A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium

Homogenization may change fundamentally the constitutive laws of materials. We show how a heterogeneous Cauchy continuum may lead to a non Cauchy continuum. We study the effective properties of a linear elastic medium reinforced periodically with thin parallel fibers made up of a much stronger linear elastic medium and we prove that, when the Lamé coefficients in the fibers and the radius of the fibers have appropriate order of magnitude, the effective material is a second gradient material, i.e. a material whose energy depends on the second gradient of the displacement.     

Asymptotic homogenization modeling of smart composite generally orthotropic grid-reinforced shells: Part II– Applications

A comprehensive micromechanical model for the analysis of thin smart composite grid-reinforced shells with an embedded periodic grid of generally orthotropic cylindrical reinforcements that may also exhibit piezoelectric properties is developed and applied to examples of practical importance. Details on derivation of a general homogenized smart shell model are provided in Part I of this work. The present paper solves the obtained unit cell problems and develops expressions for the effective elastic, piezoelectric and thermal expansion coefficients for the grid reinforced smart composite shell. Thus obtained effective coefficients are universal in nature and can be used to study a wide variety of boundary value problems. The applicability of the model is illustrated by means of several examples including cylindrical reinforced smart composite shells, and multi-layer smart shells. The derived expressions allow tailoring the effective properties of a smart grid-reinforced shell to meet the requirements of a particular application by changing certain geometric or physical parameters.     

SURFACE EFFECTS ON ELASTIC FIELDS AROUND SURFACE DEFECTS

<正>There are always severe stress concentrations around surface defects like grooves or bugles,which might induce the failure of solid materials and structures.In the present paper,we consider the elastic fields around nanosized bugles and grooves on solid surfaces.The influence of surface elasticity on the elastic deformation is addressed through a finite element method.It is found that when the size of defects shrinks to nanometer,the stress fields around such defects will be affected significantly by surface effects.     

Scattering of elastic waves by an arbitrary small imperfection in the surface of a half-space

T , the first of two articles, is concerned with the scattering of elastic waves by arbitrary surface-breaking or near surface defects in an isotropic half-plane. We present an analytical solution, by the method of matched asymptotic expansions, when the parameter , which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The problem is formulated for a general class of small defects, including cracks, surface bumps and inclusions, and for arbitrary incident waves. As a straightforward example of the asymptotic scheme we specialize the defect to a two-dimensional circular void or protrusion, which breaks the free surface, and assume Rayleigh wave excitation ; this inner problem is exactly solvable by conformal mapping methods. The displacement field is found uniformly to leading order in , and the Rayleigh waves which are scattered by the crack are explicitly determined. In the second article we use the method given here to tackle the important problem of an inclined edge-crack. In that work we show that the scattered field can be found to any asymptotic order in a straightforward manner, and in particular the Rayleigh wave coefficients are given to O(²).     

Role of discrete intra-granular slip bands on the strain-hardening of polycrystals

This work investigates a new micromechanical modeling of polycrystal plasticity, accounting slip bands for physical plastic heterogeneities considered as periodically distributed within grains. These intra-granular plastic heterogeneities are modeled by parallel flat ellipsoidal sub-domains, each of them may have a distinct uniform plastic slip. To capture the morphology of slip bands occurring in plastically deforming polycrystals, these interacting sub-domains are considered as oblate spheroids periodically distributed and constrained by spherical grain boundaries. In this paper, we focus the study on the influences of internal length scale parameters related to grain size, spatial period and thickness of slip bands on the overall material’s behavior. In a first part, the Gibbs free energy accounting for elastic interactions between plastic heterogeneities is calculated thanks to the Green function’s method in the case of an isolated spherical grain with plastic strain occurring only in slip bands embedded in an infinite elastic matrix. In a second part, the influence of discrete periodic distributions of intra-granular slip bands on the polycrystal’s behavior is investigated considering an aggregate with random crystallographic orientations. When the spatial period of slip bands is on the same order as the grain radius, the polycrystal’s mechanical behavior is found strongly dependent on the ratio between the spatial period of slip bands and the grain size, as well as the ratio between the slip band thickness and the grain size, which cannot be captured by classic length scale independent Eshelby-based micromechanics.     

Comparison of a Rapidely Converging Phase Field Model for Interfaces in Solids with the Allen-Cahn Model

We compare two phase field models for interfaces in elastic solids carrying low surface energy. One model has hybrid properties of a Hamilton-Jacobi and a parabolic equation, the other is the Allen-Cahn model. For vanishing width of the interface we construct asymptotic solutions of second order for the hybrid model and of first order for the Allen-Cahn model. These constructions show that the width of the diffused interface necessary for tracking accurately the sharp interface can be chosen much larger for the hybrid model than for the Allen-Cahn model, and that moreover the hybrid model can describe interfaces with nonlinear kinetic relation. This explains why numerical simulations based on the hybrid model are considerably more effective. These simulations are discussed in the last section.     

Numerical implementation of composite Koiter shells including membrane/bending coupling coefficientsImplémentation numérique des coques composites de Koiter prenant en considération les coefficients de couplage membrane/flexions

We propose a way for determining the generalized coefficients of rigidity – some of which are membrane/bending coupling coefficients – which appear in the deformation energy of the Koiter model of thin shells. This is concerned with a heterogeneous material in the thickness direction. A new program to compute these coefficients is implemented in the finite element code Modulef, in order to simulate problems of thin multilayered shells with linearly elastic anisotropic layers. We propose an example of an inhibited multilayered thin shell, with hyperbolic middle surface, involving a composite material with unidirectional fibres. To cite this article: H. Ranarivelo, C. R. Mecanique 330 (2002) 273–278.     

Propagation and interaction of non-linear elastic plane waves in soft solids

Using the perturbation method of weakly non-linear asymptotics we analyze the propagation and interaction of elastic plane waves in a model of a soft solid proposed by Hamilton et al. [Separation of compressibility and shear deformation in the elastic energy density, J. Acoust. Soc. Am. 116 (2004) 41-44]. We derive the evolution equations for the wave amplitudes and find analytical formulas for all interaction coefficients of quadratically non-linear interacting waves. We show that in spite of the assumption of almost incompressibility used in Hamilton et al. [Separation of compressibility and shear deformation in the elastic energy density, J. Acoust. Soc. Am. 116 (2004) 41-44], the model behaves essentially like that of a compressible isotropic material. Both the structure of the equations and the interaction patterns are similar. The models differ, however, in the elastic constants that characterize them, and hence the values of the coefficients in the evolution equations and the values of the interaction coefficients differ.     

Global behavior of compressible three-phase flow in heterogeneous porous media

The homogenization method is used to analyze the equivalent behavior of a compressible three-phase flow model in heterogeneous porous media with periodic microstructure, including capillary effects. Asymptotic expansions lead to the definition of a global or effective model of an equivalent homogeneous reservoir. The resulting equations are of the same type as the points equations, with effective coefficients. The method allows the determination of these effective coefficients from a knowledge of the geometrical structure of the basic cell and its heterogeneities. Numerical computations to obtain the homogenized coefficients of the entire reservoir have been carried out via a finite element method.     

Vertical correlations in superlattices using the Grinfeld method

In this paper, we analyse the energetics of a multilayered structures like, for instance, B/A/B/A_substrate. It is well-known that a coherent pre-strained B layer on an A substrate will generally results in a corrugation of the free-surface of the B layer. This behavior is the result of stress relaxation in the B-layer and the phenomenon is known as the Asaro-Tiller-Grinfeld instability. We extend the methods used for a two-layer structure to a multilayered structure and the main application is the vertical correlation in superlattices. We analyse the energetics of a corrugated B layer which is grown on a A/B/A_substrate, where the A layers are flat but the intermediate B layer is already corrugated. We show that the self-organization of the second B layer, due to elastic interactions in the bulk, depends on the corrugation of the first B layer and the generic best situation is that of a top-on-top (also called correlated layers) vertical alignment. We also prove that the interaction energy between two successive B layers attains a maximum at a critical thickness of the intermediate A layer. This interaction energy has the same order of magnitude as the elastic energy release due to free-surface corrugation at each upper surface of a B layer.     

Iteration method in linear elasticity of random structure composites containing heterogeneities of noncanonical shape

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of heterogeneities of arbitrary shape. The general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced and statistical averages are obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. Effective elastic moduli and the first statistical moments of stresses in the heterogeneities are estimated for statistically homogeneous composites with the general case of both the shape and inhomogeneity of the heterogeneities moduli. The explicit new representations of the effective moduli and stress concentration factors are built by the iteration method in the framework of the quasicristallite approximation but without basic hypotheses of classical micromechanics such as both the EFH and “ellipsoidal symmetry” assumption. Numerical results are obtained for some model statistically homogeneous composites reinforced by aligned identical homogeneous heterogeneities of noncanonical shape. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.     

Identification of a constitutive law for trabecular bone samples under remodeling in the framework of irreversible thermodynamics

We construct in the present paper constitutive models for bone remodeling based on micromechanical analyses at the scale of a representative unit cell (RUC) including a porous trabecular microstructure. The time evolution of the microstructure is simulated as a surface remodeling process by relating the surface growth remodeling velocity to a surface driving force incorporating a (surface) Eshelby tensor. Adopting the framework of irreversible thermodynamics, a 2D constitutive model based on the setting up of the free energy density and a dissipation potential is identified from FE simulations performed over a unit cell representative of the trabecular architecture obtained from real bone microstructures. The static and evolutive effective properties of bone at the scale of the RUC are obtained by combining a methodology for the evaluation of the average kinematic and static variables over a prototype unit cell and numerical simulations with controlled imposed first gradient rates. The formulated effective growth constitutive law at the scale of the homogenized set of trabeculae within the RUC is of viscoplastic type and relates the average growth strain rate to the homogenized stress tensor. The postulated model includes a power law function of an effective stress chosen to depend on the first and second stress invariants. The model coefficients are calibrated from a set of virtual testing performed over the RUC subjected to a sequence of loadings. Numerical simulations show that overall bone growth does not show any growth kinematic hardening. The obtained results quantify the strength and importance of different types of external loads (uniaxial tension, simple shear, and biaxial loading) on the overall remodeling process and the development of elastic deformations within the RUC.     

Strain Relaxation in an Alloy Film with a Rough Free Surface

We study strain relief by surface roughness and composition variation in a stressed alloy film. Instead of using common perturbation techniques, we derive a rigorous relaxation formula based on the energy approach in the case of slightly undulating surface and fluctuating composition. We do not require any a priori assumption of elastic isotropy or identical material properties between film and substrate in deriving our result. We show that the change of elastic energy is negative, giving rise to energy relief due to the presence of free surface. We apply our result to the study of compositional and morphological instabilities of a stressed thin layer with a free surface. The critical wave number of instability is determined by the competition between the destabilizing influence of elastic strain energy and the stabilizing influence of chemical and surface energies.     

M积分与夹杂/缺陷弹性模量的显式关系简

M积分在材料构型力学中表征着缺陷自相似扩展的能量释放率,而有效弹性模量下降量在传统损伤力学中是一个具有内变量属性的损伤参数. 探讨了两者之间的特定关系,以此为材料构型力学与损伤力学搭建桥梁.借助穆斯海里什维利（Muskhelishvili）复势函数方法获取无限大弹性平面含圆形夹杂的弹性场解,根据M 积分的复势函数解析表达式得到M 积分与夹杂弹性模量的显式表达式. 随后通过有限元分析,对含复杂缺陷群的弹塑性材料进行数值模拟,结果表明内部缺陷区域的有效弹性模量下降与M 积分存在着特定关系. 基于此,提出利用材料构型力学中的外变量参数（M 积分）来替代损伤力学中的内变量参数（弹性模量下降量）描述材料的缺陷演化.     

M积分与夹杂/缺陷弹性模量的显式关系

M积分在材料构型力学中表征着缺陷自相似扩展的能量释放率,而有效弹性模量下降量在传统损伤力学中是一个具有内变量属性的损伤参数. 探讨了两者之间的特定关系,以此为材料构型力学与损伤力学搭建桥梁.借助穆斯海里什维利（Muskhelishvili）复势函数方法获取无限大弹性平面含圆形夹杂的弹性场解,根据M 积分的复势函数解析表达式得到M 积分与夹杂弹性模量的显式表达式. 随后通过有限元分析,对含复杂缺陷群的弹塑性材料进行数值模拟,结果表明内部缺陷区域的有效弹性模量下降与M 积分存在着特定关系. 基于此,提出利用材料构型力学中的外变量参数（M 积分）来替代损伤力学中的内变量参数（弹性模量下降量）描述材料的缺陷演化.      

Experimental Investigation of Concrete Packages for Radioactive Waste Management: Permeability and Influence of Junctions

We studied the feasibility of a concrete package for radioactive waste management in a joint program involving Andra (the French agency for radioactive waste management) and CEA (the French atomic energy commission). The package??s long-term durability and radionuclides?? containment were the major concerns. The presence of junctions between the prefabricated body and the poured-in-place lids was identified as a major weakness. The first objective of this study was to characterize the permeability of the selected concrete and of the package itself (that is to say accounting for the junctions influence). We used special specimens including a junction, and tested three different surface preparation methods. The second objective was to assess the influence of the manufacturing conditions (laboratory and industrial) on permeability.     

Analytical procedure for determining the linear and nonlinear effective properties of the elastic composite cylinder

In this work we consider a cylindrical structure composed of a nonlinear core (inhomogeneity) surrounded by a different nonlinear shell (matrix). We elaborate a technique for determining its linear elastic moduli (second order elastic constants) and the nonlinear elastic moduli, which are called Landau coefficients (third order elastic constants). Firstly, we develop a nonlinear perturbation method which is able to turn the initial nonlinear elastic problem into a couple of linear problems. Then, we prove that only the solution of the first linear problem is necessary to calculate the linear and nonlinear effective properties of the heterogeneous structure. The following step consists in the exact solution of such a linear problem by means of the complex elastic potentials. As result we obtain the exact closed forms for the linear and nonlinear effective elastic moduli, which are valid for any volume fraction of the core embedded in the external shell.     

Curvature-dependent surface energy and implications for nanostructures

At small length scales, several size-effects in both physical phenomena and properties can be rationalized by invoking the concept of surface energy. Conventional theoretical frameworks of surface energy, in both the mechanics and physics communities, assume curvature independence. In this work we adopt a simplified and linearized version of a theory proposed by Steigmann–Ogden to capture curvature-dependence of surface energy. Connecting the theory to atomistic calculations and the solution to an illustrative paradigmatical problem of a bent cantilever beam, we catalog the influence of curvature-dependence of surface energy on the effective elastic modulus of nanostructures. The observation in atomistic calculations that the elastic modulus of bent nanostructures is dramatically different than under tension – sometimes softer, sometimes stiffer – has been a source of puzzlement to the scientific community. We show that the corrected surface mechanics framework provides a resolution to this issue. Finally, we propose an unambiguous definition of the thickness of a crystalline surface.     

Micromechanical modeling of the elastic–viscoplastic behavior of polycrystalline steels

A self-consistent model developed to describe the elastic–viscoplastic behavior of heterogeneous materials is applied to low carbon steels to simulate tensile tests at various strain rates in the low temperature range. The choice of crystalline laws implemented in the model is discussed through the viscoplastic flow rule and several strain-hardening laws. Comparisons between three work-hardening models show that the account of dislocation annihilation improves the results on simulations at large strains. The evolution of the Lankford coefficients and texture development are also successfully simulated. Some microstructural aspects of deformation such as the stored energy and the evolution of the flow rates are discussed. By including the dislocation density on each slip system as internal variable, intragranular heterogeneities are underscored.