executes transformations of a compliance matrix given in Cartesian coordinates into its form in various orthogonal curvilinear coordinates

2.15 Principal Directions of Anisotropy

In this section we shall develop the analytical background required for the determination of the principal directions of anisotropy. These directions are useful for general material

classifica-tion, and in particular for the task of comparing anisotropic materials. Conceptually, the follow-ing discussion is of “local nature” and deals with the “local principal directions of anisotropy”, i.e. at a given point.

To illustrate the main issue under discussion, suppose that two GEN21 anisotropic materials given by two sets of 21 independent coefficients, need to be compared. Clearly, two materials are identical if all their 21 coefficients are equal, see also Remark 2.5, but there is a possibility, that two materials that apparently seem different, differ only by rigid body rotation (i.e. a rota-tion that may depend on one, two or three Euler angles). To clarify and study this possibility, there is a need to define unique canonical principal directions of anisotropy, in which there is only one way to present a given material regardless of its orientation with respect to the coordinate system. The common criteria for defining the principal directions of anisotropy is based on the following statement: A coordinate system is said to coincide with the principal directions of anisotropy if the material, when subjected to “all-around uniform pure extension state”, forms a “pure tension state”.

To develop the required transformation of a given stiffness matrix to the principal directions of anisotropy, we first introduce the symmetric bulk modulus tensor

K=

⎡

⎢⎢

⎣

K11 K12 K13

K12 K22 K23

K13 K23 K33

⎤

⎥⎥

⎦. (2.56)

K is one of two tensors that may be obtained by contraction of the stiffness tensor S, and its elements are written as

Ki j=

∑

∑

where the index m corresponds to the pair(i, j) by the index transformation (2.38).

We may now examine the special case of strain principal directions and write the state of

“all-round uniform extension” asεi j=
εδi j (i, j = 1,2,3), where
εis a reference (constant) strain. According to (2.57), (2.2:b), the stress components are given by σi j= Ki j
ε. Hence, the principal directions of the tensor K coincide with the stress principal directions for the case under discussion, which as stated above, constitute the principal directions of anisotropy.

Quantitatively, in these principal directions,σi j= Ki^P
εδi j, where{Ki^P} are the three eigenval-ues of the tensor K. Extracting the rotation angles out of the eigenvalue analysis of the bulk modulus tensor K is identical to that discussed for the eigenvalue analysis of the stress tensor σ — see Example 1.1 of S.1.3.3.1.

Strictly speaking, to ensure a canonical (unique) principal system of anisotropy, the eigen-values of K should always be put in a certain order before calculating the resulting transforma-tion matrix. In the present case we select the order K₃^P≥ K₂^P≥ K₁^Pand for the moment assume that all eigenvalues of K are different (cases with multiple eigenvalues are discussed in Re-mark 2.6). This order force the material to orient its “strongest” direction in the z-direction and the “weakest” one in the x-direction. In such a case, the transformation is completely unique since the eigenvalue analysis itself is unique.

To illustrate the above concept, suppose that we deal with an orthotropic material that has been rotated so its stiffness matrix is populated as MON13z material (see Fig. 2.8), and we wish to restore the material principal directions of anisotropy, in which A16= A26= A36= A45= 0.

The condition K12= 0 only guarantees that A16+ A26+ A36= 0, see (2.57) (also note that K13= K23= 0 in this case). However, since the transformation is unique, there is no more than one solution to the equation K12= 0, and thus, this solution must yield a rotation that will bring the material back to its orthotropic population. Furthermore, in this case we employ the angleψonly. The relevant terms in the transformed system may be written as functions of the

coefficients in the original system as

A₁₆= −A26sin⁴ψ+ (A22− 2A66− A12)cosψsin³ψ+ 3(A26− A16)cos²ψsin²ψ + (2A66− A11+ A12)cos³ψsinψ+ A16cos⁴ψ,

A26= −A16sin⁴ψ+ (2A66− A11+ A12)cosψsin³ψ+ 3(A16− A26)cos²ψsin²ψ + (A22− 2A66− A12)cos³ψsinψ+ A26cos⁴ψ,

A36= A36cos 2ψ+(A23−A13)cosψsinψ, A45= A45cos 2ψ+(A44−A55)cosψsinψ. (2.58) Thus, by substitution of the expressions of (2.58) in K12= A16+ A26+ A36and utilization of elementary trigonometric identities, one obtains

K12= (A16+ A26+ A36)cos2ψ+1

2(A22− A11− A13+ A23)sin2ψ, (2.59) and the condition K12(ψ) = 0 yields

ψ=1 2 arctan

2(A36+ A26+ A16) A13− A23− A22+ A11



. (2.60)

Hence, the coordinate system should be rotated about the z-axis in the amount given above in order to convert the coordinate system into its principal directions of anisotropy, see also Remark 2.7. Note that a generic MON13z material will remain of MON13z-type despite the fulfillment of the condition K12= 0.

As already indicated, due to its symmetry properties, the fourth-order stiffness tensor S has another (in addition to K) symmetric second-order contracted tensor which is given by

L=

⎡

⎢⎢

⎣

L11 L12 L13

L21 L22 L23

L31 L32 L33

⎤

⎥⎥

⎦, (2.61)

where

Li j=

∑

∑

and again, the indices m and n correspond to the pairs(i,k) and ( j,k), respectively, by the index transformation of (2.38).

K and L, and hence their six invariants may be used for “fast” comparisons between two sets of stiffness matrices. Identical invariants is of course a necessary (but not sufficient) condition for declaring identity between two matrices. The invariants may be expressed (analogously to the stress tensor invariants) by their tensor trace and determinant as

I₁^K = tr K, I₂^K=1 2

(tr K)²−tr K²

, I₃^K= detK, (2.63)

while those of L are obtained analogously. Hence I_i^K, Ii^L(i= 1,2,3), are the six invariants of the stiffness matrix, while explicit expressions for the first two linear invariants are

I₁^K= A11+A22+A33+2(A12+A13+A23), I1^L= A11+A22+A33+2(A44+A55+ A66). (2.64) Note that in the isotropic case the bulk modulus, see S.2.8, is K= ¹₉I₁^K or ¹₃I₂^K/I₁^K or I₃^K/I₂^K. The above invariants could also be obtained by replacing C with S in (2.45a,b), see also P.2.9.

Remark 2.5 Strictly speaking, two materials behave in an identical manner if the expression