By Peter Haupt

The new version comprises extra analytical tools within the classical conception of viscoelasticity. This results in a brand new concept of finite linear viscoelasticity of incompressible isotropic fabrics. Anisotropic viscoplasticity is totally reformulated and prolonged to a basic constitutive thought that covers crystal plasticity as a distinct case.

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**Example text**

The transformation of material line, surface and volume elements by means of the deformation gradient is described in the following Theorem 1. 59) 3) dv = (detF)dV . 60) o Proof Statement 1) has already been established. 61) valid for any second order tensor A and vectors a, b, c E y3. This identity is one version of the multiplication theorem for determinants. 61) for any two fixed vectors a, • b and all c E y3. 9 See TRUESDELL & TOUPIN [1960], Sect. 20 for references. 29 1. 3 Deformation Gradient Since line elements, surface elements and volume elements occur in line integrals, surface integrals and volume integrals, the theorem enables us to trace the temporal development of integrals over these material areas.

2 The polar decomposition holds for the deformation gradient F (as in the case of every invertible second order tensor):10 10 See TRUESDELL & NOLL [1965], Sect. 23. 63) in which factors V and V are symmetric and positive definite, V = V T , V· Vv > 0, V = VT, V· Vv > 0, and R is orthogonal: RRT = RTR = 1. V, V and R are uniquely defined (see Fig. 1. 9). 2) V and V have the same eigenvalues: if e is an eigenvector of V, then Re is an eigenvector of V. 0 The proof of this theorem is based on Theorem 1.

O A deformation gradient is assigned to every (X, t) E R[g(J] x II: (X, t) ~ F(X, t) . 44), is the Frechet derivative of XR(X, t) with respect to X, a linear mapping from V: onto V 3. F is calculated as the Gateaux derivative. 6 For all H E V~ we have the identity 6 According to a well·known theorem of functional analysis, the Gateaux derivative is equal to the Frechet derivative, if sufficient conditions of continuity hold. Conversely, if a map is Frechet-differentiable, it is also Gateaux-differentiable, and the two derivatives coincide.