The generalized Hookes law

The most general linear relationship between stress and strain is obtained by assuming that each of the six independent components of stress is linearly related to each of the six independent components of strain. Thus 

To develop the generalized Hooke s law for isotropic materials it is convenient to construct equations for the strains Cyy, etc. in terms of the applied stresses o xxj ( yy, etc and so define Young s modulus E and Poisson s ratio v. An applied stress Oxx will produce a strain 

A shear strain Cxz is related to the corresponding shear stress Oxz by the relationship exz = OxJG, where G is the shear modulus. 

Thus we obtain the stress-strain relationships that are the starting point in many elementary textbooks of elasticity ([1], pp. 7-9)  

A bulk modulus K, related to the fractional change in volume, can also be defined, but only two of the quantities E, v, G and K are independent. For example 

The law of linear proportionality between uniaxial strain and uniaxial stress discovered by Hooke in 1676 can be generalized to a linear connection relating all nine elements of the strain tensor and all nine elements of the stress tensor, implying the existence of 81 constants of proportionality, or elastic compliances, Sijki, relating generically the strain tensor component sy to the stress tensor component in an expression of the type 

in our consideration, both the stress and the strain tensors are symmetrical about the diagonal, there are only six independent stress and strain elements, making many of the elastic compliances equal to each other. This makes it possible to simplify the generalized Hooke s law so that it involves at most 36 elastic compliances or constants, but requires the introduction of a shorthand notation both for stress and for strain for a unique representation that is referred to as the Voigt notation that we state as follows, for stresses and strains  

In eqs. (4.6)-(4.8) the strain elements y,- are referred to as the tangential shear-strain elements, the use of which permits certain convenient economies in representation. 

On the basis of the eontracted Voigt notation of stresses and strains, the generalized Hooke s law ean be written out in the form of a set of linear relations, which for the strain-stress relationships are 

These two alternatives give the formal connection of the elastic constants and elastic compliances. They can be abbreviated as matrix products as 

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