Elastic fields of 2D and 3D misfit particles in an infinite medium

In micromechanics, accurate quantification of the elastic field (stress, strain, and displacement) caused by the presence of an inclusion in an infinite body is desired for both the particle and matrix materials. Ideally, the solution should be applicable to any particle geometry or shape and for any distribution of misfit along the interface (i.e. misfit profile). This work presents a dislocation-based numerical method, that is an extension to earlier work in this journal [Lerma, J.D., Khraishi, T., Shen, Y.L., Wirth, B.D., 2003. The elastic fields of misfit cylindrical particles: a dislocation-based numerical approach. Mech. Res. Commun. 30, 325–334], for determining the elastic fields of volume misfit particles with arbitrary misfit distribution or particle shape.     

Modeling the viscoplastic micromechanical response of two-phase materials using Fast Fourier Transforms

A viscoplastic approach using the Fast Fourier Transform (FFT) method for obtaining local mechanical response is utilized to study microstructure-property relationships in composite materials. Specifically, three-dimensional, two-phase digital materials containing isotropically coarsened particles surrounded by a matrix phase, generated through a Kinetic Monte Carlo Potts model for Ostwald ripening, are used as instantiations in order to calculate the stress and strain-rate fields under uniaxial tension. The effects of the morphology of the matrix phase, the volume fraction and the contiguity of particles, and the polycrystallinity of matrix phase, on the stress and strain-rate fields under uniaxial tension are examined. It is found that the first moments of the stress and strain-rate fields have a different dependence on the particle volume fraction and the particle contiguity from their second moments. The average stresses and average strain-rates of both phases and of the overall composite have rather simple relationships with the particle volume fraction whereas their standard deviations vary strongly, especially when the particle volume fraction is high, and the contiguity of particles has a noticeable effect on the mechanical response. It is also found that the shape of stress distribution in the BCC hard particle phase evolves as the volume fraction of particles in the composite varies, such that it agrees with the stress field in the BCC polycrystal as the volume of particles approaches unity. Finally, it is observed that the stress and strain-rate fields in the microstructures with a polycrystalline matrix are less sensitive to changes in volume fraction and contiguity of particles.     

Inclusion of an arbitrary polygon with graded eigenstrain in an anisotropic piezoelectric half plane

This paper presents an exact closed-form solution for the Eshelby problem of a polygonal inclusion with graded eigenstrains in an anisotropic piezoelectric half plane with traction-free on its surface. Using the line-source Green’s function, the line integral is carried out analytically for the linear eigenstrain case, with the final expression involving only elementary functions. The solutions are applied to the semiconductor quantum wire (QWR) of square, triangular, and rectangular shapes, with results clearly illustrating various influencing factors on the induced fields. The exact closed-form solution should be useful to the analysis of nanoscale QWR structures where large strain and electric fields could be induced by the non-uniform misfit strain.     

纳米线涂层中失配位错偶极子分析

研究了纳米线环形晶体薄膜涂层中失配位错偶极子与纳米线的干涉效应,并考虑纳米尺度应力效应及纳米线晶格失配的影响。运用弹性复势方法,分别获得了涂层和纳米线区域复势函数的精确解析解。利用求得的应力场和Peach-Koehler公式,得到了作用在螺型位错偶极子上像力和失配应力的精确表达式。算例结果表明,涂层纳米线界面应力和失配应力对涂层中失配位错偶极子的作用影响很大,由于界面应力和失配应力的存在,可以改变涂层内位错偶极子与纳米线干涉的引斥规律。与宏观尺度下的线型材料相比,纳米线材料由于界面效应的影响,位错偶极子在涂层中平衡位置的数量增加,涂层中更容易产生失配位错偶极子。考虑纳米线晶格失配的影响后,位错偶极子在涂层中平衡位置的数量增加。由此可知,减小纳米线晶格失配的影响,可以控制失配位错偶极子的产生。     

Stiffness properties of particulate composites containing debonded particles

This paper considers the problem of determining the nonlinear bimodular stiffness properties, i.e., the tensile and compressive Young’s moduli and Poisson’s ratios, and the shear modulus, of particulate composite materials with particle–matrix interfacial debonding. It treats the general case in which some of the particles are debonded while the others remain intact. The Mori–Tanaka approach is extended to formulate the method of solution for the present problem. The resulting auxiliary problem of a single debonded particle in an infinite matrix subjected to a remote stress equal to the average matrix stress, for which Eshelby’s solution does not exist, is solved by the finite element method accounting for the particle–matrix separation and contact at the debonded particle–matrix interface. Because of the nonlinear nature of the problem, an iterative process is employed in calculating the stiffness properties. The predicted stiffness properties are compared to the exact solutions of the stiffness properties of particulate composites with body-centered cubic packing arrangement.     

Three-dimensional reduced Navier–Stokes solutions for subsonic separated and non-separated flows,using a global pressure relaxation procedure

The reduced Navier–Stokes and thin layer approximations to the Navier–Stokes equations are used to obtain solutions for viscous subsonic three-dimensional flows. A spatial marching method is combined with a direct sparse matrix solver to obtain successive solutions in a global relaxation process. Results have been obtained for flow fields with and without regions of flow reversal.     

Elastic fields and effective moduli of particulate nanocomposites with the Gurtin–Murdoch model of interfaces

A complete solution has been obtained for periodic particulate nanocomposite with the unit cell containing a finite number of spherical particles with the Gurtin–Murdoch interfaces. For this purpose, the multipole expansion approach by Kushch et al. [Kushch, V.I., Mogilevskaya, S.G., Stolarski, H.K., Crouch, S.L., 2011. Elastic interaction of spherical nanoinhomogeneities with Gurtin–Murdoch type interfaces. J. Mech. Phys. Solids 59, 1702–1716] has been further developed and implemented in an efficient numerical algorithm. The method provides accurate evaluation of local fields and effective stiffness tensor with the interaction effects fully taken into account. The displacement vector within the matrix domain is found as a superposition of the vector periodic solutions of Lamé equation. By using local expansion of the total displacement and stress fields in terms of vector spherical harmonics associated with each particle, the interface conditions are fulfilled precisely. Analytical averaging of the local strain and stress fields in matrix domain yields an exact, closed form formula (in terms of expansion coefficients) for the effective elastic stiffness tensor of nanocomposite. Numerical results demonstrate that elastic stiffness and, especially, brittle strength of nanoheterogeneous materials can be substantially improved by an appropriate surface modification.     

横观各向同性介质中椭圆夹杂受非弹性剪切变形引起的弹性场的确定

本文求解了横观各向同性介质中椭圆夹杂内受非弹性剪切变形引起的弹性场。采用各向异性弹性力学平面问题的复变函数解法,结合保角变换,获得夹杂内应变能和基体内边界的应力分布和相应的应变能的表达式。进一步,根据最小应变能原理,获得表征夹杂平衡边界的两个特征剪切应变,从而得到了弹性场的解析解。通过应力转换关系,验证了应力解满足夹杂边界上法向正应力和剪应力的连续条件,表明了该解的正确性。本文解可用于复合材料断裂强度的分析中。     

裂纹扩展独立于网格的一种有限元单元子划分方法

提出了一种有限元模拟裂纹扩展的单元子划分结合子结构的方法。本方法中,裂纹可以进入或穿过一个单元,或沿单元的边界扩展,因此裂纹可以沿任意路径扩展而不受初始网格的限制。对上述几类包含裂纹的单元按照裂纹的路径进行子划分,覆盖一条裂纹的所有子划分单元就组成了一个子结构,子结构规模随裂纹的扩展而增大。子结构中因单元子划分而新增的结点自由度,通过自由度的凝聚用初始网格结点的自由度表示,因此结构整体分析的总自由度不变。以上述方法为基础建立了裂纹萌生和扩展的准则。用本文的方法分析了单（双）材料无限大平面中心（界面）裂纹的裂尖场,验证了本文方法的精度,并模拟了颗粒复合材料中微裂纹在颗粒、基体和界面中逐步扩展的过程,考核了本文方法对复杂裂纹扩展问题模拟的适用性。     

Inclusion of arbitrary polygon with graded eigenstrain in an anisotropic piezoelectric full plane

In this paper, an exact closed-form solution for the Eshelby problem of a polygonal inclusion with a graded eigenstrain in an anisotropic piezoelectric full plane is presented. For this electromechanical coupling problem, by virtue of Green’s function solutions, the induced elastic and piezoelectric fields are first expressed in terms of line integrals on the boundary of the inclusion. Using the line-source Green’s function, the line integral is then carried out analytically for the linear eigenstrain case, with the final expression involving only elementary functions. Finally, the solution is applied to the semiconductor quantum wire (QWR) of square, triangle, circle and ellipse shapes within the GaAs (0 0 1) substrate. It is demonstrated that there exists significant difference between the induced field by the uniform eigenstrain and that by the linear eigenstrain. Since the misfit eigenstrain in most QWR structures is actually non-uniform, the present solution should be particularly appealing to nanoscale QWR structure analysis where strain and electric fields are coupled and are affected by the non-uniform misfit strain.     

Energetics of misfit dislocation dipoles in anisotropic epitaxial films with nanoscale compositional modulation

The elastic strain and stress fields associated with nanoscale compositional modulation in an anisotropic epitaxial film on an anisotropic substrate are obtained by using Stroh formalism and the Eshelby-type inclusion method. The composition of the epitaxial film is considered to periodically fluctuate in a surface soft mode, with the amplitude of the composition modulation maximal near the growing surface and decreasing exponentially into the film. It has been experimentally observed that the composition modulation affects the formation of a new type of crystal defects, i.e., misfit dislocation dipoles, in III–V compound semiconductor materials. The formation energy of a misfit dislocation dipole under the elastic fields due to the composition modulation is calculated in this study. It is composed of the core and self energies of two dislocations, the interaction energy between two dislocations, and the interaction energies between the composition modulation and two dislocations. Numerical calculations are performed for a dislocation dipole in a lattice-matched Ga_0.5In_0.5P film on a GaAs substrate.     

复合材料中矩形夹杂角端部力学行为分析

提出了一种分析矩形夹杂角端部奇异应力场的新型杂交有限元方法,该方法在分析矩形夹杂角端部奇异应力场时,需要在夹杂端部构造一个超级单元。超级单元的刚度矩阵可以通过夹杂端部特征问题数值解建立。我们用这种方法计算了单向载荷作用下无限大均质板中单个矩形夹杂角端部奇异应力场,并与现有的数值解进行了比较。比较结果表明:本文提出的方法是可行的、有效的,而且数值结果精度高。为说明本文方法适用范围更广,文章最后讨论了各向异性弹性材料和横观各向同性压电材料中矩形夹杂角端部电弹性场行为。     

含界面效应纳米尺度圆环形涂层中螺型位错分析

研究了纳米尺度圆环形涂层(界面层)中螺型位错与圆形夹杂以及无限大基体材料的干涉效应.涂层与夹杂的界面和涂层与基体的界面均考虑界面应力效应.运用复势方法,获得了三个区域复势函数的解析解答.利用求得的应力场和Peach-Koehler公式,得到了作用在螺型位错上位错力的精确表达式.主要讨论了界面应力对涂层(界面层)中螺型位错运动和平衡稳定的影响规律.结果表明,界面应力对界面附近位错的运动有大的影响,由于界面应力的存在,可以改变涂层内位错与夹杂/基体干涉的引斥规律,并使位错在涂层内部产生三个稳定或非稳定的平衡点.考虑界面效应后,有一个额外的排斥力或吸引力作用在位错上,使原有的位错力增加或减小.     

非理想界面对三相复合材料应力场的影响

对非理想界面的三相复合材料,提出了计算弹性应力场的微观力学模型,在适当的简化假设下,对带界相的颗粒增强和纤维增强复合材料,得到了应力场的计算公式。以剪切载荷为例给出了数值例子。给出的数值结果表明非理想界面对三相复合材料应力场的影响。     

Drag forces of interacting spheres in power-law fluids

Drag forces of interacting particles suspended in power-law fluid flows were investigated in this study. The drag forces of interacting spheres were directly measured by using a micro-force measuring system. The tested particles include a pair of interacting spheres in tandem and individual spheres in a cubic matrix of multi-sphere in flows with the particle Reynolds number from 0.7 to 23. Aqueous carboxymethycellulose (CMC) solutions and glycerin solutions were used as the fluid media in which the interacting spheres were suspended. The range of power-law index varied from 0.6 to 1.0. In conjunction to the drag force measurements, the flow patterns and velocity fields of power-law flows over a pair of interacting spheres were also obtained from the laser assisted flow visualization and numerical simulation.

Both experimental and computational results suggest that, while the drag force of an isolated sphere depends on the power-index, the drag coefficient ratio of an interacting sphere is independent from the power-law index but strongly depends on the separation distance and the particle Reynolds number. Our study also shows that the drag force of a particle in an assemblage is strongly positions dependent, with a maximum difference up to 38%.     

Elastic field perturbation by a misfitting ring inclusion

The plane elasticity problem of a circular ring inhomogeneity, with either a hollow or a rigid core, is confronted. A solution is obtained under a wide class of loading conditions, the main limitation being that internal stress sources (if any) are located in the matrix. In order to take into account interfacial residual stresses, misfit between matrix and inhomogeneity is allowed. Loading by misfit alone, by uniform remote stresses, and by an edge dislocation, are explicitly treated as special cases.     

Deformation field of a misfitting inclusion

J.D. Eshelby (1957, 1959) has calculated the deformation field associated with an ellipsoidal inclusion in a state of homogeneous strain within an infinite matrix. Since most real precipitates occur with facets, the strain within such an inclusion is not uniform. Thus, plate precipitates of θ′ in Al-Cu and η in Al-Au have coherent broad faces with mismatches of 1.34 and 4.95 % respect- ively and semicoherent or disordered interfaces at the edges with residual mismatches of about ?4.3 and ?1.00% normal to the broad faces. The deformation field in the matrix around such precipitates has been calculated using Kelvin's (1848) result for the stress field due to a point force. The calculations show the existence of high stresses near the edges of the precipitates where they have an appreciable misfit. Unlike the case of an ellipsoidal inclusion, the stress fields of these precipitates have dilatational components which can affect the diffusion of solute atoms to them and, thus, the kinetics of interface migration. The behavior of alloys containing these precipitates indicates that the moduli of the precipitates are somewhat greater than those of the matrices. The present calculations, based on the assumption that the two moduli are the same, underestimate the actual deformation field in the matrix. In real systems, therefore, the effects of the deformation field on misfit dislocation nucleation and kinetics of interface migration are likely to be somewhat greater in general.     

Non-linear temporal stability analysis of viscoelastic plane channel flows using a fully-spectral method

A non-linear analysis of the temporal evolution of finite, two-dimensional disturbances is conducted for plane Poiseuille and Couette flows of viscoelastic fluids. A fully-spectral method of solution is used with a stream-function formulation of the problem. The upper-convected Maxwell (UCM), Oldroyd-B and Giesekus models are considered. The bifurcation of solutions for increasing elasticity is investigated both in the high and low Reynolds number regimes. The transition mechanism is discussed in terms of both the transient linear growth of misfit disturbances due to non-normality, and their possible saturation into finite-amplitude periodic solutions due to non-linear effects.     

压电圆柱形夹杂对邻近裂纹三维应力场的影响

研究了压电复合材料薄板中压电圆柱形夹杂与邻近宏观钝裂纹间的相互作用。重点分析了外加电场，裂尖与压电圆柱形夹杂间韧带长度对裂尖三维应力场的影响。计算结果表明：在不同的外加电场作用下，压电体不仅能改变裂尖张开应力的大小，还能改变其分布。所得结果对进一步探讨线弹性介质中裂纹的启裂控制有参考价值。     

Particle Lagrangian Tracking with Hydrodynamic Interactions and Collisions

Hydrodynamic interactions between particle pairs are included in a Lagrangian approach for the simulation of the turbulent dispersion of a discrete particle. Several particles are simultaneously followed, and the set of equations of motion involves a coupling between particles through the use of a resistance matrix pertaining to the sedimentation theory. Results are presented for particle behavior in low turbulence fields, where collisions between particle pairs are assumed to be perfectly elastic. The influence of the turbulence anisotropy is considered and the anisotropy of the particle fluctuating motion is shown to be reduced by collisions.