A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part II: Finite deformations

This paper generalizes to finite deformations our companion paper [Gurtin, M.E., Anand, L., 2004. A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. Journal of the Mechanics and Physics of Solids, submitted]. Specifically, we develop a gradient theory for finite-deformation isotropic viscoplasticity in the absence of plastic spin. The theory is based on the Kröner–Lee decomposition F = F^eF^p of the deformation gradient into elastic and plastic parts; a system of microstresses consistent with a microforce balance; a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; a constitutive theory that allows:

• the microstresses to depend on D^p, the gradient of the plastic stretching,

• the free energy ψ to depend on the Burgers tensor G = F^pCurlF^p.

The microforce balance when augmented by constitutive relations for the microstresses results in a nonlocal flow rule in the form of a tensorial second-order partial differential equation for F^p. The microstresses are strictly dissipative when ψ is independent of the Burgers tensor, but when ψ depends on G the microstresses are partially energetic, and this, in turn, leads to backstresses and (hence) Bauschinger-effects in the flow rule. The typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with viscoplastic flow, and, as an aid to numerical solution, a weak (virtual power) formulation of the nonlocal flow rule is derived. Finally, the dependences of the microstresses on D^p are shown, analytically, to result in strengthening and possibly weakening of the body induced by viscoplastic flow.     

A one-dimensional theory of strain-gradient plasticity: Formulation, analysis, numerical results

This study develops a one-dimensional theory of strain-gradient plasticity based on: (i) a system of microstresses consistent with a microforce balance; (ii) a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; (iii) a constitutive theory that allows

•

the free-energy to depend on the gradient of the plastic strain, and

•

the microstresses to depend on the gradient of the plastic strain-rate.

The constitutive equations, whose rate-dependence is of power-law form, are endowed with energetic and dissipative gradient length-scales L and l, respectively, and allow for a gradient-dependent generalization of standard internal-variable hardening. The microforce balance when augmented by the constitutive relations for the microstresses results in a nonlocal flow rule in the form of a partial differential equation for the plastic strain. Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and properties of the resulting boundary-value problem are studied both analytically and numerically. The resulting solutions are shown to exhibit three distinct physical phenomena:

(i)

standard (isotropic) internal-variable hardening;

(ii)

energetic hardening, with concomitant back stress, associated with plastic-strain gradients and resulting in boundary layer effects;

(iii)

dissipative strengthening associated with plastic strain-rate gradients and resulting in a size-dependent increase in yield strength.

     

A large-deformation strain-gradient theory for isotropic viscoplastic materials

This study develops a thermodynamically consistent large-deformation theory of strain-gradient viscoplasticity for isotropic materials based on: (i) a scalar and a vector microstress consistent with a microforce balance; (ii) a mechanical version of the two laws of thermodynamics for isothermal conditions, that includes via the microstresses the work performed during viscoplastic flow; and (iii) a constitutive theory that allows:     

An alternative treatment of phenomenological higher-order strain-gradient plasticity theory

Phenomenological higher-order strain-gradient plasticity is here presented through a formulation inspired by previous work for strain-gradient crystal plasticity. A physical interpretation of the phenomenological yield condition that involves an effect of second gradient of the equivalent plastic strain is discussed, applying a dislocation theory-based consideration. Then, a differential equation for the equivalent plastic strain-gradient is introduced as an additional governing equation. Its weak form makes it possible to deduce and impose extra boundary conditions for the equivalent plastic strain. A connection between the present treatment and strain-gradient theories based on an extended virtual work principle is discussed. Furthermore, a numerical implementation and analysis of constrained simple shear of a thin strip are presented.     

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

A phenomenological, flow theory version of gradient plasticity for isotropic and anisotropic solids is constructed along the lines of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain-gradient plasticity. J. Mech. Phys. Solids 52, 1379-1406]. Both energetic and dissipative stresses are considered in order to develop a kinematic hardening theory, which in the absence of gradient terms reduces to conventional J₂ flow theory with kinematic hardening. The dissipative stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition with associated plastic flow. The theory includes interfacial terms: elastic energy is stored and plastic work is dissipated at internal interfaces, and a yield surface is postulated for the work-conjugate stress quantities at the interface. Uniqueness and extremum principles are constructed for the solution of boundary value problems, for both the rate-dependent and the rate-independent cases. In the absence of strain gradient and interface effects, the minimum principles reduce to the classical extremum principles for a kinematically hardening elasto-plastic solid. A rigid-hardening version of the theory is also stated and the resulting theory gives rise to an extension to the classical limit load theorems. This has particular appeal as previous trial fields for limit load analysis can be used to generate immediately size-dependent bounds on limit loads.     

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

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

A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin

This study develops a gradient theory of small-deformation viscoplasticity based on: a system of microforces consistent with its peculiar balance; a mechanical version of the second law that includes, via the microforces, work performed during viscoplastic flow; a constitutive theory that accounts for the Burgers vector through a free energy dependent on , with H^p the plastic part of the elastic-plastic decomposition of the displacement gradient. The microforce balance and the constitutive equations, restricted by the second law, are shown to be together equivalent to a nonlocal flow rule in the form of a coupled pair of second-order partial differential equations. The first of these is an equation for the plastic strain-rate in which the stress T plays a basic role; the second, which is independent of T, is an equation for the plastic spin. A consequence of this second equation is that the plastic spin vanishes identically when the free energy is independent of, but not generally otherwise. A formal discussion based on experience with other gradient theories suggests that sufficiently far from boundaries solutions should not differ appreciably from classical solutions, but close to microscopically hard boundaries, boundary layers characterized by a large Burgers vector and large plastic spin should form.Because of the nonlocal nature of the flow rule, the classical macroscopic boundary conditions need be supplemented by nonstandard boundary conditions associated with viscoplastic flow. As an aid to solution, a variational formulation of the flow rule is derived.Finally, we sketch a generalization of the theory that allows for isotropic hardening resulting from dissipative constitutive dependences on .     

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.     

Thermodynamics applied to gradient theories involving the accumulated plastic strain: The theories of Aifantis and Fleck and Hutchinson and their generalization

We discuss the physical nature of flow rules for rate-independent (gradient) plasticity laid down by Aifantis and by Fleck and Hutchinson. As a central result we show that:

•

the flow rule of Fleck and Hutchinson is incompatible with thermodynamics unless its nonlocal term is dropped. If the underlying theory is augmented by a general defect energy dependent on γ^p and ∇γ^p, then compatibility with thermodynamics requires that its flow rule reduce to that of Aifantis.

We establish this result (and others) within a general framework obtained by combining a virtual-power principle of Fleck and Hutchinson with the first two laws of thermodynamics—balance of energy and the Clausius-Duhem inequality—under isothermal conditions.     

A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part I: Formulation

In this Part I, of a two-part paper, we present a detailed continuum-mechanical development of a thermo-mechanically coupled elasto-viscoplasticity theory to model the strain rate and temperature dependent large-deformation response of amorphous polymeric materials. Such a theory, when further specialized (Part II) should be useful for modeling and simulation of the thermo-mechanical response of components and structures made from such materials, as well as for modeling a variety of polymer processing operations.     

A theory for grain boundaries with strain-gradient plasticity

In this work, the effect of the material microstructural interface between two materials (i.e., grain boundary in polycrystalls) is adopted into a thermodynamic-based higher order strain gradient plasticity framework. The developed grain boundary flow rule accounts for the energy storage at the grain boundary due to the dislocation pile up as well as energy dissipation caused by the dislocation transfer through the grain boundary. The theory is developed based on the decomposition of the thermodynamic conjugate forces into energetic and dissipative counterparts which provides the constitutive equations to have both energetic and dissipative gradient length scales for the grain and grain boundary. The numerical solution for the proposed framework is also presented here within the finite element context. The material parameters of the gradient framework are also calibrated using an extensive set of micro-scale experimental measurements of thin metal films over a wide range of size and temperature of the samples.     

A finite deformation theory of higher-order gradient crystal plasticity

For higher-order gradient crystal plasticity, a finite deformation formulation is presented. The theory does not deviate much from the conventional crystal plasticity theory. Only a back stress effect and additional differential equations for evolution of the geometrically necessary dislocation (GND) densities supplement the conventional theory within a non-work-conjugate framework in which there is no need to introduce higher-order microscopic stresses that would be work-conjugate to slip rate gradients. We discuss its connection to a work-conjugate type of finite deformation gradient crystal plasticity that is based on an assumption of the existence of higher-order stresses. Furthermore, a boundary-value problem for simple shear of a constrained thin strip is studied numerically, and some characteristic features of finite deformation are demonstrated through a comparison to a solution for the small deformation theory. As in a previous formulation for small deformation, the present formulation applies to the context of multiple and three-dimensional slip deformations.     

A theory for amorphous viscoplastic materials undergoing finite deformations, with application to metallic glasses

This study develops a finite-deformation, Coulomb-Mohr type constitutive theory for the elastic-viscoplastic response of pressure-sensitive and plastically-dilatant isotropic materials. The constitutive model has been implemented in a finite element program, and the numerical capability is used to study the deformation response of amorphous metallic glasses. Specifically, the response of an amorphous metallic glass in tension, compression, strip-bending, and indentation is studied, and it is shown that results from the numerical simulations qualitatively capture major features of corresponding results from physical experiments available in the literature.     

Generalization of strain-gradient theory to finite elastic deformation for isotropic materials

This paper concerns finite deformation in the strain-gradient continuum. In order to take account of the geometric nonlinearity, the original strain-gradient theory which is based on the infinitesimal strain tensor is rewritten given the Green–Lagrange strain tensor. Following introducing the generalized isotropic Saint Venant–Kirchhoff material model for the strain-gradient elasticity, the boundary value problem is investigated in not only the material configuration but also the spatial configuration building upon the principle of virtual work for a three-dimensional solid. By presenting one example, the convergence of the strain-gradient and classical theories is studied.     

A nonlocal strain gradient plasticity theory for finite deformations

Strain gradient plasticity for finite deformations is addressed within the framework of nonlocal continuum thermodynamics, featured by the concepts of (nonlocality) energy residual and globally simple material. The plastic strain gradient is assumed to be physically meaningful in the domain of particle isoclinic configurations (with the director vector triad constant both in space and time), whereas the objective notion of corotational gradient makes it possible to compute the plastic strain gradient in any domain of particle intermediate configurations. A phenomenological elastic–plastic constitutive model is presented, with mixed kinematic/isotropic hardening laws in the form of PDEs and related higher order boundary conditions (including those associated with the moving elastic/plastic boundary). Two fourth-order projection tensor operators, functions of the elastic and plastic strain states, are shown to relate the skew-symmetric parts of the Mandel stress and back stress to the related symmetric parts. Consistent with the thermodynamic restrictions therein derived, the flow laws for rate-independent associative plasticity are formulated in a six-dimensional tensor space in terms of symmetric parts of Mandel stresses and related work-conjugate generalized plastic strain rates. A simple shear problem application is presented for illustrative purposes.     

A general theory for nonlocal softening plasticity of integral-type

This paper deals with a comparison of several models, proposed in the literature, of softening plasticity with internal variables regularized by nonlocal averaging of integral type.     

Gradient plasticity modeling of geomaterials in a meshfree environment. Part I: Theory and variational formulation

Deformation and strength behavior of geomaterials in the pre- and post-failure regimes are of significant interest in various geomechanics applications. To address the need for development of a realistic constitutive framework, which allows for an accurate simulation of pre-failure response as well as an objective and meaningful post-failure response, a strain gradient plasticity model is formulated by incorporating the spatial gradients of elastic strain in the evolution of stress and gradients of plastic strain in the evolution of the internal variables. In turn, gradients of only kinematic variables are included in the constitutive equations. The resulting constitutive equations along with the balance of linear momentum for the continuum are cast as a coupled system of equations, with displacements and plastic multiplier appearing as the primary unknowns in the final governing integral equations. To avoid singular stress fields along element boundaries, a finite element discretization of the governing equations would require C² continuous displacements and C¹ continuous plastic multiplier, which is undesirable from a numerical implementation point of view. This issue is naturally resolved when a meshfree discretization is used. Hence the developed model is formulated within the framework of a meshfree environment. The new constitutive model allows an analysis of grain size effects on strength and dilatancy of rocks. The role and effectiveness of the new gradient terms on regularizing the underlying boundary value problems of geomechanics beyond the initiation of strain localization will be assessed in a future paper.