An Overview of Stress-Strain 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 are discussed.


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 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][18][19][20][21][22]. The problem of the determination of stress and strain fields in the elastic solids are discussed by many researchers [23][24][25][26][27][28][29][30][31][32][33]. Belfield et al. [34] discussed the stresses in elastic plates reinforced by fibers lying in concentric circles. Biot [35][36][37][38] gave the theory for the propagation of elastic waves in an initially stressed and fluid saturated transversely isotropic media.  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][43][44][45][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][53][54][55][56][57]. Kaplunov, Pichugin and Rogersion [58][59][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].
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.

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 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.
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: i. body forces and ii. surface forces.
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. shear stress component acting on the X-face in the Y-direction is identified as τ xy :

Concept of normal stress and shear stress
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 X-face at point O are defined as follows in terms of force intensity ratios (2) and the above stress components are illustrated in Figure 2(b).

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 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.

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 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.

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.

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 0 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 Similarly, write for the y' normal stress and shear stress. The transformed stresses are given as and τ x 0 y 0 ¼ σ y À σ x 2 sin 2θ þ τ xy cos 2θ where σ x 0 and τ x 0 y 0 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 and maximum shear stress is with τ max ¼ σ 1 À σ 2 2 : 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 2θ ¼ is for the principal shear plane.

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.

Equations of Mohr's circle
Rearranging the terms of Eq. (5), we get Squaring and adding the Eqs. (9.1) and (9.2), result in For simple representation of Eq. (10), the following notations are used Thus, the simplified form of Eq. (10) can be written as Eq. (12) represents the equation of a circle in a standard form. This circle has σ x 0 as its abscissa and τ x 0 y 0 as its ordinate with radius r. The coordinate for the center of the circle is σ av ; 0 ð Þ. Mohr's circle is drawn by considering the stress coordinates σ x as its abscissa and τ xy 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 x and τ xy 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 y and τ yx : 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 0 . The coordinates of M 0 in stress plane define the stresses acting on plane M 0 in physical plane and it can be easily verified.
Eq. (14) is same as the first equation of Eq. (5). This way it can be proved for shear stress τ x 0 y 0 on plane M 0 (do yourself).

Stress equilibrium relation
Let σ x , τ yx , τ zx are the stress components acting along the x-direction, τ xy , σ y , τ zy are the stress components acting along the y-direction and τ xz , τ yz , σ z are the stress components acting along the z-direction. The body forces F x , F y , F z acting along x, y, z direction respectively. Then the stress equilibrium relation or equation of motion in terms of stress components are given by

Analysis of strain
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.

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.

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 0 B 0 , having undergone a translation, extension and rotation.
The particle that was originally at x has undergone a displacement u x x; y ð Þ and the other end of the line element has undergone a displacement u x x þ Δx; y ð Þ : By the definition of normal strain In the limit Δx ! 0, Eq. (16) becomes 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

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 A similar relation can be derived by considering a line element initially lying in the y-direction. From the Figure 7( provided that (i) θ is small and (ii) the displacement gradient ∂u x =∂x is small. A similar expression for the angle λ can be derived as  and hence the shear strain can be written in terms of displacement gradients as In similar manner, the strain-displacement relation for three dimensional body is given by

Compatibility of strain
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.

Compatibility relations
Let us suppose that the point P which is act (x,y) before straining and it will be at P 0 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  we get Eqs. (25) and (26) result in which is the compatibility condition in two dimension.

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.

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 where E is known as Young's modulus.
In general, each strain is dependent on each stress. For example, the strain ε xx written as a function of each stress as ε xx ¼ C 11 σ x þ C 12 σ y þ C 13 σ z þ C 14 τ xy þ C 15 τ yz þ C 16 τ zx þ C 17 τ xz þ C 18 τ zy þ C 19 τ yx : (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 where D ijkl À Á is elasticity matrix, σ ij À Á is stress components, ε kl ð Þ is strain components.
Since both stress σ ij and strain ε ij are second-order tensors, it follows that D ijkl is a fourth order tensor, which consists of 3 4 ¼ 81 material constants if symmetry is not assumed. Now, from σ ij ¼ σ ji and ε ij ¼ ε ji , the number of 81 material constants is reduced to 36 under symmetric conditions of D ijkl ¼ D jikl ¼ D ijlk ¼ D jilk which provides stress-strain relation for most general form of anisotropic material.

Stress-strain relation for triclinic material
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.