含大量随机分布细小纤维的单向纤维增强复合材料横向弹性性能参数计算的内嵌区域模型法

利用有限元方法求取单向纤维增强复合材料的横向弹性性能参数的计算模型包括三维模型、两维平面应变模型、单胞模型等等.由于单胞模型仅仅适用于纤维规则排列情况.在纤维随机分布且纤维大小亦为随机时,单向纤维增强复合材料横向弹性性能参数必须通过对于复合材料块体的计算才能获得.同时在随机分布纤维的数量增大时,三维模型和二维平面应变模型的计算量急剧增加,模型的处理能力不强.该文提出一种利用内嵌区域模型来计算含大量随机大小、随机分布细小纤维的单向纤维增强复合材料块体的横向弹性性能参数的方法,有效降低了计算量.在较低的计算费用下,能够快速获得单向纤维增强复合材料的横向弹性性能参数.     

Reflection and refraction of plane waves on the interface of two anisotropic media

We use generalized functionally invariant solutions [1] of the equations of motion to obtain and analytically study solutions of the plane problem of reflection and refraction of plane waves on the interface of two anisotropic media with four elastic constants depending on the angles of incidence of primary waves for various relations between the elastic constants of the contacting media. For the primary waves, we find the ranges of incidence angles for which real and complex secondary waves are excited. We study all possible combinations of the distribution of phase velocities and reflection and refraction angles in detail and obtain conditions characterizing the directions of the energy flux vectors for the primary and secondary waves depending on the incidence angles of the primary waves for different relations between the elastic constants of the contacting media, which satisfy the necessary and sufficient conditions for the elastic energy form to be positive definite.     

单向纤维增强复合材料中剪切波多重散射问题的复变函数法

基于弹性波的多体散射理论和复变函数方法,利用波函数展开法和保角映射方法,研究了任意形纤维增强复合材料中剪切波的传播。根据相应的边界条件确定弹性波模式系数。给出了复合材料中基体区和纤维核区的波场。作为算例,分析了相同入射频率,不同纤维核的尺寸和形状的情况下,弹性波的传播特性。通过分析发现,低频长波区,弹性波以近似正弦函数的形式传播,纤维核的尺寸和形状对波场影响不明显;随着频率的增加,影响变得明显,大于一定的频率,纤维核的形状可明显地辨别出。最后对结果进行了分析讨论。     

The Wave Energy in Nonlinearly Deformable Composites

The energy characteristics of waves propagating in composites are discussed. To describe the deformation of materials, two models are used — the classical model of an elastic body and the microstructural model of a two-phase elastic mixture. Both models take into account the quadratic nonlinearity of deformation based on the Murnaghan elastic potential. Analytical expressions for the velocity at which the energy of travelling plane longitudinal waves propagates are derived. It is shown that the nonlinearity of composite deformation decreases the velocities of energy propagation of both nondispersive and dispersive waves     

Reflection and transmission of elastic waves at an elastic/porous solid saturated by two immiscible fluids

Wave propagation in a porous elastic medium saturated by two immiscible fluids is investigated. It is shown that there exist three dilatational waves and one transverse wave propagating with different velocities. It is found that the velocities of all the three longitudinal waves are influenced by the capillary pressure, while the velocity of transverse wave does not at all. The problem of reflection and refraction phenomena due to longitudinal and transverse wave incident obliquely at a plane interface between uniform elastic solid half-space and porous elastic half-space saturated by two immiscible fluids has been analyzed. The amplitude ratios of various reflected and refracted waves are found to be continuous functions of the angle of incidence. Expression of energy ratios of various reflected and refracted waves are derived in closed form. The amplitude ratios and energy ratios have been computed numerically for a particular model and the results obtained are depicted graphically. It is verified that during transmission there is no dissipation of energy at the interface. Some particular cases have also been reduced from the present formulation.     

Scattering of elastic waves in an elastic matrix containing an inclusion with interfaces

Based on the theory of elastic dynamics, the scattering of elastic waves and dynamic stress concentration in fiber-reinforced composite with interfaces are studied. Analytical expressions of elastic waves in different medium areas are presented and an analytic method of solving this problem is established. The mode coefficients are determined by means of the continuous conditions of displacement and stress on the boundary of the interfaces. The influence of material properties and structural size on the dynamic stress concentration factors near the interfaces is analyzed. It indicates that they have a great influence on the dynamic properties of fiber-reinforced composite. As examples, numerical results of dynamic stress concentration factors near the interfaces are presented and discussed. This paper provides reliable theoretical evidence for the study of dynamic properties in fiber-reinforced composite. Project supported by the National Natural Science Foundation of China (No. 19972018).     

单向纤维加强复合材料的总体弹粘塑性响应

应用GMC方法计算了单向纤维加强复合材料在承受不同载荷之后的总体松驰响应。Bodner和Partom的弹－粘塑性统一模型用于描述非弹性相的本构关系，该模型不假设存在屈服条件，也就不必指定加载和卸载条件，可以在加载和卸载的任何时刻使用相同的公式，计算表明，与有限元方法相比，利用GMC方法和统一模型计算单向纤维加强复合材料的总体非弹性响应简单快捷。     

Elastic waves in a fiber-reinforced composite

The propagation of time-harmonic elastic waves in a fiber-reinforced composite is studied. The circular fibers are assumed to be parallel to each other and randomly distributed with a statistically uniform distribution. The direction of propagation and the associated particle motion are considered to be normal to the fibers. It is shown that the average waves in the composite separate into compressional and shear types. General formulae for the complex wave number giving the phase velocity and the damping are obtained. It is shown that these formulae lead to the Hashin-Rosen expressions for the transverse bulk modulus and the lower bound for the transverse rigidity, if the correlation in the positions of the fibers can be ignored. The correlation terms, for exponential correlation, are shown to have a significant effect on the damping property of the composite, especially at high frequencies and concentrations.     

Static Model Describing the Inhomogeneous Deformation of Unidirectional Fibrous Composites under Plane Strain

A mathematical model is proposed to describe the static deformation of a unidirectional fiber-reinforced composite under plane-strain conditions in a plane perpendicular to the fiber direction. The model is based on the stochastic static equations for an elastic homogeneous two-component medium with nonzero body forces. Applying the method of conditional moments and double Fourier transform yields a nonlocal model in the form of integro-differential equations. Expanding the integral kernels into series in expansion coefficients yields differential equations whose order depends on the number of terms in the series. The zero-order approximation leads to the theory of effective moduli, the first-order approximation to the refined theory of effective moduli, and the second-order approximation to fourth-order equilibrium equations for mean displacements (mathematical expectations) and formulas for displacements, strains, and stresses averaged over the composite and its components. All the coefficients in theses formulas are expressed in terms of the elastic constants of the components and the geometric parameters of the structure. The model is valid for heavy stress gradients. In a particular case, the static theory of two-component elastic mixtures follows from the model__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 41–51, February 2005.     

Cubically non-linear effects of plane waves in isotropic soft solid materials

In this paper we analyze mathematical properties of an isotropic soft solid model which is characterized by three elastic constants. The model was proposed to interpret measurements of weakly non-linear shear waves in gel-like and tissue-like media. In our analysis we are particularly interested in third order non-linear terms. We present for the first time the full equations of elastodynamics, as well as the equations for plane waves for this model, with cubically non-linear terms. Next, the interaction coefficients for non-linear interactions of three plane waves to produce the fourth wave are explicitly calculated. These coefficients show which of the three waves interact with each other and determine how strong the effect of interaction is on the produced fourth wave. It turns out that these coefficients are expressed in terms of some combinations of three elastic constants. The obtained results can be helpful in experimental determination of elastic constants which describe the model.     

Features of plane wave propagation along the layers of a prestrained nanocomposite

Features of the propagation of longitudinal and transverse plane waves along the layers of nanocomposites with process-induced initial stresses are studied. The composite has a periodic structure: it is made by repeating two highly dissimilar layers. The layers exhibit nonlinear elastic behavior in the range of loads under consideration. A Murnaghan-type elastic potential dependent on the three invariants of the strain tensor is used to describe the mechanical behavior of the composite constituents. To simulate the propagation of waves, finite-strain theory is used for developing a problem statement within the framework of the three-dimensional linearized theory of elasticity assuming finite initial strains. The dependence of the relative velocities of longitudinal and transverse waves on two components of small initial stresses in each layer and on the volume fraction of the constituents is studied. It is established that there are thickness ratios of layers in some nanocomposites such that the wave velocities are independent of the initial stresses and equal to the respective wave velocities in composites without initial stresses __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 3–26, April 2007.     

纤维排列方式对单向纤维加强复合材料弹性常数的影响

本文应用GMC方法计算了纤维在规则排列和随机排列时单向纤维加强复合材料的总体弹性系数，结果表明，在一定纤维体积比范围内，虽然纵向弹性模量不受纤维排列方式的影响，但是复合材料的横向弹性模量和横向剪切模量受纤维排列方式的影响较大，所以，在考察复合材料的横向弹性常数时，应该考虑纤维排列方式的影响，在相同纤维比的情况下，正方形排列和正方形对角排列所得的弹性常数是上下界，而六角形排列和随机排列的结果位于以上两者之间，当纤维比较小时（小于20％），纤维的排列方式对各弹性常数均无明显影响，本文的结果在工程应用中有指导意义。     

A forced-vibration technique for measurement of material damping

This article describes a technique for measuring material damping in specimens under forced flexural vibration. Although the method was developed for testing fiber-reinforced composite materials, it could be used for any structural material. The test specimen is a double-cantilever beam clamped at its midpoint and excited in resonant flexural vibration by an electromagnetic shaker. Under steady state conditions, material damping is defined in terms of the ratio of input energy to strain energy stored in the specimen. If external losses are negligible, the input energy must equal the energy dissipated in the specimen. Input energy and strain energy are found from measured specimen dimensions, resonant frequency, input acceleration and bending strain. Problems associated with minimization of external energy losses in the apparatus and verification of measurements are discussed in detail. Measured damping of aluminum-alloy calibration specimens shows good agreement with calculated thermoelastic damping. Examples of measured damping showing amplitude and frequency dependence in fiber-reinforced plastic materials are presented.     

Elasto-optic technique for measurement of elastic wave propagation in solids

The modulation of the optical path of the beam of a laser vibrometer in a specimen under acoustic excitation is measured at two planes, separated by a precisely known distance. The phase shift and the decrease in magnitude are used to calculate the phase velocity and attenuation, respectively. The method is demonstrated for a homogeneous specimen, and the results compare favorably with those obtained by a conventional ultrasonic technique. The method is then applied to measure specular and first diffraction-order reflection from a coplanar periodic array of particles in an elastic matrix and phase velocity spectra in a tetragonal periodic particulate composite. As expected, in a periodic composite the establishment of dispersive Floquet-type waves is observed throughout the entire periodic particulate composite.     

Bidirectional Whitham equations as models of waves on shallow water

Hammack & Segur (1978) conducted a series of surface water-wave experiments in which the evolution of long waves of depression was measured and studied. This present work compares time series from these experiments with predictions from numerical simulations of the KdV, Serre, and five unidirectional and bidirectional Whitham-type equations. These comparisons show that the most accurate predictions come from models that contain accurate reproductions of the Euler phase velocity, sufficient nonlinearity, and surface tension effects. The main goal of this paper is to determine how accurately the bidirectional Whitham equations can model data from real-world experiments of waves on shallow water. Most interestingly, the unidirectional Whitham equation including surface tension provides the most accurate predictions for these experiments. If the initial horizontal velocities are assumed to be zero (the velocities were not measured in the experiments), the three bidirectional Whitham systems examined herein provide approximations that are significantly more accurate than the KdV and Serre equations. However, they are not as accurate as predictions obtained from the unidirectional Whitham equation.     

Electroacoustic waves in piezoelectric crystals: Certain limiting cases

The Christoffel equations for electroacoustic waves in unbounded piezoelectric crystals are solved in the limits of weak and strong electromechanical coupling and for the case where the unstiffened elastic constants satisfy the conditions of elastic isotropy. Lyubimov's proof that piezoelectric stiffening of the elastic constants increases acoustic velocities is extended to cover degenerate modes. It is shown that when the elastic contribution to the stiffened elastic constants is zero, only one acoustic branch survives with a finite velocity. When the unstiffened elastic constants satisfy the conditions of elastic isotropy one of the acoustic branches is unaffected by the piezoelectric stiffening and, moreover, Lyubimov's theorem holds nonperturbatively.     

Thermoviscoelastic surface waves of an assigned wavelength

This paper deals with the propagation of surface waves of an assigned wavelength on a thermoviscoelastic half-space. It is shown that a unique surface wave of an assigned wavelength, which satisfies the adopted criteria for behaviour at infinity, always exists. This wave is interpreted as a superposition of three dispersive inhomogeneous plane waves. The superposed waves have different directions of propagation and different phase velocities. Their directions of propagation are not parallel to the stress-free surface. The plane of constant amplitude that corresponds to each of these superposed waves is parallel to the stress-free surface and moves to it with a constant velocity, which is different for each of the superposed waves. The numerical computations refer to some typical values of the material and thermal constants at different values of the wavelength when the half-space is thermally insulated.     

Seismic Reflection from an Interface Between an Elastic Solid and a Fractured Porous Medium with Partial Saturation

The theory of Tuncay and Corapcioglu (Transp Porous Media 23:237–258, 1996a) has been employed to investigate the possibility of plane wave propagation in a fractured porous medium containing two immiscible fluids. Solid phase of the porous medium is assumed to be linearly elastic, isotropic and the fractures are assumed to be distributed isotropically throughout the medium. It has been shown that there can exist four compressional waves and one rotational wave. The phase speeds of these waves are found to be affected by the presence of fractures, in general. Of the four compressional waves, one arises due to the presence of fractures in the medium and the remaining three are those encountered by Tuncay and Corapcioglu (J Appl Mech 64:313–319, 1997). Reflection and transmission phenomena at a plane interface between a uniform elastic half-space and a fractured porous half-space containing two immiscible fluids, are analyzed due to incidence of plane longitudinal/transverse wave from uniform elastic half-space. Variation of modulus of amplitude and energy ratios with the angle of incidence are computed numerically by taking the elastic half-space as granite and the fractured porous half-space as sandstone material containing non-viscous wetting and non-wetting fluid phases. The results obtained in case of porous half-space with fractures, are compared graphically with those in case of porous half-space without fractures. It is found that the presence of fractures in the porous half-space do affect the reflection/transmission of waves, which is responsible for raising the reflection and lowering the transmission coefficients.     

Plane Horizontal Transverse Waves in Elastic Piezopowders

A procedure and results of computer simulation of plane horizontal transverse waves are described. Three materials — gallium arsenide, bismuth germanate, and lead zirconate–titanate ceramics — are selected as the piezoelectric phase. The second phase of the powder is always lead. To describe waves in the powder, the microstructural theory of two-phase mixtures is used. Therefore, the computer simulation was intended to study the influence of the lead content by volume on the wave velocities and the microstructural wave-propagation pattern — decomposition of a wave into two modes, simultaneous propagation of both modes in each phase of the powder, etc. First, sets of physical constants (elastic, piezoelectric, and dielectric) of mixture theory were evaluated for three types of powders (with the piezoelectric phase as one of the above-mentioned materials) with the volume piezoelectric-phase content varying from 0.01 to 0.5 with step 0.005. Further, dispersion curves for both modes and 3D-graphs of amplitudes as functions of the wave propagation time and distance were plotted for 300 compositions of powders (three types, each of 100 modifications). Of the phenomena described, we should first of all point out that all the phase velocities increase twice upon changing the content of the powder in the piezoelectric phase from a very small amount to the maximum possible     

纤维增强复合材料弹性性能预测的域分解方法及应用

提出了新的有限元建模方法,即域分解方法,用于预测纤维增强复合材料单向带T300/BSL914C(环氧树脂)和AS4/3501-6(环氧树脂)的弹性性能。域分解方法基于区域叠合技术,分别建立单胞的整体域与纤维域模型用于代替传统有限元建模方法中单胞的基体域与纤维域模型。整体域是真实基体体积与纤维体积的叠加,两区域网格独立划分,互不影响。采用MSC.Nastran中的多节点约束Explicit单元,在整体域与纤维域节点之间建立位移连接属性模拟单胞基体域与纤维域之间的位移约束关系,从而实现两区域的耦合计算。计算结果表明:域分解方法单胞模型纤维增强方向弹性模量Ez预测值与试验值误差在7%以内,其余弹性常数也都与试验值吻合较好。域分解方法不仅可以大大简化纤维增强复合材料的细观力学建模,而且可以准确地预测纤维增强复合材料的弹性性能。