ABSTRACT
The following are functional representations for isotropy, transverse isotropy, and orthotropy for scalar, vector, symmetric tensors, and skew symmetric tensors. The representations are given for functions of at most three symmetric tensors A, B, and C, two skew symmetric tensors W and X, and one vector v. A scalar invariant function ψ, a vector invariant function q, and a tensor invariant function T of these variables are an invariant function under the tensor transformation M if they have the property
ψ(A,B,C,W,X,v) = ψ(MAMT ,MBMT ,MCMT ,MWMT ,MXMT ,Mv), (A.1)
q(A,B,C,W,X,v) = MTq(MAMT ,MBMT ,MCMT ,MWMT ,MXMT ,Mv), (A.2)
T(A,B,C,W,X,v) = MTT(MAMT ,MBMT ,MCMT ,MWMT ,MXMT ,Mv)M. (A.3)
In general there is a group of transformations under which a function is invariant. Here we consider the groups associated with isotropy, transverse isotropy, and orthotropy. We have selected to use the full orthogonal group of tensors, as opposed to the proper group of orthogonal tensors, to represent the symmetry transformations for isotropy. If e1, e2, and e3 are three orthogonal directions, and τ = e3, then transverse isotropy can be represented by five groups of tensors given by
T1 : Rτ , T2 : Rτ ,R2, T3 : Rτ ,R1, T4 : Rτ ,R2,R1, T5 : Rτ ,−R3, (A.4) where τ is the preferred axis of transverse isotropy, Rτ represents all rotations about τ , and Ri represent the reflection about the plane perpendicular to ei. The most common representation of transverse isotropy is T2, which is used for the following representations. There are three representations for orthotropy given by
O1 : R2,R3, O2 : −R2,−R3, O3 : R2,R3,−I. (A.5) The most commonly used representation for orthotropy is O3, which is used in the following representations.
