three strain-displacement relations Eqs. (17), (18) and (22) but only two displacement components. This implies that the strains are not independent but are related in some way. The relations between

Let us suppose that the point P which is act (x,y) before straining and it will be at P<sup>0</sup> after straining on the co-ordinate plane Oxy as depicted in Figure 8. Then (u,v) is a displacement corresponding to the point P. The variable u and v are the functions

<sup>∂</sup><sup>y</sup> , <sup>ε</sup>xy <sup>¼</sup> <sup>1</sup>

2

∂ux ∂y þ ∂uy ∂x

(24)

2

<sup>∂</sup><sup>y</sup> , <sup>ε</sup>zz <sup>¼</sup> <sup>∂</sup>uz

An Overview of Stress-Strain Analysis for Elasticity Equations

is given by

<sup>ε</sup>xx <sup>¼</sup> <sup>∂</sup>ux

<sup>ε</sup>xy <sup>¼</sup> <sup>1</sup> 2

3.3 Compatibility of strain

3.3.1 Compatibility relations

of x and y.

Figure 8.

21

Deformation of line element.

<sup>∂</sup><sup>x</sup> , <sup>ε</sup>yy <sup>¼</sup> <sup>∂</sup>uy

DOI: http://dx.doi.org/10.5772/intechopen.82066

the strains are called compatibility conditions.

Using the fundamental notation

<sup>ε</sup>xx <sup>¼</sup> <sup>∂</sup>ux

<sup>∂</sup><sup>x</sup> , <sup>ε</sup>yy <sup>¼</sup> <sup>∂</sup>uy

∂ux ∂y þ ∂uy ∂x , <sup>ε</sup>xz <sup>¼</sup> <sup>1</sup>

A similar relation can be derived by considering a line element initially lying in the y-direction. From the Figure 7(b), we have

$$
\theta \approx \tan \theta = \frac{\partial u\_\circ / \partial \mathbf{x}}{\mathbf{1} + \partial u\_\mathbf{x} / \partial \mathbf{x}} \approx \frac{\partial u\_\mathbf{y}}{\partial \mathbf{x}} \tag{20}
$$

provided that (i) θ is small and (ii) the displacement gradient ∂ux=∂x is small. A similar expression for the angle λ can be derived as

$$
\lambda \approx \frac{\partial u\_{\infty}}{\partial \mathbf{y}} \tag{21}
$$

Figure 6. Deformation of a line element.

Figure 7. (a) Deformation of a line element and (b) strains in terms of displacement gradients.

and hence the shear strain can be written in terms of displacement gradients as

$$
\varepsilon\_{xy} = \frac{1}{2} \left( \frac{\partial u\_x}{\partial y} + \frac{\partial u\_y}{\partial \mathbf{x}} \right). \tag{22}
$$

In similar manner, the strain-displacement relation for three dimensional body is given by

$$\begin{split} \varepsilon\_{\text{xx}} &= \frac{\partial u\_{\text{x}}}{\partial \mathbf{x}}, \varepsilon\_{\text{yy}} = \frac{\partial u\_{\text{y}}}{\partial \mathbf{y}}, \varepsilon\_{\text{xx}} = \frac{\partial u\_{\text{x}}}{\partial \mathbf{z}}, \\ \varepsilon\_{\text{xy}} &= \frac{1}{2} \left( \frac{\partial u\_{\text{x}}}{\partial \mathbf{y}} + \frac{\partial u\_{\text{y}}}{\partial \mathbf{x}} \right), \varepsilon\_{\text{xx}} = \frac{1}{2} \left( \frac{\partial u\_{\text{x}}}{\partial \mathbf{z}} + \frac{\partial u\_{\text{z}}}{\partial \mathbf{x}} \right), \varepsilon\_{\text{yx}} = \frac{1}{2} \left( \frac{\partial u\_{\text{y}}}{\partial \mathbf{z}} + \frac{\partial u\_{\text{z}}}{\partial \mathbf{y}} \right). \end{split} \tag{23}$$

#### 3.3 Compatibility of strain

3.2.2 Shear strain

Figure 6.

Figure 7.

20

Deformation of a line element.

The particles A and B in Figure 6 also undergo displacements in the y-direction

A similar relation can be derived by considering a line element initially lying in

provided that (i) θ is small and (ii) the displacement gradient ∂ux=∂x is small. A

<sup>λ</sup> <sup>≈</sup>∂ux ∂y

<sup>1</sup> <sup>þ</sup> <sup>∂</sup>ux=∂<sup>x</sup> <sup>≈</sup>∂uy

∂x

Δx: (19)

(20)

(21)

<sup>B</sup><sup>∗</sup>B<sup>0</sup> <sup>¼</sup> <sup>∂</sup>uy ∂x

<sup>θ</sup> <sup>≈</sup>tan <sup>θ</sup> <sup>¼</sup> <sup>∂</sup>uy=∂<sup>x</sup>

and this is shown in Figure 7(a). In this case, we have

Elasticity of Materials ‐ Basic Principles and Design of Structures

the y-direction. From the Figure 7(b), we have

similar expression for the angle λ can be derived as

(a) Deformation of a line element and (b) strains in terms of displacement gradients.

As seen in the previous section, there are three strain-displacement relations Eqs. (17), (18) and (22) but only two displacement components. This implies that the strains are not independent but are related in some way. The relations between the strains are called compatibility conditions.

#### 3.3.1 Compatibility relations

Let us suppose that the point P which is act (x,y) before straining and it will be at P<sup>0</sup> after straining on the co-ordinate plane Oxy as depicted in Figure 8. Then (u,v) is a displacement corresponding to the point P. The variable u and v are the functions of x and y.

Using the fundamental notation

$$
\varepsilon\_{\infty} = \frac{\partial u\_{\varepsilon}}{\partial x}, \varepsilon\_{\mathcal{V}} = \frac{\partial u\_{\varepsilon}}{\partial \mathcal{V}}, \varepsilon\_{\mathcal{X}} = \frac{1}{2} \left( \frac{\partial u\_{\varepsilon}}{\partial \mathcal{Y}} + \frac{\partial u\_{\varepsilon}}{\partial \mathcal{X}} \right) \tag{24}
$$

\$\mathcal{Y}\$}

\$\mathcal{Q}(\mathbf{x} + \mathbf{x}', \mathbf{y} + \mathbf{y}')\$

$$
\mathcal{Q}'\$

$$
\mathcal{Q} \xrightarrow{P(\mathbf{x}, \mathbf{y})} \mathcal{Q}'(\mathbf{x} + \mathbf{u}, \mathbf{y} + \mathbf{v}) \tag{35}
$$
$$

Figure 8. Deformation of line element.

we get

$$\frac{\partial^2 \varepsilon\_{\text{xx}}}{\partial \mathbf{y}^2} = \frac{\partial^3 u\_{\text{x}}}{\partial \mathbf{x} \partial \mathbf{y}^2}, \frac{\partial^2 \varepsilon\_{\text{yy}}}{\partial \mathbf{x}^2} = \frac{\partial^3 u\_{\text{y}}}{\partial \mathbf{x}^2 \partial \mathbf{y}} \tag{25}$$

In general, each strain is dependent on each stress. For example, the strain εxx

εxx ¼ C11σ<sup>x</sup> þ C12σ<sup>y</sup> þ C13σ<sup>z</sup> þ C14τxy þ C15τyz þ C16τzx þ C17τxz þ C18τzy þ C19τyx:

Similarly, stresses can be expressed in terms of strains which state that at each point in a material, each stress component is linearly related to all the strain com-

For the most general case of three-dimensional state of stress, Eq. (28) can be

where Dijkl � � is elasticity matrix, <sup>σ</sup>ij � � is stress components, ð Þ <sup>ε</sup>kl is strain com-

Since both stress σij and strain εij are second-order tensors, it follows that Dijkl is a fourth order tensor, which consists of 3<sup>4</sup> <sup>¼</sup> 81 material constants if symmetry is

Now, from σij ¼ σji and εij ¼ εji, the number of 81 material constants is reduced

The stress-strain relation for triclinic material will consist 21 elastic constants

D<sup>11</sup> D<sup>12</sup> D<sup>13</sup> D<sup>14</sup> D<sup>15</sup> D<sup>16</sup> D<sup>12</sup> D<sup>22</sup> D<sup>23</sup> D<sup>24</sup> D<sup>25</sup> D<sup>26</sup> D<sup>13</sup> D<sup>23</sup> D<sup>33</sup> D<sup>34</sup> D<sup>35</sup> D<sup>36</sup> D<sup>14</sup> D<sup>24</sup> D<sup>34</sup> D<sup>44</sup> D<sup>45</sup> D<sup>46</sup> D<sup>15</sup> D<sup>25</sup> D<sup>35</sup> D<sup>45</sup> D<sup>55</sup> D<sup>56</sup> D<sup>16</sup> D<sup>26</sup> D<sup>36</sup> D<sup>46</sup> D<sup>56</sup> D<sup>66</sup>

The stress-strain relation for monoclinic material will consist 13 elastic constants

D<sup>11</sup> D<sup>12</sup> D<sup>13</sup> 0 D<sup>15</sup> 0 D<sup>12</sup> D<sup>22</sup> D<sup>23</sup> 0 D<sup>25</sup> 0 D<sup>13</sup> D<sup>23</sup> D<sup>33</sup> 0 D<sup>35</sup> 0 000 D<sup>44</sup> 0 D<sup>46</sup> D<sup>15</sup> D<sup>25</sup> D<sup>35</sup> 0 D<sup>55</sup> 0 000 D<sup>46</sup> 0 D<sup>66</sup>

to 36 under symmetric conditions of Dijkl ¼ Djikl ¼ Dijlk ¼ Djilk which provides

stress-strain relation for most general form of anisotropic material.

<sup>9</sup>�<sup>1</sup> <sup>¼</sup> Dijkl � �

(29)

<sup>9</sup>�9ð Þ <sup>ε</sup>kl <sup>9</sup>�<sup>1</sup> (30)

εxx εyy εzz εxy εyz εzx

εxx εyy εzz εxy εyz εzx

: (31)

: (32)

written as a function of each stress as

DOI: http://dx.doi.org/10.5772/intechopen.82066

written as

ponents.

not assumed.

which is given by

which is given by

23

ponents. This is known as generalized Hook's law.

An Overview of Stress-Strain Analysis for Elasticity Equations

4.1.1 Stress-strain relation for triclinic material

σx σy σz τxy τyz τzx

¼

4.1.2 Stress-strain relation for monoclinic material

σx σy σz τxy τyz τzx

¼

σij � �

$$\frac{\partial^2 \varepsilon\_{\text{xy}}}{\partial \mathbf{x} \partial \mathbf{y}} = \frac{1}{2} \left( \frac{\partial^3 u\_{\text{y}}}{\partial \mathbf{x}^2 \partial \mathbf{y}} + \frac{\partial^3 u\_{\text{x}}}{\partial \mathbf{x} \partial \mathbf{y}^2} \right). \tag{26}$$

Eqs. (25) and (26) result in

$$\frac{\partial^2 \varepsilon\_{\text{xy}}}{\partial \mathbf{x} \partial \mathbf{y}} = \frac{1}{2} \left( \frac{\partial^2 \varepsilon\_{\text{xx}}}{\partial \mathbf{y}^2} + \frac{\partial^2 \varepsilon\_{\text{yy}}}{\partial \mathbf{x}^2} \right) \tag{27}$$

which is the compatibility condition in two dimension.

#### 4. Stress-strain relation

In the previous section, the state of stress at a point was characterized by six components of stress, and the internal stresses and the applied forces are accompanied with the three equilibrium equation. These equations are applicable to all types of materials as the relationships are independent of the deformations (strains) and the material behavior.

Also, the state of strain at a point was defined in terms of six components of strain. The strains and the displacements are related uniquely by the derivation of six strain-displacement relations and compatibility equations. These equations are also applicable to all materials as they are independent of the stresses and the material behavior and hence.

Irrespective of the independent nature of the equilibrium equations and straindisplacement relations, usually, it is essential to study the general behavior of materials under applied loads including these relations. Strains will be developed in a body due to the application of a load, stresses and deformations and hence it is become necessary to study the behavior of different types of materials. In a general three-dimensional system, there will be 15 unknowns namely 3 displacements, 6 strains and 6 stresses. But we have only 9 equations such as 3 equilibrium equations and 6 strain-displacement equations to achieve these 15 unknowns. It is important to note that the compatibility conditions are not useful for the determination of either the displacements or strains. Hence the additional six equations relating six stresses and six strains will be developed. These equations are known as "Constitutive equations" because they describe the macroscopic behavior of a material based on its internal constitution.

#### 4.1 Linear elasticity generalized Hooke's law

Hooke's law provides the unique relationship between stress and strain, which is independent of time and loading history. The law can be used to predict the deformations used in a given material by a combination of stresses.

The linear relationship between stress and strain is given by

$$
\sigma\_{\mathfrak{x}} = E \epsilon\_{\mathfrak{x}} \tag{28}
$$

where E is known as Young's modulus.

we get

∂2 εxx <sup>∂</sup>y<sup>2</sup> <sup>¼</sup> <sup>∂</sup><sup>3</sup>

Elasticity of Materials ‐ Basic Principles and Design of Structures

∂2 εxy <sup>∂</sup>x∂<sup>y</sup> <sup>¼</sup> <sup>1</sup> 2

∂2 εxy <sup>∂</sup>x∂<sup>y</sup> <sup>¼</sup> <sup>1</sup> 2

which is the compatibility condition in two dimension.

Eqs. (25) and (26) result in

4. Stress-strain relation

the material behavior.

material behavior and hence.

on its internal constitution.

22

4.1 Linear elasticity generalized Hooke's law

where E is known as Young's modulus.

mations used in a given material by a combination of stresses. The linear relationship between stress and strain is given by

ux <sup>∂</sup>x∂y<sup>2</sup> ,

> ∂3 uy ∂x<sup>2</sup>∂y þ ∂3 ux ∂x∂y<sup>2</sup>

∂2 εxx ∂y<sup>2</sup> þ

In the previous section, the state of stress at a point was characterized by six components of stress, and the internal stresses and the applied forces are accompanied with the three equilibrium equation. These equations are applicable to all types of materials as the relationships are independent of the deformations (strains) and

Also, the state of strain at a point was defined in terms of six components of strain. The strains and the displacements are related uniquely by the derivation of six strain-displacement relations and compatibility equations. These equations are also applicable to all materials as they are independent of the stresses and the

Irrespective of the independent nature of the equilibrium equations and strain-

Hooke's law provides the unique relationship between stress and strain, which is independent of time and loading history. The law can be used to predict the defor-

σ<sup>x</sup> ¼ Eεxx (28)

displacement relations, usually, it is essential to study the general behavior of materials under applied loads including these relations. Strains will be developed in a body due to the application of a load, stresses and deformations and hence it is become necessary to study the behavior of different types of materials. In a general three-dimensional system, there will be 15 unknowns namely 3 displacements, 6 strains and 6 stresses. But we have only 9 equations such as 3 equilibrium equations and 6 strain-displacement equations to achieve these 15 unknowns. It is important to note that the compatibility conditions are not useful for the determination of either the displacements or strains. Hence the additional six equations relating six stresses and six strains will be developed. These equations are known as "Constitutive equations" because they describe the macroscopic behavior of a material based

∂2 εyy <sup>∂</sup>x<sup>2</sup> <sup>¼</sup> <sup>∂</sup><sup>3</sup>

∂2 εyy ∂x<sup>2</sup>

uy ∂x<sup>2</sup>∂y

(25)

(27)

: (26)

In general, each strain is dependent on each stress. For example, the strain εxx written as a function of each stress as

$$\boldsymbol{\sigma}\_{\rm xx} = \mathbf{C}\_{11}\boldsymbol{\sigma}\_{\rm x} + \mathbf{C}\_{12}\boldsymbol{\sigma}\_{\rm y} + \mathbf{C}\_{13}\boldsymbol{\sigma}\_{\rm x} + \mathbf{C}\_{14}\boldsymbol{\tau}\_{\rm xy} + \mathbf{C}\_{15}\boldsymbol{\tau}\_{\rm yx} + \mathbf{C}\_{16}\boldsymbol{\tau}\_{\rm xx} + \mathbf{C}\_{17}\boldsymbol{\tau}\_{\rm xx} + \mathbf{C}\_{18}\boldsymbol{\tau}\_{\rm xy} + \mathbf{C}\_{19}\boldsymbol{\tau}\_{\rm yx} \tag{29}$$

Similarly, stresses can be expressed in terms of strains which state that at each point in a material, each stress component is linearly related to all the strain components. This is known as generalized Hook's law.

For the most general case of three-dimensional state of stress, Eq. (28) can be written as

$$(\sigma\_{\vec{\eta}})\_{g\times \mathbf{1}} = \left(D\_{\vec{\eta}kl}\right)\_{g\times g} (\varepsilon\_{kl})\_{g\times \mathbf{1}} \tag{30}$$

where Dijkl � � is elasticity matrix, <sup>σ</sup>ij � � is stress components, ð Þ <sup>ε</sup>kl is strain components.

Since both stress σij and strain εij are second-order tensors, it follows that Dijkl is a fourth order tensor, which consists of 3<sup>4</sup> <sup>¼</sup> 81 material constants if symmetry is not assumed.

Now, from σij ¼ σji and εij ¼ εji, the number of 81 material constants is reduced to 36 under symmetric conditions of Dijkl ¼ Djikl ¼ Dijlk ¼ Djilk which provides stress-strain relation for most general form of anisotropic material.

#### 4.1.1 Stress-strain relation for triclinic material

The stress-strain relation for triclinic material will consist 21 elastic constants which is given by

$$
\begin{bmatrix}
\sigma\_{\mathbf{x}} \\
\sigma\_{\mathbf{y}} \\
\sigma\_{z} \\
\tau\_{\mathbf{xy}} \\
\tau\_{\mathbf{xy}} \\
\tau\_{\mathbf{yz}} \\
\tau\_{\mathbf{xz}}
\end{bmatrix} = \begin{bmatrix}
D\_{11} & D\_{12} & D\_{13} & D\_{14} & D\_{15} & D\_{16} \\
D\_{12} & D\_{22} & D\_{23} & D\_{24} & D\_{25} & D\_{26} \\
D\_{13} & D\_{23} & D\_{33} & D\_{34} & D\_{35} & D\_{36} \\
D\_{14} & D\_{24} & D\_{34} & D\_{44} & D\_{45} & D\_{46} \\
D\_{15} & D\_{25} & D\_{35} & D\_{45} & D\_{55} & D\_{56} \\
D\_{16} & D\_{26} & D\_{36} & D\_{46} & D\_{56} & D\_{66} \\
\end{bmatrix} \begin{bmatrix}
\varepsilon\_{\mathbf{xx}} \\
\varepsilon\_{\mathbf{yy}} \\
\varepsilon\_{\mathbf{xz}} \\
\varepsilon\_{\mathbf{xy}} \\
\varepsilon\_{\mathbf{yz}} \\
\varepsilon\_{\mathbf{xz}} \\
\end{bmatrix}.
\tag{31}
$$

#### 4.1.2 Stress-strain relation for monoclinic material

The stress-strain relation for monoclinic material will consist 13 elastic constants which is given by

$$
\begin{bmatrix}
\sigma\_{\mathbf{x}} \\
\sigma\_{\mathbf{y}} \\
\sigma\_{\mathbf{z}} \\
\tau\_{\mathbf{xy}} \\
\tau\_{\mathbf{yz}} \\
\tau\_{\mathbf{z}x} \\
\tau\_{\mathbf{z}x}
\end{bmatrix} = \begin{bmatrix}
D\_{11} & D\_{12} & D\_{13} & \mathbf{0} & D\_{15} & \mathbf{0} \\
D\_{12} & D\_{22} & D\_{23} & \mathbf{0} & D\_{25} & \mathbf{0} \\
D\_{13} & D\_{23} & D\_{33} & \mathbf{0} & D\_{35} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & D\_{44} & \mathbf{0} & D\_{46} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & D\_{45} & \mathbf{0} & D\_{55} \\
D\_{15} & D\_{25} & D\_{35} & \mathbf{0} & D\_{55} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & D\_{46} & \mathbf{0} & D\_{66}
\end{bmatrix} \begin{bmatrix}
\varepsilon\_{\mathbf{x}} \\
\varepsilon\_{\mathbf{y}} \\
\varepsilon\_{\mathbf{z}} \\
\varepsilon\_{\mathbf{xy}} \\
\varepsilon\_{\mathbf{yz}} \\
\varepsilon\_{\mathbf{z}}
\end{bmatrix}. \tag{32}$$

#### 4.1.3 Stress-strain relation for orthotropic material

A material that exhibits symmetry with respect to three mutually orthogonal planes is called an orthotropic material. The stress-strain relation for orthotropic material will consist 9 elastic constants which is given by

$$
\begin{bmatrix}
\sigma\_{\mathbf{x}} \\
\sigma\_{\mathbf{y}} \\
\sigma\_{\mathbf{z}} \\
\tau\_{\mathbf{xy}} \\
\tau\_{\mathbf{yz}} \\
\tau\_{\mathbf{z}x} \\
\tau\_{\mathbf{z}x}
\end{bmatrix} = \begin{bmatrix}
D\_{11} & D\_{12} & D\_{13} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
D\_{12} & D\_{22} & D\_{23} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
D\_{13} & D\_{23} & D\_{33} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & D\_{44} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & D\_{55} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & D\_{66}
\end{bmatrix} \begin{bmatrix}
\varepsilon\_{\mathbf{x}\tau} \\
\varepsilon\_{\mathcal{T}\tau} \\
\varepsilon\_{\mathcal{E}\tau} \\
\varepsilon\_{\mathcal{E}\tau} \\
\varepsilon\_{\mathcal{E}\tau} \\
\varepsilon\_{\mathcal{E}\tau}
\end{bmatrix}.
\tag{33}
$$

σx σy σz τxy τyz τzx

follow

D<sup>11</sup> D<sup>12</sup> D<sup>12</sup> 000 D<sup>12</sup> D<sup>11</sup> D<sup>12</sup> 000 D<sup>12</sup> D<sup>12</sup> D<sup>11</sup> 000 ð Þ D<sup>11</sup> � D<sup>12</sup> =20 0 000 0 ð Þ D<sup>11</sup> � D<sup>12</sup> =2 0

εxx εyy εzz εxy εyz εzx

(36)

(37)

000 0 0 ð Þ D<sup>11</sup> � D<sup>12</sup> =2

which consists only two independent elastic constants. Replacing D<sup>12</sup> and D<sup>12</sup> ð Þ D<sup>11</sup> � D<sup>12</sup> =2 by λ and μ which are called Lame's constants and in particular μ is

> <sup>σ</sup><sup>x</sup> <sup>¼</sup> ð Þ <sup>2</sup><sup>μ</sup> <sup>þ</sup> <sup>λ</sup> <sup>ε</sup>xx <sup>þ</sup> λ εyy <sup>þ</sup> <sup>ε</sup>zz � �, σ<sup>y</sup> ¼ ð Þ 2μ þ λ εyy þ λ εð Þ xx þ εzz , <sup>σ</sup><sup>z</sup> <sup>¼</sup> ð Þ <sup>2</sup><sup>μ</sup> <sup>þ</sup> <sup>λ</sup> <sup>ε</sup>zz <sup>þ</sup> λ εyy <sup>þ</sup> <sup>ε</sup>xx � �, τxy ¼ μεxy, τyz ¼ μεyz, τzx ¼ μεzx:

Also, from the above relation some important terms are induced which are as

K ¼ λ þ

to withstand changes in length when under lengthwise tension or

(1) Bulk modulus: Bulk modulus is the relative change in the volume of a body produced by a unit compressive or tensile stress acting uniformly over its

> 

(2) Young's modulus: Young's modulus is a measure of the ability of a material

<sup>E</sup> <sup>¼</sup> <sup>μ</sup>ð Þ <sup>3</sup><sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup>

(3) Poisson's ratio: The ratio of transverse strain and longitudinal strain is

<sup>ν</sup> <sup>¼</sup> <sup>λ</sup>

This chapter dealt the analysis of stress, analysis of strain and stress-strain relationship through particular sections. Concept of normal and shear stress, principal stress, plane stress, Mohr's circle, stress invariants and stress equilibrium relations are discussed in analysis of stress section while strain-displacement

 >>>>=

>>>>;

μ: (38)

<sup>λ</sup> <sup>þ</sup> <sup>μ</sup> : (39)

ð Þ <sup>λ</sup> <sup>þ</sup> <sup>μ</sup> : (40)

also called shear modulus of elasticity, we get

An Overview of Stress-Strain Analysis for Elasticity Equations

DOI: http://dx.doi.org/10.5772/intechopen.82066

surface. Symbolically

compression. Symbolically

called Poisson's ratio. Symbolically

5. Conclusions

#### 4.1.4 Stress-strain relation for transversely isotropic material

Transversely isotropic material exhibits a rationally elastic symmetry about one of the coordinate axes x, y and z. In such case, the material constants reduce to 5 as shown below

$$
\begin{bmatrix}
\sigma\_{\mathbf{x}} \\
\sigma\_{\mathbf{y}} \\
\sigma\_{\mathbf{z}} \\
\tau\_{\mathbf{xy}} \\
\tau\_{\mathbf{yz}} \\
\tau\_{\mathbf{xz}} \\
\tau\_{\mathbf{xz}}
\end{bmatrix} = \begin{bmatrix}
D\_{11} & D\_{12} & D\_{13} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
D\_{12} & D\_{11} & D\_{13} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
D\_{13} & D\_{13} & D\_{33} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & (D\_{11} - D\_{12})/2 & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & D\_{44} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & D\_{44}
\end{bmatrix} \begin{bmatrix}
\varepsilon\_{\mathbf{x}\mathbf{x}} \\
\varepsilon\_{\mathbf{yy}} \\
\varepsilon\_{\mathbf{xz}} \\
\varepsilon\_{\mathbf{xy}} \\
\varepsilon\_{\mathbf{yz}} \\
\varepsilon\_{\mathbf{xz}}
\end{bmatrix}.
\tag{34}$$

#### 4.1.5 Stress-strain relation for fiber-reinforced material

The constitutive equation for a fiber-reinforced material whose preferred direction is that of a unit vector a ! is

$$\begin{aligned} \pi\_{\vec{\eta}} &= \beta e\_{kk} \delta\_{\vec{\eta}} + 2\mu\_T e\_{\vec{\eta}} + a \left( a\_k a\_m e\_{km} \delta\_{\vec{\eta}} + e\_{kk} a\_l a\_j \right) + 2(\mu\_L - \mu\_T) \left( a\_l a\_k e\_{\vec{\eta}} + a\_j a\_k e\_{\vec{\alpha}} \right) \\ &+ \beta a\_k a\_m e\_{km} a\_l a\_j; \quad i, j, k, m = 1, 2, 3 \end{aligned} \tag{35}$$

where τij are components of stress, eij are components of infinitesimal strain, and ai the components of a ! , which are referred to rectangular Cartesian co-ordinates xi. The vector a ! may be a function of position. Indices take the value 1, 2 and 3, and the repeated suffix summation convention is adopted. The coefficients λ, μL, μT, α and β are all elastic constant with the dimension of stress.

#### 4.1.6 Stress-strain relation for isotropic material

For a material whose elastic properties are not a function of direction at all, only two independent elastic material constants are sufficient to describe its behavior completely. This material is called isotropic linear elastic. The stress-strain relationship for this material is written as

An Overview of Stress-Strain Analysis for Elasticity Equations DOI: http://dx.doi.org/10.5772/intechopen.82066

4.1.3 Stress-strain relation for orthotropic material

σx σy σz τxy τyz τzx

shown below

σx σy σz τxy τyz τzx

tion is that of a unit vector a

ai the components of a

xi. The vector a

material will consist 9 elastic constants which is given by

Elasticity of Materials ‐ Basic Principles and Design of Structures

4.1.4 Stress-strain relation for transversely isotropic material

4.1.5 Stress-strain relation for fiber-reinforced material

τij ¼ λekkδij þ 2μTeij þ α akamekmδij þ ekkaiaj

4.1.6 Stress-strain relation for isotropic material

ship for this material is written as

! is

and β are all elastic constant with the dimension of stress.

A material that exhibits symmetry with respect to three mutually orthogonal planes is called an orthotropic material. The stress-strain relation for orthotropic

> D<sup>11</sup> D<sup>12</sup> D<sup>13</sup> 000 D<sup>12</sup> D<sup>22</sup> D<sup>23</sup> 000 D<sup>13</sup> D<sup>23</sup> D<sup>33</sup> 000 D<sup>44</sup> 0 0 D<sup>55</sup> 0 D<sup>66</sup>

Transversely isotropic material exhibits a rationally elastic symmetry about one of the coordinate axes x, y and z. In such case, the material constants reduce to 5 as

The constitutive equation for a fiber-reinforced material whose preferred direc-

<sup>þ</sup> <sup>β</sup>akamekmaiaj; i, j, k, m <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup> (35)

where τij are components of stress, eij are components of infinitesimal strain, and

For a material whose elastic properties are not a function of direction at all, only two independent elastic material constants are sufficient to describe its behavior completely. This material is called isotropic linear elastic. The stress-strain relation-

the repeated suffix summation convention is adopted. The coefficients λ, μL, μT, α

� � <sup>þ</sup> <sup>2</sup> <sup>μ</sup><sup>L</sup> � <sup>μ</sup><sup>T</sup> ð Þ aiakekj <sup>þ</sup> ajakeki

! , which are referred to rectangular Cartesian co-ordinates

! may be a function of position. Indices take the value 1, 2 and 3, and

D<sup>11</sup> D<sup>12</sup> D<sup>13</sup> 0 00 D<sup>12</sup> D<sup>11</sup> D<sup>13</sup> 0 00 D<sup>13</sup> D<sup>13</sup> D<sup>33</sup> 0 00 ð Þ D<sup>11</sup> � D<sup>12</sup> =20 0 000 0 D<sup>44</sup> 0 000 0 0 D<sup>44</sup>

εxx εyy εzz εxy εyz εzx

� �

: (34)

εxx εyy εzz εxy εyz εzx

: (33)

$$
\begin{bmatrix}
\sigma\_{\mathbf{x}} \\
\sigma\_{\mathbf{y}} \\
\sigma\_{x} \\
\tau\_{\mathbf{xy}} \\
\tau\_{\mathbf{yz}} \\
\tau\_{\mathbf{xz}} \\
\tau\_{\mathbf{xz}}
\end{bmatrix} = \begin{bmatrix}
D\_{11} & D\_{12} & D\_{12} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
D\_{12} & D\_{11} & D\_{12} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
D\_{12} & D\_{12} & D\_{11} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & (D\_{11} - D\_{12})/2 & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & (D\_{11} - D\_{12})/2 & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & (D\_{11} - D\_{12})/2
\end{bmatrix} \begin{bmatrix}
\varepsilon\_{\mathbf{xx}} \\
\varepsilon\_{\mathbf{yy}} \\
\varepsilon\_{\mathbf{xz}} \\
\varepsilon\_{\mathbf{xy}} \\
\varepsilon\_{\mathbf{yz}} \\
\varepsilon\_{\mathbf{xz}} \\
\varepsilon\_{\mathbf{xx}}
\end{bmatrix} \tag{36}$$

which consists only two independent elastic constants. Replacing D<sup>12</sup> and D<sup>12</sup> ð Þ D<sup>11</sup> � D<sup>12</sup> =2 by λ and μ which are called Lame's constants and in particular μ is also called shear modulus of elasticity, we get

$$\begin{cases} \sigma\_{\mathbf{x}} = (2\mu + \lambda)\varepsilon\_{\mathbf{x}\mathbf{x}} + \lambda(\varepsilon\_{\mathbf{y}\mathbf{y}} + \varepsilon\_{\mathbf{z}\mathbf{z}}), \\ \sigma\_{\mathbf{y}} = (2\mu + \lambda)\varepsilon\_{\mathbf{y}\mathbf{y}} + \lambda(\varepsilon\_{\mathbf{x}\mathbf{x}} + \varepsilon\_{\mathbf{z}\mathbf{z}}), \\ \sigma\_{\mathbf{z}} = (2\mu + \lambda)\varepsilon\_{\mathbf{z}\mathbf{z}} + \lambda(\varepsilon\_{\mathbf{y}\mathbf{y}} + \varepsilon\_{\mathbf{x}\mathbf{x}}), \\ \tau\_{\mathbf{xy}} = \mu\varepsilon\_{\mathbf{xy}}, \tau\_{\mathbf{yz}} = \mu\varepsilon\_{\mathbf{yz}}, \tau\_{\mathbf{z}\mathbf{x}} = \mu\varepsilon\_{\mathbf{z}\mathbf{x}}. \end{cases} \tag{37}$$

Also, from the above relation some important terms are induced which are as follow

(1) Bulk modulus: Bulk modulus is the relative change in the volume of a body produced by a unit compressive or tensile stress acting uniformly over its surface. Symbolically

$$K = \lambda + \frac{2}{3}\mu.\tag{38}$$

(2) Young's modulus: Young's modulus is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression. Symbolically

$$E = \frac{\mu(\Im \lambda + \Im \mu)}{\lambda + \mu}. \tag{39}$$

(3) Poisson's ratio: The ratio of transverse strain and longitudinal strain is called Poisson's ratio. Symbolically

$$
\omega = \frac{\lambda}{2(\lambda + \mu)}.\tag{40}
$$

#### 5. Conclusions

This chapter dealt the analysis of stress, analysis of strain and stress-strain relationship through particular sections. Concept of normal and shear stress, principal stress, plane stress, Mohr's circle, stress invariants and stress equilibrium relations are discussed in analysis of stress section while strain-displacement

relationship for normal and shear strain, compatibility of strains are discussed in analysis of strain section through geometrical representations. Linear elasticity, generalized Hooke's law and stress-strain relation for triclinic, monoclinic, orthotropic, transversely-isotropic, fiber-reinforced and isotropic materials with some important relations for elasticity are discussed mathematically.

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An Overview of Stress-Strain Analysis for Elasticity Equations

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### Acknowledgements

The authors convey their sincere thanks to Indian Institute of Technology (ISM), Dhanbad, India for facilitating us with best research facility and provide a Senior Research Fellowship to Mr. Pulkit Kumar and also thanks to DST Inspire India to provide Senior Research Fellowship to Ms. Moumita Mahanty.

### Conflict of interest

There is no conflict of interest to declare.

### Author details

Pulkit Kumar\*, Moumita Mahanty and Amares Chattopadhyay Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, India

\*Address all correspondence to: pulkitkumar.maths@gmail.com

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An Overview of Stress-Strain Analysis for Elasticity Equations DOI: http://dx.doi.org/10.5772/intechopen.82066

#### References

relationship for normal and shear strain, compatibility of strains are discussed in analysis of strain section through geometrical representations. Linear elasticity, generalized Hooke's law and stress-strain relation for triclinic, monoclinic, orthotropic, transversely-isotropic, fiber-reinforced and isotropic materials with

The authors convey their sincere thanks to Indian Institute of Technology (ISM), Dhanbad, India for facilitating us with best research facility and provide a Senior Research Fellowship to Mr. Pulkit Kumar and also thanks to DST Inspire

some important relations for elasticity are discussed mathematically.

Elasticity of Materials ‐ Basic Principles and Design of Structures

India to provide Senior Research Fellowship to Ms. Moumita Mahanty.

Pulkit Kumar\*, Moumita Mahanty and Amares Chattopadhyay

\*Address all correspondence to: pulkitkumar.maths@gmail.com

Department of Applied Mathematics, Indian Institute of Technology (Indian School

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Acknowledgements

Conflict of interest

Author details

26

of Mines), Dhanbad, Jharkhand, India

provided the original work is properly cited.

There is no conflict of interest to declare.

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Malden, U.S.A: Blackwell Publishing. 2009

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DOI: http://dx.doi.org/10.5772/intechopen.82066

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[53] Sengupta PR, Nath S. Surface waves in fibre-reinforced anisotropic elastic media. Sadhana. 2001;26(4):363-370

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[56] Kolsky H. Stress Waves in Solids. New York: Dover Publication; 1963

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Society of America. 2005;118(5):

[60] Pichugin AV, Rogerson GA. Extensional edge waves in pre-stressed incompressible plates. Mathematics and Mechanics of Solids. 2012;17(1):27-42

[61] Hool GA, Kinne WS, Zipprodt RR. Reinforced Concrete and Masonry Structures. New York: McGraw-Hill;

[62] Spencer AJM. Boundary layers in highly anisotropic plane elasticity. International Journal of Solids and Structures. 1974;10(10):1103-1123

[59] Kaplunov J, Pichugin AV, Zernov V. Extensional edge modes in elastic plates and shells. The Journal of the Acoustical Society of America. 2009;125(2):

2975-2983

621-623

1924

[54] Shearer PM. Introduction to Seismology. 2nd ed. Cambridge: Cambridge University Press; 2009

[44] Chapman C. Fundamentals of Seismic Wave Propagation. Cambridge, UK: Cambridge University Press; 2004

[45] Carcione JM. Wave propagation in anisotropic linear viscoelastic media: Theory and simulated wavefields. Geophysical Journal International. 1990;

[46] Chadwick P. Wave propagation in transversely isotropic elastic media—I. Homogeneous plane waves. Proceedings of the Royal Society of London A. 1989;

[47] Chang SJ. Diffraction of plane dilatational waves by a finite crack. The Quarterly Journal of Mechanics and Applied Mathematics. 1971;24(4):

[48] Chattopadhyay A, Choudhury S. Magnetoelastic shear waves in an infinite self-reinforced plate.

International Journal for Numerical and Analytical Methods in Geomechanics.

[49] Das SC, Dey S. Edge waves under initial stress. Applied Scientific Research. 1970;22(1):382-389

[50] Fu YB. Existence and uniqueness of edge waves in a generally anisotropic elastic plate. Quarterly Journal of Mechanics and Applied Mathematics.

[51] Fu YB, Brookes DW. Edge waves in asymmetrically laminated plates. Journal of the Mechanics and Physics of

[52] Sneddon IN. The distribution of stress in the neighbourhood of a crack in an elastic solid. Proceedings of the Royal Society A. 1946;187(1009):229-260

University Press; 1985

101(3):739-750

422(1862):23-66

1995;19(4):289-304

2003;56(4):605-616

Solids. 2006;54(1):1-21

29

423-443

[32] Press F. Seismic wave attenuation in the crust. Journal of Geophysical Research. 1964;69(20):4417-4418

[33] Ben-Menahem A, Singh SJ. Seismic Waves and Sources. New York: Springer Verlag; 1981

[34] Belfield AJ, Rogers TG, Spencer AJM. Stress in elastic plates reinforced by fibres lying in concentric circles. Journal of the Mechanics and Physics of Solids. 1983;31(1):25-54

[35] Biot MA. The influence of initial stress on elastic waves. Journal of Applied Physics. 1940;11(8):522-530

[36] Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range. The Journal of the Acoustical Society of America. 1956;28(2):168-178

[37] Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. The Journal of the Acoustical Society of America. 1956;28(2):179-191

[38] Biot MA, Drucker DC. Mechanics of incremental deformation. Journal of Applied Mechanics. 1965;32(1):957.

[39] Borcherdt RD. Rayleigh-type surface wave on a linear viscoelastic half-space. The Journal of the Acoustical Society of America. 1974;55(1):13-15

[40] Borcherdt RD. Viscoelastic Waves in Layered Media. Cambridge, New York: Cambridge University Press; 2009

[41] Brekhovskikh L. Waves in Layered Media. New York: Academic Press; 1980

[42] Bullen KE. Compressibility-pressure hypothesis and the Earth's interior. Geophysical Journal International. 1949; 5:335-368

#### An Overview of Stress-Strain Analysis for Elasticity Equations DOI: http://dx.doi.org/10.5772/intechopen.82066

[43] Bullen KE, Bullen KE, Bolt BA. An Introduction to the Theory of Seismology. Cambridge, New York: Cambridge University Press; 1985

[22] Kumar P, Chattopadhyay A, Singh AK. Shear wave propagation due to a point source. Procedia Engineering.

Elasticity of Materials ‐ Basic Principles and Design of Structures

Malden, U.S.A: Blackwell Publishing.

[32] Press F. Seismic wave attenuation in

[33] Ben-Menahem A, Singh SJ. Seismic Waves and Sources. New York: Springer

[34] Belfield AJ, Rogers TG, Spencer AJM. Stress in elastic plates reinforced by fibres lying in concentric circles. Journal of the Mechanics and Physics of

[35] Biot MA. The influence of initial stress on elastic waves. Journal of Applied Physics. 1940;11(8):522-530

[36] Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range. The Journal of the Acoustical Society of America. 1956;28(2):168-178

[37] Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. The Journal of the Acoustical Society of America. 1956;28(2):179-191

[38] Biot MA, Drucker DC. Mechanics of incremental deformation. Journal of Applied Mechanics. 1965;32(1):957.

[39] Borcherdt RD. Rayleigh-type surface wave on a linear viscoelastic half-space. The Journal of the Acoustical Society of America. 1974;55(1):13-15

[40] Borcherdt RD. Viscoelastic Waves in Layered Media. Cambridge, New York: Cambridge University Press; 2009

[41] Brekhovskikh L. Waves in Layered Media. New York: Academic Press; 1980

[42] Bullen KE. Compressibility-pressure hypothesis and the Earth's interior. Geophysical Journal International. 1949;

5:335-368

Solids. 1983;31(1):25-54

the crust. Journal of Geophysical Research. 1964;69(20):4417-4418

2009

Verlag; 1981

[23] Chattopadhyay A, Singh P, Kumar P, Singh AK. Study of Love-type wave propagation in an isotropic tri layers elastic medium overlying a semi-infinite elastic medium structure. Waves in Random and Complex Media. 2017;28

[24] Chattopadhyay A, Saha S, Chakraborty M. Reflection and transmission of shear waves in

monoclinic media. International Journal for Numerical and Analytical Methods in Geomechanics. 1997;21(7):495-504

[25] Chattopadhyay A. Wave reflection and refraction in triclinic crystalline media. Archive of Applied Mechanics.

[26] Chattopadhyay A, Singh AK. Propagation of magnetoelastic shear waves in an irregular self-reinforced

[27] Udias A, Buforn E. Principles of Seismology. Cambridge, New York: Cambridge University Press; 2017

[28] Novotny O. Seismic Surface Waves.

[29] Brillouin L. Wave Propagation and Group Velocity. Academic Press, New

[30] Badriev IB, Banderov VV, Makarov MV, Paimushin VN. Determination of stress-strain state of geometrically nonlinear sandwich plate. Applied Mathematical Sciences. 2015;9(77–80):

[31] Jaeger JC, Cook NG, Zimmerman R. Fundamentals of Rock Mechanics.

layer. Journal of Engineering Mathematics. 2012;75(1):139-155

Bahia, Salvador: Instituto de

Geociencias; 1999

York; 1960

3887-3895

28

2017;173:1544-1551

(4):643-669

2004;73(8):568-579

[44] Chapman C. Fundamentals of Seismic Wave Propagation. Cambridge, UK: Cambridge University Press; 2004

[45] Carcione JM. Wave propagation in anisotropic linear viscoelastic media: Theory and simulated wavefields. Geophysical Journal International. 1990; 101(3):739-750

[46] Chadwick P. Wave propagation in transversely isotropic elastic media—I. Homogeneous plane waves. Proceedings of the Royal Society of London A. 1989; 422(1862):23-66

[47] Chang SJ. Diffraction of plane dilatational waves by a finite crack. The Quarterly Journal of Mechanics and Applied Mathematics. 1971;24(4): 423-443

[48] Chattopadhyay A, Choudhury S. Magnetoelastic shear waves in an infinite self-reinforced plate. International Journal for Numerical and Analytical Methods in Geomechanics. 1995;19(4):289-304

[49] Das SC, Dey S. Edge waves under initial stress. Applied Scientific Research. 1970;22(1):382-389

[50] Fu YB. Existence and uniqueness of edge waves in a generally anisotropic elastic plate. Quarterly Journal of Mechanics and Applied Mathematics. 2003;56(4):605-616

[51] Fu YB, Brookes DW. Edge waves in asymmetrically laminated plates. Journal of the Mechanics and Physics of Solids. 2006;54(1):1-21

[52] Sneddon IN. The distribution of stress in the neighbourhood of a crack in an elastic solid. Proceedings of the Royal Society A. 1946;187(1009):229-260

[53] Sengupta PR, Nath S. Surface waves in fibre-reinforced anisotropic elastic media. Sadhana. 2001;26(4):363-370

[54] Shearer PM. Introduction to Seismology. 2nd ed. Cambridge: Cambridge University Press; 2009

[55] Sheriff RE, Geldart LP. Exploration Seismology. Cambridge, New York: Cambridge University Press; 1995

[56] Kolsky H. Stress Waves in Solids. New York: Dover Publication; 1963

[57] Gubbins D. Seismology and Plate Tectonics. Cambridge, New York: Cambridge University Press; 1990

[58] Kaplunov J, Prikazchikov DA, Rogerson GA. On three-dimensional edge waves in semi-infinite isotropic plates subject to mixed face boundary conditions. The Journal of the Acoustical Society of America. 2005;118(5): 2975-2983

[59] Kaplunov J, Pichugin AV, Zernov V. Extensional edge modes in elastic plates and shells. The Journal of the Acoustical Society of America. 2009;125(2): 621-623

[60] Pichugin AV, Rogerson GA. Extensional edge waves in pre-stressed incompressible plates. Mathematics and Mechanics of Solids. 2012;17(1):27-42

[61] Hool GA, Kinne WS, Zipprodt RR. Reinforced Concrete and Masonry Structures. New York: McGraw-Hill; 1924

[62] Spencer AJM. Boundary layers in highly anisotropic plane elasticity. International Journal of Solids and Structures. 1974;10(10):1103-1123

Section 2

Engineering Applications

in Theory of Elasticity

31
