A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media

A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of three-dimensional elasticity, but, since the assumption of strict separation of scales is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which higher-order terms, (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. An energy based micro–macro transition is then proposed for upscaling and constitutes, in fact, a generalization of the Hill–Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. As an illustration purpose, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed in the last part of the paper. Its efficiency is shown in the particular case of composites reinforced by long fibers.     

Multi-Scale Asymptotic Expansion for a Small Inclusion in Elastic Media

The aim of this paper is to present an asymptotic expansion of the influence of a small inclusion of different stiffness in an elastic media. The applicative interest of this study is to provide tools which take into account this influence and correct the deformation without inclusion by additive terms that can be precalculated and which depend only on the shape of the inclusion. We treat two problems: an anti-plane linearized elasticity problem and a plane strain problem. On every expansion order we provide corrective terms modeling the influence of the inclusion using techniques of scaling and multi-scale asymptotic expansions. The resulting expansion is validated by comparing it to a test case obtained by solving the Poisson transmission problem in the case of an inclusion of circular shape using the separation of variables method. Proofs of existence and uniqueness of our fields on unbounded domains are also adapted to the bidimensional Poisson problem and the linear elasticity problem.     

Balance laws and energy release rates for cracks in dipolar gradient elasticity

It is the purpose of this work to derive the balance laws (in the Günther–Knowles–Sternberg sense) pertaining to dipolar gradient elasticity. The theory of dipolar gradient (or grade 2) elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain–energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the gradient of both strain and rotation (additional terms). The balance laws are derived here through a more straightforward procedure than the one usually employed in classical elasticity (i.e. Noether’s theorem). Indeed, the pertinent balance laws are obtained through the action of the standard operators of vector calculus (grad, curl and div) on appropriate forms of the Hamiltonian of the system under consideration. These laws are directly related to the energy release rates in the processes of crack translation, rotation and self-similar expansion. Under certain conditions, they are identified with conservation laws and path-independent integrals are obtained.     

An asymptotic strain gradient Reissner-Mindlin plate model

In this paper we derive a strain gradient plate model from the three-dimensional equations of strain gradient linearized elasticity. The deduction is based on the asymptotic analysis with respect of a small real parameter being the thickness of the elastic body we consider. The body is constituted by a second gradient isotropic linearly elastic material. The obtained model is recognized as a strain gradient Reissner-Mindlin plate model. We also provide a mathematical justification of the obtained plate model by means of a variational weak convergence result.     

基于迭代法的非线性弹性均质化研究

微观结构对复合材料的宏观力学性能具有至关重要的影响, 通过合理设计复合材料微观结构可以得到期望的宏观性能. 均质化方法作为一种有效的设计方法, 它从微观结构的角度出发, 利用均匀化的概念, 实现了对复合材料宏观力学性能的预测和设计. 而当考虑非线性因素, 均质化的实现就非常困难. 本文利用双渐近展开方法, 将位移按照宏观位移和微观位移展开, 推导了非线性弹性均质化方程. 通过直接迭代法, 对非线性弹性均质化方程进行了求解, 并给出了具体的迭代方法和实现步骤. 本文基于迭代步骤和非线性弹性均质化方程编写MATLAB 程序, 对3种典型本构关系的周期性多孔材料平面问题进行了计算, 对比细致模型的应变能、最大位移和等效泊松比, 对程序及迭代方法的准确性进行了验证. 之后对一种三元橡胶基复合材料进行多尺度均质化, 将其分为芯丝尺度和层间尺度. 用线弹性的均质化方法得到了芯丝尺度的等效弹性参数, 并将其作为层间尺度的材料参数. 在层间尺度应用非线性弹性均质化方法对结构进行计算, 得到材料的宏观等效性能, 并以实验结果为基准进行评价.      

The flow of closely fitting particles in tapered tubes

The flow of rigid spheres, truncated cones and elastic incompressible spheres in tapered tubes is investigated assuming that the Reynolds equation is valid in the fluid and the linear theory of elasticity is applicable in the solid. It is shown that leading terms in the asymptotic expansion of pressure drop in terms of minimum fluid film thickness for neutrally buoyant rigid spheres and truncated cones are of higher order of magnitude compared to the corresponding terms for the flow of these particles in circular cylindrical tubes. The effect of taper angle on pressure drop is reduced in the case of soft elastic particles because of particle deformations and significant velocities at the particle surface.     

Dislocations in second strain gradient elasticity

A second strain gradient elasticity theory is proposed based on first and second gradients of the strain tensor. Such a theory is an extension of first strain gradient elasticity with double stresses. In particular, the strain energy depends on the strain tensor and on the first and second gradient terms of it. Using a simplified but straightforward version of this gradient theory, we can connect it with a static version of Eringen’s nonlocal elasticity. For the first time, it is used to study a screw dislocation and an edge dislocation in second strain gradient elasticity. By means of this second gradient theory it is possible to eliminate both strain and stress singularities. Another important result is that we obtain nonsingular expressions for the force stresses, double stresses and triple stresses produced by a straight screw dislocation and a straight edge dislocation. The components of the force stresses and of the triple stresses have maximum values near the dislocation line and are zero there. On the other hand, the double stresses have maximum values at the dislocation line. The main feature is that it is possible to eliminate all unphysical singularities of physical fields, e.g., dislocation density tensor and elastic bend-twist tensor which are still singular in the first strain gradient elasticity.     

Higher order model for soft and hard elastic interfaces

The present paper deals with the derivation of a higher order theory of interface models. In particular, it is studied the problem of two bodies joined by an adhesive interphase for which “soft” and “hard” linear elastic constitutive laws are considered. For the adhesive, interface models are determined by using two different methods. The first method is based on the matched asymptotic expansion technique, which adopts the strong formulation of classical continuum mechanics equations (compatibility, constitutive and equilibrium equations). The second method adopts a suitable variational (weak) formulation, based on the minimization of the potential energy. First and higher order interface models are derived for soft and hard adhesives. In particular, it is shown that the two approaches, strong and weak formulations, lead to the same asymptotic equations governing the limit behavior of the adhesive as its thickness vanishes. The governing equations derived at zero order are then put in comparison with the ones accounting for the first order of the asymptotic expansion, thus remarking the influence of the higher order terms and of the higher order derivatives on the interface response. Moreover, it is shown how the elastic properties of the adhesive enter the higher order terms. The effects taken into account by the latter ones could play an important role in the nonlinear response of the interface, herein not investigated. Finally, two simple applications are developed in order to illustrate the differences among the interface theories at the different orders.     

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ?; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.     

The use of displacement potentials in second order elasticity

This paper is concerned with the use of a representation in terms of displacement potentials in second order elasticity for equilibrium problems of homogeneous and isotropic materials. After justifying the adoption of an existing representation for linear elasticity for the purpose at hand, appropriate representations for solutions of second order elasticity problems in terms of displacement potentials (for both compressible and incompressible materials) are discussed. The use of the representations in obtaining complete solutions for equilibrium boundary-value problems is then illustrated by application to two examples of plane strain problems of compressible materials.     

Wedge and twist disclinations in second strain gradient elasticity

The aim of this paper is to study disclinations in the framework of a second strain gradient elasticity theory. This second strain gradient elasticity has been proposed based on the first and second gradients of the strain tensor by Lazar et al. [Lazar, M., Maugin, G.A., Aifantis, E.C., 2006. Dislocations in second strain gradient elasticity. Int. J. Solids Struct. 43, 1787–1817]. Such a theory is an extension of the first strain gradient elasticity [Lazar, M., Maugin, G.A., 2005. Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity. Int. J. Eng. Sci. 43, 1157–1184] with triple stress. By means of the stress function method, the exact analytical solutions for stress and strain fields of straight disclinations in an infinitely extended linear isotropic medium have been found. An important result is that the force stress, double stress and triple stress produced by wedge and twist disclinations are nonsingular. Meanwhile, the corresponding elastic strain and its gradients are also nonsingular. Analytical results indicate that the second strain gradient theory has the capacity of eliminating all unphysical singularities of physical fields.     

Strain gradient elastic homogenization of bidimensional cellular media

The present paper aims at introducing an homogenization scheme for the determination of strain gradient elastic coefficients. This scheme is based on a quadratic extension of homogeneous boundary condition (HBC). It allows computing strain elastic effective tensors. This easy-to-handle computational procedure will then be used to construct overall behaviors and to verify some theoretical predictions on strain gradient elasticity.     

纤维排列方式对复合材料总体粘弹性常数的影响

对于金属基或高分子聚合物基复合材料，在特定情况下会表现出明显的粘弹性特性。本文采用Riemann—Liouville形式的分数阶导数模型描述基体的粘性特性，通过渐进均匀化方法给出了预测纤维加强复合材料整体本构关系的解析表达式，给出应用于基体具有Makris粘弹性关系的具体形式。最后，考察了圆截面纤维正方形排列和对角排列时的总体粘弹性弹性常数随纤维比的变化曲线。结果表明，这类复合材料仍具有粘弹性特性，其整体粘弹性本构关系的弹性部分综合了纤维弹性和基体弹性的贡献，粘性部分来自基体粘性的贡献，复合材料具有和基体相同的粘性系数和分数阶。为分析微结构特征对整体特性的贡献，须求解两类局部问题。在相同纤维体积比情况下，正方形排列的总体弹性系数大于正方形对角排列，而粘性常数相反。     

Asymptotic Analysis of an Elasticity Equation for a Finger-Like Hydraulic Fracture

We derive a novel integral equation relating the fluid pressure in a finger-like hydraulic fracture to the fracture width. By means of an asymptotic analysis in the small height to length ratio limit we are able to establish the action of the integral operator for receiving points that lie within three distinct regions: (1) an outer expansion region in which the dimensionless pressure is shown to be equal to the dimensionless width plus a small correction term that involves the second derivative of the width, which accounts for the nonlocal effects of the integral operator. The leading order term in this expansion is the classic local elasticity equation in the PKN model that is widely used in the oil and gas industry; (2) an inner expansion region close to the fracture tip within which the action of the elastic integral operator is shown to be the same as that of a finite Hilbert transform associated with a state of plane strain. This result will enable pressure singularities and stress intensity factors to be incorporated into analytic models of these finger-like fractures in order to model the effect of material toughness; (3) an intermediate region within which the action of the Fredholm integral operator of the first kind is reduced to a second kind operator in which the integral term appears as a small perturbation which is associated with a convergent Neumann series. These results are important for deriving analytic models of finger-like hydraulic fractures that are consistent with linear elastic fracture mechanics. Submitted to Journal of Elasticity on February 5, 2007. Re-submitted with revisions on May 30, 2007.     

Cavity expansion of a gradient-dependent solid cylinder

This paper presents an elasto-plastic analysis for cavity expansion in a solid cylinder. The solid is modelled using a strain gradient plasticity model to account for the influence of microstructures on the macroscopic mechanical behaviour. A numerical shooting method, together with Broyden’s iteration procedure, is developed to solve the resulting fourth-order ordinary differential equation with two-point boundary conditions for the gradient-dependent problem. Fully elastic-plastic solutions to the cavity expansion are obtained and they are compared with conventional results for a number of examples. The effects of microstructure on macroscopic behaviour for the cavity expansion problem are analysed. It is demonstrated that, with consideration of microstructural effects, the deformation and stress distributions in the cylinder are highly inhomogeneous during both the initial loading and the subsequent elastic and plastic expansion stages. The gradient effects can result in a stiffer response in the elastic regime (as compared with the corresponding conventional prediction), but a weaker response in the plastic regime. As expected, the overall elasto-plastic behaviour of the gradient-dependent cylinder depends on the material parameters as well as the cylinder thickness. It is shown that the strain gradient theory solutions reduce to the conventional ones as a special case when the dimension of the microstructures is negligible compared with the cylinder size. The results in this paper can be used as a benchmark for further numerical investigations of the cavity expansion problem.     

Numerical homogenization of elastic and thermal material properties for metal matrix composites (MMC)

A two-scale material modeling approach is adopted in order to determine macroscopic thermal and elastic constitutive laws and the respective parameters for metal matrix composite (MMC). Since the common homogenization framework violates the thermodynamical consistency for non-constant temperature fields, i.e., the dissipation is not conserved through the scale transition, the respective error is calculated numerically in order to prove the applicability of the homogenization method. The thermomechanical homogenization is applied to compute the macroscopic mass density, thermal expansion, elasticity, heat capacity and thermal conductivity for two specific MMCs, i.e., aluminum alloy Al2024 reinforced with 17 or 30 % silicon carbide particles. The temperature dependency of the material properties has been considered in the range from 0 to \(500{\,}^\circ \mathrm {C}\), the melting temperature of the alloy. The numerically determined material properties are validated with experimental data from the literature as far as possible.     

Hydrodynamics of active polar systems in a (Visco)elastic background

We derive the full set of macroscopic equations necessary to describe the dynamics of systems with active polar order in a viscoelastic or elastic background. The active polar order is manifested by a second velocity, whose non-zero modulus is the polar order parameter and whose direction is the polar preferred direction. Viscoelasticity is described by a relaxing strain field allowing for a straightforward change from transient to permanent elasticity. Relative rotations of the elastic structure with respect to the polar direction are taken into account. The intricate coupling between active polar order and (transient) elasticity leads to a combined relaxation of the polar order parameter and the strains. The rather involved sound spectrum contains a specific excitation due to a reversible coupling between elasticity and polar order. Effects of chirality are also considered.     

Interface Conditions for a Phase Field Model with Anisotropic and Non-Local Interactions

An alternative formulation of the phase field method is utilized from an integral equation perspective. The technique allows one to derive macroscopic conditions at the interface from the microscopic potentials. Differential geometry and asymptotic analysis yield interface conditions, in arbitrary spatial dimension, for interactions that may include anisotropy as well as non-local potentials. The interface conditions can be expressed in various formulations, for example, in terms of the principal curvature directions of the interface, or the second order directional derivatives of the (signed) distance function and the Hessian of the surface tension.