The Cauchy Relations in Linear Elasticity Theory

c = c = c . (2) 1 We use the notation of Schouten [9]. Symmetrization over two indices is denoted by parentheses, A(ij) := (Aij + Aji)/2!, antisymmetrization by brackets, B[ij] := (Bij − Bji)/2!. The analogous is valid for more indices, as, e.g., in (13): c i[jk]l = (c − c)/2!. If one or more indices are exempt from (anti)symmetrization, they are enclosed by vertical bars, as, e.g., in (12), c = (c + c)/2!, or, more complicated, in (6), c = (c + c)/2 = (c + c + c + c)/4. Symmetrization over more than 2 indices is, by definition, the normalized sum over all possible permutations of the indices involved. Thus, c := (c + c + c + 21 more terms)/4!, see (5). The totally antisymmetric Levi-Civita symbol is denoted by ǫijk and ǫ , with ǫ123 = ǫ 123 = +1; for all even (odd) permutations of 1, 2, 3, we have +1 (−1), otherwise 0. The partial derivative ∂/∂x we denote by ∂k.