On multiscale significance of Rice’s normality structure

The normality structure proposed by [Rice, J.R., 1971. Inelastic constitutive relations for solids: an integral variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455.] provides a minimal framework of multiscale thermodynamics. As shown in this paper, Rice’s multiscale thermodynamic formalism is exactly consistent with Ziegler’s essential notion [Ziegler, H., 1977. An Introduction to Thermomechanics, North-Holland, Amsterdam.] that the entire constitutive response is determined by the knowledge of two scalar potential functions: an energy function and a dissipation function. In Rice’s multiscale thermodynamic formulation, the variational equation relating macroscale and microscale thermodynamic fluxes and forces plays a central role and ensures the equality between microscale and macroscale dissipation rate. The variational equation can be further reformulated into a principle of maximum equivalent dissipation. Based on the variation equation, the transformation from microscale to macroscale is characterized by two linear transformations with the same corresponding matrix.     

Relationship between normality structure and orthogonality condition

The nonlinear generalization of the phenomenological equations along with Onsager [Reciprocal relations in irreversible processes, I, II. Physical Review 37 (1931) 405] reciprocal relations includes the normality structure of Rice [Inelastic constitutive relations for solids: an integral variable theory and its application to metal plasticity. Journal of the Mechanics and Physics of Solids 19 (1971) 433; Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In: Argon, A.S. (Ed.), Constitutive Equations in Plasticity. MIT Press, Cambridge, MA, 1975, pp. 23–79] and the orthogonality condition of Ziegler [An Introduction to Thermomechanics. North-Holland, Amsterdam, 1977] and Ziegler and Wehrli [The derivation of constitutive relations from the free-energy and the dissipation function. Advances in Applied Mechanics 25 (1987a) 183–238; On a principle of maximal rate of entropy production. Journal of Non-Equilibrium Thermodynamics 12 (1987b) 229–243]. The normality structure is generally not consistent with the orthogonality condition due to the different potential functions. The aim of this paper is to examine whether or not the two theories conflict with each other and to establish the consistency condition between the two theories. It is shown in this paper that monotonic increasing and homogeneous kinetic rate laws of local internal variables are thermo-dynamically admissible within the normality structure. The homogeneity property of the kinetic rate laws is the necessary and sufficient consistency condition between the normality structure and orthogonality condition. The thermodynamic processes obeying the orthogonality condition belong to a subset of these processes covered by the normality structure, so the normality structure possesses much more generality than the orthogonality condition.     

J-integral and crack driving force in elastic–plastic materials

This paper discusses the crack driving force in elastic–plastic materials, with particular emphasis on incremental plasticity. Using the configurational forces approach we identify a “plasticity influence term” that describes crack tip shielding or anti-shielding due to plastic deformation in the body. Standard constitutive models for finite strain as well as small strain incremental plasticity are used to obtain explicit expressions for the plasticity influence term in a two-dimensional setting. The total dissipation in the body is related to the near-tip and far-field J-integrals and the plasticity influence term. In the special case of deformation plasticity the plasticity influence term vanishes identically whereas for rigid plasticity and elastic-ideal plasticity the crack driving force vanishes. For steady state crack growth in incremental elastic–plastic materials, the plasticity influence term is equal to the negative of the plastic work per unit crack extension and the total dissipation in the body due to crack propagation and plastic deformation is determined by the far-field J-integral. For non-steady state crack growth, the plasticity influence term can be evaluated by post-processing after a conventional finite element stress analysis. Theory and computations are applied to a stationary crack in a C(T)-specimen to examine the effects of contained, uncontained and general yielding. A novel method is proposed for evaluating J-integrals under incremental plasticity conditions through the configurational body force. The incremental plasticity near-tip and far-field J-integrals are compared to conventional deformational plasticity and experimental J-integrals.     

Implicit finite element formulations for multi-phase transformation in high carbon steel

An anomalous plastic deformation observed during the phase transformation of steels was implemented into the finite element modeling. The constitutive equations for the transformation plasticity originally proposed by Greenwood and Johnson [Greenwood, G.W., Johnson, R.H., 1965. The deformation of metals under small stresses during phase transformation. Proc. Roy. Soc. A 283, 403] and further extended by Leblond et al. [Leblond, J.B., Mottet, G., Devaux, J.C., 1986a. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, I. Derivation of general relations. J. Mech. Phys. Solids 34, 395–409; Leblond, J.B., Mottet, G., Devaux, J.C., 1986b. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, II. Study of classical plasticity for ideal-plastic phases. J. Mech. Phys. Solids 34, 411–432; Leblond, J.B., Devaux, J., Devaux, J.C., 1989a. Mathematical modeling of transformation plasticity in steels, I: case of ideal-plastic phases. Int. J. Plasticity 5, 511–572; Leblond, J.B., 1989b. Mathematical modeling of transformation plasticity in steels, II: coupling with strain hardening phenomena. Int. J. Plasticity 5, 573–591] were modified to consider the thermo-mechanical response of generalized multi-phase steel during phase transformations from austenite at high temperature. An implicit numerical solution procedure to calculate the plastic deformation of each constituent phase was newly proposed and implemented into the general purpose implicit finite element program via user material subroutine. The new algorithms include efficient calculation of consistent tangent modulus for the transformation plasticity and application of general anisotropic yield functions without limitation to the isotropic yield function. Besides the thermo-elastic–plastic constitutive equations, non-isothermal transformation kinetics was characterized by the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation and additivity relationship for the diffusional transformation, while the model proposed by Koistinen and Marburger was used for the diffusionless transformation. Numerical verifications for the continuous cooling experiments under various loading conditions were conducted to demonstrate the applicability of the developed numerical algorithms to the high carbon steel SK5.     

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.     

Energy dissipation and contour integral characterizing fracture behavior of incremental plasticity

Jep -integral is derived for characterizing the frac- ture behavior of elastic-plastic materials. The J ep -integral differs from Rice’s J-integral in that the free energy density rather than the stress working density is employed to define energy-momentum tensor. The J ep -integral is proved to be path-dependent regardless of incremental plasticity and deformation plasticity. The J epintegral possesses clearly clear physical meaning: (1) the value J ep tip evaluated on the infinitely small contour surrounding the crack tip represents the crack tip energy dissipation; (2) when the global steadystate crack growth condition is approached, the value of J ep farss calculated along the boundary contour equals to the sum of crack tip dissipation and bulk dissipation of plastic zone. The theoretical results are verified by simulating mode I crack problems.     

The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity

A general set of flow laws and associated variational formulations are constructed for small-deformation rate-independent problems in strain-gradient plasticity. The framework is based on the thermodynamically consistent theory due to Gurtin and Anand (J Mech Phys Solids 53:1624–1649, 2005), and includes as variables a set of microstresses which have both energetic and dissipative components. The flow law is of associative type. It is expressed as a normality law with respect to a convex but otherwise arbitrary yield function, or equivalently in terms of the corresponding dissipation function. Two cases studied are, first, an extension of the classical Hill-Mises or J ₂ flow law and second, a form written as a linear sum of the magnitudes of the plastic strain and strain gradient. This latter form is motivated by work of Evans and Hutchinson (Acta Mater 57:1675–1688, 2009) and Nix and Gao (J Mech Phys Solids 46:411–425, 1998), who show that it leads to superior correspondence with experimental results, at least for particular classes of problems. The corresponding yield function is obtained by a duality argument. The variational problem is based on the flow rule expressed in terms of the dissipation function, and the problem is formulated as a variational inequality in the displacement, plastic strain, and hardening parameter. Dissipative components of the microstresses, which are indeterminate, are absent from the formulation. Existence and uniqueness of solutions are investigated for the generalized Hill-Mises and linear-sum dissipation functions, and for various combinations of defect energy. The conditions for well-posedness of the problem depend critically on the choice of dissipation function, and on the presence or otherwise of a defect energy in the plastic strain or plastic strain gradient, and of internal-variable hardening.     

The equivalent axisymmetric model for Berkovich indenters in power-law hardening materials

Nix and Gio [Nix, W.D., Gao, H.J., 1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. Journal of the Mechanics and Physics of Solids 46, 411–425] established an important relation between the micro-indentation hardness and indentation depth for axisymmetric indenters. For the Berkovich indenter, however, this relation requires an equivalent cone angle. Qin et al. [Qin, J., Huang, Y., Xiao, J., Hwang, K.C., 2009. The equivalence of axisymmetric indentation model for three-dimensional indentation hardness. Journal of Materials Research 24, 776–783] showed that the widely used equivalent cone angle from the criterion of equal base area leads to significant errors in micro-indentation, and proposed a new equivalence of equal cone angle for iridium. It is shown in this paper that this new equivalence holds for a wide range of plastic work hardening materials. In addition, the prior equal-base-area criterion does not hold because the Berkovich indenter gives much higher density of geometrically necessary dislocations than axisymmetric indenter. The equivalence of equal cone angle, however, does not hold for Vickers indenter.     

Alternative formulations of isotropic hardening for Mises materials, and associated variational inequalities

This work provides insight into aspects of classical Mises–Hill plasticity, its extension to the Aifantis theory of gradient plasticity, and the formulations of both theories as variational inequalities. Firstly, it is shown that the classical isotropic hardening rule, which is dissipative in nature, may equally well be characterized via a defect energy—and, what is striking, this energetically based hardening rule mimics dissipative behavior by describing loading processes that are irreversible. A second aspect concerns the equivalence between the conventional form of the flow rule and its formulation in terms of dissipation. This equivalence has been previously established using the tools of convex analysis (cf., e.g., Han and Reddy, Plasticity: mathematical theory and numerical analysis, Springer, New York, 1999)—in the current work this equivalence is derived directly from the constitutive equations and the specific form of the dissipation, without recourse to such machinery. Variational inequalities corresponding to the dissipative and energetic forms of the flow rule are derived; these inequalities involve only the displacement and plastic strain and are well suited to computational studies. Finally, it is shown that the framework developed for the classical theory is easily extended to incorporate the gradient-plasticity theory of Aifantis (Trans ASME J Eng Mater Technol 106:326–330, 1984).      

Elastic compliances of non-flat cracks

Cracks of non-flat geometries are common in materials science applications. Their effect on the overall elastic properties is addressed. Applications of Rice’s [Rice, J.R., 1975. Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In: Argon, A. (Ed.), Constitutive Equations in Plasticity, MIT Press, pp. 23–75.] theorem that, in particular, relates stress intensity factors to crack compliances, are discussed and illustrated by a 2-D example of a circular arc crack. Then we conduct computational studies of several non-flat patterns and suggest a simple approximation for compliances of non-flat cracks. Implications for anisotropy due to non-random orientation distributions of cracks of non-flat geometries are discussed.     

On perfectly plastic flow in porous material

Experimental observations suggest that for perfectly-plastic materials containing pores, the (small) strain at which significant macroscopic yielding occurs is relatively insensitive to porosity, for volume fractions below approximately 15–20% (although the yield stress drops significantly with increasing porosity). Another observation is that, at these porosity levels, the stress–strain curve remains approximately linear almost up to the yield point. Based on these observations, Sevostianov and Kachanov constructed yield surfaces that explicitly reflect the shapes of the pores and their orientation. The underlying microscale mechanism is that local plastic “pockets” near pores blunt the stress concentrations; as a result, they remain limited in size and well contained in the elastic field until they connect and almost the entire matrix plasticizes within a narrow interval of stresses that can be idealized as the yield point. The present paper provides direct insight into the micromechanics of poroplasticity through direct microscale numerical simulation. Besides confirming the basic microscale mechanism, these simulations reveal that the reduction of the macroscopic poroplastic yield stress is approximated quite closely by 1−v₂ times the dense nonporous yield stress, where v₂ is the volume fraction of the pores.     

Large Eddy Simulations: How to evaluate resolution

The present paper gives an analysis of fully developed channel flow at Reynolds number of Re=u_τδ/ν=4000 based on the friction velocity, u_τ, and half the channel height, δ. Since the Reynolds number is high, the LES is coupled to a URANS model near the wall (hybrid LES–RANS) which acts as a wall model. It it found that the energy spectra is not a good measure of LES resolution; neither is the ratio of the resolved turbulent kinetic energy to the total one (i.e. resolved plus modelled turbulent kinetic energy). It is suggested that two-point correlations are the best measures for estimating LES resolution. It is commonly assumed that SGS dissipation takes place at high wavenumbers. Energy spectra of the fluctuating velocity gradients show that this is not true; the major part of the SGS dissipation takes place at low to midrange wavenumbers. Furthermore, the energy spectra of the fluctuating velocity gradients reveals that the accuracy of the predicted velocity gradients at the highest resolved wavenumbers is very poor.     

On internal dissipation inequalities and finite strain inelastic constitutive laws: Theoretical and numerical comparisons

Internal dissipation always occurs in irreversible inelastic deformation processes of materials. The internal dissipation inequalities (specific mathematical forms of the second law of thermodynamics) determine the evolution direction of inelastic processes. Based on different internal dissipation inequalities several finite strain inelastic constitutive laws have been formulated for instance by Simo [Simo, J.C., 1992. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering 99, 61–112]; Simo and Miehe [Simo, J.C., Miehe, C., 1992. Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Computer Methods in Applied Mechanics and Engineering 98, 41–104]; Lion [Lion, A., 1997. A physically based method to represent the thermo-mechanical behavior of elastomers. Acta Mechanica 123, 1–25]; Reese and Govindjee [Reese, S., Govindjee, S., 1998. A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures 35, 3455–3482]; Lin and Schomburg [Lin, R.C., Schomburg, U., 2003. A finite elastic–viscoelastic–elastoplastic material law with damage: theoretical and numerical aspects. Computer Methods in Applied Mechanics and Engineering 192, 1591–1627]; Lin and Brocks [Lin, R.C., Brocks, W., 2004. On a finite strain viscoplastic theory based on a new internal dissipation inequality. International Journal of Plasticity 20, 1281–1311]; and Lin and Brocks [Lin, R.C., Brocks, W., 2005. An extended Chaboche’s viscoplastic law at finite strains: theoretical and numerical aspects. Journal of Materials Science and Technology 21, 145–147]. These constitutive laws are consistent with the second law of thermodynamics. As the internal dissipation inequalities are described in different configurations or coordinate systems, the related constitutive laws are also formulated in the corresponding configurations or coordinate systems. Mathematically, these constitutive laws have very different formulations. Now, a question is whether the constitutive laws provide identical constitutive responses for the same inelastic constitutive problems. In the present work, four types of finite strain viscoelastic and endochronically plastic laws as well as three types of J₂-plasticity laws are formulated based on four types of dissipation inequalities. Then, they are numerically compared for several problems of homogeneous or complex finite deformations. It is demonstrated that for the same inelastic constitutive problem the stress responses are identical for deformation processes without rotations. In the simple shear deformation process with large rotation, the presented viscoelastic and endochronically plastic laws also show almost identical stress responses up to a shear strain of about 100%. The three laws of J₂-plasticity also produce the same shear stresses up to a shear strain of 100%, while different normal stresses are generated even at infinitesimal shear strains. The three J₂-plasticity laws are also compared at three complex finite deformation processes: billet upsetting, cylinder necking and channel forming. For the first two deformation processes similar constitutive responses are obtained, whereas for the third deformation process (with large global rotations) significant differences of constitutive responses can be observed.     

Consequences of the dissipative restrictions in finite anisotropic elasto-plasticity

The paper is devoted to the dissipation postulate in anisotropic finite elastoplasticity, properly formulated in terms of the total strain histories, on small cycles only. The equivalence between the dissipation postulate and the existence of the stress potential together with the dissipation inequality is proved. The modified flow rules compatible with the dissipation postulate follow as a necessary condition. The convexity and normality properties can be treated as an equivalent issue of the dissipation postulate only within the framework of Σ models. We identify such classes of Σ-models based on the pre-image theorem. The difficulties arise from the non-injectivity of the Mandel's stress measure, as dependent on the elastic strain. We define the yield stress function and the admissible elastic stress range in Σ-space. The equivalence is achieved only if it possible to construct the elastic range in strain space, having just the topological properties originally assumed as a basis of the dissipation postulate. The normality to the admissible elastic stress range does not mean an associative flow rule. The results are exemplified for transversely isotropic elasto–plastic materials as well as for models with small elastic strains.     

Validation of a discrete element model using magnetic resonance measurements

The discrete element model （DEM） is a very promising modelling strategy for two-phase granular systems. However, owing to a lack of experimental measurements, validation of numerical simulations of two-phase granular systems is still an important issue. In this study, a small two-dimensional gas- fluidized bed was simulated using a discrete element model. The dimensions of the simulated bed were 44mm × 10mm × 120 mm and the fluidized particles had a diameter dp = 1.2 mm and density ρp = 1000 kg/m^3. The comparison between DEM simulations and experiments are performed on the basis of time-averaged voidage maps. The drag-law of Beetstra et al. [Beetstra, R., van der Hoef, M.A., ＆ Kuipers,J. A. M. （2007b）. Drag force of intermediate Reynolds number flow past mono- and bidispersed arrays of spheres. AIChE Journal, 53,489-501 ] seems to give the best results. The simulations are fairly insensitive to the coefficient of restitution and the coefficient of friction as long as some route of energy dissipation during particle-particle and particle-wall contact is provided. Changing the boundary condition of the gas phase at the side-walls from zero-slip to full-slip does not affect the simulation results. Care is to be taken that the cell sizes are chosen so that a reasonable number of particles can be found in a fluid cell.     

A unified treatment of strain gradient plasticity

A theoretical framework is presented that has potential to cover a large range of strain gradient plasticity effects in isotropic materials. Both incremental plasticity and viscoplasticity models are presented. Many of the alternative models that have been presented in the literature are included as special cases. Based on the expression for plastic dissipation, it is in accordance with Gurtin (J. Mech. Phys. Solids 48 (2000) 989; Int. J. Plast. 19 (2003) 47) argued that the plastic flow direction is governed by a microstress q_ij and not the deviatoric Cauchy stress σ_ij′ that has been assumed by many others. The structure of the governing equations is of second order in the displacements and the plastic strains which makes it comparatively easy to implement in a finite element programme. In addition, a framework for the formulation of consistent boundary conditions is presented. It is shown that there is a close connection between surface energy of an interface and boundary conditions in terms of plastic strains and moment stresses. This should make it possible to study boundary layer effects at the interface between grains or phases. Consistent boundary conditions for an expanding elastic-plastic boundary are as well formulated. As examples, biaxial tension of a thin film on a thick substrate, torsion of a thin wire and a spherical void under remote hydrostatic tension are investigated.     

Meshless analysis of plasticity with application to crack growth problems

The governing equations for classical rate-independent plasticity are formulated in the framework of meshless method. The special J₂ flow theory for three-dimensional, two-dimensional plane strain and plane stress problems are presented. The numerical procedures, including return mapping algorithm, to obtain the solutions of boundary-value problems in computational plasticity are outlined. For meshless analysis the special treatment of the presence of barriers and mirror symmetries is formulated. The crack growth process in elastic–plastic solid under plane strain and plane stress conditions is analyzed. Numerical results are presented and discussed.     

Progress in subgrid-scale transport modelling for continuous hybrid non-zonal RANS/LES simulations

The partially integrated transport modelling (PITM) method can be viewed as a continuous approach for hybrid RANS/LES modelling allowing seamless coupling between the RANS and the LES regions. The subgrid turbulence quantities are thus calculated from spectral equations depending on the varying spectral cutoff location [Schiestel, R., Dejoan, A., 2005. Towards a new partially integrated transport model for coarse grid and unsteady turbulent flow simulations. Theoretical and Computational Fluid Dynamics 18, 443–468; Chaouat, B., Schiestel, R., 2005. A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows. Physics of Fluids, 17 (6)] The PITM method can be applied to almost all statistical models to derive its hybrid LES counterpart. In the present work, the PITM version based on the transport equations for the turbulent Reynolds stresses together with the dissipation transport rate equation is now developed in a general formulation based on a new accurate energy spectrum function E(κ) valid in both large and small eddy ranges that allows to calibrate more precisely the c_sgs2 function involved in the subgrid dissipation rate _sgs transport equation. The model is also proposed here in an extended form which remains valid in low Reynolds number turbulent flows. This is achieved by considering a characteristic turbulence length-scale based on the total turbulent energy and the total dissipation rate taking into account the subgrid and resolved parts of the dissipation rate. These improvements allow to consider a large range of flows including various free flows as well as bounded flows. The present model is first tested on the decay of homogeneous isotropic turbulence by referring to the well known experiment of Comte-Bellot and Corrsin. Then, initial perturbed spectra E(κ) with a peak or a defect of energy are considered for analysing the model capabilities in strong non-equilibrium flow situations. The second test case is the classical fully turbulent channel flow that allows to assess the performance of the model in non-homogeneous flows characterised by important anisotropy effects. Different simulations are performed on coarse and refined meshes for checking the grid independence of solutions as well as the consistency of the subgrid-scale model when the filter width is changed. A special attention is devoted to the sharing out of the energy between the subgrid-scales and the resolved scales. Both the mean velocity and the turbulent stress computations are compared with data from direct numerical simulations.     

Critical thought experiment to choose the driving force for interface propagation in inelastic materials

The problem of defining the driving force for interface propagation in inelastic materials is discussed. In most publications, the driving force coincides with the Eshelby driving force, i.e. it represents a total dissipation increment on the moving interface due to all the dissipative processes (phase transition (PT) and plasticity). Recently (Levitas, V.I., 1992a. Post-bifurcation Behaviour in Finite Elastoplasticity. Applications to Strain Localization and Phase Transitions. Universität Hannover. Insititut für Baumecharik and Numerische Mechanik, [BNM-Bencht 1JP 585-LC, 92/5, Hannover; Int. J. Eng. Sci. 33 (1995) 921; Mech. Res. Commun. 22 (1995) 87; J. de Physique III 5 (1995) 173; J. De Physique III 5 (1995) 41; Int. J. Solids Struct. 35 (1998) 889], an alternative approach was developed in which the driving force represents the dissipation increment due to PT only, i.e. total dissipation minus plastic dissipation. The aim of this paper is to prove the contradictory character of application of the Eshelby driving force to inelastic materials. For this purposes, a problem on the interface propagation in a rigid–plastic half-space under homogeneous normal and shear stresses is solved using both definitions, along with the principle of the maximum the driving force. Finite strain theory is used. It appears that the first approach exhibits some qualitative contradictions, which are not observed in our approach. In particular, even for shape memory alloys, when transformation strain can be accommodated elastically (or even without internal stresses), maximization of the Eshelby driving force requires as much plasticity as possible. When applied shear stress tends to the yield stress in shear of a new phase, the driving force tends to infinity, i.e. PT has to always occur at the beginning of plastic flow. Note that in this paper plasticity means dislocation plasticity rather than plasticity due to twinning. Twinning during martensitic PT is the appearance of several martensitic variants which are in twin relation to each other. Consequently, for twinned martensite one has microheterogeneous transformation strain without plastic dissipation term, i.e. both approaches coincide.     

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.