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.     

基于修正高精度广义胞元法的复合材料细观力学分析

高精度广义胞元法是多尺度分析复合材料模量和微观应力应变场的有效方法之一．然而，由于位移插值函数中缺少二次耦合项，很大程度上影响了复合材料局部应力、应变场，特别是剪切场的计算精度．本文通过引入二次方向耦合项，提出了一种修正的高精度广义胞元法插值函数．在施加周期性边界条件、平均应力和平均位移连续性条件后，可以确定位移插值函数中的系数．通过对多相复合材料弹性模量和局部场分析，并且与有限元分析和实验测量结果比较，验证了修正高精度广义胞元法的准确性．与高精度广义胞元相比，本文提出的修正高精度广义胞元法在不需要引入额外未知变量，不影响计算效率的前提下，对复合材料的局部应力场计算得更加准确．     

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).     

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.     

Finite-volume direct averaging micromechanics of heterogeneous materials with elastic–plastic phases

In this communication, we extend the recently re-constructed micromechanics model called high-fidelity generalized method of cells (Bansal, Y., Pindera, M.-J., 2005. A second look at the higher-order theory for periodic multiphase materials. J. Appl. Mech. 72 (2), 177–195.) by incorporating inelastic response capability for the individual phases. The re-construction, based on the local/global stiffness matrix approach, has simplified the model’s theoretical framework and substantially increased its computational efficiency as well as implementability, enabling analysis of unit cells with realistic multiphase microstructures previously unattainable in the original formulation developed by Aboudi et al. (Aboudi, J., Pindera, M-J., Arnold, S.M., 2001. Linear thermoelastic higher-order theory for periodic multiphase materials. J. Appl. Mech. 68 (5), 697–707; Aboudi, J., Pindera, M-J, Arnold, S.M., 2003. Higher-order theory for periodic multiphase materials with inelastic phases. Int. J. Plasticity 19 (6), 805–847.) with an accuracy approaching finite-element solutions. Just as importantly, the re-construction has revealed the model to be based on a finite-volume, direct averaging approach with clearly discernible similarities to, and differences with, the finite-element method and the finite-volume technique used in computational fluid mechanics. Herein, easily programmable closed-form expressions have been derived for the thermo-inelastic contributions to the local stiffness matrix equations that facilitate incorporation of different inelastic constitutive theories for the phase response. The re-constructed model is then employed to investigate orientational and architectural effects in unidirectional metal matrix composites characterized by multi-inclusion unit cells. Classical incremental plasticity theory with isotropic hardening is employed for the matrix response for consistency and comparison with previously reported results by Aboudi et al. (2003). Unit cells representative of a square array of fibers rotated by an angle about the fiber axis, which lack planes of material symmetry in the rotated coordinate system in which the micromechanical analysis is performed, belong in the first category. New results are presented for such rotated unidirectional porous composites which suggest guidelines for optimizing stiffness and ductility of this class of light-weight materials relative to dominant loading directions. Strengthening effects due to fiber clustering, which require highly discretized multi-inclusion unit cells, fall in the second category. It is demonstrated that the previously documented results for particulate composites, explained by the clustering-induced alteration of stress invariants which govern plastic strain evolution, are recovered for unidirectional composites as well.     

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.     

极限下限分析的正交基无单元Galerkin法

基于极限分析的下限定理,建立了用正交基无单元Galerkin法进行理想弹塑性结构极 限分析的整套求解算法．下限分析所需的虚拟弹性应力场可由正交基无单元Galerkin法直接 得到,所需的自平衡应力场由一组带有待定系数的自平衡应力场基矢量的线性组合进行模 拟．这些自平衡应力场基矢量可由弹塑性增量分析中的平衡迭代得到．通过对自平衡应力场 子空间的不断修正,整个问题的求解将化为一系列非线性数学规划子问题,并通过复合形法 进行求解．算例表明该方法有效地克服了维数障碍问题,使计算效率得到了充分的提高,是 切实可行的．     

Nonlinear analysis of thick plates on an elastic foundation by HT FE with p-extension capabilities

The paper is concerned with the development of hybrid-Trefftz (HT) p-element for nonlinear analysis of Reissner-Mindlin plates resting on an elastic foundation. The foundation may be of Winkler-type or Pasternak-type. Exact solutions of the Lame-Navier equations are used for the in-plane intraelement displacement field and an incremental form of the basic equations is adopted. With the aid of incremental form of these equations, all nonlinear terms may be taken as pseudo-loads. Moreover, some modifications have been made on the nonlinear boundary equations to simplify the ensuing derivation. As a result, the in-plane and out-of-plane equations are uncoupled, and then the derivation for the HT finite element (FE) formulation becomes very simple. The practical efficiency of the new element model has been assessed through several examples.     

结构安定分析的Galerkin边界元方法

基于Melan静力安定定理,利用Galerkin边界元方法建立了多组交变载荷作用下结构安定分析的下限计算格式.在给定载荷域的载荷角点所对应载荷作用下,采用Galerkin边界元法计算相应的虚拟弹性应力场,并且利用结构在Galerkin边界元弹塑性增量计算中同一增量步中不同迭代步之间的应力差作为自平衡应力场的基矢量,通过这些基矢量的线性组合构造了自平衡应力场,大大降低了所形成的数学规划问题的未知变量数.并通过复合形法对非线性规划问题直接进行求解,得到了结构在交变载荷作用下的下限安定乘子.计算结果表明,所采用的方法具有较高的精度和计算效率.     

周期性结构热动力时间-空间多尺度分析

研究一种时间-空间多尺度渐近均匀化分析方法，模拟不同的极端热和动力载荷下微尺度多 相周期性结构中热动力响应问题，并建立一个广义的波动函数场控制方程描述热动力响应. 通过引入一个放大空间尺度和两个缩小时间尺度，在不同时间尺度上获得由空间非均匀性引 起的波动效应和非局部效应. 根据高阶均匀化理论在空间和时间上进行均匀化，获得高阶非 局部函数场波动方程. 并进一步用C0连续修正了高阶非局部函数场波动方程的有限元近 似解，使问题的求解避免了对有限元离散的C1连续性要求. 并与经典的空间均匀化方法 相比较，指出了经典的空间均匀化方法的局限性，进一步以一维非傅立叶热传导和热动力问 题为例，讨论了各种情况下方法的正确性与有效性.     

Elasto-plastic postbuckling of damaged orthotropic plates

Based on the elasto-plastic mechanics and continuum damage theory, a yield criterion related to spherical tensor of stress is proposed to describe the mixed hardening of damaged orthotropic materials. Its dimensionless form is isomorphic with the Mises criterion for isotropic materials. Furthermore, the incremental elasto-plastic damage constitutive equations and damage evolution equations are established. Based on the classical nonlinear plate theory, the incremental nonlinear equilibrium equations of orthotropic thin plates considering damage effect are obtained, and solved with the finite difference and iteration methods. In the numerical examples, the effects of damage evolution and initial deflection on the elasto-plastic postbuckling of orthotropic plates are discussed in detail.     

A homogenization theory for time-dependentnonlinear composites with periodic internal structures

A homogenization theory for time-dependent deformation such as creep andviscoplasticity of nonlinear composites with periodic internal structures is developed. To beginwith, in the macroscopically uniform case, a rate-type macroscopic constitutive relation betweenstress and strain and an evolution equation of microscopic stress are derived by introducing twokinds of Y-periodic functions, which are determined by solving two unit cell problems.Then, the macroscopically nonuniform case is discussed in an incremental form using thetwo-scale asymptotic expansion of field variables. The resulting equations are shown to beeffective for computing incrementally the time-dependent deformation for which the history ofeither macroscopic stress or macroscopic strain is prescribed. As an application of the theory,transverse creep of metal matrix composites reinforced undirectionally with continuous fibers isanalyzed numerically to discuss the effect of fiber arrays on the anisotropy in such creep.     

Numerical solution of solidification in a square prism using an algebraic grid generation technique

The solidification of an infinitely long square prism was analyzed numerically. A front fixing technique along with an algebraic grid generation scheme was used, where the finite difference form of the energy equation is solved for the temperature distribution in the solid phase and the solid–liquid interface energy balance is integrated for the new position of the moving solidification front. Results are given for the moving solidification boundary with a circular phase change interface. An algebraic grid generation scheme was developed for two-dimensional domains, which generates grid points separated by equal distances in the physical domain. The current scheme also allows the implementation of a finer grid structure at desired locations in the domain. The method is based on fitting a constant arc length mesh in the two computational directions in the physical domain. The resulting simultaneous, nonlinear algebraic equations for the grid locations are solved using the Newton-Raphson method for a system of equations. The approach is used in a two-dimensional solidification problem, in which the liquid phase is initially at the melting temperature, solved by using a front-fixing approach. The difference of the current study lies in the fact that front fixing is applied to problems, where the solid–liquid interface is curved such that the position of the interface, when expressed in terms of one of the coordinates is a double valued function. This requires a coordinate transformation in both coordinate directions to transform the complex physical solidification domain to a Cartesian, square computational domain. Due to the motion of the solid–liquid interface in time, the computational grid structure is regenerated at every time step.     

含任意椭圆核各向异性板杂交应力有限元

基于含椭圆核有限大各向异性板弹性问题的复变函数级数解,应用杂交变分原理建立了一种与常规有限元相协调的含任意椭圆核各向异性板杂交应力有限元.单元内的应力场和位移场采用满足平衡方程、几何方程与物理方程的复变函数级数解,假设的复变函数级数解精确满足椭圆核边界处的位移协调条件和应力连续条件,单元外边界上的位移场按常规有限元位移场假设,单元内椭圆核的长轴可以与材料主轴不重合.单元刚度矩阵采用Gauss积分求得,并给出了建立刚度矩阵的主要公式和推倒过程.数值计算结果表明该单元具有计算精度高、计算工作量小等优点.     

求解非线性结构动力方程的预估校正-辛时间子域法

提出了求解非线性结构动力方程的预估校正-辛时间子域法。首先,将结构非线性动力方程转换为状态空间方程,在任一时间子域内利用改进的欧拉法对各离散时刻的状态变量值进行预估和校正。然后,将离散的非线性项用Lagrange插值多项式展开并视为外荷载,结合辛时间子域法即可求解非线性动力系统的响应。这种方法不必对状态矩阵求逆,无需计算高阶导数,计算简单,格式统一,易于编程。算例结果表明,本文方法具有较高的计算精度、效率和稳定性,是一种求解非线性结构动力方程的有效方法。     

形状记忆合金管接头空间轴对称有限元分析

本文采用形状记忆合金（ＳＭＡ）的三维本构方程和有限变形理论，考虑拉、压不同应力状态对相变点移动的规律，编制了ＳＭＡ轴对称大变形的有限元程序，与单向拉伸下解析所得的应力、应变曲线相比，证实程序的正确性．文末计算一ＳＭＡ管接头，并指出按空间轴对称计算的必要性．     

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．     

Fractional four-step finite element method for analysis of thermally coupled fluid-solid interaction problems

An integrated fluid-thermal-structural analysis approach is presented. In this approach, the heat conduction in a solid is coupled with the heat convection in the viscous flow of the fluid resulting in the thermal stress in the solid. The fractional four-step finite element method and the streamline upwind Petrov-Galerkin (SUPG) method are used to analyze the viscous thermal flow in the fluid. Analyses of the heat transfer and the thermal stress in the solid are performed by the Galerkin method. The second-order semiimplicit Crank-Nicolson scheme is used for the time integration. The resulting nonlinear equations are linearized to improve the computational efficiency. The integrated analysis method uses a three-node triangular element with equal-order interpolation functions for the fluid velocity components, the pressure, the temperature, and the solid displacements to simplify the overall finite element formulation. The main advantage of the present method is to consistently couple the heat transfer along the fluid-solid interface. Results of several tested problems show effiectiveness of the present finite element method, which provides insight into the integrated fluid-thermal-structural interaction phenomena.     

Verification of deforming polarized structure computation by using a closed-form solution

Various engineering systems exploit the conversion between electromagnetic and mechanical work. It is important to compute this coupling accurately, and we present a method for solving the governing equations simultaneously (at once) without a staggering scheme. We briefly present the theory for coupling the elecgoverning equations as well as the variational formulation that leads to the weak form. This weak form is nonlinear and couples various fields. In order to solve the weak form, we use the finite element method in space and the finite difference method in time for the discretization of the computational domain. Numerical problems are circumvented by selecting the field equations carefully, and the weak form is assembled using standard shape functions. In order to examine the accuracy of the method, for the case of a linear elastic material under small deformations, we present and use an analytic solution. Comparison of the computation to the closed-form solution shows that the computational approach is reliable and models the jump of the electromagnetic fields across the interface between two different materials.     

基于特征正交分解的一类瞬态非线性热传导问题的新型快速分析方法

提出了一种基于特征正交分解(POD)和有限元法的瞬态非线性热传导问题的模型降阶快速分析方法, 建立了导热系数随温度变化的一类瞬态非线性热传导问题有限元格式的POD降阶模型. 在隐式时间推进方法的基础上有效结合单元预转换方法和多级线性化方法发展了一种加速求解瞬态非线性热传导降阶模型的新型计算方法,并通过二维和三维算例验证了该方法的准确性和高效性. 研究结果表明: (1)降阶模型解的均方根误差在经过初始时段轻微的脉动后稳定于0.01%以下, 而其计算效率比有限元全阶模型提高2$\sim $3个数量级, 并且自由度数量(DOFs)愈大提高的幅度也愈加显著; (2)新型算法解决了常规算法在计算非线性降阶模型时加速性能差的问题, 即使是在DOFs比较小的时候也能够明显提高计算效率; (3)常数边界条件下得到的POD模态可以用来建立相同求解域在各种复杂时变边界条件下的瞬态非线性热传导降阶模型, 并对其传热过程和温度场进行快速准确的分析与预测, 具有很好的工程应用价值.