A continuum micromechanical theory of overall plasticity for particulate composites including particle size effect

Classical continuum micromechanics cannot predict the particle size dependence of the overall plasticity for composite materials, a simple analytical micromechanical method is proposed in this paper to investigate this size dependence. The matrix material is idealized as a micropolar continuum, an average equivalent inclusion method is advanced and the Mori–Tanaka's method is extended to a micropolar medium to evaluate the effective elastic modulus tensor. The overall plasticity of composites is predicted by a new secant moduli method based on the second order moment of strain and torsion of the matrix in a framework of micropolar theory. The computed results show that the size dependence is more pronounced when the particle's size approaches to the matrix characteristic length, and for large particle sizes, the prediction coincides with that predicted by classical micromechanical models. The method is analytical in nature, and it can capture the particle size dependence on the overall plastic behavior for particulate composites, and the prediction agrees well with the experimental results presented in literature. The proposed model can be considered as a natural extension of the widely used secant moduli method from a heterogeneous Cauchy medium to a micropolar composite.     

颗粒增强复合材料的界面开裂与尺度效应

采用基于Huang等提出的塑性应变梯度传统理论发展的有限元方法,模拟了颗粒增强金属基复合材料的界面开裂与颗粒尺度效应.分别针对考虑颗粒与基体间界面开裂和不开裂两种情况进行分析,并将考虑界面开裂的模拟结果与实验结果进行比较,证明了模型的有效性,同时也获得应变梯度理论中所包含的材料特征尺度参量的取值.     

SIZE EFFECT AND GEOMETRICAL EFFECT OF SOLIDS IN MICRO—INDENTATION TEST

Micro-indentation tests at scales of the order of sub-micron show that the measured hardness increases strongly with decreasing indent depth or indent size,which is frequently referred to as the size effect.At the same time,at micron or sub-micron scale,another effect,which is referred to as the geometrical size effects such as crystal grain size effect,thin flim thickness effect,etc.,also influences the measured material hardness.However,the trends are at odds with the size-independence implied by the conventional elastic-plastic theory.In the present research,the strain gradient plasticity theory(Fleck and Hutchinson)is used to model the composition effects(size effect and geometrical effect) for polycrystal material and metal thin film/ceramic substrate systems when materials undergo micro-indenting.The phenomena of the “pile-up“ and “sink-in“ apeared in the indentation test for the polycrystal materials are also discussed.Meanwhile,the micro-indentation experiments for the polycrystal Al and for the Ti/Si3N4 thin film/substrate system are carried out.By comparing the theoretical predictions with experimental measuremtns.the values and the variation trends of the micro-scale parameter included in the strain gradient plasticity theory are predicted.     

Strain gradient effects on microscopic strain field in a metal matrix composite

The purpose of this paper is to demonstrate the improved modeling accuracy of a finite-deformation strain gradient crystal plasticity formulation over its classical counterpart by conducting a joint experimental and numerical investigation of the microscopic details of the deformation of a whisker-reinforced metal-matrix composite. The lattice rotation distribution around whiskers is obtained in thin foils using a TEM technique and is then correlated with numerical predictions based on finite element analyses of a unit-cell of a single crystal matrix containing a rigid whisker. The matrix material is first characterized by a classical, scale-independent crystal plasticity theory. It is found that the classical theory predicts a lattice rotation distribution with a spatial gradient much higher than experimentally measured. A strain gradient crystal plasticity formulation is then applied to model the matrix. The strain gradient formulation accounts for both strain hardening and strain gradient hardening. The deformation thus predicted exhibits a strong dependence on the size of the whisker. For a constitutive length scale comparable to the whisker diameter, the spatial gradient of the lattice rotation is several times lower than that predicted by the classical theory, and hence correlates significantly better with the experimental results.     

基于能量等效原理的应变局部化分析:Ⅰ.一维解析解

基于热力学第一定律和非局部塑性理论,提出了一种求解应变局部化问题的非局部方法.对材料的每一点定义了局部和非局部两种状态空间,局部状态空间的内变量通过非局部权函数映射到非局部空间,成为非局部内变量.在应变软化过程中,局部状态空间中的塑性变形服从正交流动法则,材料的软化律在非局部状态空间中被引入.通过两个状态空间的塑性应变能耗散率的等效,得到了应变软化过程中明确定义的局部化区域以及其中的塑性应变分布.应用本方法导出了一维应变局部化问题的解析解.解析解表明,应变局部化区域的尺寸只与材料内尺度有关;对于高斯型非局部权函数,局部化区域的尺寸大约是材料内尺度的6倍.一维算例表明,局部化区域的塑性应变分布以及载荷-位移曲线仅与材料参数和结构几何尺寸有关,变形局部化区域的尺寸随着材料内尺度的减小而减小,同时塑性应变也随着材料内尺度的减小变得更加集中.当内尺度趋近于零时,应用本文方法得到的解与采用传统的局部塑性理论得到的解相同.     

Thermo–elasto–plastic stresses in functionally graded materials under consideration of the fabrication process

Summary  The present study analyzes elasto–plastic thermal stresses in some particle-reinforced functionally graded material plates (FGP) by taking into consideration residual stresses of the fabrication process. For the FGP, the region near the cooling metal surface consists of distributed ceramic particles in a metal matrix, while the region near the heating ceramic surface contains distributed metal particles in a ceramic matrix. We use the thermo–elasto–plastic constitutive equation of a particle-reinforced composite, taking into consideration temperature changes and damage as well as the reinforcing effect of particles. Elasto–plastic thermal stresses are discussed here with the goal of reducing the thermal stresses. Two kinds of particle-reinforced FGP are considered: the first kind (FGP1) represents distributed ceramic particles in the metal matrix, and the second one (FGP2) represents distributed metal particles in the ceramic matrix. We modify the thermo–elasto–plastic constitutive equation of a particle-reinforced composite for the FGP2 by taking into consideration temperature changes and damage as well as the reinforcing effect of particles. Using the temperature-dependent material properties, three cases of temperature conditions are studied. The first one is the cooling from the fabrication temperature to the room temperature, the second one is the heating from the room temperature, and the last one is the heating after cooling from the fabrication temperature. The particle volume fraction is assumed to vary according to a power function in the thickness direction of the FGPs. Using the finite element method, the effects of the distribution parameter of the composition on the macroscopic stress, the stress in the matrix and the stress in the particle in the FGPs are discussed. Also, the effects of the particle volume fraction and the fabrication temperature on the maximum tensile matrix stress are discussed. Received 22 November 2000; accepted for publication 24 April 2001     

Void growth to coalescence in a non-local material

The size-effect in metals containing distributed spherical voids is analyzed numerically using a finite strain generalization of a length scale dependent plasticity theory. Results are obtained for stress-triaxialities relevant in front of a crack tip in an elastic-plastic metal. The influence of different material length parameters in a multi-parameter theory is studied, and it is shown that the important length parameter is the same as under purely hydrostatic loading. It is quantified how micron scale voids grow less rapidly than larger voids, and the implications of this in the overall strength of the material is emphasized. The size effect on the onset of coalescence is studied, and results for the void volume fraction and the strain at the onset of coalescence are presented. It is concluded that for cracked specimens not only the void volume fraction, but also the typical void size is of importance to the fracture strength of ductile materials.     

Strain gradient plasticity effects in whisker-reinforced metals

A metal reinforced by fibers in the micron range is studied using the strain gradient plasticity theory of Fleck and Hutchinson (J. Mech. Phys. Solids 49 (2001) 2245). Cell-model analyses are used to study the influence of the material length parameters numerically, for both a single parameter version and the multiparameter theory, and significant differences between the predictions of the two models are reported. It is shown that modeling fiber elasticity is important when using the present theories. A significant stiffening effect when compared to conventional models is predicted, which is a result of a significant decrease in the level of plastic strain. Moreover, it is shown that the relative stiffening effect increases with fiber volume fraction. The higher-order nature of the theories allows for different higher-order boundary conditions at the fiber-matrix interface, and these boundary conditions are found to be of importance. Furthermore, the influence of the material length parameters on the stresses along the interface between the fiber and the matrix material is discussed, as well as the stresses within the elastic fiber which are of importance for fiber breakage.     

The uniaxial tension of particulate composite materials with nonlinear interface debonding

Debonding of particle/matrix interfaces can significantly affect the macroscopic behavior of composite material. We have used a nonlinear cohesive law for particle/matrix interfaces to study interface debonding and its effect on particulate composite materials subject to uniaxial tension. The dilute solution shows that, at a fixed particle volume fraction, small particles lead to hardening behavior of the composite while large particles yield softening behavior. Interface debonding of large particles is unstable since the interface opening (and sliding) displacement(s) may have a sudden jump as the applied strain increases, which is called the catastrophic debonding. A simple estimate is given for the critical particle radius that separates the hardening and softening behavior of the composite.     

A size-dependent Reddy–Levinson beam model based on a strain gradient elasticity theory

A size-dependent Reddy–Levinson beam model is developed based on a strain gradient elasticity theory. Governing equations and boundary conditions are derived by using Hamilton’s principle. The model contains three material length scale parameters, which may effectively capture the size effect in micron or sub-micron. This model can degenerate into the modified couple stress model or even the classical model if two or all material length scale parameters are taken to be zero respectively. In addition, the present model recovers the micro scale Timoshenko and Bernoulli–Euler beam models based on the same strain gradient elasticity theory. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Reddy–Levinson beam are solved respectively; the results are compared with the reduced models. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Reddy–Levinson models are getting larger as the beam thickness is comparable to the material length scale parameters. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. This study may be helpful to characterize the mechanical properties of small scale beam-like structures for a wide range of potential applications.     

Suppressed plastic deformation at blunt crack-tips due to strain gradient effects

Large deformation gradients occur near a crack-tip and strain gradient dependent crack-tip deformation and stress fields are expected. Nevertheless, for material length scales much smaller than the scale of the deformation gradients, a conventional elastic–plastic solution is obtained. On the other hand, for significant large material length scales, a conventional elastic solution is obtained. This transition in behaviour is investigated based on a finite strain version of the Fleck–Hutchinson strain gradient plasticity model from 2001. The predictions show that for a wide range of material parameters, the transition from the conventional elastic–plastic to the elastic solution occurs for length scales ranging from 0.001 times the size of the plastic zone to a length scale of the same order of magnitude as the plastic zone.     

Comparison of computational predictions of material failure using nonlocal damage models

In computational analysis of damage failure the strain delocalizations are of great importance in predicting assessment of structure integrity. In this paper we are investigating effects of the intrinsic material length on computational prediction of material failure using both cell model, i.e. the conventional micro-mechanical damage model with the constant–sized finite elements for the damage zones, and nonlocal damage model based on the gradient plasticity. The corresponding experiments performed for an engineering steel are taken as reference for verification. The experimental observation has revealed that reducing the specimen size will arise the specific strength of small notched specimen which cannot be predicted using the cell damage model. The nonlocal damage model based on the strain gradient-dependent constitutive plasticity theory reproduces the experimental records. The material length affects evolution of the material porosity and gives an understandable explanation of the size effect.     

A unified model for polystyrene–nanorod and polystyrene–nanoplatelet melt composites

We present a unified constitutive model capable of predicting the steady shear rheology of polystyrene (PS)–nanoparticle melt composites, where particles can be rods, platelets, or any geometry in between, as validated against experimental measurements. The composite model incorporates the rheological properties of the polymer matrix, the aspect ratio and characteristic length scale of the nanoparticles, the orientation of the nanoparticles, hydrodynamic particle–particle interactions, the interaction between the nanoparticles and the polymer, and flow conditions of melt processing. We demonstrate that our constitutive model predicts both the steady rheology of PS–carbon nanofiber composites and the steady rheology of PS–nanoclay composites. Along with presenting the model and validating it against experimental measurements, we evaluate three different closure approximations, an important constitutive assumption in a kinetic theory model, for both polymer–nanoparticle systems. Both composite systems are most accurately modeled with a quadratic closure approximation.     

Mechanism-based strain gradient plasticity in C0 axisymmetric element

Non-uniform plastic deformation of materials exhibits a strong size dependence when the material and deformation length scales are of the same order at micro- and nano-metre levels. Recent progresses in testing equipment and computational facilities enhancing further the study on material characterization at these levels confirmed the size effect phenomenon. It has been shown that at this length scale, the material constitutive condition involves not only the state of strain but also the strain gradient plasticity. In this study, C⁰ axisymmetric element incorporating the mechanism-based strain gradient plasticity is developed. Classical continuum plasticity approach taking into consideration Taylor dislocation model is adopted. As the length scale and strain gradient affect only the constitutive relation, it is unnecessary to introduce either additional model variables or higher order stress components. This results in the ease and convenience in the implementation. Additional computational efforts and resources required of the proposed approach as compared with conventional finite element analyses are minimal. Numerical results on indentation tests at micron and submicron levels confirm the necessity of including the mechanism-based strain gradient plasticity with appropriate inherent material length scale. It is also interesting to note that the material is hardened under Berkovich compared to conical indenters when plastic strain gradient is considered but softened otherwise.     

A micro scale Timoshenko beam model based on strain gradient elasticity theory

A micro scale Timoshenko beam model is developed based on strain gradient elasticity theory. Governing equations, initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle. The new model incorporated with Poisson effect contains three material length scale parameters and can consequently capture the size effect. This model can degenerate into the modified couple stress Timoshenko beam model or even the classical Timoshenko beam model if two or all material length scale parameters are taken to be zero respectively. In addition, the newly developed model recovers the micro scale Bernoulli–Euler beam model when shear deformation is ignored. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Timoshenko beam are solved respectively. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Timoshenko models are large as the beam thickness is comparable to the material length scale parameter. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. In addition, Poisson effect on the beam deflection, rotation and natural frequency possesses an interesting “extreme point” phenomenon, which is quite different from that predicted by the classical Timoshenko beam model.     

考虑尺度影响的平面正交各向异性功能梯度微梁的屈曲分析

基于新修正偶应力理论，建立了能描述尺度效应的各向异性功能梯度微梁的屈曲分析模型。基于最小势能原理推导了控制方程及边界条件，并以简支梁为例分析了屈曲载荷及尺度效应受材料尺度参数和几何尺寸的影响。算例结果表明，在材料几何尺寸较小时，本文模型预测到的屈曲载荷明显大于传统理论的结果，有效地反映了尺度效应。几何尺寸较大时，尺度效应消失，本文模型将自动退化为传统宏观模型。模型反映出不同方向上的尺度参数对各向异性材料影响的效果不同。     

A simple theory of strain gradient plasticity based on stress-induced anisotropy of defect diffusion

A new strain gradient plasticity theory is formulated to accommodate more than one material length parameter. The theory is an extension of the classical J₂ flow theory of metal plasticity to the micron scale. Distinctive features of the proposed theory as compared to other existing theories are the simplicities of mathematical formulation, numerical implementation and physical interpretation.     

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.