2D dynamic analysis of cracks and interface cracks in piezoelectric composites using the SBFEM

The dynamic stress and electric displacement intensity factors of impermeable cracks in homogeneous piezoelectric materials and interface cracks in piezoelectric bimaterials are evaluated by extending the scaled boundary finite element method (SBFEM). In this method, a piezoelectric plate is divided into polygons. Each polygon is treated as a scaled boundary finite element subdomain. Only the boundaries of the subdomains need to be discretized with line elements. The dynamic properties of a subdomain are represented by the high order stiffness and mass matrices obtained from a continued fraction solution, which is able to represent the high frequency response with only 3–4 terms per wavelength. The semi-analytical solutions model singular stress and electric displacement fields in the vicinity of crack tips accurately and efficiently. The dynamic stress and electric displacement intensity factors are evaluated directly from the scaled boundary finite element solutions. No asymptotic solution, local mesh refinement or other special treatments around a crack tip are required. Numerical examples are presented to verify the proposed technique with the analytical solutions and the results from the literature. The present results highlight the accuracy, simplicity and efficiency of the proposed technique.     

Moiré-numerical hybrid analysis of cracks in orthotropic media

The effectiveness of a new photomechanical-numerical hybrid method for evaluating stress intensity factors (SIFs) in orthotropic composites is demonstrated. Reliable results are obtained from few moiré-measured displacements and they originate well away from the crak tip. The method is illustrated for the case of a uniaxially loaded orthotropic glass/epoxy composite containing a central, transverse crack.     

任意多孔多裂纹板的裂纹闭合接触分析

基于摩擦接触问题的数学规划解法，采用各向异性体平面弹性理论中的复势方法，建立了含多椭圆孔及裂纹群有限大各向异性板，在任意载荷作用下裂纹闭合或局部闭合问题的有效分析方法。通过在可能闭合的裂纹边界引入互补变量函数并将其展成Fourier级数形式，以Faber级数为工具，应用保角映射技术和最小二乘边界配点法，导出无卸载情况下裂纹面摩擦接触的线性互补模型，并通过算例验证了方法的有效性。数值结果表明，由于采用级数解描述板应力场和位移场，该方法具有较高的计算精度和效率，便于研究裂纹闭合对应力强度因子等断裂参数的影响。     

The evolution of caustics from holes to cracks

The method of caustics was used for the study of the evolution of stress concentration around a circular hole, which progressively changes in shape and becomes an elliptic hole, tending to an internal crack. The influence of the amount of ellipticity of the holes and their orientation relative to the axis of the applied external loads at infinity on the form of caustics created around the discontinuity was studied, as the elliptic holes tended to become internal cracks. A series of experiments with tension specimens containing small elliptic holes of any ellipticity and orientation was performed. Comparison of experimentally obtained caustics with theory yielded a good agreement of both results. Finally, the use of small elliptic holes drilled all over a biaxial stress field for the determination of the individual principal stresses and the principal directions at the area of the holes was outlined.     

Electromagnetoelastic problem for a plate with holes and cracks

The methods of complex potentials, conformal mappings, and least squares are used to solve a plane problem of electromagnetoelasticity for a plate with holes and cracks. Numerical solutions are found for a ring and a disk with a crack under internal concentrated forces or electric charges as well as electric potentials applied to the boundaries     

正交异性平面内的椭圆孔或裂纹系问题

应用Ｆａｂｅｒ级数展开和各向异性体平面问题复应力函数的方法，对于含有任意个椭圆孔或裂纹的正交异性平面，给出了孔周应力场解或孔附近裂纹应力强度因子解，其特例与前人结果一致．     

Dynamic penny-shaped cracks in multilayer sandwich composites

A point force method is proposed for obtaining the dynamic elastic response of a multilayer sandwich composite in the presence of a penny-shaped crack under a harmonic loading. The sandwich composite is a multilayered solid whose lower half is the mirror image of the upper half with the center plane as the mirror. The crack is lying on the mirror plane of the composite. The solution of the mode I dynamic crack problem is formulated by integrating the Green’s function of a time-harmonic surface normal point force over the crack surface with an unknown point force distribution. The dual integral equations of the unknown point force distribution are established by considering the boundary conditions, which can be reduced to a Fredholm integral equation of the second kind. A complete solution of the crack problem under consideration can be obtained by solving this Fredholm integral equation. It will be shown that the results obtained by this approach are the same as some existing solutions.     

Interaction between parallel cracks in layered composites

The plane strain problem of a multi-layered composite with parallel cracks is considered. The main objective of this paper is to study the interaction between parallel and collinear cracks. The problem is formulated in terms of a set of simultaneous singular integral equations which are solved numerically. The effect of material properties on the interaction between cracks is also demonstrated.     

Fracture mechanics of composites in compression along cracks

Conclusions Thus, based on the concepts in [5–7], the beginning of fracture of a composite in compression along a plane containing cracks can be represented as follows. If the material has not fractured already, then local fracture necessarily occurs next to the cracks at a value of compressive load corresponding to surface instability. Here, the upper limit of the ultimate theoretical strength corresponds to the minimum value of the shear modulus. This fracture mechanism can be seen in composites reinforced with high-modulus fibers. It should be noted that fracture can also take place on free surfaces of the material at these load values as a result of surface instability [1]. If the area of the free surface (lateral surface) of the material is substantially less than the total area of the cracks present throughout the volume of the material, then obviously the cracks will be the deciding factor in the fracture mechanism. The above-noted conclusions and quantitative abalysis were made only for a linearly elastic, orthotropic material with a high shear stiffness (brittle fracture), while the general results obtained in the present article also pertain to nonlinearly elastopiastic models. The results can be refined for more complex models. It must also be noted that the theoretical ultimate strength may be reduced substantially if the interaction of cracks located in parallel planes during instability is considered. This feature of the fracture mechanism was noted in note 6 in the article [6]. Composite materials generally have a fairly large number of cracks in planes and surfaces along the reinforcing elements. In connection with this, for composites it is best to allow for interaction of cracks located in parallel planes, which in turn should lead to a substantial reduction in the theoretical ultimate strength value obtained in the present article.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 18, No. 6, pp. 3–9, June, 1982.     

反平面圆形夹杂和多圆孔多裂纹相互作用问题

动用复变函数及积分方法方法求解了反平面圆形夹杂和多圆孔多裂纹相互作用问题。为解决该问题,建立了两种类型的基本解。利用叠加原理和所得的基本解没圆孔和裂纹表面取待定的基本解密度函数,可得一组Ｆｒｅｄｈｏｌｍ积分方程,通过积分方程组的数值求解,可以得到密度函数的离散值,进而得到应力强度因子。     

Energy release rate for cracks in ideal composites

Infinitesimal plane deformations of ideal fiber-reinforced composites with elastic shearing stress response are considered. The fibers are straight and parallel, and there is a straight crack perpendicular to the fibers. A general expression for the energy release rate per unit length of crack advance is obtained. Explicit expressions in terms of the body geometry and loading are obtained for three special classes of body shapes: bodies symmetrical about a fiber, bodies bounded on the cracked side by a fiber, and bodies bounded on the opposite side by a fiber. The results also apply to cracks parallel to the fibers, and to cracks in compressible materials reinforced by two orthogonal families of inextensible fibers.     

Damping in ceramic matrix composites with matrix cracks

The paper presents an analytical solution capable of predicting the effect of matrix cracking in ceramic matrix composites (CMC) on damping. The cracking scenarios considered in the paper include through-the-thickness cracks and cracks terminating at the layer interfaces. The increase in damping associated with matrix cracking is mostly due to the frictional energy dissipation along the damaged fiber–matrix interfaces adjacent to the bridging cracks whose plane of propagation intersects the fiber axis. Damping increases with a higher density of matrix cracks. The loss factor is affected by the angle of the lamina relative to the direction of the applied load. The loss factor is also influenced by the frequency and magnitude of local dynamic stresses. Examples of distributions of the local loss factor along the axis of a CMC beam subject to pulsating loads of various frequencies are shown in the paper.     

Molecular modeling of cracks at interfaces in nanoceramic composites

Toughness in Ceramic Matrix Composites (CMCs) is achieved if crack deflection can occur at the fiber/matrix interface, preventing crack penetration into the fiber and enabling energy-dissipating fiber pullout. To investigate toughening in nanoscale CMCs, direct atomistic models are used to study how matrix cracks behave as a function of the degree of interfacial bonding/sliding, as controlled by the density of C interstitial atoms, at the interface between carbon nanotubes (CNTs) and a diamond matrix. Under all interface conditions studied, incident matrix cracks do not penetrate into the nanotube. Under increased loading, weaker interfaces fail in shear while stronger interfaces do not fail and, instead, the CNT fails once the stress on the CNT reaches its tensile strength. An analytic shear lag model captures all of the micromechanical details as a function of loading and material parameters. Interface deflection versus fiber penetration is found to depend on the relative bond strengths of the interface and the CNT, with CNT failure occurring well below the prediction of the toughness-based continuum He–Hutchinson model. The shear lag model, in contrast, predicts the CNT failure point and shows that the nanoscale embrittlement transition occurs at an interface shear strength scaling as _τs~_εf,CNTσCNT${\tau }_s~{\epsilon }_{f,CNT}{\sigma }_{CNT}$ rather than _τs~_σCNT${\tau }_s~{\sigma }_{CNT}$ typically prevailing for micron scale composites, where _εf,CNT${\epsilon }_{f,CNT}$ and _σCNT${\sigma }_{CNT}$ are the CNT failure strain and stress, respectively. Interface bonding also lowers the effective fracture strength in SWCNTs, due to formation of defects, but does not play a role in DWCNTs having interwall coupling, which are weaker than SWCNTs but less prone to damage in the outerwall.     

Problems of electromagnetoelasticity for a half-plane and a strip with holes and cracks

The methods of complex potentials, conformal mappings, Cauchy integrals, and least-squares are used to develop a method for determining the electromagentoelastic state (EMES) of a multiply connected half-plane, with the boundary conditions on the straight-line boundary satisfied exactly. The method underlies an approximate method for determining the EMES of a strip with arbitrarily arranged holes and cracks. The dependence of the EMES on the geometrical parameters of a strip with a circular hole or a crack is analyzed     

Integral equations for any configuration of curved cracks and holes in an elastic strip

Summary A method is presented to deal with the problems of an elastic strip weakened by cracks and holes of any configuration and geometry. The solution in terms of complex potentials is given by integrals over the cracks and holes with integrands expressed in terms of determined functions (Green) and an unknown density function (t). A singular integral equation for the complex density function (t) is derived for the problem. The appropriate Green's functions are derived from the solution for the problem of an uncracked strip subjected to a concentrated force or a dislocation. The integral equation is solved numerically for a strip in tension with an internal circular arc crack.

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Integralgleichungen für beliebige Konfigurationen von gekrümmten Rissen und Löchern in einer elastischen Scheibe

Übersicht Eine Methode wird dargestellt, welche zur Lösung von Problemen einer unendlichen elastischen Scheibe endlicher Breite mit Rissen und Löchern beliebiger Konfiguration und Geometrie geeignet ist. Die Lösung in Form von komplexen Spannungsfunktionen wurde mit Hilfe von Integralen über den Rissen und Löchern gegeben. Die Integranden wurden mit Hilfe von bekannten Funktionen (Green) und einer unbekannten Dichtefunktion (t) ausgedrückt, für welche eine singuläre Integralgleichung abgeleitet wurde. Die geeigneten Greenschen Funktionen wurden von der Lösung des Problems der entsprechenden Scheibe ohne Risse, welche mit einer Einzelkraft belastet wird oder eine Versetzung enthält, hergeleitet. Die singuläre Integralgleichung wurde numerisch für eine Scheibe endlicher Breite mit einem Kreisbogenriß im einachsigen Zugspannungsfeld gelöst.

     

The effective properties of composites with fiber-end cracks

The paper describes use of self-consistent finite element method (SCFEM) for predicting effective properties of fiber composite with partially debonded interface. The effective longitudinal Young's modulus and shear modulus for unidirectional fiber reinforced composites with fiber-end cracks are calculated. Numerical results show that the effective properties are considerably influenced by the fiber-end cracks. The effects of microstructural parameters, such as fiber volume fraction, modulus ratio of the constituents and fiber aspect, on the effective properties of the composites were discussed. The project supported by the National Natural Science Foundation of China