非饱和颗粒材料的多孔连续体有效压力与有效广义Biot应力

多孔连续体理论框架下的非饱和多孔介质广义有效压力定义和Bishop参数的定量表达式长期以来存在争议,这也影响了对与其直接相关联的非饱和多孔介质广义Biot有效应力的正确预测.基于随时间演变的离散固体颗粒-双联液桥-液膜体系描述的Voronoi胞元模型,利用由模型获得的非饱和颗粒材料表征元中水力-力学介观结构和响应信息,文章定义了低饱和度多孔介质局部材料点的有效内状态变量:非饱和多孔连续体的广义Biot有效应力和有效压力,导出了其表达式.所导出的有效压力公式表明,非饱和多孔连续体的有效压力张量为各向异性,它不仅对非饱和多孔连续体广义Biot有效应力张量的静水应力分量的影响呈各向异性,同时也对其剪切应力分量有影响.文章表明,非饱和多孔连续体中提出的广义Biot理论和双变量理论的基本缺陷在于它们均假定反映非混和两相孔隙流体对固相骨架水力-力学效应的有效压力张量为各向同性.此外,为定义各向同性有效压力张量和作为加权系数而引入的Bishop参数并不包含对非饱和多孔连续体中局部材料点水力-力学响应具有十分重要效应的基质吸力.所导出的非饱和多孔介质广义Biot有效应力和有效压力公式(包括反映有效压力...     

Thermomechanical Multiscale Constitutive Modeling: Accounting for Microstructural Thermal Effects

In this work we present a thermomechanical multiscale constitutive model for materials with microstructure. In these materials thermal effects at microscale have an impact on the effective macroscopic stress. As a result, it turns out that the homogenized stress depends upon the macroscopic temperature and its gradient. In order to allow this interplay to be thermodynamically valid, we resort to a macroscopic extended thermodynamics whose elements are derived from the microscopic behavior using homogenization concepts. Hence, the thermodynamics implications of this new class of multiscale models are discussed. A variational approach based on the Hill–Mandel Principle of Macro-homogeneity, and which makes use of the volume averaging concept over a local representative volume element (RVE), is employed to derive the thermal and mechanical equilibrium problems at the RVE level and the corresponding homogenization expressions for the effective heat flux and stress. The material behavior at the RVE level is described through standard phenomenological constitutive models. To sum up, the novel contribution of the model presented here is that it allows to include the microscopic temperature fluctuation field, obtained from the multiscale thermal analysis, in the micro-mechanical problem at the RVE level while keeping thermodynamic consistency.     

Macroscopic strain potentials in nonlinear porous materials

By taking a hollow sphere as a representative volume element (RVE), the macroscopic strain potentials of porous materials with power-law incompressible matrix are studied in this paper. According to the principles of the minimum potential energy in nonlinear elasticity and the variational procedure, static admissible stress fields and kinematic admissible displacement fields are constructed, and hence the upper and the lower bounds of the macroscopic strain potential are obtained. The bounds given in the present paper differ so slightly that they both provide perfect approximations of the exact strain potential of the studied porous materials. It is also found that the upper bound proposed by previous authors is much higher than the present one, and the lower bounds given by Cocks is much lower. Moreover, the present calculation is also compared with the variational lower bound of Ponte Castañeda for statistically isotropic porous materials. Finally, the validity of the hollow spherical RVE for the studied nonlinear porous material is discussed by the difference between the present numerical results and the Cocks bound.     

Coupled modeling of damage growth and permeability variation in brittle rocks

This paper presents a coupled model for anisotropic damage and permeability evolution by using a micro–macro approach. The damage state is represented by a second order tensor. The evolution of damage is determined from a crack propagation criterion. The free enthalpy function of cracked material is obtained by using micromechanical considerations. It is assumed that cracks exhibit normal aperture which is coupled with the crack growth due to asperities of crack faces. By using Darcy’s law for macroscopic fluid flow and assuming laminar flow in microcracks, the overall permeability of the RVE is obtained by a volume averaging procedure taking into account crack aperture in each orientation.     

A thermodynamically-consistent microplane model for shape memory alloys

In microplane theory, it is assumed that a macroscopic stress tensor is projected to the microplane stresses. It is also assumed that 1D constitutive laws are defined for associated stress and strain components on all microplanes passing through a material point. The macroscopic strain tensor is obtained by strain integration on microplanes of all orientations at a point by using a homogenization process. Traditionally, microplane formulation has been based on the Volumetric–Deviatoric–Tangential split and macroscopic strain tensor was derived using the principle of complementary virtual work. It has been shown that this formulation could violate the second law of thermodynamics in some loading conditions. The present paper focuses on modeling of shape memory alloys using microplane formulation in a thermodynamically-consistent framework. To this end, a free energy potential is defined at the microplane level. Integrating this potential over all orientations provides the macroscopic free energy. Based on this free energy, a new formulation based on Volumetric–Deviatoric split is proposed. This formulation in a thermodynamic-consistent framework captures the behavior of shape memory alloys. Using experimental results for various loading conditions, the validity of the model has been verified.     

基于介观力学信息的颗粒材料损伤——愈合与塑性宏观表征

本文在二阶计算均匀化框架下提出了颗粒材料损伤--愈合与塑性的多尺度表征方法. 颗粒材料结构在宏观尺度模型化为梯度Cosserat连续体,在其有限元网格的每个积分点处定义具有离散颗粒介观结构的表征元. 建立了表征元离散颗粒系统的非线性增量本构关系. 表征元周边介质作用于表征元边界颗粒的增量力与增量力偶矩以表征元边界颗粒的增量线位移与增量转动角位移、当前变形状态下表征元离散介观结构弹性刚度、以及凝聚到表征元边界颗粒的增量耗散摩擦力表示. 基于平均场理论与Hill定理,导出了基于介观力学信息的梯度Cosserat连续体增量非线性本构关系. 在等温热动力学框架下定义了表征颗粒材料各向异性损伤--愈合和塑性的损伤、愈合张量因子与综合损伤、愈合效应的净损伤张量因子和塑性应变. 此外,定义了损伤和塑性耗散能密度与愈合能密度,以定量比较材料损伤、愈合、塑性对材料失效的效应. 应变局部化数值例题结果显示了所建议的颗粒材料损伤--愈合--塑性表征方法的有效性.      

The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach

After recalling the constitutive equations of finite strain poroelasticity formulated at the macroscopic level, we adopt a microscopic point of view which consists of describing the fluid-saturated porous medium at a space scale on which the fluid and solid phases are geometrically distinct. The constitutive equations of poroelasticity are recovered from the analysis conducted on a representative elementary volume of porous material open to fluid mass exchange. The procedure relies upon the solution of a boundary value problem defined on the solid domain of the representative volume undergoing large elastic strains. The macroscopic potential, computed as the integral of the free energy density over the solid domain, is shown to depend on the macroscopic deformation gradient and the porous space volume as relevant variables. The corresponding stress-type variables obtained through the differentiation of this potential turn out to be the macroscopic Boussinesq stress tensor and the pore pressure. Furthermore, such a procedure makes it possible to establish the necessary and sufficient conditions to ensure the validity of an ‘effective stress’ formulation of the constitutive equations of finite strain poroelasticity. Such conditions are notably satisfied in the important case of an incompressible solid matrix.     

Design sensitivity analysis for the homogenized elasticity tensor of a polymer filled with rubber particles

The main purpose of this work is the computational simulation of the sensitivity coefficients of the homogenized tensor for a polymer filled with rubber particles with respect to the material parameters of the constituents. The Representative Volume Element (RVE) of this composite contains a single spherical particle, and the composite components are treated as homogeneous isotropic media, resulting in an isotropic effective homogenized material. The sensitivity analysis presented in this paper is performed via the provided semi-analytical technique using the commercial FEM code ABAQUS and the symbolic computation package MAPLE. The analytical method applied for comparison uses the additional algebraic formulas derived for the homogenized tensor for a medium filled with spherical inclusions, while the FEM-based technique employs the polynomial response functions recovered from the Weighted Least-Squares Method. The homogenization technique consists of equating the strain energies for the real composite and the artificial isotropic material characterized by the effective elasticity tensor. The homogenization problem is solved using ABAQUS by the application of uniform deformations on specific outer surfaces of the composite RVE and the use of tetrahedral finite elements C3D4. The energy approach will allow for the future application of more realistic constitutive models of rubber-filled polymers such as that of Mullins and for RVEs of larger size that contain an agglomeration of rubber particles.     

复合材料刚度性能表征的协同损伤力学模型

针对复合材料层合板的弥散型损伤,提出一个刚度性能表征的协同损伤力学模型. 该模型兼顾了微观物理损伤响应和宏观材料刚度性能表征. 从微观角度,建立细观RVE 模型求解裂纹表面张开位移和滑开位移,以此定义损伤张量,并在宏观上通过对材料应变和损伤表面位移进行均匀化处理,建立单向板或层合板的损伤刚度矩阵和损伤张量之间的联系. 以基体裂纹为例,详细分析并建立了横向裂纹和纵向裂纹的损伤本构. 计算了[±θ/90₄]_S 铺层层合板中基体横向裂纹对刚度性能的影响,结果表明该方法能够准确地预测复合材料层合板由损伤导致的刚度性能衰减.      

Multi-scale constitutive modeling of the small deformations of semi-crystalline polymers

A multi-scale constitutive model for the small deformations of semi-crystalline polymers such as high density Polyethylene is presented. Each macroscopic material point is supposed to be the center of a representative volume element which is an aggregate of randomly oriented composite inclusions. Each inclusion consists of a stack of parallel crystalline lamellae with their adjacent amorphous layers.Micro-mechanically based constitutive equations are developed for each phase. A viscoplastic model is used for the crystalline lamellae. A new nonlinear viscoelastic model for the amorphous phase behavior is proposed. The model takes into account the fact that the presence of crystallites confines the amorphous phase in extremely thin layers where the concentration of chain entanglements is very high. This gives rise to a stress contribution due to elastic distortion of the chains. It is shown that the introduction of chains’ elastic distortion can explain the viscoelastic behavior of crystalline polymers. The stress contribution from elastic stretching of the tie molecules linking the neighboring lamellae is also taken into account.Next, a constitutive model for a single inclusion considered as a laminated composite is proposed. The macroscopic stress-strain behavior for the whole RVE is found via a Sachs homogenization scheme (uniform stress throughout the material is assumed).Computational algorithms are developed based on fully implicit time-discretization schemes.     

梯度增强Cosserat连续体的广义Hill定理

基于经典Cauchy连续体的Hill定理,在平均场理论的框架下导出了梯度增强Cosserat连续体细、宏观均匀化方法的广义Hill定理。在梯度增强Cosserat连续体中,不仅宏观样条点上的应变和应力张量,而且它们的梯度均作用于与该样条点相关联的细观表征元（RVE）。依据此广义Hill定理,对梯度增强Cosserat连...     

Differential constitutive equations of incompressible media with finite deformations

A method is proposed for constructing a system of constitutive equations of an incompressible medium with nonlinear dissipative properties with finite deformations. A scheme of the mechanical behavior of a material is used, in which the points are connected by horizontally aligned elastic, viscous, plastic, and transmission elements. The properties of each element of the scheme are described with the use of known equations of the nonlinear elasticity theory, the theory of nonlinear viscous fluids, and the theory of plastic flow of the material under conditions of finite deformations of the medium. The system of constitutive equations is closed by equations that express the relation between the deformation rate tensor of the material and the deformation rate tensor of the plastic element. Transmission elements are used to take into account a significant difference between macroscopic deformations of the material and deformations of elements of the medium at the structural level. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 158–170, May–June, 2009.     

Strain gradient solution for the Eshelby-type polyhedral inclusion problem

The Eshelby-type problem of an arbitrary-shape polyhedral inclusion embedded in an infinite homogeneous isotropic elastic material is analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains a material length scale parameter. The Eshelby tensor for a polyhedral inclusion of arbitrary shape is obtained in a general analytical form in terms of three potential functions, two of which are the same as the ones involved in the counterpart Eshelby tensor based on classical elasticity. These potential functions, as volume integrals over the polyhedral inclusion, are evaluated by dividing the polyhedral inclusion domain into tetrahedral duplexes, with each duplex and the associated local coordinate system constructed using a procedure similar to that employed by Rodin (1996. J. Mech. Phys. Solids 44, 1977–1995). Each of the three volume integrals is first transformed to a surface integral by applying the divergence theorem, which is then transformed to a contour (line) integral based on Stokes' theorem and using an inverse approach different from those adopted in the existing studies based on classical elasticity. The newly derived SSGET-based Eshelby tensor is separated into a classical part and a gradient part. The former contains Poisson's ratio only, while the latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. This SSGET-based Eshelby tensor reduces to that based on classical elasticity when the strain gradient effect is not considered. For homogenization applications, the volume average of the new Eshelby tensor over the polyhedral inclusion is also provided in a general form. To illustrate the newly obtained Eshelby tensor and its volume average, three types of polyhedral inclusions – cubic, octahedral and tetrakaidecahedral – are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the SSGET-based Eshelby tensor for each of the three inclusion shapes vary with both the position and the inclusion size, while their counterparts based on classical elasticity only change with the position. It is found that when the inclusion is small, the contribution of the gradient part is significantly large and should not be neglected. It is also observed that the components of the averaged Eshelby tensor based on the SSGET change with the inclusion size: the smaller the inclusion, the smaller the components. When the inclusion size becomes sufficiently large, these components are seen to approach (from below) the values of their classical elasticity-based counterparts, which are constants independent of the inclusion size.     

Micro-macro homogenization of gradient-enhanced Cosserat media

Based on the Hill’s lemma for classical Cauchy continuum, a generalized Hill’s lemma for micro-macro homogenization modeling of heterogeneous gradient-enhanced Cosserat continuum is presented in the frame of the average-field theory. In this context not only the strain and stress tensors defined in classical Cosserat continuum but also their gradients at each macroscopic sampling point are attributed to associated microstructural representative volume element (RVE). The admissible boundary conditions required to prescribe on the RVE for the modeling are extracted as a corollary of the presented generalized Hill’s lemma and discussed to ensure the satisfaction of the enhanced Hill–Mandel energy condition and the average-field theory.     

Periodic Homogenization and Material Symmetry in Linear Elasticity

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demonstrates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.     

Modelling matrix damage and fibre–matrix interfacial decohesion in composite laminates via a multi-fibre multi-layer representative volume element (MRVE)

A three-dimensional multi-fibre multi-layer micromechanical finite element model was developed for the prediction of mechanical behaviour and damage response of composite laminates. Material response and micro-scale damage mechanism of cross-ply, [0/90]_ns, and angle-ply, [±45]_ns, glass-fibre/epoxy laminates were captured using multi-scale modelling via computational micromechanics. The framework of the homogenization theory for periodic media was used for the analysis of the proposed ‘multi-fibre multi-layer representative volume element’ (M²RVE). Each layer in M²RVE was represented by a unit cube with multiple randomly distributed, but longitudinally aligned, fibres of equal diameter and with a volume fraction corresponding to that of each lamina (equal in the present case). Periodic boundary conditions were applied to all the faces of the M²RVE. The non-homogeneous stress–strain fields within the M²RVE were related to the average stresses and strains by using Gauss’ theorem in conjunction with the Hill–Mandal strain energy equivalence principle. The global material response predicted by the M²RVE was found to be in good agreement with experimental results for both laminates. The model was used to study effect of matrix friction angle and cohesive strength of the fibre–matrix interface on the global material response. In addition, the M²RVE was also used to predict initiation and propagation of fibre–matrix interfacial decohesion and propagation at every point in the laminae.     

Non-quadratic Kelvin modes based plasticity criteria for anisotropic materials

Novel (non-quadratic) plasticity criteria based on Kelvin modes are formulated here for anisotropic materials. As an example, such a macroscopic criterion is applied with success to the case of FCC nickel-base single crystals. Indeed, relying on the cubic symmetry of the material, the Kelvin decomposition of elasticity tensor easily allows for the definition of an objective and loading independent criterion. The criterion identification is performed from different loading cases for CMSX2 single crystal superalloy. Tension-torsion yield surfaces at room temperature and yield stress dependence on crystal orientation are modeled. The Kelvin modes based criterion is compared to experimental data, to Hill and Barlat and coworkers macroscopic criteria and to Schmid law predictions. The results show that a simple three-parameter yield function built thanks to von Mises equivalent Kelvin stresses accounts for a satisfying plasticity criterion for such alloys.Non-quadratic norm ∥·∥_a plasticity framework is addressed. Intrinsic generalizations of Hershey-Hosford criterion are proposed for cubic material symmetry.     

Closed-form expressions for the macroscopic flexural rigidity coefficients of periodic brickwork

Approximate expressions for the macroscopic out-of-plane elastic coefficients of brick masonry with a regular pattern are derived in closed form using a homogenization approach for periodic media. Following an approach similar to the Method of Cells for fiber reinforced composites, a (piecewise-)differentiable expression depending on very a limited number of degrees of freedom and fulfilling suitable periodicity conditions is proposed for the microscopic transverse displacement field over any Representative Volume Element (RVE). Some of the equilibrium conditions at the interfaces between bricks and mortar joints are also fulfilled. By averaging the moment and curvature fields over the RVE, the macroscopic bending stiffness coefficients can be explicitly obtained. Using the FE solution of a masonry panel subjected to elementary load conditions as a benchmark, the proposed approach is found to accurately match the numerically obtained stiffness coefficients, for masonry elements of different geometry and different mechanical properties. In several instances, the proposed expressions agree with the numerical predictions better than other analytical expressions available in the literature.