Transversely isotropic sequentially laminated composites in finite elasticity

A general expression for the energy-density function of sequentially laminated composites is derived. For the class of neo-Hookean composites in the limit of small deformations well-known results for linear transversely isotropic composites are recovered. However, it is shown that under large deformations these composites are not isotropic. Transversely isotropic composites are obtained with sequentially-coated composites in which the next rank composite is constructed by lamination of the previous composite with thin layers of the matrix phase. The transverse behavior of this sequentially-coated composite is neo-Hookean with shear modulus in the form of the Hashin-Shtrikman bounds for the corresponding class of linear composites. Comparison of the behaviors of these composites with recent estimates for transversely isotropic composites reveals good agreement up to relatively large deformations and volume fractions of the inclusion phase.     

Micromechanics-based constitutive modeling for unidirectional laminated composites

A micromechanics-based constitutive model is developed to predict the effective mechanical behavior of unidirectional laminated composites. A newly developed Eshelby’s tensor for an infinite circular cylindrical inclusion [Cheng, Z.Q., Batra, R.C., 1999. Exact Eshelby tensor for a dynamic circular cylindrical inclusion. J. Appl. Mech. 66, 563–565] is adopted to model the unidirectional fibers and is incorporated into the micromechanical framework. The progressive loss of strength resulting from the partial fiber debonding and the nucleation of microcracks is incorporated into the constitutive model. To validate the proposed model, the predicted effective stiffness of transversely isotropic composites under far field loading conditions is compared with analytical solutions. The constitutive model incorporating the damage models is then implemented into a finite element code to numerically characterize the elastic behavior of laminated composites. Finally, the present predictions on the stress–strain behavior of laminated composite plate containing an open hole is compared with experimental data to verify the predictive capability of the model.     

Determining the effective thermoelastic characteristics of regularly laminated composite in the asymmetric theory of elasticity

The asymmetric theory of elasticity is used to model a hybrid laminated composite of regular structure with all phases isotropic. The effective thermoelastic characteristics of the composite are determined. It is shown that the equations derived can be used to determine stress–strain state in all the phases of the composite using the average components of the tensors of force stresses, couple stresses, strains, and wryness in a layered material, which is of fundamental importance for the design of composites based on structural theories of failure     

Anisotropic strength of composites

The present investigation is concerned with the development of a theory of strength of anisotropic composite materials and the establishment of sound experimental procedures for the confirmation of the predicted results. A general theory is stated whereby the strength of laminated as well as unidirectional composite materials subjected to any state of combined stress can be predicted once the basic strength characteristics of a unidirectional layer have been determined. The transversely isotropic layer is treated in detail and, based on the understanding of the mechanical behavior of laminated composites, the procedure outlined for determining the strength of laminated systems. An experimental program was conducted, using glass-filament-reinforced resin test spceimens, and data obtained confirmed the results predicted for the strength of unidirectional composites. Based on the results of the analysis presented, composites may then be designed to incorporate an optimum utilization of the inherent strength characteristics of the constitutent materials.     

A general hyperelastic model for incompressible fiber-reinforced elastomers

This work presents a new constitutive model for the effective response of fiber-reinforced elastomers at finite strains. The matrix and fiber phases are assumed to be incompressible, isotropic, hyperelastic solids. Furthermore, the fibers are taken to be perfectly aligned and distributed randomly and isotropically in the transverse plane, leading to overall transversely isotropic behavior for the composite. The model is derived by means of the “second-order” homogenization theory, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” Compared to other constitutive models that have been proposed thus far for this class of materials, the present model has the distinguishing feature that it allows consideration of behaviors for the constituent phases that are more general than Neo-Hookean, while still being able to account directly for the shape, orientation, and distribution of the fibers. In addition, the proposed model has the merit that it recovers a known exact solution for the special case of incompressible Neo-Hookean phases, as well as some other known exact solutions for more general constituents under special loading conditions.     

Design of three dimensional isotropic microstructures for maximized stiffness and conductivity

The level-set method of topology optimization is used to design isotropic two-phase periodic multifunctional composites in three dimensions. One phase is stiff and insulating whereas the other is conductive and mechanically compliant. The optimization objective is to maximize a linear combination of the effective bulk modulus and conductivity of the composite. Composites with the Schwartz primitive and diamond minimal surfaces as the phase interface have been shown to have maximal bulk modulus and conductivity. Since these composites are not elastically isotropic their stiffness under uniaxial loading varies with the direction of the load. An isotropic composite is presented with similar conductivity which is at least 23% stiffer under uniaxial loading than the Schwartz structures when loaded uniaxially along their weakest direction. Other new near-optimal isotropic composites are presented, proving the capablities of the level-set method for microstructure design.     

A one-parameter generalized self-consistent model for isotropic multiphase composites

The classical generalized self-consistent model (GSCM) is recognized to be suitable and efficient for estimating the effective moduli of an isotropic composite consisting of an isotropic host matrix and an isotropic inclusion phase. The present work aims to enlarge the scope of the GSCM so that it becomes applicable to a good number of important situations where the phases cannot be differentiated as the host matrix and inclusions. This objective is achieved first by inserting into the unknown effective medium a coated composite sphere whose core is made of the unknown effective medium and whose coatings are formed of the constituent phases and then by imposing an energy equivalency condition. The equations thus obtained to characterize the effective bulk and shear moduli involve a microstructural parameter which turns out to be capable of describing in some sense how far a microstructure is from the host matrix/inclusion morphology. The important case of two-phase composites is studied in detail to illustrate the salient features of the proposed model.     

Characterization of short duration stress pulses generated by impacting laminated carbon-fiber/epoxy composites with magnetic flyer plates

Short duration stress pulses are of particular interest in determining the interfacial crack tip instability criteria for the dynamic fracture behavior of laminated carbon-fiber/epoxy composites. However, the heterogeneous architectures of laminated composites can alter the characteristics of a stress pulse as it propagates toward a crack tip. This makes it difficult to use standard dynamic testing techniques for characterizing these materials, since these techniques assume the characteristics of the stress pulse do not change as a result of propagation and can therefore be unambiguously determined from impact conditions. This paper presents a novel experimental technique that has been developed for characterizing short duration stress pulse propagation in laminated composite materials. In this technique, a dynamic moiré interferometer is used to capture fringe patterns corresponding to displacement fields associated with short duration stress pulses that were generated by impacting 0° and 90°/0°/90° carbon-fiber/epoxy composites with a magnetic flyer plate. Appropriate dynamic testing conditions for capturing high fidelity fringe patterns were determined using the recently developed dynamic moiré fringe contrast factor. The effects of the composite architecture on the propagation of short duration stress pulses observed with the dynamic moiré interferometer were confirmed by transient dynamic finite element analysis. From comparisons of experimental and numerical data, it was determined that the impact conditions for the magnetic flyer plate and laminated composite will not necessarily be planar, which has a significant effect on the intensity and duration of the propagating stress pulse.     

Modeling the macroscopic behavior of two-phase nonlinear composites by infinite-rank laminates

A new approach is proposed for estimating the macroscopic behavior of two-phase nonlinear composites with random, particulate microstructures. The central idea is to model composites by sequentially laminated constructions of infinite rank whose macroscopic behavior can be determined exactly. The resulting estimates incorporate microstructural information up to the two-point correlation functions, and require the solution to a Hamilton–Jacobi equation with the inclusion concentration and the macroscopic fields playing the role of ‘time’ and ‘spatial’ variables, respectively. Because they are realizable, by construction, these estimates are guaranteed to be convex, to satisfy all pertinent bounds, to exhibit no duality gap, and to be exact to second order in the heterogeneity contrast. Sample results are provided for two- and three-dimensional power-law composites, and are compared with other homogenization estimates, as well as with numerical simulations available from the literature. The estimates are found to give physically sensible predictions for all the cases considered, even for extreme values of the nonlinearity and heterogeneity contrast. Interestingly, in the case of isotropic porous materials under hydrostatic loadings, the estimates agree exactly with standard Gurson-type models for viscoplastic porous media.     

Neo-Hookean fiber-reinforced composites in finite elasticity

The response of a transversely isotropic fiber-reinforced composite made out of two incompressible neo-Hookean phases undergoing finite deformations is considered. An expression for the effective energy-density function of the composite in terms of the properties of the phases and their spatial distribution is developed. For the out-of-plane shear and extension modes this expression is based on an exact solution for the class of composite cylinder assemblages. To account for the in-plane shear mode we incorporate an exact result that was recently obtained for a special class of transversely isotropic composites. In the limit of small deformation elasticity the expression for the effective behavior agrees with the well-known Hashin-Shtrikman bounds. The predictions of the proposed constitutive model are compared with corresponding numerical simulation of a composite with a hexagonal unit cell. It is demonstrated that the proposed model accurately captures the overall response of the periodic composite under any general loading modes.     

Large amplitude free flexural vibrations of laminated composite skew plates

Here, the large amplitude free flexural vibration behaviors of thin laminated composite skew plates are investigated using finite element approach. The formulation includes the effects of shear deformation, in-plane and rotary inertia. The geometric non-linearity based on von Karman's assumptions is introduced. The non-linear governing equations obtained employing Lagrange's equations of motion are solved using the direct iteration technique. The variation of non-linear frequency ratios with amplitudes is brought out considering different parameters such as skew angle, number of layers, fiber orientation, boundary condition and aspect ratio. The influence of higher vibration modes on the non-linear dynamic behavior of laminated skew plates is also highlighted. The present study reveals the redistribution of vibrating mode shape at certain amplitude of vibration depending on geometric and lamination parameters of the plate. Also, the degree of hardening behavior increases with the skew angle and its rate of change depends on the level of amplitude of vibration.     

Exactly solvable spherically anisotropic thermoelastic microstructures

Microstructures possessing local spherical anisotropy are considered in this paper. An example is a spherulitic polymer which can be modelled by an assemblage of spheres of all sizes in which a radial direction in every sphere is an axis of local transverse isotropy. Our purpose is to construct effectively isotropic microstructures, with spherically anisotropic and thermoelastic constituents, whose effective bulk modulus, thermal stress and specific heat can be exactly determined. The basic microstructure for which this is achieved in the present paper is the nested composite sphere assemblage of Milgrom and Shtrikman (J. Appl. Phys. 66 (1989) 3429) which was originally formulated for isotropic constituents, in the settings of conductivity and coupled fields with scalar potentials. Here, we allow the phases of this microstructure to be spherically thermoelastic with a symmetry class which can be trigonal, tetragonal, transversely isotropic, cubic or isotropic with respect to a local spherical coordinate system. A rich class of new exact results for two-phase composites and polycrystals are obtained.     

Three-phase plane composites of minimal elastic stress energy: High-porosity structures

The paper establishes exact lower bound on the effective elastic energy of two-dimensional, three-material composite subjected to the homogeneous, anisotropic stress. It is assumed that the materials are mixed with given volume fractions and that one of the phases is degenerated to void, i.e., the effective composite is porous. Explicit formula for the energy bound is obtained using the translation method enhanced with additional inequality expressing certain property of stresses. Sufficient optimality conditions of the energy bound are used to set the requirements which have to be met by the stress fields in each phase of optimal effective material regardless of the complexity of its microstructural geometry. We show that these requirements are fulfilled in a special class of microgeometries, so-called laminates of a rank. Their optimality is elaborated in detail for structures with significant amount of void, also referred to as high-porosity structures. It is shown that geometrical parameters of optimal multi-rank, high-porosity laminates are different in various ranges of volume fractions and anisotropy level of external stress. Non-laminate, three-phase microstructures introduced by other authors and their optimality in high-porosity regions is also discussed by means of the sufficient conditions technique. Conjectures regarding low-porosity regions are presented, but full treatment of this issue is postponed to a separate publication. The corresponding “G-closure problem” of a three-phase isotropic composite is also addressed and exact bounds on effective isotropic properties are explicitly determined in these regions where the stress energy bound is optimal.     

复合材料层合板在动集中力作用下的结构声强特性

本文研究了复合材料正交异性层合板在动集中力作用下的结构声强特性。应用MSC/-NASTRAN商业软件计算了复合材料正交异性层合板在动集中力作用下各单元的内力和速度,再应用MATLAB软件得出复合材料层合板的结构声强。算例表明,复合材料正交异性层合板的结构声强流线图与各向同性板存在明显不同的特性。复合材料正交异性层合板的结构声强流线图受边界条件、层合板叠层顺序和层数的影响。从结构声强向量图和流线图可获得关于能量传递路径、源位置和能量汇合点的许多信息。进一步,结构振动产生的噪声可根据上述信息加以控制。     

采用瓣叶复合材料的生物心脏瓣膜的力学分析

采用微观组织结构分析及宏观复合材料分析结合的方法,分析了猪主动脉瓣的非线性复合材料性质,提出了一种适用于猪主动脉瓣的非线性复合材料本构模型,用提出的非线性复合材料本构模型,对闭合承载状态下的等厚度与变厚度几何模型的猪主动脉瓣的应力分布及变形进行了有限元数值模拟,发现：与各向同性瓣叶相比,单向增强复合材料的瓣叶不但具有较强的承载能力,而且具有较大的柔软性。     

Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers

A micromechanically based constitutive model for the elasto-viscoplastic deformation and texture evolution of semi-crystalline polymers is developed. The model idealizes the microstructure to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of finite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic flow occurring via crystallographic slip. The amorphous phase is modeled as isotropic elastic with plastic flow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic fields by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress-strain behavior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the effect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading-reloading cycles are presented.     

Effective elastic properties of matrix composites with transversely-isotropic phases

The present work addresses the problem of calculation of the macroscopic effective elastic properties of composites containing transversely isotropic phases. As a first step, the contribution of a single inhomogeneity to the effective elastic properties is quantified. Relevant stiffness and compliance contribution tensors are derived for spheroidal inhomogeneities. The limiting cases of spherical, penny-shaped and cylindrical shapes are discussed in detail. The property contribution tensors are used to derive the effective elastic moduli of composite materials formed by transversely isotropic phases in two approximations: non-interaction approximation and effective field method. The results are compared with elastic moduli of quasi-random composites.     

Inelastic behavior of an AS4/PEEK composite under combined transverse compressionand shear.Part II: modeling

A micromechanical model is presented for simulating the nonlinearities exhibited by AS4/PEEK composites in shear and transverse compression, their interaction, and their rate dependence at room temperature. The fibers are assumed to be transversely isotropic and to be distributed in a hexagonal pattern in the matrix. The PEEK matrix is modeled as an elastic–powerlaw viscoplastic, isotropic solid through two related models. Model I is the simple J₂–type viscoplasticity; Model II is a rate dependent version of the non-associative Drucker-Prager model. Both models are calibrated so that they reproduce the shear response of the composite. Model II is also calibrated to its transverse compression response. Both models capture the rate dependence of the composite well. Model I is significantly less stiff in transverse compression than the experimental data. However, it does a reasonable job of predicting other aspects of the biaxial experiments and captures the important trends of behavior observed. Model II does better in transverse compression, but shearing in the presence of transverse compression is found to be stiffer than the measured responses. The unit cell model allows us to examine the stresses in the composite, providing an explanation for the lack of interaction between the constant stress and the increasing stress observed experimentally for certain loading paths.     

具有曲率和惯性非线性以及材料内阻的旋转复合材料轴的主共振

旋转复合材料轴作为一类典型的转子动力学系统,在先进直升机和汽车动力驱动系统中有着广阔的应用前景.研究旋转复合材料轴的非线性振动特性具有重要的理论与实用价值.然而,目前有关旋转轴的非线性振动研究仅限于各向同性金属材料轴,很少考虑材料内阻的影响.本文研究具有材料内阻的旋转非线性复合材料轴的主共振.非线性来源于不可伸长复合材料轴的大变形引起的非线性曲率和非线性惯性,材料内阻来源于复合材料的黏弹性.动力学建模计入转动惯量和陀螺效应.基于扩展的Hamilton原理,导出具有偏心激励的旋转复合材料轴的弯-弯耦合非线性振动偏微分方程组.采用Galerkin法将偏微分方程离散化为常微分方程,采用多尺度法对常微分方程进行摄动分析,导出主共振响应的解析表达式.对内阻、外阻、铺层角、长径比、铺层方式和偏心距进行数值分析,研究上述参数对旋转非线性复合材料轴的稳态受迫振动响应行为的影响.研究发现,角铺设复合材料轴的内阻系数随着铺层角的增大而增大;内阻对主共振响应特性的影响主要体现在对抑制振幅和改变频率响应的稳定性方面;发生在正进动固有频率附近的主共振响应具有典型的硬弹簧非线性特性.本文提出的模型能够用于描述旋转复合材料轴的主共振特性,是对不可伸长旋转金属轴非线性动力学模型的重要推广.     

A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model

A new modified couple stress theory for anisotropic elasticity is proposed. This theory contains three material length scale parameters. Differing from the modified couple stress theory, the couple stress constitutive relationships are introduced for anisotropic elasticity, in which the curvature (rotation gradient) tensor is asymmetric and the couple stress moment tensor is symmetric. However, under isotropic case, this theory can be identical to modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The differences and relations of standard, modified and new modified couple stress theories are given herein. A detailed variational formulation is provided for this theory by using the principle of minimum total potential energy. Based on the new modified couple stress theory, composite laminated Kirchhoff plate models are developed in which new anisotropic constitutive relationships are defined. The First model contains two material length scale parameters, one related to fiber and the other related to matrix. The curvature tensor in this model is asymmetric; however, the couple stress moment tensor is symmetric. Under isotropic case, this theory can be identical to the modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The present model can be viewed as a simplified couple stress theory in engineering mechanics. Moreover, a more simplified model of couple stress theory including only one material length scale parameter for modeling the cross-ply laminated Kirchhoff plate is suggested. Numerical results show that the proposed laminated Kirchhoff plate model can capture the scale effects of microstructures.