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.     

A thermodynamic based higher-order gradient theory for size dependent plasticity

A physically motivated and thermodynamically consistent formulation of small strain higher-order gradient plasticity theory is presented. Based on dislocation mechanics interpretations, gradients of variables associated with kinematic and isotropic hardenings are introduced. This framework is a two non-local parameter framework that takes into consideration large variations in the plastic strain tensor and large variations in the plasticity history variable; the equivalent (effective) plastic strain. The presence of plastic strain gradients is motivated by the evolution of dislocation density tensor that results from non-vanishing net Burgers vector and, hence, incorporating additional kinematic hardening (anisotropy) effects through lattice incompatibility. The presence of gradients in the effective (scalar) plastic strain is motivated by the accumulation of geometrically necessary dislocations and, hence, incorporating additional isotropic hardening effects (i.e. strengthening). It is demonstrated that the non-local yield condition, flow rule, and non-zero microscopic boundary conditions can be derived directly from the principle of virtual power. It is also shown that the local Clausius–Duhem inequality does not hold for gradient-dependent material and, therefore, a non-local form should be adopted. The non-local Clausius–Duhem inequality has an additional term that results from microstructural long-range energy interchanges between the material points within the body. A detailed discussion on the physics and the application of proper microscopic boundary conditions, either on free surfaces, clamped surfaces, or intermediate constrained surfaces, is presented. It is shown that there is a close connection between interface/surface energy of an interface or free surface and the microscopic boundary conditions in terms of microtraction stresses. Some generalities and utility of this theory are discussed and comparisons with other gradient theories are given. Applications of the proposed theory for size effects in thin films are presented.     

Determination of the effective elastic–plastic response of metal–ceramic composites

The mechanical response of metal–ceramic composites is analysed through a homogenization model accounting for the mechanical behaviour of the constituent materials. In order to achieve this purpose a nonlinear homogenization method based on the phase field approach has been suitably implemented into a numerical code. A prescribed homogenized strain state is applied to a unit volume element of a metal–ceramic composite with proportional loading in which all components of the strain tensor are proportional to one scalar parameter. The mechanical response of the material has been modeled by considering a von Mises plasticity model for the metal phase and a Drucker–Prager associative elastic–plastic material model for the ceramic phase. A two stages plasticity has been obtained in which inelastic strain develops in the metal phase followed by a fully plastic response. A comparison with a finite element model of the stress–strain response of an axisymmetric unit cell has been carried out with the purpose to validate the homogenization based modeling presented in the paper. Plastic parameters of a Drucker–Prager yield surface for the homogenized composite have been calculated at different materials compositions. Associative Drucker–Prager plasticity has been found to be accurate for high ceramic content.     

Plastic deformation modes in perforated sheets and their relation to yield and limit surfaces

Construction of mechanism-based plasticity theories for the homogenized response of heterogeneous materials requires identification of plastic deformation modes as a function of loading direction relative to the microstructural details. Herein, we employ an efficient homogenization theory to identify for the first time such deformation modes in plates under plane stress with hexagonal arrays of circular holes at small and intermmediate pore volume fractions, and establish their relation to the branches of initial and subsequent yield and limit surfaces. Newly introduced maps of the intrinsic geometric features of point-wise yield surfaces provide full-field picture of the investigated microstructures’ propensity for plastic strain initiation and localization. The identified characteristic plastic modes provide a rational explanation for the evolving geometric features of subsequent yield and limit surfaces whose branches represent different plastic flow mechanisms, as well as a basis for the construction of a mechanism-based homogenized plasticity theory for use in structural analysis algorithms. The results suggest the need for composite yield surfaces comprised of multiple branches in the construction of mechanism-based homogenized plasticity theory for the investigated class of porous materials.     

Size-dependent yield strength of thin films

Biaxial strain and pure shear of a thin film are analysed using a strain gradient plasticity theory presented by Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52, 1379–1406]. Constitutive equations are formulated based on the assumption that the free energy only depends on the elastic strain and that the dissipation is influenced by the plastic strain gradients. The three material length scale parameters controlling the gradient effects in a general case are here represented by a single one. Boundary conditions for plastic strains are formulated in terms of a surface energy that represents dislocation buildup at an elastic/plastic interface. This implies constrained plastic flow at the interface and it enables the simulation of interfaces with different constitutive properties. The surface energy is also controlled by a single length scale parameter, which together with the material length scale defines a particular material.Numerical results reveal that a boundary layer is developed in the film for both biaxial and shear loading, giving rise to size effects. The size effects are strongly connected to the buildup of surface energy at the interface. If the interface length scale is small, the size effect vanishes. For a stiffer interface, corresponding to a non-vanishing surface energy at the interface, the yield strength is found to scale with the inverse of film thickness.Numerical predictions by the theory are compared to different experimental data and to dislocation dynamics simulations. Estimates of material length scale parameters are presented.     

Effects of microscopic boundary conditions on plastic deformations of small-sized single crystals

The finite deformation version of the higher-order gradient crystal plasticity model proposed by the authors is applied to solve plane strain boundary value problems, in order to obtain an understanding of the effect of the higher-order boundary conditions. Numerical solutions are carried out for uniaxial plane strain compression of a single crystal block and for uniform pure bending of a single crystal foil. The compressed block has loading surfaces that are penetrable or impenetrable to dislocations. This allows for a study of the two types of higher-order boundaries available, and a significant effect of higher-order boundary conditions on the overall deformation mode of the block is observed. The bent foil has free surfaces through which dislocations can go out of the material, and we observe a strong size-dependent mechanical response resulting from the surface condition assumed.     

A material model for finite elasto-plastic deformations considering a substructure

Developing further the substructure models proposed by Mandel and Dafalias a thermodynamically consistent system of differential and algebraic equations is derived to describe anisotropic elasto-plastic material behavior at finite deformations. Based on the multiplicative split of the deformation gradient an appropriate material law is formulated applying the principle of the maximum of plastic dissipation. Generalized basic relations of this material model containing a relation of hyperelasticity, evolutional equations for the internal variables describing different kinds of hardening, and the yield condition are presented. The capacity of the proposed material model is demonstrated on the example of a sheet with a hole. Presenting the evolution of yield surfaces the capability of the model to describe anisotropic hardening behavior is shown.     

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.     

Strain gradient plasticity,strengthening effects and plastic limit analysis

Within the framework of isotropic strain gradient plasticity, a rate-independent constitutive model exhibiting size dependent hardening is formulated and discussed with particular concern to its strengthening behavior. The latter is modelled as a (fictitious) isotropic hardening featured by a potential which is a positively degree-one homogeneous function of the effective plastic strain and its gradient. This potential leads to a strengthening law in which the strengthening stress, i.e. the increase of the plastically undeformed material initial yield stress, is related to the effective plastic strain through a second order PDE and related higher order boundary conditions. The plasticity flow laws, with the role there played by the strengthening stress, are addressed and shown to admit a maximum dissipation principle. For an idealized elastic perfectly plastic material with strengthening effects, the plastic collapse load problem of a micro/nano scale structure is addressed and its basic features under the light of classical plastic limit analysis are pointed out. It is found that the conceptual framework of classical limit analysis, including the notion of rigid-plastic behavior, remains valid. The lower bound and upper bound theorems of classical limit analysis are extended to strengthening materials. A static-type maximum principle and a kinematic-type minimum principle, consequences of the lower and upper bound theorems, respectively, are each independently shown to solve the collapse load problem. These principles coincide with their respective classical counterparts in the case of simple material. Comparisons with existing theories are provided. An application of this nonclassical plastic limit analysis to a simple shear model is also presented, in which the plastic collapse load is shown to increase with the decreasing sample size (Hall–Petch size effects).     

A finite strain constitutive model for metal powder compaction using a unique and convex single surface yield function

Constitutive modelling of metal powder compaction processes is a challenge in view of realistic simulations. To this end, the article under consideration has two objectives: the first goal is to present a new unique and convex single surface yield function for pressure dependent materials, which is also applicable to other areas of granular materials such as soils or concrete. The flexibility is shown at various materials. The yield function is based on a log-interpolation of two known simple yield functions. A convexity proof of the new yield function is provided. The second objective is to propose a new rate-independent finite strain plasticity model for metal powder compaction, which is based on the multiplicative decomposition of the deformation gradient into an elastic and a plastic part with evolution equations for internal variables representing the basic behaviour of powder materials under compaction conditions. These variables are used for the evolution of the yield function in order to represent the compressible hardening behaviour of powder materials. On the basis of the constitutive model, the material parameters are identified at experimental data of copper powder.     

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.     

Mechanics of indentation of plastically graded materials—I: Analysis

The introduction of controlled gradients in plastic properties is known to influence the resistance to damage and cracking at contact surfaces in many tribological applications. In order to assess potentially beneficial effects of plastic property gradients in tribological applications, it is essential first to develop a comprehensive and quantitative understanding of the effects of yield strength and strain hardening exponent on contact deformation under the most fundamental contact condition: normal indentation. To date, however, systematic and quantitative studies of plasticity gradient effects on indentation response have not been completed. A comprehensive parametric study of the mechanics of normal indentation of plastically graded materials was therefore undertaken in this work by recourse to finite element method (FEM) computations. On the basis of a large number of computational simulations, a general methodology for assessing instrumented indentation response of plastically graded materials is formulated so that quantitative interpretations of depth-sensing indentation experiments could be performed. The specific case of linear variation in yield strength with depth below the indented surface is explored in detail. Universal dimensionless functions are extracted from FEM simulations so as to predict the indentation load versus depth of penetration curves for a wide variety of plastically graded engineering metals and alloys for interpretation of, and comparisons with, experimental results. Furthermore, the effect of plasticity gradient on the residual indentation pile-up profile is systematically studied. The computations reveal that pile-up of the graded alloy around the indenter, for indentation with increasing yield strength beneath the surface, is noticeably higher than that for the two homogeneous reference materials that constitute the bounding conditions for the graded material. Pile-up is also found to be an increasing function of yield strength gradient and a decreasing function of frictional coefficient. The stress and plastic strain distributions under the indenter tip with and without plasticity gradient are also examined to rationalize the predicted trends. In Part II of this paper, we compare the predictions of depth-sensing indentation and pile-up response with experiments on a specially made, graded model Ni-W alloy with controlled gradients in nanocrystalline grain size.     

基于应变率阶跃测试的晶体铜压入应变率敏感性研究

纳米压入测试可以原位获取材料的诸多力学性能,包括弹性模量,硬度,屈服应力,应变率敏感指数等。本文利用应变率阶跃测试技术对多晶铜试样的应变率敏感性进行测试分析,硬度-位移曲线表明压头下方所存在的变形梯度对各阶跃应变率下的硬度值存在明显影响;采用基于晶体细观机制的塑性应变梯度理论对压入变形梯度效应予以修正,比较了修正与未修正数据所得的应变率敏感指数,在有效剔除压入变形梯度影响的基础上,应变率阶跃测试可实现单次压入下材料应变率敏感性的测试表征。     

Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structured metals

The effect of the material microstructural interfaces increases as the surface-to-volume ratio increases. It is shown in this work that interfacial effects have a profound impact on the scale-dependent yield strength and strain hardening of micro/nano-systems even under uniform stressing. This is achieved by adopting a higher-order gradient-dependent plasticity theory [Abu Al-Rub, R.K., Voyiadjis, G.Z., Bammann, D.J., 2007. A thermodynamic based higher-order gradient theory for size dependent plasticity. Int. J. Solids Struct. 44, 2888–2923] that enforces microscopic boundary conditions at interfaces and free surfaces. Those nonstandard boundary conditions relate a microtraction stress to the interfacial energy at the interface. In addition to the nonlocal yield condition for the material’s bulk, a microscopic yield condition for the interface is presented, which determines the stress at which the interface begins to deform plastically and harden. Hence, two material length scales are incorporated: one for the bulk and the other for the interface. Different expressions for the interfacial energy are investigated. The effect of the interfacial yield strength and interfacial hardening are studied by analytically solving a one-dimensional Hall–Petch-type size effect problem. It is found that when assuming compliant interfaces the interface properties control both the material’s global yield strength and rates of strain hardening such that the interfacial strength controls the global yield strength whereas the interfacial hardening controls both the global yield strength and strain hardening rates. On the other hand, when assuming a stiff interface, the bulk length scale controls both the global yield strength and strain hardening rates. Moreover, it is found that in order to correctly predict the increase in the yield strength with decreasing size, the interfacial length scale should scale the magnitude of both the interfacial yield strength and interfacial hardening.     

A viscoplastic strain gradient analysis of materials with voids or inclusions

A finite strain viscoplastic nonlocal plasticity model is formulated and implemented numerically within a finite element framework. The model is a viscoplastic generalisation of the finite strain generalisation by Niordson and Redanz (2004) [Journal of the Mechanics and Physics of Solids 52, 2431–2454] of the strain gradient plasticity theory proposed by Fleck and Hutchinson (2001) [Journal of the Mechanics and Physics of Solids 49, 2245–2271]. The formulation is based on a viscoplastic potential that enables the formulation of the model so that it reduces to the strain gradient plasticity theory in the absence of viscous effects. The numerical implementation uses increments of the effective plastic strain rate as degrees of freedom in addition to increments of displacement. To illustrate predictions of the model, results are presented for materials containing either voids or rigid inclusions. It is shown how the model predicts increased overall yield strength, as compared to conventional predictions, when voids or inclusions are in the micron range. Furthermore, it is illustrated how the higher order boundary conditions at the interface between inclusions and matrix material are important to the overall yield strength as well as the material hardening.     

Size effects on the plastic collapse limit load of thin foils in bending and thin wires in torsion

Following a previous paper by the author [Strain gradient plasticity, strengthening effects and plastic limit analysis, Int. J. Solids Struct. 47 (2010) 100–112], a nonconventional plastic limit analysis for a particular class of micron scale structures as, typically, thin foils in bending and thin wires in torsion, is here addressed. An idealized rigid-perfectly plastic material is considered, which is featured by a strengthening potential degree-one homogeneous function of the effective plastic strain and its spatial gradient. The nonlocal (gradient) nature of the material resides in the inherent strengthening law, whereby the yield strength is related to the effective plastic strain through a second order PDE with associated higher order boundary conditions. The peculiarity of the considered structures stems from their geometry and loading conditions, which dictate the shape of the collapse mechanism and make the higher order boundary conditions on the (microscopically) free boundary be accommodated by means of a boundary singularity mechanism. This consists in the formation of thin boundary layers with unbounded stresses, but bounded stress resultants which —together with the regular bulk stresses— contribute to the value of the collapse load. Closed-form solutions are provided for thin foils in pure bending and thin wires in pure torsion, and in particular the limit bending and torque moments are given as functions of an adimensionalized internal length parameter.     

Photovisco-elasto-plastic analysis tested on polyester by the scattered-light method

A three-dimensional photovisco-elasto-plastic model considering the strain rate effect was investigated by the scattered-light method using polyester as a model material. To examine the mechanical and optical properties of the material, tension and torsion tests were carried out on cylindrical specimens under various strain rates at 30°C. The effects of strain rate on the stress-strain relation and scattered-light fringe appearance were evaluated. The equivalent shearing stress-strain relation can be approximated by the Ramberg-Osgood equation with rate-dependent modulus and yield stress. The fringe gradient, when normalized by a rate-dependent yield gradient, can be related to an equivalent strain in the same form regardless of the strain rate. The strain rate can be evaluated from the measurement of the rate of increase of the fringe gradient. Hence, the relation between the fringe gradient and its rate of increase was derived as a function of strain rate. Finally, a method is proposed for the estimation of the visco-elasto-plastic stress and strain in a three-dimensional specimen from the measurement of only the fringe gradient and its rate of increase. The method was successfully applied not only to uniaxial tension but also to pure torsion.     

Analysis of plate in second strain gradient elasticity

The bending analysis of a thin rectangular plate is carried out in the framework of the second gradient elasticity. In contrast to the classical plate theory, the gradient elasticity can capture the size effects by introducing internal length. In second gradient elasticity model, two internal lengths are present, and the potential energy function is assumed to be quadratic function in terms of strain, first- and second-order gradient strain. Second gradient theory captures the size effects of a structure with high strain gradients more effectively rather than first strain gradient elasticity. Adopting the Kirchhoff’s theory of plate, the plane stress dimension reduction is applied to the stress field, and the governing equation and possible boundary conditions are derived in a variational approach. The governing partial differential equation can be simplified to the first gradient or classical elasticity by setting first or both internal lengths equal to zero, respectively. The clamped and simply supported boundary conditions are derived from the variational equations. As an example, static, stability and free vibration analyses of a simply supported rectangular plate are presented analytically.