A finite deformation theory of strain gradient plasticity

Plastic deformation exhibits strong size dependence at the micron scale, as observed in micro-torsion, bending, and indentation experiments. Classical plasticity theories, which possess no internal material lengths, cannot explain this size dependence. Based on dislocation mechanics, strain gradient plasticity theories have been developed for micron-scale applications. These theories, however, have been limited to infinitesimal deformation, even though the micro-scale experiments involve rather large strains and rotations. In this paper, we propose a finite deformation theory of strain gradient plasticity. The kinematics relations (including strain gradients), equilibrium equations, and constitutive laws are expressed in the reference configuration. The finite deformation strain gradient theory is used to model micro-indentation with results agreeing very well with the experimental data. We show that the finite deformation effect is not very significant for modeling micro-indentation experiments.     

On lower order strain gradient plasticity theories

By way of numerical examples, this paper explores the nature of solutions to a class of strain gradient plasticity theories that employ conventional stresses, equilibrium equations and boundary conditions. Strain gradients come into play in these modified conventional theories only to alter the tangent moduli governing increments of stress and strain. It is shown that the modification is far from benign from a mathematical standpoint, changing the qualitative character of solutions and leading to a new type of localization that is at odds with what is expected from a strain gradient theory. The findings raise questions about the physical acceptability of this class of strain gradient theories.     

The role of macroscopic hardening and individual length-scales on crack tip stress elevation from phenomenological strain gradient plasticity

This paper quantifies the effect of strain gradient plasticity (SGP) on crack tip stress elevation for a broad range of applied loading conditions and constitutive model parameters, including both macroscopic hardening parameters and individual material length-scales controlling gradient effects. Finite element simulations incorporating the Fleck-Hutchinson SGP theory are presented for an asymptotically sharp stationary crack. Results identify fundamental scaling relationships describing (i) the physical length-scales over which strain gradients are prominent, and (ii) the degree of stress elevation over conventional Hutchinson-Rice-Rosengren (HRR) fields. Results illustrate that the three length-scale theory predicts much larger SGP effects than the single length-scale theory. Critically, the first length-scale parameter dominates SGP stress elevation: this suggests that SGP effects in fracture can be predicted using the length-scales extracted from nanoindentation, which exhibits similar behavior. Transitional loading/material parameters are identified that establish regimes of SGP relevance: this provides the foundation for the rational application of SGP when developing new micromechanical models of crack tip damage mechanisms and associated subcritical crack propagation behavior in structural alloys.     

Generalizing J 2 flow theory: Fundamental issues in strain gradient plasticity

It has not been a simple matter to obtain a sound extension of the classical J₂ flow theory of plasticity that in- corporates a dependence on plastic strain gradients and that is capable of capturing size-dependent behaviour of metals at the micron scale. Two classes of basic extensions of clas- sical J₂ theory have been proposed: one with increments in higher order stresses related to increments of strain gradi- ents and the other characterized by the higher order stresses themselves expressed in terms of increments of strain gra- dients. The theories proposed by Muhlhaus and Aifantis in 1991 and Fleck and Hutchinson in 2001 are in the first class, and, as formulated, these do not always satisfy ther- modynamic requirements on plastic dissipation. On the other hand, theories of the second class proposed by Gudmundson in 2004 and Gurtin and Anand in 2009 have the physical deficiency that the higher order stress quantities can change discontinuously for bodies subject to arbitrarily small load changes. The present paper lays out this background to the quest for a sound phenomenological extension of the rate- independent J₂ flow theory of plasticity to include a de- pendence on gradients of plastic strain. A modification of the Fleck-Hutchinson formulation that ensures its thermo- dynamic integrity is presented and contrasted with a compa- rable formulation of the second class where in the higher or- der stresses are expressed in terms of the plastic strain rate. Both versions are constructed to reduce to the classical J₂ flow theory of plasticity when the gradients can be neglected and to coincide with the simpler and more readily formulated J₂ deformation theory of gradient plasticity for deformation histories characterized by proportional straining.     

A homogenization theory of strain gradient single crystal plasticity and its finite element discretization

In this study, a homogenization theory based on the Gurtin strain gradient formulation and its finite element discretization are developed for investigating the size effects on macroscopic responses of periodic materials. To derive the homogenization equations consisting of the relation of macroscopic stress, the weak form of stress balance, and the weak form of microforce balance, the Y-periodicity is used as additional, as well as standard, boundary conditions at the boundary of a unit cell. Then, by applying a tangent modulus method, a set of finite element equations is obtained from the homogenization equations. The computational stability and efficiency of this finite element discretization are verified by analyzing a model composite. Furthermore, a model polycrystal is analyzed for investigating the grain size dependence of polycrystal plasticity. In this analysis, the micro-clamped, micro-free, and defect-free conditions are considered as the additional boundary conditions at grain boundaries, and their effects are discussed.     

Isotropic hardening in micropolar plasticity

Experimental evidence for length scale effects in plasticity has been provided, e.g., by Fleck et al. (Acta Metall. Mater. 42:475–487, 1994). Results from torsional loadings on copper wires, when appropriately displayed, indicated that, for the same shear at the outer radius, the normalized torque increased with decreasing specimen radius. Modeling of the constitutive behavior in the framework of micropolar plasticity is a possibility to account for length scale effects. The present paper is concerned with this possibility and deals with the theory developed by Grammenoudis and Tsakmakis (Contin. Mech. Thermodyn. 13:325–363, 2001; Int. J. Numer. Methods Eng. 62:1691–1720, 2005; Proc. R. Soc. 461:189–205, 2005). Both isotropic and kinematic hardening are present in that theory, with isotropic hardening being captured in a unified manner. Here, we discuss isotropic hardening composed of two parts, responsible for strain and gradient effects, respectively.     

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.     

Fracture of bicrystal metal/ceramic interfaces: A study via the mechanism-based strain gradient crystal plasticity theory

Two continuum mechanical models of crystal plasticity theory namely, conventional crystal plasticity theory and mechanism-based crystal plasticity theory, are used to perform a comparative study of stresses that are reached at and ahead of the crack tip of a bicrystal niobium/alumina specimen. Finite element analyses are done for a stationary crack tip and growing cracks using a cohesive modelling approach. Using mechanism-based strain gradient crystal plasticity theory the stresses reached ahead of the crack tip are found to be two times larger than the stresses obtained from conventional crystal plasticity theory. Results also show that strain gradient effects strongly depend on the intrinsic material length to the size of plastic zone ratio (l/R₀). It is found that the larger the (l/R₀) ratio, the higher the stresses reached using mechanism-based strain gradient crystal plasticity theory. An insight into the role of cohesive strength and work of adhesion in macroscopic fracture is also presented which can be used by experimentalists to design better bimaterials by varying cohesive strength and work of adhesion.     

Strain gradient effects on cyclic plasticity

Size effects on the cyclic shear response are studied numerically using a recent higher order strain gradient visco-plasticity theory accounting for both dissipative and energetic gradient hardening. Numerical investigations of the response under cyclic pure shear and shear of a finite slab between rigid platens have been carried out, using the finite element method. It is shown for elastic-perfectly plastic solids how dissipative gradient effects lead to increased yield strength, whereas energetic gradient contributions lead to increased hardening as well as a Bauschinger effect. For linearly hardening materials it is quantified how dissipative and energetic gradient effects promote hardening above that of conventional predictions. Usually, increased hardening is attributed to energetic gradient effects, but here it is found that also dissipative gradient effects lead to additional hardening in the presence of conventional material hardening. Furthermore, it is shown that dissipative gradient effects can lead to both an increase and a decrease in the dissipation per load cycle depending on the magnitude of the dissipative length parameter, whereas energetic gradient effects lead to decreasing dissipation for increasing energetic length parameter. For dissipative gradient effects it is found that dissipation has a maximum value for some none zero value of the material length parameter, which depends on the magnitude of the deformation cycles.     

On the homogenization of metal matrix composites using strain gradient plasticity

The homogenized response of metal matrix composites（MMC） is studied using strain gradient plasticity.The material model employed is a rate independent formulation of energetic strain gradient plasticity at the micro scale and conventional rate independent plasticity at the macro scale. Free energy inside the micro structure is included due to the elastic strains and plastic strain gradients. A unit cell containing a circular elastic fiber is analyzed under macroscopic simple shear in addition to transverse and longitudinal loading. The analyses are carried out under generalized plane strain condition. Micro-macro homogenization is performed observing the Hill-Mandel energy condition,and overall loading is considered such that the homogenized higher order terms vanish. The results highlight the intrinsic size-effects as well as the effect of fiber volume fraction on the overall response curves, plastic strain distributions and homogenized yield surfaces under different loading conditions. It is concluded that composites with smaller reinforcement size have larger initial yield surfaces and furthermore,they exhibit more kinematic hardening.     

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.     

Modelling of the interface between a thin film and a substrate within a strain gradient plasticity framework

Interfaces play an important role for the plastic deformation at the micron scale. In this paper, two types of interface models for isotropic materials are developed and applied in a thin film analysis. The first type, which can also be motivated from dislocation theory, assumes that the plastic work at the interface is stored as a surface energy that is linear in plastic strain. In the second model, the plastic work is completely dissipated and there is no build-up of a surface energy. Both formulations introduce one length scale parameter for the bulk material and one for the interface, which together control the film behaviour. It is demonstrated that the two interface models give equivalent results for a monotonous, increasing load. The combined influence of bulk and interface is numerically studied and it is shown that size effects are obtained, which are controlled by the length scale parameters of bulk and interface.     

Strain hardening in bent copper foils

A series of systematic tensile and microbend tests were conducted on copper foil specimens with different thicknesses. The specimens were made of a copper foil having almost unidirectional crystal orientations that was considered to be nearly single-crystal. In order to investigate the effects of slip system interactions, two different crystal orientations relative to the tensile direction were considered in the tests: one is close to coplanar double-slip orientation, and the other is close to the ideal cube orientation (the tensile direction nearly coincides to [0 0 1]) that yields multi-planar multi-slip deformation. We extended the microbend test method to include the reversal of bending, and we attempted to divide the total amount of strain-hardening into isotropic and kinematic hardening components. In the tensile tests, no systematic tendency of size dependence was observed. In the microbend tests, size-dependent kinematic hardening behavior was observed for both the crystal orientations, while size dependence of isotropic hardening was observed only for the multi-planar multi-slip case. We introduce an extended crystal plasticity model that accounts for the effects of the geometrically necessary dislocations (GNDs), which correspond to the spatial gradients of crystallographic slips. Through numerical simulations performed using the model, the origin of the size-dependent behavior observed in the microbend tests is discussed.     

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.     

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.     

A generic stress-strain model for metallic materials with two-stage strain hardening behaviour

Constitutive equations are often used to describe the stress-strain behaviour of metallic materials. This allows the execution of parametric studies for various purposes. Despite the large number of developed stress-strain equations, all frequently applied ones fail to accurately describe a strain hardening behaviour in two distinct stages, which many metallic materials tend to show. For this purpose, the authors developed a new stress-strain model, based on the well-known Ramberg-Osgood equation, which focuses on this two-stage strain hardening behaviour. This article describes the model and its analytical background, along with a graphical method to derive suited model parameters. To validate the proposed methodology, it is applied on stress-strain curves of two high-strength steels, an aluminium alloy and a duplex stainless-steel alloy. Whereas a good correspondence for the stainless-steel alloy is confined to limited plastic strains, excellent agreements are observed for the steels and the aluminium alloy. Following the proposed method, it was possible to obtain model parameter values that give a good correspondence within a detectable strain range.     

叠加型A-F类随动强化模型塑性应变的数值计算法

以Chaboche随动强化模型为例,在M isses屈服准则及正交流动准则的前提下,推导了叠加型A rm-strong-F rederick(A-F)类随动强化模型塑性应变的数值计算法,联合利用四阶龙格-库塔法与径向返回法实现数值计算中的内部平衡迭代。同时推导了统一切向矩阵以便确定每一平衡迭代后的试算应变。利用AN SY S提供的U PF s将算法嵌入到AN SY S有限元程序,实现了叠加型A-F类随动强化模型塑性应变的数值计算,并利用四边形单元模拟了单轴循环加载时的棘轮应变,计算结果能够很好地与实验值吻合。     

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.     

The evolution of voids in the plasticity strain hardening gradient materials

The main aim of this paper is to opens out the meso-mechanism of void growth and coalescence in the matrix materials with graded strain-hardening exponent distribution. For this end, detailed finite element computations of a representative cylindrical cell containing a spherical void have been carried out. According to the FE analyses, significant effects of the strain-hardening exponent gradient (SEG) in the matrix on the void growth and coalescence are revealed: (1) In the homogeneous materials, the void growth and coalescence are slightly dependent on the strain-hardening exponent, however, the SEG distribution in the matrix can increase remarkably the void growth rate and decrease seriously the void coalescence strain. (2) The critical void shapes in the homogeneous materials are mainly governed by the macroscopic stress triaxiality, but due to earlier plastic flow localization in the softer matrix layer, the SEG distribution in the matrix has very significant effects on the deformed void shapes, especially when the stress triaxiality is lower. (3) When the triaxial stress levels are lower, in the homogeneous materials, the shape change mode of the void evolution is dominate so the void growth rate is very low; however, the SEG distribution in the matrix can bring the volume change mode out, as a result of increasing the void growth rate. (4) Comparisons of the numerical results with the existing damage model indicate that the classic damage model cannot give satisfying prediction to the void growth in both the homogeneous strain-hardening matrix and the SEG materials. On the basis of large numbers of numerical computations, a new damage model, which can uniformly describe the void growing in the homogeneous and plasticity gradient materials, is suggested. A mass of element computations have validated that the new damage model can give satisfying agreement with the FE results of cell model.