An effective numerical method for an interfacial crack in a finite bi-material plate

In this paper an effective numerical method is presented for analyzing the stress intensity factors associated with the stress field near a partially debonded interface in a finite bi-material plate. The strees functions are assumed such that they can represent the stress singularity at the crack tips, satisfying not only the equilibrium equations in the domain, but also the stress and displacement conditions on the crack surfaces and across the interface. Therefore, only the boundary conditions of the plate need be considered, and they can be satisfied approximately by the Boundary Collocation Method. Numerical examples demonstrated that the proposed method gives satisfactory results and has many advantages compared to other methods.     

Measurement of displacement field around an interfacial crack in a bimaterial sheet

The displacement field around a bimaterial interfacial crack was investigated using a laser speckle technique. The two half-plates of the bimaterial specimen were made of two different polyester resins cast together with a central crack embedded during casting. The bimaterial constant of the plate was close to the value of aluminium-iron combination. The specimen was loaded in remote pure shear by a specially designed loading jig. In contrast to the predictions by many of the theoretical models, the present measurements revealed no crack closure at either crack tip. Furthermore, a comparison was made between the present measured values and the Rice and Sih's theoretical solution of displacement jumps across the crack face. The result showed that while tangential displacement jump was in very good agreement with the theoretical predictions, the opening displacement jump measured was much larger than that predicted by the theory.     

An estimation of the plastic zone size for a bimaterial interfacial crack

This note deals with the size of the plastic zone ahead of an interfacial crack between two dissimilar isotropic elastic materials. Dugdale's concept of finite stress at the tip of the crack is used in the study. The plastic zone size is determined for plane stress problems under the von Mises yield condition.     

An interfacial edge crack in anisotropic bimaterial under anti-plane singularity

An interfacial edge crack of a length a in two bonded dissimilar anisotropic quarter planes having an anti-plane singularity such as a screw dislocation or anti-plane line force is analyzed. The energy release rate and stress intensity factor are explicitly expressed in terms of complex potentials. The interaction problems in two bonded orthotropic quarter planes are exemplified. It is confirmed that the present solution reduces to the known solutions dealing with particular cases of the present problem.     

Propagation of a transient shear wave from a crack in an elastic plate

A simple analytical technique based on the methods of transverse resonance and group velocity is developed in order to predict the propagation of shear (SH) waves emitted from a crack in elastic plates. Since a total solution of the Rayleight Lamb waves propagation inside the plate is very complicated, the simple SH mode was used to demonstrate the argument of this paper. The effects of varying distances between the crack and the sensor are also discussed.

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Résumé On développe une technique analytique simple basée sur les procédés de résonance transversale et de vitesse de trains d'onde en vue de prédire la propagation d'ondes SH émises à partir d'une fissure dans une plaque élastique.Comme une solution complète pour la propagation des ondes de Rayleight-Lamb est très compliquée à mettre en oeuvre, on s'est restreint à utiliser le mode SH pour démontrer la validité de l'étude.On discute également des effets d'une distance variable entre la fissure et le palpeur.

     

Kinking of an interface crack in an orthotropic bimaterial

We investigated the asymptotic problem of a kinked interface crack in an orthotropic bimaterial under in‐plane loading conditions. The stress intensity factors at the tip of the kinked interface crack are described in terms of the stress intensity factors of the interface crack prior to the kink combined with a dimensionless matrix function. Using a modified Stroh formalism and an orthotropy rescaling technique, the matrix function was obtained from the solutions of the corresponding problem in transformed bimaterial. The effects of orthotropic and bimaterial parameters on the matrix function were examined. A reduction in the number of dependent material parameters on the matrix function was made using the modified Stroh formalism. Moreover, the explicit dependence of one orthotropic parameter on the matrix function was determined using an orthotropic rescaling technique. The effects of the other material parameters on the matrix function were numerically examined. The energy release rate was obtained for a kinked interface crack in an orthotropic bimaterial.     

Dependence of stress on elastic constants in an anisotropic bimaterial under plane deformation; and the interfacial crack

Nondimensional parameters that are needed to describe the stress field in an anisotropic bimaterial with tractions prescribed on its boundary are investigated. Two real matrices, which play an important role in representing the stress field, are proposed. They are shown to be an extension of the Dundurs parameters to the anisotropic bimaterial. A general solution of the stress and displacement fields in an anisotropic bimaterial with a straight interface is also obtained by using the complex function theory. In particular, a complete solution for the stress and displacement fields in the vicinity of the tip of an interfacial crack, between two dissimilar anisotropic elastic media, is also derived.This research is supported by the Office of Naval Research. The first author also acknowledges the support from the Korea Science and Engineering Foundation.     

A numerical study on the crack growth behavior of a low and a high strength steel

Geometry and size effects on the crack growth resistance curves in CT-specimens of a low and high strength material are numerically investigated. The fracture process is controlled by experimentally determined critical values of the crack tip opening displacement (CTOD_i for crack growth initiation and the crack tip opening angle (CTOA_C) for stable crack growth. The results show that for the low and the high strength material the geometry and size effects are different. This is mainly due to the fact that the load vs. load line displacement records are basically different for the two materials which results from the large difference in the scaled fracture toughness parameters, CTOD_i/_O and CTOA_C/_O, of the two materials.Presented at the Far East Fracture Group (FEFG) International Symposium on Fracture and Strength of Solids, 4–7 July 1994 in Xi'an, China.     

A novel singular finite element on mixed-mode bimaterial interfacial cracks with arbitrary crack surface tractions

This paper investigates interfacial cracks with arbitrary crack surface tractions. A novel singular finite element which is constructed with the analytical solution around interfacial cracks is presented. Interfacial crack problems can be analyzed numerically using the singular finite element, and Mode I and/or Mode II stress intensity factors can be obtained directly. Unlike other enriched elements for cracks, neither extra unknowns nor transition elements are required. Numerical examples are given to illustrate the validity of present method.     

Elastodynamics of a crack on the bimaterial interface

The paper is an application of boundary integral equations to the problem of a crack located on the bimaterial interface under time-harmonic loading. A system of linear algebraic equations is derived for solving the problem numerically. The distributions of the displacements and tractions at the bimaterial interface are obtained and analysed for the case of a penny-shaped crack under normal tension-compression wave. The dynamic stress intensity factors (normal and shear modes) are also computed. The results are compared with those obtained for the static case.     

A multi-scale numerical modelling of crack propagation in a 2D metallic plate

A new multi-scale model of brittle fracture growth in an Ag plate with macroscopic dimensions is proposed in which the crack propagation is identified with the stochastic drift-diffusion motion of the crack-tip atom through the material. The model couples molecular dynamics simulations, based on many-body interatomic potentials, with the continuum-based theories of fracture mechanics. The Ito stochastic differential equation is used to advance the tip position on a macroscopic scale before each nano-scale simulation is performed. Well-known crack characteristics, such as the roughening transitions of the crack surfaces, as well as the macroscopic crack trajectories are obtained.     

Numerical modelling of plane strain plasticity induced crack closure effects for bimaterial interfacial cracks

The effects of plane strain plasticity induced crack closure on fatigue cracks located at the interface of dissimilar steel materials are presented using finite element modelling. Based on the study, it has been observed that bimaterial cracks produced unsymmetrical residual plastic strains and crack profiles in the crack wakes. It is seen that Young’s modulus and yield stress mismatch have profound effects on the development of unsymmetrical residual plastic strain and crack profiles, whereas the effect of Poisson’s ratio is insignificant. However, it has been found that for the material properties considered, low value of crack closure levels have been identified.     

Stress intensity factors of an inclined elliptical crack near a bimaterial interface

In this paper the stress intensity factors are discussed for an inclined elliptical crack near a bimaterial interface. The solution utilizes the body force method and requires Green’s functions for perfectly bonded semi-infinite bodies. The formulation leads to a system of hypersingular integral equation whose unknowns are three modes of crack opening displacements. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of stress intensity factors along the crack front accurately. Distributions of stress intensity factors are presented in tables and figures with varying the shape of crack, distance from the interface, and elastic modulus ratio. It is found that the inclined crack can be evaluated by the models of vertical and parallel cracks within the error of 24% even for the cracks very close to the interface.     

A numerical study of inclusion-matrix debonding in the presence of a nearby crack

Debonding of an inclusion from the surrounding matrix in the presence of a nearby crack tip is studied numerically. Finite element models of symmetric three-point bend configurations are implemented in conjunction with a stiffness degradation method and the maximum tensile stress criteria to investigate the influence of debonding on crack tip parameters. The geometry considered is a single edge cracked beam having a symmetrically positioned stiff or soft cylindrical inclusion ahead of the crack tip. The numerical model is first validated by interferometric measurements on an edge cracked epoxy specimen with a glass inclusion. The measured quantities namely, the crack mouth opening displacement, crack mouth compliance, energy release rate, dominant strain, are all successfully captured by the adopted numerical methodology, before and after the inclusion debonds from the matrix. Subsequently, the effects of parameters such as the distance between the crack tip and inclusion center (L), the inclusion diameter (d), and the Young’s modulus ratio between the inclusion and the matrix, are studied using the model. A stiff inclusion of constant (d/L) ratio debonds from the matrix at higher load levels when the inclusion interface is farther away from the crack tip. The increase in the crack driving force due to debonding is the highest when the inclusion proximity parameter ρ is approximately 0.4 and it decreases when ρ is increased or decreased relative to this value. However, when d/L ratio is varied, higher crack driving forces due to debonding are observed for larger size inclusions due to a greater loss of crack tip shielding and reinforcement. The influence of the modulus ratio (E_i/E_m) due to debonding is most prominent in the range 0-1 for fixed d/L and ρ values. Additionally, a stiffer inclusion tends to debond from the matrix at lower loads for a constant interfacial strength.     

Stress behaviour in the vicinity of a crack approaching a bimaterial interface

The integral approach of Cook and Erdogan was used to obtain the asymptotic behaviour of a crack close to a bimaterial interface. It was found that, in addition to the square-root singularity which dominates the stresses in the vicinity of a conventional crack, there are two additional contributions to stress which should not be neglected. One contribution is due to an “image singularity” and the second is due to an O(1) term which can become very large for small tip to interface distances. It is shown how these features of the stress distribution can be used to advantage in the numerical calculation of stress intensity factors.     

Mixed mode interface crack in a pure power-hardening bimaterial

Analytical and numerical analysis of the dominant singularity solutions of the stress and strain field near an interface crack in a pure power-hardening bimaterial indicates that the crack stress singularity is –1/(n _II+1) for hardening power of n _I and n _II(n _I<n _II). This result is obtained by solving the non-linear eigenvalue equations of the stress field near a plane crack while observing the conservation properties of the J-integral and the continuity conditions of the interface. Numerical results are presented for the distribution of stress, strain and displacement field of various mixed mode interface cracks when n _I=1 and n _II=5. Possible further destruction of the bimaterial with interface cracks is also discussed.

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Résumé On procède à una analyse théorique et numérique des solutions de singularité dominantes dans les champs de contraintes et de dilatation voisins d'une fissure d'interface dans un complexe bimétallique répondant à une loi simple de durcissement parabolique. On montre que la singularité de la contrainte vaut –1/n _II+1 pour des modules d'écrouissage n _I et n _II(n _I<n _II).On obtient ce résultat en résolvant les équations non linéaires d'eigenvalues du champ de contraintes au voisinage d'une fissure plane, tout en respectant les propriétés de conservation de l'intégrale J, et les conditions de continuité numérique pour la distribution des champs de contraintes, de dilatations et de déplacements correspondant à diverses fissures d'interfaces sollicitées selon des modes mixtes, avec n _I=1 et n _II=5.On discute également d'une destruction ultérieure possible du bimatériau en présence de fissures d'interface.

     

Evaluation of interfacial fracture toughness of a flip-chip package and a bimaterial system by a combined experimental and numerical method

In this paper, the interfacial fracture toughness of a flip-chip package subjected to a constant concentrated line load and a bimaterial system under thermal loading condition were evaluated using a unique six-axis submicron tester, a thermal vacuum chamber and FEM modeling coupled with a high density laser moiré interferometry. The six-axis submicron tester was used to provide a constant concentrated line load, whereas the moiré interferometry technique was used to monitor the crack length during the test. In addition, a finite element technique was simultaneously used to determine the near crack tip displacement fields of the specimens. The interfacial fracture toughness and phase angle were computed by using these near tip displacement variables through the analytical energy release rate and phase angle expressions derived by authors. The interfacial fracture toughness and the phase angle of the flip-chip package considered at the interface where the passivated silicon chip meets the underfill are 35 J/m² and −65°, respectively, while the interfacial fracture toughness and the phase angle of the tested bimaterial specimen at the interface of the molding compound/silicon with the crack length of 3.3 mm under the temperature rise thermal load from room temperature (20°C) to 138°C are 20.02 J/m² and −54.8°, respectively.     

Study on crack propagation behavior in a quenched glass plate

Crack pattern transition and crack propagation behavior in a quenched glass plate are investigated. Theoretical analysis indicates that the distance between the crack tip and the cold front is closely related to the crack pattern transition. This theoretical result is examined experimentally using instantaneous phase-stepping photoelasticity. As expected theoretically, when the crack tip remains close enough to the cold front, crack propagation remains straight. When this distance reaches a given value, the crack oscillates. These experimental results are in good agreement with the theory of crack pattern transition. Therefore, present theoretical analysis is valid in predicting the instability of crack propagation. The crack tip stress field is also examined by the present experimental method. In particular, in the oscillating regime, the mode-I stress intensity factor frequently becomes larger than the fracture toughness, and the mode-II stress intensity factor has a nonzero value during propagation. For the former result, some reasons are discussed, but the cause of this problem is still unknown. However, the latter result can be explained by the theoretical analysis of an infinitesimal kinked edge crack just after crack initiation.     

Accelerated fatigue crack growth simulation in a bimaterial interface

A method for accelerated simulation of fatigue crack growth in a bimaterial interface (e.g. in a face/core sandwich interface) is proposed. To simulate fatigue crack growth, a routine is incorporated in the commercial finite element program ANSYS and a method to accelerate the simulation is implemented. The proposed method (the cycle jump technique) is based on conducting finite element analysis for a set of cycles to establish a trend line, extrapolating the trend line spanning many cycles, and use the extrapolated state as initial state for additional finite element simulations. A control criterion is utilized to ensure the accuracy of the cycle jumps. The inputs of the developed scheme are the crack growth rate as a function of energy release rate for discrete mode-mixities. If these relationships are available for a specific interface, interface fatigue crack growth in any structure with the same interface can be simulated. Using this approach, fatigue crack growth in the face/core interface of a sandwich beam is simulated. Results of the simulation show that with fair accuracy, using the cycle jump technique, more than 65% reduction in computation time can be achieved. Results show that in highly nonlinear problems the control parameter needs to be chosen with care.