The present contribution provides a tensor formalism for fourth-order tensors in the so-called absolute notation and focusses in particular on the use of this notation in the process of tensor differentiation with respect to a second-order tensor. Three tensor products, two new double contraction rules and a set of well-defined notations are introduced which in combination with the tensor differentiation rules simplify analytical derivation procedures considerably and provide significant advantages for various tasks in continuum mechanics. The suitability of the proposed rules and definitions is demonstrated in a number of relevant problems of continuum mechanics such as linearization of the generalized midpoint-rule and the exponential function. Special attention is given to the differentiation with respect to symmetric, skew-symmetric and inverse second-order tensors.where I = g i ⊗ g i = g i ⊗ g i = g ij g i ⊗ g j = g ij g i ⊗ g j is the second-order identity tensor.
Key words fourth-order tensors, tensor differentiation, tangent operators, anisotropic hyperelasticity MSC (2000) 03E25In Part II of this contribution the theory of fourth-order tensors and tensor differentiation introduced in Part I will be applied to finite deformation problems using a mathematical framework which is mainly borrowed from tensor analysis on manifolds. The main aspects are the explicit consideration of component variance, the introduction of different configurations and its associated metric tensors. Some relevant problems of continuum mechanics like the construction of tangent operators are discussed. Special attention is given to the conjugated formulation of the corresponding tangent operator in terms of the left Cauchy-Green-tensor.In the first part of this contribution a tensor formalism for fourth-order tensors has been introduced which has been used in the process of tensor differentiation with respect to a second-order tensor according to a recent approach of Itskov [8]. The proposed rules allow an easy application of the tensor differentiation law in absolute tensor notation which is in particular useful for the linearization of nonlinear functionals, the differentation of isotropic tensor functions e.g. in pow er series or the construction of tangent operators. In the first part the laws have been applied in a mathematical framework referred to as classical tensor analysis [1,16,11,12]. However, as far as finite deformation or large strain problems are concerned, the mathematical framework of tensor analysis on manifolds [2,13,24] is more appropriate since it delivers a rigorous and profound mathematical theory for the study of such kinds of problems. This mathematical framework has attained popularity especially through the work of Marsden & Hughes [13]. In a number of overview articles the topic of tensor algebra on manifolds has been addressed in a comprehensive way (see e.g. [4,5,22]).Due to the relevance of this mathematical field the present contribution will focus on the algebra of tensor analysis on manifolds and aims at a unified presentation considering the theory of fourth-order tensors and the rules of tensor differentiation. The consideration of large strain problems rests upon the introduction of different configurations of a body and its associated metric tensors. Following the typical approach of tensor algebra in dual spaces these metric tensors, which are used to construct invariants of tensors, are introduced as independent argument tensors in the corresponding tensor functions since the push-forward or pull-back of a metric tensor is not trivially the identity. According to this approach the derivative with respect to a metric tensor is a well-established quantity. We introduce the well-known Doyle-Ericksenformula [17,19], which is given as derivative of the strain energy function with respect to the spatial metric tensor g, and discuss its conjugated formulation in terms of the left Cauchy-Green-tensor b. The latter problem has been addressed in earlier contribution...
The current paper deals with the assessment and the numerical simulation of low cycle fatigue of an aluminum 2024 alloy. According to experimental observations, the material response of Al2024 is highly direction-dependent showing a material behavior between ductile and brittle. In particular, in its corresponding (small transversal) S-direction, the material behavior can be characterized as quasi-brittle. For the modeling of such a mechanical response, a novel, fully coupled isotropic ductile-brittle continuum damage mechanics model is proposed. Since the resulting model shows a large number of material parameters, an efficient, hybrid parameter identification strategy is discussed. Within this strategy, as many parameters as possible have been determined a priori by exploiting analogies to established theories (like P' law), while the remaining free unknowns are computed by solving an optimization problem. Comparisons between the experimentally observed and the numerically simulated lifetimes reveal the prediction capability of the proposed model.
In analytical analyses for single-screw extruders the simplified approach of rotating the barrel and keeping the screw fixed is often used instead of rotating the screw and fixing the barrel. Although the flow field is independent of the reference frame, as has already been shown to a satisfactory degree (Rauwendaal et al., 1997), the question of the dependence of the melt temperature rise on the reference frame is still being challenged, e.g. Campbell et al. (2001, 2008). In this work we develop a finite-volume CFD code allowing for the three-dimensional simulation of the flow and temperature rise in both reference frames. The question of frame invariance is addressed by simulating the flow of a Newtonian-like polycarbonate both in a two-dimensional cross-section of a single-screw extruder and in a three-dimensional model with two full turns of the screw. Our results show that the kinematics and the melt temperature rise are equal for screw- and barrel-rotation and thus independent of the reference frame. Furthermore, the presence of a clearance flow has a negligible influence on the temperature rise.
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