Analysis of the decay of edge effects in laminated materials on the basis of a representative element

A model and design model for representative-element analysis of the decay of the Saint-Venant edge effects in laminated materials are proposed. The decay of the edge effect in a matrix with laminated covering (a composite material of irregular structure) is considered for the case of symmetric deformation. The source of the edge effect is simulated by a piecewise-constant surface periodic load normal to the layers. This load is local for the working domain and is varied within a boundary section that is commensurable with the characteristic size of the heterogeneity of the material structure. The equations of the linear theory of elasticity, a model of a piecewise-homogeneous medium, and quantitative criteria for identification of edge effects are used. A discrete model of the problem and its solution are constructed within the concept of basic schemes. Data on the zone of the edge effect and the character of its decay for a representative element of the material are analyzed. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 135–144, June, 2000.     

Influence of Young's Moduli of a Laminated Composite with a Periodic System of Cracks on Edge Effects

A problem on edge effects in a laminated composite with a periodic system of symmetric contact cracks is studied. Loading of the composite induces constant deformation in the reinforcement direction beyond the edge effect. The problem is solved in an exact formulation under the linear theory of elasticity for piecewise-homogeneous media. An approximate solution is found by the mesh approach using the concept of a base scheme. For stresses _ij , the shape and size of the edge effect zone are studied depending on the ratio of Young's moduli of the composite components, which varies over a wide range     

Micro-stress prediction in composite laminates with high stress gradients

The objective of this research is to develop a macroscopic theory, which can provide the connection between macro-mechanics and micro-mechanics in characterizing the micro-stress of composite laminates in regions of high macroscopic stress gradients. The micro-polar theory, a class of higher-order elasticity theory, of composite laminate mechanics is implemented in a well-known Pipes–Pagano free edge boundary problem. The micro-polar homogenization method to determine the micro-polar anisotropic effective elastic moduli is presented. A displacement-based finite element method based on micro-polar theory in anisotropic solids is developed in analyzing composite laminates. The effects of fiber volume fraction and cell size on the normal stress along the artificial interface resulting from ply homogenization of the composite laminate are also investigated. The stress response based on micro-polar theory is compared with those deduced from the micro-mechanics and classical elasticity theory. Special attention of the investigation focuses on the stress fields near the free edge where the high macro-stress gradient occurs. The normal stresses along the artificial interface and especially, the micro-stress along the fiber/matrix interface on the critical cell near the free edge where the high macro-stress gradient detected are the focus of this investigation. These micro-stresses are expected to dominate the failure initiation process in composite laminate. A micro-stress recovery scheme based on micro-polar analysis for the prediction of interface micro-stresses in the critical cell near the free edge is found to be in very good agreement with “exact” micro-stress solutions. It is demonstrated that the micro-polar theory is able to capture the micro-stress accurately from the homogenized solutions.     

The Edge Effect in a Laminated Composite with a Transversally Isotropic Filler

The edge effects in a laminated composite with a regular structure are considered. The equations of the linear theory of elasticity of piecewise-homogeneous media are used as a mathematical and mechanical model. An approximate solution is obtained on the basis of the mesh approach. An example of solving a specific problem is given and the results are analyzed     

A layered composite containing a crack in its lower layer loaded by a rigid stamp

The elastostatic plane problem of a layered composite containing an internal or edge crack perpendicular to its boundaries in its lower layer is considered. The layered composite consists of two elastic layers having different elastic constants and heights and rests on two simple supports. Solution of the problem is obtained by superposition of solutions for the following two problems: The layered composite subjected to a concentrated load through a rigid rectangular stamp without a crack and the layered composite having a crack whose surface is subjected to the opposite of the stress distribution obtained from the solution of the first problem. Using theory of Elasticity and Fourier transform technique, the problem is formulated in terms of two singular integral equations. Solving these integral equations numerically by making use of Gauss–Chebyshev integration, numerical results related to the normal stress σ_x(0,y), the stress-intensity factors, and the crack opening displacements are presented and shown graphically for various dimensionless quantities.     

材料非线性对复合材料层合板热自由边界效应的影响

本文用准三维有限元法研究了材料非线性对复合材料层合板热自山边界效应的影响,给出了修正型Hahn-Tsai非线性应力-应变关系的三维形式。由本文非线性分析方法得到的层间应力与以往由线性分析方法得到的层间应力做了比较,结果表明:材料非线性能显著降低层间剪应力的集中程度,但对层间正应力影响不太明显。     

Edge effect in a laminated composite with longitudinally compressed laminas

The paper is concerned with the determination of edge effect zone in a laminated composite with laminas subject to longitudinal compression. The dependence of the maximum decay length on the ratio between the period of external loading and the structure parameter is studied. The load period depends on the number of unloaded laminas. The decay of the edge effect is analyzed by numerically solving a boundary-value problem of elasticity for piecewise-homogeneous materials and using a quantitative decay criterion for the near-edge normal stresses __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 89–96, August 2006.     

Analysis of mode Ⅲ crack perpendicular to the interface between two dissimilar strips

The present work is concerned with the problem of mode Ⅲ crack perpendicular to the interface of a bi-strip composite. One of these strips is made of a functionally graded material and the other of an isotropic material, which contains an edge crack perpendicular to and terminating at the interface. Fourier transforms and asymptotic analysis are employed to reduce the problem to a singular integral equation which is numerically solved using Gauss-Chebyshev quadrature formulae. Furthermore, a parametric study is carried out to investigate the effects of elastic and geometric characteristics of the composite on the values of stress intensity factor.     

Thermal shock analyses for a strip containing an infinite row of periodically distributed cracks normal to its edge

The transient thermal stress problem of a semi-infinite plate containing an infinite row of periodically distributed cracks normal to its edge is investigated in this paper. The elastic medium is assumed to be cooled suddenly on the crack-containing edge. By the superposition principle, the formulation leads to a mixed boundary value problem, with the negating tractions arisen from the thermal stresses for a crack-free semi-infinite plate. The resulting singular integral equation is solved numerically. The effects on the stress intensity factors due to the presence of periodically distributed cracks in a semi-infinite plate are illustrated. For both the edge crack and the embedded crack arrays, the stress intensity factors increase, due to the reduction of the shielding effect, as the stacking cracks are more separated. For the case of embedded crack array, one has the further conclusion that the stress intensity factors decline as the crack array shifts from the plate edge.     

Edge Effects in a Composite Weakly Reinforced with Fibers of Rectangular Cross Section

This paper deal with the edge effect in a composite weakly reinforced with fibers of rectangular cross section and subjected to biaxial uniform loading. The edge effects due to the difference between Poisson's ratios of the composite components are studied. Numerical results are presented     

Influence of the Inclusion Shape on the Edge Effects in a Ribbon-Reinforced Composite

The elastic state and edge-effect zones in a weakly ribbon-reinforced composite under plane strain and uniform loading are determined. It is studied how the shape factor of the ribbon affects the nature of edge effects     

Influence of a thin isotropic coating on the edge effect zone in isotropic and transversely isotropic materials

The effect of a thin isotropic coating on the edge effect zone in a representative element of a coated material is examined. Isotropic and transversely isotropic materials are considered. The transversely isotropic material has the elastic properties of unidirectional glass-fiber-reinforced plastic. The decay of the edge effect in the directions perpendicular to the coating plane and to the plane of isotropy is studied. A boundary-value problem of elasticity for piecewise-homogeneouse orthotropic bodies and a quantitative edge effect decay criterion for normal stresses are used as a design model. The problem is solved using the finite-difference method and base schemes. The results of evaluation of the edge effect zone in homogeneous and inhomogeneous materials are presented. It is shown that the presence of a thin isotropic coating blocks the edge effect, that is, decreases the edge effect zone in both isotropic and transversely isotropic materials __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 61–67, December 2007.     

Determination of the edge-effect zone with the help of the holographic MoirÉ method

In parts made from materials with a distinct anisotropy, it is necessary to take into account edge effects which, as is well known [1], can slowly decay away from the edge. The size of the edge-effect zone in composite materials has been estimated many times and, in addition, the problem has been solved theoretically and experimentally [1–3]. In this work such estimates are made with the help of the method of holographic interferometry.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 132–135, September–October, 1986.     

功能梯度界面相对周期分布纤维增强复合材料反平面剪切的影响

基于复变函数理论和边界配点法,探索了功能梯度界面相在周期均匀分布纤维增强复合材料反平面剪切问题中所起的作用。由于纤维在复合材料基体中的周期分布是均匀的,将其简化成含一功能梯度界面相夹杂的方形单胞。采用分层均匀化方法,将功能梯度界面相离散成K层界面层。当K足够大时,每个界面层可视为匀质材料,同时计算得到的复合材料宏观性能趋于定值。根据单胞内的基体、界面相和夹杂的几何外形特点,分别给出复势函数的级数形式,这些复势函数在各组分的相邻界面应满足连续性条件,在单胞的外边界应满足周期性边界条件和远场加载条件,从而确定复势函数中的待定系数,进而根据平均场理论确定复合材料有效模量。主要探讨了夹杂体积分数、各组分模量、功能梯度界面相的模量渐变形式等因素对纤维增强复合材料性能的影响。结果表明：不管基体模量相对于夹杂模量是大还是小,都有对应的界面相模量渐变形式可使夹杂周围的等效应力集中系数减小;另外还发现仅当夹杂模量较大时,功能梯度界面相模量的变化方式对复合材料有效模量产生不可忽视的影响。     

Failure instability of a debonded interface in the presence of a main crack

The problem of fracture initiating from an edge crack in a nonhomogeneous beam made of two dissimilar linear elastic materials that are partially bonded along a common interface is studied by the strain energy density theory. The beam is subjected to three-point bending and the unbonded part of the interface is symmetrically located with regard to the applied loading. The applied load acts on the stiffer material, while the edge crack lies in the softer material. Fracture initiation from the tip of the edge crack and global instability of the composite beam are studied by considering both the local and global stationary values of the strain energy density function, dW/dV. A length parameter l defined by the relative distance between the maximum of the local and global minima of dW/dV is determined for evaluating the stability of failure initiation by fracture. Predictions on critical loads for fracture initiation from the tip of the edge crack, crack trajectories and fracture instability are made. In the analysis the load, the length of the edge crack and the length and position of the interfacial crack remained unchanged. The influence of the ratio of the moduli of elasticity of the two materials, the position of the edge crack and the width of the stiffer material on the local and global instability of the beam was examined. A general trend is that the critical load for crack initiation and fracture instability is enhanced as the width and the modulus of elasticity of the stiffer material increase. Thus, the stiffer material acts as a barrier in load transfer.     

Antiplane internal crack normal to the edge of a functionally graded piezoelectric/piezomagnetic half plane

This paper deals with the antiplane magnetoelectroelastic problem of an internal crack normal to the edge of a functionally graded piezoelectric/piezomagnetic half plane. The properties of the material such as elastic modulus, piezoelectric constant, dielectric constant, piezomagnetic coefficient, magnetoelectric coefficient and magnetic permeability are assumed in exponential forms and vary along the crack direction. Fourier transforms are used to reduce the impermeable and permeable crack problems to a system of singular integral equations, which is solved numerically by using the Gauss-Chebyshev integration technique. The stress, electric displacement and magnetic induction intensity factors at the crack tips are determined numerically. The energy density theory is applied to study the effects of nonhomogeneous material parameter β, edge conditions, location of the crack and load ratios on the fracture behavior of the internal crack.     

Opening‐Mode Crack in a Unidirectional Composite Material

The plane static elastic problem of stress concentration in a unidirectional discrete infinite composite weakened by fiber breaks on a line normal to the reinforcement direction (an analog of the Griffith problem of elasticity theory) is considered. The composite is subjected to uniform stresses at infinity, and the crack edges are loaded symmetrically by the normal pressure. The problem reduces to constructing a polynomial with known values at the points of fiber breaks. The stress distribution along the line of breaks is obtained in the form of a fractional rational function of fiber number.     

Symmetrical contact problem of a thick transversely isotropic plate

The symmetrical problem of the contact between a spherical indenter and a thick transversely isotropic plate is solved using the techniques of the Hankel transform. Solutions are written as the sums of the associated half-space solution and plate thickness effect terms. The normalized normal contact stress and the surface radial stress are obtained and calculated numerically for both composite materials and metallic substances. The example materials give both real and complex characteristic roots. A general method of calculation is described to determine the maximum tensile radial stress and the maximum compressive normal contact stress.The plate thickness effects on the contact stress and on the tensile radial stress are shown to be increasingly important with decreasing thickness. The effects that the material anisotropy has on the magnification of the contact stress and the maximum tensile radial stress are clearly revealed in the numerical results presented. The effects of material anisotropy for the composite are compared to those for the metallic substances.     

The 3-D stress intensity factor for a half plane crack in a transversely isotropic solid due to the motion of loads on the crack faces

Three-dimensional analysis of a half plane crack in a transversely isotropic solid is performed. The crack is subjected to a pair of normal point loads moving in a direction perpendicular to the crack edge on its faces. Transform methods are used to reduce the boundary value problem to a single integral equation that can be solved by the Wiener-Hopf technique. The Cagniard-de Hoop method is employed to invert the transforms. An exact expression is derived for the mode I stress intensity factor as a function of time and position along the crack edge. Some features of the solution are discussed through numerical results. The project supported by the Guangdong Provincial Natural Science Foundation and the Science Foundation of Shantou University     

刚性圆形压头作用下热电材料半平面的无摩擦接触问题

热电材料可以将热能转化为电能,反之亦然,这一优良的性质将有助于研发更具成本效益的设备和器件。本文研究了刚性圆形压头作用在热电材料半平面的无摩擦接触问题。假定压头为电导体、热导体,且压头压入深度及与材料的接触区域宽度未知。首先求解电场和温度场,利用傅里叶变换得到了电势函数、温度、电流密度和能量通量的解析表达式。然后求解弹性场,利用积分变换和边界条件,将该热弹性接触问题转化为第一类奇异积分方程并数值求解。数值结果讨论了压头半径和热电载荷对法向接触应力、电流强度因子和能量通量强度因子的影响。结果表明,对于圆压头,热电材料的法向电流密度、法向能量通量在接触边缘表现出奇异性,而表面法向接触应力在接触边缘为零。本文建立的研究模型有助于更深层次的了解热电材料的接触行为。