Thermal fracture of a functionally graded/homogeneous bimaterial with system of cracks

The thermal fracture of a bimaterial consisting of a homogeneous material and a functionally graded material (FGM) with a system of internal cracks and an interface crack is investigated. The bimaterial is subjected to a heat flux. The thermal properties of FGM are assumed to be continues functions of the thickness coordinate, while the elastic properties are constants. The method of the solution is based on the singular integral equations. For a special case where the interface crack is much larger than the internal cracks in the FGM the asymptotic analytical solution of the problem is obtained as series in a small parameter (the ratio between sizes of the internal and interface crack) and the thermal stress intensity factors (TSIFs) are derived as functions of geometry of the problem and material characteristics. A parametric analysis of the effects of the location and orientation of the cracks and of the inhomogeneity parameter of FGM’s thermal conductivity on the TSIFs is performed. The results are applicable to such kinds FGMs as ceramic/ceramic FGMs, e.g., TiC/SiC, MoSi₂/Al₂O₃ and MoSi₂/SiC, and also some ceramic/metal FGMs.     

Size dependence and orientation dependence of elastic properties of ZnO nanofilms

The elastic properties of ZnO nanofilms with different film thickness, surface orientations and loading directions are investigated by using molecular mechanics (MM) method. The size dependence of elastic properties is relevant to both the film surface crystallographic orientation and loading direction. Both atomic structure analysis and energy calculation are employed to identify the mechanisms of the size-dependent elastic properties, under different loading directions and surface orientations. Upon small axial deformation, the relationship between intralayer and interlayer bond length variation and film elastic stiffness is established; it is found that the atomic layers with larger bond length variation have higher elastic stiffness. The strain energies of atomic layers of ZnO nanofilm and bulk are decoupled, from which the stiffness of film surface, intralayers, and interlayers are derived and compared with their bulk counterparts. The surface stiffness is found to be much lower than that of the interior layers and bulk counterpart, and with the decrease of film thickness, the residual tension-stiffened interior atomic layers are the main contributions of the increased elastic modulus of ZnO nanofilms.     

Size effects on stress concentration induced by a prolate ellipsoidal particle and void nucleation mechanism

There generally exist two void nucleation mechanisms in materials, i.e. the breakage of hard second-phase particle and the separation of particle–matrix interface. The role of particle shape in governing the void nucleation mechanism has already been investigated carefully in the literatures. In this study, the coupled effects of particle size and shape on the void nucleation mechanisms, which have not yet been carefully addressed, have been paid to special attention. To this end, a wide range of particle aspect ratios (but limited to the prolate spheroidal particle) is considered to reflect the shape effect; and the size effect is captured by the Fleck–Hutchinson phenomenological strain plasticity constitutive theory (Advance in Applied Mechanics, vol. 33, Academic Press, New York, 1997, p. 295). Detailed theoretical analyses and computations on an infinite block containing an isolated elastic prolate spheroidal particle are carried out to light the features of stress concentrations and their distributions at the matrix–particle interface and within the particle. Some results different from the scale-independent case are obtained as: (1) the maximum stress concentration factor (SCF) at the particle–matrix interface is dramatically increased by the size effect especially for the slender particle. This is likely to trigger the void nucleation at the matrix–particle interface by cleavage or atomic separation. (2) At a given overall effective strain, the particle size effect significantly elevates the stress level at the matrix–particle interface. This means that the size effect is likely to advance the interface separation at a smaller overall strain. (3) For scale-independent cases, the elongated particle fracture usually takes place before the interface debonding occurs. For scale-dependent cases, although the SCF within the particle is also accentuated by the particle size effect, the SCF at the interface rises at a much faster rate. It indicates that the probability of void nucleation by the interface separation would increase.     

含界面相Voronoi单元的等效弹性常数的计算

采用含界面相Voronoi单元有限元法,根据广义胡克定律,计算了在给定边界条件下,颗粒增强复合材料的等效弹性常数。建立了含多个随机分布的椭圆形夹杂及界面相的VCFEM模型,分析了夹杂体分比,界面相厚度和界面相弹性模量等因素对颗粒增强复合材料等效弹性常数的影响,并利用普通有限元方法对比验证。结果表明,当界面相弹性模量小于基体与夹杂时,材料的等效弹性模量会随着界面相厚度的增大而减小,随着夹杂体分比的增大而减小,并且界面过薄时,材料的等效弹性模量会随着夹杂体分比的增大而增大;当界面相弹性模量大于基体或夹杂时,材料的等效弹性模量会随着夹杂体分比和界面相厚度的增大而增大。而界面相的厚度和弹性模量对材料的等效泊松比的影响较小,材料的等效泊松比主要受夹杂体分比的影响,与其呈反比关系。     

Reprint of “Size dependence and orientation dependence of elastic properties of ZnO nanofilms” [In. J. Solids Struct. 45 (2008) 1730–1753]

The elastic properties of ZnO nanofilms with different film thickness, surface orientations and loading directions are investigated by using molecular mechanics (MM) method. The size dependence of elastic properties is relevant to both the film surface crystallographic orientation and loading direction. Both atomic structure analysis and energy calculation are employed to identify the mechanisms of the size-dependent elastic properties, under different loading directions and surface orientations. Upon small axial deformation, the relationship between intralayer and interlayer bond length variation and film elastic stiffness is established; it is found that the atomic layers with larger bond length variation have higher elastic stiffness. The strain energies of atomic layers of ZnO nanofilm and bulk are decoupled, from which the stiffness of film surface, intralayers, and interlayers are derived and compared with their bulk counterparts. The surface stiffness is found to be much lower than that of the interior layers and bulk counterpart, and with the decrease of film thickness, the residual tension-stiffened interior atomic layers are the main contributions of the increased elastic modulus of ZnO nanofilms.     

Determination of the size of the representative volume element for random composites: statistical and numerical approach

The representative volume element (RVE) plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties. A quantitative definition of its size is proposed in this work. A RVE size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume V of microstructure that one is able to consider. It is shown to depend on the investigated morphological or physical property, the contrast in the properties of the constituents, and their volume fractions. The methodology is applied to a specific random microstructure, namely a two-phase three-dimensional Voronoı̈ mosaic. Finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity. The volumes are subjected to homogeneous strain, stress or periodic boundary conditions. The effective properties can be determined for large volumes and a small number of realizations. Conversely, smaller volumes can be used providing that a sufficient number of realizations are considered. A bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions. The variance of computed apparent properties for each volume size is used to define the precision of the estimation. The key-notion of integral range is introduced to relate this error estimation and the definition of the RVE size. For given wanted precision and number of realizations, one is able to provide a minimal volume size for the computation of effective properties. The results can also be used to predict the minimal number of realizations that must be considered for a given volume size in order to estimate the effective property for a given precision. The RVE sizes found for elastic and thermal properties, but also for a geometrical property like volume fraction, are compared.     

Influence of plasticity on interface toughness in a layered solid with residual stresses

For crack growth along an interface between two adjacent elastic–plastic materials in a layered solid, the resistance curve behaviour is analysed by approximating the behaviour in terms of a bi-material interface under small scale yielding conditions. Thus, it is assumed that the layers are thick enough so that the extent of the plastic regions around the crack tip are much smaller than the thickness of the nearest layers. The focus is on the effect of initial residual stresses in the layered material, or on T-stress components induced during loading. The fracture process is represented in terms of a cohesive zone model. It is found that the value of the T-stress component in the softer material adjacent to the interface crack plays a dominant role, such that a negative value of this T-stress gives a significant increase of the interface fracture toughness, while a positive value gives a reduction of the fracture toughness.     

A micromechanical model for effective elastic properties of particulate composites with imperfect interfacial bonds

A micromechanical model for effective elastic properties of particle filled acrylic composites with imperfect interfacial bonds is proposed. The constituents are treated as three distinct phases, consisting of agglomerate of particles, bulk matrix and interfacial transition zone around the agglomerate. The influence of the interfacial transition zone on the overall mechanical behavior of composites is studies analytically and experimentally. Test data on particle filled acrylic composites with three different interfacial properties are also presented. The comparison of analytical simulation with experimental data demonstrated the validity of the proposed micromechanical model with imperfect interface. Both the experimental results and analytical prediction show that interfacial conditions have great influence on the elastic properties of particle filled acrylic composites.     

Use of a Downscaling Procedure for Macroscopic Heat Source Modeling in Porous Media

Many porous media such as rocks have mesoscale inhomogeneities. The characteristic sizes of such inhomogeneities are much larger than the pore size but much less than the characteristic scale of the problem, such as the length of the sample on which measurements are taken. In this paper, we have solved the one-particle problem for depolarization of an ellipsoidal particle located in a porous medium with electrokinetic effect. To calculate the effective physical properties of a porous medium with many ellipsoidal inclusions, we have applied the effective field method. The application of this method allows us to take into account the texture of an inhomogeneous medium. The analysis performed has shown that three effective properties of inhomogeneous media (permeability, electroosmotic coupling coefficient and electrical conductivity) are not completely independent variables. General theory is illustrated by calculations of the effective properties of media containing spherical and spheroidal inclusions.     

A moving interface crack between two dissimilar functionally graded piezoelectric layers under electromechanical loading

The dynamic propagation of an interface crack between two dissimilar functionally graded piezoelectric material (FGPM) layers under anti-plane shear is analyzed using the integral transform method. The properties of the FGPM layers vary continuously along the thickness. The properties of the FGPM layers vary differently and the two layers are connected weak-discontinuously. A constant velocity Yoffe-type moving crack is considered. Fourier transform is used to reduce the problem to two sets of dual integral equations, which are then expressed to the Fredholm integral equations of the second kind. Numerical values on the dynamic energy release rate (DERR) are presented for the FGPM to show the effects on the electric loading, gradient of material properties, crack moving velocity, and thickness of layers. Followings are helpful to increase of the resistance of the interface crack propagation of FGPM: (a) certain direction and magnitude of the electric loading; (b) increase of the gradient of material properties; (c) increase of the material properties from the interface to the upper and lower free surface; (d) increase of the thickness of FGPM layer. The DERR increases or decreases with increase of the crack moving velocity.     

Size-dependent effective elastic moduli of particulate composites with interfacial displacement and traction discontinuities

This work aims at estimating the size-dependent effective elastic moduli of particulate composites in which both the interfacial displacement and traction discontinuities occur. To this end, the interfacial discontinuity relations derived from the replacement of a thin uniform interphase layer between two dissimilar materials by an imperfect interface are reformulated so as to considerably simplify the characteristic expressions of a general elastic imperfect model which is adopted in the present work and include the widely used Gurtin–Murdoch and spring-layer interface models as particular cases. The elastic fields in an infinite body made of a matrix containing an imperfectly bonded spherical particle and subjected to arbitrary remote uniform strain boundary conditions are then provided in an exact, coordinate-free and compact way. With the aid of these results, the elastic properties of a perfectly bonded spherical particle energetically equivalent to an imperfectly bonded one in an infinite matrix are determined. The estimates for the effective bulk and shear moduli of isotropic particulate composites are finally obtained by using the generalized self-consistent scheme and discussed through numerical examples.     

反倾层状岩体斜坡弯曲-拉裂两种失稳破坏之判据探讨

对反倾层状岩体斜坡弯曲-拉裂的失稳破坏判据,已有研究分别基于两种力学模型进行推导,即竖直压杆弹性屈曲稳定和平直梁弯折破坏模型,但对层间错动阻力及挠度产生附加弯矩等因素未加以考虑,不尽合理。在反倾斜坡岩层受力分析基础上,建立考虑了板侧层间错动阻力的下端嵌固、上端自由的斜置等厚弹性悬臂板梁模型,统一地通过瑞利-里兹能量方法,推导了弹性屈曲临界条件和嵌固端弯折破坏临界条件。实例计算及讨论表明,弹性屈曲判据适用于陡立岩层;而中-陡反倾岩层应主要为弯折破坏,但层间的力学性质对弯折临界判据值具有较大影响。     

On quantitative characterization of microstructures and effective properties

Proper quantitative characterization of microstructures, for the purpose of modeling the effective properties, is discussed. This is a broad subject that covers different physical properties (elastic, conductive, transport, etc.), as well as various types of microstructures. The present work focuses on microstructures that can be characterized as continuous matrices containing isolated inhomogeneities of diverse shapes, properties and orientations. We address their proper quantitative characterization in the context of elastic and conductive properties (transport and fracture-related properties are also briefly discussed).Proper microstructural parameters must correctly represent the individual inhomogeneity contributions to the considered property. They may differ for different physical properties. The key problem is to identify the mentioned individual contributions. For the elastic properties, we demonstrate, on a number of microstructures, how the proper parameters are implied by the elastic potential. Relative importance of various “irregularity factors” (shape irregularities, orientation scatter) is analyzed.We discuss similarities and differences between microstructural parameters intended for different physical properties. The possibility of explicit cross-property connections between two physical properties depends on whether the proper microstructural parameters for these two properties are sufficiently similar. We outline such explicit connections between the elastic and the conductive properties.The micromechanical approach is compared with the one based on an a priori introduced “fabric” tensors and general tensor representations that contain a number of uncertain factors. Various problems arising in this context are discussed.     

Cahn–Hilliard modeling of particles suspended in two‐phase flows

In this paper, we present a model for the dynamics of particles suspended in two‐phase flows by coupling the Cahn–Hilliard theory with the extended finite element method (XFEM). In the Cahn–Hilliard model the interface is considered to have a small but finite thickness, which circumvents explicit tracking of the interface. For the direct numerical simulation of particle‐suspended flows, we incorporate an XFEM, in which the particle domain is decoupled from the fluid domain. To cope with the movement of the particles, a temporary ALE scheme is used for the mapping of field variables at the previous time levels onto the computational mesh at the current time level. By combining the Cahn–Hilliard model with the XFEM, the particle motion at an interface can be simulated on a fixed Eulerian mesh without any need of re‐meshing. The model is general, but to demonstrate and validate the technique, here the dynamics of a single particle at a fluid–fluid interface is studied. First, we apply a small disturbance on a particle resting at an interface between two fluids, and investigate the particle movement towards its equilibrium position. In particular, we are interested in the effect of interfacial thickness, surface tension, particle size and viscosity ratio of two fluids on the particle movement towards its equilibrium position. Finally, we show the movement of a particle passing through multiple layers of fluids. Copyright © 2011 John Wiley & Sons, Ltd.     

Effect of crack position on stress intensity factor in particle-reinforced metal-matrix composites

In this study, a numerical model was developed to study the effects of mechanical properties of the particle and matrix materials, the crack position (in particle/in matrix) and loading conditions (mode 1 and mixed-mode) in particle-reinforced metal-matrix composites. The finite element technique was used to calculate the stress intensity factors for crack at and near-interface. The Displacement Correlation Method was used to calculate the stress intensity factors K₁ and K₂. In the present model, the particle and matrix materials were modeled in linear elastic conditions. The interface crack was considered between the particle and matrix, without the presence of the interface. For near-interface crack problem, two different crack positions (in particle/in matrix) were selected. The obtained results show the key role on the stress intensity factors played by the relative elastic properties of the particle and matrix. The results also show that loading condition has an important effect on the K₂ stress intensity factor and the crack deflection angle.     

Mode III effects on interface delamination

For crack growth along an interface between dissimilar materials the effect of combined modes I, II and III at the crack-tip is investigated. First, in order to highlight situations where crack growth is affected by a mode III contribution, examples of material configurations are discussed where mode III has an effect. Subsequently, the focus is on crack growth along an interface between an elastic-plastic solid and an elastic substrate. The analyses are carried out for conditions of small-scale yielding, with the fracture process at the interface represented by a cohesive zone model. Due to the mismatch of elastic properties across the interface the corresponding elastic solution has an oscillating stress singularity, and this solution is applied as boundary conditions on the outer edge of the region analyzed. For several combinations of modes I, II and III crack growth resistance curves are calculated numerically in order to determine the steady-state fracture toughness. For given values of K_I and K_II the minimum fracture toughness corresponds to K_III=0 in most of the range analyzed, but there is a range where the minimum occurs for a nonzero value of K_III.     

Axially compressed buckling of pressured multiwall carbon nanotubes

This paper studies axially compressed buckling of an individual multiwall carbon nanotube subjected to an internal or external radial pressure. The emphasis is placed on new physical phenomena due to combined axial stress and radial pressure. According to the radius-to-thickness ratio, multiwall carbon nanotubes discussed here are classified into three types: thin, thick, and (almost) solid. The critical axial stress and the buckling mode are calculated for various radial pressures, with detailed comparison to the classic results of singlelayer elastic shells under combined loadings. It is shown that the buckling mode associated with the minimum axial stress is determined uniquely for multiwall carbon nanotubes under combined axial stress and radial pressure, while it is not unique under pure axial stress. In particular, a thin N-wall nanotube (defined by the radius-to-thickness ratio larger than 5) is shown to be approximately equivalent to a single layer elastic shell whose effective bending stiffness and thickness are N times the effective bending stiffness and thickness of singlewall carbon nanotubes. Based on this result, an approximate method is suggested to substitute a multiwall nanotube of many layers by a multilayer elastic shell of fewer layers with acceptable relative errors. Especially, the present results show that the predicted increase of the critical axial stress due to an internal radial pressure appears to be in qualitative agreement with some known results for filled singlewall carbon nanotubes obtained by molecular dynamics simulations.     

Image decomposition method for the analysis of a mixed dislocation in a general multilayer

The elastic solutions for a mixed dislocation in a general multilayer with N dissimilar anisotropic layers are obtained via a generalized image decomposition method. The original problem is decomposed into N homogeneous subproblems with strategically placed continuously distributed image (virtual) dislocations which satisfy the consistency conditions for degenerate N − M (M < N) layer problems. The image dislocations are used to satisfy the interface or free surface conditions, and represent the unknowns of the problem. The resulting singular Cauchy integral equations are transformed into non-singular Fredholm integral equations of the second kind using certain H- and I-integral transforms. The Fredholm integral equations are then solved via the classical Nyström method. The general decomposition and the elimination of all singular integrals yield an exact formulation of the problem; the approximation arises only in the Nyström method. The dislocation mixity and the number of layers dissimilar in thickness and elastic anisotropy can be handled without difficulty, constrained only by the number of linear algebraic equations in the Nyström method for large N. For the numerical study, image forces on a dislocation in two- and three-layer systems are calculated. The accuracy of the results is verified by checking the boundary conditions and by comparison with previous results. The dependence of the image force on the dislocation position and mixity, and on the layer thicknesses and elastic anisotropies, is also illustrated via numerical investigations.     

Structural interfaces in linear elasticity. Part II: Effective properties and neutrality

The model of structural interfaces developed in Part I of this paper allows us to analytically attack and solve different problems of stress concentration and composites. In particular, (i) new formulae are given for effective properties of composite materials containing dilute suspensions of (randomly oriented) reinforced elliptical voids or inclusions; (ii) a new definition is proposed for inclusion neutrality (to account for the fact that the matrix is always ‘overstressed’, and thus non-neutral in a classical sense, at the contacts with the interfacial structure), which is shown to provide interesting stress optimality conditions. More generally, it is shown that the incorporation of an interfacial structure at the contact between two elastic solids exhibits properties that cannot be obtained using the more conventional approach of the zero-thickness, linear interface. For instance: contrary to the zero-thickness interface, both bulk and shear effective moduli can be optimized for a structural interface; effective properties higher that those possible with a perfect interface can be attained with a structural interface; and neutrality holds with a structural interface for a substantially broader range of parameters than for a zero-thickness interface.     

Plane acoustic wave propagation through a composite of elastic and Kelvin–Voigt viscoelastic material layers

The problem of plane wave propagation through a plane composite layer of thickness h is considered. The composite consists of periodically repeated elastic and Kelvin–Voigt viscoelastic material layers, and all layers are either parallel or perpendicular to the incident wave front. Moreover, it is assumed that the thickness of each separate layer of the composite is much less than the acoustic wave length and the thickness h of the entire composite. We study the problem by using a homogenized model of the composite, which allows us to find the reflection and transmission factors and the variation in the sound intensity level as it propagates though the composite layer of thickness h.