Stress tensor transformation properties

The quantities cyku describing the material properties, are called elastic stiffness constants or stiffiiesses. The transformation properties of the cyki under a rotation of the coordinate system is uniquely determined by the requirement of a coordinate invariant formulation of Eq. (3.51). The rank of this material tensor is equal to the sum of the ranks of strain and stress tensor. Accordingly, cyki are the coordinates of a fourth rank tensor. The symmetry of strain and stress tensors reqitires the symmetry of the stiffness tensor with regard to an interchange of i and j as well as k and /... 

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

Thus, in order to describe completely the state of stress at a point in a continuum, we must specify the stress tensor T. A key property of a tensor is the tranrformation law of its components. This law expresses the way in which the tensor components in one coordinate system are related to its components in another coordinate system. The precise form of this transformation law is a consequence of the physical or geometric meaning of the tensor. 

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

It is worth recalling here that each tensor has an order (I, II, III, IV, etc.). Tensor order reflects the physical properties of a tensor and is determined by the power of the direction cosines product, that is, the power of the product of linear transformation coefficients. The tensor order physically reflects the possibility of visualizing the various properties of a field or a body from different viewpoints. Tensor order is also an indicator of the different ways in which spatial anisotropy is revealed. Scalar quantities, that is, temperature, mass, and amount of heat, are zeroth-order tensors the vectors of velocity or force are the first-order tensors mechanical stresses and strains are second-order tensors, while the elasticity modulus is a fourth-order tensor, as will be shown in the following text. 

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