On the Algebra of Two‐Point Tensors on Manifolds with Applications in Nonlinear Solid Mechanics

Kollmann, Franz; Hackenberg, Hans-Peter

doi:10.1002/zamm.19930731107

Cited by 8 publications

(5 citation statements)

References 6 publications

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“…As an illustration, a special class of tensors, called one- and two-point tensors, will be introduced being essential in continuum mechanics [26]. A tensor is called a two-point tensor if it is defined on two different vector spaces V , W , whereas a one-point tensor is solely defined either on frakturV or on frakturW .…”

Section: Tensor Representations On Linear Spacesmentioning

confidence: 99%

“…If some of the base (co-)vectors are replaced by base (co-)vectors of a different (dual-)vector space, the tensor is called a second-order two-point tensor, cf. [5, 26].…”

Section: Tensor Representations On Linear Spacesmentioning

confidence: 99%

“…The expressions for push-forwards and pull-backs are summarized in Table 1. In contrast to prior approaches [26], the symmetry-preserving transformation rules of a contra-variant tensor require a correction in the definition. The present approach results in push-forwards and pull-backs being unambiguous, e.g.…”

Section: Symmetry and Transformation Rules Of High-order Tensorsmentioning

confidence: 99%

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Representations of m-linear functions on tensor spaces: Duals and transpositions with applications in continuum mechanics

Holopainen

2012

Mathematics and Mechanics of Solids

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The main goal of the present paper is to introduce certain duals and transpositions of a high-order tensor playing an important role in continuum mechanics. Emphasis is also placed on the comparison of the duals and the transpositions. In contrast to the duals, the transpositions depend on a metric of an underlying metric space. A high-order tensor is itself a representation of a multilinear function on a tensor space, obtained by means of the multilinear extension. The duals and the transpositions of a high-order tensor are identified as multilinear maps defined by the generalized scalar and the inner product, respectively. Consequently, the duals and the transpositions distinguish and define symmetries and symmetry-preserving transformation rules of a co-, contra- and mixed-variant tensor, respectively. As an application in continuum mechanics, the duals and the transpositions of a usual fourth-order tensor are defined and are employed to the determination of symmetries involved.

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Section: Tensor Representations On Linear Spacesmentioning

confidence: 99%

“…If some of the base (co-)vectors are replaced by base (co-)vectors of a different (dual-)vector space, the tensor is called a second-order two-point tensor, cf. [5, 26].…”

Section: Tensor Representations On Linear Spacesmentioning

confidence: 99%

Section: Symmetry and Transformation Rules Of High-order Tensorsmentioning

confidence: 99%

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Representations of m-linear functions on tensor spaces: Duals and transpositions with applications in continuum mechanics

Holopainen

2012

Mathematics and Mechanics of Solids

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“…By raising and lowering the components of A * 3 or A * 4 according to (10) it is ensured that the transpose maps vectors onto vectors or co-vectors onto co-vectors, respectively. For more detail of this special topic please refer to [4,9] and [22].…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

Fourth-order tensors – tensor differentiation with applications to continuum mechanics. Part II: Tensor analysis on manifolds

Kintzel

2006

Z. angew. Math. Mech.

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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...

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“…the constant metric of the Euclidean space, it is convenient to identify dual space and vector space by means of the metric (see WANG and TRUESDELL [61] (p. 19)). This case corresponds to the classical tensor algebra with the operation on vectors given by (21).…”

Section: Classical Tensor Algebra On Inner Product Spacesmentioning

confidence: 99%

The Application of Tensor Algebra on Manifolds to Nonlinear Continuum Mechanics — Invited Survey Article

Stumpf

Hoppe

1997

Z Angew Math Mech

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Some properties of the tensor algebra on manifolds are discussed with respect to the classical tensor algebra of continuum mechanics. Basic definitions and relations of linear maps are briefly recalled and applied to tensor spaces, where special attention is focused to the definition of dual and transposed maps. As application to nonlinear continuum mechanics, algebraic pull‐back and push‐forward maps, and Lie‐like time derivatives associated with linear maps are defined and used to build up commutative schemes for deformation measures, objective deformation rates, stresses, and frame‐invariant forms of the stress power. The transition to the classical formulation of tensor algebra by identifying vector spaces and their duals is examined. It is shown how this concept enables a straight‐forward derivation of the Doyle‐Ericksen formula and its possible variants.

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On the Algebra of Two‐Point Tensors on Manifolds with Applications in Nonlinear Solid Mechanics

Cited by 8 publications

References 6 publications

Representations of m-linear functions on tensor spaces: Duals and transpositions with applications in continuum mechanics

Representations of m-linear functions on tensor spaces: Duals and transpositions with applications in continuum mechanics

Fourth-order tensors – tensor differentiation with applications to continuum mechanics. Part II: Tensor analysis on manifolds

The Application of Tensor Algebra on Manifolds to Nonlinear Continuum Mechanics — Invited Survey Article

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