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

The one-dimensional free energy model for ferroelectric materials developed by Smith et al. [29–31] is generalized to two dimensions. The two-dimensional free energy potential proposed in this paper consists of four energy wells that correspond to four variants of the material. The wells are separated by four saddle points, representing the barriers for 90°-switching processes, and a local maximum, across which 180°-switching processes take place. The free energy potential is combined with evolution equations for the variant fractions based on the theory of thermally activated processes. The model is compared to recent measurements on BaTiO₃ single crystals by Burcsu et al. [8], and predicitions are made concerning the response to the application of in-plane multi-axial electric fields at various frequencies and loading directions. The kinetics of the 90°- and 180°-switching processes are discussed in detail.     

Three-dimensional finite element simulations of ferroelectric polycrystals under electrical and mechanical loading

Complex, non-linear, irreversible, hysteretic behavior of polycrystalline ferroelectric materials under a combined electro-mechanical loading is a result of domain wall motion, causing simultaneous expansion and contraction of unlike domains, grain sub-divisions that have distinct spontaneous polarization and strain. In this paper, a 3-dimensional finite element method is used to simulate such a polycrystalline ferroelectric under electrical and mechanical loading. A constitutive law due to Huber et al. [1999. A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids 47, 1663-1697] for switching by domain wall motion in multidomain ferroelectric single crystals is employed in our model to represent each grain, and the finite element method is used to solve the governing conditions of mechanical equilibrium and Gauss's law. The results provide the average behavior for the polycrystalline ceramic. We compare the outcomes predicted by this model with the available experimental data for various electromechanical loading conditions. The qualitative features of ferroelectric switching are predicted well, including hysteresis and butterfly loops, the effect on them of mechanical compression, and the response of the polycrystal to non-proportional electrical loading.     

Crack propagation in a four-parameter piezoelectric medium

The simplified three-parameter formulation of a piezoelectric medium proposed by Gao et al. [Gao, H.J., Zhang, T.Y., Tong, P., 1997. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 45, 491–510] is extended to a four-parameter modified model in order to point out the features of a steadily propagating Griffith crack. It is assumed that the crack is electro-elastically free and the medium is subjected to a generalized electro-mechanical loading applied at infinity. The complete solution is provided under impermeable and permeable boundary conditions and results are presented in order to show the main dynamical features.     

The influence of a compressive stress on the nonlinear response of ferroelectric crystals

The origin of nonlinearity in a ferroelectric crystal is domain reorientation, and such a process can be affected by the presence of a compressive stress. In this article we examine how a superimposed compression affects the evolution of new domain and how it changes the shape of the hysteresis loop. We start out by considering the thermodynamic driving force for domain reorientation, and then use a dual-phase homogenization theory to calculate the overall response. To uncover the influence of a compressive stress, the theory is used to calculate the hysteresis loop between the electric displacement D and the electric field E of a BaTiO₃ crystal, first without and then with a compression, using a two-consecutive 90° switch model (i.e. 0° → 90° → 180°). It is found that, from the initial 0° position, the compressive stress will increase the thermodynamic driving force and promote an earlier onset of the 90° domain, but its presence will cause a significant delay for the reorientation process to pass through the intermediate 90° state in route to its final 180° configuration. The D vs. E loop then exhibits a more round shape and a lesser steep slope near the coercive field. The delayed passage and more rounded shape are found to be consistent with a recent experimental observation [Burcsu et al., 2004. J. Mech. Phys. Solids 52, 823–846].     

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.     

Finite element simulation of a polycrystalline ferroelectric based on a multidomain single crystal switching model

In this paper, we compute the constitutive behavior of a ferroelectric ceramic by a plane strain finite element model, where each element represents a single grain in the polycrystal. The properties of a grain are described by the microscopic model for switching in multidomain single crystals of ferroelectric materials presented by Huber et al. [J. Mech. Phys. Solids 47 (1999) 1663]. The poling behavior of the polycrystal is obtained by employing the finite element formulation for electromechanical boundary value problems developed by Landis [Int. J. Numer. Meth. Eng. 55 (2002) 613]. In particular, we address the influence of the single grain properties and the interaction between grains, respectively.     

Two-subsystem thermodynamics for the mechanics of aging amorphous solids

The effect of physical aging on the mechanics of amorphous solids as well as mechanical rejuvenation is modeled with nonequilibrium thermodynamics, using the concept of two thermal subsystems, namely a kinetic one and a configurational one. Earlier work (Semkiv and Hütter in J Non-Equilib Thermodyn 41(2):79–88, 2016) is extended to account for a fully general coupling of the two thermal subsystems. This coupling gives rise to hypoelastic-type contributions in the expression for the Cauchy stress tensor, that reduces to the more common hyperelastic case for sufficiently long aging. The general model, particularly the reversible and irreversible couplings between the thermal subsystems, is compared in detail with models in the literature (Boyce et al. in Mech Mater 7:15–33, 1988; Buckley et al. in J Mech Phys Solids 52:2355–2377, 2004; Klompen et al. in Macromolecules 38:6997–7008, 2005; Kamrin and Bouchbinder in J Mech Phys Solids 73:269–288 2014; Xiao and Nguyen in J Mech Phys Solids 82:62–81, 2015). It is found that only for the case of Kamrin and Bouchbinder (J Mech Phys Solids 73:269–288, 2014) there is a nontrivial coupling between the thermal subsystems in the reversible dynamics, for which the Jacobi identity is automatically satisfied. Moreover, in their work as well as in Boyce et al. (Mech Mater 7:15–33, 1988), viscoplastic deformation is driven by the deviatoric part of the Cauchy stress tensor, while for Buckley et al. (J Mech Phys Solids 52:2355–2377, 2004) and Xiao and Nguyen (J Mech Phys Solids 82:62–81, 2015) this is not the case.     

A variational model for fracture mechanics: Numerical experiments

In the variational model for brittle fracture proposed in Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319-1342], the minimum problem is formulated as a free discontinuity problem for the energy functional of a linear elastic body. A family of approximating regularized problems is then defined, each of which can be solved numerically by a finite element procedure. Here we re-formulate the minimum problem within the context of finite elasticity. The main change is the introduction of the dependence of the strain energy density on the determinant of the deformation gradient. This change requires new, more general existence and Γ-convergence results. The results of some two-dimensional numerical simulations are presented, and compared with corresponding simulations made in Bourdin et al. [2000. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797-826] for the linear elastic model.     

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.     

Generalized self-consistent estimation of the apparent isotropic elastic moduli and minimum representative volume element size of heterogeneous media

Under investigation is a heterogeneous material consisting of an elastic homogeneous isotropic matrix in which layered elastic isotropic inclusions or pores are embedded. The generalized self-consistent model (GSCM) is extended so as to be capable of estimating the apparent elastic properties of a finite-size specimen smaller than a representative volume element (RVE). The kinematical or static apparent shear modulus is determined as a root of a cubic polynomial equation instead of a quadratic polynomial equation as in the classical GSCM of Christensen and Lo [Christensen, R.M., Lo, K.H., 1979. Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids 27, 315–330]. It turns out that the extended GSCM establishes a link between the composite sphere assemblage model (CSAM) of Hashin [Hashin, Z., 1962. The elastic moduli of heterogeneous materials. J. Appl. Mech. 29, 143–150] and the classical GSCM. Demanding that the normalized distance between the kinematical and static apparent moduli of a finite-size specimen be smaller than a certain tolerance, the minimum RVE size is estimated in a closed form.     

Porous materials with two populations of voids under internal pressure: I. Instantaneous constitutive relations

This study is devoted to the mechanical behavior of uranium dioxide (UO₂) which is a porous material with two populations of voids of very different size subjected to internal pressure. The smallest voids are intragranular and spherical in shape whereas the largest pores located at the grain boundary are ellipsoidal and randomly oriented. In this first part of the study, attention is focused on the effective properties of these materials with fixed microstructure. In a first step, the poro-elastic properties of these doubly voided materials are studied. Then two rigorous upper bounds are derived for the effective poro-plastic constitutive relations of these materials. The first bound, obtained by generalizing the approach of Gologanu et al. (Gologanu, M., Leblond, J., Devaux, J., 1994. Approximate models for ductile metals containing non-spherical voids-case of axisymmetric oblate ellipsoidal cavities. ASME J. Eng. Mater. Technol. 116, 290–297) to compressible materials, is accurate at high stress-triaxiality. The second one, which derives from the variational method of Ponte Castañeda (Ponte Castañeda, P., 1991. The effective mechanical properties of non-linear isotropic composites. J. Mech. Phys. Solids 39, 45–71), is accurate when the stress triaxiality is low. A N-phase model, inspired by Bilger et al. (Bilger, N., Auslender, F., Bornert, M., Masson, R., 2002. New bounds and estimates for porous media with rigid perfectly plastic matrix. C.R. Mécanique 330, 127–132), is proposed which matches the best of the two bounds at low and high triaxiality. The effect of internal pressures is discussed. In particular it is shown that when the two internal pressures coincide, the effective flow surface of the saturated biporous material is obtained from that of the drained material by a shift along the hydrostatic axis. However, when the two pressures are different, the modifications brought to the effective flow surface in the drained case involve not only a shift along the hydrostatic axis but also a change in shape and size of the surface.     

Constitutive and damage model for 63Sn37Pb solder under uniaxial and torsional cyclic loading

A constitutive model with Ohno–Wang kinematic hardening rule is developed and employed to simulate the isothermal cyclic behavior of Sn–Pb solder under uniaxial and torsional loading. An implicit constitutive integration scheme is presented for inelastic flow of solder. Then a modified low cycle fatigue life prediction model is put forward in which the sum of maximum shear strain range and normal strain range based on the critical plane concept is adopted to replace the uniaxial strain range used by Stolkarts et al. [Stolkarts, V., Keer, L.M., Fine, M.E., 1999. Damage evolution governed by microcrack nucleation with application to the fatigue of 63Sn–37Pb solder. J. Mech. Phys. Solids 47, 2451–2468]. Comparison of the experimental results and simulation verifies that the stress strain hysteresis loops and peak stress decline curve of solder can be reasonably modeled over a wide range of loading conditions with implement of damage coupled constitutive model, and the lifetime estimations of 63Sn37Pb solder based on the assumption of microcrack nucleation governed damage is effective to provide a conservative prediction.     

Experimental and numerical study of flat momentumless wake

Flat momentumless wake behind wing profile has been studied both experimentally and with numerical simulation based on the second order Reynolds stress turbulence model. Some obtained regular trends of momentumless wake decay are mainly consistent with previous experiments by Dmitrenko et al. (Dmitrenko, Yu.M., Kovalev, I.I., Luchko, N.N., Cherepanov, P.Ya., 1987. J. Eng. Phys. 52, 743–751 (in Russian)), Cimbala and Park (Cimbala, J.M., Park, W.J., 1990. J. Fluid Mech. 213, 479–509; Park, W.J., Cimbala, J.M., 1991. J. Fluid Mech. 224, 29–47). Transverse distributions of statistical characteristics and its evolution along the flow axis depend on initial conditions and body shape. It was shown that characteristic turbulence parameters associated with the wake's width, mean velocity deficit, turbulence kinetic energy, and its dissipation rate are not constant as some of the theories predict.     

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.     

Strip electric-magnetic breakdown model in a magnetoelectroelastic medium

Extending the polarization saturation model [Gao et al., 1997. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 45, 491-510] and the dielectric breakdown (DB) model [Zhang et al., 2005. The strip dielectric breakdown model. Int. J. Fract. 132, 311-327] in piezoelectric materials, the Strip Electric-Magnetic Breakdown (SEMB) model is proposed for electrically and magnetically impermeable crack in a magnetoelectroelastic medium to study the effect of the nonlinear character of electric field and magnetic field on fracture of magnetoelectroelastic materials. In the SEMB model, the electric field in the strip of the electric breakdown zone ahead of the crack tip is equal to the electric breakdown strength, while the magnetic filed in the strip of the magnetic breakdown zone is equal to the magnetic breakdown strength. By using the extended Stroh formalism and the extended dislocation modeling of a crack, the Griffith crack problem under the electrically and magnetically elastic-plastic condition in a magnetoelectroelastic medium is reduced to a set of dual integral equations. The sizes of the electric breakdown zone and the magnetic breakdown zone, the extended intensity factors and the local J-integral are obtained. The effect of the combined mechanical-electric-magnetic loadings on the local J-integral is studied.     

A three-dimensional progressive damage model for fibre-composite materials

This note presents a damage model for fibre-composite materials based in the approach by Matzenmiller et al. [Matzenmiller, A., Lubliner, J., Taylor, R.L., 1995. A constitutive model for anisotropic damage in fiber-composites. Mech. Mater. 20, 125]. In this work, the model is developed in a three-dimensional context with modified formulation for the constitutive law and damage evolution. An orthotropic composite subjected to mixed failure modes is assumed in this development. Its formulation and implementation details are provided.     

Numerical simulations of Richtmyer-Meshkov instabilities using conservative front-tracking method

This paper presents the numerical simulations of two Richtmyer-Meshkov (RM) instability experiments using the conservative front-tracking method developed in (Mao, D. Towards front-tracking based on conservation in two space dimensions II, tracking discontinuities in capturing fashion. J. Comput. Phys., 226, 1550–1588 (2007)). The numerical results are compared with those obtained in (Holmes, R. L., Grove, J. W., and Sharp, D. H. Numerical investigation of Richtmyer-Meshkov instability using front-tracking. J. Fluid Mech., 301, 51–64 (1995)). The present simulations are generally in good agreement with those obtained by Holmes et al., and also capture the nonlinear and compessive phenomenon, i.e., the self-interactions of the transmitted and reflected wave edges, which was pointed out by Holmes et al. as the cause of the deceleration of the interfaces. However, the perturbation amplitudes and the amplitude growth rates of the interfaces obtained with the present conservative front-tracking method are a bit larger than those obtained by Holmes et al.     

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.     

Heuristic search for a predictive strain-energy function in nonlinear elasticity

In this work, a new, quasi-structural model – bootstrapped eight-chain model – is proposed as a modification to the strain energy of eight-chain model [Arruda, E.M., Boyce, M.C., 1993. A three-dimensional constitutive model for the large stretch behaviour of rubber elastic materials. J. Mech. Phys. Solids 41, 389—412] that invokes the Langevin chain statistics. This development has been led to by our heuristic search into how the strain energy of eight-chain model may be adapted in order to account better for the mechanical behaviour of elastomeric materials in both linear and nonlinear elastic regimes [Treloar, L.R.G., 1944. Stress–strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc. 40, 59–70]. The eight-chain model appears to produce very similar results in predicting biaxial stress to those of a first stretch-invariant model that gives a good fit in uniaxial extension and, thus, it is shown that the former can not be significantly enhanced within the limitation of the latter. Evaluation of predictive capability for an additive invariant-separated form of strain energy shows that an explicit inclusion of a second stretch-invariant function would not work and that any thus added term ought to be dependent on both the first and second stretch-invariants of deformation tensor, and hints that an improvement is possibly needed at low strain. The composite and filament models [Miroshnychenko, D., Green, W.A., Turner, D.M., 2005. Composite and filament models for the mechanical behaviour of elastomeric materials. J. Mech. Phys. Solids 53 (4), 748–770] have their strain-energy functions in that suggested form and cope very well with predicting the experimental data of Treloar (1944). We use the form of strain energy for the filament model, that proved to be successful, to bootstrap the strain energy of eight-chain model in order to improve the performance of the latter at low strain. Thus, we derive a new model – bootstrapped eight-chain model – that requires only two material parameters – a rubber modulus and a limiting chain extensibility. The proposed model is quasi-structural due to bootstrapping and it retains the best traits and corrects the faults of the eight-chain model, conforming more closely to the classical experimental data of Treloar (1944).     

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.