Conjugate Stress Tensor Research Articles

Basis-free expressions for families of objective strain tensors, their rates, and conjugate stress tensors

The objective (Lagrangian and Eulerian) strain tensors, their rates, and conjugate stress tensors used in continuum mechanics equations are considered. These tensors are represented in the eigenprojections of the right and left stretch tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . The novelty of the research lies in the simultaneous derivation of expressions for Lagrangian versions of tensors and their Eulerian counterparts with decomposition of the obtained expressions into components coaxial and orthogonal to the tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . We consider the Lagrangian and Eulerian Doyle–Ericksen families of strain tensors, which, in turn, are subfamilies of the well-known Hill family of strain tensors. Basis-free expressions are obtained for the material rates of Lagrangian tensors from the previously introduced family of generalized strain tensors, whose subfamilies are the Lagrangian and Eulerian Hill families of strain tensors, and similar expressions are obtained for the Green–Naghdi rates of Eulerian strain tensors, which are Eulerian counterparts of the material rates of Lagrangian strain tensors. Basis-free expressions for stress tensors are derived from the classical definitions of the conjugacy of stress and strain tensors. Expressions are found for the Lagrangian and Eulerian symmetric Atluri stress tensors that do not have conjugate strain tensors from the Hill family, and the obtained expressions are decomposed into components coaxial and orthogonal to the tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . The ranges of allowed ratios of different principal stretches at which the Lagrangian and Eulerian Hencky and Atluri stress tensors approximate the rotated/standard Kirchhoff stress tensors are determined. It is shown that the range of allowed ratios for the Hencky stress tensors is wider than that for the Atluri stress tensors.

Constitutive theories for thermoelastic solids in Lagrangian description using Gibbs potential

This paper presents constitutive theories for finite deformation of homogeneous, isotropic thermoelastic solids in Lagrangian description using Gibbs potential. Since conservation of mass, balance of momenta and the energy equation are independent of the constitution of the matter, the second law of thermodynamics, that is, entropy inequality, must form the basis for all constitutive theories of the deforming matter to ensure thermodynamic equilibrium during the evolution (Surana and Reddy in Continuum mechanics, 2012; Eringen in Nonlinear theory of continuous media. McGraw-Hill, New York, 1962). The entropy inequality expressed in terms of Helmholtz free energy is recast in terms of Gibbs potential. The conditions resulting from the entropy inequality expressed in terms of Gibbs potential permit the derivation of the constitutive theory for the strain tensor in terms of the conjugate stress tensor and the constitutive theory for the heat vector. In the work presented here, it is shown that using the conditions resulting from the entropy inequality, the constitutive theory for the strain tensor can be derived using three different approaches: (i) assuming the Gibbs potential to be a function of the invariants of the conjugate stress tensor and then using the conditions resulting from the entropy inequality, (ii) using theory of generators and invariants and (iii) expanding Gibbs potential in the conjugate stress tensor using Taylor series about a known configuration and then using the conditions resulting from the entropy inequality. The constitutive theories resulting from these three approaches are compared for equivalence between them as well as their merits and shortcomings. The constitutive theory for the heat vector can also be derived either directly using the conditions resulting from the entropy inequality or using the theory of generators and invariants. The derivation of the constitutive theory for the heat vector using the theory of generators and invariants with the complete set of argument tensors yields a more comprehensive constitutive theory for the heat vector. In the work, we consider both approaches. Summaries of the constitutive theories using parallel approaches (as described above) resulting from the entropy inequality expressed in terms of Helmholtz free-energy density are also presented and compared for equivalence with the constitutive theories derived using Gibbs potential.

Stresses, Singularities, and a Complementary Energy Principle for Large Strain Elasticity

Abstract This article reviews three aspects of large strain elasticity. First, various conjugate stress tensors to strain tensors are reviewed. Many researchers have studied the theory of large strain elasticity. Many stress tensors including the Cauchy stress tensor, the first and the second Piola–Kirchhoff stress tensor, and the Jaumann stress tensor have been proposed to describe the stress state at a point. Recently, the first author of this article proposed a concept of “base forces” to reveal the essence of stress state. By the concept of base forces, the description of the stress state becomes clearer than other stress tensors. We attempt to take base forces as a basic point of view to deal with a review in which different descriptions of stress state are discussed and compared. The governing equations and boundary conditions expressed by the base forces are given. Second, this article reviews the solution of some singularity problems for large strain elasticity, i.e., problems of stress singularity at a crack or a notch tip, at the point of application of a concentrated force and at the vertex of contact in rubberlike materials. Methods of getting the singularity index of stress by using base forces are introduced and compared to earlier work. Complementary energy principles for large strain elasticity have eluded researchers for nearly 100years. A review of some important advances in this is also given, and a new complementary energy principle related to base forces is introduced.

Formulation, validation and numerical procedures for Hencky’s elasticity model

Hencky’s elasticity model based on the logarithmic strain tensor is discussed. Continuum formulation makes use of the conjugate stress tensor derived by Hoger. Constitutive equations are cast in two settings: General Lagrangean and isotropic Eulerean forms. Recent experimental data for tension, compression, simple shear and torsion loadings with the inclusion of Poynting’s effect are employed to validate this constitutive model. Subsequently, the performance of numerical methods in the finite element method context is tested. The paper closes with numerical examples, involving 60% compression test, which exhibits barreling caused by friction.

Conjugate stresses of the Seth–Hill strain tensors

In this paper, conjugate stresses of the Seth–Hill strain measures are treated by means of the definition of energy conjugacy and Hill’s principal axis method. This approach results in relations between components of two different conjugate stress tensors. The results are valid for distinct as well as coalescent principal stretches. Illustrative examples are solved to demonstrate the simplicity and usefulness of the derived relations between conjugate stress tensors.

A hyperelastic‐based large strain elasto‐plastic constitutive formulation with combined isotropic‐kinematic hardening using the logarithmic stress and strain measures

AbstractThis paper addresses the formulation of a set of constitutive equations for finite deformation metal plasticity. The combined isotropic‐kinematic hardening model of the infinitesimal theory of plasticity is extended to the large strain range on the basis of three main assumptions: (i) the formulation is hyperelastic based, (ii) the stress‐strain law preserves the elastic constants of the infinitesimal theory but is written in terms of the Hencky strain tensor and its elastic work conjugate stress tensor, and (iii) the multiplicative decomposition of the deformation gradient is adopted. Since no stress rates are present, the formulation is, of course, numerically objective in the time integration. It is shown that the model gives adequate physical behaviour, and comparison is made with an equivalent constitutive model based on the additive decomposition of the strain tensor.

A new variational principle for finite elastic displacements

Variational principles for finite elastic displacements have been formulated in terms of Green strain and Kirchhoff-Trefftz stress tensors. The first is a functional of the displacement field only and implies stationarity of the total potential. The second is a canonical principle, in the sense of Friedrichs [1], involving both stresses and displacements and generalizing Reissner's principle [5]. In contrast with the geometrically linearized elasticity theory, it cannot be reduced to a complementary energy principle involving equilibrium stresses only [9,10]. The paper discusses the Levinson[9] and Zubov[11] formulation in terms of displacement gradients and the Piola stress tensor, which, however interesting from a theoretical viewpoint, does not appear suitable for practical applications. A new set of variational principles, of displacement, canonical or complementary energy types, is found to derive from the use of the polar decomposition theorem of the jacobian. It involves the engineering strain tensor and its conjugate stress tensor is to be regarded as a function of the Piola tensor and the material rotation. The complementary energy formulation is discussed in terms of first order stress functions. The presence of the rotational degrees of freedom opens the possibility of discretizing the rotational equilibrium equations in approximate solutions.