Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment

A methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial differential equation loses hyperbolicity. The approach is limited to rate‐independent materials, so that the transition occurs on a set of measure zero. The discrete discontinuity is treated by the extended finite element method (XFEM) whereby arbitrary discontinuities can be incorporated in the model without remeshing. Loss of hyperbolicity is tracked by a hyperbolicity indicator that enables both the crack speed and crack direction to be determined for a given material model. A new method was developed for the case when the discontinuity ends within an element; it facilitates the modelling of crack tips that occur within an element in a dynamic setting. The method is applied to several dynamic crack growth problems including the branching of cracks. Copyright © 2003 John Wiley & Sons, Ltd.     

Modelling brittle crack formation at geometrical and material discontinuities using a finite fracture mechanics approach

In this study, finite fracture mechanics procedures are employed to predict crack formation at geometrical and material discontinuities in brittle elastic structures. A hybrid failure model is utilised taking into account the stress field in the undamaged structure and the energy balance for the formation of cracks. Asymptotic formulations are compared to a direct numerical implementation. Experiments carried out on notched brittle specimens exhibiting various geometries and loading-modes are analysed by means of both approaches. Additionally, free-edge effects in composite laminates are analysed. It is found that the predictions from the model agree well with experimental results.     

Effects of mean stress and crack closure on fatigue life of spot welds

The effects of mean stress and crack closure on fatigue life of spot welds were investigated. A review showed that several of the previously proposed mean stress corrections give similar results. Fatigue tests on shear and peel loaded specimens were carried out, and the results agreed with the corrections reviewed. The present study shows that crack closure explains the mean stress effects observed. The crack opening force for spot welds was obtained, both experimentally from F–N curves with different load ratios and analytically from the available mean stress corrections. This was verified with detailed finite element simulations. Finally, the experiments and simulations indicate that the use of linear damage accumulation in fatigue life prediction of spot welds can be non‐conservative.     

Mechanics of two-stage crack growth in fretting fatigue

Motivated by experimental observations, we carry out a numerical analysis of the two-stage crack growth under fretting fatigue by using an efficient and accurate boundary element method. To start with, the variation of stress field during a loading cycle is analyzed. Various values of friction coefficient in the contact zone are considered, which is shown to considerably affect the stress field. Then, by assuming crack initiation to occur in the shear mode, a surface-breaking crack is introduced to the specimen at the location of highest shear-stress amplitude. The crack-tip stress intensity factors (SIFs) are calculated for various crack lengths and at various crack angles ranging from 25° to 45° about the contact surface. It is shown that, for a loading ratio of 0.5, the cyclic mode-II SIF amplitude decreases with increasing crack length, whilst its mean value increases. It suggests that the (first-stage) shear crack would sooner or later become dormant, or switch to another mode that can provide continuous support of growth. Then, the first-stage shear crack is manually kinked into a second-stage opening crack, and the follow-on driving force is analyzed. It is shown that the kinking event is only favored after the first-stage crack has grown to a certain length. The present study thus provides insights in the mechanics of two-stage crack growth that has been frequently observed in a typical dovetail joint under fretting fatigue. It also suggests an improved experimental setup to quantitatively investigate the fretting fatigue in dovetail joints.     

Fracture mechanics analysis of a crack in a residual stress field

The standard definition of the J integral leads to a path dependent value in the presence of a residual stress field, and this gives rise to numerical difficulties in numerical modelling of fracture problems when residual stresses are significant. In this work, a path independent J definition for a crack in a residual stress field is obtained. A number of crack geometries containing residual stresses have been analysed using the finite element method and the results demonstrate that the modified J shows good path-independence which is maintained under a combination of residual stress and mechanical loading. It is also shown that the modified J is equivalent to the stress intensity factor, K, under small scale yielding conditions and provides the intensity of the near crack tip stresses under elastic-plastic conditions. The paper also discusses two issues linked to the numerical modelling of residual stress crack problems-the introduction of a residual stress field into a finite element model and the introduction of a crack into a residual stress field.     

Stress intensity factors for an inclined edge crack in a semiplane

Mode I and II Stress Intensity Factors under uniform general biaxial loadings were derived for an inclined edge crack in a semiplane. By interpolating Finite Element results in the angular range [0°÷80°], analytical expressions were obtained for both K_I and K_II with an accuracy better than 1%. Influence coefficients were defined in the crack reference frame thus highlighting the coupling effects between Modes I and II due to the loss of symmetry when the crack is not normal to the surface.     

Prediction of crack initiation at blunt notches and cavities - size effects

Crack initiation at corners, V-notches and other situations such as interfaces breaking a free surface (delamination initiation) cannot be correctly predicted by the usual brittle fracture criteria (either Griffith or maximum stress). They give contradictory results and neither one nor the other agrees with the experiments. An additional characteristic length is required to define a satisfying criterion giving rise to the so-called “Finite fracture mechanics”. The crack is supposed to jump this length which depends both on the material properties and the local geometry of the structure; it is not a material parameter. In most cases this crack increment is small. The size effect arises with the interaction between the crack increment and another length characterising a microstructure such as a pore diameter, a notch root radius or an interface layer thickness. The remote load at failure depends on the actual value of this microstructure parameter whereas it was not expected in all cases. Assuming that the two interacting lengths remain small compared to the size of the global structure, an asymptotic procedure allows bringing into evidence the change in the apparent resistance of the structure due to this phenomenon. Results are compared with experiments in various domains: polymers, ceramics and rocks.     

A mixed augmented Lagrangian-extended finite element method for modelling elastic–plastic fatigue crack growth with unilateral contact

The complete modelling of fatigue crack growth is still an industrial challenging issue for numerical methods. A new technique for the finite element modelling of elastic–plastic fatigue crack growth with unilateral contact on the crack faces is presented. The extended finite element method (X-FEM) is used to discretize the equations, allowing for the modelling of arbitrary cracks whose geometries are independent of the finite element mesh. This paper presents an augmented Lagrangian formulation in the X-FEM framework that is able to deal with elastic–plastic crack growth with treatment of contact. An original formulation, which takes advantages of two powerful numerical methods, is presented. Next the numerical issues such as contact treatment and numerical integration are addressed, and finally numerical examples are shown to validate the method. Copyright © 2007 John Wiley & Sons, Ltd.     

XFEM analysis of the effects of voids,inclusions and other cracks on the dynamic stress intensity factor of a major crack

This paper is devoted to the extraction of the dynamic stress intensity factor (DSIF) for structures containing multiple discontinuities (cracks, voids and inclusions) by developing the extended finite element method (XFEM). In this method, four types of enrichment functions are used in the framework of the partition of unity to model interface discontinuity within the classical finite element method. In this procedure, elements that include a crack segment, the boundary of a void or the boundary of an inclusion are not required to conform to discontinuous edges. The DSIF is evaluated by the interaction integral. After the effectiveness of the implemented XFEM program is verified, the effects of voids, inclusions and other cracks on the DSIF of a stationary major crack are investigated by using XFEM. The results show that the dynamic effects have an influence on the path independence of the interaction integral, and these voids, inclusions and other cracks have a significant effect on the DSIF of the major crack.     

Investigation of crack tunneling in ductile materials

Crack tunneling has been commonly observed in crack growth experiments on specimens made of ductile materials such as steel and aluminum alloys. The objective of this study is to investigate the crack tunneling phenomenon and study the effects of crack tunneling on the distribution of several mechanics parameters controlling ductile fracture. Three-dimensional (3D) elastic-plastic finite element analyses of stable tearing experiments involving tunneling fracture are carried out. Two model problems based on stable tearing experiments are considered. The first model problem involves a plate specimen containing a stationary, single-edge crack with a straight or tunneled crack front, under remote mode I loading. In the numerical analyses, the crack tip opening displacement, the von Mises effective stress, the mean stress, the stress constraint and the effective plastic strain around straight and tunneled crack fronts are obtained and compared. It is found that crack tunneling produces significant changes in the stress and deformation fields around the crack front. The second model problem involves a specimen containing a stably growing single-edge crack with a straight or tunneled crack front, under remote mode I loading. Crack growth events with a straight or tunneled crack front are simulated using the finite element method, and the effect of crack tunneling on the prediction of the load-crack-extension response based on a CTOD fracture criterion is investigated.     

Probabilistic crack growth analyses using a boundary element model: Applications in linear elastic fracture and fatigue problems

This paper addresses the numerical solution of random crack propagation problems using the coupling boundary element method (BEM) and reliability algorithms. Crack propagation phenomenon is efficiently modelled using BEM, due to its mesh reduction features. The BEM model is based on the dual BEM formulation, in which singular and hyper-singular integral equations are adopted to construct the system of algebraic equations. Two reliability algorithms are coupled with BEM model. The first is the well known response surface method, in which local, adaptive polynomial approximations of the mechanical response are constructed in search of the design point. Different experiment designs and adaptive schemes are considered. The alternative approach direct coupling, in which the limit state function remains implicit and its gradients are calculated directly from the numerical mechanical response, is also considered. The performance of both coupling methods is compared in application to some crack propagation problems. The investigation shows that direct coupling scheme converged for all problems studied, irrespective of the problem nonlinearity. The computational cost of direct coupling has shown to be a fraction of the cost of response surface solutions, regardless of experiment design or adaptive scheme considered.     

蠕变断裂局部损伤方法的有限元模型

提出了蠕变断裂局部损伤方法的有限元模型，通过有局部薄弱环到引入损伤单元，较好地不了多向应力条件下蠕变损伤发生和发展的规律，应用一有限元模型分析缺口圆棒试样的蠕变变形与损伤，得到的计算结果与试验结果能够较好的相吻合。     

Coarse‐graining of multiscale crack propagation

A method for coarse graining of microcrack growth to the macroscale through the multiscale aggregating discontinuity (MAD) method is further developed. Three new features are: (1) methods for treating nucleating cracks, (2) the linking of the micro unit cell with the macroelement by the hourglass mode, and (3) methods for recovering macrocracks with variable crack opening. Unlike in the original MAD method, ellipticity is not retained at the macroscale in the bulk material, but we show that the element stiffness of the bulk material is positive definite. Several examples with comparisons with direct numerical simulations are given to demonstrate the effectiveness of the method. Copyright © 2009 John Wiley & Sons, Ltd.     

结构损伤识别中灵敏度法的改进

针对原有灵敏度法对复杂结构损伤识别中存在的不足（误判损伤位置和高估损伤程度），本文对其进行了改进。改进后的方法仅利用结构损伤前后的前两阶模态参数即可确定结构损伤的位置并评估结构损伤的程度。为验证改进后方法的有效性，利用二维钢架结构有限元模型数值模拟了五种典型的损伤工况，用原有灵敏度法和改进后的方法进行了识别。结果表明，与原有灵敏度法相比改进后的方法提高了结构损伤识别的精度，并且该方法简便，有望应用于实际结构中。     

The perturbation method and the extended finite element method. An application to fracture mechanics problems

The extended finite element method has been successful in the numerical simulation of fracture mechanics problems. With this methodology, different to the conventional finite element method, discretization of the domain with a mesh adapted to the geometry of the discontinuity is not required. On the other hand, in traditional fracture mechanics all variables have been considered to be deterministic (uniquely defined by a given numerical value). However, the uncertainty associated with these variables (external loads, geometry and material properties, among others) it is well known. This paper presents a novel application of the perturbation method along with the extended finite element method to treat these uncertainties. The methodology has been implemented in a commercial software and results are compared with those obtained by means of a Monte Carlo simulation.     

Numerical simulation of crack deflection and penetration at an interface in a bi-material under dynamic loading by time-domain boundary element method

The hybrid time-domain boundary element method (BEM), together with the multi-region technique, is applied to simulate the dynamic process of crack deflection/ penetration at an interface in a bi-material. The whole bi-material is divided into two regions along the interface. The traditional displacement boundary integral equations (BIEs) are employed with respect to the exterior boundaries; meanwhile, the non-hypersingular traction BIEs are used with respect to the part of the crack in the matrix. Crack propagation along the interface is numerically modelled by releasing the nodes in the front of the moving crack tip and crack propagation in the matrix is modeled by adding new elements of constant length to the moving crack tip. The dynamic behaviours of the crack deflection/penetration at an interface, propagation in the matrix or along the interface and kinking out off the interface, are controlled by criteria developed from the quasi-static ones. The numerical results of the crack growth trajectory for different inclined interface and bonded strength are computed and compared with the corresponding experimental results. Agreement between numerical and experimental results implies that the present time-domain BEM can provide a simulation for the dynamic propagation and deflection of a crack in a bi-material.     

Size independent fracture energy evaluation for plain cement concrete

The present work deals with the investigation of a robust analytical scheme to assess the size‐independent fracture energy of concrete. The study involves the numerical modelling of three‐point bend (TPB) concrete beams that are geometrically similar, having constant length to depth ratio with varying range of notch to depth (a/W) ratios. The unique nonlinear behaviour of concrete 1material is incorporated through fracture energy‐based strain‐softening model in the finite‐element numerical simulation. The International Union of Laboratories and Experts in Construction Materials, Systems and Structures (RILEM) fracture energy values are evaluated through numerical simulation of several set of experimentally observed load‐load line displacement response. The RILEM fracture energy values associated with geometrically similar beams have been utilised to develop a methodology for assessment of the size‐independent fracture energy of concrete. The numerically predicted and experimentally evaluated size‐independent fracture energy using the RILEM fracture energy values are found to be in close agreement.