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By Elmer Franklin Bruhn

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192) we recall that for any vector field ϕ = ϕ(x) the curl operator is defined by [see Eq. 10 Problems and Solutions Related to the Mathematical Preliminaries 27 Since φ,k j is a second order tensor that is symmetric with respect to the indexes k and j, while εi jk is asymmetric with respect to k, j, then by Eq. 192). 210) Since u k,i j is a third order tensor that is symmetric with respect to the indices i and j, while εi jk is asymmetric with respect to those indices, by Eq. 193). 211) Since, by the ε − δ relation [see Eq.

32) provides an alternative definition of the displacement vector u = u(x,t) related to the stress tensor S = S(x,t). 36) The tensor C = C(x) is called the elasticity tensor field on B. It follows from Eq. 38) The elasticity tensor C is also assumed to be invertible, that means that a restriction of C to the space of all symmetric tensors is invertible. The elasticity tensor on the space of all tensors cannot be invertible since its value on every skew tensor is zero. 40) The tensor K = K(x) is called the compliance tensor.

7. 140) Note. The matrix [Tˆi j ] is called the cofactor of the matrix [Ti j ], while [TˆiTj ] is called the adjoint of [Ti j ]. Solution. 144) and therefore, Eq. 141). 139). 140). 10 Problems and Solutions Related to the Mathematical Preliminaries Fig. 8. The xi system is obtained by rotating the xi system about the x3 axis through an angle 0 < θ < π/2, as shown in Fig. 1. Let T be a symmetric second-order tensor referred to the xi system. 159) T13 − i T23 = exp (i θ ) (T13 − T23 ) T33 = T33 √ where i = −1.