No matter what stresses are imposed on an elastic body, provided the material does not rupture, displacement at any point can have only one value. Therefore the displacement at any point can be completely given by the three single valued components u, v and w along the three co-ordinate axes x, y and z respectively. The normal and shear strains may be derived in terms of these displacements.
Consider an element AB of length δx (figure-22.214.171.124). If displacement of end A is u, that of end B is . This gives an increase in length of and therefore the strain in x-direction is Similarly, strains in y and z directions are .Therefore, we may write the three normal strain components as
In the same way we may define the shear strains. For this purpose consider an element ABCD in x-y plane and let the displaced position of the element be A′B′C′D′ ( Figure-126.96.36.199). This gives shear strain in xy plane as where α is the angle made by the displaced line B′C′ with the vertical and β is the angle made by the displaced line A′D′ with the horizontal. This gives
188.8.131.52F- Shear strain associated with the distortion of an infinitesimal element.
We may therefore write the three shear strain components as
Therefore, the complete strain matrix can be written as
The state of strain at a point can be completely described by the six strain components and the strain components in their turns can be completely defined by the displacement components u, v, and w. The constitutive equations relate stresses and strains and in linear elasticity we simply have where E is modulus of elasticity. It is also known that σx produces a strain of in xdirection, in y-direction and in z-direction. Therefore we may write the generalized Hooke’s law as
It is also known that the shear stress where G is the shear modulus and γ is shear strain. We may thus write the three strain components as
In general each strain is dependent on each stress and we may write
For isotropic material
Rest of the elements in K matrix are zero.
On substitution, this reduces the general constitutive equation to equations for isotropic materials as given by the generalized Hooke’s law. Since the principal stress and strains axes coincide, we may write the principal strains in terms of principal stresses as
From the point of view of volume change or dilatation resulting from hydrostatic pressure we also have
These equations allow the principal strain components to be defined in terms of principal stresses. For isotropic and homogeneous materials only two constants viz. E and ν are sufficient to relate the stresses and strains. The strain transformation follows the same set of rules as those used in stress transformation except that the shear strains are halved wherever they appear.
Relations between E, G and K
The largest maximum shear strain and shear stress can be given by
Considering now the hydrostatic state of stress and strain we may write
So far the state of strain at a point was considered entirely due to applied forces. Changes in temperature may also cause stresses if a thermal gradient or some external constraints exist. Provided that the materials remain linearly elastic, stress pattern due to thermal effect may be superimposed upon that due to applied forces and we may write
It is important to note that the shear strains are not affected directly by temperature changes. It is sometimes convenient to express stresses in terms of strains. This may be done using the relation Substituting the above expressions for εx , εy and εz we have.
and substituting we have
Combining this with we have
Substituting we may write the normal and shear stresses as
These equations are considered to be suitable in thermoelastic situations.