**The relationship of stress and strain in a three dimensional, linearly elastic, isotropic body**

*It may be seen from the third of Eqs (1.47) that the conditions of plane stress and plane strain do not necessarily describe identical situations. Changes in the linear dimensions of a strained body may lead to a change in volume. Suppose that a small element of a body has dimensions d _{x}, d_{y} and d_{z}_{.} When subjected to a three-dimensional stress system the element will sustain a volumetric strain e (change in volume/unit volume) equal to *

_{}

_{}

*Lame constants:*

In linear elasticity, the Lamé parameters are the two parameters

· λ, also called Lamé’s first parameter

· μ, the shear modulus or Lamé’s second parameter (also referred to as G)

In homogenous and isotropic materials, these satisfy Hooke’s law in 3D,

where σ is the stress, ε the strain tensor, the identity matrix and the trace function.

The first parameter λ is related to the bulk modulus and the shear modulus via in three-dimensions and for two-dimensional solids, and serves to simplify the stiffness matrix in Hooke’s law. Although the shear modulus, μ, must be positive, the Lamé’s first parameter, λ, can be negative, in principle; however, for most materials it is also positive. The two parameters together constitute a parameterization of the elastic moduli for homogeneous isotropic media, popular in mathematical literature, and are thus related to the other elastic moduli.

*Compatibility equation:*

** I**n Section 1.9 we expressed the six components of strain at a point in a deformable body in terms of the three components of displacement at that point, u, v and w.We have supposed that the body remains continuous during the deformation so that no voids are formed. It follows that each component, u, v and w, must be a continuous, single-valued function or, in quantitative terms

If voids were formed then displacements in regions of the body separated by the voids would be expressed as different functions of x, y and z. The existence, therefore, of just three single-valued functions for displacement is an expression of the continuity or compatibility of displacement which we have presupposed.

Since the six strains are defined in terms of three displacement functions then they must bear some relationship to each other and cannot have arbitrary values. These relationships are found as follows. Differentiating g_{xy} from Eqs (1.20) with respect to x and y gives

or since the functions of u and v are continuous

which may be written, using Eq. (1.18)

Principal stress:

For given values of σ_{x,} σ_{y} and τ_{xy} in other words given loading conditions, σ_{n} varies with the angle Ѳ and will attain a maximum or minimum value when dσ_{n}/dѲ = 0. From Eq. (1.8)

Two solutions, Ѳ and Ѳ + P/2, are obtained from Eq. (1.10) so that there are two mutually perpendicular planes on which the direct stress is either a maximum or a minimum. Further, by comparison of Eqs (1.10) and (1.9) it will be observed that these planes correspond to those on which there is no shear stress. The direct stresses on these planes are called principal stresses and the planes themselves, principal planes. From Eq. (1.10)

where σ_{I} is the maximum or major principal stress and σ_{II} is the minimum or minor principal stress. Note that σ_{I} is algebraically the greatest direct stress at the point while aI1 is algebraically the least. Therefore, when aII is negative, i.e. compressive, it is possible for σ_{II} to be numerically greater than σ_{I}.

The maximum shear stress at this point in the body may be determined in an identical manner. From Eq. (1.9)

Equations (1.14) and (1.15) give the maximum shear stress at the point in the body in the plane of the given stresses. For a three-dimensional body supporting a two dimensional stress system this is not necessarily the maximum shear stress at the point.

Since Eq. (1.13) is the negative reciprocal of Eq. (1.10) then the angles 2Ѳ given by these two equations differ by 90″ or, alternatively, the planes of maximum shear stress are inclined at 45″ to the principal planes.

Mohr’s circle of stress:

The state of stress at a point in a deformable body may be determined graphically by Mohr’s circle of stress.

In Section 1.6 the direct and shear stresses on an inclined plane were shown to be given by

respectively. The positive directions of these stresses and the angle 8 are defined in Fig. 1.9(a). Equation (1.8) may be rewritten in the form

*Principal Strain:*

Saints Venant’s Principle:

In the examples of Section 2.3 we have seen that a particular stress function form may be applicable to a variety of problems. Different problems are deduced from a given stress function by specifying, in the first instance, the shape of the body and then assigning a variety of values to the coefficients. The resulting stress functions give stresses which satisfy the equations of equilibrium and compatibility at all points within and on the boundary of the body. It follows that the applied loads must be distributed around the boundary of the body in the same manner as the internal stresses at the boundary. Thus, in the case of pure bending (Fig. 2.2(a)) the applied bending moment must be produced by tensile and compressive forces on the ends of the plate, their magnitudes being dependent on their distance from the neutral axis. If this condition is invalidated by the application of loads in an arbitrary fashion or by preventing the free distortion of any section of the body then the solution of the problem is no longer exact. As this is the case in practically every structural problem it would appear that the usefulness of the theory is strictly limited. To surmount this obstacle we turn to the important principle of St. Venant which may be summarized as stating:

that while statically equivalent systems of forces acting on a body produce substantially different local effects the stresses at sections distant from the surface of loading are essentially the same.

Thus, at a section AA close to the end of a beam supporting two point loads P the stress distribution varies as shown in Fig. 2.3, whilst at the section BB, a distance usually taken to be greater than the dimension of the surface to which the load is applied, the stress distribution is uniform.

We may therefore apply the theory to sections of bodies away from points of applied loading or constraint. The determination of stresses in these regions requires, for some problems, separate calculation.