Double virtual crack extension method for crack growth stability assessment

The second variation of energy corresponding to crack length is required in the stability analysis of crack growth. For determining such an energy gradient, an efficient finite element method extending the classical virtual crack extension concept is described in this paper. In elasticity, the method can be used for the prediction of the growth pattern of one single crack, and especially a system of interacting cracks as well from the results of a single strain-stress analysis. Example computations are performed for (1) a center-cracked plate and (2) a finite width strip containing two interacting cracks. Close agreement between the numerical results given by our method and reference solutions has been found in all testing cases.     

A substructuring application of the virtual crack extension method

The virtual crack extension method (VCE) is a very effective tool with finite elements for the accurate evaluation of stress intensity factors in two- or three-dimensional structures. The method utilizes the concept of extending any node on the crack profile in any required direction and by any required amount. A new facility stores a relevant substructure of crack tip stiffnesses so that any number of restarts, each with a new set of extensions, can be performed. Different nodes around three-dimensional crack profiles can be considered, either at vertex or midside node positions. The extension of the vertex nodes, in particular, also involves moving adjacent midside nodes along the profile, and different ways of effecting this, together with the calculation of the relevant area change, have been introduced and are described along with illustrative examples.     

The virtual crack extension method for evaluation of J- and

The equation for evaluating the nonlinear fracture mechanics parameters J- and Ĵ-integrals are derived using the virtual crack extension method. The validity of the equations derived here are checked by solving several numerical examples, that is, the J-integral analyses of compact tension specimen and three-point bend specimen, and the Ĵ-integral analysis of centrally cracked plate. Reasonably good agreement is found between the virtual crack extension method and the line integral method.     

Maximum energy release rate distribution from a generalized 3D virtual crack extension method

This paper presents a generalized 3D virtual crack extension (VCE) technique for determining the distribution of the maximum energy release rate along a general 3D crack front. The method allows for VCEs at any inclination to the local crack plane. By taking the component of the extension in the crack front's local normal plane it evaluates the local energy release rate, G, in that component's direction. Repeated VCEs applied to the crack front at different inclinations enable the maximum G and its associated direction in the normal plane to be determined. This technique has been applied to a quarter-circular crack in a square cross-section bar under axial and torsional loading. The evaluated maximum energy release rates show the expected antisymmetric direction and symmetric magnitude distributions. The test case also demonstrates a sinusoidal G distribution within the normal plane which would imply that the maximum G and its direction could be evaluated from only two local G values.     

Variational approach of displacement discontinuity method and application to crack problems

This paper deals with a variational approach of the displacement discontinuity method. This method is an indirect boundary element technique which uses the double layer potential representation of displacements and stresses. The variational approach presented here is based upon the theorem of minimum potential energy in elasticity. In the numerical procedure, the global shape function used to approximate the displacement discontinuity distribution is the continuous piecewise linear function. Regular displacements and resultant force expressions are obtained from these shape functions and these expressions are used to build the system of linear equations. The method is applied to crack problems in both infinite and finite bodies. The stress intensity factors are then calculated and high accurate numerical results are obtained.     

Analysis of surface cracks using the line-spring boundary element method and the virtual crack extension technique

The authors have developed a new line-spring boundary element method which couples the line-spring model with the boundary element method to deal with the problem of a surface cracked plate. However, the drawback of the line-spring model is that a reliable stress intensity factor could not be directly obtained near the free surface intersection. Therefore, the virtual crack extension technique is employed in a post-processor of the line-spring boundary element method to obtain the stress intensity factor at the crack front-free surface intersection. Theoretical analysis is described. Stress intensity factors for surface cracks are calculated to verify the proposed method. The interaction of two surface cracks is also investigated. The solutions obtained by the line-spring boundary element method show that the method proposed is efficient and reasonably accurate.     

On the method of virtual crack extensions

The measurement of energy release rates using virtual crack extensions has been made using finite element techniques. An economic and accurate technique for calculating energy changes due to any number of virtual tip changes is presented. Mixed-mode situations can be dealt with by observing the direction of maximum energy release rate. Examples are given including various cracks in a plate in tension, a curving crack in a general two-dimensional shape, and a three-dimensional crack in a plate.     

Weight function calculations for mixedmode fracture problems with the virtual crack extension technique

An efficient finite element method is presented for calculating the stress intensity factors (K_I and K_II) and the weight functions for mixed-mode cracks with one virtual crack extension. The computational efficiency is enhanced through the use of singular elements and the application of colinear virtual crack extension (VCE) technique to symmetric mesh in cracktip neighborhood. This symmetric mesh in crack-tip vicinity permits the analytical separation of strain energy release rate into G_I for Mode I and G_II for Mode II for the mixed fracture problems with the colinear virtual crack extension.

Rice's displacement derivative representation of weight function vector for symmetric crack has been extended to the mixed fracture mode at nodal location (x_i,y_i) with crack length (a) and inclination angle (β) as h_I(II)(x_i, y_i, a, β) = (H/2K_I(II)(∂U_I(II)(x_i, y_i, a, β/∂a).

This equation permits explicit determination of weight functions for the entire structure of a given asymmetric crack geometry with colinear VCE technique. The explicit weight functions for mixed fracture mode depend strongly on the constraint conditions. The method of obtaining the required stress intensity factors of a given asymmetric crack geometry, from the weight function concept under the selected constraint conditions, which are different from constraint conditions used in the available weight functions for the same crack geometry, is also presented in this paper. This is accomplished by combining the predetermined explicit weight functions with the self-equilibrium forces at their application locations. These self-equilibrium forces include both the applied surface tractions and the reaction forces induced from the constraint conditions.     

Non-symmetric extension of a crack

Summary The dynamic in-plane problem of the non-symmetric extension of a crack in an infinite, isotropic elastic medium under normal stress is analyzed. Following Cherepanov [8], Cherepanov and Afanas'ev [9] the general solution of the problem is derived in terms of an analytic function of complex variable. The results include the expressions for the stress intensity factors at the crack tips and the rate of energy flux into the cxtending crack edges. For a particular case, numerical calculations for the stress intensity factor and the energy flux rate are carried out.     

Comparison of virtual crack extension and strain energy density methods applied to contact surface crack growth

The paper is concerned with comparison of two crack propagation methods applied to a two-dimensional computational model of the surface initiated crack growth in the lubricated contact area of meshing gears. The virtual crack extension method and the minimum strain energy density criterion are used for simulation of the crack propagation in the framework of the finite element analysis. The discretised equivalent contact model, with the assumed size and orientation of the initial crack, is subjected to contact loading conditions, accounting for the elasto-hydro-dynamic lubrication effects, tangential loading due to sliding and the influence of lubricating fluid, driven into the crack by hydraulic mechanism. The computational results show that both crack propagation methods give comparable results, although the virtual crack extension method has some clear advantages due to its theoretical superiority in dealing with mixed-mode short crack propagation close to the loading boundary.     

A mesh objective method for modeling crack propagation using the smeared crack approach

This paper presents a mesh objective method for modeling crack propagation in brittle materials using a conventional finite element formulation. The primary shortcoming of the smeared crack approach is its pathological sensitivity to the mesh orientation, which is manifested by shear locking and stress field misalignment around the crack tip. Such undesirable characteristics preclude the ability to model arbitrary crack propagation at an angle through the mesh. Several techniques are developed to address these shortcomings. First, to preclude shear locking, a modified failure constitutive model is developed, which projects out the spurious stress increments as the crack opens. If a crack exists in an element, a crack tracking algorithm is used to identify the neighboring elements most likely to show crack continuation. This algorithm also identifies a crossover element when a crack passes through adjacent sides of an element. Then, the characteristic element length used in the constitutive equation is changed with the objective of providing the correct failure energy per unit crack length, a procedure called crossover scaling. The examples provided demonstrate that the developed methods work collectively to provide a simple and efficient method for modeling failure in brittle materials without mesh bias.     

Obtaining mode mixity for a bimaterial interface crack using the virtual crack closure technique

We review, unify and extend work pertaining to evaluating mode mixity of interfacial fracture utilizing the virtual crack closure technique (VCCT). From the VCCT, components of the strain energy release rate (SERR) are obtained using the forces and displacements near the crack tip corresponding to the opening and sliding contributions. Unfortunately, these components depend on the crack extension size, Δ, used in the VCCT. It follows that a mode mixity based upon these components also will depend on the crack extension size. However, the components of the strain energy release rate can be used for determining the complex stress intensity factors (SIFs) and the associated mode mixity. In this study, we show that several—seemingly different—suggested methods presented in the literature used to obtain mode mixity based on the stress intensity factors are indeed identical. We also present an alternative, simpler quadratic equation to this end. Moreover, a Δ-independent strain energy release based mode mixity can be defined by introducing a “normalizing length parameter.” We show that when the reference length (used for the SIF-based mode mixity) and the normalizing length (used for Δ-independent SERR-based mode mixity) are equal, the two mode mixities are only shifted by a phase angle, depending on the bimaterial parameter ε.