复合材料应力分析的均匀化方法

建立了基于均匀化理论的确定复合材料结构应力场的方法．其实质是用均质的宏观结构和非均质的具有周期性分布的细观结构描述原结构；将力学量表示成关于宏观坐标和细观坐标的函数，并用细观和宏观两种尺度之比为小参数展开，用摄动技术将原问题化为一细观均匀化问题和一宏观均匀化问题．这两个问题的解确定了包含等效位移和一阶近似位移的位移场，由此获得应力场．利用该方法给出了圆柱形孔隙材料和单向纤维复合材料在单向拉伸时的应力场以及空隙材料简支梁的局部应力场，说明了该方法的有效性     

Micromechanical approach to the strength properties of frictional geomaterials

The present paper describes a micromechanics-based approach to the strength properties of composite materials with a Drucker–Prager matrix in the situation of non-associated plasticity. The concept of limit stress states for such materials is first extended to the context of homogenization. It is shown that the macroscopic limit stress states can theoretically be obtained from the solution to a sequence of viscoplastic problems stated on the representative elementary volume. The strategy of resolution implements a non-linear homogenization technique based on the modified secant method. This procedure is applied to the determination of the macroscopic strength properties and plastic flow rule of materials reinforced by rigid inclusions, as well as for porous media. The role of the matrix dilatancy coefficient is in particular discussed in both cases. Finally, finite element solutions are derived for a porous medium and compared to the micromechanical predictions.     

基于均匀化理论的页岩微观多孔黏土强度特性

页岩强度是页岩油气开发所必需的基础技术参数之一,对页岩强度的研究贯穿于钻完井、压裂工艺施工的全过程.常规宏观室内实验存在试样获取困难、耗时较长,受井下工矿制约,地球物理方法获取资料品质欠佳且增加了井下设备卡、埋风险.因此,提出基于均匀化理论评价页岩微观多孔黏土强度的方法,进行多孔黏土组成与力学分析.基于耗散能原理和Drucker-Prager准则,开展了微观多孔黏土的强度与$\pi$函数的应变求解分析;讨论黏土颗粒与粒间孔隙的力学特性,建立多孔黏土的均匀化应变能;采用强度均匀化理论构建微观非线性函数模型,建立与多孔黏土组成、摩擦系数、内聚系数等参数相关的均匀化函数模型;基于纳米力学实验、量纲分析和有限元模拟,分析多孔黏土硬度、强度与组成的内在关系.研究结果表明,页岩微观多孔黏土的弹性模量和硬度与黏土堆积密度正相关,当黏土堆积密度一定时,硬度与内聚系数的比值受摩擦系数影响较大,为非线性递增;通过量纲分析和有限元模拟,求解页岩微观多孔黏土关于硬度--强度--堆积密度的$\pi$函数,揭示页岩微观黏土矿物的组成与力学性质的关系,为进一步深入研究页岩细观强度参数和宏观强度预测奠定基础.      

A homogenization-based constitutive model for isotropic viscoplastic porous media

An approximate model based on the “second-order” nonlinear homogenization method is proposed to estimate the effective behavior of isotropic, viscoplastic, porous materials. The model is constructed in such a way that it reproduces exactly the behavior of a “composite-sphere assemblage” in the limit of hydrostatic loadings, and therefore coincides with the hydrostatic limit of Gurson’s criterion in the special case of ideal plasticity. As a consequence, the new model improves on earlier homogenization estimates, which have been found to be quite accurate for low triaxialities but overly stiff for sufficiently high triaxialities and nonlinearities. Additionally, the estimates delivered by the model exhibit a dependence on the third invariant of the macroscopic stress tensor, which has a nontrivial effect on the effective response of the material at moderate triaxialities. The proposed model is compared with exact results obtained for a special class of porous materials with sequentially laminated microstructures. The agreement is found to be quite good for the entire range of stress triaxialities, and all values of the porosity and nonlinearity considered.     

Asymptotic analysis of the equilibrium equation of a fluid-saturated porous medium by the homogenization method

The homogenization of static elasticity equations describing the stress strain state of fluid-saturated porous medium is considered. In this paper, the homogenization method is used to determine the pore pressure transfer tensor, which (a coefficient in the isotropic case) is an important parameter influencing the stress-strain state of fluid-saturated rocks. It shows what a part of the pressure in the fluid is “active” in the formation of macroscopic strains.The pore pressure transfer tensor is calculated for model and real geological specimens. The dependence of this tensor on the porosity, pore shape, and Poisson ratio is investigated. The use of the computational technique for determining the effective properties of rocks shows that it is practically important in the engineering geology.     

Analysis of Composite Molding Using the Homogenization Method

In this paper, we present an application of the homogenization method to the analysis of Resin Transfer Molding (RTM) and Structural Reaction Injection Molding (SRIM). RTM and SRIM are relatively new molding processes for manufacturing continuous fiber reinforced polymer composites. First, the mold flow is analyzed. In the molding process, the resin experiences significant temperature changes as it fills the mold and forms a free boundary with air as it pushes out the air. In addition, the flow domain is the mold cavity packed with fiber perform, which is a porous medium. Here, the homogenization method is used to model the non-isothermal flow through porous media with free boundaries. A computer program is developed which is capable of simulating a three-dimensional mold flow using the finite element approximation. An example is provided for a three-dimensional part. Then, an analysis of the residual stress developed in the curing stage is given. The curing stage starts when the mold is completely filled and it involves chemical reaction and large temperature variation. In curing, the resin part undergoes larger volume shrinkage than the fiber part, and the residual stresses are developed due to this volume mismatch. In some cases, these stresses are large enough to cause micro-cracking and to exhaust the strength of the material. Here, a brief discussion of the application of the homogenization method to a residual stress analysis is given and one example is provided.     

Détermination du critère de rupture macroscopique d'un milieu poreux par homogénéisation non linéaireDetermination of the macroscopic strength criterion of a porous medium by nonlinear homogenization

A porous medium, which matrix is a perfectly plastic solid, is considered. This paper proposes a method to determine the macroscopic admissible stress states. The method is based on a homogenization technique which takes advantage of the equivalence, under certain conditions, between a problem of limit analysis and a ficticious nonlinear elastic problem. The particular case of a Drucker–Prager solid matrix is considered. The method provides an analytical expression for the complete macroscopic strength criterion. To cite this article: J.-F. Barthélémy, L. Dormieux, C. R. Mecanique 331 (2003).     

Homogenization Modeling and Parametric Study of Moisture Transfer in an Unsaturated Heterogeneous Porous Medium

The classical mass balance equation is usually used to model the transfer of humidity in unsaturated macroscopically homogeneous porous media. This equation is highly non-linear due to the pressure-dependence of the hydrodynamic characteristics. The formal homogenization method by asymptotic expansions is applied to derive the upscaled form of this equation in case of large-scale heterogeneities of periodic structure. The nature of such heterogeneities may be different, resulting in locally variable hydrodynamic parameters. The effective capillary capacity and the effective hydraulic conductivity are defined as functions of geometry and local characteristics of the porous medium. A study of a two-dimensional stone-mortar system is performed. The effect of the second medium (the mortar), on the global behavior of the system is investigated. Numerical results for the Brooks and Corey hydrodynamic model are provided. The sensitivity analysis of the parameters of the model in relation to the effective hydrodynamic parameters of the porous structure is presented.     

Stress equivalence principle for saturated porous mediaPrincipe d'équivalence en contrainte pour milieux poreux saturés

The stress equivalence principle for saturated porous media is studied in the plastic domain using a homogenization approach. The skeleton is composed of a micro-isotropic and micro-homogeneous material. The stress localization law in saturated porous media is first obtained. This makes it possible to define an appropriate effective stress tensor in the sense of the stress equivalence principle. The form of the effective stress tensor is examined for two particular yield functions of skeleton material. To cite this article: D. Lydzba, J.-F. Shao, C. R. Mecanique 330 (2002) 297–303.