Stress tensor matrix representation

The stress tensor may be represented as a 3 x 3 matrix, with components Oy, where i and j both go from 1 to 3. The diagonal elements represent normal stresses, whereas the off-diagonal ones represent shear stresses. Positive normal stresses are tensile, while negative ones are compressive (but an opposite sign convention is sometimes used, most notably in the soil mechanics literature). Finally, from the balance of angular momentum (or torque in the static case), it follows that the stress tensor and its matrix representation are symmetric (ay = aji), meaning that only six out of the nine components are in fact independent. 

What remains to be shown is that the matrix xap is actually a representation of a second-rank tensor r to whidi we shall henceforth refer as the stress tensor. We need to demonstrate that the matrix representing t satisfies transfonnation properties under rotation of the coordinate system that constitute a second-rank tensor. To this end consider an infinitesimally... 

Deformation is measured by a quantity known as strain (strain is a relative extension or contraction of dimension). Strain is similarly a tensor of the second rank having nine components (3x3 matrix). The relation between stress and strain in the elastic regime is given by the classical Hooke s law. It is therefore obvious that the Hooke s proportionality constant, known as the elastic modulus, is a tensor of 4 rank and is represented by a (9 x 9) matrix. Before further discussion we note the following. The stress tensor consists of 9 elements of which stability conditions require cjxy=(jyx, stress components in the symmetric stress matrix are only six. Similarly there are only six independent strain components. Therefore there can only be six stress and six strain components for an elastic body which has unequal elastic responses in x, y and z directions as in a completely anisotropic solid. The representation of elastic properties become simple by following the well known Einstein convention. The subscript xx, yy, zz, yz, zx and xy are respectively represented by 1, 2, 3, 4, 5 and 6. Therefore Hooke s law may now be written in a generalized form as. 

If we change the coordinate system, the components of the stress tensor a (its matrix representation) change, but it still describes the same state of stress. The transformation rules are detailed in appendix A.5. 

The symmetry of tensors for strain and stress and the related reduction in the number of coordinates has given rise to the introduction of an abbreviated notation and a representation of these quantities by matrices. This abbreviated matrix notation is almost exclusively in use in the literature on piezoelectricity and in the... 

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