A micromechanical study of drying and carbonation effects in cement-based materials

This paper is devoted to a micromechanical study of mechanical properties of cement-based materials by taking into account effects of water saturation degree and carbonation process. To this end, the cement-based materials are considered as a composite material constituted with a cement matrix and aggregates (inclusions). Further, the cement matrix is seen as a porous medium with a solid phase (CSH) and pores. Using a two-step homogenization procedure, a closed-form micromechanical model is first formulated to describe the basic mechanical behavior of materials. This model is then extended to partially saturated materials in order to account for the effects of water saturation degree on the mechanical properties. Finally, considering the solid phase change and porosity variation related to the carbonation process, the micromechanical model is coupled with the chemical reaction and is able to describe the consequences of carbonation on the macroscopic mechanical properties of material. Some comparisons between numerical results and experimental data are presented.     

Distribution-enhanced homogenization framework and model for heterogeneous elasto-plastic problems

Multi-scale computational models offer tractable means to simulate sufficiently large spatial domains comprised of heterogeneous materials by resolving material behavior at different scales and communicating across these scales. Within the framework of computational multi-scale analyses, hierarchical models enable unidirectional transfer of information from lower to higher scales, usually in the form of effective material properties. Determining explicit forms for the macroscale constitutive relations for complex microstructures and nonlinear processes generally requires numerical homogenization of the microscopic response. Conventional low-order homogenization uses results of simulations of representative microstructural domains to construct appropriate expressions for effective macroscale constitutive parameters written as a function of the microstructural characterization. This paper proposes an alternative novel approach, introduced as the distribution-enhanced homogenization framework or DEHF, in which the macroscale constitutive relations are formulated in a series expansion based on the microscale constitutive relations and moments of arbitrary order of the microscale field variables. The framework does not make any a priori assumption on the macroscale constitutive behavior being represented by a homogeneous effective medium theory. Instead, the evolution of macroscale variables is governed by the moments of microscale distributions of evolving field variables. This approach demonstrates excellent accuracy in representing the microscale fields through their distributions. An approximate characterization of the microscale heterogeneity is accounted for explicitly in the macroscale constitutive behavior. Increasing the order of this approximation results in increased fidelity of the macroscale approximation of the microscale constitutive behavior. By including higher-order moments of the microscale fields in the macroscale problem, micromechanical analyses do not require boundary conditions to ensure satisfaction of the original form of Hill's lemma. A few examples are presented in this paper, in which the macroscale DEHF model is shown to capture the microscale response of the material without re-parametrization of the microscale constitutive relations.     

A semi-analytical model for the behavior of saturated viscoplastic materials containing two populations of voids of different sizes

This paper presents a micromechanical model for a porous viscoplastic material containing two populations of pressurized voids of different sizes. Three scales are distinguished: the microscopic scale (corresponding to the size of the small voids), the mesoscopic scale (corresponding to the size of the large voids) and the macroscopic scale. It is assumed that the first homogenization step is performed at the microscopic scale, and, at the mesoscopic scale, the matrix is taken to be homogeneous and compressible. At the mesoscopic scale, the second homogenization step, on which the present study focuses, is based on a simplified representative volume element: a hollow sphere containing a pressurized void surrounded by a nonlinear viscoplastic compressible matrix. The nonlinear behavior of the matrix, which is expressed using the results obtained in the first homogenization step, is approached using a modified secant linearization procedure involving the discretization of the hollow sphere into concentric layers. Each layer has uniform secant moduli. The predictions of the model are compared with the more accurate numerical results obtained using the finite element method. Good agreement is found to exist with all the macroscopic stress triaxialities and all the porosity and nonlinearity values studied.     

Equivalent mechanical properties of textile monolayers from discrete asymptotic homogenization

The determination of the effective mechanical moduli of textiles from mechanical measurements is usually difficult due to their discrete architecture, which makes micromechanical analyses a relevant alternative to access those properties. Micropolar continuum models describing the effective mechanical behavior of woven fabric monolayers are constructed from the homogenization of an identified repetitive pattern of the textile within a representative unit cell. The interwoven yarns within the textile are represented as a network of trusses connected by nodes at their crossover points. These trusses have extensional and bending rigidities to allow for yarn stretching and flexion, and a transverse shear deformation is additionally considered. Interactions between yarns at the crossover points are captured by beam segments connecting the nodes. The woven fabric is modeled after homogenization as an anisotropic planar continuum with two preferred material directions in the mean plane of the textile. Based on the developed methodology, the effective mechanical properties of plain weave and twill are evaluated, including their bending moduli and characteristic flexural lengths. A satisfactory agreement is obtained between the effective moduli obtained by homogenization and numerical values obtained by finite element simulations performed over periodic unit cells.     

A nonlinear homogenization procedure for periodic masonry

The present paper deals with the problem of the determination of the in-plane behavior of masonry material. The masonry is considered as a composite material composed by a regular distribution of blocks connected by horizontal and vertical mortar joints. The overall constitutive relationships of the regular masonry are derived by a rational micromechanical and homogenization procedure. Linear elastic constitutive relationship is considered for the blocks, while a new special nonlinear constitutive law is proposed for the mortar joints. In particular, a mortar constitutive law, which accounts for the coupling of the damage and friction phenomena occurring during the loading history, is proposed; the developed model is based on an original micromechanical analysis of the damage process of the mortar joint. Then, an effective nonlinear homogenization procedure, representing the main novelty of the paper, is proposed; it is based on the transformation field analysis, using the technique of the superposition of the effects and the finite element method. The presented methodology is implemented in a numerical code. Finally, numerical applications are performed in order to assess the performances of the proposed procedure in reproducing the mechanical behavior of masonry material.     

A micromechanical model for porous materials with a reinforced matrix

In this study a micromechanical model is proposed for ductile porous material whose matrix is reinforced by small inclusions. The solid phase is described by a pressure sensitive plastic model. Based on works of Maghous et al. [6], a macroscopic plastic criterion is firstly obtained by using a two-step homogenization procedure. The effect of porosity at the mesoscale and the influence of inclusions at the microscale are taken into account simultaneously by this criterion. With a non-associated plastic flow rule, the micro-macro model is applied to modeling of mechanical behavior of a cement paste. In particular, we have considered at the microscopic scale the formation of calcite grains by carbonation process in the solid matrix. The studied cement paste is then seen as a reinforced matrix–pore system. Comparisons between numerical results and experimental data show that the proposed model is able to capture the main features of the mechanical behavior of the studied material.     

A multi-phase micromechanical model for unsaturated concrete repaired using the electrochemical deposition method

Most concrete structures repaired using the electrochemical deposition method (EDM) are not fully saturated in reality. To theoretically illustrate the deposition healing process by micromechanics and quantitatively describe the effective properties of unsaturated concrete during the EDM healing process, a multi-phase multi-level micromechanical framework is proposed based on the microstructure of unsaturated concrete and the EDM’s healing mechanism. In the proposed model, the volume fractions of water and deposition products, the water effect (including further hydration and viscosity in pores) and the shapes of pores in the concrete are comprehensively considered. Moreover, multi-level homogenization procedures are employed to predict the effective properties of unsaturated concrete repaired using the EDM. For the first-level homogenization of this model, a modified function is presented to correct the Mori–Tanaka (M–T) method, which is used to predict the effective properties of equivalent inclusions composed of deposition products and water. To demonstrate the feasibility of the proposed micromechanical model, predictions obtained via the proposed multi-phase micromechanical model are compared with the experimental data, including results from extreme states during the EDM healing process. Finally, the influences of equivalent aspect ratios and deposition product properties on the healing effectiveness of EDM are discussed based on the proposed micromechanical model.     

Multi-scale analysis in elasto-viscoplasticity coupled with damage

The purpose of this study is to present a micromechanical approach, based on the transformation field analysis (TFA), proposed by Dvorak, which has been generalized at Onera in order to analyze the nonlinear behavior of heterogeneous materials in elasto-viscoplasticity coupled with damage. In such analysis, the macroscopic constitutive equations are not purely phenomenological but are built up from multi-scale approaches starting from the knowledge of the properties of the constituents at the microscopic or mesoscopic scales. The model can take into account some local characteristics that can evolve during the thermo-mechanical applied loads or the manufacturing process, like the grain size for metallic alloys or the fiber volume fraction for composites.The determination of some specific tensors which are present in this formulation is closely linked to the microstructure morphology of heterogeneous materials constituting the macroscopic structure. For example, an Eshelby’s based approach is more appropriate to characterize polycrystalline materials with a random microstructure, while the homogenization of periodic media technique can be used for composite materials with a sufficiently regular microstructure. The proposed methodologies allowing to perform this nonlinear analysis across the scales are illustrated with examples based on the behavior of structures reinforced with a long fiber unidirectional metal matrix composite.     

基于自洽法的电化学沉积修复饱和混凝土细观描述

针对当前电化学沉积修复混凝土缺乏细观层次上的理论描述,首先,以饱和混凝土的细观结构和电化学沉积修复的主要机理为基础, 提出了含电化学沉积产物、水和混凝土基体在内的三相复合材料细观力学模型. 其次,为了从细观层次定量描述电化学沉积修复对混凝土宏观性能的影响,基于自洽法对上述细观力学模型进行多层次均匀化处理,获取电化学沉积修复饱和混凝土的有效性能,其中,第一层均匀化是采用广义自洽模型获取等效夹杂的有效性能,第二层均匀化是采用自洽方法和沃伊特(Voigt) 上限相结合的方式获取修复混凝土的有效性能. 最后,为了验证该文所提出模型和方法的有效性,对比了该文的预测结果、试验数据和已有模型,结果表明了该文模型和方法是合理的.      

Micromechanical analysis of the finite elastic–viscoplastic response of multiphase composites

Nonlinear thermoelastic–viscoplastic constitutive equations for large deformations with isotropic and directional hardening, are incorporated into a micromechanical finite strain analysis. As a result of this analysis, which is based on the homogenization technique for periodic microstructures, a global thermoinelastic constitutive law is established that governs the overall response of multiphase materials under finite deformations. This constitutive law is expressed in terms of the instantaneous effective mechanical and thermal stress tangent tensors together with the instantaneous global inelastic stress tensor that represents the viscoplastic effects. Results for a thermoinelastic matrix reinforced by a hyperelastic compressible material are given that illustrate the response of fibrous and particulate composites to various types of loading.     

Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales: a coupled thermo-viscoelastic example problem

A systematic approach for analyzing multiple physical processes interacting at multiple spatial and temporal scales is developed. The proposed computational framework is applied to the coupled thermo-viscoelastic composites with microscopically periodic mechanical and thermal properties. A rapidly varying spatial and temporal scales are introduced to capture the effects of spatial and temporal fluctuations induced by spatial heterogeneities at diverse time scales. The initial-boundary value problem on the macroscale is derived by using the double scale asymptotic analysis in space and time. It is shown that an extra history-dependent long-term memory term introduced by the homogenization process in space and time can be obtained by solving a first order initial value problem. This is in contrast to the long-term memory term obtained by the classical spatial homogenization, which requires solutions of the initial-boundary value problem in the unit cell domain. The validity limits of the proposed spatial–temporal homogenized solution are established. Numerical example shows a good agreement between the proposed model and the reference solution obtained by using a finite element mesh with element size comparable to that of material heterogeneity.     

Optimal internal architectures of femoral bone based on relaxation by homogenization and isotropic material design

The influence of the loading conditions on the trabecular architecture of a femur is investigated by using topology optimization methods. The response of the bone to physiological loads results in changes of the internal architecture of bone, reflected by a modification of internal effective density and mechanical properties. The homogenization based optimization model is developed for predicting optimal bone density distribution, wherein bone tissue is assumed to be a composite material consisting of a mixture of material and void. The homogenization scheme treats the geometric parameters of the microstructures and their orientation as design variables and homogenizes the properties in that microstructure, which is generally anisotropic. The penalization of the optimal material density then leads to a classical optimal structure which consists of regions with bone material and regions without bone material. The IMD (Isotropic Material Design) approach is next applied to determine the optimal elasticity tensor in terms of the bulk and shear moduli for the present loading applied to the femoral bone sample. IMD is able to provide both the external shape and topology together with the optimal layout of the isotropic moduli. Both topology optimization methods appear to be complementary. Simulations of the internal bone architecture of the human proximal femur results in a density distribution pattern with good consistency with that of the real bone.     

平纹机织碳纤维复合材料的多尺度随机力学性能预测研究

平纹机织碳纤维复合材料在结构上具有多尺度特性和空间随机性. 同时, 组分材料会因存储条件和组成相成分、批次的不同导致力学性能有所差异. 当考虑各尺度结构和组分性能参数不确定性进行随机力学性能预测时, 存在以下两个难点: 一是随机变量众多, 使得对不确定性传递方法的精度和效率提出了要求; 二是由于随机参数之间存在高维相关性, 需要建立高精度的相关性模型. 针对以上问题, 本文提出了基于混沌多项式展开和Vine Copula的平纹机织复合材料多尺度随机力学性能预测方法, 综合考虑了平纹机织碳纤维复合材料微观及介观尺度的材料、结构随机参数, 基于自下而上层级传递的策略逐尺度地研究力学性能不确定性. 该方法采用Vine Copula理论构造相关随机变量的高维联合概率分布, 并运用非嵌入式混沌多项式展开法实现不确定性传递. 结果显示, 本方法构造的相关性模型几乎与原模型一致, 且能够高效准确地实现各尺度力学性能的随机预测.      

Dual-stage nested homogenization for rate-dependent anisotropic elasto-plasticity model of dendritic cast aluminum alloys

This paper proposes a nested dual-stage homogenization method for developing microstructure based continuum elasto-viscoplastic models for large secondary dendrite arm spacing or SDAS cast aluminum alloys. Microstructures of these alloys are characterized by extremely inhomogeneous distribution of inclusions along the dendrite cell boundaries. Traditional single-step homogenization methods are not suitable for this type of microstructure due to the size of the representative volume element (RVE) and the associated computations required for micromechanical analyses. To circumvent this limitation, two distinct RVE’s or statistically equivalent RVE’s are identified, corresponding to the inherent scales of inhomogeneity in the microstructure. The homogenization is performed in multiple stages for each of the RVE’s identified. The macroscopic behavior is described by a rate-dependent, anisotropic homogenization based continuum plasticity (HCP) model. Anisotropy and viscoplastic parameters in the HCP model are calibrated from homogenization of micro-variables for the different RVE’s. These parameters are dependent on microstructural features such as morphology and distribution of different phases. The uniqueness of the nested two-stage homogenization is that it enables evaluation of the overall homogenized model parameters of the cast alloy from limited experimental data, but also material parameters of constituents like inter-dendritic phase and pure aluminum matrix. The capabilities of the HCP model are demonstrated for a cast aluminum alloy AS7GU having a SDAS of 30 μm.     

多相材料有效性能预测的高精度方法

有效介质方法是常用的细观力学方法之一.其可用于计算多相材料的有效性能,并建立材料微细观结构和宏观性能的定量关系;有助于指导新材料设计,减少试验工作量等.然而,当夹杂含量升高时,传统有效介质方法的计算精度下降.本文以两相材料为研究对象,提出一种新的参考介质,即:为更合理考虑不同夹杂颗粒间的相互作用,假定参考介质的应变是基体相平均应变和某一修正张量的双点积.在此基础上,推导了新参考介质下两相材料的有效模量表达式,并给出该修正张量的近似计算方法;通过反复更新参考介质,采用多层次均匀化思路,将本文方法进一步用于多相材料性能的预测.为验证方法的有效性,将预测结果与已有模型结果和试验数据进行对比.结果表明本文方法较已有方法更为合理、有效.当夹杂含量升高时,本文方法较传统有效介质方法的计算精度有所提升.     

Multiscale modeling of a fluid saturated medium with double porosity: Relevance to the compact bone

In this paper, we develop a model of a homogenized fluid-saturated deformable porous medium. To account for the double porosity the Biot model is considered at the mesoscale with a scale-dependent permeability in compartments representing the second-level porosity. This model is treated by the homogenization procedure based on the asymptotic analysis of periodic “microstructure”. When passing to the limit, auxiliary microscopic problems are introduced, which provide the corrector basis functions that are needed to compute the effective material parameters. The macroscopic problem describes the deformation-induced Darcy flow in the primary porosities whereas the microflow in the double porosity is responsible for the fading memory effects via the macroscopic poro-visco-elastic constitutive law. For the homogenization procedure, we use the periodic unfolding method. We discuss also the stress and flow recovery at multiple scales characterizing the heterogeneous material. The model is proposed as a theoretical basis to describe compact bone behavior on multiple scales.     

An analytical homogenization method for heterogeneous porous materials

The objective of this work is to develop an analytical homogenization method to estimate the effective mechanical properties of fluid-filled porous media with periodic microstructure. The method is based on the equivalent inclusion concept of homogenization applied earlier for solid–solid mixture. It is assumed that porous media are described by the poroelastic constitutive law developed by Biot where porosity is a material parameter. By solving the governing equations of poroelasticity in Fourier transformed domain, the relation between periodic strain and eigenstrain in porous media is established. This relation is subsequently used in an average consistency condition involving both solid and fluid phase stresses and strains. The geometry of the porous microstructure is captured in the g-integral. This homogenization method can also be applied to estimate the equivalent properties of solid–fluid mixture where a pure solid and fluid can be modeled by assuming very low and high porosity, respectively. Several examples are considered to establish this new method by comparing with other existing analytical and numerical methods of homogenization. As an application, poroelastic properties of cortical bone fibril are estimated and compared with previously computed values.     

Micro–macro analysis of shape memory alloy composites

The aim of the paper is to develop a micro–macro approach for the analysis of the mechanical behavior of composites obtained embedding long fibers of Shape Memory Alloys (SMA) into an elastic matrix. In order to determine the overall constitutive response of the SMA composites, two homogenization techniques are proposed: one is based on the self-consistent method while the other on the analysis of a periodic composite. The overall response of the SMA composites is strongly influenced by the pseudo-elastic and shape memory effects occurring in the SMA material. In particular, it is assumed that the phase transformations in the SMA are governed by the wire temperature and by the average stress tensor acting in the fiber. A possible prestrain of the fibers is taken into account in the model.Numerical applications are developed in order to analyze the thermo-mechanical behavior of the SMA composite. The results obtained by the proposed procedures are compared with the ones determined through a micromechanical analysis of a periodic composite performed using suitable finite elements.Then, in order to study the macromechanical response of structural elements made of SMA composites, a three-dimensional finite element is developed implementing at each Gauss point the overall constitutive laws of the SMA composite obtained by the proposed homogenization procedures. Some numerical applications are developed in order to assess the efficiency of the proposed micro–macro model.     

Analysis of uncertainty effects due to microstructural disorder in cellular or porous materials

Aim of the present study is an analysis of the effect of microstructural uncertainties on the scatter in the macroscopic material properties of highly porous materials consisting of metallic or other constituents. For the numerical analysis of the uncertainty effects, a probabilistic homogenization scheme is proposed. In contrast to direct Monte-Carlo approaches, the thermomechanical response of a limited number of pre-selected cases throughout the range of possible microstructures is analyzed. Their effective properties are determined by means of an energy based homogenization procedure. In a stochastic evaluation, the results of the individual computations are weighted with the probability of the occurrence of the underlying microstructures. As a result, the probability distributions for the effective properties are obtained. The basic uncertain microstructural properties considered in the investigation are the microstructural geometry and orientation, the local relative density and the local pore size distribution. In an application to an experimental data base from other sources, the approach proves to be accurate and numerically efficient compared to direct Monte-Carlo approaches. Parameter studies reveal that uncertainties in the local relative density are the most important factor leading to scatter in the macroscopic material properties of cellular materials.