Novel variational formulations for nonlocal plasticity

A nonlocal structural model of softening plasticity is considered in the framework of the internal variable theories of inelastic behaviours of associative type. The finite-step nonlocal structural problem in a geometrically linear range is formulated according to a backward difference scheme for time integration of the flow rule. The related finite-step variational formulation in the complete set of local and nonlocal state variables is recovered. A family of mixed nonlocal variational formulations, with different combinations of state variables, is provided starting from the general variational formulation. The specialization of a mixed variational formulation to existing nonlocal models of softening plasticity, assuming both linear and nonlinear constitutive behaviour, is provided to show the effectiveness of the theory.     

Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals

The paper presents new continuous and discrete variational formulations for the homogenization analysis of inelastic solid materials undergoing finite strains. The point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro-structure as a generalized standard medium in terms of an energy storage and a dissipation function. Consistent with this type of finite inelasticity we develop a new incremental variational formulation of the local constitutive response, where a quasi-hyperelastic micro-stress potential is obtained from a local minimization problem with respect to the internal variables. It is shown that this local minimization problem determines the internal state of the material for finite increments of time. We specify the local variational formulation for a distinct setting of multi-surface inelasticity and develop a numerical solution technique based on a time discretization of the internal variables. The existence of the quasi-hyperelastic stress potential allows the extension of homogenization approaches of finite elasticity to the incremental setting of finite inelasticity. Focussing on macro-deformation-driven micro-structures, we develop a new incremental variational formulation of the global homogenization problem for generalized standard materials at finite strains, where a quasi-hyperelastic macro-stress potential is obtained from a global minimization problem with respect to the fine-scale displacement fluctuation field. It is shown that this global minimization problem determines the state of the micro-structure for finite increments of time. We consider three different settings of the global variational problem for prescribed displacements, non-trivial periodic displacements and prescribed stresses on the boundary of the micro-structure and develop numerical solution methods based on a spatial discretization of the fine-scale displacement fluctuation field. Representative applications of the proposed minimization principles are demonstrated for a constitutive model of crystal plasticity and the homogenization problem of texture analysis in polycrystalline aggregates.     

Variational formulations,convergence and stability properties in nonlocal elastoplasticity

A thermodynamically consistent formulation of nonlocal plasticity in the framework of the internal variable theories of inelastic behaviors of associative type is presented. A family of mixed variational formulations, with different combinations of state variables, is provided starting from the finite-step nonlocal elastoplastic structural problem. It is shown that a suitable minimum principles provides a rational basis to exploit the iterative elastic predictor-plastic corrector algorithm in terms of the dissipation functional. A sufficient condition is proved for the convergence of the iterative elastic predictor-plastic corrector algorithm based on a suitable choice of the elastic operator in the prediction phase and a necessary and sufficient condition for the existence of a unique solution (if any) of the nonlocal problem at hand is then provided. The nonlinear stability analysis of the nonlocal problem is carried out following the concept of nonexpansivity proposed in local plasticity.     

Variational formulations and a consistent finite-element procedure for a class of nonlocal elastic continua

The structural boundary-value problem in the context of nonlocal (integral) elasticity and quasi-static loads is considered in a geometrically linear range. The nonlocal elastic behaviour is described by the so-called Eringen model in which the nonlocality lies in the constitutive relation. The diffusion processes of the nonlocality are governed by an integral relation containing a recently proposed symmetric spatial weight function expressed in terms of an attenuation function. A firm variational basis to the nonlocal model is given by providing the complete set of variational formulations, composed by ten functionals with different combinations of the state variables. In particular the nonlocal counterpart of the classical principles of the total potential energy, complementary energy and mixed Hu–Washizu principle and Hellinger–Reissner functional are recovered. Some examples concerning a piecewise bar in tension are provided by using the Fredholm integral equation and the proposed nonlocal FEM.     

Stress gradient versus strain gradient constitutive models within elasticity

A stress gradient elasticity theory is developed which is based on the Eringen method to address nonlocal elasticity by means of differential equations. By suitable thermodynamics arguments (involving the free enthalpy instead of the free internal energy), the restrictions on the related constitutive equations are determined, which include the well-known Eringen stress gradient constitutive equations, as well as the associated (so far uncertain) boundary conditions. The proposed theory exhibits complementary characters with respect to the analogous strain gradient elasticity theory. The associated boundary-value problem is shown to admit a unique solution characterized by a Hellinger–Reissner type variational principle. The main differences between the Eringen stress gradient model and the concomitant Aifantis strain gradient model are pointed out. A rigorous formulation of the stress gradient Euler–Bernoulli beam is provided; the response of this beam model is discussed as for its sensitivity to the stress gradient effects and compared with the analogous strain gradient beam model.     

A new nonlocal bending model for Euler–Bernoulli nanobeams

This paper is concerned with the bending problem of nanobeams starting from a nonlocal thermodynamic approach. A new coupled nonlocal model, depending on two nonlocal parameters, is obtained by using a suitable definition of the free energy. Unlike previous approaches which directly substitute the expression of the nonlocal stress into the classical equilibrium equations, the proposed approach provides a methodology to recover nonlocal models starting from the free energy function. The coupled model can then be specialized to obtain a nanobeam formulation based on the Eringen nonlocal elasticity theory and on the gradient elastic model. The variational formulations are consistently provided and the differential equations with the related boundary conditions are thus derived. Nanocantilevers are solved in a closed-form and numerical results are presented to investigate the influence of the nonlocal parameters.     

A nonlocal model with strain-based damage

A thermodynamically consistent formulation of nonlocal damage in the framework of the internal variable theories of inelastic behaviours of associative type is presented. The damage behaviour is defined in the strain space and the effective stress turns out to be additively splitted in the actual stress and in the nonlocal counterpart of the relaxation stress related to damage phenomena. An important advantage of models with strain-based loading functions and explicit damage evolution laws is that the stress corresponding to a given strain can be evaluated directly without any need for solving a nonlinear system of equations. A mixed nonlocal variational formulation in the complete set of state variables is presented and is specialized to a mixed two-field variational formulation. Hence a finite element procedure for the analysis of the elastic model with nonlocal damage is established on the basis of the proposed two-field variational formulation. Two examples concerning a one-dimensional bar in simple tension and a two-dimensional notched plate are addressed. No mesh dependence or boundary effects are apparent.     

A multi-field incremental variational framework for gradient-extended standard dissipative solids

The paper presents a constitutive framework for solids with dissipative micro-structures based on compact variational statements. It develops incremental minimization and saddle point principles for a class of gradient-type dissipative materials which incorporate micro-structural fields (micro-displacements, order parameters, or generalized internal variables), whose gradients enter the energy storage and dissipation functions. In contrast to classical local continuum approaches to inelastic solids based on locally evolving internal variables, these global micro-structural fields are governed by additional balance equations including micro-structural boundary conditions. They describe changes of the substructure of the material which evolve relatively to the material as a whole. Typical examples are theories of phase field evolution, gradient damage, or strain gradient plasticity. Such models incorporate non-local effects based on length scales, which reflect properties of the material micro-structure. We outline a unified framework for the broad class of first-order gradient-type standard dissipative solids. Particular emphasis is put on alternative multi-field representations, where both the microstructural variable itself as well as its dual driving force are present. These three-field settings are suitable for models with threshold- or yield-functions formulated in the space of the driving forces. It is shown that the coupled macro- and micro-balances follow in a natural way as the Euler equations of minimization and saddle point principles, which are based on properly defined incremental potentials. These multi-field potential functionals are outlined in both a continuous rate formulation and a time-space-discrete incremental setting. The inherent symmetry of the proposed multi-field formulations is an attractive feature with regard to their numerical implementation. The unified character of the framework is demonstrated by a spectrum of model problems, which covers phase field models and formulations of gradient damage and plasticity.     

Strain gradient crystal plasticity effects on flow localization

In metal grains one of the most important failure mechanisms involves shear band localization. As the band width is small, the deformations are affected by material length scales. To study localization in single grains a rate-dependent crystal plasticity formulation for finite strains is presented for metals described by the reformulated Fleck–Hutchinson strain gradient plasticity theory. The theory is implemented numerically within a finite element framework using slip rate increments and displacement increments as state variables. The formulation reduces to the classical crystal plasticity theory in the absence of strain gradients. The model is used to study the effect of an internal material length scale on the localization of plastic flow in shear bands in a single crystal under plane strain tension. It is shown that the mesh sensitivity is removed when using the nonlocal material model considered. Furthermore, it is illustrated how different hardening functions affect the formation of shear bands.     

Strain gradient plasticity by internal-variable approach with normality structure

This paper presents a constitutive formulation for materials with strain gradient effects by internal-variable approach with normality structure. Specific micro-structural rearrangements are assumed to account for the inelasticity deformations for this class of materials, and enter the constitutive formulations in form of internal variables. It is further assumed that the kinetic evolution of any specific micro-structural rearrangement may be fully determined by the thermodynamic forces associated with that micro-structural rearrangement, by normality relations via a flow potential. Macroscopic gradient-enhanced inelastic behaviours may then be predicted in terms of the microscopic internal variables and their conjugate forces, and thus a micro–macro bridging formulation is available for strain-gradient-characterised materials. The obtained formulations are first applied to crystallographic materials, and a crystal gradient plasticity model is developed to account for the influence of microscopic slip rearrangements on the macroscopic gradient-dependent mechanical behaviour for this class of materials. Micro-cracked geomaterials are also treated with these formulations and a gradient-enhanced damage constitutive model is developed to address the impacts of the evolutions of micro-cracks on the macroscopic inelastic deformations with strain gradient effects for these materials. The available formulations are further compared with other thermodynamic approaches of constitutive developing.     

Boundary element approach to creep deformation of edge notched and cracked specimens

A direct formulation of the boundary element method using a complex variable numerical approach is presented for the time-dependent inelastic stresses in edge notched and cracked creeping metallic structural components subject to high temperature gradients. Particular attention is focused on the numerical evaluation of energy rate contour integrals in single edge cracked specimens in tension. The constitutive models used in the numerical calculation are internal state variable creep–plasticity or elastic power law creep model. Numerical results are compared with solution obtained from other methods for different loading rates.     

Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour

An improved gradient-enhanced approach for softening elasto-plasticity is proposed, which in essence is fully nonlocal, i.e. an equivalent integral nonlocal format exists. The method utilises a nonlocal field variable in its constitutive framework, but in contrast to the integral models computes this nonlocal field with a gradient formulation. This formulation is considered ‘implicit’ in the sense that it strictly incorporates the higher-order gradients of the local field variable indirectly, unlike the common (explicit) gradient approaches. Furthermore, this implicit gradient formulation constitutes an additional partial differential equation (PDE) of the Helmholtz type, which is solved in a coupled fashion with the standard equilibrium condition. Such an approach is particularly advantageous since it combines the long-range interactions of an integral (nonlocal) model with the computational efficiency of a gradient formulation. Although these implicit gradient approaches have been successfully applied within damage mechanics, e.g. for quasi-brittle materials, the first attempts were deficient for plasticity. On the basis of a thorough comparison of the gradient-enhancements for plasticity and damage this paper rephrases the problem, which leads to a formulation that overcomes most reported problems. The two-dimensional finite element implementation for geometrically linear plain strain problems is presented. One- and two-dimensional numerical examples demonstrate the ability of this method to numerically model irreversible deformations, accompanied by the intense localisation of deformation and softening up to complete failure.     

Nonlocal gradient-dependent modeling of plasticity with anisotropic hardening

This work addresses the formulation of the thermodynamics of nonlocal plasticity using the gradient theory. The formulation is based on the nonlocality energy residual introduced by Eringen and Edelen (1972). Gradients are introduced for those variables associated with isotropic and kinematic hardening. The formulation applies to small strain gradient plasticity and makes use of the evanescent memory model for kinematic hardening. This is accomplished using the kinematic flux evolution as developed by Zbib and Aifantis (1988). Therefore, the present theory is a four nonlocal parameter-based theory that accounts for the influence of large variations in the plastic strain, accumulated plastic strain, accumulated plastic strain gradients, and the micromechanical evolution of the kinematic flux. Using the principle of virtual power and the laws of thermodynamics, thermodynamically-consistent equations are derived for the nonlocal plasticity yield criterion and associated flow rule. The presence of higher-order gradients in the plastic strain is shown to enhance a corresponding history variable which arises from the accumulation of the plastic strain gradients. Furthermore, anisotropy is introduced by plastic strain gradients in the form of kinematic hardening. Plastic strain gradients can be attributed to the net Burgers vector, while gradients in the accumulation of plastic strain are responsible for the introduction of isotropic hardening. The equilibrium between internal Cauchy stress and the microstresses conjugate to the higher-order gradients frames the yield criterion, which is obtained from the principle of virtual power. Microscopic boundary conditions, associated with plastic flow, are introduced to supplement the macroscopic boundary conditions of classical plasticity. The nonlocal formulation developed here preserves the classical assumption of local plasticity, wherein plastic flow direction is governed by the deviatoric Cauchy stress. The theory is applied to the problems of thin films on both soft and hard substrates. Numerical solutions are presented for bi-axial tension and simple shear loading of thin films on substrates.     

A unified residual-based thermodynamic framework for strain gradient theories of plasticity

A unified thermodynamic framework for gradient plasticity theories in small deformations is provided, which is able to accommodate (almost) all existing strain gradient plasticity theories. The concept of energy residual (the long range power density transferred to the generic particle from the surrounding material and locally spent to sustain some extra plastic power) plays a crucial role. An energy balance principle for the extra plastic power leads to a representation formula of the energy residual in terms of a long range stress, typically of the third order, a macroscopic counterpart of the micro-forces acting on the GNDs (Geometrically Necessary Dislocations). The insulation condition (implying that no long range energy interactions are allowed between the body and the exterior environment) is used to derive the higher order boundary conditions, as well as to ascertain a principle of the plastic power redistribution in which the energy residual plays the role of redistributor and guarantees that the actual plastic dissipation satisfies the second thermodynamics principle. The (nonlocal) Clausius-Duhem inequality, into which the long range stress enters aside the Cauchy stress, is used to derive the thermodynamic restrictions on the constitutive equations, which include the state equations and the dissipation inequality. Consistent with the latter inequality, the evolution laws are formulated for rate-independent models. These are shown to exhibit multiple size effects, namely (energetic) size effects on the hardening rate, as well as combined (dissipative) size effects on both the yield strength (intrinsic resistance to the onset of plastic strain) and the flow strength (resistance exhibited during plastic flow). A friction analogy is proposed as an aid for a better understanding of these two kinds of strengthening effects. The relevant boundary-value rate problem is addressed, for which a solution uniqueness theorem and a minimum variational principle are provided. Comparisons with other existing gradient theories are presented. The dissipation redistribution mechanism is illustrated by means of a simple shear model.     

Towards variational constitutive updates for non-associative plasticity models at finite strain: Models based on a volumetric-deviatoric split

In this paper, an enhanced variational constitutive update suitable for a class of non-associative plasticity theories at finite strain is proposed. In line with classical numerical formulations for plasticity models, such as the by now established return-mapping algorithm, variational constitutive updates represent a numerical method for computing the unknown state variables. However, in contrast to conventional algorithms, variational constitutive updates are fully variational, i.e., all unknown variables follow jointly from minimizing a certain potential. In addition to the physical and mathematical elegance of these variational schemes, they show several practical advantages as well. For instance, numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem. Since mathematically, plasticity is a non-smooth problem and often, it leads to highly singular systems of equations as known from single crystal plasticity, a robust implementation is of utmost importance. So far, variational constitutive updates have been developed for different classes of standard dissipative solids, i.e., solids characterized by associative evolution equations and flow rules. In the present paper, this framework is extended to a certain class of non-associative plasticity models at finite strain. All models falling into this class show a volumetric-deviatoric split of the Helmholtz energy and the yield function. Typical prototypes are Drucker-Prager or Mohr-Coulomb models playing an important role in soil mechanics. The efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples.     

Formulation of strain gradient plasticity with interface energy in a consistent thermodynamic framework

In this work, the strain gradient formulation is used within the context of the thermodynamic principle, internal state variables, and thermodynamic and dissipation potentials. These in turn provide balance of momentum, boundary conditions, yield condition and flow rule, and free energy and dissipative energies. This new formulation contributes to the following important related issues: (i) the effects of interface energy that are incorporated into the formulation to address various boundary conditions, strengthening and formation of the boundary layers, (ii) nonlocal temperature effects that are crucial, for instance, for simulating the behavior of high speed machining for metallic materials using the strain gradient plasticity models, (iii) a new form of the nonlocal flow rule, (iv) physical bases of the length scale parameter and its identification using nano-indentation experiments and (v) a wide range of applications of the theory. Applications to thin films on thick substrates for various loading conditions and torsion of thin wires are investigated here along with the appropriate length scale effect on the behavior of these structures. Numerical issues of the theory are discussed and results are obtained using Matlab and Mathematica for the nonlinear ordinary differential equations (NODE) which constitute the boundary value problem.This study reveals that the micro-stress term has an important effect on the development of the boundary layers and hardening of the material at both hard and soft interface boundary conditions in thin films. The interface boundary conditions are described by the interfacial length scale and interfacial strength parameters. These parameters are important to control the size effect and hardening of the material. For more complex geometries the generalized form of the boundary value problem using the nonlocal finite element formulation is required to address the problems involved.     

Evolutionary model of finite-strain thermoelasticity

The equation of state of finite-strain thermoelasticity is obtained using a formalized approach to constructing constitutive relations for complex media under the assumption of closeness of intermediate and current configurations. A variational formulation of the coupled thermoelastic problem is proposed. The constitutive equation, the heat-conduction equation, the relations for internal energy, free energy, and entropy, and the variational formulation of the coupled problem of finite-strain thermoelasticity are tested on the problem of uniaxial extension of a bar. The model adequately describes experimental data for elastomers, such as entropic elasticity, temperature inversion, and temperature variation during an adiabatic process. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 184–196, May–June, 2008.     

On nonlocal rate-independent plasticity

A nonlocal rate-independent large strain theory for elastic-plastic bodies consistent with thermodynamic theory is derived. The theory is based on a strain space formulation, where plastic strain is regarded as a primitive variable, characterised by an appropriate constitutive equation for its rate. Stress and free energy are assumed to be functions of a set of nonlocal variables, constructed from a collection of basic state functions, constituted by strain, plastic strain and a scalar measure of strain hardening. A yield function is introduced depending on the same set of independent, nonlocal variables. Yield criteria, flow rules, and loading conditions are formulated. The consistency condition is not, as in local theory, expressed by an algebraic equation, but by an integral equation defined throughout the region of plastic loading.     

A symmetric nonlocal damage theory

The paper presents a thermodynamically consistent formulation for nonlocal damage models. Nonlocal models have been recognized as a theoretically clean and computationally efficient approach to overcome the shortcomings arising in continuum media with softening. The main features of the presented formulation are: (i) relations derived by the free energy potential fully complying with nonlocal thermodynamic principles; (ii) nonlocal integral operator which is self-adjoint at every point of the solid, including zones near to the solid’s boundary; (iii) capacity of regularizing the softening ill-posed continuum problem, restoring a meaningful nonlocal boundary value problem. In the present approach the nonlocal integral operator is applied consistently to the damage variable and to its thermodynamic conjugate force, i.e. nonlocality is restricted to internal variables only. The present model, when associative nonlocal damage flow rules are assumed, allows the derivation of the continuum tangent moduli tensor and the consistent tangent stiffness matrix which are symmetric. The formulation has been compared with other available nonlocal damage theories.Finally, the theory has been implemented in a finite element program and the numerical results obtained for 1-D and 2-D problems show its capability to reproduce in every circumstance a physical meaningful solution and fully mesh independent results.