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.     

Interfacial excess energy, excess stress and excess strain in elastic solids: Planar interfaces

In this paper, interfacial excess energy and interfacial excess stress for coherent interfaces in an elastic solid are reformulated within the framework of continuum mechanics. It is shown that the well-known Shuttleworth relationship between the interfacial excess energy and interfacial excess stress is valid only when the interface is free of transverse stresses. To account for the transverse stress, a new relationship is derived between the interfacial excess energy and interfacial excess stress. Dually, the concept of transverse interfacial excess strain is also introduced, and the complementary Shuttleworth equation is derived that relates the interfacial excess energy to the newly introduced transverse interfacial excess strain. This new formulation of interfacial excess stress and excess strain naturally leads to the definition of an in-plane interfacial stiffness tensor, a transverse interfacial compliance tensor and a coupling tensor that accounts for the Poisson's effect of the interface. These tensors fully describe the elastic behavior of a coherent interface upon deformation.     

Influences of particle size and interface energy on the stress concentration induced by the oblate spheroidal particle and the void nucleation mechanism

Separation of the particle–matrix interface and breakage of the second-phase particle are two main void nucleation mechanisms, which are directly associated with the stress concentration factors (SCFs) at the interface and within the particle, respectively. This work investigates the coupled effects of particle size and particle shape on these stress concentrations by solving an infinite solid containing an oblate spheroidal particle under remote stress boundary condition. The phenomenological strain plasticity theory by Fleck–Hutchinson [Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.), Advance in Applied Mechanics, vol. 33. Academic Press, New York, pp. 295–361] is adopted to capture the size effect, various particle aspect ratios are considered to depict the particle shape effect and an interfacial energy concept is introduced to settle the double-traction equilibrium problem at the matrix–particle interface. By using a Ritz procedure, solutions about the stress concentrations are numerically achieved and three main results are found. First, the interfacial normal stress near the particle pole, the interfacial shear stress and the particle opening stress are dramatically elevated and their distributions are significantly modified by decrease in the particle size. Second, this particle size effect is influenced by the remote effective strain, remote stress triaxiality and the interfacial energy to different extent. Finally, the particle shape effect is coupled with this particle size effect, and the more oblate the particle is, the more significant the size effect on SCF elevation is. These findings are helpful for us to understand deeply the void nucleation mechanism at the micron scale.     

A Finite Strain Analysis of Cavity Formation at a Rigid Inhomogeneity

The formation of a cavity by inclusion-matrix interfacial separation is examined by analyzing the response of a plane rigid inclusion embedded in an unbounded incompressible matrix subject to remote equibiaxial dead load traction. A vanishingly thin interfacial cohesive zone, characterized by normal and tangential interface force-separation constitutive relations, is assumed to govern separation behavior. Rotationally symmetric cavity shapes (circles) are shown to be solutions of an interfacial integral equation depending on the strain energy density of the matrix, the interface force constitutive relation and the remote loading. Nonsymmetrical cavity formation, under rotationally symmetric conditions of geometry and loading, is treated within the theory of infinitesimal strain superimposed on a given finite strain state. Rotationally symmetric and nonsymmetric bifurcations are analyzed and detailed results, for the Mooney–Rivlin strain energy density and for an exponential interface force-separation law, are presented. For the nonsymmetric rigid body displacement mode, a simple formula for the critical load is presented. The effect on bifurcation behavior of interfacial shear stiffness and other interface parameters is treated as well. In particular we demonstrate that (i) for the smooth interface nonsymmetric bifurcation always precedes rotationally symmetric bifurcation, (ii) unlike rotationally symmetric bifurcation, there is no threshold value of interface parameter for which nonsymmetric bifurcation will not occur and (iii) interfacial shear may significantly delay the onset of nonsymmetric bifurcation. Also discussed is the range of validity of a nonlinear infinitesimal strain theory previously presented by the author (Levy [1]). This revised version was published online in July 2006 with corrections to the Cover Date.     

Debonding failure and size effects in micro-reinforced composites

Failure in micro-reinforced composites is investigated numerically using the strain-gradient plasticity theory of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52 (6) 1379–1406] in a plane strain visco-plastic formulation. Bi-axially loaded unit cells are used and failure is modeled using a cohesive zone at the reinforcement interface. During debonding a sudden stress drop in the overall average stress–strain response is observed. Adaptive higher-order boundary conditions are imposed at the reinforcement interface for realistically modeling the restrictions on moving dislocations as debonding occurs. It is found that the influence of the imposed higher-order boundary conditions at the interface is minor. If strain-gradient effects are accounted for a void with a smooth shape develops at the reinforcement interface while a smaller void having a sharp tip nucleates if strain-gradient effects are excluded. Using orthogonalization of the plastic strain gradient with three corresponding material length scales it is found that, the first length scale dominates the evaluated overall average stress–strain response, the second one only has a small effect and the third one has an intermediate effect. Finally, studies of reinforcement having elliptical cross-sections show rather significant gradients of stress which is not seen for the corresponding circular cross-sections. Also, an increased drop in the overall load carrying capacity is observed for cross-sections elongated perpendicular to the principal tensile direction compared to the corresponding circular cross-sections.     

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.     

偶应力理论下拉力型锚固体的受力特性

锚固体的受力特征及其影响因素是锚固体设计的重要依据,直接影响锚固效果。传统的经典弹性理论没有考虑应变梯度的影响。偶应力理论引进弯曲曲率,考虑了弯曲效应对介质变形特性的影响。基于偶应力理论,建立了平面应变问题的有限元计算模型,研究锚固体锚固段界面上的剪应力分布、锚固体轴力分布、偶应力的尺度效应以及弹性模量和围压对锚固力的影响,并将偶应力理论的计算结果和经典弹性理论的计算结果进行了比较。结果表明,在偶应力理论下,锚固体锚固段界面的剪应力有所减小,特别是峰值处的剪应力减小明显;岩土的弹性模量越大,锚固界面局部剪应力越大;锚固力随着围压的增大而增大,偶应力尺度效应明显。     

集中荷载作用下的压电压磁双材料III型界面裂纹问题

研究了压电压磁双材料的III型界面裂纹问题,裂纹表面和界面电磁绝缘,且作用有集中荷载．通过解析延拓和复变函数技术,得到该问题的全场解,并给出了裂纹尖端的场强因子．还讨论了Wan等所定义的用以表征界面上下压电压磁材料失配比的无量纲参数G,发现G实际表征的是界面两侧应变的比值,也即界面两侧应变的不相容性．     

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.     

Stable crack growth along a brittle\ductile interface—II. Small scale yielding solutions and interfacial toughness predictions

The problem of a crack growing steadily and quasi-statically along a brittle\ductile interface under plane strain, mixed mode, and small scale yielding conditions is considered. The ductile material is assumed to be characterized by the J₂-flow theory of plasticity with linear strain hardening, while the brittle material is assumed to be linear elastic. A displacement-based finite element method, exploiting the convective nature of the problem, is utilized to solve the relevant boundary value problem. In Part I of this work, the corresponding asymptotic problem was solved. This paper addresses the full-field problem in order to validate the asymptotic solutions, and to explore the physical implications of the results. The numerical full-field results are found to be in good agreement with the analytical asymptotic solutions. In particular, the full-field results strongly suggest that the stress fields in the vicinity of the crack tip are variable-separable of the power singular type; and also that the mode mix of the near-tip stress fields is, to a large extent, independent of the applied elastic mode mix. The amplitude (the plastic stress intensity factor) and the regions of validity of the asymptotic fields are estimated from the full-field results, and are observed to be strongly dependent on the applied mode mix. The remote elastic loading fields appear to influence the near-tip fields, primarily, through the plastic stress intensity factor. The present work also explores the suggestion made by Bose and Ponte Castaneda, 1992 that the solutions to the small scale yielding problem may be used in the context of a standard crack growth criterion, requiring that continued growth take place with a fixed near-tip crack opening profile, to obtain theoretical predictions for the dependence of interfacial toughness on the applied mode mix. Based on the numerical results, predictions for mixed mode toughness of the brittle\ductile interface are reported. The results, which are in qualitative agreement with available experimental data and also with some recent theoretical results, predict a strong dependence of interfacial toughness on mode mix. This suggests that ductility provides the main operating mechanism for explaining the dependence of interfacial toughness on the mode mix of the applied loading fields, during steady crack growth.     

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.     

基于Hellinger-Reissner变分原理的应变梯度杂交元设计

从一般的偶应力理论出发,基于Hellinger-Reissner变分原理,通过对有限元 离散体系的位移试解引入非协调位移函数,得到了偶应力理论下有限元离散系统的能量相容 条件,并由此建立了应变梯度杂交元的应力函数优化条件. 根据该优化条件,构造了一 个C0类的平面4节点梯度杂交元,数值结果表明,该单元对可压缩和不可压缩状态的 梯度材料均可给出合理的数值结果,再现材料的尺度效应.     

A mathematical basis for strain-gradient plasticity theory—Part I: Scalar plastic multiplier

Strain-gradient plasticity theories are reviewed in which some measure of the plastic strain rate is treated as an independent kinematic variable. Dislocation arguments are invoked in order to provide a physical basis for the hardening at interfaces. A phenomenological, flow theory version of gradient plasticity is constructed in which stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition. Plastic work is also done at internal interfaces and a yield surface is postulated for the work-conjugate stress quantities at the interface. Thereby, the theory has the potential to account for grain size effects in polycrystals. Both the bulk and interfacial stresses are taken to be dissipative in nature and due attention is paid to ensure that positive plastic work is done. It is shown that the mathematical structure of the elasto-plastic strain-gradient theory has similarities to conventional rigid-plasticity theory. Uniqueness and extremum principles are constructed for the solution of boundary value problems.     

Separation phenomena at the interface of a finitely deformed composite sphere

The problem of the finite deformation of a composite sphere subjected to a spherically symmetric dead load traction is revisited focusing on the formation of a cavity at the interface between a hyperelastic, incompressible matrix shell and a rigid inhomogeneity. Separation phenomena are assumed to be governed by a vanishingly thin interfacial cohesive zone characterized by uniform normal and tangential interface force–separation constitutive relations. Spherically symmetric cavity shapes (spheres) are shown to be solutions of an interfacial integral equation depending on the strain energy density of the matrix, the interface force constitutive relation, the dead loading and the volume concentration of inhomogeneity. Spherically symmetric and non-symmetric bifurcations initiating from spherically symmetric equilibrium states are analyzed within the framework of infinitesimal strain superimposed on a given finite deformation. A simple formula for the dead load required to initiate the non-symmetrical rigid body mode is obtained and a detailed examination of a few special cases is provided. Explicit results are presented for the Mooney–Rivlin strain energy density and for an interface force–separation relation which allows for complete decohesion in normal separation.     

纤维增强韧性基体界面力学行为

分析了纤维增强韧性基体的界面力学行为及其失效机理,按剪滞理论和应变理化规律研究微复合材料的弹塑性变形和应力状态,讨论了幂硬化和线性硬化基体的弹塑性变形和界面应力分布,并给出纤维应力和位移的表达式。按最大剪应力强度理论建立了纤维／基体界面失效准则,推导出弹塑性界面失效的平均剪应力随纤维埋入长度的变化关系。     

基于弹性-塑性内聚力模型的纤维拔出界面宏观行为分析

用解析法分析了单纤维从聚合物基体中的拔出过程,采用弹性—塑性内聚力模型模拟裂纹的扩展和界面失效,确定了临界纤维埋入长度,该值区分两种不同长度的纤维拔出过程. 在纤维拔出过程,界面经历不同的阶段. 纤维埋长小于临界长度时,界面的脱粘载荷与纤维的埋长成正比;超过临界长度后,界面的脱粘载荷近似为常数. 分析了界面参数对脱粘载荷的影响:增加界面的剪切强度和界面的断裂韧性,或减小界面裂纹萌生位移,均能提高界面的脱粘载荷;界面脱粘后无界面摩擦应力时,拔出载荷—位移曲线的峰值载荷等于界面的脱粘载荷;界面摩擦应力存在时,使峰值载荷大于脱粘载荷,需要较长的纤维埋入长度和较大的界面摩擦应力.      

纤维段裂试验的界面端应力奇异性研究

纤维段裂试验是测定纤维复合材料界面剪切强度的细观实验方法之一，其试验结果与其他三种细观试验方法(纤维拔出、纤维压人和微珠脱粘)测得的结果各不相符，相差较大。针对该问题，仔细研究了纤维段裂试验过程，可发现如下两个问题，首先是试件中纤维断裂造成的界面端应力奇异性问题；其次是纤维断成临界长度时界面是否脱粘的问题。针对界面端应力奇异性问题，本文建立了界面端轴对称分析模型，运用渐近展开法，推导出求解界面端特征值的特征方程，并由此得到应力奇异性指数随Dundurs常数的变化规律；采用文献[5]所用试件的纤维／基体性能数据，计算出了界面端的应力奇异性指数，并与文献[7]得到的其他三种试验的界面端应力奇异性指数进行比较，发现纤维段裂试件也存在界面端应力奇异性，而且应力奇异性最强，也说明了与其他三种试验结果不具可比性。本文还对纤维断成临界长度时界面是否脱粘的问题，进行了讨论。     

Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory

In this paper, a size-dependent first-order shear deformable shell model is developed based upon the modified strain gradient theory (MSGT) for the axial buckling analysis of functionally graded (FG) circular cylindrical microshells. It is assumed that the material properties of FG materials, which obey a simple power-law distribution, vary through the thickness direction. The principle of virtual work is utilized to formulate the governing equations and corresponding boundary conditions. Numerical results are presented for the axial buckling of FG circular cylindrical microshells subject to simply-supported end conditions and the effects of material length scale parameter, material property gradient index, length-to-radius ratio and circumferential mode number on the size-dependent critical buckling load are extensively studied. For comparison purpose, the critical buckling loads predicted by modified couple stress theory (MCST) and classical theory (CT) are also presented. Results show that the size effect plays an important role for lower values of dimensionless length scale parameter. Moreover, it is observed that the critical buckling loads obtained based on MSGT are greater than those obtained based on MCST and CT.     

Scale-dependent constitutive relations and the role of scale on nominal properties

Size effects in strength and fracture energy of heterogeneous materials is considered within a context of scale-dependent constitutive relations. Using tools of wavelet analysis, and considering the failure state of a one-dimensional solid, constitutive relations which include scale as a parameter are derived from a ‘background’ gradient formulation. In the resulting theory, scale is not a fixed quantity independent of deformation, but rather directly dependent on the global deformation field. It is shown that strength or peak nominal stress (maximum point at the engineering stress–strain diagram) decreases with specimen size while toughness or total work to fracture per nominal area (area under the curve in the engineering stress–strain diagram integrated along the length of the considered one-dimensional specimen) increases. This behavior is in agreement with relevant experimental findings on heterogeneous materials where the overall mechanical response is determined by variations in local material properties. The scale-dependent constitutive relations are calibrated from experimental data on concrete specimens.