On strong ellipticity for isotropic hyperelastic materials based upon logarithmic strain

This paper discusses various constitutive restrictions on the strain energy function for an isotropic hyperelastic material derived from the condition of strong ellipticity. The strain energy function is assumed to be a function of a novel set of invariants of the Hencky (logarithmic or natural) strain tensor introduced by Criscione et al. (J. Mech. Phys. Solids 48 (2000) 2445). A key step in the analysis is the derivation of an expression for the Fréchet derivative of the Hencky strain with respect to the deformation gradient that is convenient for analyzing the quadratic form over the space of second order tensors central to establishing strong ellipticity. The theory is illustrated by applying the restrictions to a model for rubber proposed by Criscione et al. (J. Mech. Phys. Solids 48 (2000) 2445) It is shown that while that model can be made to violate strong ellipticity, it does so only for very large strains.     

Problems in determining the elastic strain energy function for rubber

Lightly crosslinked natural rubber can be stretched by 600% or more, and recovers almost completely. It is often regarded as a model highly elastic material and characterized by a strain energy function to describe its stress-strain behavior under various types of deformation. A number of such functions have been proposed; some of them appear in current finite element programs. They are usually validated by comparison with measured stress-strain relations by Treloar [7] [L.R.G. Treloar, Stress-strain data for vulcanized rubber under various types of deformation, Trans. Faraday Soc. 40 (1944) 59-70] and Jones and Treloar [15] [D.F. Jones, L.R.G. Treloar, The properties of rubber in pure homogeneous strain, J. Phys. D Appl. Phys. 8 (1975) 1285-1304]. But Treloar pointed out that the relations at high strains became markedly irreversible, and he did not assign a strain energy function for strains greater than about 300%. Rivlin's universal relation between torsional stiffness and tensile stress [14] [R.S. Rivlin, Large elastic deformations of isotropic materials. Part V1: further results in the theory of torsion, shear and flexure, Philos. Trans. R. Soc. A 243 (1949) 251-288] is applied here to show that a typical elastic solid cannot be described by any strain energy function at strains greater than about 300%. Elastic strain energy functions for higher strains, or for other rubbery materials, are thus of doubtful value unless evidence for reversibility of stress-strain relations is adduced or the applicability of a strain energy function is demonstrated.     

A comparison of limited-stretch models of rubber elasticity

In this paper we describe various limited-stretch models of non-linear rubber elasticity, each dependent on only the first invariant of the left Cauchy–Green strain tensor and having only two independent material constants. The models are described as limited-stretch, or restricted elastic, because the strain energy and stress response become infinite at a finite value of the first invariant. These models describe well the limited stretch of the polymer chains of which rubber is composed. We discuss Gent׳s model which is the simplest limited-stretch model and agrees well with experiment. Various statistical models are then described: the one-chain, three-chain, four-chain and Arruda–Boyce eight-chain models, all of which involve the inverse Langevin function. A numerical comparison between the three-chain and eight-chain models is provided. Next, we compare various models which involve approximations to the inverse Langevin function with the exact inverse Langevin function of the eight-chain model. A new approximate model is proposed that is as simple as Cohen׳s original model but significantly more accurate. We show that effectively the eight-chain model may be regarded as a linear combination of the neo-Hookean and Gent models. Treloar׳s model is shown to have about half the percentage error of our new model but it is much more complicated. For completeness a modified Treloar model is introduced but this is only slightly more accurate than Treloar׳s original model. For the deformations of uniaxial tension, biaxial tension, pure shear and simple shear we compare the accuracy of these models, and that of Puso, with the eight-chain model by means of graphs and a table. Our approximations compare extremely well with models frequently used and described in the literature, having the smallest mean percentage error over most of the range of the argument.     

A comparison between crystal and isotropic strain gradient plasticity theories with accent on the role of the plastic spin

In the small deformation range, we consider crystal and isotropic “higher-order” theories of strain gradient plasticity, in which two different types of size effects are accounted for: (i) that dissipative, entering the model through the definition of an effective measure of plastic deformation peculiar of the isotropic hardening function and (ii) that energetic, included by defining the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., [Gurtin, M.E., 2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]). In order to compare the two modellings, we recast both of them into a unified deformation theory framework and apply them to a simple boundary value problem for which we can exploit the Γ-convergence results of [Bardella, L., Giacomini, A., 2008. Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity. J. Mech. Phys. Solids 56 (9), 2906–2934], in which the crystal model is made isotropic by imposing that any direction be a possible slip system. We show that the isotropic modelling can satisfactorily approximate the behaviour described by the isotropic limit obtained from the crystal modelling if the former constitutively involves the plastic spin, as in the theory put forward in Section 12 of [Gurtin, M.E., 2004. A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J. Mech. Phys. Solids 52, 2545–2568]. The analysis suggests a criterium for choosing the material parameter governing the plastic spin dependence into the relevant Gurtin model.     

A multiscale (visco)-hyperelastic modelling for particulate composites

The aim of this work is to pursue, in the wake of the paper [Martin, C., Dragon, A., Trumel, H., 1999. Mechanics Research Communications 26, 327], a non-classical micromechanical study and scale transition for highly filled particulate composites with a viscoelastic matrix. The present extension of a morphologically-based approach due to Christoffersen [Christoffersen, J., 1983. J. Mech. Phys. Solids 31, 55] carried forward to viscoelastic small strain context by Martin et al. [Martin, C., Dragon, A., Trumel, H., 1999. Mechanics Research Communications 26, 327], Nadot-Martin et al. [Nadot-Martin, C., Trumel, H., Dragon, A., 2003. Eur. J. Mech. A/Solids 22, 89], consists in introducing large strain (visco)-hyperelastic behaviour of the constituents (notably the matrix). The form of a local problem is analytically stated for compressive constituents. Numerical simulation for simplified hyperelastic behaviour and regular microstructure, employing different grain/matrix contrast parameters, is discussed in order to illustrate salient features of the advanced approach.     

Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity

In the context of single-crystal strain gradient plasticity, we focus on the simple shear of a constrained strip in order to study the effects of the material parameters possibly involved in the modelling. The model consists of a deformation theory suggested and left undeveloped by Bardella [(2007). Some remarks on the strain gradient crystal plasticity modelling, with particular reference to the material length scales involved. Int. J. Plasticity 23, 296–322] in which, for each glide, three dissipative length scales are considered; they enter the model through the definition of an effective slip which brings into the isotropic hardening function the relevant plastic strain gradients, averaged by means of a p-norm. By means of the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]), the model further involves an energetic material length scale. The application suggests that two dissipative length scales may be enough to qualitatively describe the size effect of metals at the microscale, and they are chosen in such a way that the higher-order state variables of the model be the dislocation densities. Moreover, we show that, depending on the crystallography, the size effect governed by the defect energy may be different from what expected (based on the findings of [Bardella, L., 2006. A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 54, 128–160] and [Gurtin et al. 2007. Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55, 1853–1878]), leading mostly to some strengthening. In order to investigate the model capability, we also exploit a Γ-convergence technique to find closed-form solutions in the “isotropic limit”. Finally, we analytically show that in the “perfect plasticity” case, should the dissipative length scales be set to zero, the presence of the sole energetic length scale may lead, as in standard plasticity, to non-uniqueness of solutions.     

Modeling of polycrystals using a gradient crystal plasticity theory that includes dissipative micro-stresses

This study investigates thermodynamically consistent dissipative hardening in gradient crystal plasticity in a large-deformation context. A viscoplastic model which accounts for constitutive dependence on the slip, the slip gradient as well as the slip rate gradient is presented. The model is an extension of that due to Gurtin (Gurtin, M. E., J. Mech. Phys. Solids, 52 (2004) 2545–2568 and Gurtin, M. E., J. Mech. Phys. Solids, 56 (2008) 640–662)), and is guided by the viscoplastic model and algorithm of Ekh et al. (Ekh, M., Grymer, M., Runesson, K. and Svedberg, T., Int. J. Numer. Meths Engng, 72 (2007) 197–220) whose governing equations are equivalent to those of Gurtin for the purely energetic case. In contrast to the Gurtin formulation and in line with that due to Ekh et al., viscoplasticity in the present model is accounted for through a Perzyna-type regularization. The resulting theory includes three different types of hardening: standard isotropic hardening is incorporated as well as energetic hardening driven by the slip gradient. In addition, as a third type, dissipative hardening associated with plastic strain rate gradients is included. Numerical computations are carried out and discussed for the large strain, viscoplastic model with non-zero dissipative backstress.     

Second-order estimate of the macroscopic behavior of periodic hyperelastic composites: theory and experimental validation

This paper deals with some theoretical and experimental aspects of the behavior of periodic hyperelastic composites. We focus here on composites consisting of an elastomeric matrix periodically reinforced by long fibers. The paper is composed of three parts. The first part deals with the theoretical aspects of compressible behavior. The second-order theory of Ponte Castañeda (J. Mech. Phys. Solids 44 (1996) 827) is considered and extended to periodic microstructures. Comparisons with results obtained by the finite element method show that the composite behavior predicted by the present model is much more accurate for compressible than for incompressible materials. The second part deals with the extension of the method to incompressible behavior. A mixed formulation (displacement-pressure) is used which improves the accuracy of the estimate given by the model. The third part presents experimental results. The composite tested is made of a rubber matrix reinforced by steel wires. Firstly, the matrix behavior is identified with a tensile test and a shear test carried out on homogeneous samples. Secondly, the composite is tested under shearing. The experimentally measured homogenized stress is then compared with the predictions of the model.     

A conventional theory of mechanism-based strain gradient plasticity

There exist two frameworks of strain gradient plasticity theories to model size effects observed at the micron and sub-micron scales in experiments. The first framework involves the higher-order stress and therefore requires extra boundary conditions, such as the theory of mechanism-based strain gradient (MSG) plasticity [J Mech Phys Solids 47 (1999) 1239; J Mech Phys Solids 48 (2000) 99; J Mater Res 15 (2000) 1786] established from the Taylor dislocation model. The other framework does not involve the higher-order stress, and the strain gradient effect come into play via the incremental plastic moduli. A conventional theory of mechanism-based strain gradient plasticity is established in this paper. It is also based on the Taylor dislocation model, but it does not involve the higher-order stress and therefore falls into the second strain gradient plasticity framework that preserves the structure of conventional plasticity theories. The plastic strain gradient appears only in the constitutive model, and the equilibrium equations and boundary conditions are the same as the conventional continuum theories. It is shown that the difference between this theory and the higher-order MSG plasticity theory based on the same dislocation model is only significant within a thin boundary layer of the solid.     

A rate-dependent three-dimensional free energy model for ferroelectric single crystals

The one-dimensional free energy model for ferroelectric materials developed by Smith et al. [Smith, R.C., Seelecke, S., Ounaies, Z., 2002. A free energy model for piezoceramic materials. In: 9th SPIE Conference on Smart Structures and Materials, San Diego, USA, pp. 17–22; Smith, R.C., Seelecke, S., Ounaies, Z., Smith, J., 2003. A free energy model for hysteresis in ferroelectric materials. J. Intell. Mater. Syst. Struct. 14, 719–739; Smith, R.C., Seelecke, S., Dapino, M.J., Ounaies, Z., 2005. A unified framework for modeling hysteresis in ferroic materials. J. Mech. Phys. Solids 54, 46–85] is generalized to three space dimensions including both polarization and strain. In the resulting nine-dimensional energy function, six free energy potentials representing the six distinct types of tetragonal variants of perovskite lattice structures are given as quadratic functions of polarization vector and strain tensor. Energy barrier expressions as functions of thermodynamic driving forces are obtained through a generalization of the one-dimensional equations derived from the model of Smith et al. This approach presents an alternative to the cumbersome determination of higher-dimensional saddle points and is attractive for a computationally efficient implementation. The energy barrier expressions are combined with evolution equations for the variant fractions based on the theory of thermally activated processes and thus allow for a natural treatment of rate-dependent effects. The predictions of the model are compared with recent measurements on BaTiO₃ single crystals by Burcsu et al. [Burcsu, E., Ravichandran, G., Bhattacharya, K., 2004. Large electrostrictive actuation of barium titanate single crystals. J. Mech. Phys. 52, 823–846]. The effects of applied stress and 90°- and 180°-switching processes are discussed in detail.     

A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations

We propose a deformation theory of strain gradient crystal plasticity that accounts for the density of geometrically necessary dislocations by including, as an independent kinematic variable, Nye's dislocation density tensor [1953. Acta Metallurgica 1, 153-162]. This is accomplished in the same fashion as proposed by Gurtin and co-workers (see, for instance, Gurtin and Needleman [2005. J. Mech. Phys. Solids 53, 1-31]) in the context of a flow theory of crystal plasticity, by introducing the so-called defect energy. Moreover, in order to better describe the strengthening accompanied by diminishing size, we propose that the classical part of the plastic potential may be dependent on both the plastic slip vector and its gradient; for single crystals, this also makes it easier to deal with the “higher-order” boundary conditions. We develop both the kinematic formulation and its static dual and apply the theory to the simple shear of a constrained strip (example already exploited in Shu et al. [2001. J. Mech. Phys. Solids 49, 1361-1395], Bittencourt et al. [2003. J. Mech. Phys. Solids 51, 281-310], Niordson and Hutchinson [2003. Euro J. Mech. Phys. Solids 22, 771-778], Evers et al. [2004. J. Mech. Phys. Solids 52, 2379-2401], and Anand et al. [2005. J. Mech. Phys. Solids 53, 1789-1826]) to investigate what sort of behaviour the new model predicts. The availability of the total potential energy functional and its static dual allows us to easily solve this simple boundary value problem by resorting to the Ritz method.     

A Strain Energy Function for Vulcanized Rubbers

A three-parameter strain energy function is developed to model the nonlinearly elastic response of rubber-like materials. The development of the model is phenomenological, based on data from the classic experiments of Treloar, Rivlin and Saunders, and Jones and Treloar on sheets of vulcanized rubber. A simple two-parameter version, similar to the Mooney-Rivlin and Gent-Thomas strain energies, provides an accurate fit with all of the data from Rivlin and Saunders and Jones and Treloar, as well as with Treloar’s data for deformations for which the principal deformation invariant I ₁ has values in the range 3≤I ₁≤20.     

Well-posedness of a model of strain gradient plasticity for plastically irrotational materials

The initial boundary value problem corresponding to a model of strain gradient plasticity due to [Gurtin, M., Anand, L., 2005. A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. J. Mech. Phys. Solids 53, 1624–1649] is formulated as a variational inequality, and analysed. The formulation is a primal one, in that the unknown variables are the displacement, plastic strain, and the hardening parameter. The focus of the analysis is on those properties of the problem that would ensure existence of a unique solution. It is shown that this is the case when hardening takes place. A similar property does not hold for the case of softening. The model is therefore extended by adding to it terms involving the divergence of plastic strain. For this extended model the desired property of coercivity holds, albeit only on the boundary of the set of admissible functions.     

Numerical modelling of the plasticity induced during diffusive transformation. An ensemble averaging approach for the case of random arrays of nuclei

The grounds of a numerical modelling of the mechanical consequences of diffusive phase transformation in solids have been established by Leblond [Leblond, J.B., Mottet, G., Devaux, J.C., 1986. A theoretical and numerical approach to the plastic behavior of steels during phase transformations I: derivation of general relations, J. Mech. Phys. Solids 34 (4) 395–409] and Ganghoffer [Ganghoffer, J.F., Denis, S., Gautier, E., Simon, A., Sjöström, S., 1993. Finite element calculation of the micromechanics of a diffusional transformation, Eur. J. Mech. A Solids 12 (1) 21–32]: this modelling resorts to the FE method to evaluate the stress and strain fields which ensure the mechanical equilibrium between a diffusionaly growing phase and its parent phase. It has been the subject of a thorough analysis in [Barbe, F., Quey, R., Taleb, L., 2007. Numerical modelling of the plasticity induced during diffusive transformation. Case of a cubic array of nuclei, Eur. J. Mech. A Solids 26, 611–625] which has evidenced the main limit underlying this modelling with regard to physics, relative to the fact that nuclei are implicitly positioned according to a periodic array. The present work proposes, in details, an extension to the case of nuclei instantaneously appearing at random positions in a quasi infinite homogeneous medium. It constitutes a fundamental step towards a numerical modelling explicitly taking into account the crystalline plasticity and the morphology of the transforming medium.     

The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 2: single-crystal plasticity

Variational formulations are constructed for rate-independent problems in small-deformation single-crystal strain-gradient plasticity. The framework, based on that of Gurtin (J Mech Phys Solids 50: 5–32, 2002), makes use of the flow rule expressed in terms of the dissipation function. Provision is made for energetic and dissipative microstresses. Both recoverable and non-recoverable defect energies are incorporated into the variational framework. The recoverable energies include those that depend smoothly on the slip gradients, the Burgers tensor, or on the dislocation densities (Gurtin et al. J Mech Phys Solids 55:1853–1878, 2007), as well as an energy proposed by Ohno and Okumura (J Mech Phys Solids 55:1879–1898, 2007), which leads to excellent agreement with experimental results, and which is positively homogeneous and therefore not differentiable at zero slip gradient. Furthermore, the variational formulation accommodates a non-recoverable energy due to Ohno et al. (Int J Mod Phys B 22:5937–5942, 2008), which is also positively homogeneous, and a function of the accumulated dislocation density. Conditions for the existence and uniqueness of solutions are established for the various examples of defect energy, with or without the presence of hardening or slip resistance.     

The coupled effects of plastic strain gradient and thermal softening on the dynamic growth of voids

This paper examines the combined effects of temperature, strain gradient and inertia on the growth of voids in ductile fracture. A dislocation-based gradient plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99] is applied, and temperature effects are incorporated. Since a strong size-dependence is introduced into the dynamic growth of voids through gradient plasticity, a cut-off size is then set by the stress level of the applied loading. Only those voids that are initially larger than the cut-off size can grow rapidly. At the early stages of void growth, the effects of strain gradients greatly increase the stress level. Therefore, thermal softening has a strong effect in lowering the threshold stress for the unstable growth of voids. Once the voids start rapid growth, however, the influence of strain gradients will decrease, and the rate of dynamic void growth predicted by strain gradient plasticity approaches that predicted by classical plasticity theories.     

Development of new constitutive equations for the Mullins effect in rubber using the network alteration theory

During cyclic loading, both natural and synthetic elastomers exhibit a stress-softening phenomenon known as the Mullins effect. In the last few years, numerous constitutive equations have been proposed. The major difficulty lies in the development of models which are both physically motivated and sufficiently mathematically well defined to be used in finite element applications. An attempt to reconcile both physical and phenomenological approaches is proposed in this paper. The network alteration theory of Marckmann et al. [Marckmann, G., Verron, E., Gornet, L., Chagnon, G., Charrier, P., Fort, P., 2002. A theory of network alteration for the Mullins effect. J. Mech. Phys. Solids 50, 2011–2028] is considered and modified. The equivalence between three different strain energy functions is then used to develop two new constitutive equations. They are founded on phenomenological strain energy densities which ensure simple numerical use, but the evolution of their material parameters during stress-softening is based on physical considerations. Basic examples illustrate the efficiency of this approach.     

Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys

The results from a systematic study of the response of a Ti–6Al–4V alloy under quasi-static and dynamic loading, at different strain rates and temperatures, are presented. The correlations and predictions using modified Khan–Huang–Liang (KHL) viscoplastic constitutive model are compared with those from Johnson–Cook (JC) model and experimental observations for this strain rate and temperature-dependent material. Overall, KHL model correlations and predictions are shown to be much closer to the observed responses, than the corresponding JC model predictions and correlations. Similar trend has been demonstrated for other titanium alloys using published experimental data [Mech. Mater. 33(8) (2001) 425; J. Mech. Phys. Solids 47(5) (1999) 1157].     

Composite and filament models for the mechanical behaviour of elastomeric materials

In this work we present a composite model, which combines the approach of Poisson's function with the filament theory and requires three material parameters. We also suggest the form for a strain-energy function that approximates the constitutive equations of the composite model. Furthermore, a simple asymptotic analysis allows us to reduce the number of material constants to only two, thus, forming a new filament model. The predictive capability of the two models to reproduce the mechanical behaviour of elastomeric materials in deformation experiments is evaluated against the extensive data of Kawabata et al. (Macromolecules 14 (1981) 154). The models give excellent agreement in not only uniaxial and equibiaxial but also non-equibiaxial extension. Although being rather more simplistic in comparison with some successful network models involving non-Gaussian chain statistics, the two models conform much more closely to the classical experimental data of Treloar (Trans. Faraday Soc. 40 (1944) 59).