用广义胞元法研究弱界面复合材料偏轴拉伸强度

本文用广义胞元法结合应力集中系数模型，从细观、宏观力学结合的角度，预测了弱界面复合材料偏轴拉伸强度值．用广义胞元法/高精度广义胞元法计算复合材料开裂前和开裂后的应力场，引入基体应力集中系数以得到基体真实应力．在计算真实应力时根据宏观试验现象考量是否对界面开裂后的复合材料进行刚度衰减，最终形成4种方案计算出复合材料的偏轴拉伸强度．通过对比芳纶纤维和亚麻纤维两种弱界面复合材料的偏轴拉伸强度试验值，找到了最可靠的预报方案并具有良好的预报精度．     

非对称铺层复合材料层合板的双稳态特性半解析研究

非对称铺层的复合材料层合板在存在热残余应力的情况下，具有双稳态性质．层合板的两个稳态之间仅需要一个适当的激励就可以互相转化，因此该结构在变体飞机上应用广泛．基于经典层合板理论，本文引入几何大变形建立了具有双稳态性质的复合材料层合板的能量泛函，提出了一个高阶的位移场函数，用瑞利里兹法推导出一组关于位移场函数系数的非线性方程组．结合牛顿迭代法和消元法求解非线性方程组，得到了层合板面外位移场．同时利用有限元软件ABAQUS，对复合材料层合板双稳态机理进行了数值模拟．选取了几组具有代表性的铺层进行计算，以有限元结果为基准，比较了本文的位移场结果与前人的结果，验证了高阶位移场函数的准确性．     

复合材料层合壳有限元分析的预测-修正法

对于复合材料层合壳的有限元分析，本文根据Reissner-Mindlin型的全局位移场给出了一个预测一修正法。首先按照一般的有限元分析过程(没有引入剪切修正系数)计算出全局响应(如挠度，频率和屈曲载荷等)的预测值；然后利用Lagrange插值构造横向剪应力的一般形式，使得满足层间连续和表面上为零的条件，通过最小二乘法拟合三维应力平衡方程获得横向剪应力；最后在单元上计算和引入剪切修正系数，再经过有限元分析计算出全局响应的修正值。     

饱和多孔介质中的混合有限元法和有限应变下应变局部化分析

对基于Biot理论的饱和多孔介质中动力-渗流耦合分析提出了一个耦合场混合元．固相位移、应变和有效应力以及流相压力、压力梯度和Darcy速度在单元内均处理为独立变量分别插值．基于胡海昌-Washizu三变量广义变分原理给出的饱和多孔介质动力-渗流耦合问题控制方程的单元弱形式，导出了单元公式．进一步导出了考虑压力相关非关联塑性的非线性单元公式和发展了相应的一致性算法．对几何非线性分析，采用了共旋公式途径．数值结果例题显示所发展耦合场混合元模拟大应变下由应变软化引起以应变局部化为特征的渐进破坏现象的性能．     

复合材料层合圆柱壳的有限元分析

本文根据Reissner-Mindlin型的全局位移场(一阶和三阶)，应用有限元预测一修正法，数值计算和分析了机械载荷作用下复合材料层合圆柱壳的挠度和横向剪应力。首先按照一般的有限元分析过程(没有引入剪切修正系数)计算出层合圆柱壳的挠度预测值；然后利用Lagrange插值构造横向剪应力的一般形式，使得满足层间连续和表面上为零的条件，通过最小二乘法拟合三维应力平衡方程获得横向剪应力；最后在单元上计算和引入剪切修正系数，再经过有限元分析计算出层合圆柱壳的挠度修正值。数值计算结果与三维线弹性解的比较表明，挠度修正值和横向剪应力的精度是十分满意的。     

改进的二维高精度通用单胞模型

基于复合材料细观力学周期性假设,利用高阶理论的改进算法,对高精度通用单胞模型的计算方法进行了改进．模型中用界面的平均量代替细观位移函数中解系数,并利用细观单元力学方程的分析与求解,建立了细观平均量与复合材料宏观平均量间的联系．改进高精度通用单胞模型的求解方程数目减少了大约60％,求解时间大大缩短;并且消除了亚子胞的概念,同时解耦了横向与纵向的方程．该模型的计算结果与试验结果及理论计算结果具有较好的一致性．     

含高温度梯度及接触热阻非线性热力耦合问题的谱元法

建立了含高温度梯度及接触热阻的非线性热力耦合问题的谱元法格式, 考虑了温度相关的热导率、弹性模量、泊松比和热膨胀系数, 以及界面应力相关的接触热阻的影响. 谱元法的插值函数基于非等距分布的Lobatto结点集或第二类Chebyshev结点集, 兼具谱方法的高精度和有限元法的灵活性. 数值算例表明, 建立的谱元法计算格式可以高效高精度地求解域内高温度梯度以及含接触热阻的非线性热力耦合问题, 不仅收敛速度快于传统有限元法, 而且用较少的自由度和较短的计算时间即可得到比传统有限元法更高精度的计算结果, 在工程实际热力耦合问题中具有广阔的应用前景.      

复合材料切口应力奇性指数计算

提出一种计算广义平面应交状态下复合材料切口应力奇性指数的新方法.在切口尖端的位移幂级数渐近展开式被引入正交各向异性材料的物理方程后,将用位移表示的应力分量代入切口端部柱状邻域的线弹性理论控制方程,切口应力奇性指数的计算被转化为常微分方程组特征值的求解.采用插值矩阵法求解该常微分方程组,可一次性地获取切口尖端多阶应力奇性指数.本法适合平面和反平面应力场耦合或解耦的情形,并可退化计算裂纹或各向同性材料切口的应力奇性指数.算例表明,所提方法对分析复合材料切口应力奇性指数是一种准确有效的手段.     

纺织复合材料和结构多尺度耦合的数值分析

研究了纺织复合材料和结构多尺度耦合的数值分析模型。建立了微、细观单胞,给出了纺织复合材料平均弹性常数的逐级分析方法,着重研究了由宏观结构、到细观纤维束、再到微观纤维三个尺度耦合的应力分析方案。对于常用的板壳状纺织复合材料结构,在面内载荷下,假设每层细观单胞的平均面内应变是一致的,在弯曲、横向剪切及扭曲等非面内载荷下,在内力等效条件下将沿厚度方向连续分布的宏观应力简化为阶梯状分布,忽略了每层细观单胞范围内宏观应力沿厚度方向的梯度变化,由此利用细观单胞模型实现宏观应力与细观应力之间的传递,再利用微观单胞可得到纤维尺度的微观应力。最后以一种三维机织复合材料为例,用上述多尺度耦合的模型逐级分析了材料的平均弹性常数,并沿相反方向,由宏观结构分析逐级计算出纤维束尺度和纤维尺度的细、微观应力的局部波动。     

不可压缩平面问题的位移-压力混合重心插值配点法

引入人工压力变量,将弹性本构方程以应力、应变和压力表达,建立求解不可压缩平面弹性问题的位移-压力方程和不可压缩条件方程的耦合偏微分方程组。利用张量积型重心Lagrange插值近似二元函数,得到计算插值节点处偏导数的偏微分矩阵。采用配点法离散不可压缩弹性控制方程,利用偏微分矩阵直接离散弹性力学控制方程为矩阵形式方程组。利用插值公式离散位移和应力边界条件,将离散边界条件与离散控制方程组合为新的方程组,得到求解弹性问题的过约束线性代数方程组;利用最小二乘法求解线性方程组,得到弹性力学问题位移数值解。数值算例验证了所提方法的数值计算精度为10-14~10-10。     

Formulation of the high-fidelity generalized method of cells with arbitrary cell geometry for refined micromechanics and damage in composites

A new parametric formulation for high-fidelity generalized method of cells (HFGMC) is presented for the micromechanical analysis of multiphase periodic composites. To this end, a linear parametric and geometric mapping is employed to transform arbitrary quadrilateral cell shapes from the physical space to an auxiliary uniform square shapes. A complete quadratic displacement expansion is performed in the mapped space. Thus, a new bilinear term is added to the quadratic displacement expansion; unlike the original HFGMC for regular array of rectangular cells where this term in not required. The continuity of displacements, tractions, together with the periodicity and equilibrium conditions are imposed in the average sense, similar to the original HFGMC formulation, using both the physical and mapping variables. However, the addition of bilinear terms requires the introduction of the first averaged moments of the equilibrium equations. In order to demonstrate the ability the new HFGMC formulation, spatial stress fields are compared with analytical and numerical solutions of circular and elliptical fibers in an infinite medium. Furthermore, two progressive damage methodologies are coupled with the new HFGMC formulation in order to predict the strain softening and elastic degrading behaviors. The first methodology employs a cell extinction approach, while the second uses cohesive interfaces between the cells. Due to the strain softening, both damage methodologies require an iterative solution approach of the governing system nonlinear equations. Damage applications are presented for the transverse loading of composites with square and hexagonal repeating unit-cells (RUC).     

Nonlinear micromechanical formulation of the high fidelity generalized method of cells

The recent High Fidelity Generalized Method of Cells (HFGMC) micromechnical modeling framework of multiphase composites is formulated in a new form which facilitates its computational efficiency that allows an effective multiscale material–structural analysis. Towards this goal, incremental and total formulations of the governing equations are derived. A new stress update computational method is established to solve for the nonlinear material constituents along with the micromechanical equations. The method is well-suited for multiaxial finite increments of applied average stress or strain fields. Explicit matrix form of the HFGMC model is presented which allows an immediate and convenient computer implementation of the offered method. In particular, the offered derivations provide for the residual field vector (error) in its incremental and total forms along with an explicit expression for the Jacobian matrix. This enables the efficient iterative computational implementation of the HFGMC as a stand alone. Furthermore, the new formulation of the HFGMC is used to generate a nested local-global nonlinear finite element analysis of composite materials and structures. Applications are presented to demonstrate the efficiency of the proposed approach. These include the behavior of multiphase composites with nonlinearly elastic, elastoplastic and viscoplastic constituents.     

Piezoresistive fiber-reinforced composites: A coupled nonlinear micromechanical–microelectrical modeling approach

Piezoresistive composites are materials that exhibit spatial and effective electrical resistivity changes as a result of local mechanical deformations in their constituents. These materials have a wide array of applications from non-destructive evaluation to sensor technology. We propose a new coupled nonlinear micromechanical-microelectrical modeling framework for periodic heterogeneous media. These proposed micro-models enable the prediction of the effective piezoresistive properties along with the corresponding spatial distributions of local mechanical–electrical fields, such as stress, strain, current densities, and electrical potentials. To this end, the high fidelity generalized method of cells (HFGMC), originally developed for micromechanical analysis of composites, is extended for the micro-electrical modeling in order to predict their spatial field distributions and effective electrical properties. In both cases, the local displacement vector and electrical potential are expanded using quadratic polynomials in each subvolume (subcell). The equilibrium and charge conservations are satisfied in an average volumetric fashion. In addition, the continuity and periodicity of the displacements, tractions, electrical potential, and current are satisfied at the subcell interfaces on an average basis. Next, a one way coupling is established between the nonlinear mechanical and electrical effects, whereby the mechanical deformations affect the electrical conductivity in the fiber and/or matrix constituents. Incremental and total formulations are used to arrive at the proper nonlinear solution of the governing equations. The micro-electrical HFGMC is first verified by comparing the stand-alone electrical solution predictions with the finite element method for different doubly periodic composites. Next, the coupled HFGMC is calibrated and experimentally verified in order to examine the effective piezoresistivity of different composites. These include conductive polymeric matrices doped with carbon nano-tubes or particles. One advantage of the proposed nonlinear coupled micro-models is its ability to predict the local and effective electro-mechanical behaviors of multi-phase periodic composites with different conductive phases.     

Discussion paper: Has renaming the high fidelity generalized method of cells been justified?

The high fidelity generalized method of cells (HFGMC) has been originally developed by Aboudi, 2001, Aboudi et al., 2001 as a micromechanical method for periodic multi-phase composite media. A computational implementation of the HFGMC equations has been proposed by Bansal and Pindera (2004) to enhance numerical efficiency, still with direct reference to the HFGMC formulation. Later, the same computational implementation is recast as a new method called “finite volume direct averaging micromechanics” (FVDAM), starting by Bansal and Pindera (2006). The current discussion paper has two aims. The first is to show that the FVDAM is not a new method and that it has the same assumptions and identical governing equations as those originally derived by the HFGMC. The only difference is in the solution procedure where intermediate dependent variables, in the form of average displacements at the interfaces, are used instead of directly solving for the unknown micro-variables; the coefficients of the displacement polynomials. Thus, renaming the HFGMC micromodel to FVDAM has not been justified. In fact, (Haj-Ali and Aboudi, 2009) have shown that the same reduction of variables can be achieved by a simple static condensation carried out at the global system of equations instead of introducing intermediate variables. The second aim of this paper is to address misrepresentations in a recent discussion paper by the FVDAM authors claiming, in part, that the HFGMC method using parametric geometry of the subcells should follow their formulation (termed parametric FVDAM). We show that the latter is limited to an incomplete quadratic expansion of the displacement and an approximation in the form of a priori constant Jacobian of the parametric mapping. However, the HFGMC with arbitrary cell geometry, (Haj-Ali and Aboudi, 2010), has been formulated in a direct and general manner, i.e. retaining the full quadratic expansion of the displacement together with the complete Jacobian. Thus, the parametric FVDAM is a special case of the parametric HFGMC, i.e. when the Jacobian is sampled and evaluated only at one point, namely the origin of the parametric coordinate system. The intended new contribution of Haj-Ali and Aboudi (2010) to refined micromechanics and progressive damage has been completely ignored by the FVDAM-discussion paper. Therefore, in order to maintain scientific clarity, it is strongly advocated to preserve the original name of the HFGMC method, regardless of the different computational implementations used for solving the governing equations for both orthogonal and parametric geometries of the subcells.     

Higher-order theory for periodic multiphase materials with inelastic phases

An extension of a recently-developed linear thermoelastic theory for multiphase periodic materials is presented which admits inelastic behavior of the constituent phases. The extended theory is capable of accurately estimating both the effective inelastic response of a periodic multiphase composite and the local stress and strain fields in the individual phases. The model is presently limited to materials characterized by constituent phases that are continuous in one direction, but arbitrarily distributed within the repeating unit cell which characterizes the material's periodic microstructure. The model's analytical framework is based on the homogenization technique for periodic media, but the method of solution for the local displacement and stress fields borrows concepts previously employed by the authors in constructing the higher-order theory for functionally graded materials, in contrast with the standard finite-element solution method typically used in conjunction with the homogenization technique. The present approach produces a closed-form macroscopic constitutive equation for a periodic multiphase material valid for both uniaxial and multiaxial loading. The model's predictive accuracy in generating both the effective inelastic stress-strain response and the local stress and inelastic strain fields is demonstrated by comparison with the results of an analytical inelastic solution for the axisymmetric and axial shear response of a unidirectional composite based on the concentric cylinder model and with finite-element results for transverse loading.     

含有多个涂层压电纤维复合材料的平面问题研究

压电纤维在未来的复合材料结构健康监测中具有重要作用．本文基于横观各向同性压电材料位移和应力连续条件以及经典的复势函数理论,讨论了同时受到平面内机械载荷和出平面电载荷作用时含有多个带涂层压电纤维的无限大线弹性基体的平面力学问题．首先将线弹性基体、涂层和压电纤维的应力场、位移场表示成复势函数,然后通过横观各向同性压电材料和线弹性材料的位移和应力连续条件确定复势函数表达式．将得到的复势函数表达式代入线弹性基体、涂层和压电纤维的的应力场、位移场公式可确定其应力场和位移场．最后,通过定量的案例讨论了涂层的材料属性对线弹性基体应力场的影响．案例分析表明涂层的材料属性对压电复合材料的应力场有重要的影响．     

Hamiltonian and Lagrangian theory of viscoelasticity

The viscoelastic relaxation modulus is a positive-definite function of time. This property alone allows the definition of a conserved energy which is a positive-definite quadratic functional of the stress and strain fields. Using the conserved energy concept a Hamiltonian and a Lagrangian functional are constructed for dynamic viscoelasticity. The Hamiltonian represents an elastic medium interacting with a continuum of oscillators. By allowing for multiphase displacement and introducing memory effects in the kinetic terms of the equations of motion a Hamiltonian is constructed for the visco-poroelasticity.

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Flux vector splitting solutions for coupling hydraulic transient of gas-liquid-solid three-phase flow in pipelines

The gas-liquid-solid three-phase mixed flow is the most general in multiphase mixed transportation. It is significant to exactly solve the coupling hydraulic transient problems of this type of multiphase mixed flow in pipelines. Presently, the method of characteristics is widely used to solve classical hydraulic transient problems. However, when it is used to solve coupling hydraulic transient problems, excessive interpolation errors may be introduced into the results due to unavoidable multiwave interpolated calculations. To deal with the problem, a finite difference scheme based on the Steger-Warming flux vector splitting is proposed. A flux vector splitting scheme is established for the coupling hydraulic transient model of gas-liquid-solid three-phase mixed flow in the pipelines. The flux subvectors are then discretized by the Lax-Wendroff central difference scheme and the Warming-Beam upwind difference scheme with second-order precision in both time and space. Under the Rankine-Hugoniot conditions and the corresponding boundary conditions, an effective solution to those points located at the boundaries is developed, which can avoid the problem beyond the calculation region directly induced by the second-order discrete technique. Numerical and experimental verifications indicate that the proposed scheme has several desirable advantages including high calculation precision, excellent shock wave capture capability without false numerical oscillation, low sensitivity to the Courant number, and good stability.     

非线性复合材料的一种杂交应力有限单元方法

建立了非线性复合材料模型的杂交应力有限元方法,并在材料主坐标系下提出直接方法计算单元非线性应力场,然后由此计算单元切线刚度矩阵和剩余载荷并转换到整体坐标系下,利用Newton-Raphson方法进行结构的位移迭代。在Hahn-Tsai非线性复合材料杂交元分析中,由位移和应力方程所导出求解单元非线性应力场的简单迭代法是条件收敛的,对较大载荷当迭代位移增加到一定程度以后无法得到应力收敛解。但是,利用本文提出的直接法由于完全避免了非线性应力场迭代,不仅很好地解决了这一问题,而且极大地提高了计算效率。数值算例说明该方法是确实有效的。     

A semi-analytic solution of a wellbore in a non-isothermal low-permeability porous medium under non-hydrostatic stresses

The transient stress, displacement, pore pressure and temperature fields around a wellbore in a thermo-poro-elastic (THM) medium subject to non-hydrostatic remote stresses are analyzed under non-isothermal plane-strain conditions. The linear THM model proposed by Coussy (1989) is adopted in the analysis with a focus on thermal effects in low-permeability saturated rocks, characterized by a latent heat associated with local changes of fluid mass content. Non-dimensionalized parameters are identified by reformulating the fully-coupled governing equations and boundary conditions. The wellbore problem is simplified by decomposing it into axisymmetric and deviatoric loading cases. The corresponding analytical solutions are obtained in Laplace space. The inverse Laplace transforms are performed numerically to find the time-dependent distributions of field variables in the rock mass around the wellbore. These numerical results show that although the pore pressure diffusion has little influence on temperature and stress, temperature changes can strongly affect the pore pressure and stress around the wellbore. The temperature change can lead to changes in near-well stresses and the resulting significant change in wellbore breakdown pressure illustrates the importance of considering the THM coupling.