Ordering tensor

Equations (8.20) are not sufficiently specific for practical purposes, so it is important to consider special cases leading to simpler relations. When the pore orientations are isotropically distributed, the second order tensors k, 3 and y are isotropic and are therefore scalar multiples of the unit tensor. Thus equation (8.20) simplifies to... 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

Components of a second-order tensor T in a three-dimensional frame of reference are written as the following 3x3 matrix... 

It follows that by using component fomis second-order tensors can also be manipulated by rules of matrix analysis. 

The unit second-order tensor is the Kronecker delta, S, whose components are given as the following matrix... 

The matrix gp, represents the components of a covariant second-order tensor called the metric tensor , because it defines distance measurement with respect to coordinates To illustrate the application of this definition in the... 

The transformation of components of a second-order tensor, given as the following matrix in the coordinate system x... 

Here c[-], which will be called the elastic modulus tensor, is a fourth-order linear mapping of its second-order tensor argument, while b[-], which will be called the inelastic modulus tensor, is a linear mapping of k whose form will depend on the specific properties assigned to k. They depend, in general, on and k. For example, if k consists of a single second-order tensor, then in component form... 

Here g [ ] may be called the elastic compliance tensor, andl [-]maybe called the inelastic compliance tensor. Note that g is a fourth-order tensor which shares the symmetries of t. Again, (5.16) may be written as... 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

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 may first be noted that the referential symmetric Piola-Kirchhoff stress tensor S and the spatial Cauchy stress tensor s are related by (A.39). Again with the back stress in mind, it will be assumed in this section that the set of internal state variables is comprised of a single second-order tensor whose referential and spatial forms are related by a similar equation, i.e., by... 

An indifferent second-order tensor is one which maps indifferent vectors into indifferent vectors. Consider the mapping A such that b = Aa where a and b are arbitrary indifferent vectors. Then under the transformation (A.50) b = Qb and fl = Qa, so that... 

Since a is an arbitrary vector, from the second relation it follows that an indifferent second-order tensor transforms as... 

In this section, well-known properties of second-order positive-definite symmetric tensors and functions involving them will be cited without proof. The principal values and principal vectors (m = 1, 2, 3) of a symmetric second-order tensor A are given by... 

A scalar-valued function/(/4) of one symmetric second-order tensor A is said to be symmetric if... 

A scalar-valued function f(A, B) of two symmetric second-order tensors A and B is said to be isotropic if... 

The stress-intensity factors are quite different from stress concentration factors. For the same circular hole, the stress concentration factor is 3 under uniaxial tension, 2 under biaxiai tension, and 4 under pure shear. Thus, the stress concentration factor, which is a single scalar parameter, cannot characterize the stress state, a second-order tensor. However, the stress-intensity factor exists in all stress components, so is a useful concept in stress-type fracture processes. For example. 

Matrix and tensor notation is useful when dealing with systems of equations. Matrix theory is a straightforward set of operations for linear algebra and is covered in Section A.I. Tensor notation, treated in Section A.2, is a classification scheme in which the complexity ranges upward from scalars (zero-order tensors) and vectors (first-order tensors) through second-order tensors and beyond. 

Cartesian tensors, i.e., tensors in a Cartesian coordinate system, will be discussed. Three Independent quantities are required to describe the position of a point in Cartesian coordinates. This set of quantities is X where X is (x, X2, X3). The index i in X has values 1,2, and 3 because of the three coordinates in three-dimensional space. The indices i and j in a j mean, therefore, that a j has nine components. Similarly, byi has 27 components, Cp has 81 components, etc. The indices are part of what is called index notation. The number of subscripts on the symboi denotes the order of the tensor. For example, a is a zero-order tensor... 

Stress and strain are both second-order tensors. 

The stiffness and compliances in stress-strain and strain-stress relations are fourth-order tensors because they relate two second-order tensors ... 

Contracted notation is a rearrangement of terms such that the number of indices is reduced although their range increases. For second-order tensors, the number of indices is reduced from 2 to 1 and the range increased from 3 to 9. The stresses and strains, for example, are contracted as in Table A-1. Similarly, the fourth-order tensors for stiffnesses and compliances in Equations (A.42) and (A.43) have 2 instead of 4 free indices with a new range of 9. The number of components remains 81 (3 = 9 ). 

A fourth-order tensor can be written as a 9x9 array in analogy to Equation (A.51) but, by use of contracted notation, is sometimes drastically simplified to a 6 x 6 symmetric array. 

The nth-order property is the nth-order derivative of the energy, d EjdX" (the factor 1 /n may or may not be included in the property). Note that the perturbation is usually a vector, and the first derivative is therefore also a vector, the second derivative a matrix, the third derivative a (third-order) tensor etc. 

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