Three coplanar Griffith cracks in an infinite elastic strip

We consider the problem of determining the stress intensity factors and crack energy in an infinitely long isotropic, homogeneous elastic strip containing three coplanar Griffith cracks. The three coplanar Griffith cracks are situated symmetrically on a line perpendicular to the edges of the strip. We assume that the cracks are opened by an internal pressure and the edges of the strip are fixed. By using the theory of Fourier series we reduce the problem to solving a set of quadruple trigonometrical series equations with a cosine kernel. Closed form solution is obtained for the quadruple series equations. Closed form analytical expressions are derived for the stress intensity factors, the shape of the deformed cracks and the crack energy. Solutions to some particular problems are derived as limiting cases.     

Two coplanar Griffith cracks in an infinitely long elastic strip

We consider the problem of determining the stress intensity factors and the crack energy in an infinitely long isotropic, homogeneous elastic strip containing two coplanar Griffith cracks. We assume that the cracks are opened by an internal pressure and the edges of the strip are rigidly fixed. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with cosine kernel and a weight function. The triple integral equations are solved exactly. Exact analytical expressions are derived for the stress intensity factors, shape of the deformed crack and the crack energy. Solutions to some particular problems are derived as limiting cases.This work was supported by the National Research Council of Canada through NRC-Grant No. A4177.     

Interaction of three interfacial Griffith cracks between bonded dissimilar orthotropic half planes

The paper deals with the interaction of a pair of outer cracks on a central crack situated at the interface of two dissimilar orthotropic half-planes. The mixed boundary value problem is reduced to solving a pair of simultaneous singular integral equations which have finally been solved numerically by using Jacobi polynomials. The analytical expressions for stress intensity factors at the central crack tip and the expression of the strain energy release rate have been derived for general loading. Numerical values of the interaction effects of the outer cracks on the central crack have been calculated through stress magnification factors. It is seen that the interaction effects are either shielding or amplification depending on the size of the outer cracks and their spacing from the central crack.     

Propagation of surface breathing cracks in shafts under quasi-static rotary bending

Nonlinear Dynamics -     

Two collinear Griffith cracks subjected to uniform tension in infinitely long strip

The problem of determining the stress field in an elastic strip of finite width when the uniform tension is applied to the faces of two collinear symmetrical cracks situated within it is considered. By using the Fourier transform, the problem can be solved with a set of triple integral equations. These equations are solved using Schmidts method. This method is suitable for solving the strips problem of arbitrary width.     

Fatigue crack growth of surface cracks in the rail web

The analytical prediction of surface cracks in the rail web is investigated by two different approaches. The first approach uses engineering fracture mechanics principles with elementary beam theory (i.e., one dimensional) stress analysis. The second approach applies the strain energy density criterion to a three dimensional finite element stress analysis of the rail web. Results are presented in terms of crack size as a function of accumulated tonnage for variations in loading (tangent or curved track), support conditions (foundation modulus), and assumed levels of residual stress. The results of both models are consistent in that the predicted growth rates are fairly slow when compared to other types of rail defects (for example, the detail fracture in the rail head).     

Steady growth of cracks in compressible elastic-plastic media

Based on the theoretical framework for crack growth analysis provided by Gao and Hwang, the 5-sector solution of near-tip fields of mode-I cracks growing quasi-statically and steadily in compressible elastic perfectly plastic materials is obtained. As Poisson's ratio ν tends to 1/2, the 5-sector solution degenerates to the 4-sector solution of near-tip fields of crack growth in incompressible elastic perfectly plastic materials. The project supported by the National Natural Science Foundation of China.     

Near tip quasi-static crack growth behavior in strain hardening and softening material

Analyzed in this work is a semi-infinite crack that grows slowly in a steady-state. The assumed constitutive relation for the material permits strain hardening and softening as it is damaged in time. Four distinct regions divided angularly are identified for the asymptotic expressions of the quasi-static crack-tip stress field. They refer to material degraded in front of the crack; undergone elastic unloading; reloading of degraded material; and material completely by exhausted in its load carrying capacity.     

Perturbation growth and localization in fluid-saturated inelastic porous media under quasi-static loadings

Instabilities in inelastic saturated porous media are investigated here for general three-dimensional states under quasi-static loadings using a perturbation approach and focussing in particular on the two limiting cases of the onset of growth and of blowing up of perturbations.For associative flow rules for the skeleton, both onset of growth and blowing up of perturbations depend only on the underlying drained properties. Unbounded growth is obtained when the condition of localization for the underlying drained deformation (singularity of the drained acoustic tensor) is approached or just passed. Onset of growth has always a divergence growth character and critical conditions are always associated to the shortwavelength regime leading to the fact that the failure mode is expected to be a localized one.For non-associative behaviour of the skeleton we show in contrast that the onset of growth and unbounded growth may be defined either by the drained or undrained properties. One or the other depends on the details of the constitutive behaviour but also on the type of loadings. In particular, unbounded growth occurs when either the condition of localization under drained or undrained conditions is first passed. Transition from decaying to growing behaviour may have a divergence character or flutter-type character. Here the critical conditions are associated either to the shortwavelength or to the longwavelength regimes and therefore the failure mode may be localized or diffuse.The hierarchy between criticality of drained and undrained properties is analysed for a general class of constitutive equations and the results are fully and explicitly illustrated for saturated porous media with skeleton obeying Drucker-Prager like constitutive model.     

Modeling quasi-static crack growth with the extended finite element method Part I: Computer implementation

The extended finite element method (X-FEM) is a numerical method for modeling strong (displacement) as well as weak (strain) discontinuities within a standard finite element framework. In the X-FEM, special functions are added to the finite element approximation using the framework of partition of unity. For crack modeling in isotropic linear elasticity, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are used to account for the crack. This enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces, and hence quasi-static crack propagation simulations can be carried out without remeshing. In this paper, we discuss some of the key issues in the X-FEM and describe its implementation within a general-purpose finite element code. The finite element program Dynaflow™ is considered in this study and the implementation for modeling 2-d cracks in isotropic and bimaterial media is described. In particular, the array-allocation for enriched degrees of freedom, use of geometric-based queries for carrying out nodal enrichment and mesh partitioning, and the assembly procedure for the discrete equations are presented. We place particular emphasis on the design of a computer code to enable the modeling of discontinuous phenomena within a finite element framework.     

Modeling quasi-static crack growth with the extended finite element method Part II: Numerical applications

In Part I [Int. J. Solids Struct., 2003], we described the implementation of the extended finite element method (X-FEM) within Dynaflow™, a standard finite element package. In our implementation, we focused on two-dimensional crack modeling in linear elasticity. For crack modeling in the X-FEM, a discontinuous function and the near-tip asymptotic functions are added to the finite element approximation using the framework of partition of unity. This permits the crack to be represented without explicitly meshing the crack surfaces and crack propagation simulations can be carried out without the need for any remeshing. In this paper, we present numerical solutions for the stress intensity factor for crack problems, and also conduct crack growth simulations with the X-FEM. Numerical examples are presented with a two-fold objective: first to show the efficacy of the X-FEM implementation in Dynaflow™; and second to demonstrate the accuracy and versatility of the method to solve challenging problems in computational failure mechanics.     

Environmentally assisted initiation and growth of multiple surface cracks

The initial stages of stress corrosion on an amorphous polymer is investigated. This is done by exposing stressed specimens of polycarbonate to an acetone and water solution. The surface develops two distinct features of degradation that appear on different length scales when subjected to tensile stress. Small pits form on the surface and make it rough. These pits are in the order of micrometers, and are found to be randomly distributed. They occur even without load and seem to slightly increase in number with increasing stress. In the millimeter domain, visible to the bare eye, surface cracks are formed transverse to the direction of loading. The occurrence of cracks is seen to have a positive stress threshold value, exceeding which, a linear increase of number of cracks with stress is found. The manners in which the cracks grow and coalesce on the surface are examined. It is seen that they do not meet crack tip to crack tip. Instead, they avoid each other initially and coalesce crack tip to crack side. The results are discussed in the light of mechanical considerations. A stress analysis for a few configurations of meeting cracks supports the experimental observations. With assumptions of stress corrosion crack growth and coalescence, a simulation of cracks growing from randomly distributed initiation sites is performed. Similar crack patterns as obtained in the experiments are found.     

On delamination growth from channel cracks in thin-film coatings

Interfacial fracture (delamination) originating from channel or tunnel cracks is a common failure mode in layered structures. While this subject has been addressed extensively, little is known on the actual process of fracture. It is this aspect which is of concern here. The evolution of delamination damage is followed in situ using an all-transparent system designed to reduce material variability and thermal stresses. The specimen is composed of two glass plates glued onto a polycarbonate slab by a RT epoxy resin. With a proper control of the glass surface, stable delamination growth from a single channel crack occurs. This growth evolves smoothly from the tip of the channel crack, although the delamination area is generally irregular and non-symmetric. The fracture resistance varies greatly between nominally identical samples, attesting to a great sensitivity to such irregularity as well as to bonding surface conditions. The effect of system variables on delamination growth is evaluated using a 2D FEA. The analysis predicts general trends observed in the tests, and it indicates means for an optimal design against premature failure. Finally, the merit of evaluating fracture toughness of ultra-thin films using the channel cracking approach is discussed.     

Selection of finite-element mesh parameters in modeling the growth of hydraulic fracturing cracks

The effect of the mesh geometry on the accuracy of solutions obtained by the finite-element method for problems of linear fracture mechanics is investigated. The guidelines have been formulated for constructing an optimum mesh for several routine problems involving elements with linear and quadratic approximation of displacements. The accuracy of finite-element solutions is estimated based on the degree of the difference between the calculated stress-intensity factor (SIF) and its value obtained analytically. In problems of hydrofracturing of oil-bearing formation, the pump-in pressure of injected water produces a distributed load on crack flanks as opposed to standard fracture mechanics problems that have analytical solutions, where a load is applied to the external boundaries of the computational region and the cracks themselves are kept free from stresses. Some model pressure profiles, as well as pressure profiles taken from real hydrodynamic computations, have been considered. Computer models of cracks with allowance for the pre-stressed state, fracture toughness, and elastic properties of materials are developed in the MSC.Marc 2012 finite-element analysis software. The Irwin force criterion is used as a criterion of brittle fracture and the SIFs are computed using the Cherepanov–Rice invariant J-integral. The process of crack propagation in a linearly elastic isotropic body is described in terms of the elastic energy release rate G and modeled using the VCCT (Virtual Crack Closure Technique) approach. It has been found that the solution accuracy is sensitive to the mesh configuration. Several parameters that are decisive in constructing effective finite-element meshes, namely, the minimum element size, the distance between mesh nodes in the vicinity of a crack tip, and the ratio of the height of an element to its length, have been established. It has been shown that a mesh that consists of only small elements does not improve the accuracy of the solution.