Mechanical model for staggered bio-structure

A generic mechanical model for bio-composites, including stiff platelets arranged in a staggered order inside a homogeneous soft matrix, is proposed. Equations are formulated in terms of displacements and are characterized by a set of non-dimensional parameters. The displacements, stress fields and effective modulus of the composite are formulated. Two analytical models are proposed, one which includes the shear deformations along the entire medium and another simplified model, which is applicable to a slender geometry and yields a compact expression for the effective modulus. The results from the models are validated by numerical finite element simulations and found to be compatible with each other for a wide range of geometrical and material properties. Finally, the models are solved for two bio-structures, nacre and a collagen fibril, and their solutions are discussed.     

Elliptical inclusion in an anisotropic plane: non-uniform interface effects

We study the plane deformation of an elastic composite system made up of an anisotropic elliptical inclusion and an anisotropic foreign matrix surrounding the inclusion. In order to capture the influence of interface energy on the local elastic field as the size of the inclusion approaches the nanoscale, we refer to the Gurtin-Murdoch model of interface elasticity to describe the inclusion-matrix interface as an imaginary and extremely stiff but zero-thickness layer of a finite stretching modulus. As opposed to isotropic cases in which the effects of interface elasticity are usually assumed to be uniform (described by a constant interface stretching modulus for the entire interface), the anisotropic case considered here necessitates non-uniform effects of interface elasticity (described by a non-constant interface stretching modulus), because the bulk surrounding the interface is anisotropic. To this end, we treat the interface stretching modulus of the anisotropic composite system as a variable on the interface curve depending on the specific tangential direction of the interface. We then devise a unified analytic procedure to determine the full stress field in the inclusion and matrix, which is applicable to the arbitrary orientation and aspect ratio of the inclusion, an arbitrarily variable interface modulus, and an arbitrary uniform external loading applied remotely. The non-uniform interface effects on the external loading-induced stress distribution near the interface are explored via a group of numerical examples. It is demonstrated that whether the nonuniformity of the interface effects has a significant effect on the stress field around the inclusion mainly depends on the direction of the external loading and the aspect ratio of the inclusion.     

片状颗粒间液桥力变化规律的计算研究

研究颗粒间液桥力有助于揭示非饱和土持水特性的内在机理. 为探究片状颗粒间液桥力演化规律, 从细观尺度研究非饱和土的水力特性机理, 使用Surface Evolver软件在两平行的片状颗粒间构建出三维液桥模型, 分析了液桥拉伸过程中接触角、液桥体积、分离距离以及固液接触线钉扎效应等对液桥力变化规律的影响. 基于圆弧假定, 计算相应条件下液桥力以及接触半径的大小, 并与上述模拟结果进行对比分析. 结果表明: 片状颗粒间液桥力随液桥体积增大而递增, 随分离距离的增大而递减, 随固液接触角的增大先增后减或一直递减; 液桥体积一定时, 在钉扎状态下, 其液桥力随着分离距离的增大迅速递增达到峰值, 而后逐渐降低; Surface Evolver模拟与液桥界面环形近似的计算结果相对比, 当固液接触角较大时(θ = 60°和θ = 80°), 二者相对误差在6%以内, 而当固液接触角减小到30°及以下时, 相对误差随之增大, 且颗粒间分离距离越大, 相对误差越大.      

Meso-scale approach to modelling the fracture process zone of concrete subjected to uniaxial tension

A meso-scale analysis is performed to determine the fracture process zone of concrete subjected to uniaxial tension. The meso-structure of concrete is idealised as stiff aggregates embedded in a soft matrix and separated by weak interfaces. The mechanical response of the matrix, the inclusions and the interface between the matrix and the inclusions is modelled by a discrete lattice approach. The inelastic response of the lattice elements is described by a damage approach, which corresponds to a continuous reduction of the stiffness of the springs. The fracture process in uniaxial tension is approximated by an analysis of a two-dimensional cell with periodic boundary conditions. The spatial distribution of dissipated energy density at the meso-scale of concrete is determined. The size and shape of the deterministic FPZ is obtained as the average of random meso-scale analyses. Additionally, periodicity of the discretisation is prescribed to avoid influences of the boundaries of the periodic cell on fracture patterns. The results of these analyses are then used to calibrate an integral-type nonlocal model.     

Contact stress analysis of an elastic half-plane containing multiple inclusions

This paper presents a two-dimensional contact stress analysis to investigate the effects of multiple inclusions on the contact pressure and subsurface stresses in an elastic half-plane. The boundary element method is used to analyze the contact problem where a set of integral equations is derived on the contact region and the matrix–inclusion interfaces. As the contact region is unknown a priori, an iterative procedure is implemented to determine the actual contact region and the contact pressure, and the tractions and displacements on the matrix–inclusion interfaces are obtained by solving the integral equations numerically. Numerical results show that the inclusions near contact surface could cause significant alterations in the contact pressure distribution. The stiff inclusions could toughen the surrounding material and reduce the internal stresses while the soft inclusions could increase the subsurface stresses.     

Rheology of polymer blends: linear model for viscoelastic emulsions

 Kerner's model for flow of composite elastic media is extended to an emulsion of viscoelastic phases with interfacial tension undergoing deformations of small amplitude. A privileged internal structure inside the suspended drops is discussed in terms of fluid circulation across the interface. It is shown that for usual drop radius and interfacial tension values of emulsions, the rheological behavior predicted by the model, with very simple expression for the complex shear modulus, is quantitatively similar to that predicted by Palierne's model. Predictions of the model are compared with experimental data obtained on a polystyrene/polyethylene blend sheared in a small-amplitude oscillatory mode. Received: 10 August 1998 Accepted: 18 December 1998     

Solutions for effective shear properties in three phase sphere and cylinder models

Solutions are presented for the effective shear modulus of two types of composite material models. The first type is that of a macroscopically isotropic composite medium containing spherical inclusions. The corresponding model employed is that involving three phases: the spherical inclusion, a spherical annulus of matrix material and an outer region of equivalent homogeneous material of unlimited extent. The corresponding two-dimensional, polar model is used to represent a transversely isotropic, fiber reinforced medium. In the latter case only the transverse effective shear modulus is obtained. The relative volumes of the inclusion phase to the matrix annulus phase in the three phase models are taken to be the given volume fractions of the inclusion phases in the composite materials at large. The results are found to differ from those of the well-known Kerner and Hermans formulae for the same models. The latter works are now understood to violate a continuity condition at the matrix to equivalent homogeneous medium interface. The present results are compared extensively with results from other related models. Conditions of linear elasticity are assumed.     

液体界面现象及表面张力的分子动力学模拟

本文利用分子动力学计算机模拟的方法,研究了氪的固-液、液-气和气-固的界面现象。显示了液体的零级径向分布是有序的,与固体相邻的液体部分有类晶结构,固体表面出现了气体吸附层。表面张力计算结果与实验值很好的吻合,证明Lennead-Jones势对球形原子的相互作用作了很好的描述,在表面张力计算中可以忽略去截断效应。     

Debonding of particles in thin films

Thin composite films consisting of a matrix with embedded particles are currently being developed both as hard, wear resistant coatings and as functional surfaces. The effect of stiff particles in the film are studied for systems where the film is under residual tensile stresses. The particles, when they are fully bonded to the matrix, increase the stiffness of the composite film. In cases where the particles debond from the matrix material, the stiffness of the composite film decreases. The conditions under which the debonding process is stable are studied. For systems properly designed, a controlled debonding process of the particles can thus be used to reduce the stress levels in composite film lowering the risk for delamination of the composite film from the substrate as well as the risk of through cracks in the film. The work includes finite element based unit cell calculations of interface debonding between spherical particles and the film, and the release of residual stresses following this. The three dimensional unit cell calculations assume a periodic distribution of particles in the plane parallel to the substrate interface with equi-biaxial tension and periodicity with zero overall stress perpendicular to the substrate interface.     

The interaction between a screw dislocation and a circular inhomogeneity in gradient elasticity

The interaction between a screw dislocation and a circular inhomogeneity in gradient elasticity is investigated. The screw dislocation is located inside either the inhomogeneity or the matrix. By using the Fourier transform method, closed analytical solutions are obtained when the inhomogeneity and the matrix have the same gradient coefficient. The explicit expressions of image forces exerted on screw dislocations are derived. The motion of the appointed screw dislocation and its equilibrium positions are discussed. The results show that the classical singularity is eliminated. Especially, for the case of a tiny inhomogeneity, the relation of dislocations and inhomogeneities become quite different. The screw dislocation may be attracted by the stiff inhomogeneity and repelled by the soft inhomogeneity when it tends to the interface. So there is an unstable equilibrium position when a dislocation tends to a tiny stiff inhomogeneity and there is a stable equilibrium position when a dislocation tends to a tiny soft inhomogeneity.     

Effective elastic moduli of composites reinforced by particle or fiber with an inhomogeneous interphase

The solution of the strain energy change of an infinite matrix due to the presence of one spherical particle or cylindrical fiber surrounded by an inhomogeneous interphase is the basis of solving effective elastic moduli of corresponding composites based on various micromechanics models. In order to find out the strain energy change, the composite sphere or cylinder, i.e., the spherical particle or cylindrical fiber together with its interphase, is replaced by an effective homogeneous particle or fiber. Independent governing differential equations for each modulus of the effective particle or fiber are derived by extending the replacement method [J. Mech. Phys. Solids 12 (1964) 199]. As far as the strain energy changes of the infinite matrix subjected to various far-field stress systems are concerned, the present model is simple. Meanwhile, FEM analysis is carried out for a verification, which shows that the model can lead to rather accurate results for most practical interphases. Besides, to check the validity of the model further when the interactions among composite cylinders exist, the two problems of an infinite matrix containing two composite cylinders and the effective moduli of composites with the equilateral triangular distribution of composite cylinders are analyzed using FEM. The FEM results show that the model is still rather accurate, especially for the case of interphase properties varying between those of fiber and matrix. Therefore, composite spheres or cylinders are assumed as the effective homogeneous particles or fibers and simple expressions of the effective moduli of composites containing the composite spheres or cylinders are obtained. Furthermore, the present model is compared with some existing models that are based on very complicated derivations.     

Interface effect on the effective bulk modulus of a particle-reinforced composite

Classical micromechanical methods for calculating the effective moduli of a heterogeneous material are generalized to include the interface (surface) effect. By using Hashin‘s Composite Sphere Assemblage (CSA) model, a new expression of the bulk modulus for a particle-reinforced composite is derived. It is emphasized that the present study is within the finite-deformation framework such that the effective properties are not influenced by the interface stress itself solely, but influenced by the change of the interface stress due to changes of the shape and size of the interface, itence some inadequacies in previous papers are pointed out.     

Contact analysis in presence of spherical inhomogeneities within a half-space

This paper presents a fast method of solving contact problems when one of the mating bodies contains multiple heterogeneous inclusions, and numerical results are presented for soft or stiff inhomogeneities. The emphasis is put on the effects of spherical inclusions on the contact pressure distribution and subsurface stress field in an elastic half-space. The computing time and allocated memory are kept small, compared to the finite element method, by the use of analytical solution to account for the presence of inhomogeneities. Eshelby’s equivalent inclusion method is considered in the contact solver. An iterative process is implemented to determine the displacements and stress fields caused by the eigenstrains of all spherical inclusions. The proposed method can be seen as an enrichment technique for which the effect of heterogeneous inclusions is superimposed on the homogeneous solution in the contact algorithm. 3D and 2D Fast Fourier Transforms are utilized to improve the computational efficiency. Configurations such as stringer and cluster of spherical inclusions are analyzed. The effects of Young’s modulus, Poisson’s ratio, size and location of the inhomogeneities are also investigated. Numerical results show that the presence of inclusions in the vicinity of the contact surface could significantly changes the contact pressure distribution. From a numerical point of view the role of Poisson’s ratio is found very important. One of the findings is that a relatively ‘soft’ and nearly incompressible inclusion – for example a cavity filled with a liquid – can be more detrimental for the stress state within the matrix than a very hard inclusion with a classical Poisson’s ratio of 0.3.     

Influence of interfaces on effective properties of nanomaterials with stochastically distributed spherical inclusions

The method of conditional moments is generalized to include evaluation of the effective elastic properties of particulate nanomaterials and to investigate the size effect in those materials. Determining the effective constants necessitates finding a stochastically averaged solution to the fundamental equations of linear elasticity coupled with surface/interface conditions (Gurtin–Murdoch model). To obtain such a solution the system of governing stochastic differential equations is first transformed to an equivalent system of stochastic integral equations. Using statistical averaging, the boundary-value problem is then converted to an infinite system of linear algebraic equations. A two-point approximation is considered and the stress fluctuations within the inclusions are neglected in order to obtain a finite system of algebraic equations in terms of component-average strains. Closed-form expressions are derived for the effective moduli of a composite consisting of a matrix and randomly distributed spherical inhomogeneities. As a numerical example a nanoporous material is investigated assuming a model in which the interface effects influence only the bulk modulus of the material. In that model the resulting shear modulus is the same as for the material without surface effects. Dependence of the bulk moduli on the radius of nanopores and on the pore volume fraction is analyzed. The results are compared to, and discussed in the context of other theoretical predictions.     

考虑界面应力时纳米涂层纤维增强复合材料的有效力学性能

基于Gurtin-Murdoch表/界面理论和广义自洽方法,获得了考虑界面应力时纳米涂层纤维增强复合材料有效反平面剪切模量的闭合形式解。讨论了涂层的壁厚、力学性能和界面性能对复合材料有效性能的影响。结果显示：在纳米尺度范围内,复合材料的有效反平面剪切模量受纳米涂层的尺寸影响显著。纤维体积分数一定时,涂层壁厚越大,纤维半径越小,有效反平面剪切模量与经典结果偏差越大。纤维刚度和涂层界面性能对复合材料有效模量的影响也取决于涂层刚度,非常软或非常硬的涂层都大大限制了纤维刚度对复合材料有效模量的贡献,过高的涂层刚度屏蔽了纳米复合材料表/界面效应的影响。     

纳米夹杂复合材料的有效反平面剪切模量研究

基于Gurtin-Murdoch表面/界面理论模型,利用复变函数方法,获得了考虑夹杂界面应力时夹杂/基体/等效介质模型的全场精确解,发展了能够预测纳米夹杂复合材料有效反平面剪切模量的广义自洽方法,给出了复合材料有效反平面剪切模量的封闭形式解。数值结果显示：当夹杂尺寸在纳米量级时,复合材料的有效反平面剪切模量具有尺度相关性,随着夹杂尺寸的增大,本文结果趋近于经典弹性理论的预测值;夹杂尺寸对于有效反平面剪切模量(本文结果)的影响范围要小于其对有效体积模量与剪切模量(各向同性材料)的影响范围;有效反平面剪切模量受夹杂的界面性能和夹杂刚度影响显著。     

不同拉压特性的厚壁圆筒极限内压统一解

厚壁圆筒在实际工程领域中应用广泛,若能精确计算出极限内压,对预防事故发生,降低风险有重要意义.工程中存在许多材料,其拉压强度和拉压模量均存在差异,这些差异对极限内压的大小有显著影响.以往研究表明,仅考虑拉压强度与拉压模量的一个方面,计算结果与实际情况存在一定的误差.本文基于双剪统一强度理论,综合考虑中间主应力效应及材料拉压强度和拉压模量的不同,推导了内压作用下厚壁圆筒的弹、塑性状态的应力分布及弹性极限内压、塑性极限内压与安定极限内压的统一解,通过与其他文献对比分析验证了本文计算结果的正确性,分析了半径比、统一强度理论参数、拉压强度比与拉压模量系数对弹性极限内压、塑性极限内压及安定极限内压的影响.结果表明:统一解均随半径比和统一强度理论参数的增大而增大,随拉压强度比的增大而减小,弹性极限内压随材料拉压模量系数的增大而减小,当壁厚增加到一定值后,安定极限内压随材料拉压模量系数的增大而减小;材料的拉压模量不同、拉压强度差异对厚壁圆筒的安定性影响显著,考虑中间主应力效应可使材料的潜能得到更充分发挥,极限内压随半径比的变化规律可为选择合理壁厚提供参考,该结论可为厚壁圆筒的工程应用提供理论依据.     

Residual strength of a fiber reinforced metal matrix composite with a crack

The residual strength of a cracked unidirectional fiver reinforced metal matrix composite is studied. We propose a bridging model based on the Dugdale strip yielding zones in the matrix ahead of the crack tips that accounts for ductile deformations of the matrix and fiber debonding and pull-out in the strip yielding zone. The bridging model is used to study the fracture of an anisotropic material and its residual strength is calculated numerically. The predicted results for a SiC/titanium composite agree well with the existing experimental data. It is found that a higher fiber bridging stress and a larger fiber pull-out length significantly contribute to the composite's residual strength. The composite's strength may be more notch-insensitive than the corresponding matrix material's strength depending on several factors such as fiber-matrix interface properties and the ratio of the matrix modulus to an ‘effective modulus’ of the composite.     

The cohesive law for the particle/matrix interfaces in high explosives

The debonding of particle/matrix interfaces has an important effect on the macroscopic behavior of composite materials. There are extensive analytical and numerical studies on interface debonding in composite materials based on cohesive zone models which assume a phenomenological relation between the normal (and shear) traction(s) and opening (and sliding) displacement(s) across the particle/matrix interface. However, there are little or no experiments to determine the cohesive law for particle/matrix interfaces in composite materials. In this paper, we develop a method to determine the cohesive law for particle/matrix interfaces in the high explosive PBX 9501. We use the digital image correlation technique to obtain the stress and displacement around a macroscopic crack tip in the modified compact tension experiment of PBX 9501. We use the extended Mori-Tanaka method (which accounts for the effect of interface debonding) and the equivalence of cohesive energy on the macroscale and microscale to link the macroscale compact tension experiment to the microscale cohesive law for particle/matrix interfaces. Such an approach enables us to quantitatively determine key parameters in the microscale cohesive law, namely the linear modulus, cohesive strength, and softening modulus of particle/matrix interfaces in the high explosive PBX 9501. The present study shows that Ferrante et al.'s [1982 Universal binding energy relations in metallic adhesion. In: J.M. Georges (Ed.), Microscopic Aspects of Adhesion and Lubrication, Elsevier, Amsterdam, pp. 19-30.] cohesive law, which is established primarily for bimetallic interfaces, is not suitable to the high explosive PBX 9501.     

Analytical model for melting in a semi-infinite PCM storage with an internal fin

The most PCMs with high energy storage density have an unacceptably low heat conductivity and hence internal heat transfer enhancement techniques such as fins or other metal structures are required in latent heat thermal storage (LHTS) applications. Previous work has concentrated on numerical and experimental examination in determining the influence of the fins in melting phase change material. This paper presents a simplified analytical model based on a quasi-linear, transient, thin-fin equation which predicts the solid-liquid interface location and temperature distribution of the fin in the melting process with a constant imposed end-wall temperature. The analytical results are compared to the numerical results and they show good agreement. Due to the assumptions made in the model, the speed of the solid-liquid interface during the melting process is slightly too slow.