1. Preliminaries on Materials with Fading Memory
Society for Industrial and Applied Mathematics eBooks · 1992
1.1. Notation. The space under consideration is the three-dimensional Euclidean point space . Vectors are elements of the associated translation space V and are denoted by boldface minuscules. Second-order tensors are meant as linear transformations of V into V. Sym is the set of symmetric (second-order) tensors while sym denotes the symmetric part of a tensor. Lin stands for the set of all tensors; Lin(Sym) is the set of all linear transformations of Sym into Sym; is the subset of Lin whose elements have a positive determinant; Psym is the set of elements of Sym which are positive definite; is the set of all rotations. Boldface majuscules usually denote elements of Lin or Lin(Sym); 1 is the identity of Lin. The symbols tr and det stand for the trace and the determinant of tensors while the superscript T, e.g., , denotes the transpose. Letting u, v ∈ V, we write u ⋅ v for the standard inner product. If L, M ∈ Lin, then L ⋅ M stands for . If M is an element of Lin or Lin(Sym), then the writing means that M is positive (negative) definite in V or in Sym. For any two vectors u, v, u ⊗ v denotes the tensor product.