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 incorporates 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 classical J2 theory have been proposed： one with increments in higher order stresses related to increments of strain gradients and the other characterized by the higher order stresses themselves expressed in terms of increments of strain gradients. The theories proposed by Muhlhans and Aifantis in 1991 and Fleck and Hutchinson in 2001 are in the first class, and, as formulated, these do not always satisfy thermodynamic 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 rateindependent 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 comparable 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.

Strain gradient plasticity: energetic or dissipative?

Abstract For an infinite slab of strain gradient sensitive material subjected to plane-strain tensile loading, compu- tation established and analysis confirmed that passivation of the lateral boundaries at some stage of loading inhibits plastic deformation upon further loading. This result is not surprising in itself except that, remarkably, if the gradient terms contribute to the dissipation, the plastic deformation is switched off completely and only resumes at a clearly defined higher load, corresponding to a total strain ec, say. The analysis presented in this paper confirms the delay of plastic deformation following passivation and determines the exact manner in which the plastic flow resumes. The plastic strain rate is continuous at the exact point ec of resumption of plastic flow and, for the first small increment Ae = e - ec in the imposed total strain, the corresponding increment in plastic strain, AeP, is proportional to （Ae）2. The constant A in the relation AeP（0） = A（Ae）2, where AeP（0） denotes the plastic strain increment at the centre of the slab, has been determined explicitly; it depends on the hardening modulus of the material. The presence of energetic gradient terms has no effect on the value of ec unless the dissipative terms are absent, in which case passivation reduces the rate of plastic deformation but introduces no delay. This qualitative effect of dissipative gradient terms opens the possibility of experimen- tal discrimination of their presence or absence. The analysisemploys an incremental variational formulation that is likely to find use in other problems.

A SEPARATED LAW OF HARDENING IN STRAIN GRADIENT PLASTICITY

This paper presents a separated law of hardening in plasticity with strain gradient effects. The value of the length parameter l contained in this model was estimated from the experimental data for copper.

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.

On dissipative gradient effect in higher-order strain gradient plasticity: the modelling of surface passivation

The phenomenological flow theory of higher-order strain gradient plasticity proposed by Fleck and Hutchinson(J.Mech.Phys.Solids,2001)and then improved by Fleck and Willis(J.Mech.Phys.Solids,2009)is used to investigate the surface-passivation problem and micro-scale plasticity.An extremum principle is stated for the theory involving one material length scale.To solve the initial boundary value problem,a numerical scheme based on the framework of variational constitutive updates is developed for the strain gradient plasticity theory.The main idea is that,in each incremental time step,the value of the effective plastic strain is obtained through the variation of a functional in regard to effective plastic strain,provided the displacement or deformation gradient.Numerical results for elasto-plastic foils under tension and bending,thin wires under torsion,are given by using the minimum principle and the numerical scheme.Implications for the role of dissipative gradient effect are explored for three non-proportional loading conditions:(1)stretch-passivation problem,(2)bending-passivation problem,and(3)torsion-passivation problem.The results indicate that,within the Fleck-Hutchinson-Willis theory,the dissipative length scale controls the strengthening size effect,i.e.the increase of initial yielding strength,while the surface passivation gives rise to an increase of strain hardening rate.

Damage and fracture with strain gradient plasticity for high-capacity electrodes of Li-ion batteries

In consideration of the high-density dislocations from the lithiation process of high-capacity electrodes in Li-ion batteries, in this paper, a new elastoplastic model is established to describe the diffusion-induced deformation and damage fracture. With the help of the relative physical quantities and state of charge, the surface damage and fracture behaviors of electrode materials are discussed based on the elastic-perfectly plastic(PP) and the strain gradient plasticity(SGP) theories, respectively. The results show that the lithiation deformation could be alleviated by reducing the electrode scale, and the plastic flow can play an essential role in the extrusion ratcheting effect relating to the upper surface fracture. Furthermore, the interface damage is more likely to appear by increasing the initial bond stiffness at the upper surface, which has little effect on the later fracture. A strong size effect is also found in the damage and fracture critical curves for the PP and SGP models.

Quantification of grain boundary effects on the geometrically necessary dislocation density evolution and strain hardening of polycrystalline Mg-4Al using in situ tensile testing in scanning electron microscope and HR-EBSD

In situ tensile testing in a scanning electron microscope(SEM)in conjunction with high-resolution electron backscatter diffraction(HR-EBSD)under load was used to characterize the evolution of geometrically necessary dislocation(GND)densities at individual grain boundaries as a function of applied strain in a polycrystalline Mg-4Al alloy.The increase in GND density was investigated at plastic strains of 0%,0.6%,2.2%,3.3% from the area including 76 grains and correlated with(i)geometric compatibility between slip systems across grain boundaries,and(ii)plastic incompatibility.We develop expressions for the grain boundary GND density evolution as a function of plastic strain and plastic incompatibility,from which uniaxial tensile stress-strain response of polycrystalline Mg-4Al are computed and compared with experimental measurement.The findings in this study contribute to understanding the mechanisms governing the strain hardening response of single-phase polycrystalline alloys and more reliable prediction of mechanical behaviors in diverse microstructures.

Generalized Aifantis strain gradient plasticity model with internal length scale dependence on grain size,sample size and strain

The internal length scale(ILS)is a dominant parameter in strain gradient plasticity(SGP)theories,which helps to successfully explain the size effect of metals at the microscale.However,the ILS is usually introduced into strain gradient frameworks for dimensional consistency and is model-dependent.Even now,its physical meaning,connection with the microstructure of the material,and dependence on the strain level have not been thoroughly elucidated.In the current work,Aifantis'SGP model is reformulated by incorporating a recently proposed power-law relation for strain-dependent ILS.A further extension of Aifantis'SGP model by including the grain size effect is conducted according to the Hall-Petch formulation,and then the predictions are compared with torsion experiments of thin wires.It is revealed that the ILS depends on the sample size and grain size simultaneously;these dependencies are dominated by the dislocation spacing and can be well described through the strain hardenmg exponent.Furthermore,both the original and generalized Aifantis models provide larger estimated values for the ILS than Fleck-Hutchinson's theory.

Modeling of Cyclic Bending of Thin Foils Using Higher-Order Strain Gradient Plasticity

The plastic behaviors of thin metallic foils,including size effect,Bauschinger effect,and passivation effect,are studied under cyclic bending condition using the strain gradient visco-plasticity theory.The finite element simulations are performed on the cyclic bending of the elasto-viscoplastic thin foils with passivated and unpassivated surfaces.The study is also conducted on the transition from a passivated surface to an unpassivated one.The roles of the dissipative and energetic gradient terms are emphasized.From the results,it is found that the dissipative gradient terms increase the yield strength,while the energetic gradient terms increase the strain hardening,resulting in an anomalous Bauschinger effect.Further,it is observed that the surface passivation effect increases both the normalized bending moment at initial yielding and strain hardening.The comparison between the numerical results of cases with and without passivation demonstrates that the switching of boundary conditions significantly affects the plastic behavior of the foils under cyclic bending.

Dynamic analysis of fault rockburst based on gradient-dependent plasticity and energy criterion

Fault rockburst is treated as a strain localization problem under dynamicloading condition considering strain gradient and strain rate. As a kind of dynamic fracturephenomena, rockburst has characteristics of strain localization, which is considered as aone-dimensional shear problem subjected to normal compressive stress and tangential shear stress.The constitutive relation of rock material is bilinear (elastic and strain softening) and sensitiveto shear strain rate. The solutions proposed based on gradient-dependent plasticity show thatintense plastic strain is concentrated in fault band and the thickness of the band depends on thecharacteristic length of rock material. The post-peak stiffness of the fault band was determinedaccording to the constitutive parameters of rock material and shear strain rate. Fault bandundergoing strain softening and elastic rock mass outside the band constitute a system and theinstability criterion of the system was proposed based on energy theory. The criterion depends onthe constitutive relation of rock material, the structural size and the strain rate. The staticresult regardless of the strain rate is the special case of the present analytical solution. Highstrain rate can lead to instability of the system.

Quantitative calculation for the dissipated energy of fault rock burst based on gradient-dependent plasticity

The capacity of energy absorption by fault bands after rock burst wascalculated quantitatively according to shear stress-shear deformation curves considering theinteractions and interplaying among microstructures due to the heterogeneity of strain softeningrock materials. The post-peak stiffness of rock specimens subjected to direct shear was derivedstrictly based on gradient-dependent plasticity, which can not be obtained from the classicalelastoplastic theory. Analytical solutions for the dissipated energy of rock burst were proposedwhether the slope of the post-peak shear stress-shear deformation curve is positive or not. Theanalytical solutions show that shear stress level, confining pressure, shear strength, brittleness,strain rate and heterogeneity of rock materials have important influence on the dissipated energy.The larger value of the dissipated energy means that the capacity of energy dissipation in the formof shear bands is superior and a lower magnitude of rock burst is expected under the condition ofthe same work done by external shear force. The possibility of rock burst is reduced for a lowersoftening modulus or a larger thickness of shear bands.

Numerical simulation on size effect of copper with nano-scale twins

To understand the tensile deformation of electro-deposited Cu with nano-scale twins, a numerical study was carried out based on a conventional theory of mechanism-based strain gradient plasticity （CMSG）. The concept of twin lamella strengthening zone was used in terms of the cohesive interface model to simulate grain-boundary sliding and separation. The model included a number of material parameters, such as grain size, elastic modulus, plastic strain hardening exponent, initial yield stress, as well as twin lamellar distribution, which may contribute to size effects of twin layers in Cu polycrystalline. The results provide information to understand the mechanical behaviors of Cu with nano-scale growth twins.

PARTICULATE SIZE EFFECTS IN THE PARTICLE-REINFORCED METAL-MATRIX COMPOSITES

The influences of I,article size on the mechanical properties of the particulate metal matrix composite;are obviously displayed in the experimental observations. However, the phenomenon can not be predicted directly using the conventional elastic-plastic theory. It is because that no length scale parameters are involved in the conventional theory. In the present research, using the strain gradient plasticity theory, a systematic research of the particle size effect in the particulate metal matrix composite is carried out. The roles of many composite factors, such as: the particle size, the Young's modulus of the particle, the particle aspect ratio and volume fraction, as well as the plastic strain hardening exponent of the matrix material, are studied in detail. In order to obtain a general understanding for the composite behavior, two kinds of particle shapes, ellipsoid and cylinder, are considered to check the strength dependence of the smooth or non-smooth particle surface. Finally, the prediction results will be applied to the several experiments about the ceramic particle-reinforced metal-matrix composites. The material length scale parameter is predicted.

The indenter tip radius effect in micro- and nanoindentation hardness experiments

Nix and Gao established an important relation between the microindentation hardness and indentation depth. Such a relation has been verified by many microindentation experiments （indentation depths in the micrometer range）, but it does not always hold in nanoindentation experiments （indentation depths approaching the nanometer range）. Indenter tip radius effect has been proposed by Qu et al. and others as possibly the main factor that causes the deviation from Nix and Gao＇s relationship. We have developed an indentation model for micro- and nanoindentation, which accounts for two indenter shapes, a sharp, conical indenter and a conical indenter with a spherical tip. The analysis is based on the conventional theory of mechanism-based strain gradient plasticity established from the Taylor dislocation model to account for the effect of geometrically necessary dislocations. The comparison between numerical result and Feng and Nix＇s experimental data shows that the indenter tip radius effect indeed causes the deviation from Nix-Gao relation, but it seems not be the main factor.

INFLUENCES OF MICROSCOPE SIZE EFFECT OF MATRIX ON PLASTIC POTENTIAL AND VOID GROWTH OF POROUS MATERIALS

Based on approximate theoretical analyses on a typical spherical cellcontaining a spherical rnicrovoid, the influences of matrix materials' microscopic scale on themacroscopic constitutive potential theory of porous material and microvoid growth have beeninvestigated in detail. By assuming that the plastic: deformation behavior of matrix materialsfollows the strain gradient (SG) plastic theory involving the stretch and rotation gradients , theratio (λ = l/a) of the matrix materials' intrinsic characteristic length l to the micro-void radiusa is introduced into the plastic constitutive potential and the void growth law. The presentresults indicate that, when the radius a of microvoids is comparable with the intrinsiccharacteristic length l of the matrix materials, the influence of microscopic size effect on neitherthe constitutive potential nor the micro-void evolution predicted can be ignored. And when the voidradius a is much lager than the intrinsic characteristic length l of the matrix materials, thepresent model can retrogress automatically to the improved Gur-son model that takes into account thestrain hardening effect of matrix materials.

金属细丝扭转中的尺寸效应与钝化效应

本文对直径在25-50微米、有无钝化层的微尺度铜丝进行了扭转测试,开展了尺度效应与钝化效应的实验研究.通过磁控溅射制备了镀钛铜丝试样,在扭转测试中观察到直径减小和钝化层存在引起的屈服强度和流动应力的增加.与未钝化铜丝相比,钝化铜丝中存在更加明显的尺度效应,钝化铜丝的规范化扭矩随着直径减小而增加更为显著.基于Fleck-Hutchinson应变梯度塑性理论对铜丝的扭转响应进行了预测,数值结算结果与实验数据吻合较好,并阐明了钝化层对铜丝扭转变形影响的物理机制。

Theoretical and experimental researches of size effect in micro-indentation test

Micro-indentation tests at scales on the order of sub-micron have shown that the measured hardness increases strongly with the indent depth or indent size decreasing, which is frequently referred to as the size effect. However, the trend is at odds with the size-independence implied by conventional elastic-plastic theory. In this paper, strain gradient plasticity theory is used to model the size effect for materials undergoing the micro-indenting. Meanwhile, the micro-indentation experiments for single crystal copper and single crystal aluminum are carried out. By the comparison of the theoretical predictions with experimental measurements, the micro-scale parameter of strain gradient plasticity theory is predicted, which is fallen into the region of 0.8—1.5 micron for the conventional metals such as copper (Cu), aluminum (Al) and silver (Ag). Moreover, the phenomena of the pile-up and sink-in near micro-indent boundary are investigated and analyzed in detail.

Study on mathematical model of cutting force in micromachining

With the development of micromachining technology,it is very important to study the mechanism of micromachining,determine the micromachining parameters and ensure the products’quality during the micromachining process.Combined with the micromechanism between tool and workpiece during micromachining process,the sources of the micro-cutting force were analyzed,the micro-cutting physical model was constructed,and the microstress model interacted between the cutting arc edge of the tool and the material of the workpiece was analyzed.Combined with the surface friction and elastic extrusion mechanism between the cutting tool and workpiece,the micro-cutting force model was constructed from two aspects.The micro-cutting depth is deeper than the minimum cutting depth and the micro-cutting depth is shallower than the minimum cutting depth,then the minimum cutting depth value was calculated.Combined with the dislocation properties and microcrystal structure of workpiece’s material,the internal stress of the micromachining force model based on the gradient plasticity theory was calculated,and the force model of the micro-cutting process was studied too.It is significant to control the precision of micromachining process during the micromachining process by constructing the micromachining process force model through studying the small deformation of the material and the mechanism of micromachining.