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.     

Numerical implementation of a thermo-elastic–plastic constitutive equation in consideration of transformation plasticity in welding

Finite element analysis of welding processes, which entail phase evolution, heat transfer and deformations, is considered in this paper. Attention focuses on numerical implementation of the thermo-elastic–plastic constitutive equation proposed by Leblond et al. [J. Mech. Phys. Solids 34(4) (1986a) 395; J. Mech. Phys. Solids 34(4) (1986b) 411] in consideration of the transformation plasticity. Based upon the multiplicative decomposition of deformation gradient, hyperelastoplastic formulation is borrowed for efficient numerical implementation, and the algorithmic consistent moduli for elastic–plastic deformations including transformation plasticity are obtained in the closed form. The convergence behavior of the present implementation is demonstrated via a couple of numerical examples.     

A model of size effects in nano-indentation

The indentation hardness-depth relation established by Nix and Gao [1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411-425] agrees well with the micro-indentation but not nano-indentation hardness data. We establish an analytic model for nano-indentation hardness based on the maximum allowable density of geometrically necessary dislocations. The model gives a simple relation between indentation hardness and depth, which degenerates to Nix and Gao [1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411-425] for micro-indentation. The model agrees well with both micro- and nano-indentation hardness data of MgO and iridium.     

The correlation of the indentation size effect measured with indenters of various shapes

Experimental results are presented which show that the indentation size effect for pyramidal and spherical indenters can be correlated. For a pyramidal indenter, the hardness measured in crystalline materials usually increases with decreasing depth of penetration, which is known as the indentation size effect. Spherical indentation also shows an indentation size effect. However, for a spherical indenter, hardness is not affected by depth, but increases with decreasing sphere radius. The correlation for pyramidal and spherical indenter shapes is based on geometrically necessary dislocations and work-hardening. The Nix and Gao indentation size effect model (J. Mech. Phys. Solids 46 (1998) 411) for conical indenters is extended to indenters of various shapes and compared to the experimental results.     

Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments

The enhanced gradient plasticity theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical plasticity and the classical continuum plasticity. They are successful in explaining the size effects encountered in many micro- and nano-advanced technologies due to the incorporation of an intrinsic material length parameter into the constitutive modeling. However, the full utility of the gradient-type theories hinges on one's ability to determine the intrinsic material length that scales with strain gradients, and this study aims at addressing and remedying this situation. Based on the Taylor's hardening law, a micromechanical model that assesses a nonlinear coupling between the statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs) is used here in order to derive an analytical form for the deformation-gradient-related intrinsic length-scale parameter in terms of measurable microstructural physical parameters. This work also presents a method for identifying the length-scale parameter from micro- and nano-indentation experiments using both spherical and pyramidal indenters. The deviation of the Nix and Gao [Mech. Phys. Solids 46 (1998) 411] and Swadener et al. [J. Mech. Phys. Solids 50 (2002) 681; Scr. Mater. 47 (2002) 343] indentation size effect (ISE) models’ predictions from hardness results at small depths for the case of conical indenters and at small diameters for the case of spherical indenters, respectively, is largely corrected by incorporating an interaction coefficient that compensates for the proper coupling between the SSDs and GNDs during indentation. Experimental results are also presented which show that the ISE for pyramidal and spherical indenters can be correlated successfully by using the proposed model.     

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.     

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.     

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.     

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].     

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.     

Finite element approximation of field dislocation mechanics

A tool for studying links between continuum plasticity and dislocation theory within a field framework is presented. A finite element implementation of the geometrically linear version of a recently proposed theory of field dislocation mechanics (J. Mech. Phys. Solids 49 (2001) 761; Proc. Roy. Soc. 459 (2003) 1343; J. Mech. Phys. Solids 52 (2004) 301) represents the main idea behind the tool. The constitutive ingredients of the theory under consideration are simply elasticity and a specification of dislocation velocity and nucleation. The set of equations to be approximated are non-standard in the context of solid mechanics applications. It comprises the standard second-order equilibrium equations, a first-order div-curl system for the elastic incompatibility, and a first-order, wave-propagative system for the evolution of dislocation density. The latter two sets of equations require special treatment as the standard Galerkin method is not adequate, and are solved utilizing a least-squares finite element strategy. The implementation is validated against analytical results of the classical elastic theory of dislocations and analytical results of the theory itself. Elastic stress fields of dislocation distributions in generally anisotropic media of finite extent, deviation from elastic response, yield-drop, and back-stress are shown to be natural consequences of the model. The development of inhomogeneity, from homogeneous initial conditions and boundary conditions corresponding to homogeneous deformation in conventional plasticity, is also demonstrated. To our knowledge, this work represents the first computational implementation of a theory of dislocation mechanics where no analytical results, singular solutions in particular, are required to formulate the implementation. In particular, a part of the work is the first finite element implementation of Kröner's linear elastic theory of continuously distributed dislocations in its full generality.     

Superelastic behavior in tension–torsion of an initially-textured Ti–Ni shape-memory alloy

A recently-developed crystal-mechanics-based constitutive model for polycrystalline shape-memory alloys [J. Mech. Phys. Solids 49 (2001) 909] is shown to quantitatively predict the superelastic response of an initially-textured Ti–Ni alloy in (i) a proportional-loading, combined tension–torsion experiment, as well as (ii) a path-change, tension–torsion experiment.     

Size effects in the conical indentation of an elasto-plastic solid

The size effect in conical indentation of an elasto-plastic solid is predicted via the Fleck and Willis formulation of strain gradient plasticity (Fleck, N.A. and Willis, J.R., 2009, A mathematical basis for strain gradient plasticity theory. Part II: tensorial plastic multiplier, J. Mech. Phys. Solids, 57, 1045–1057). The rate-dependent formulation is implemented numerically and the full-field indentation problem is analyzed via finite element calculations, for both ideally plastic behavior and dissipative hardening. The isotropic strain-gradient theory involves three material length scales, and the relative significance of these length scales upon the degree of size effect is assessed. Indentation maps are generated to summarize the sensitivity of indentation hardness to indent size, indenter geometry and material properties (such as yield strain and strain hardening index). The finite element model is also used to evaluate the pertinence of the Johnson cavity expansion model and of the Nix–Gao model, which have been extensively used to predict size effects in indentation hardness.     

Evaluation of the Lazarus-Leblond constants in the asymptotic model of the interfacial wavy crack

The paper addresses the problem of a semi-infinite plane crack along the interface between two isotropic half-spaces. Two methods of solution have been considered in the past: Lazarus and Leblond [1998a. Three-dimensional crack-face weight functions for the semi-infinite interface crack-I: variation of the stress intensity factors due to some small perturbation of the crack front. J. Mech. Phys. Solids 46, 489-511, 1998b. Three-dimensional crack-face weight functions for the semi-infinite interface crack-II: integrodifferential equations on the weight functions and resolution J. Mech. Phys. Solids 46, 513-536] applied the “special” method by Bueckner [1987. Weight functions and fundamental fields for the penny-shaped and the half-plane crack in three space. Int. J. Solids Struct. 23, 57-93] and found the expression of the variation of the stress intensity factors for a wavy crack without solving the complete elasticity problem; their solution is expressed in terms of the physical variables, and it involves five constants whose analytical representation was unknown; on the other hand, the “general” solution to the problem has been recently addressed by Bercial-Velez et al. [2005. High-order asymptotics and perturbation problems for 3D interfacial cracks. J. Mech. Phys. Solids 53, 1128-1162], using a Wiener-Hopf analysis and singular asymptotics near the crack front.The main goal of the present paper is to complete the solution to the problem by providing the connection between the two methods. This is done by constructing an integral representation for Lazarus-Leblond's weight functions and by deriving the closed form representations of Lazarus-Leblond's constants.     

Multidimensional stability of martensite twins under regular kinetics

This paper considers phase boundaries governed by regular kinetic relations as first proposed by Abeyaratne and Knowles [1990. On the driving traction acting on a surface of strain discontinuity in a continuum. J. Mech. Phys. Solids 38 (3), 345-360; 1991. Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114, 119-154]. It shows that static configurations of hyperelastic materials, in which two different martensitic (monoclinic) states meet along a planar interface, are dynamically stable towards fully three-dimensional perturbations. For that purpose, the reduced stability (or reduced Lopatinski) function associated to the static twin [Freistühler and Plaza, 2007. Normal modes and nonlinear stability behavior of dynamic phase boundaries in elastic materials. Arch. Ration. Mech. Anal. 186 (1), 1-24] is computed numerically. The results show that the interface is weakly stable under Maxwellian kinetics expressing conservation of energy across the boundary, whereas it is uniformly stable with respect to linearly dissipative kinetic rules of Abeyaratne and Knowles type.     

Mircopolar Model of Fracture for Composite Material

Failure of a composite is a complex process accompanied by irreversible changes in the microstructure of the material. Microscopic mechanisms are known of the accumulation of damage and failure of the type of localized and multiple ruptures of the fibers delamination along interphase boundaries, and also mechanisms associated with fracture of fibers. In this work, we propose a mathematical model of the local mechanism of failure of a composite material randomly reinforced with a system of short fibers. We implement the Cosserat moment model of crack tip for filament material, reinforced with whiskers or in fiber- reinforced polycrystalline materials. It is assumed that the angular distribution of the fibers is isotropic and the elastic characteristics of the fibers are considerably higher than the elastic constants of the matrix. We implement the homogenization procedure for the effective Cosserat constants similarly to the effective elastic constants. The singular solution in the vicinity of the crack tip in the Cosserat moment model is found. Using this solution, we examine the bending stresses in the filaments due to effective moment stresses in the material. The constructed model describes the phenomenon of fracture of the fibers occurring during crack propagation in those composites. The following assumptions are used as the main hypotheses for the micromechanical model. The matrix contains a nucleation crack. When the load is increased the crack grows and its boundary comes into contact with the reinforcing fibers. A further increase of the stress causes bending of the fiber. When~the fiber curvature reaches a specific critical value, the fiber ruptures. If the stress at infinity is given, the fibers no longer delay the development of failure during crack propagation The degree of bending distortion of the fiber in the vicinity of the boundary of the crack is determined by the moment model of the material. The necessity to take into account the moment stresses in the failure theory of the reinforced material was stressed in [Muki and Sternberg (1965) Zeitschrift f angew Math und Phys 16:611–615; Garajeu and Soos (2003) Math Mech Solids 8(2):189–218; Ostoja-Starzewski et al (1999) Mech Res Commun 26:387–396]. The moment Cosserat stresses were accounted also for inhomogeneous biomechanical materials by Buechner and Lakes (2003) Bio Mech Model Mechanobiol 1: 295–301. We should also mention the important methodological studies [Sternberg and Muki (1967) J Solids Struct 1:69–95; Atkinson and Leppington (1977) Int J Solids Struct 13: 1103–1122] concerned with the moment stresses in homogeneous fracture mechanics.     

The finite deformation theory of Taylor-based nonlocal plasticity

Recent experiments have shown that metallic materials display significant size effect at the micron and sub-micron scales. This has motivated the development of strain gradient plasticity theories, which usually involve extra boundary conditions and possibly higher-order governing equations. We propose a finite deformation theory of nonlocal plasticity based on the Taylor dislocation model. The theory falls into Rice's theoretical framework of internal variables [J Mech Phys Solids 19 (1971) 433], and it does not require any extra boundary conditions. We apply the theory to study the micro-indentation hardness experiments, and it agrees very well with the experimental data over a wide range of indentation depth.     

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.     

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.