Energy release rates of bi-material interface crack including residual thermal stresses: Application of crack tip element method

Crack tip element method is applied to the formulation of the energy release rate associated with interfacial crack growth of laminates with residual thermal stresses using the Timochenko beam model. Special attention is paid to the energy release rates of double cantilever beam and mixed-mode bending tests of bi-material specimens, and mode-I and mode-II energy release rates are formulated including residual thermal stresses. The derived results are verified by the comparison to finite element analysis, and the effect of residual thermal stresses on the mode mixity of the double cantilever beam and mixed-mode bending tests is discussed.     

Two practical applications of crack kinking

This paper illustrates on two examples of practical significance the usefulness of the crack kinking analysis pioneered by Sia Nemat-Nasser and his colleagues. The first example deals with the calculation of the pull-out force of axisymmetric headed anchors embedded deep in concrete blocks and the second with the interaction between the crack deflection and phase transformation mechanisms of toughening in ceramics containing dispersed tetragonal zirconia precipitates.     

Finite element computation of the strain energy release rate for kinking of a crack

The approximate energy release rate for a crack which extends, or kinks, away from its original direction has been computed by finite element analysis using three different methods: stiffness derivative procedure, J integral method and crack closure integral procedure. A case study is presented to compare the computational accuracies. It suggests that a satisfactory approximation can be obtained provided that the ratio of kink length to original crack length, l(0)/a0.04, despite the interaction of the crack tip and knee stress singularities.

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Résumé En utilisant trois méthodes différentes (dérivation de la raideur, intégrale J et intégrale de fermeture de la fissure), on a calculé par éléments finis la vitesse approchée de relaxation de l'energie dans le cas d'une fissure qui diverge ou qui part en arborescence par rapport à sa direction initiale.On présente une étude de cas de manière à comparer les précisions respectives obtenus dans les calculs.Cette étude suggère qu'une approximation satisfaisante peut être obtenue, pour autant que le rapport de la longueur déviée à la longueur initiale de la fissure soit inférieure ou égale à 0,04, en dépit de l'interaction entre l'extrémité de la fissure et la singularité de la contrainte.

     

Creep crack growth by grain boundary cavitation: crack tip fields and crack growth rates under transient conditions

Transient creep crack growth due to grain boundary cavitation, and under plane strain and small scale creep conditions, is investigated. Full account is taken of the finite geometry changes accompanying crack tip blunting and the material is characterized as an elastic-power law creeping solid with an additional contribution to the creep rate arising from a given density of cavitating grain boundary facets. All voids are assumed present from the outset, distributed on a given density of cavitating grain boundary facets. Our analyses show the competing effects of stress relaxation due to creep, diffusion and crack tip blunting, and the stress increase due to crack growth. Another outcome of our analyses is the crack growth rate under various conditions of loading and for various values of material properties and for various characterizations of the failure process. Prior to crack growth, Hutchinson-Rice-Rosengren type singular fields dominate over the crack tip region, outside of a finite strain zone that has dimensions of the order of the crack opening displacement. These singular fields scale with the path integral C(t), which to a good approximation decays as K _I ²/t, with t being the elapsed time since load application and K _Ithe imposed stress intensity factor. When the crack growth rate is faster than the growth rate of the creep zone, our finite element results show that Hui-Riedel singular fields dominate over the crack tip region and the magnitude of the Hui-Riedel fields scales with the crack growth rate. For a crack that grows more slowly than the creep zone, Hutchinson-Rice-Rosengren type fields dominate over the crack tip region. In these circumstances, the crack growth rate is found to scale as C(t) to a power. Regardless of which of the two singular fields dominates for the growing crack, finite strain effects are found to be significant over a size scale of the order of the crack opening displacement at crack growth initiation. The effect of increased mesh refinement is also considered and very little mesh dependence is found.

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Résumé On étudie la croissance d'une fissure en fluage transitoire, associée à la cavitation aux frontières des grains, sous des conditions d'état plan de déformation et de fluage à petite échelle. On tient compte des modifications finies de géométrie accompagnant l'arrondisement de l'extrémité de la fissure, et le matériau suit une loi de fluage elasto-parabolique, avec une contribution additionnelle à la vitesse de fluage venant d'une densité donnée de facettes de joints de grains comportant de la cavitation. On suppose que toutes les cavités sont présentes dès le début, et qu'elles sont distribuées selon une densité déterminée de ces facettes. L'analyse montre les effets rivaux d'une relaxation des contraintes associée au fluage, à la diffusion et à l'arrondisement des extrémités de fissure, et d'une augmentation de contraintes due à la croissance de fissure. Un autre résultat de l'analyse est l'établissement de la vitesse de croissance de la fissure sous diverses conditions de mise en charge, pour diverses valeurs des propriétés du matériau, et pour divers modes de caractérisation du processus de rupture. Avant croissance de la fissure, ce sont des champs singuliers de type Hutchinson-Rice-Rosengren (HDR) qui prédominent sur la région de l'extrémité de la fissure, à l'extérieur d'une zone de déformations finies dont la taille est de l'ordre de grandeur du COD. Ces champs singulier sont proportionnels à l'intégrale de parcours C(t) laquelle, avec une bonne approximation, s'atténue en fonction de K _I ²/t, où t est le temps qui s'est écoulé depuis la mise en charge et K _Ile facteur d'intensité de contraintes imposés. Lorsque la vitesse de croissance de la fissure dépasse la vitesse de croissance de la zone en fluage, les résultats de l'analyse par éléments finis montre que ce sont les champs singuliers de Hui-Riedel qui prédominent sur la zone de l'extrémité de la fissure, et que l'amplitude de ces champs est proportionnelle à la vitesse de croissance de la fissure. Pour une fissure qui croit moins vite que la vitesse de fluage, es champs de type HRR sont prédominants et on trouve que la vitesse de croissance de la fissure est proportionelle C(t) à une certaine puissance. Quel que soit le type de champs singulier qui détermine la croissance de la fissure, on trouve que les effets de déformation finies sont significatifs sur une échelle de dimension de l'ordre du COD à l'amorçage de la fissuration. On considère également l'effet d'un affinage plus important du réseau, et l'on trouve me très faible dépendance par rapport à ce paramètre.

     

超声吸收体边界条件的分析

超声吸收体的边界条件分析对于求解超声吸收体表面温升与入射波声强的函数关系至关重要。通过分析在超声吸收体与水和与空气界面处超声辐照的物理过程,分别得到超声吸收体与水界面处的平均传热系数与焦点声强之间的函数关系式和超声吸收体与空气界面处的复合传热表面系数与温差的函数关系式。通过仿真分析了不同条件下两个界面处的传热系数对超声吸收体与空气界面温升的影响。实验结果表明,当辐照时间较短时,对于超声吸收体与空气界面的温度变化,超声吸收体与水界面可以认为是一个无限远且温度恒定的边界,超声吸收体与空气界面可以认为是一个符合第一类边界条件的连续热传导。     

Energy rates and crack stability in small scale yielding

Crack stability in small scale yielding is traditionally analysed using the R-curve approach with toughness indexed by either of the linear elastic fracture mechanics parameters K or G. In ductile materials stable tearing commences well before crack instability and progresses under increasing G_R. This is often assumed to mean that toughness is increasing with crack growth. It is shown in this paper that a rising G_R curve is generated even when a crack propagates with constant toughness (constant energy dissipation rate). The paper demonstrates that this apparent anomaly occurs because G does not represent the energy input rate for a crack advancing under increasing load in an elastic-plastic material. The constant energy dissipation rate model is consistent with a size independent G_R curve; also crack instability predictions are identical with both theories. The G_R curve approach has practical advantages, but use of energy dissipation rate offers better physical insight and greater versatility when analysing tough materials.     

Curvilinear fatigue crack reliability analysis by stochastic boundary element method

In this paper, the stochastic boundary element method, which combines the mixed boundary integral equations method explored in Reference 1 with the first-order reliability method, is developed to study probabilistic fatigue crack growth. Due to the high degree of complexity and non-linearity of the response, direct differentiation coupied with the response-surface method is employed to determine the response gradient. Three random processes, the mode I and mode II. stress intensity factors and the crack direction angle, are included in the expression of the response gradient. The sensitivity of these random processes is determined using a first-order response model. An iteration scheme based on the HL-RF method² is applied to locate the most probable failure point on the limit-state surface. The accuracy and efficiency of the stochastic boundary element method are evaluated by comparing the cumulative distribution function of the fatigue life obtained with Monte Carlo simulation. The reliability index and the corresponding probability of failure are calculated for a fatigue crack growth problem with randomness in the crack geometry, defect geometry, fatigue parameters and external loads. The response sensitivity of each primary random variable at the design point is determined to show its role in the fatigue failure. The variation of each primary random variable at the design point with the change of probability of failure is also presented in numerical examples.     

Explicit expressions for energy release rates using virtual crack extensions

A new finite element technique for calculating energy release rates is presented. An explicit expression for energy changes due to virtual crack extensions is formulated based on a variation of isoparametric element mappings. Energy release rates are calculated directly from integral expressions evaluated over singular quarter-point isoparametric elements surrounding the crack tip. Since the energy release rates are expressed in variational form, there is no need for the analyst to select a small finite crack extension to simulate a virtual crack extension. The method is shown to produce very accurate solutions even with fairly coarse element meshes. A similar technique for mixed-mode fracture based on mutual potential energy release rates is described.     

Three-dimensional crack analysis using singular boundary elements

A multi-domain method of solving three-dimensional elastic crack problems in an infinite elastic body using the boundary element method is proposed. The displacement and traction behaviours near a crack front are incorporated in special crack elements. The elimination of singularities arising from the term combined with Kelvin's kernel for displacement in the integrals is discussed in detail. Stress intensity factors of modes I, II and III are obtained directly from crack-front nodal values, without any extrapolation as in some other methods. No differentiation of conventional boundary integral equations (with Kelvin's tensor kernels) is necessary in the current approach. This method is applicable to cracks of arbitrary shape. Infinite bodies are modelled precisely as such, not approximated as large finite bodies. Numerical solutions of stress intensity factors are given for several problems involving a penny-shaped crack.     

Elastoplastic crack analysis by boundary integral method and surface spectrum measurement

The paper describes a hybrid experimental-numerical technique for elastoplastic crack analysis. It consists of the experimental surface spectrum measurement of plastic strains ahead the crack tip and the boundary element method (BEM). The light scattering method is used to measure the power density spectrum from which the values of plastic strains are obtained by comparison with a calibration experiment on the same material. Plastic strains obtained experimentally are conveniently used for the calculation of unknown boundary displacement or traction vectors by the boundary element method. Instead of an iterative solution of the boundary integral equations in pure numerical solution, the boundary unknowns are computed once for a required loading level. Also asymptotic distribution of strains or stresses is not needed in the evaluation of the domain integral for the BEM formulation in the vicinity of the crack tip. Significant CPU time saving is achieved in comparison with the pure BEM solution. The method presented is illustrated by the example for a three point bending specimen with an edge crack.     

Prediction of fatigue crack growth rates using crack closure finite element analysis

A three-dimensional finite element fatigue crack closure model of a corner crack and of a through thickness crack has been developed to evaluate the range of effective stress intensity factor from the distribution of the range of stress ahead of the crack tip. The corresponding fatigue crack growth rate was evaluated from a Paris law fit to experimental data from high stress ratio tests. The point of origin for the range of stress distribution was adjusted in accordance with Irwin’s plastic zone correction. Encouraging comparisons of finite element predictions of fatigue crack growth rate incorporating closure effects with experimental measurements were obtained.     

Dual boundary element method for axisymmetric crack analysis

In this paper a dual boundary element formulation is developed and applied to the evaluation of stress intensity factors in, and propagation of, axisymmetric cracks. The displacement and stress boundary integral equations are reviewed and the asymptotic behaviour of their singular and hypersingular kernels is discussed. The modified crack closure integral method is employed to evaluate the stress intensity factors. The combination of the dual formulation with this method requires the adoption of an interpolating function for stresses after the crack tip. Different functions are tested under a conservative criterion for the evaluation of the stress intensity factors. A crack propagation procedure is implemented using the maximum principal stress direction rule. The robustness of the technique is assessed through several examples where results are compared either to analytical ones or to BEM and FEM formulations.     

Complex variables boundary element analysis of three-dimensional crack problems

This paper presents a new boundary element-based approach for solving three-dimensional problems of an elastic medium containing multiple cracks of arbitrary shapes. The medium could be loaded by far-field stress (for infinite domains), surface tractions (including those at the cracks surfaces), or point loads. Constant body forces are also allowed. The elastic fields outside of the cracks are represented by integral identities. Triangular elements are employed to discretize the boundaries. Integration over each element is performed analytically. In-plane components of the fields are combined in various complex combinations to simplify the integration. No singular integrals are involved since the limit, as the field point approaches the boundary, is taken after the integration. The collocation method is used to set up the system of linear algebraic equations to find the unknown boundary displacements and tractions. No special procedure is required to evaluate the fields outside of the boundaries, as the integration is performed before the limit is taken. Several numerical examples are presented to demonstrate the capacity of the method.     

Design sensitivity analysis with hypersingular boundary elements

The finite difference load method for shape design sensitivity analysis requires the calculation of stress and stress gradient on the boundary. In the standard boundary element method, the basic state variables-displacement and traction are continuous, and are considered as very accurate. However, the boundary stress and stress gradient, derived from the differentiation of the state variables and Hooke's law, are discontinuous and have relatively lower accuracy than the basic state variables. The hypersingular boundary integral equation is introduced in this paper to determine the stress and stress gradient in the design sensitivity analysis. The numerical examples demonstrate the accuracy of the design sensitivity using the hypersingular boundary elements.     

The kinking behaviour of a bimaterial interface crack under indentation loading

The aims of this paper are twofold. The first is to evaluate the applicability of the formula for the crack kink angle—based on the maximum principle stress criterion—for predicting the interface kink angle in a bimaterial sample undergoing indentation loading. This formula was developed for cracks in homogenous materials but in this paper, it is used to predict the kink angle using the mode mixity at the tip of a crack lying on a bimaterial interface. The second aim is to examine the behaviour of the system, in terms of the crack kink angle and contact radius, for various coating thickness', crack lengths and combinations of properties of the coating and substrate. The system that is analysed consists of a planar bimaterial sample undergoing indentation with a tungsten-carbide spherical indenter. Two-dimensional, axisymmetric models are created to represent the system, with subdomains used for modelling the cracks. In order to determine the applicability of the kink angle formula, the angle predicted is compared to the angle that is directly calculated using boundary element method models that establish the angle of the kink which yields the maximum mechanical energy release rate. The second aim of the paper is achieved by varying the material property combinations and coating thickness of the bimaterial sample and observing the effect on the kink angle of the interface crack and the contact radius. The methodologies employed are initially verified on homogenous samples with known solutions.     

A model on twinning-induced crack kinking out of a metal-ceramics interface

A continuum model is proposed to study the effects of deformation twinning on interface crack kinking in metal/ceramics layered materials. At the final stage of material failure, plastic work hardening exhausts and lattice rotation becomes main mechanism after competing with dislocation gliding. The crack-tip plasticity is established in terms of the second gradient of microrotation due to the coupling effect of the twins. The formed twinning structures not only shield the crack tip, but inhibit further dislocation emission by increasing the near-tip stress levels. A Dislocation-Free Zone (DFZ) can exist in the immediate vicinity of the tip. The model is based on the equivalence of the stresses derived from twin-based crack-tip plasticity, macroscopic plasticity and elasticity on the boundary. The two-parameter characterization of near-tip stress fields is used for the outer plastic zone to account for constraint effects. Crack kinking out of the interface follows the direction of the maximum flow stress from the crack-tip plasticity. The DFZ size and the crack-tip shielding ratio, as well as the kink angle, are obtained for various values of low hardening exponents and crack-tip constraints.     

Prediction of energy release rates for crack growth using FEM-based energy derivative technique

This paper presents a new method, named energy derivative technique, to calculate energy release rate for a variety of crack growth scenarios. The new method is based on energy conservation principle for crack growth, and is applicable to crack development in any quasi-static condition in which dynamic energy for crack growth is negligible. The method has the advantage over existing finite element-based methods in that the former does not require an elaborate fine mesh in the vicinity of a crack tip, and is not limited to linear deformation behaviour. Several case studies are presented to demonstrate validity of the method, which are (i) growth of penny-shaped crack for linear elastic fracture behaviour, (ii) crack growth in rubber sheet under tension for nonlinear elastic fracture behaviour, (iii) delamination in end-notched flexure specimen with friction, and (iv) crack growth with plastic deformation in double-edge-notched plate under tension. Results from these case studies show excellent agreement with data available in the literature, which were determined using either analytical or other FEM-based techniques.     

On mode I and mode II energy release rates of an interface crack

The opening (mode I) and sliding (mode II) components of the energy that is released during an incremental extension of an interface crack between two different elastic materials are evaluated by the Irwin's crack closure method. Each component of the energies (G _I and G _II ) is expressed in terms of the functions of the length of the incremental crack extension (a) and the real and imaginary part of the complex stress intensity factor defined by Malyshev and Salganik. It is found that values of G _I /a and G _II /a oscillate violently when a approaches zero and that, hence, in contrast with the case for homogeneous materials, each energy release rate should be defined as G _I /a and G _II /a for an actual crack growth step size.