Isotropic Tensor Functions Research Articles

On the theory of fourth-order tensors and their applications in computational mechanics

Many problems concerned with the mathematical treatment of fourth-order tensors still remain open in the literature. In the present paper they will be considered in the framework of a complete theory involving a set of notations and definitions, a tensor operation algebra, differentiation rules, eigenvalue problems, applications of fourth-order tensors to isotropic tensor functions and some other relevant aspects. A tensor is understood to be an invariant quantity with respect to any co-ordinate system transformation, which justifies the use of absolute notation preferred in this work. As a most important application field of fourth-order tensors, elastic moduli are formulated in a material and a spatial description and given for various hyperelastic material models.

On the theory of fourth‐order tensors and their application in large strain elasticity

AbstractThis paper briefly presents some important aspects of the theory of fourth‐order tensors proposed in ]I[. The theory involves an operation algebra, differentiation rules, eigenvalue problems, an application to isotropic tensor functions and some other relevant problems. A most important application field the theory finds in large strain elasticity, where fourth‐order tensors appear as elastic moduli.

Representing Tensor Functions with a Cholesky Transform

The tensor representation theorem provides a powerful tool for evaluating isotropic tensor-valued tensor functions. Efficient computational techniques exist when an analysis is being done in rectangular Cartesian coordinates. Introducing a Cholesky decomposition of the metric tensor permits the tensor polynomial of the tensor representation theorem, when expressed in curvilinear coordinates, to be transformed into a pseudo-Cartesian frame where these efficient algorithms apply. It also leads to accurate descriptions for the physical components of a tensor.

A new computational model for Tresca plasticity at finite strains with an optimal parametrization in the principal space

A new computational model for the rate-independent elasto-plastic solids characterized by yield surfaces containing singularities and general nonlinear isotropic hardening is presented. Within the context of fully implicit return mapping algorithms, a numerical scheme for integration of the constitutive equations is formulated in the space of principal stresses. As a direct consequence of the principal stress approach, the representation of a yield surface is cast in terms of ‘optimal’ parameterization, which for the Tresca yield criterion takes a simple linear form. The associated return mapping equations then reduce to a remarkably simple format. In addition, due to assumed isotropy of the models, the associated algorithmic (incremental) constitutive functionals can be identified as particular members of a class of isotropic tensor functions of one tensor in which the function eigenvalues are expressed in terms of the eigenvalues of the argument. This observation leads to a simple closed form derivation of the consistent tangent moduli associated with the described integration algorithms. The extension of the present model to finite strains is carried out following standard multiplicative plasticity described in terms of logarithmic stretches and exponential approximation to the flow rule. The efficiency and robustness of the computational model are illustrated on a range of numerical examples.

The Representation Theorem for Linear, Isotropic Tensor Functions in Even Dimensions

In this note we give a proof of the representation theorem for linear, isotropic, tensor functions, which only assumes invariance under proper orthogonal tensors.

The Representation Theorem for Linear, Isotropic Tensor Functions Revisited

Two recent notes [1–2] express ideas that can be combined in order to get an elementary proof of the representation theorem of the title, a proof stressing the geometrical action of 4-tensors in subspaces. Thus we examine direct consequences, for linear maps C of Lin into Lin, of the notion of invariance (we recall this concept and we recall other standard notations in the first section). Suppose we classify the invariant (under Orth) subspaces of Lin, that an invariant linear C maps invariant subspaces into invariant subspaces and that Lin = Skw ⊕ Sph ⊕ Dev expresses the unique decomposition of Lin into a direct sum of three-, one- and five-dimensional invariant subspaces. Then, if we assume that C is also invertible, the structure of C comes from the knowledge of the invariant linear maps in Skw and Dev. As they must be, in each case, multiples of the identity, CE=C(E_3+E_1+ E_5)=λ_3E_3 +λ_1E_1+λ_5E_5 for (nonzero) constants λ_3, λ_1 and λ_5. The noninvertible case results from considering C+λI.

A constitutive frame of elastoplasticity at large strains based on the notion of a plastic metric

The article presents a constitutive framework of large-strain elastoplasticity in both the Lagrangian and the Eulerian geometric setting which takes into account anisotropic material response. In summary, the key ingredients of this framework are: (i) the introduction of a plastic metric which is assumed to describe locally the history-dependent inelastic material response in the sense of an internal variable formulation. (ii) The definition of a convex elastic domain in the space of the local stress-like variable conjugate to the plastic metric, denoted as the plastic force. (iii) An equivalent Lagrangian and Eulerian representation of all constitutive functions as isotropic tensor functions in terms of an extended set of arguments, denoted as anisotropy variables. (iv) The set-up of normality rules for the evolution of the plastic metric and the anisotropy variables, yielding a canonical symmetric form of the elastoplastic tangent moduli. (v) A geometrically exact decomposition of the set of constitutive equations into possibly decoupled volumetric and isochoric contributions. Applications of the constitutive framework are demonstrated by means of several conceptual model problems which cover isotropic, initial anisotropic and induced anisotropic elastoplastic response.

Einige Bemerkungen zur Kettenregel bei Zeitableitungen isotroper Tensorfunktionen

In this paper the restrictions of the so-called chain rule on objective time-derivatives are examined. A generalization of the chain rule on scalar valued isotropic tensor functions with more than one tensorial arguments (second order tensors) is also introduced. The analysis shows that there are different solutions associated with the chain rule. Not all of them have a physical significance. With respect to the physical reasonable solutions and a so-called invariance condition the chain rule gives an additional restriction if different objective time-derivatives for different arguments in scalar isotropic tensor functions are used. In such a situation the antimetric part of the tensor of convective velocity has to be the same in each objective time-derivative.

Comparison of two algorithms for the computation of fourth-order isotropic tensor functions

This paper compares two possible representations of a certain class of fourth-order isotropic tensor functions. It focusses on the derivatives of symmetric second-order isotropic tensor functions by a symmetric second-order tensor argument. This tensor argument is assumed to be given in spectral form. The first representation explicitly needs the eigenvectors of the argument tensor and is well-known in the literature. The second representation proposed here is based on the knowledge of second-order eigenvalue bases and avoids the computation of eigenvectors. The evaluation of a typical model problem shows that the second representation needs less computer time than the first one. The representations discussed here are of high importance for numerical solvers of nonlinear initial-boundary-value problems in large-strain elasticity and elastoplasticity.

Representation Theorems for Hemitropic and Tranversely Isotropic Tensor Functions

A representation theorem for transversely isotropic tensor-valued functions of a symmetric tensor variable is proved. The theorem holds in any finite dimension. The proof is based on the decomposition of a symmetric tensor of dimension N into a scalar, a vector, and a symmetric tensor of dimension N-1, and on the fact that the transverse isotropy of the original function is equivalent to the hemitropy of three functions, one scalar-valued, one vector-valued, and one tensor-valued, of the last two terms in the decomposition. Representation theorems for the three functions are obtained as generalizations of two theorems of W. Noll on isotropic functions. The proofs make use of an appropriate algebraic structure based on alternating forms. The three-dimensional case, as well as those of linear and of hyperelastic functions, are treated as special cases.

A unified invariant description of micromechanically-based effective elastic properties for two-dimensional damaged solids

This paper is devoted to an unified invariant description of the effective elastic compliance, say S, of a two-dimensional (2D) damaged solid with an isotropic matrix. A macro- and micromechanically combined approach is proposed, which consists of three major steps. First, the internal variables, i.e., the damage tensors are correlated with the microscopical geometry and distribution of the damage. It is shown that the damage due to microcracks and holes affect S through a symmetric second-order tensor, d, and an irreducible fourth-order tensor, D. Second, the general formulation of the effective compliance S as an isotropic tensor function of d and D is phenomenologically given by making use of several results in the theory of tensor function representation. The linearization and second-rank nonlinearization from the general formulation are derived, where one and six new parameters (named damage constants), respectively, are consistently introduced. It is shown that the linearization provides the most general invariant formulation for micromechanically-based non-interacting or dilute estimations of the effective compliance, and nonlinear models are accounted for the interacting effect of damage. Third, we relate these damage constants to shape and morphology factors of the damage via micromechanical analyses or computer numerical experiments for problems with simple and regular damage. The above-mentioned approach allows invariant consistent models of effective elastic compliance which are not only capable of describing complex and irregular damage, but also indicate in detail the role of the real damage. Thus, these invariant models may further constitute a solid basis of predicating damage evolution. As an application of the proposed approach, the most general second-rank interacting model for damage due to microcracks is established, which is available for any distribution of microcracks.

Two General Representation Theorems for Arbitrary‐Order‐Tensor‐Valued Isotropic and Anisotropic Tensor Functions of Vectors and Second Order Tensors

AbstractIn this paper, it is shown that irreducible representations for r‐th‐order‐tensor‐valued isotropic or anisotropic tensor functions depending on an arbitrary number of vectors and second order tensors as variables may be formed by the union of irreducible representations for certain r‐th‐order‐tensor‐valued isotropic or anisotropic tensor functions merely depending on not more than three variables (for r ≧ 1) or four variables (for r = 0) and hence that the representation problem for the former may be reduced to that for the latter.

Differentiability of the scalar coefficients in two representation formulae for isotropic tensor functions in two dimensions

Differentiability of the scalar coefficients in two representation formulae for isotropic tensor functions in two dimensions

On the representation theorem for linear, isotropic tensor functions

On the representation theorem for linear, isotropic tensor functions

On the Tensor Function Representations of 2nd‐Order and 4th‐Order Tensors. Part I

AbstractIn this paper, we establish the complete representations for isotropic scalar‐, 2nd‐order tensor‐ and 4th‐order tensor‐valued functions of any finite number of 2nd‐ and 4th‐order tensors in both 2‐ and 3‐dimensional spaces, and prove the irreducibility of these representations. The fourth‐order tensors under consideration are those regarded as skew‐symmetric linear transformations of both symmetric and skew‐symmetric 2nd‐order tensors in 3‐dimensional space, symmetric linear transformations of skew‐symmetric 2nd‐order tensors in 3‐dimensional space, and both symmetric and skew‐symmetric linear transformations of both symmetric and skew‐symmetric 2nd‐order tensors in 2‐dimensional space. Complete and irreducible isotropic tensor function representations in 3‐dimensional space involving 4th‐order tensors as symmetric linear transformations of symmetric 2nd‐order tensors are derived in a continued paper (Part II). These representations allow general forms of constitutive laws involving 4th‐order tensor agencies (damage tensors, internal variables) of isotropic materials to be developed.

Theory of Representations for Tensor Functions—A Unified Invariant Approach to Constitutive Equations

Representations in complete and irreducible forms for tensor functions allow general consistent invariant forms of the nonlinear constitutive equations and specify the number and type of the scalar variables involved. They have proved to be even more pertinent in attempts to model mechanical behavior of anisotropic materials, since here invariant conditions predominate and the number and type of independent scalar variables cannot be found by simple arguments. In the last few years, the theory of representations for tensor functions has been well established, including three fundamental principles, a number of essential theorems and a large amount of complete and irreducible representations for both isotropic and anisotropic tensor functions in three- and two-dimensional physical spaces. The objective of the present monograph is to summarize and recapitulate the up-to-date developments and results in the theory of representations for tensor functions for the convenience of further applications in contemporary applied mechanics. Some general topics on unified invariant formulation of constitutive laws are investigated.

Remarks on the continuity of the scalar coefficients in the representation H(A) = αI + βA + γA2 for isotropic tensor functions

Remarks on the continuity of the scalar coefficients in the representation H(A) = αI + βA + γA2 for isotropic tensor functions

Computation of isotropic tensor functions

AbstractAn explicit formulation and a procedure for the computation of isotropic tensor‐valued tensor functions is discussed. The formulation is based on a spectral decomposition in terms of second‐order eigenvalue bases, which avoids the costly computation of eigenvectors. As an important result a compact structure of the fourth‐order derivatives of general second‐order isotropic tensor functions is presented.

Numerically well-conditioned expressions for isotropic tensor functions

Abstract A numerically well-conditioned expression for an isotropic tensor function of a symmetric second order tensor is derived in the cases of two and three dimensions, and the result is also stated for n dimensions. The derivation starts with the expression of the tensor function in its spectral representation in terms of projection operators onto eigenspaces of the tensor. The spectral representation, however, is ill-conditioned when any eigenvalue is too close to any differing eigenvalue. A rearrangement and recombination of terms gives the desired result, which is a well-conditioned expression.

Conjugate stress and tensor equation [formula omitted

In this paper we seek an explicit tensorial expression which, for each integer m ≠ 0, gives the stress T(m) conjugate to the strain E(m) in the Seth-Hill class. The preceding problem leads us to find an intrinsic (i.e. coordinate-free) representation of the solution of the tensor equation displayed in the title. We obtain the requisite representation by using Hill's principal axis method and the representation theorem of isotropic tensor functions. We illustrate the general procedure to obtain an explicit lensorial expression for T(m) by working out the instance m = −3 in detail.