Interface crack extension at any constant speed in orthotropic or transversely isotropic bimaterials––I. General exact solutions

A semi-infinite crack along the interface of two dissimilar half-spaces extends under in-plane loading. Each half-space belongs to a class of orthotropic or transversely isotropic elastic materials, the crack can extend at any constant speed, and all six possible relations between the four body wave speeds are considered. A steady dynamic situation is treated, and exact full displacement fields derived. A key step is a factorization that produces, despite anisotropy, simple solution forms and compact crack speed-dependent functions that exhibit the Rayleigh and Stoneley speeds as roots. These roots are calculated for various representative bimaterials.Closed-form crack opening displacement gradient and interface stress fields are also derived from a general set of coupled singular integral equations. The equation eigenvalues can, depending on crack speed, be complex/imaginary conjugates, purely real, or zero. This suggests possibilities observed in other studies: oscillations and square-root singular behavior at the crack edge, non-singular behavior, singular behavior not of square-root order, and the radiation of displacement gradient discontinuities at crack speeds beyond the purely sub-sonic range.These possibilities are explored further in terms of two important special cases in Part II of this study [Int. J. Solids Struct., 39, 1183–1198].     

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.     

弹性波与单侧界裂纹相互作用问题的边界元法

措助边界元法设计了一种迭代修正方法来求解单侧界面裂纹模型与弹性波的相互作用问题,作为对算法的检验,用这种方法我们具体地分析了平面简谐弹性波对一个则界面裂纹的入射,给出了裂纹面的接触形态及应力场。     

In-plane wave motion and resonance phenomena in periodically layered composites with a crack

This paper investigates the transmission and propagation of two-dimensional (2D) time-harmonic plane waves in periodically multilayered elastic composites with a strip-like crack. The total wave field in the composite structure is represented as a sum of the incident wave field determined by the transfer matrix method and the scattered wave field described by integral representations in terms of the Green’s matrices and the crack-opening-displacements. A numerical scheme is developed to compute the wave propagation characteristics and the crack-characterizing quantities. The effects of the crack location and size as well as the angle of wave incidence are investigated using the averaged crack-opening-displacements and the stress intensity factors. Special attention of the paper is devoted to resonance wave motion and wave localization phenomena in a stack of periodical elastic layers weakened by a single strip-like crack. Numerical results are presented and discussed to reveal the usual and the resonant wave transmission by using the power-density vector and the energy streamlines in the vicinity of the crack. Wave localization due to interior and interface cracks is analyzed by considering the energy captured by a crack, and resonance induced crack growth is also discussed.     

Transition of mode II cracks from sub-Rayleigh to intersonic speeds in the presence of favorable heterogeneity

Understanding sub-Rayleigh-to-intersonic transition of mode II cracks is a fundamental problem in fracture mechanics with important practical implications for earthquake dynamics and seismic radiation. In the Burridge-Andrews mechanism, an intersonic daughter crack nucleates, for sufficiently high prestress, at the shear stress peak traveling with the shear wave speed in front of the main crack. We find that sub-Rayleigh-to-intersonic transition and sustained intersonic propagation occurs in a number of other models that subject developing cracks to intersonic loading fields. We consider a spontaneously expanding sub-Rayleigh crack (or main crack) which advances, along a planar interface with linear slip-weakening friction, towards a place of favorable heterogeneity, such as a preexisting subcritical crack or a small patch of higher prestress (similar behavior is expected for a small patch of lower static strength). For a range of model parameters, a secondary dynamic crack nucleates at the heterogeneity and acquires intersonic speeds due to the intersonic stress field propagating in front of the main crack. Transition to intersonic speeds occurs directly at the tip of the secondary crack, with the tip accelerating rapidly to values numerically equal to the Rayleigh wave speed and then abruptly jumping to an intersonic speed. Models with favorable heterogeneity achieve intersonic transition and propagation for much lower prestress levels than the ones implied by the Burridge-Andrews mechanism and have transition distances that depend on the position of heterogeneity. We investigate the dependence of intersonic transition and subsequent crack propagation on model parameters and discuss implications for earthquake dynamics.     

Dynamic crack deflection and penetration at interfaces in homogeneous materials: experimental studies and model predictions

We examine the deflection/penetration behavior of dynamic mode-I cracks propagating at various speeds towards inclined weak planes/interfaces of various strengths in otherwise homogeneous isotropic plates. A dynamic wedge-loading mechanism is used to control the incoming crack speeds, and high-speed photography and dynamic photoelasticity are used to observe, in real-time, the failure mode transition mechanism at the interfaces. Simple dynamic fracture mechanics concepts used in conjunction with a postulated energy criterion are applied to examine the crack deflection/penetration behavior and, for the case of interfacial deflection, to predict the crack tip speed of the deflected crack. It is found that if the interfacial angle and strength are such as to trap an incident dynamic mode-I crack within the interface, a failure mode transition occurs. This transition is characterized by a distinct, observable and predicted speed jump as well as a dramatic crack speed increase as the crack transitions from a purely mode-I crack to an unstable mixed-mode interfacial crack.     

Asymptotic solutions for buckling delamination induced crack propagation in the thin film-compliant substrate system

In a thin film-substrate system in-plane compressive stress is commonly generated in the film due to thermal mismatch in operation or fabrication process. If the stress exceeds a critical value, part of the film may buckle out of plane along the defective interface. After buckling delamination, the interface crack at the ends may propagate. In the whole process, the compliance of the substrate compared with the film plays an important role. In this work, we study a circular film subject to compressive stress on an infinitely thick substrate. We study the effects of compliance of the substrate by modeling the system as a plate on an elastic foundation. The critical buckling condition is formulated. The asymptotic solutions of post-buckling deformation and the corresponding energy release rate of the interface crack are obtained with perturbation methods. The results show that the more compliant the substrate is, the easier for the film to buckle and easier for the interface crack to propagate after buckling.     

Numerical analysis of macrocrack propagation along a bimaterial interface under dynamic loading processes

Advances in computing as well as measurement instrumentation have recently allowed for the investigation of a wider spectrum of physical phenomena in dynamic failure than previously possible. With increasing demand for specialized lightweight, high strength structures, failure of inhomogeneous solids has been receiving increased attention. Such inhomogeneous solids include structural composites such as bonded and sandwich structures, layered and composite materials as well as functionally graded solids. Many of such solids are composed of brittle constituents possessing substantial mismatch in wave speeds, and are bonded together with weak interfaces, which may serve as sites for catastrophic failure (Rosakis and Ravichandran (2000)).In the present study numerical analysis of macrocrack propagation along a bimaterial interface under dynamic loading processes is presented. A general constitutive model of elasto-viscoplastic damaged polycrystalline solids is developed within the thermodynamic framework of the rate type covariance structure with finite set of the internal state variables. A set of the internal state variables is assumed and interpreted such that the theory developed takes account of the effects as follows: (i) plastic non-normality; (ii) softening generated by microdamage mechanisms; (iii) thermomechanical coupling (thermal plastic softening and thermal expansion); (iv) rate sensitivity.To describe suitably the time and temperature dependent effects observed experimentally during dynamic loading processes the kinetics of microdamage has been modified. The relaxation time is used as a regularization parameter. By assuming that the relaxation time tends to zero, the rate independent elastic–plastic response can be obtained. The identification procedure is developed basing on the experimental observations. The finite difference method for regularized elasto-viscoplastic model is used. The edge-cracked bimaterial specimen is considered. In the initial configuration, the height of the specimen is equal to 30 cm, width is 12.5 cm and the length of the initial crack is equal to 2.5 cm. The length of the boundary over which impact is applied is equal to 5 cm, the rise time is fixed at 0.1 μs and the impact velocity is varied. The impact area is localized symmetrically or asymmetrically to the shorter axis of the specimen (symmetry axis of the cohesive band). Basing on the available data of recent experimental observation Rosakis et al. (1999) that have been carried out for relatively thin specimens both the plane stress and plane strain conditions are considered. The material of the specimen is AISI 4340 steel, while PMMA is the cohesive band, both modelled by thermo-elasto-viscoplastic constitutive equations with effects of isotropic hardening and softening generated by microdamage mechanisms and thermomechanical coupling. Fracture criterion based on the evolution of microdamage is assumed. Both, isothermal and adiabatic processes are considered.Particular attention is focused on the investigation of the interactions and reflections of stress waves and the influence of these waves on the propagation of macrocrack within the interface band. The propagation of the macroscopic crack within the material of the interface band for both symmetrical and asymmetrical impact cases has been investigated. It has been found that macrocrack-tip speeds vary from the shear wave speed to the dilatational wave speed of the material and is higher than the Rayleigh surface wave speed. This result is in accord with the experimental observations performed by Rosakis et al. (1999).     

An Interface Crack for a Functionally Graded Strip Sandwiched Between Two Homogeneous Layers of Finite Thickness

In this paper, the behavior of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness subjected to an uniform tension is resolved using a somewhat different approach, named the Schmidt method. The Fourier transform technique is applied and a mixed boundary value problem is reduced to two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surface. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. This process is quite different from those adopted in previous works. 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 cracks. It can be obtained that the results of the present paper are the same as ones of the same problem that was solved by the singular integral equation method. As a special case, when the material properties are not continuous through the crack line, an approximate solution of the interface crack problem is also given under the assumption that the effect of the crack surface interference very near the crack tips is negligible. Contrary to the previous solution of the interface crack, it is found that the stress singularities of the present interface crack solution are the same as ones of the ordinary crack in homogenous materials.     

Stress intensity factors for a moving cylindrical crack in a nonhomogeneous cylindrical layer in composite materials

Stresses are determined for a finite cylindrical crack that is propagating with a constant velocity in a nonhomogeneous cylindrical elastic layer, sandwiched between an infinite elastic medium and a circular elastic cylinder made from another material. The Galilean transformation is employed to express the wave equations in terms of coordinates that are attached to the moving crack. An internal gas pressure is then applied to the crack surfaces. The solution is derived by dividing the nonhomogeneous interfacial layer into several homogeneous cylindrical layers with different material properties. The boundary conditions are reduced to two pairs of dual integral equations. These equations are solved by expanding the differences in the crack surface displacements into a series of functions that are equal to zero outside the crack. The Schmidt method is then used to solve for the unknown coefficients in the series. Numerical calculations for the stress intensity factors were performed for speeds and composite material combinations.     

Assessment of filtered gas–solid momentum transfer models via a linear wave propagation speed test

The propagation speeds of linear waves in gas–solid suspensions depend strongly on the solids volume fraction and the wave frequency. The latter is due to gas–solid momentum transfer and allows a simple test on filtered gas–solid momentum transfer models. Such models may predict linear wave propagation speeds different from those obtained with the non-filtered model at wave frequencies higher than the filter frequency, but not at wave frequencies lower than the filter frequency.     

On the plastic wave speeds in rate-independent elastic–plastic materials with anisotropic elasticity

This paper presents a theoretical study of the speeds of plastic waves in rate-independent elastic–plastic materials with anisotropic elasticity. It is shown that for a given propagation direction the plastic wave speeds are equal to or lower than the corresponding elastic speeds, and a simple expression is provided for the bound on the difference between the elastic and the plastic wave speeds. The bound is given as a function of the plastic modulus and the magnitude of a vector defined by the current stress state and the propagation direction. For elastic–plastic materials with cubic symmetry and with tetragonal symmetry, the upper and lower bounds on the plastic wave speeds are obtained without numerically solving an eigenvalue problem. Numerical examples of materials with cubic symmetry (copper) and with tetragonal symmetry (tin) are presented as a validation of the proposed bounds. The lower bound proposed here on the minimum plastic wave speed may also be used as an efficient alternative to the bifurcation analysis at early stages of plastic deformation for the determination of the loss of ellipticity.     

The scattering of general SH plane wave by interface crack between two dissimilar viscoelastic bodies

The scattering of general SH plane wave by an interface crack between two dissimilar viscoelastic bodies is studied and the dynamic stress intensity factor at the crack-tip is computed. The scattering problem can be decomposed into two problems: one is the reflection and refraction problem of general SH plane waves at perfect interface (with no crack); another is the scattering problem due to the existence of crack. For the first problem, the viscoelastic wave equation, displacement and stress continuity conditions across the interface are used to obtain the shear stress distribution at the interface. For the second problem, the integral transformation method is used to reduce the scattering problem into dual integral equations. Then, the dual integral equations are transformed into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Finally, the singular integral equation is solved by Kurtz's piecewise continuous function method. As a consequence, the crack opening displacement and dynamic stress intensity factor are obtained. At the end of the paper, a numerical example is given. The effects of incident angle, incident frequency and viscoelastic material parameters are analyzed. It is found that there is a frequency region for viscoelastic material within which the viscoelastic effects cannot be ignored. This work was supported by the National Natural Science Foundation of China (No.19772064) and by the project of CAS KJ 951-1-20     

Crack tip shielding and anti-shielding effects of the imperfect interface in a layered piezoelectric sensor

The purpose of the present work is to study the effect of an imperfect interface on the fracture behavior of a layered piezoelectric sensor. For mathematical convenience, the problem is investigated under mode III-a simple case in fracture mechanics. Fracture analysis is performed by the methods of Fourier integral transform and Cauchy singular integral equation. Parametric studies on the numerical results of energy release rate reveal the crack tip shielding and anti-shielding effects of the imperfect interface. When the inclined angle of crack is less than 0.1π, the imperfect interface may shield the crack, however, when it is larger than 0.2π, the crack may be anti-shielded. If the distance between a crack tip and the imperfect interface is less than two times of the crack length, the shielding or anti-shielding effect is remarkable, and otherwise it is negligible. Finally, the crack tip shielding and anti-shielding effects of the mechanical imperfection are generally more remarkable than those of the dielectric imperfection.     

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。     

Dynamic stress concentration around two interacting coated nanowires with surface/interface effect

In coated nanowires, the surface/interface effects are particularly prominent due to a larger ratio of surface area to volume. In this paper, the effect of surface/interface stress on the macroscopic dynamic stress concentration around two nanowires under anti-plane shear waves is studied. The analytical solutions of displacements around the coated nanowires, in the coating layers, and inside the nanowires are expressed by wave function expansion method. The expanded mode coefficients are determined by satisfying the boundary conditions at the two interfaces around the nanowires. To accomplish the superposition of displacement fields, the addition theorem for cylindrical wave function is employed. Analyses show that the effect of interface properties on the dynamic stress is significantly related to the wave frequency, the material properties of nanowires and coating layers, and the relative position and distance between the two nanowires. If the wires and coating layers are softer, the dynamic stress decreases greatly, and the interface effect on the dynamic stress is also little. The results may be potentially useful for providing information on the mechanical properties and interactions among array of nanowires under different external mechanical stimulus.     

Sudden deceleration or acceleration of an intersonic shear crack

Sudden jumps in the crack tip velocity were revealed by numerical simulation (in both continuum/cohesive element and molecular dynamics approaches) and experiments for rapid shear cracking. The cracking velocity may accelerate from a sub-Rayleigh speed to the intersonic range, or from an intersonic speed to a higher one, when the reflected impact wave reloads the crack tip. On the other hand, the cracking velocity may decelerate from an intersonic speed to a lower one or recede to the sub-Rayleigh range when the fracture driving force declines. The velocity change encountered during intersonic cracking plays a different role from that in the acceleration or deceleration of a subsonic crack. A crack propagating at an intersonic speed would leave a shear wave trailing behind. When the crack decelerates or accelerates, the effect of the trailing wave will lead to a transition period from one steady-state solution of crack tip singularity to another. This investigation aims at quantifying these processes. The full field solution of an intersonic mode II crack whose speed changed suddenly from one velocity (intersonic or subsonic) to another (intersonic or subsonic) is given in closed form. The solution is facilitated via superposing a series of propagating crack problems that are loaded by dislocations to seal the unwanted crack-face sliding or by concentrated forces moving at various speeds to negate the crack-face traction. In contrast to the subsonic solution, the results in the intersonic case indicate that the elastic fields around the crack tip depend on the deceleration or acceleration history that is traced back over a long time. Singularity matching dictates the jump that may actually take place.     

The deformation and fracture of balloons

Inflation of balloons provides a straightforward way of achieving large biaxial deformations. Previous studies have shown that when a balloon bursts, crack propagation occurs at very high speed – much higher than would be expected from the low strain modulus and elastic wave velocity of the rubber. The present paper is concerned with studies of the deformation and fracture of cylindrical balloons. On inflation, the deformations of such a balloon pass through an unstable region but subsequently increase monotonically with pressure. In this relatively high pressure region, the ratio of the longitudinal and circumferential extension ratios is broadly in accord with expectations from high-strain elasticity theory when the ratio of the corresponding stresses is taken into account. On bursting, crack speeds up to around 300 m/s in this region. It is shown that these speeds are in accord with large increase in incremental moduli for the highly-strained rubber. Marked changes in crack tip profile observed at very high crack speeds are consistent with control of the rate of growth by inertia rather than by the viscoelastic properties of the rubber (as is believed to be the case at lower speeds). Consistent with this, various elastomers having different glass transition temperatures show similar crack growth behaviour in the very high speed region.     

Crack propagation in an elastic solid subjected to general loading— IV. Obliquely incident stress pulse

The stress intensity factors of a half-plane crack extending nonuniformly in an isotropic elastic solid subjected to stress wave loading are considered. A plane stress pulse is obliquely incident on the crack, and the wavefront strikes the crack at some initial time. At some arbitrary later time, the crack begins to extend at a nonuniform rate. It is found that the mode I and mode II stress intensity factors each have the form of the product of a universal function of instantaneous cracktip speed with the stress intensity factor for an equivalent stationary crack. An energy-rate balance fracture criterion is applied to obtain an equation of motion for the crack tip and to determine the actual delay time between the arrival of the incident wave and the onset of fracture as a function of angle of incidence of the loading wave.     

Influence of plasticity on interface toughness in a layered solid with residual stresses

For crack growth along an interface between two adjacent elastic–plastic materials in a layered solid, the resistance curve behaviour is analysed by approximating the behaviour in terms of a bi-material interface under small scale yielding conditions. Thus, it is assumed that the layers are thick enough so that the extent of the plastic regions around the crack tip are much smaller than the thickness of the nearest layers. The focus is on the effect of initial residual stresses in the layered material, or on T-stress components induced during loading. The fracture process is represented in terms of a cohesive zone model. It is found that the value of the T-stress component in the softer material adjacent to the interface crack plays a dominant role, such that a negative value of this T-stress gives a significant increase of the interface fracture toughness, while a positive value gives a reduction of the fracture toughness.