An effective medium model for elastic waves in microcrack damaged media

Direct numerical simulations of waves traveling through microcrack-damaged media are conducted and the results are compared to effective medium calculations to determine the applicability of the latter for studying wave propagation. Both tensile and compressive waves and various angular distributions of randomly-located cracks are considered. The relationships between the input wavelength and the output wave speed and output signal strength are studied. The numerical simulations show that the wave speed is nearly constant when 1/ka > 60 for tensile waves and 1/ka > 10 for compressive waves, where k is the wave number and a is the average half-crack length. The direct simulations also show that when the input wavelength is much longer than the crack length, 1/ka > 60, the wave can pass through the damaged medium relatively unattenuated. On the other hand, when the input wavelength is shorter than a “cut off” wave length, the output wave magnitude decreases linearly with the input wavelength. The effective medium wave speed and magnitude calculations are not dependent on the input wavelength and therefore the results correspond well with the numerical simulations for large 1/ka. This suggests a minimum wavelength for which the homogenized methods can be used for studying these problems.     

Anisotropic effective moduli of microcracked materials under antiplane loading

This study focuses on the prediction of the anisotropic effective elastic moduli of a solid containing microcracks with an arbitrary degree of alignment by using the generalized self-consistent method (GSCM). The effective elastic moduli pertaining to anti-plane shear deformation are discussed in detail. The undamaged solid can be isotropic as well as anisotropic. When the undamaged solid is isotropic, the GSCM can be realized exactly. When the undamaged solid is anisotropic it is difficult to provide an analytical solution for the crack opening displacement to be used in the GSCM, thus an approximation of the GSCM is pursued in this case. The explicit expressions of coupled nonlinear equations for the unknown effective moduli are obtained. The coupled nonlinear equations are easily solved through iteration.     

Mechanics of layered anisotropic poroelastic media with applications to effective stress for fluid permeability

The mechanics of vertically layered porous media has some similarities to and some differences from the more typical layered analysis for purely elastic media. Assuming welded solid contact at the solid-solid interfaces implies the usual continuity conditions, which are continuity of the vertical (layering direction) stress components and the horizontal strain components. These conditions are valid for both elastic and poroelastic media. Differences arise through the conditions for the pore pressure and the increment of fluid content in the context of fluid-saturated porous media. The two distinct conditions most often considered between any pair of contiguous layers are: (1) an undrained fluid condition at the interface, meaning that the increment of fluid content is zero (i.e., δζ = 0), or (2) fluid pressure continuity at the interface, implying that the change in fluid pressure is zero across the interface (i.e., δp_f = 0). Depending on the types of measurements being made on the system and the pertinent boundary conditions for these measurements, either (or neither) of these two conditions might be directly pertinent. But these conditions are sufficient nevertheless to be used as thought experiments to determine the expected values of all the poroelastic coefficients. For quasi-static mechanical changes over long time periods, we expect drained conditions to hold, so the pressure must then be continuous. For high-frequency wave propagation, the pore-fluid typically acts as if it were undrained (or very nearly so), with vanishing of the fluid increment at the boundaries being appropriate. Poroelastic analysis of both these end-member cases is discussed, and the general equations for a variety of applications to heterogeneous porous media are developed. In particular, effective stress for the fluid permeability of such poroelastic systems is considered; fluid permeabilities characteristic of granular media or tubular pore shapes are treated in some detail, as are permeabilities of some of the simpler types of fractured materials.     

On the effective electroelastic properties of microcracked generally anisotropic solids

In this study we first obtain the explicit expressions for the 15 effective reduced elastic compliances of an elastically anisotropic solid containing multiple microcracks with an arbitrary degree of alignment under two-dimensional deformations within the framework of the non-interaction approximation (NIA). Under special situations, our results can reduce to the classical ones derived by Bristow (J Appl Phys 11: 81–85, 1960), and Mauge and Kachanov (J Mech Phys Solids 42(4):561–584, 1994). Some interesting phenomena are also observed. For example, when the undamaged solid is orthotropic, the effective in-plane shear modulus is dependent on the degree of the crack alignment. The NIA method is then extended to obtain the effective electroelastic properties of an anisotropic piezoelectric solid containing two-dimensional insulat- ing, permeable or conducting microcracks with an arbitrary degree of alignment. We also derive a set of fifteen coupled nonlinear equations for the unknown effective reduced elastic compliances of a microcrac- ked, anisotropic, elastic solid by using the generalized self-consistent method (GSCM). The set of coupled nonlinear equations can be solved through iteration.     

A singular edge-based smoothed finite element method (ES-FEM) for crack analyses in anisotropic media

The recently developed edge-based smoothed finite element method (ES-FEM) is extended to fracture problems in anisotropic media using a specially designed five-node singular crack-tip (T5) element. In the formulation of singular ES-FEM, only the assumed displacement values (not the derivatives) on the boundaries of the smoothing domains are needed. Thus, a layer of T5 crack-tip element is devised to construct “singular” shape functions via a simple point interpolation with a fractional order basis, without mapping procedure. The effectiveness of the present singular ES-FEM is demonstrated by intensive examples for a wide range of degrees of anisotropy.     

Variational bounds for the effective electroelastic moduli of piezoelectric composites with electromechanical coupling spring-type interfaces

The present work focuses on variational bounds for the effective electroelastic moduli of multiphase piezoelectric composites with thin piezoelectric interphase. Both the inhomogeneities and the matrix are assumed to be piezoelectric and transversely isotropic. The piezoelectric interphase is modeled as the spring-type interface with electromechanical coupling. The inhomogeneities are assumed to be spheroidal so that the reinforcement geometry is able to range from thin flake to continuous fiber. The effective properties of the piezoelectric composite with interfacial imperfection are defined and the principles of minimum internal energy and enthalpy are derived. These principles are applied to analytically obtain the upper and lower bounds for the effective electroelastic moduli. Unlike the Voigt–Reuss-type bounds for perfect interface, the present bounds depend not only on the material properties and volume fraction, but also on the interface parameters, inhomogeneity shape and orientation. An example of a two-phase composite is given for detailed discussion, where dependence of the electroelastic moduli and their bounds on the inhomogeneity shapes and orientations as well as the interface properties is provided and discussed. To qualitatively account for the dependence, analysis based on two possible mechanisms, i.e., the simple mixture rule of composite and the weakening effect by imperfect interface, are also provided.     

Two-scale homogenization of elastic layered composites with interfaces oscillating in two directions

In a variety of situations of practical interest, the interface between two phases in a composite cannot be reasonably assumed to be smooth but has to be taken as being rough at the microscopic scale. How to determine the effective properties of such a composite remains a largely open problem in micromechanics. The present work is concerned with layered composites in which the interface between two neighboring layers oscillates quickly and periodically along two directions in the plane normal to the layering direction. In this case, the classical homogenization theory of layered composites is no longer applicable, since the interfacial oscillations prevent the layered composite in question from being homogeneous in the plane perpendicular to the layering direction. To overcome this difficulty, a two-scale homogenization method is proposed in the present work. First, at the mesoscopic scale, each zone in which an interface oscillates is homogenized as an interphase by an asymptotic analysis. The effective elastic properties of this interphase are determined by using a numerical method based on the fast Fourier transform (FFT) or estimated by applying the generalized self-consistent scheme (GSCS). Then, at the macroscopic scale, the effective elastic moduli of the composite made of the resulting plane layers and interphases are calculated with the help of the classical homogenization theory of layered composites. Finally, numerical examples are provided to illustrate the results for the effective elastic moduli of a layered composite obtained by the two-scale homogenization method proposed and to compare them with the corresponding numerical results given by the finite element method (FEM).     

Dislocation tri-material solution in the analysis of bridged crack in anisotropic bimaterial half-space

The problem of an edge-bridged crack terminating perpendicular to a bimaterial interface in a half-space is analyzed for a general case of elastic anisotropic bimaterials and specialized for the case of orthotropic bimaterials. The edge crack lies in the surface layer of thickness h bonded to semi-infinite substrate. It is assumed that long fibres bridge the crack. Bridging model follows from the assumption of “large” slip lengths adjacent to the crack faces and neglect of initial stresses. The crack is modelled by means of continuous distribution of dislocations, which is assumed to be singular at the crack tip. With respect to the bridged crack problems in finite dissimilar bodies, the reciprocal theorem (Ψ-integral) is demonstrated as to compute, in the present context, the generalized stress intensity factor through the remote stress and displacement field for a particular specimen geometry and boundary conditions using FEM. Also the application of the configurational force mechanics is discussed within the context of the investigated problem.     

计算复合材料有效弹性模量的重心有限元方法

采用几何法构造出任意边数多边形单元的重心插值形函数, 应用Galerkin法提出了求解弹性力学问题的重心有限元方法。用重心有限元方法对SiC/Ti和B/Al 2种纤维复合材料横向截面的有效弹性模量进行了预报。计算模型取纤维呈六边形排列且为各向同性的代表性单胞, 对其杨氏模量、 剪切模量和体积模量在较大的体积分数范围内进行了数值模拟。通过与解析公式和传统有限元的计算结果对比, 重心有限元方法的计算结果符合解析公式解的趋势, 与传统有限元的计算结果吻合较好。与传统有限元方法相比, 重心有限元方法的单元划分不受三角形或四边形的形状限制, 能够再现材料的真实结构。由于单元较大且数目较少, 本文方法具有很高的计算效率。      

The effect of honeycomb relative density on its effective in-plane elastic moduli: An experimental study

Honeycombs are discrete materials at the macro-scale level but their mechanical properties need to be calculated as a continuum material in order to simplify their design in engineering applications. The effective mechanical behavior of hexagonal honeycombs was studied by analytical means and correlated with experimental results for aluminum honeycombs. In particular, the effective in-plane elastic moduli of the honeycombs were studied as a function of their relative density. The effect of the bending, shear and axial deformations on various existing beam models was analyzed for both in-plane honeycomb directions. An experimental program was performed with honeycombs of densities ranging from the commercial low-density ones to the solid construction material. It is shown experimentally that the beam models describe well the material response in the direction of the honeycomb double wall. However, it is concluded that the effective elastic moduli for honeycombs with low relative densities are not similar in the two in-plane directions as predicted by previous studies.     

Probabilistic bounds on effective elastic moduli for the superconducting coils

The main purpose of the paper presented is probabilistic characterization of the effective elastic characteristics of a superconducting four-component composite. The probabilistic moments of these characteristics up to the fourth-order have been estimated on the basis of experimentally measured expected values and standard deviations of component materials elastic properties (Young moduli and Poisson coefficients). The Monte-Carlo simulation technique has been used to carry out all computational experiments. The results computed have been discussed in details in order to verify the sensitivity of respective probabilistic moments to input random data and to total number of random trials used to estimate these values.     

Influence of imperfect interface on the elastic moduli of syntactic foams

Syntactic foams are manufactured by dispersing microspheres in a polymeric matrix, and the macroscale material properties of these foams are estimated by analyzing a periodic distribution of the inclusions. The analysis in the simplest form, further assume that the inclusions are perfectly bonded to the matrix material. It has been shown in a previous study [P.R. Marur, Mater. Lett. 59 (2005) 1954–1957.] that analytical model overestimated the experimentally determined elastic moduli, and that the morphology of particle distribution has negligible influence on the elastic moduli. In this paper, the assumption of perfect adhesion between the inclusion and the matrix is relaxed to allow for possible localized slip and separation at the particle interface. The analytical results obtained considering imperfect interface well agree with the measured elastic modulus reported in the literature.     

Quantitative prediction of effective material properties of heterogeneous media

Effective electrical conductivity and electrical permittivity of water-saturated natural sandstones are evaluated on the basis of local porosity theory (LPT). In contrast to earlier methods, which characterize the underlying microstructure only through the volume fraction, LPT incorporates geometric information about the stochastic microstructure in terms of local porosity distribution and local percolation probabilities. We compare the prediction of LPT and of traditional effective medium theory with the exact results. The exact results for the conductivity and permittivity are obtained by solving the microscopic mixed boundary value problem for the Maxwell equations in the quasistatic approximation. Contrary to the predictions from effective medium theory, the predictions of LPT are in better quantitative agreement with the exact results.     

Effect of anisotropic plasticity on mixed mode interface crack growth

Crack growth along an interface between a solid with plastic anisotropy and an elastic substrate is modelled by representing the fracture process in terms of a traction-separation law specified on a crack plane. A phenomenological elastic-viscoplastic material model is applied, using one of two different anisotropic yield criteria to account for the plastic anisotropy. Conditions of small-scale yielding are assumed, and due to the mismatch of elastic properties across the interface the corresponding oscillating stress singularity field is applied as boundary conditions on the outer edge of the region analyzed. Crack growth resistance curves are calculated numerically, and based on these results the dependence of the steady-state fracture toughness on the near-tip mode mixity is determined. Different initial orientations of the principal axes relative to the interface are considered and it is found that the steady-state fracture toughness is quite sensitive to this orientation of the anisotropy.     

On the third-order bounds of the effective shear modulus of two-phase composites

Formulation of variational bounds for properties of inhomogeneous media constitutes one of the most fundamental parts of mechanics. The earliest work on multiphase media is the so-called Voigt’s upper bound and Reuss’ lower bound, corresponding to the simple rule of mixture or first-order bounds. The second-order bounds were formulated by Hashin and Shtrikman for macroscopically isotropic random composites. The third-order bounds of the bulk modulus were derived by Beran, which contain a pair of third-order bulk parameters. The third-order bounds of the shear modulus first derived by McCoy were improved by Milton and Phan-Thien, which further involve a pair of third-order shear parameters. In this study, by applying the stochastic variational principle of Xu (2009) the third-order bounds of the shear modulus are derived in an analytically most extensive trial function space. By further modifying Milton’s definition of shear parameters, the third-order bounds are finalized into a symmetric form, exactly like the Beran’s bounds of the bulk modulus. Since the bounds of the shear modulus play an essential role in plasticity theory of composites, the finalization of the third-order bounds also paves the way for further formulation of variational principles and bounds of nonlinear composites.     

Evaluation of the three-dimensional elastic Green's function in anisotropic cubic media

By using the Fourier transforms method, the three-dimensional Green's function solution for a unit force applied in an infinite cubic material is evaluated in this paper. Although the elastic behavior of a cubic material can be characterized by only three elastic constants, the explicit solutions of Green's function for a cubic material are not available in the literatures. The central problem for explicitly solving the elastic Green's function of anisotropic materials depends upon the roots of a sextic algebraic equation, which results from the inverse Fourier transforms and is composed of the material constants and position vector parameters. The close form expression of Green's function is presented here in terms of roots of the sextic equation. The sextic equation for an anisotropic cubic material is discussed thoroughly and specific results are given for possible explicit solutions.     

An FE-analysis of anisotropic creep damage and deformation in the single crystal SRR99 under multiaxial loads

Multidimensional stress–strain and damage analyses of engineering structural components with the help of numerical simulations are of great interest. These can only be done by using adequate material models and suitable numerical methods. Bertram and Olschewski (Computational modelling of anisotropic materials under creep conditions, Math. Modelling Sci. Comp. 5 (1995) 100–109; Anisotropic creep modeling of the single crystal superalloy SRR99, J. Comp. Mater. Sci. 5 (1996) 12–16), proposed a three-dimensional creep model for single crystals. An anisotropic creep damage model for single crystals was also suggested by Qi and Bertram (W. Qi, A. Bertram, Anisotropic creep damage modeling of single crystal superalloys, Tech. Mech. 17 (1997) 313–322; W. Qi, Modellierung der Kriechschadigung einkristalliner Superlegierungen in Hochtemperaturbereich, Ph.D. dissertation, Technical University Berlin, VDI Verlag, Düsseldorf, 1998; W. Qi, A. Bertram, Damage modeling of the single crystal superalloy SRR99 under monotonous creep, Comput. Mater. Sci. 13 (1998) 132–141). The coupled model has been used to predict the creep deformation and the lifetime of the single crystal SRR99 under uniaxial creep loads at 760°C. The purpose of this work is the application of the coupled model to the simulation of multiaxial creep behavior and damage development, and its dependence upon non-proportional loading paths of SRR99 at 760°C.     

An anisotropic model of the Mullins effect

The Mullins effect in rubber-like materials is inherently anisotropic. However, most constitutive models developed in the past are isotropic. These models cannot describe the anisotropic stress-softening effect, often called the Mullins effect. In this paper a phenomenological three-dimensional anisotropic model for the Mullins effect in incompressible rubber-like materials is developed. The terms, damage function and damage point, are introduced to facilitate the analysis of anisotropic stress softening in rubber-like materials. A material parametric energy function which depends on the right stretch tensor and written explicitly in terms of principal stretches and directions is postulated. The material parameters in the energy function are symmetric second-order damage and shear-history tensors. A class of energy functions and a specific form for the constitutive equation are proposed which appear to simplify both the analysis of the three-dimensional model and the calculation of material constants from experimental data. The behaviour of tensional and compressive ground-state Young’s moduli in uniaxial deformations is discussed. To further justify our model we show that the proposed model produces a transversely anisotropic non-virgin material in a stress-free state after a simple tension deformation. The proposed anisotropic theory is applied to several types of homogenous deformations and the theoretical results obtained are consistent with expected behaviour and compare well with several experimental data.     

Dependence of a crack growth path on the elastic moduli of an anisotropic solid

One of the ways to increase the resistance of a structure to catastrophic fracture is to force a main line crack to deviate from its path. For this reason the influence of the elastic moduli of an anisotropic material on crack rotation are studied. In particular a linear elastic problem for a straight Mode I crack, located on a symmetry axis of an orthotropic plane is considered. The strength properties of the material are assumed to be isotropic. Several crack models are considered for studying the direction of a crack growth path. It is shown that a crack modeled as a thin, elongated, elliptical hole leads to more plausible results concerning crack rotation conditions than an ideal cut model. The maximal tensile stresses are taken as a crack growth criterion. It is shown that for a class of orthotropic materials a crack deviates from the straight path just after it starts to grow, even in the conditions of uniaxial normal tension. The problem of the stability of a straight crack path under Mode I loadings is also considered. This problem is reduced to the problem of the fracture direction determination for thin, elongated, elliptical cavities slightly inclined to the initial direction. The conditions of instability are obtained within the framework of the proposed approach. It is shown that for a class of orthotropic materials a straight crack path is unstable in the conditions of uniaxial normal tension. This class of materials is larger than the one for which a crack deviates from the straight crack path just after its start.