On the energy release rate and the J-integral for 3-D crack configurations

In this paper an analytical expression for the energy release rate has been derived and put in a form suitable for a numerical analysis of an arbitrary 3-D crack configuration. The virtual crack extension method can most conveniently be used for such a derivation. This method was originally developed from finite element considerations and the resulting expressions were, therefore, based on the finite element matrix formulation [1–5]. In this paper the derivation of the energy release rate leads to an expression which is independent of any specific numerical procedure. The formulation is valid for general fracture behavior including nonplanar fracture and shear lips and applies to elastic materials as well as materials following the deformation theory of plasticity. The body force effect is also included. For 3-D fracture problems it is of advantage to use both an average and a local form of the energy release rate and definitions for both forms are suggested. For certain restrictions on the crack geometry it is shown that the energy release rate reduces to the 3-D form of the J-integral.

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Résumé Dans le mémoire, on a établi une expression analytique pour la vitesse de relaxation de l'énergie et on l'a mise sous une forme convenable pour une analyse numérique de configurations de fissures arbitrairement à trois dimensions. La méthode d'extension virtuelle de la fissure est celle qui convient le mieux pour un tel traitement. Cette méthode a été, à l'origine, développée à partir de considérations d'éléments finis et les expressions qui en résultaient ont dès lors été basées sur une formulation de matrice d'éléments finis [1, 5]. Dans le présent mémoire, la dérivation de la vitesse de relaxation de l'énergie conduit à une expression indépendante de toute procédure numérique spécifique. La formulation est applicable au comportement général à la rupture comprenant des ruptures non coplanaires et des lèvres de cisaillement, et s'applique à des matériaux aussi bien élastiques que redevables de la théorie des déformations en plasticité. On tient compte également de l'effet des forces appliquées sur un corps. Dans le cas de problèmes de rupture à trois dimensions, il est avantageux d'utiliser à la fois une forme moyenne et une forme locale de taux de relaxation d'énergie et l'on suggère des définitions pour ces deux formes. Dans le cas de certaines restrictions relatives à la géométrie de la fissure, on montre que le taux de relaxation de l'énergie se ramène à une expression générale à trois dimensions de l'intégrale J.

     

Efficient integral equation method for the solution of 3-D magnetostatic problems

Nonlinear three-dimensional (3-D) magnetostatic field problems are solved using integral equation methods (IEM). Only the nonlinear material itself has to be discretized. This results in a system of nonlinear equations with a relative small number of unknowns. To keep computational costs low the fully dense system matrix is compressed with the fast multipole method. The accuracy of the applied indirect IEM formulation is improved significantly by the use of a difference field concept and a special treatment of singularities at edges. An improved fixed point solver is used to ensure convergence of the nonlinear problem.     

A High‐order extended finite element method for extraction of mixed‐mode strain energy release rates in arbitrary crack settings based on Irwin's integral

An analytical formulation based on Irwin's integral and combined with the extended finite element method is proposed to extract mixed‐mode components of strain energy release rates in linear elastic fracture mechanics. The proposed formulation extends our previous work to cracks in arbitrary orientations and is therefore suited for crack propagation problems. In essence, the approach employs high‐order enrichment functions and evaluates Irwin's integral in closed form, once the linear system is solved and the algebraic degrees of freedom are determined. Several benchmark examples are investigated including off‐center cracks, inclined cracks, and two crack growth problems. On all these problems, the method is shown to work well, giving accurate results. Moreover, because of its analytical nature, no special post‐processing is required. Thus, we conclude that this method may provide a good and simple alternative to the popular J‐integral method. In addition, it may circumvent some of the limitations of the J‐integral in 3D modeling and in problems involving branching and coalescence of cracks. Copyright © 2013 John Wiley & Sons, Ltd.     

Singularity/integral equation method development for 3-D electromagnetic field calculation

A new integral equation formulation for solving general 3-D electromagnetic field problems using the method of singularities is briefly described. Detailed expressions of integral equations for axially symmetrical open and confined field cases derived from the basic general 3-D formula are given. Mathematical singularities encountered in the equations are analytically removed and an iterative solution procedure is adapted. Numerical results obtained for several test cases during the developmental stage of this method are presented.     

Formulation of 3-D dynamic SSI analysis involving contact nonlinearities by time domain BEM-FEM

The formulation of a hybrid time domain BEM-FEM scheme is developed for the solution of three-dimensional dynamic soil-structure interaction problems involving contact nonlinearities at the soil-structure interface. Both uplift and sliding are considered at the interface through special thin-layer interface elements. The linear elastic structure, the geometrically nonlinear interface and the linear elastic soil medium are coupled together through equilibrium and compatibility and the analysis proceeds stepwise in time with iterations at each time step in order to define the contact area. Both external dynamic loads and seismic waves are considered.     

On the calculation of energy release rates for cracked laminates

A general method is given for calculating the energy release rate G from the local values of bending moments and loads in a cracked laminate. This total value is then partitioned into mode I and II components. Examples are given of the analysis of several test geometries including both variable and constant ratio mixed mode tests. Solutions for compression failures with buckling are also given. Finally there is some discussion of specimen compliances and stability criteria for fixed load and fixed displacement.

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Résumé On fournit une méthode générale de calcul de la vitesse G de relaxation de l'énergie, à partir des valeurs locales des moments de flexion et des charges agissant sur un composite lamellaire fissurée. On divise selon les composantes de Mode I et de Mode II la valeur totale obtenue. Des exemples d'analyse appliquée à diverses géométries d'essai sont fournis, qui sont relatifs à des essais de mode mixte à ratios variable ou constant. On fournit également les solutions relatives à de la ruine par compression avec flambage. Enfin, on discute de la compliance des éprouvettes et des critères de stabilité dans le cas de charge imposées et de déplacements imposés.

     

A direct analytical method to extract mixed‐mode components of strain energy release rates from Irwin's integral using extended finite element method

A new analytical approach, within the extended finite element framework, is proposed to compute mixed‐mode components of strain energy release rates directly from Irwin's integral. Crack tip enrichment functions in extended FEM allow for evaluation of integral quantities in closed form (for some crack configurations studied) and therefore resulting in a simple and accurate method. Several benchmark examples on pure and mixed‐mode problems are studied. In particular, we analyze the effects of high‐order enrichments, mesh refinement, and the integration limits of Irwin's integral. The results indicate that high‐order enrichment functions have significant effect on the convergence, in particular when the integral limits are finite. When the integral limits tend to zero, simpler strain energy release rate expressions are obtained, and high‐order terms vanish. Nonetheless, these terms contribute indirectly via coefficients of first‐order terms. The numerical results show that high accuracy can be achieved with high‐order enrichment terms and mesh refinement. However, the effect of the integral limits remains an open question, with finite integration intervals chosen as h ∕ 2 tending to give more accurate results. Copyright © 2013 John Wiley & Sons, Ltd.     

Virtual crack extension method for calculating the second order derivatives of energy release rates for multiply cracked systems

In this paper, we further generalize the work of Lin and Abel [Lin SC, Abel JF. Variational approach for a new direct-integration form of the virtual crack extension method. Int J Fract 1988;38:217-35.] to the case of higher order derivatives of energy release rates for two-dimensional, multiply cracked systems. The direct integral expressions are presented for the energy release rates and their first and second order derivatives. The salient feature of this numerical method is that the energy release rates and their first and second order derivatives can be computed in a single analysis. It is demonstrated through a set of examples that the proposed method gives expectedly decreasing, but acceptably accurate results for the energy release rates and their first and second order derivatives. The computed errors were approximately 0.5% for the energy release rates, 3-5% for their first order derivatives and 10-20% for their second order derivatives for the mesh densities used in the examples. Potential applications of the present method include a universal size effect model and a probabilistic fracture analysis of cracked structures.     

Structural scale decomposition of energy release rates for delamination propagation

A kinematic based decomposition of energy release rates for structural scale delamination propagation is established for a general class of 1-D propagation in both curved and flat composite structures. The energy release rates, and the associated decomposition, allow for geometric nonlinearities, are self consistent, and are compatible with typical thin structure models employed in the analysis of composite structures. The energy release rates are expressed explicitly in terms of the curvatures and membrane strains (forces) of the sublaminates at the delamination edge, are independent of the particular loading of the structure, and allow for simple computation, application, evaluation and comparison of the critical delamination energy (toughness) based on simple experiments. The decomposition established herein is applied to several limiting cases to check proper convergence and is then applied to several selected examples.     

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.     

A 3-D approach to the calculation of the energy release rate in some fracture problems

A 3-D ellipsoidal flaw model is sufficiently versatile to cover a wide variety of flaw shapes: existing 2-D flaw models are special cases. The energy release rate from an ellipsoidal flaw in an infinite medium may be calculated by both a strain method and a displacement method. Solution techniques for both tension and compression are presented. The results calculated by both methods are in excellent agreement with available explicit results. The simpler and more efficient strain method is preferred in the calculation of the energy release rate for various flaws, except for line cracks and flat cracks subject to tensile stress normal to the crack plane. The 3-D formulation has considerable promise for providing understanding of the effects of various parameters on the energy release rate under triaxial stress states.     

A 3-D Cell Method Formulation for Coupled Electric and Thermal Problems

A finite formulation is presented for the solution of coupled problems of steady-state electric and transient thermal conductions in 3-D regions. The model is based on the cell method approach and takes advantage of the very agile algebraic formulations that it can provide for field theories. Dual barycentric cell complexes are used for both space and time domains, the latter inducing a Crank-Nicolson time integration scheme. An example of application is given, consisting in the simulation of a treatment for the thermal ablation of liver cancer. Comparisons are given with a commercial finite-element method program     

Comparison of static and dynamic energy release rates for different fracture specimens

Static and dynamic energy release rates were utilised to study energy losses away from the crack tip in three different fracture specimens during a dynamic fracture event. The method of analysis utilizes an energy balance in the system. The results show that a substantial amount of energy is lost away from the crack tip. Moreover the energy loss is found to be dependent on the specimen geometry and material.     

An efficient implementation of the radial basis integral equation method

An efficient and accurate implementation of the meshless radial basis integral equation method (RBIEM) is proposed. The proposed implementation does not involve discretization of the subdomains’ boundaries. By avoiding the boundary discretization, it was hypothesised that a significant source of error in the numerical scheme is avoided. The proposed numerical scheme was tested on two problems governed by the Poisson and Helmholtz equations. The test problems were selected such that the spatial gradients of the solutions were high to examine the robustness of the numerical scheme. The dual reciprocity method (DRM) and the cell integration technique were used to treat the domain integrals arising from the source terms in the partial differential equations. The results showed that the proposed implementation is more accurate and more robust than the previously suggested implementation of the RBIEM. Though the CPU time usage of the proposed scheme is lower, the difference to the previously proposed scheme is not significant. The proposed scheme is easier to implement, since the task of keeping track of boundary elements and boundary nodes is not needed. The proposed implementation of the RBIEM is promising and opens up possibilities for efficient implementation in three-dimensional problems. This is currently under investigation.     

A method for approximate calculation of the energy integral for notched and cracked bodies

We propose an approximate method for the calculation of the energyJ-integral for bodies with notches (cracks) subjected to elastoplastic deformations based on an analysis of stress and stress concentration at the tip of the notch (crack). The formulas for theJ-integral are obtained in terms of the theoretical stress concentration factor (stress intensity factor), nominal stresses, radius of the notch tip (crack length), and elastoplastic properties of the material. These formulas enable one to representJ-based design curves with account of the effect for material hardening.Blagonravov Institute of Mechanical Engineering, Russian Academy of Sciences, Moscow; Moscow Institute of Engineering Physics, Moscow. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 30, No. 3, pp. 82–87, May–June, 1994.     

On the calculation of energy release rates for cracked laminates with residual stresses

Prior methods for calculating energy release rate in cracked laminates were extended to account for heterogeneous laminates and residual stresses. The method is to partition the crack tip stresses into local bending moments and normal forces. A general equation is then given for the total energy release rate in terms of the crack-tip moments and forces and the temperature difference experienced by the laminate. The analysis method is illustrated by several example test geometries. The examples were verified by comparison to numerical calculations. The residual stress term in the total energy release rate equation was found to be essentially exact in all example calculations.     

3-D schapery representation for non-linear viscoelasticity and finite element implementation

On the basis of the one-dimensional Schapery representation for non-linear viscoelasticity, a three-dimensional constitutive model incorporating the effects of temperature and physical ageing is developed for isotropic non-linear viscoelastic materials. Adopting the assumption that the hydrostatic and deviatoric responses are uncoupled, the contitutive equation is expressed in incremental form for both compressible and incompressible materials, with the hereditary integral updated at the end of each time increment by recursive computation. The proposed model is implemented in the finite element package MARC. Numerical examples are given to demonstrate the effectiveness of the model and the numerical algorithms.Laboratory for Engineering Mechanics, Delft University of Technology, P. O. Box 5033, 2600 GA Delft, The Netherlands     

A boundary integral equation method for plate flexure with conditions inside the domain

A method for solving boundary value problems for thin plate flexure as described by Kirchhoff's theory is proposed. An integral formulation leads to a system of boundary integral equations involving values of deflection, slope, bending moment and transverse shear force along the edge. A discretization leading to a matrix formulation is proposed. To solve problems with inner conditions in the plate domain, an elimination of boundary unknowns proves successful. The degenerate case where the boundary is free (which leads to a non-invertible matrix) is investigated. Three examples illustrate the efficiency of the method.