Isotropic Tensor Functions Research Articles

On Itskov (2024) counterexample to the functional basis of isotropic vector and tensor functions by Shariff (2023)

On Itskov (2024) counterexample to the functional basis of isotropic vector and tensor functions by Shariff (2023)

On the functional basis of isotropic vector and tensor functions by Shariff (2023)

In the paper Shariff (Q. J. Mech. Appl. Math. 76:143–161, 2023) a functional basis of a system of vectors and symmetric tensors is proposed. The functional basis is expressed in terms of eigenvalues and eigenvectors of the first tensor and includes a smaller number of terms in comparison to the classical irreducible representation (see, e.g., Boehler, J. Appl. Math. Mech. 57:323–327, 1977; Pennisi and Trovato, Int. J. Eng. Sci. 25:1059–1065, 1987). In the present contribution, we show that elements of the functional basis by Shariff (Q. J. Mech. Appl. Math. 76:143–161, 2023) do not represent isotropic invariants of the vector and tensor arguments and cannot thus be referred to as the functional basis. To this end, a counterexample with two symmetric tensors is considered. Under an arbitrary orthogonal transformation the functional basis (Shariff, Q. J. Mech. Appl. Math. 76:143–161, 2023) of these two tensors should remain constant but it does change in contrast to the classical representation.

Addendum to “fourth-order tensor calculus operations and application to continuum mechanics”

The authors provided a novel explicit expression for the dot derivative of an isotropic tensor function in the work by Kellermann et al. An alternate proof is provided in this addendum without the power series assumption.

Fourth-order tensor calculus operations and application to continuum mechanics

This paper provides a presentation of differential calculus involving fourth-order tensors in consideration of the cyclic class-based tensor operations introduced in an earlier paper on fourth-order tensor algebraic operations. Three classes of cyclic second-order tensor operators, denoted by dot, cross, and star, with separate definitions for double contraction, quadruple contraction, and tensor products were recognized. The present paper examines the differentiation of second-order isotropic tensor functions with respect to a second-order tensor, taking into account the three classes of second-order tensor operators. Product rules, chain rules, and identities necessary to maintain consistency are defined. In combining the proposed calculus rules with the second-order tensor operators, a comprehensive set of proofs for the power derivatives is first presented. A closed-form solution for the second-order tensor derivative of an isotropic tensor function is also detailed. Some of the new solutions are an outcome of capabilities offered by new operators and identities provided in the literature. The paper is arranged so as to be a useful reference work with regard to the mathematics of fourth-order tensors in absolute notation, particularly for application in the field of continuum mechanics in engineering.

Isotropic Tensor Functions with Quasipolynomial Scalar Potential in Nonlinear Elasticity Theory

Isotropic Tensor Functions with Quasipolynomial Scalar Potential in Nonlinear Elasticity Theory

Conditions for the Division of the Deviator and Spherical Properties for Nonlinear Isotropic Tensor Functions

Conditions for the Division of the Deviator and Spherical Properties for Nonlinear Isotropic Tensor Functions

An elementary proof for the representation theorem of analytic isotropic tensor functions of a second-order tensor

Based on the Cayley-Hamilton theorem and fixed-point method, we provide an elementary proof for the representation theorem of analytic isotropic tensor functions of a second-order tensor in a three-dimensional (3D) inner-product space, which avoids introducing the generating function and Taylor series expansion. The proof is also extended to any finite-dimensional inner-product space.

A generalized strain energy function using fractional powers: Application to isotropy, transverse isotropy, orthotropy, and residual stress symmetry

In this paper, we propose a generalized strain energy density function based on invariants of stretch tensor with arbitrary exponents. We employ polynomial, logarithmic and exponential functions of these invariants to develop the strain energy functions. We also study characteristics and applications of the proposed model for isotropy, transverse isotropy, orthotropy with a special focus on initial/residual stress symmetry. The proposed invariants generalize the existing strain energy potentials constructed with invariants of Cauchy stretch. We construct generalized strain energy functions for initial stress problems using initial stress symmetries. We obtain objectivity of strain energy for initially stressed transversely isotropic solids to derive the invariants and study resulting symmetry. By extending Dunford–Taylor integration based approach and tensor diagonalization approach, we obtain stress through differentiation of anisotropic scalar invariants with respect to a tensor. These approaches are usually applicable for derivative of isotropic tensor functions with respect to tensors. Using initial stress compatibility, we derive constraint equations for material parameters by evaluating the limits of Cauchy stress in the reference configuration. In order to apply the proposed model for initial stress problems, we further investigate bending and unbending of hyperelastic structures. We study bending of a rectangle to a cylinder and unbending of a cylinder to a rectangle in presence of initial/residual stress and observe both magnifying and moderating effects of initial stress for stress distribution and flexural characteristics. Using experimental data we substantiate the proposed model for seat foam, bovine pericardium and rabbit skin which represent compressible isotropic and incompressible orthotropic materials. For demonstration, we use both polynomial and exponential functions of these invariants. Furthermore, we develop a nonlinear finite element computational model for thermo-hyperelastic structure where thermal stress represents the initial stress. We corroborate this stress–strain data with the proposed model for initial stress symmetry and observe nice agreements.

An Approach to Isotropic Tensor Functions and Their Derivatives Via Omega Matrix Calculus

In this work we show how to obtain a closed form expression of any isotropic tensor function $F\left (\boldsymbol{A}\right )$ and their associated derivatives with $\boldsymbol{A}$ a second order tensor in a finite dimensional space. Our approach is based on a recent work of the author (SIAM Rev. 62(1):264–280, 2020) extending the Omega operator calculus, originally devised by MacMahon to describe partitions of natural numbers, to the realm of matrix analysis, namely, the Omega Matrix Calculus (OMC). The OMC is conceptually simple and useful in practice. Indeed, we show that the Cayley-Hamilton theorem and an improvement for low-rank second order tensors due to Segercrantz (Am. Math. Mon. 99(1):42–44, 1992), the representation of isotropic tensor functions and their first derivative of Itskov (Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 459(2034):1449–1457, 2003), and Theorem 1.1 of Norris (Q. Appl. Math. 66(4):725–741, 2008) are all special cases of a general Omega expression introduced in this work.

A commutative-symmetrical multiplicative decomposition of left and right stretch tensors

Symmetric positive definite left or right stretch tensors are decomposed multiplicatively into Eulerian or Lagrangean triple tensor products of symmetrizing rotations in the middle between two symmetric positive definite partial stretches. The proper orthogonal rotation tensors in the middle are determined from the symmetry conditions of the whole triple tensor products. The substitutions of the symmetrizing rotation tensors yield two commutative-symmetrical partial-stretch tensor products, which are (isotropic tensor) functions of the partial-stretch tensors of either proper Eulerian type defined with respect to a present configuration or proper Lagrangean type defined with respect to a reference configuration and which are equal to the symmetric total stretch tensors. Commutative-symmetrical partial-stretch tensor products do not rely on intermediate (stress-free) configurations. The eigenbase vector orientations of their proper Eulerian or proper Lagrangean multiplicative-elastic stretch tensors are well-defined, which is essential in order to model constitutive equations properly. Finite material orthotropy can be modeled simultaneously for both constituents and without the interference of their deformation-induced anisotropies when the partial-stretch tensors of the Lagrangean commutative-symmetrical products are defined with respect to the same reference configuration of orthotropy. The commutative-symmetrical partial-stretch tensor products are applicable to the constitutive modeling of finite anisotropy, and they constitute a novel approach to the kinematics of multiplicatively coupled total and partial stretch tensors.

Dunford-Taylor Integral and the Isotropic Tensor Valued Functions Having the Commutative Property with their Tensor Argument

Isotropic tensor valued functions of tensor arguments play an important role in the formulation of the equations governing the behavior of solid materials in the field of continuum mechanics. When the tensor argument is non-symmetric, the complexity and the difficulty in dealing with the tensor functions are high. In this work, the issue of expressing an isotropic tensor valued tensor function of a non-symmetric tensor argument is attention by utilizing the Dunford-Taylor integral. An important subclass of the isotropic tensor functions is considered with the commutative property.

Families of continuous spin tensors and applications in continuum mechanics

A new method is proposed for generating families of continuous spin tensors associated with families of corotational rates of second-order tensors using isotropic tensor functions of the same tensor arguments and different forms of continuous antisymmetric scalar spin functions of scalar arguments. Tensor functions are represented in terms of eigenprojections of a symmetric tensor S, which is one of the arguments of these functions. Each member of the generated family is represented as the sum of some basic spin tensor associated with the basic corotational tensor rate and the above-mentioned tensor function, whose structure is matched to the structure of the tensor function required to construct the twirl tensor of the triad of orthonormal eigenvectors of the tensor S (but this twirl tensor itself does not belong to the family of continuous spin tensors). The developed method is used in continuum mechanics to generate two families of continuous spin tensors associated with two families of objective corotational rates: Lagrangian and Eulerian. In these families, isotropic tensor functions are constructed using Lagrangian and Eulerian tensor arguments of the kinematic type, respectively. It is shown that if the same scalar spin function is used in deriving tensor functions of Lagrangian and Eulerian tensor arguments, then the corotational tensor rates associated with the generated spin tensors are objective (Lagrangian and Eulerian) counterparts of each other. It is shown that the spin tensors associated with the classical Eulerian corotational tensor rates (Zaremba–Jaumann, Green–Naghdi, d-rate) and their Lagrangian counterparts (including material rate) belong to the generated families of continuous spin tensors. It is also shown that both of these families of continuous spin tensors are subfamilies of the families of material spin tensors derived by Xiao et al. (J Elast 52:1–41, 1998). It is noted that the twirl tensors of the Lagrangian and Eulerian triads associated with the Gurtin–Spear corotational rates of tensors belong to the families of material spin tensors but do not belong to the families of continuous spin tensors. The final section gives expressions of continuous spin tensors from families associated with the families of Lagrangian and Eulerian corotational tensor rates which are appropriate for applications.

On potential isotropic tensor functions of two tensor arguments in mechanics of solids

In solid mechanics, the phenomenological description of processes that occur at microand nanolevel means that new material parameters modeling some characteristics of the object structure are introduced in the mathematical model and, first of all, in the constitutive relations. These parameters can be either of scalar or tensor nature.

Perturbation of Constitutive Relations in Tensor Non-Linear Materials

Isotropic both vector and tensor functions in the most general case of scalar and tensor nonlinearity are investigated from a position of stability with respect to perturbations of material functions. These vector and tensor functions mean constitutive relations connecting stress tensor and strain rate tensor in continuum with complex rheology (advanced materials). The constitutive relations involve scalar material functions of invariants. Perturbations proportional to some low physical parameter α may be imposed on these material functions. They provoke as a consequence, a perturbation of tensor function itself. Both linear and quadratic in α components of such perturbations are obtained. In each approximation the closed system consisting of the motion equations and incompressibility condition is recorded.

Perturbations of flows of incompressible nonlinearly viscous and viscoplastic fluids caused by variations in material functions

Material functions are necessary element of the constitutive relations determining any model of continuum. These functions can be defined as a collection of objects from which the operator of constitutive relations can be reconstructed completely. The material functions are found in test experiments and show the differences between a given medium and other media in the framework of the same model [1]. The “test experiment theory” is an important part of modern experimental mechanics. Just as in any experiment, from determining the viscosity coefficient by using the rotational viscosimeters to constructing the yield surface by using machines combined loading, the material functions are determined with an unavoidable error. For example, experimenters know that, in experiments with arbitrary accuracy, the moduli of elasticity can only be measured with an unimprovable tolerance of about 7%. Starting already from [2], the investigators’ attention has been repeatedly drawn to the fact that it is necessary to take into account this tolerance in determining the material constants, functions, and functionals in problems of mechanics and especially in analyzing the stability of deformation processes. Mathematically, this means that problems of stability under perturbations of the initial data, external constantly acting forces, domain boundaries, etc. should be supplemented with the assumption that the material functions have unknown perturbations of a certain class [3]. The variations of material functions in the framework of the linearized stability theory were considered in [2, 4, 5]. In what follows, we study isotropic tensor functions in the most general case of scalar and tensor nonlinearity. These functions are assigned the meaning of constitutive relations between the stress and strain rate tensors in continuum. These constitutive relations contain scalar material functions of invariants on which, as follows from the above, some variations proportional to a small physical parameter α can be imposed. These variations imply perturbations of the tensor function itself. The components of such perturbations linear and quadratic in α are determined. In each of the approximations, we write out a closed system of equations consisting of the equations of motion (linear in the variables of the respective approximation) and the incompressibility condition. We analyze tensor-linear functions with arbitrary scalar rheology inmore detail. Materials with such constitutive relations include non-Newtonian viscous fluids and viscoplastic materials. Viscoplastic materials are characterized by the existence of rigidity zones, where the stress intensity is less than the yield strength. We derive equations for the boundaries of the rigidity zones in the perturbed motion, in particular, for the case in which the unperturbed medium is a viscous Newtonian fluid. Throughout the paper, index-free notation is used.

On the derivatives of a subclass of isotropic tensor functions of a nonsymmetric tensor

This paper studies a subclass of isotropic tensor-valued functions of a nonsymmetric tensor, which satisfy the commutative condition, and their derivatives. This subclass of tensor functions includes tensor power series, exponential tensor function, etc., and is more general than those investigated before. In the case of three distinct eigenvalues, the derivatives of these tensor functions are constructed by solving a tensor equation, which is acquired by differentiating the commutative condition. By taking limits, the results are extended to the cases of repeated eigenvalues.

Explicit formulations of tangent stiffness tensors for isotropic materials

AbstractA compact explicit expression for the tangent stiffness tensor is presented. Throughout the analysis, the formulation holds for general isotropic elastic materials and does not require solving eigenvector problems. On the theoretical side, a very simple solution of a tensor equation is obtained. Then the expressions for the derivatives of general symmetric isotropic tensor functions of a symmetric tensor are developed. On the computational side, particular attention is given to the consideration of the special case, Green elastic materials, in which the strain energy does not admit a closed‐form expression in terms of principal invariants. Finally, a simple formulation of the tangent stiffness tensor for Ogden material model is supplied. Copyright © 2006 John Wiley & Sons, Ltd.

Derivatives on the isotropic tensor functions

The derivative of the isotropic tensor function plays an important part in continuum mechanics and computational mechanics, and also it is still an opening problem. By means of a scalar response function f (Λ i I 1 , I 2 ) and solving a tensor equation, this problem is well studied. A compact explicit expression for the derivative of the isotropic tensor function is presented, which is valid for both distinct and repeated eigenvalue cases. Throughout the analysis, the formulation holds for general isotropic tensor functions without need to solve eigenvector problems or determine coefficients. On the theoretical side, a very simple solution of a tensor equation is obtained. As an application to continuum mechanics, a base-free expression for the Hill’s strain rate is given, which is more compact than the existent results. Finally, with an example we compute the derivative of an exponent tensor function. And the efficiency of the present formulations is demonstrated.

Aspects of the simulation of the ferroelectric hysteresis

AbstractIn this paper we present a thermodynamically consistent phenomenological model for an assumed transversely isotropic ferroelectric crystal. Such materials become spontaneously polarized within a certain range of temperature. On the macro scale a remanent polarization and remanent strains, due to the reorientation of the polarization vectors, are observed, if an electric field above the so‐called coercive field is applied. The goal of this work is to construct a thermodynamically consistent formulation of the electro‐mechanically coupled ceramic on a meso scale, that takes into account the orientation of the assumed transversely isotropic unit cell, see [4]. The anisotropic behavior is governed by isotropic tensor functions, which are formulated in terms of a finite set of invariants, see [3]. Considering a simple model problem we discuss the linearization of discrete weak forms, resulting from the electromechanical boundary value problem. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A thermodynamically consistent mesoscopic model for transversely isotropic ferroelectric ceramics in a coordinate-invariant setting

In this contribution we present a phenomenological mesoscopic thermodynamically consistent model for the description of switching processes in ferroelectric materials that is able to describe the fundamental electromechanical hysteresis effects. The main goal is to develop a representation using the set of independent variables, the strains and the electric field, in a coordinate-invariant setting. This formulation is particularly suitable for the treatment of a variety of complex boundary-value problems (BVP) with regard to the essential boundary conditions. Here we restrict ourselves to transversely isotropic solids. The anisotropic behavior is governed by isotropic tensor functions that depend on a finite set of invariants. Thus the material symmetry requirements are automatically fulfilled.