Meso cell model of fiber reinforced composite: Interface stress statistics and debonding paths

The primary goal of this work is to develop an efficient analytical tool for the computer simulation of progressive damage in the fiber reinforced composite (FRC) materials and thus to provide the micro mechanics-based theoretical framework for a deeper insight into fatigue phenomena in them. An accurate solution has been obtained for the micro stress field in a meso cell model of fibrous composite. The developed method combines the superposition principle, Kolosov–Muskhelishvili’s technique of complex potentials and Fourier series expansion. By using the properly chosen periodic potentials, the primary boundary-value problem stated on the multiple-connected domain has been reduced to an ordinary, well-posed set of linear algebraic equations. The meso cell can include up to several hundred inclusions which is sufficient to account for the micro structure statistics of composite. The presented numerical examples demonstrate an accuracy and high numerical efficiency of the method which makes it to be a promising tool for studying progressive damage in FRCs. By averaging over a number of random structure realizations, the statistically meaningful results have been obtained for both the local stress and effective elastic moduli of disordered fibrous composite. A special attention has been paid to the interface stress statistics and the fiber debonding paths development, which appear to correlate well with the experimental observations.     

Transverse conductivity and longitudinal shear of elliptic fiber composite with imperfect interface

The paper addresses the problem of calculating the local fields and effective transport properties and longitudinal shear stiffness of elliptic fiber composite with imperfect interface. The Rayleigh type representative unit cell approach has been used. The micro geometry of composite is modeled by a periodic structure with a unit cell containing multiple elliptic inclusions. The developed method combines the superposition principle, the technique of complex potentials and certain new results in the theory of special functions. An appropriate choice of the potentials provides reducing the boundary-value problem to an ordinary, well-posed set of linear algebraic equations. The exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the local gradient and flux fields. The convergence of solution has been verified and the parametric study of the model has been performed. The obtained accurate, statistically meaningful results illustrate a substantial effect of imperfect interface on the effective behavior of composite.     

Elastic interaction of partially debonded circular inclusions. II. Application to fibrous composite

A complete analytical solution has been obtained of the elasticity problem for a plane containing periodically distributed, partially debonded circular inclusions, regarded as the representative unit cell model of fibrous composite with interface damage. The displacement solution is written in terms of periodic complex potentials and extends the approach recently developed by Kushch et al. (2010) to the cell type models. By analytical averaging the local strain and stress fields, the exact formulas for the effective transverse elastic moduli have been derived. A series of the test problems have been solved to check an accuracy and numerical efficiency of the method. An effect of interface crack density on the effective elastic moduli of periodic and random structure FRC with interface damage has been evaluated. The developed approach provides a detailed analysis of the progressive debonding phenomenon including the interface cracks cluster formation, overall stiffness reduction and damage-induced anisotropy of the effective elastic moduli of composite.     

Effect of crack orientation statistics on effective stiffness of mircocracked solid

This paper addresses the problem of calculating effective elastic properties of a solid containing multiple cracks with prescribed orientation statistics. To do so, the representative unit cell approach has been used. The microgeometry of a cracked solid is modeled by a periodic structure with a unit cell containing multiple cracks; a sufficient number is taken to account for the microstructure statistics. The developed method combines the superposition principle, the technique of complex potentials and certain new results in the theory of special functions. A proper choice of potentials provides reducing the boundary-value problem to an ordinary, well-posed set of linear algebraic equations. The exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the strain and stress fields. The convergence study has been performed; the statistically meaningful results obtained show dependence of the effective elastic stiffness on angular scattering of cracks. Comparison has been made with the selected simple micromechanical models, namely, non-interaction approximation, differential scheme and modified differential scheme. It is found that, among these models, the differential scheme provides the best fit of the numerical data.     

Elastic equilibrium of a half plane containing a finite array of elliptic inclusions

An accurate analytical method has been proposed to solve for stress in a half plane containing a finite array of elliptic inclusions, the last being a model of near-surface zone of the fibrous composite part. The method combines the Muskhelishvili’s method of complex potentials with the Fourier integral transform technique. By accurate satisfaction of all the boundary conditions, a primary boundary-value elastostatics problem for a piece-homogeneous domain has been reduced to an ordinary well-posed set of linear algebraic equations. A properly chosen form of potentials provides a remarkably simple form of equations and thus an efficient computational algorithm. The theory developed is rather general and can be applied to solve a variety of elastostatics problems. Up to several hundred interacting inclusions can be considered in this way in practical simulations which makes the model of composite half plane realistic and flexible enough to account for the microstructure statistics. The stress concentration factors and effective thermoelastic properties of random structure composites with dilute concentration of fibers are estimated in the vicinity of a free edge. The numerical examples are given showing accuracy and numerical efficiency of the developed method and disclosing the way and extent to which the nearby free or loaded boundary influences the local and mean stress concentration in the fibrous composite.     

Elastic interaction of partially debonded circular inclusions. I. Theoretical solution

A complete solution has been obtained of the elasticity problem for a plane containing a finite array of partially debonded circular inclusions, regarded as the open-crack model of fibrous composite with interface damage. A general displacement solution of the single-inclusion problem has been derived by combining the complex potentials technique with the newly derived series expansions. This solution is valid for any non-uniform far load and is finite and exact in the case of polynomial far field. Applying the superposition principle expands this theory to the multiple inclusion problem and provides a simple and rapidly convergent iterative algorithm. The presented numerical data show an accuracy and numerical efficiency of the proposed method and discover the way and extent to which the elastic interaction between the partially debonded inclusions affects the local fields, stress intensity factors and the energy release rate at the interface crack tips.     

Influence of fiber’s shape and size on overall elastoplastic property for micropolar composites

A general micromechanical method is developed for a micropolar composite with ellipsoidal fibers, where the matrix material is idealized as a micropolar material model. The method is based on a special micro–macro transition method, and the classical effective moduli for micropolar composites can be determined in an analytical way. The influence of both fiber’s shape and size can be analyzed by the proposed method. The effective moduli, initial yield surface and effective nonlinear stress and strain relation for a micropolar composite reinforced by ellipsoidal fibers are examined, it is found that the prediction on the effective moduli and effective nonlinear stress and strain curves are always higher than those based on classical Cauchy material model, especially for the case where the size of fiber approaches to the characteristic length of matrix material. As expected, when the size of fiber is sufficiently large, the classical results (size-independence) can be recovered.     

Effective balance equations for elastic composites subject to inhomogeneous potentials

We derive the new effective governing equations for linear elastic composites subject to a body force that admits a Helmholtz decomposition into inhomogeneous scalar and vector potentials. We assume that the microscale, representing the distance between the inclusions (or fibers) in the composite, and its size (the macroscale) are well separated. We decouple spatial variations and assume microscale periodicity of every field. Microscale variations of the potentials induce a locally unbounded body force. The problem is homogenizable, as the results, obtained via the asymptotic homogenization technique, read as a well-defined linear elastic model for composites subject to a regular effective body force. The latter comprises both macroscale variations of the potentials, and nonstandard contributions which are to be computed solving a well-posed elastic cell problem which is solely driven by microscale variations of the potentials. We compare our approach with an existing model for locally unbounded forces and provide a simplified formulation of the model which serves as a starting point for its numerical implementation. Our formulation is relevant to the study of active composites, such as electrosensitive and magnetosensitive elastomers.     

Hahn-Tsai复合材料的非线性杂交应力有限元方法

成功建立了Hahn-Tsai复合材料模型的非线性杂交应力有限元方程,采用Newton-Raphson迭代法求解结构的非线性位移方程。在迭代过程中,为了提高计算效率可采用简单迭代法由节点位移求解单元应力场。但是,当载荷增加到一定程度以后,非线性应力场由于循环迭代而无法收敛,显然,一般的加速方法不能解决这种循环迭代的发散问题。因此,本文发展了一种确实有效的非线性应力场迭代新方法,在不增加计算工作量的情况下,不仅极大地提高了收敛速度,而且对于较大载荷也能够很好地收敛,从而解决了大载荷下非线性杂交元方法失败的关键问题。数值算例表明该方法是确实可行的。     

Stress intensity factors for an arbitrarily oriented crack near a hole in longeron web

Damage-tolerant design of composite components in aerospace structures requires computationally effective stress and failure analysis methods. This study introduces an analytical/numerical method to determine the stress field and the stress intensity factors in a composite longeron web with an arbitrarily oriented straight crack near a hole. Typical of webs in wing longerons with massive belts, the tapered web is loaded in bending and shear. The solution method makes use of the complex potentials in conjunction with the boundary collocation technique. The present results are in close agreement with those obtained by finite element.     

Three-dimensional continuum analysis for a unidirectional composite with a broken fiber

The three-dimensional problem of a periodic unidirectional composite with a penny-shaped crack traversing one of the fibers is analyzed by the continuum equations of elasticity. The solution of the crack problem is represented by a superposition of weighted unit normal displacement jump solutions, everyone of which forms a Green’s function. The Green’s functions for the unbounded periodic composite are obtained by the combined use of the representative cell method and the higher-order theory. The representative cell method, based on the triple discrete Fourier transform, allows the reduction of the problem of an infinite domain to a problem of a finite one in the transform space. This problem is solved by the higher-order theory according to which the transformed displacement vector is expressed by a second order expansion in terms of local coordinates, in conjunction with the equilibrium equations and the relevant boundary conditions. The actual elastic field is obtained by a numerical evaluation of the inverse transform. The accuracy of the suggested approach is verified by a comparison with the exact analytical solution for a penny-shaped crack embedded in a homogeneous medium. Results for a unidirectional composite with a broken fiber are given for various fiber volume fractions and fiber-to-matrix stiffness ratios. It is shown that for certain parameter combinations the use of the average stress in the fiber, as it is employed in the framework of the shear lag approach, for the prediction of composite’s strength, leads to an over estimation. To this end, the concept of “point stress concentration factor” is introduced to characterize the strength of the composite with a broken fiber. Several generalizations of the proposed approach are offered.     

复合材料应力分析的均匀化方法

建立了基于均匀化理论的确定复合材料结构应力场的方法．其实质是用均质的宏观结构和非均质的具有周期性分布的细观结构描述原结构；将力学量表示成关于宏观坐标和细观坐标的函数，并用细观和宏观两种尺度之比为小参数展开，用摄动技术将原问题化为一细观均匀化问题和一宏观均匀化问题．这两个问题的解确定了包含等效位移和一阶近似位移的位移场，由此获得应力场．利用该方法给出了圆柱形孔隙材料和单向纤维复合材料在单向拉伸时的应力场以及空隙材料简支梁的局部应力场，说明了该方法的有效性     

半空间饱和土中圆柱形衬砌内荷载引起的动力响应解答

地下衬砌结构经常会受到内部动荷载的作用,内荷载引起的衬砌结构的动应力集中备受关注。采用Laplace变换和波函数展开法,对半空间饱和土中均布突加荷载作用下圆柱形衬砌结构的动力响应进行了研究。对以大半径凸圆弧来近似半空间表面,将半空间饱和土中的波动方程展开成无穷级数的形式,应用Graff加法公式进行坐标变换,根据连续条件和边界条件,确定波函数展开式中的未知系数,求得了半空间饱和土中均布突加荷载作用下圆柱形衬砌的动力响应解答,讨论了衬砌埋深对圆柱形衬砌动力响应的影响。该问题的求解,为地下结构的动力分析提供了一种有效方法。     

Intensity of singular stress at the fiber end in a hexagonal array of fibers

To evaluate the mechanical strength of fiber-reinforced composites it is necessary to consider singular stresses at the end of fibers because they cause crack initiation, propagation, and final failure. The singular stress field is controlled by generalized stress intensity factor (GSIF) defined at the fiber end. In this study, periodic and zigzag arrays of cylindrical inclusions under longitudinal tension are considered in comparison with the results for a single fiber. The unit cell region is approximated as an axi-symmetric cell; then, the body force method is applied, which requires the stress and displacement fields due to ring forces in infinite bodies having the same elastic constants as those of the matrix and inclusions. The given problem is solved on the superposition of two auxiliary problems under different boundary conditions. To obtain the GSIF accurately, the unknown body force densities are expressed as piecewise smooth functions using fundamental densities and power series. Here, the fundamental densities are chosen to represent the symmetric stress singularity, and the skew-symmetric stress singularity. The GSIFs are systematically calculated with varying the elastic modulus ratio and spacing of fibers. The effects of volume fraction and spacing of fibers are discussed in fiber-reinforced plastics.     

Antiplane interaction of line crack with an arbitrarily located elliptical inclusion

The antiplane problem of the interaction between a main crack and an arbitrarily located elastic elliptical inclusion near its tip is addressed in the current study. The analysis is based on the use of the complex potentials for the antiplane problem, Laurent series expansion method and an appropriate superposition scheme. The stress intensity factor at the main crack is obtained in a general series form. Explicit asymptotic solutions are also derived by using a perturbation technique and retaining the leading order terms in series expansion. The present solutions are shown to coincide with the Taylor expansion of exact solutions for special cases available in the literature. Discussed are changes in the crack tip stress intensity which can be enhanced or suppressed depending on the location of the elliptical inclusion. The explicit solutions provided herein are well suited for the further quantitative analysis of toughening mechanisms in ceramic composite materials.     

STATISTIC MODELING OF THE CREEP BEHAVIOR OF METAL MATRIX COMPOSITES BASED ON FINITE ELEMENT ANALYSIS

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Effective properties of elastic periodic composite media with fibers

A series solution to obtain the effective properties of some elastic composites media having periodically located heterogeneities is described. The method uses the classical expansion along Neuman series of the solution of the periodic elasticity problem in Fourier space, based on the Green's tensor, and exact expressions of factors depending on the shape of the inclusions. Some properties of convergence of the solution are presented, more specifically concerning the elasticity tensor of the reference medium, showing that the convergence occurs even for empty fibers. The solution is extended for rigid inclusions. A comparison is made with previous exact solutions for a fiber composite made of cylindrical fibers with circular cross-sections and with previous estimates. Different examples are presented for new situations concerning the study of fiber composites: composites with elliptic cross-sections and multi-phase fibrous composites.     

新修正偶应力层合板理论中的几个基本假设对自由振动计算的影响

基于新修正偶应力理论,在对微细观尺度的复合材料层合梁/板进行力学响应计算时,往往采用一系列假设来简化模型。现有文献都全部或部分应用了这些假设,但对这些假设是否会对计算结果造成影响尚未进行充分讨论分析。本文建立了未经简化的新修正偶应力Reddy层合板模型,并对其自由振动进行了分析。通过数值算例的对比,讨论了常用的几个简化假设对微细观复合材料四边简支方板自振频率的影响以及适用范围。算例结果表明,常用的几个简化假设对于微尺度层合薄板自由振动的影响很小,对于厚板的低阶频率影响也很小,但对厚板的高阶频率影响显著。     

Tikhonov regularization with Incremental Hole-Drilling and the Integral Method in Cross-Ply Composite Laminates

Background: Incremental hole-drilling with the integral method has been extensively used in composite laminates but is sensitive to small measurement errors. Error sensitivity can be reduced by limiting the number of depth increments used in the calculation procedure. This approach is limited if a rapidly varying residual stress distribution exists since the calculated stress in each incremental depth is considered constant. Distortion of stress results can consequently occur due to averaging effects if the depth increments become too large. Tikhonov regularization is usually applied in isotropic materials to smooth the resulting residual stress distribution and reduce stress uncertainties, but has only been applied to composite laminates using the slitting technique. Objective: The intention of this work is to extend the use of Tikhonov regularization to incremental hole-drilling of composite laminates using the integral method. Methods: Finite element modelling is used to calculate the necessary calibration coefficients for unit pulses of uniform stress. Monte Carlo simulation is used to the determine uncertainties in the calculated residual stress distributions. Tikhonov regularization is optimised to reduce the stress uncertainty, while ensuring that the stress solution is not distorted. Results: The method is demonstrated on a GFRP (Glass fibre reinforced plastic) laminate of [0₂/90₂]_s construction and the calculated residual stress field is compared with those obtained by the standard integral method and series expansion. Conclusions: It is found that Tikhonov regularization significantly improves the accuracy of the standard integral method in composite laminates and shows good agreement with the series expansion method.

     

SMA长纤维铝基复合材料的热力学性能

从作者建立的增量型的SMA本构关系出发 ,借助于细观力学的方法 ,推导出了新的长纤维SMA复合材料的增量型细观本构模型 ;应用此模型分析了该复合材料的在循环热载作用下的力学性能 ,尤其是计算了复合材料在不同条件下残余应力的变化 ,这对智能复合材料的设计提供了很大帮助。