Work-conjugate Stress Research Articles

Laplace stretch: Eulerian and Lagrangian formulations

Two triangular factorizations of the deformation gradient tensor are studied. The first, termed the Lagrangian formulation, consists of an upper-triangular stretch premultiplied by a rotation tensor. The second, termed the Eulerian formulation, consists of a lower-triangular stretch postmultiplied by a different rotation tensor. The corresponding stretch tensors are denoted as the Lagrangian and Eulerian Laplace stretches, respectively. Kinematics (with physical interpretations) and work-conjugate stress measures are analyzed and compared for each formulation. While the Lagrangian formulation has been used in prior work for constitutive modeling of anisotropic and hyperelastic materials, the Eulerian formulation, which may be advantageous for modeling isotropic solids and fluids with no physically identifiable reference configuration, does not seem to have been used elsewhere in a continuum mechanical setting for the purpose of constitutive development, though it has been introduced before in a purely kinematic setting.

Work conjugate stress and strain variables for unsaturated frozen soils

In general, most of the foundations of structures in the cold regions rest on frozen soils which are often under unsaturated state and subject to mechanical and thermal loads. The thermal loads will induce water migration and water-ice phase change, as a result, the dynamic phase transition between unfrozen water and ice occurs and degree of saturation in frozen soils is affected. These complicate the understanding of the strength and deformation characteristics of frozen soils under mechanical and thermal loads. Furthermore, it leads to a necessity to develop a coupling thermo-hydro-mechanical model for practical permafrost engineering. Up to date, there are limited studies to quantify this complicated problem by a framework with proper stress and strain variables for unsaturated frozen soils. This study derives the expression of input work rate for unsaturated frozen soils based on the multiphase porous media theory. Work conjugate stress and strain variables have been determined based on the expression of input work rate. This set of variables not only can smoothly transit from unsaturated frozen state to unsaturated unfrozen state when temperature is above freezing point, but also can smoothly transit from unsaturated state to saturate state. The proposed framework can be consistently applied to multi-phase soils with unsaturated frozen soils, unsaturated soils and saturated soils. Moreover, the application of the framework can be further extended to other porous mediums. Additionally, the selected variables in this framework are validated against published experimental results. This framework can subsequently be used to establish the coupled thermo-hydro-mechanical models for unsaturated and saturated frozen soils.

Objective Symmetrically Physical Strain Tensors, Conjugate Stress Tensors, and Hill’s Linear Isotropic Hyperelastic Material Models

We introduce a new family of strain tensors—a family of symmetrically physical (SP) strain tensors—which is also a subfamily of the well-known Hill family of strain tensors. For the further analysis, five scale functions are chosen which generate strain tensors belonging to the families of strain tensors previously introduced by other authors (i.e., the Doyle–Ericksen, Curnier–Rakotomanana, Curnier–Zysset, Itskov, and Darijani–Naghdabadi families) and to the new family of SP strain tensors. In particular, these five scale functions include the scale function generating the Lagrangian and Eulerian Hencky strain tensors. We introduce the family of SPH models of isotropic hyperelastic materials (with Hill’s linear relations) which are generated by SP strain tensors and work-conjugate stress tensors based on Hill’s natural generalization of Hooke’s law. Five SPH models of isotropic hyperelastic materials are generated on the basis of chosen SP strain tensors and work-conjugate stress tensors. These models are tested by solving two problems with homogeneous strain and stress tensors fields: the simple elongation and simple shear problems. Analysis of these solutions shows that the solutions of both problems for the Hencky isotropic hyperelastic material model (one of the five generated SPH models of isotropic hyperelastic materials) are qualitatively different from the solutions for the remaining four material models. That is, the solutions using the Hencky isotropic hyperelastic material model are of yielding nature typical of inelastic deformation of metals whereas the solutions for the other four material models reproduce strain diagrams typical of rubber.

Stress and strain mapping tensors and general work-conjugacy in large strain continuum mechanics

In this paper we show that mapping tensors may be constructed to transform any arbitrary strain measure in any other strain measure. We present the mapping tensors for many usual strain measures in the Seth–Hill family and also for general, user-defined ones. These mapping tensors may also be used to transform their work-conjugate stress measures. These transformations are merely geometric transformations obtained from the deformation gradient and, hence, are valid regardless of any constitutive equation employed for the solid. Then, advantage of this fact may be taken in order to simplify the form of constitutive equations and their numerical implementation and thereafter, perform the proper geometric mappings to convert the results –stresses, strains and constitutive tangents– to usually employed measures and to user-selectable ones for input and output. We herein provide the necessary transformations. Examples are the transformation of small strains formulations and algorithms to large deformations using logarithmic strains.

A constitutive model for unsaturated soil–structure interfaces

SummaryA constitutive model for simulation of the behavior of unsaturated interfaces is presented here. The model is an extension of an existing critical state compatible interface model for dry and saturated interfaces that was already proposed by one of the authors [Lashkari, A. 2013. Int. J. Numer. Anal. Meth. Geomech. 37(8): 904–931]. For a proper simulation of the behavior of partially saturated interfaces, the extended model is formulated in terms of two pairs of work conjugate stress–strain‐like variables. The modified model simulations are compared with the existing data of dry, unsaturated, and saturated interfaces. For each interface type, it is shown that the proposed model can capture the essential elements of the behavior using a unique set of parameters. Copyright © 2015 John Wiley & Sons, Ltd.

Logarithmic strain and its material derivative for a QR decomposition of the deformation gradient

The well-known polar decomposition of the deformation gradient decomposes it into an orthogonal rotation and a symmetric stretch. We consider a Gram–Schmidt factorization of the deformation gradient, which decomposes it into a different orthogonal rotation with a right-triangular field that we call distortion. Properties of this distortion tensor are discussed, and a work-conjugate stress tensor is derived for this Lagrangian frame. The logarithm of distortion and its material derivative are then introduced, and their components are quantified, resulting in a new logarithmic measure for strain and its rate, distinct from Hencky strain (the logarithm of stretch) and its rate. An eigenprojection analysis and a first-order, differential, matrix equation solved using the technique of back substitution both produce the same matrix components describing the logarithm of distortion. Three homogeneous deformations illustrate similarities and differences between the logarithms of distortion and stretch. They are distinct measures of strain. The new logarithmic strain measure shows monotonic behavior under simple shear as opposed to the non-monotonic behavior of Hencky strain (see Fig. 2).

Integrating a logarithmic-strain based hyperelastic formulation into a three-field mixed finite element formulation to deal with incompressibility in finite-strain elastoplasticity

This paper deals with the treatment of incompressibility in solid mechanics in finite-strain elastoplasticity. A finite-strain model proposed by Miehe, Apel and Lambrecht, which is based on a logarithmic strain measure and its work-conjugate stress tensor is chosen. Its main interest is that it allows for the adoption of standard constitutive models established in a small-strain framework. This model is extended to take into account the plastic incompressibility constraint intrinsically. In that purpose, an extension of this model to a three-field mixed finite element formulation is proposed, involving displacements, a strain variable and pressure as nodal variables with respect to standard finite element. Numerical examples of finite-strain problems are presented to assess the performance of the formulation. To conclude, an industrial case for which the classical under-integrated elements fail is considered.

On the operating critical friction ratio in general stress states

Stress dilatancy provides a fundamental flow rule for constitutive models and several variations on the framework have similar operational performance. However, regardless of idealisation adopted for stress dilatancy, the scaling coefficient evolves and only attains the critical friction ratio at large strain. The scaling coefficient also depends on the proportion of intermediate principal stress. Data on stress dilatancy in triaxial compression and plane strain are reviewed. Based on these data, a simple function is proposed for stress-dilatancy scaling using an operational friction ratio Mi based on work-conjugate stress and strain invariants. The function is calibrated to soil properties measured in conventional drained triaxial tests (about three, suitably chosen, tests are needed in addition to those used to define the critical state locus) and predicts strength in plane strain to an accuracy of ∆η = ±0·03 at 80% confidence for the Brasted and Changi sands considered (i.e. about ± 3% of typical peak strength). The proposed function is suitable for any soil constitutive model that includes a critical state line and which invokes stress dilatancy.

A mathematical basis for strain-gradient plasticity theory. Part II: Tensorial plastic multiplier

A phenomenological, flow theory version of gradient plasticity for isotropic and anisotropic solids is constructed along the lines of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain-gradient plasticity. J. Mech. Phys. Solids 52, 1379–1406]. Both energetic and dissipative stresses are considered in order to develop a kinematic hardening theory, which in the absence of gradient terms reduces to conventional J 2 flow theory with kinematic hardening. The dissipative stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition with associated plastic flow. The theory includes interfacial terms: elastic energy is stored and plastic work is dissipated at internal interfaces, and a yield surface is postulated for the work-conjugate stress quantities at the interface. Uniqueness and extremum principles are constructed for the solution of boundary value problems, for both the rate-dependent and the rate-independent cases. In the absence of strain gradient and interface effects, the minimum principles reduce to the classical extremum principles for a kinematically hardening elasto-plastic solid. A rigid-hardening version of the theory is also stated and the resulting theory gives rise to an extension to the classical limit load theorems. This has particular appeal as previous trial fields for limit load analysis can be used to generate immediately size-dependent bounds on limit loads.

A mathematical basis for strain-gradient plasticity theory—Part I: Scalar plastic multiplier

Strain-gradient plasticity theories are reviewed in which some measure of the plastic strain rate is treated as an independent kinematic variable. Dislocation arguments are invoked in order to provide a physical basis for the hardening at interfaces. A phenomenological, flow theory version of gradient plasticity is constructed in which stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition. Plastic work is also done at internal interfaces and a yield surface is postulated for the work-conjugate stress quantities at the interface. Thereby, the theory has the potential to account for grain size effects in polycrystals. Both the bulk and interfacial stresses are taken to be dissipative in nature and due attention is paid to ensure that positive plastic work is done. It is shown that the mathematical structure of the elasto-plastic strain-gradient theory has similarities to conventional rigid-plasticity theory. Uniqueness and extremum principles are constructed for the solution of boundary value problems.

On the incremental plastic work and related aspects of invariance – Part I

Taking for granted that the free energy function is invariant under a change of a finite strain measure and/or the reference configuration, Hill's transformation rules for selected fundamental constitutive quantities (such as tangent elastic modulus, plastic increments of total strain and work conjugate stress, the work of work-conjugate stress, the work expended in the plastic part of incremental strain etc.) are derived in a manner different from that of Hill. On this background distinguished by Hill [6] subtle aspects of invariance in mechanics of elastic plastic solids are discussed. It is shown that the plastic part of the increment of elastic strain energy (when taken with reverse sign) defines the true invariant incremental plastic work which in general is not equal to the work expended in the plastic part of the strain increment. It plays the role of a potential for the plastic part of the increment of work-conjugate stress. This fundamental fact has not found proper account in the literature. The analytical interrelations between two apparently different theoretical frameworks, Hill-Rice (fixed reference configuration) and Eckart-Mandel (mobile unloaded configuration) are discussed showing their equivalence. Since the transformation rules are complex in the general 3D case, the first part of the paper illustrates instructively the discussed aspects in a 1D situation (simple tension or simple extension).

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part II – the formulation of elastoplastic coupling at large strain

The two key phenomena occurring in the process of ceramic powder compaction are the progressive gain in cohesion and the increase of elastic stiffness, both related to the development of plastic deformation. The latter effect is an example of ‘elastoplastic coupling’, in which the plastic flow affects the elastic properties of the material, and has been so far considered only within the framework of small strain assumption (mainly to describe elastic degradation in rock-like materials), so that it remains completely unexplored for large strain. Therefore, a new finite strain generalization of elastoplastic coupling theory is given to describe the mechanical behaviour of materials evolving from a granular to a dense state. The correct account of elastoplastic coupling and of the specific characteristics of materials evolving from a loose to a dense state (for instance, nonlinear – or linear – dependence of the elastic part of the deformation on the forming pressure in the granular – or dense – state) makes the use of existing large strain formulations awkward, if even possible. Therefore, first, we have resorted to a very general setting allowing general transformations between work-conjugate stress and strain measures; second, we have introduced the multiplicative decomposition of the deformation gradient and, third, employing isotropy and hyperelasticity of elastic response, we have obtained a relation between the Biot stress and its ‘total’ and ‘plastic’ work-conjugate strain measure. This is a key result, since it allows an immediate achievement of the rate elastoplastic constitutive equations. Knowing the general form of these equations, all the specific laws governing the behaviour of ceramic powders are finally introduced as generalizations of the small strain counterparts given in Part I of this paper.

Hencky's elasticity model and linear stress-strain relations in isotropic finite hyperelasticity

Hencky's elasticity model is an isotropic finite elasticity model assuming a linear relation between the Kirchhoff stress tensor and the Hencky or logarithmic strain tensor. It is a direct generalization of the classical Hooke's law for isotropic infinitesimal elasticity by replacing the Cauchy stress tensor and the infinitesmal strain tensor with the foregoing stress and strain tensors. A simple, straightforward proof is presented to show that Hencky's elasticity model is exactly a hyperelasticity model, derivable from a quadratic potential function of the Hencky strain tensor. Generally, Hill's isotropic linear hyperelastic relation between any given Doyle-Ericksen or Seth-Hill strain tensor and its work-conjugate stress tensor is studied. A straightforward, explicit expression of this general relation is derived in terms of the Kirchhoff stress and left Cauchy-Green strain tensors. Certain remarkable properties of Hencky's model are indicated from both theorectical and experimental points of view.

A framework for large strain elastic–plastic damage mechanics based on metric transformations

A continuum elastic–plastic damage model employing irreversible thermodynamics and internal state variables is presented. The approach is based on a kinematic description using multiplicative decomposition of the metric transformation tensor into elastic and damage-plastic parts. Furthermore, undamaged configurations are introduced which are related to the damaged ones via associated metric transformations which allow for the interpretation as damage tensors. Thus, the damage tensor is explicitly characterized in terms of a kinematic measure of damage. This leads to the definition of appropriate logarithmic strain measures. Strain rates are shown to be additively decomposed into elastic, plastic and damage strain rate tensors. Moreover, based on the standard dissipative material approach and a generalized effective stress concept, work-conjugate stress tensors are introduced. Elastic and plastic constitutive equations are formulated in an effective stress space. A generalized macroscopic yield condition is employed and the evolution of the effective plastic part of the strain rate tensor is determined via a non-associated flow rule. Considering the damaged configurations a generalized damage criterion is formulated using stress components referred to the elastically unloaded (stress free) damaged configuration. The evolution of the damage part of the strain rate tensor is discussed in some detail. It is based on a damage potential function and takes into account isotropic as well as anisotropic effects.

A nonlinear generalization of the Koiter–Sanders–Budiansky bending strain measure

A physically motivated kinematical property to be demanded of a bending strain measure, beyond the ones that are conventionally agreed upon, is proposed. Guided by the criterion, a bending strain measure is proposed for a first-order, nonlinear, elastic shell theory that has the property of vanishing in rigid and pure-stretch deformations of the shell. The measure is motivated by the Koiter–Sanders–Budiansky (KSB) bending measure of the first-order, linear shell theory, and is shown to linearize to the KSB measure about the undeformed shell. On retaining the standard measure for membrane straining, work-conjugate stress and stress-couple resultants to the membrane and bending strains are derived, as is the expression for the internal virtual work in terms of the proposed strains, stress and stress-couple resultants. The bending strain is calculated for some simple deformations and compared with other generalizations of the KSB measure of the linear theory.

Strictly conjugate stress and strain tensors

a subclass of strictly conjugate tensors, namely, the tensors that satisfy the requirement for transformation by the same law upon rigid motion of the neighborhood of a material particle, is separated into the class of work-conjugate stress and strain tensors. The advantage of the use of strictly conjugate stress and strain tensors in formulating the variational principles for bodies from a hyperelastic material is shown.

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.

Theory and numerics of thin elastic shells with finite rotations

A bending theory for thin shells undergoing finite rotations is presented, and its associated finite element model is described. The kinematic assumption is based on a Reissner-Mindlin theory. The starting point for the derivation of the strain measures is the polar decomposition of the material deformation gradient. The work-conjugate stress resultants and stress couples are integrals of the Biot stress tensor. This tensor is invariant with respect to rigid body motions and therefore appropriate for the formulation of constitutive equations. The rotations are described by using Eulerian angles. The finite element discretization of arbitrary shells is performed using isoparametric elements. The advantage of the proposed shell formulation and its numerical model is shown by application to different non-linear plate and shell problems. Finite rotations can be calculated within one load increment. Thus the step size of the load increment is only imited by the local convergence behaviour of Newton's method or the appearance of stability phenomena.