Dynamic crack growth along a polymer composite-Homalite interface

Dynamic crack growth along the interface of a fiber-reinforced polymer composite-Homalite bimaterial subjected to impact shear loading is investigated experimentally and numerically. In the experiments, the polymer composite-Homalite specimens are impacted with a projectile causing shear dominated interfacial cracks to initiate and subsequently grow along the interface at speeds faster than the shear wave speed of Homalite. Crack growth is observed using dynamic photoelasticity in conjunction with high-speed photography. The calculations are carried out for a plane stress model of the experimental configuration and are based on a cohesive surface formulation that allows crack growth, when it occurs, to emerge as a natural outcome of the deformation history. The effect of impact velocity and loading rate is explored numerically. The experiments and calculations are consistent in identifying discrete crack speed regimes within which crack growth at sustained crack speeds is possible. We present the first conclusive experimental evidence of interfacial crack speeds faster than any characteristic elastic wave speed of the more compliant material. The occurrence of this crack speed was predicted numerically and the calculations were used to design the experiments. In addition, the first experimental observation of a mother-daughter crack mechanism allowing a subsonic crack to evolve into an intersonic crack is documented. The calculations exhibit all the crack growth regimes seen in the experiments and, in addition, predict a regime with a pulse-like traction distribution along the bond line.     

Mechanics and fracture of crack tip deformable bi-material interface

Novel interface deformable bi-layer beam theory is developed to account for local effects at crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable sub-layers with consideration of crack tip deformation. Unlike the sub-layer model in the literature in which the crack tip deformations under the interface peel and shear stresses are ignored and thus a “rigid” joint is used, the present study introduces two interface compliances to account for the effect of interface stresses on the crack tip deformation which is referred to as the elastic foundation effect; thus a flexible condition along the interface is considered. Closed-form solutions of resultant forces, deformations, and interface stresses are obtained for each sub-layer in the bi-layer beam, of which the local effects at the crack tip are demonstrated. In this study, an elastic deformable crack tip model is presented for the first time which can improve the split beam solution. The present model is in excellent agreements with analytical 2-D continuum solutions and finite element analyses. The resulting crack tip rotation is then used to calculate the energy release rate (ERR) and stress intensity factor (SIF) of interface fracture in bi-layer materials. Explicit closed-form solutions for ERR and SIF are obtained for which both the transverse shear and crack tip deformation effects are accounted. Compared to the full continuum elasticity analysis, such as finite element analysis, the present solutions are much explicit, more applicable, while comparable in accuracy. Further, the concept of deformable crack tip model can be applied to other bi-layer beam analyses (e.g., delamination buckling and vibration, etc.).     

Analysis of transient wave scattering from an inclusion with an unilateral smoothly contact interface by BEM

Summary A 2D time-domain Boundary Element Method (BEM) is applied to solve the problem of transient scattering of plane waves by an inclusion with a unilateral smooth contact interface. The incident wave is assumed strong enough so that localized separations take place along the interface. The present problem is indeed a nonlinear boundary value problem since the mixed boundary conditions involve unknown intervals (separation and contact regions). In order to determine the unknown intervals, an iterative technique is developed. As an example, we consider the scattering of plane waves by the cross section of a circular cylinder embedded in an infinite solid. Numerical results for the near field solutions are presented. The distortion of the response waves and the variation of the interface states are discussed. The financial support by the China National Natural Science Foundation under Grant No. 19872001 and No. 59878004 is gratefully acknowledged. The second author is also grateful to the support of the National Science Fund for Distinguished Young Scholars under Grant No. 10025211.     

Interaction of a screw dislocation with a semi-infinite interfacial crack in a magneto-electro-elastic bi-material

The interaction between a screw dislocation and a semi-infinite interfacial crack in a transversely isotropic magneto-electro-elastic bi-material is investigated. The dislocation line is perpendicular to the isotropic basal plane of the bi-material. The elastic and electromagnetic fields induced by the dislocation are obtained through the use of the complex variable method together with the superposition scheme. The stress, electric displacement and magnetic intensity factors as well as the image exerted on the dislocation are given explicitly. We find that the intensity factors are expressed in terms of the so-called effective materials and the radial component of the image force is only dependent on the elastic modulus of the material with the dislocation. As an illustrative example, the bi-material that consists of piezoelectric and piezomagnetic phases is analyzed.     

基于有限断裂法和比例边界有限元法的裂纹扩展模拟

基于有限断裂法和比例边界有限元法提出了一种裂缝开裂过程模拟的数值模型。采用基于有限断裂法的混合断裂准则作为起裂及扩展的判断标准,当最大环向应力和能量释放率同时达到其临界值时,裂缝扩展。结合多边形比例边界有限元法,可以半解析地求解裂尖区域附近的应力场和位移场,在裂尖附近无需富集即可获得高精度的解。计算能量释放率时,只需将裂尖多边形内的裂尖位置局部调整,无需改变整体网格的分布,网格重剖分的工作量降至最少。裂缝扩展步长通过混合断裂准则确定,避免了人为假设的随意性,并可以实现裂缝变步长扩展的模拟,更符合实际情况。通过对四点剪切梁的复合型裂缝扩展过程的模拟,对本文模型进行了验证,并应用于重力坝模型的裂缝扩展模拟,计算结果表明,本文提出的模型简单易行且精度较高。     

Interface crack between two shear deformable elastic layers

An improved method based on the first-order shear deformable plate theory is developed to calculate the energy release rate and stress intensity factor for a crack at the interface of a bi-layer structure. By modeling the uncracked region of the structure as two separate Reissner-Mindlin plates bonded perfectly along the interface, this method is able not only to take into account the shear deformation in the cracked region, but also to capture the shear deformation in the uncracked region of the structure. A closed form solution of energy release rate and mode decomposition at the interface crack is obtained for a general loading condition, and it indicates that the energy release rate and stress intensity factor are determined by two independent loading parameters. Compared to the approach based on the classical plate theory, the proposed method provides a more accurate prediction of energy release rate as well as mode decomposition. The computational procedures introduced are relatively straightforward, and the closed form solution can be used to predict crack growth along the layered structures.     

A general solution method for an anti-plane shear crack dynamically accelerating along a bimaterial interface

We develop a general solution method for a dynamically accelerating crack under anti-plane shear conditions along the interface between two different homogeneous isotropic elastic materials. The crack is initially at rest, and after loading is applied the crack-tip speed which may accelerate up to the shear wave speed of the more compliant material. The analysis includes an exact, closed-form expression for the stress intensity factor for an arbitrary time-dependent crack-face traction, as well as expressions for computing the crack-face displacements and the stress in front of the crack. We also present some numerical examples for fixed loads and for loads moving with the crack tip, using a stress intensity factor fracture criterion, in order to examine the predicted effect of material mismatch on interfacial fracture.     

Comparison of energy release rate and minimum strain energy density for potential failure of composite with bi-material interface

An analytical method is developed to describe the fields of stress and displacement in a bi-material strip specimen with an edge interfacial crack. All of the basic governing equations, boundary conditions on crack surfaces and conditions of continuity along the interface are satisfied by the eigenfunction expansion method. The other boundary conditions are satisfied by the generalized variational principle. The stress intensity factors are calculated for determining the energy release rate and minimum strain energy density factor S_min that is used the strain energy density criterion. Problems with oscillatory singularity and contact zone are discussed. Not only the effects of bi-material modulus ratio, thickness ratio, Poisson's ratio and crack length to S_min, but also the influences of bi-material modulus ratio, thickness ratio to phase angle are presented. Among these parameters, particular situations where S_min become jeopardously high and lead to failure are discussed.     

An analytically-numerical approach for the analysis of an interface crack with a contact zone in a piezoelectric bimaterial compound

An interface crack with an artificial contact zone at the right-hand side crack tip between two dissimilar finite-sized piezoelectric materials is considered under remote mixed-mode loading. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical loads, the stress intensity factors at the singular points are obtained. As a particular case of this solution, the contact zone model (in Comninou’s sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are derived. The dependencies of the contact zone lengths on external load coefficients are illustrated in graphical form. For a particular case of a short crack with respect to the dimensions of the bimaterial compound, the numerical results are compared to the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.     

A penny-shaped interface crack between a functionally graded piezoelectric layer and a homogeneous piezoelectric layer

The problem of a penny-shaped interface crack between a functionally graded piezoelectric layer and a homogeneous piezoelectric layer is investigated. The surfaces of the composite structure are subjected to both mechanical and electrical loads. The crack surfaces are assumed to be electrically impermeable. Integral transform method is employed to reduce the problem to a Fredholm integral equation of the second kind. The stress intensity factor, electric displacement intensity factor and energy release rate are derived, some typical numerical results are plotted graphically. The effects of electrical loads, material nonhomogeneity and crack configuration on the fracture behaviors of the cracked composite structure are analyzed in detail.     

SH 波入射时半空间界面裂纹与圆形衬砌的相互作用

利用复变函数和Green函数法研究了双相介质半空间界面裂纹及界面附近圆形衬砌对SH 波的散射与动应力集中问题。首先,采用映像思想构造满足自由边界条件的散射波表达式,进而求解所需的Green函数;其次,采用裂纹切割技术构造裂纹,并根据连续性条件建立了求解该问题的无穷代数方程组;最后,给出了不同入射波数时界面裂纹与衬砌的相互作用。结果表明,裂纹的存在显著放大了衬砌界面的动应力集中。     

Opening and contact zones of an interface crack in a piezoelectric bimaterial under combined compressive-shear loading

A plane problem for a tunnel electrically permeable interface crack between two semi-infinite piezoelectric spaces is studied. A remote mechanical and electrical loading is applied. Elastic displacements and potential jumps as well as stresses and electrical displacement along the interface are presented using a sectionally holomorphic vector function. It is assumed that the interface crack includes zones of crack opening and frictionless contact. The problem is reduced to a combined Dirichlet–Riemann boundary value problem which is solved analytically. From the obtained solution, simple analytical expressions are derived for all mechanical and electrical characteristics at the interface. A quite simple transcendental equation, which determines the point of separation of open and close sections of the crack, is found. For the analysis of the obtained results, the main attention is devoted to the case of compressive-shear loading. The analytical analysis and numerical results show that, even if the applied normal stress is compressive, a certain crack opening zone exists for all considered loading values provided the shear field is present. It is found that the shear stress intensity factor at the closed crack tip and the energy release rates at the both crack tips depend very slightly on the magnitude of compressive loading.     

Intersonic crack propagation in homogeneous media under shear-dominated loading: theoretical analysis

The mechanics of cohesive failure under mixed-mode loading is investigated for the case of a steadily propagating subsonic and intersonic dynamic crack subjected to a follower tensile and shear distributed load. The cohesive failure model chosen in this study is rate independent but accounts for the coupling between normal and tangential damage. Special emphasis is placed here on mixed-mode cases with predominantly shear loading. The analysis shows that the size of the mixed-mode cohesive zone is smaller than that obtained in the pure shear case. The relative extent of the shear and tensile cohesive damage zones depends on the crack speed and the mode mixity. In the intersonic regime, the failure process takes place exclusively in shear, even under remote mixed-mode loading conditions.     

Crack growth of an interface crack between two dissimilar magneto-electro-elastic materials under anti-plane mechanical and in-plane electric magnetic impact

This paper examines the dynamic response of an interface crack between two dissimilar magneto-electro-elastic materials subjected to the mechanical and electric magnetic impacts. The magneto-electric impermeable boundary conditions are adopted. Laplace and Fourier transforms and dislocation density functions are employed to reduce the mixed boundary value problem to Cauchy singular integral equations in Laplace transform domain, which are solved numerically. Lots of numerical results are given graphically in time domain. The effects of electric impact loading and magnetic impact loading on dynamic energy density factors are discussed. Crack growth and propagation is predicted. The study of this problem is expected to have applications to the investigation of dynamic fracture properties of magneto-electro-elastic materials with cracks.     

Model of the fracture process zone at the tip of a crack reaching the nonsmooth interface between elastic media

The paper is concerned with the fracture process zone at the tip of a crack at the nonsmooth interface between isotropic elastic media. A plane symmetric problem is formulated. The zone is modeled by lines of discontinuity of the normal displacement at the interface. The exact solution of the elastic problem is found by the Wiener-Hopf method Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 13–22, October 2008.     

Exact solution of plane isolated crack normal to a bimaterial interface of infinite extent

This paper presents an exact solution for the transverse interface crack in the plane strain case. The crack is perpendicular to the interface and in one material. The exact complex stress functions are first obtained with some unknown constants. The satisfactions of all boundary conditions are then checked, the condition at infinity is considered and the unknown constants are determined. Further study may focus on the case with different shear moduli and the influence of the large deformation.The English text was polished by Keren Wang.     

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.     

Investigation of the behavior of a Griffith crack at the interface of a layer bonded to a half plane using the Schmidt method for the opening crack mode

In this paper, the behavior of a Griffith crack at the interface of a layer boned to a half plane subjected to a uniform tension is investigated by use of the Schmidt method under the assumptions that the effect of the crack surface overlapping very near the crack tips is negligible and also there is a sufficiently large component of mode-I loading so that the crack essentially remains open. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the thickness of the material layer and the materials constants upon the stress intensity factor of the crack. As a special case in our solution, we also give the solution of the ordinary crack in homogeneous materials. Contrary to the previous solution of the interface crack problem, it is found that the stress singularities of the present interface crack solution are similar with ones for the ordinary crack in homogeneous materials.     

Time-harmonic P-waves engulfing a rectangular limited-permeable crack in piezoelectric medium: Energy density and energy release

The dynamic behavior of a limited-permeable rectangular crack in a transversely isotropic piezoelectric material is impinged by to a P-wave. The generalized Almansi theorem and the Schmidt method are used to determine the stress intensity factor and energy density factor as the primary fracture criterion of failure. The mixed boundary value problem entails the evaluation of the appropriate crack edge stress singularities that are characteristics of the fundamental functions. The stress and electric displacement intensity factors are also used to find the energy release rate that can be computed numerically and compared with the results corresponding to those of the stress intensity factor, and energy density factor. Graphical presentation shows that the energy release rate is always negative for the boundary conditions considered while the energy density factors always remain positive. Under certain conditions, the stress and electric displacement intensity factors can be negative and subject to physical limitations. Piezoelectric material boundary value problem solutions should therefore be qualified by the application of failure criteria by fracture of otherwise, particularly when the mechanical and electrical energy can release by creating free surface at the macroscopic and microscopic scales. Negative energy release rate found for the piezoelectric medium in this work can be a case in point.Positive definiteness of the energy density factor can be applied to mutliscale fracture. This is not true for the stress intensity factor nor the energy release rate. Hence, crack initiation behavior for the permittivity of a rectangular crack due to the wave propagation effects may be studied. In particular, the initiation of micro-cracks may be identified with certain critical stress wave frequency band. Negative stress intensity factor may not enhance macrocracking but it does not exclude microcrack initiation.