Calculation of deformative and thermal properties of multiphase composite materials filled with composite or hollow spherical inclusions by the effective medium method

A theoretical investigation was carried out to examine the possibilities of a structural approach to prediction of elastic constants, creep functions, and thermal properties of multiphase polymer composite materials filled with composite or hollow spherical Inclusions of several types. The problem of determining effective properties of the composite was solved by generalizing the effective medium method, a variant of the self-consistent method, for the case of a four-phase kernel-shell-matrix-equivalent homogeneous medium model. Exact analytical expressions for the bulk modulus thermal expansion coefficient, thermal conductivity coefficient, and specific heat were obtained. The solution for the shear modulus is given in the form of a nonlinear equation whose coefficients are the solution of a system of 12 linear equations.To be presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October 1995.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 462–472, July–August, 1995.     

球面各向同性颗粒复合材料膨胀系数的界限

本文致力于球面各向同性颗粒增强复合材料膨胀行为的研究.首先,在所提出的基体均匀场概念的基础上导出了该种复合材料等效膨胀系数与体积模量的精确关系,其次,由能量极值原理给出了等效体积模量的Paul型界限,从而得到等效膨胀系数的界限.     

Elastic properties of two-dimensional two-phase composites with isotropic phases

Formal series of powers of Fourier coefficients for the effective elastic constants of a heterogeneous material (Herring’s series) are considered. It is demonstrated that, on their basis, all the known exact solutions of an elastic problem for a two-dimensional two-phase composite can be found. It is also shown how a full renormalization of the series for the inverse bulk modulus can be carried out. A general expression for Young’s modulus is deduced, leading to considerable simplifications in some special cases. All results have been obtained without any restrictions on the Fourier coefficients of local parameters of the composite.     

预测纳米纤维复合材料有效弹性性能的界面模型和界面相模型北大核心CSCD

基于广义自洽法,同时采用Gurtin-Murdoch界面模型和界面相模型研究了纳米纤维复合材料的有效弹性性能,获得了两种模型下有效体积模量的封闭解析解和计算有效面内剪切模量数值解的全部公式.基于界面模型的解答,讨论了有效体积模量和有效面内剪切模量的界面效应.证明了界面模型的解答可由界面相模型的解答退化得到,其中有效体积模量可以实现解析退化,有效面内剪切模量则可以数值退化.以含纳米孔洞的金属铝为例,比较了两种模型计算结果的差异.结果表明,当纳米孔洞半径较小时,两个模型的结果存在很大差异,而当半径较大时两个模型的结果差别不大.     

球形涂层粒子增强复合材料的有效模量

本文通过四相球模型和复合材料的等效介质理论,研究了球形涂层粒子增强复合材料的有效模量性质,得到了这种增强复合材料的有效体积模量和有效剪切模量的理论预测公式。这些结果在特殊情况下,可退化到三相球模型确定的球形粒子增强复合材料的有效模量公式。     

On subcritical development of a shear crack in a composite with viscoelastic components

We present results of an investigation of the development of a transverse shear crack in a composite material with linearly viscoelastic components under external shear load. The solution is divided into the following two main stages: determination of the time dependence of the crack tip opening displacement and determination of the crack-growth kinetics as a result of the solution of integral equations. In the first stage, we use the solution of the corresponding elastic problem of determination of the crack opening displacement and the problem of determination of the effective moduli of the composite reinforced with unidirectional discrete fibers. Using the theoretically proved principle of elasto-viscoelastic analogy and the method of Laplace inverse transformation, we obtain a solution in a time domain. In the second stage, using the criterion of critical crack opening displacement for a transverse shear crack and an equation for the viscoelastic crack opening displacement of this crack, we construct an equation of crack growth. We present results of the numerical solution, which illustrate the influence of relations between the relaxation parameters of the materials of the components on the durability of the body with a crack.     

A Comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai Equations

In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the Hashin-Shtrikman bounds and the Halpin-Tsai equations. Optimal bounds on the fitting parameters in the Halpin-Tsai equations have been formulated.     

Effective bulk moduli of materials containing stochastically distributed nano-inhomogeneities with surface stresses

In the present contribution, a mathematical model for the investigation of the effective properties of a material with randomly distributed nano-particles is proposed. The surface effect is introduced via Gurtin-Murdoch equations describing properties of the matrix/nano-particle interface. They are added to the system of stochastic differential equations formulated within the framework of linear elasticity. The homogenization problem is reduced to finding a statistically averaged solution of the system of stochastic differential equations. These equations are based on the fundamental equations of linear elasticity, which are coupled with surface/interface elasticity accounting for the presence of surface tension. Using Green's function this system is transformed to a system of statistically non-linear integral equations. It is solved by the method of conditional moments. Closed-form expressions are derived for the effective moduli of a composite consisting of a matrix with randomly distributed spherical inhomogeneities. The radius of the nano-particles is included in the expression for the bulk moduli. As numerical examples, nano-porous aluminum and nano-porous gold are investigated assuming that only the influence of the interface effects on the effective bulk modulus is of interest. The dependence of the normalized bulk moduli of nano-porous aluminum on the pore volume fraction (for certain radii of nano-pores) are compared to and discussed in the context of other theoretical predictions. The effective Young's modulus of nano-porous gold as a function of pore radius (for fixed void volume fraction) and the normalized Young's modulus vs. the pore volume fraction for different pore radii are analyzed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parametric vibrations of cylindrical shells with a viscoelastic core

The method of calculating the axisymmetric and nonaxisymmetric parametric vibrations of a cylindrical shell bonded to an elastic core [2] is extended to the case of hollow and solid viscoelastic cores by substituting for the material moduli in the equations of motion of the core integral operators with kernels in the form of an exponential and a sum of exponentials. Expressions are given for the reaction of the viscoelastic core, together with the equation of the boundaries of the spectrum of principal regions of dynamic instability. The effect of relaxation time and the long-term modulus of elasticity of the core on the shape and location of the regions of dynamic instability is analyzed.     

A non-linear law of the deformation of soft soils

Using a model of an elastic and viscoelastic (standard-linear) solid, a non-linear equation of state is constructed to model the volume deformation of soft soils. The moduli of bulk compression are assumed to be non-linear functions of a parameter which characterizes the structural failure of the soil. It is shown that this law describes observed effects of the bulk compression of soil. The results of a parametec analysis of the equation of state are given.     

粘弹性介质中波场重建方法

传统的波动方程波场重建基于完全弹性介质，不能获得满意的地震资料分辨率，本基于能描述大地吸收弹性介质中地震波传播的斯托克斯波动方程．提出了一种新的多尺度粘弹性波动方程波场重建的方法，根据地震波传播核函数的物理特性，研究了一种新的物理小波，提出了小波多尺度波场重建方法。达到对吸收信息的补偿，提高地震资料的分辨率。     

被动约束层阻尼圆柱壳振动和阻尼分析的一种新矩阵方法

基于线弹性薄壳理论和线粘弹性理论,考虑粘弹性层的剪切耗能作用和各层间的相互作用力,导出了被动约束层阻尼层合圆柱壳在谐激励作用下的一阶常微分矩阵控制方程．然后,借助作者提出的齐次扩容精细积分技术建立了一种新的矩阵方法,并利用该方法研究了层合圆柱壳的振动特性和阻尼特性．该方法与已提出的以位移及其导数作为状态向量的传统传递矩阵法的根本区别在于,控制方程中的状态向量中包含了层合壳的全部位移和整合内力变量,因此,可以方便地适用于各种位移和内力边界条件以及部分环状覆盖约束层阻尼圆柱壳的动态分析．数值算例与解析解和有限元解的结果比较有力说明了该方法的正确性和有效性．     

Exact relations for effective tensors of composites: Necessary conditions and sufficient conditions

Typically the elastic and electrical properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for effective moduli that are universally valid no matter what the microstructure. Such exact formulae provide useful benchmarks for testing numerical and actual experimental data and for evaluating the merit of various approximation schemes. They can also be regarded as fundamental invariances existing in a given physical context. Classic examples include Hill's formulae for the effective bulk modulus of a two‐phase mixture when the phases have equal shear moduli, Levin's formulae linking the effective thermal expansion coefficient and effective bulk modulus of two‐phase mixtures, and Dykhne's result for the effective conductivity of an isotropic two‐dimensional polycrystalline material. Here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones. The search for exact relations is reduced to a search for matrix subspaces having a structure of special Jordan algebras. One of many new exact relations is for the effective shear modulus of a class of three‐dimensional polycrystalline materials. We present complete lists of exact relations for three‐dimensional thermoelectricity and for three‐dimensional thermopiezoelectric composites that include all exact relations for elasticity, thermoelasticity, and piezoelectricity as particular cases. © 2000 John Wiley & Sons, Inc.     

具有分数导数型本构关系的粘弹性柱的动力稳定性

研究简支的受轴向周期激励的粘弹性柱动力稳定性,柱的材料满足分数导数型本构关系.建立了描述粘弹性柱动力学行为的弱奇异性Volterra积分-偏微分方程,利用Galerkin方法将其化归为弱奇异性Volterra积分-常微分方程.利用平均化方法的思想给出了粘弹性柱运动稳定状态的存在性条件.给出一种新的计算方法,克服了存储整个响应历史数据的困难,并给出了数值算例,计算结果与解析方法的结论比较吻合.     

Main features of the viscoelastic behavior of carbon fiber-reinforced plastics with a polymer matrix: Model study and calculation

It was shown that below the glass transition temperature of the amorphous matrix, its viscoelastic behavior in a composite is described by the Halpin-Tsao equation for composites, as modified by Nielsen, with a sufficient degree of reliability. However, when the matrix passes into the highly elastic and especially the viscous flow state characteristics of thermoplastic matrices, a significant difference is observed in the experimental and calculated temperature curves of the elastic and matrix loss moduli. It is possible to hypothesize that this is due to hindrance of plastic deformation of the matrix in the composite not accounted for in the Halpin-Tsao model. Using simple models of the one-sided coating and sandwich type, it was shown that consideration of the stress state of a matrix in contact with a rigid coating can result in the same kind of effects as in a composite, namely, apparent toughening of the matrix in the highly elastic region and shifting of the maximum of the mechanical loss modulus to the high-temperature region.Institute of Macromolecular Compounds. Russian Academy of Sciences, St. Petersburg, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 5, 690–703, September–October, 1997.     

About the stress-strain state of growing viscoelastic systems

A problem of calculation of the stress-strain state of viscoelastic heterogeneously ageing growing systems at the various moments of time is considered. The efficient numerical technique based on the Finite Element Method (FEM) is suggested. The constitutive equation between stresses and strains in the viscoelastic growing solid is corresponded with the Maslov-Arutyunyan theory. The general system of equations of the FEM-based technique is being formed for nodal displacement increments on the basis of variation of the total energy functional by components of the vector of nodal displacement increments. The numerical technique allows to take into account linear and nonlinear creep of the material, shrinkage (if it is necessary) and the instantly-elastic modulus time-dependence. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reliability Estimation of Optimal Viscoelastic Composite Shells in Critical-Time Calculations

The problem of reliability estimation for optimal viscoelastic composite shells in critical-time calculations is considered. The weight minimization of a viscoelastic composite shell, with constraints on deflections at the critical time, leads to a system deforming unstably, whose deflections grow in time with an increasing rate. A method for estimating the reliability of such shells in calculating the critical time is discussed. This time is regarded as a random variable depending on many, roughly equivalent, factors. An analysis of the reliability is carried out for different values of coefficients of variation, occurring in practice.     

聚合物时温等效模型有限元应用研究

为更好地描述聚合物材料力学性能的温度相关性问题,对目前广泛应用的WLF模型进行改进研究,并引入“零时间”因子提高了粘弹性材料变温松弛模量的获取精度.在此基础上基于ABAQUS用户材料子程序UTRS将时温等效模型应用到数值计算中.根据不同温度水平下的应力松弛实验获得模型参数,并通过等速拉伸实验与数值结果的对比验证了该模型及其有限元方法的可行性及正确性.结果表明:引入“零时间”因子的变温松弛模量精度更高;改进WLF模型对复合推进剂具有更好的适用性和更高的精确度.     

Nonlinear modeling and analysis of electrically actuated viscoelastic microbeams based on the modified couple stress theory

A nonclassical nonlinear continuum model of electrically actuated viscoelastic microbeams is presented based on the modified couple stress theory to consider the microstructure effect in the framework of viscoelasticity. The nonlinear integral-differential governing equation and related boundary conditions of are derived based on the extended Hamilton's principle and Euler–Bernoulli hypothesis for viscoelastic microbeams with clamped-free, clamped-clamped, simply-supported boundary conditions. The proposed model accounts for system nonlinearities including the axial residual stress, geometric nonlinearity due to midplane stretching, electrical forcing with fringing effect. The behavior of the microbeam is simulated using generalized Maxwell viscoelastic model. A new generalized differential/integral quadrature method is developed to solve the resulting governing equation. The developed model is verified against elastic behavior and a favorable agreement is obtained. Efficiency of the developed model is demonstrated by analyzing the quasistatic pull-in phenomena of electrically actuated viscoelastic microbeams with different boundaries at various material length scale parameters and axial residual stresses in the framework of linear viscoelasticity.     

Nonlinear evolution equation for describing waves in a viscoelastic tube

Propagation of the nonlinear waves in a viscoelastic tube filled with a liquid is studied. The bending of the tube wall is taken into account. By using the reductive perturbation method the fifth order nonlinear evolution equation is derived. Exact solutions for this equation is obtained by means of the simplest equation method. Properties of nonlinear waves in a viscoelastic tube filled with a liquid are discussed.