Numerical study on deformations in a cracked viscoelastic body with the extended finite element method

Deformations such as crack opening and sliding displacements in a cracked viscoelastic body are numerically investigated by the extended finite element method (XFEM). The solution is carried out directly in time domain with a mesh not conforming to the crack geometry. The generalized Heaviside function is used to reflect the displacement discontinuity across a crack surface while the basis functions extracted from the viscoelastic asymptotic fields are used to manifest the gradient singularity at a crack tip. With these treatments, the XFEM formulations are derived in an incremental form. In evaluating the stiffness matrix, a selective integration scheme is suggested for problems with high Poisson ratios often encountered in viscoelastic materials over different element types in the XFEM. Numerical examples show that the crack opening displacement and crack sliding displacement are satisfactory.     

Natural frequencies of cracked functionally graded material plates by the extended finite element method

In this paper, the linear free flexural vibration of cracked functionally graded material plates is studied using the extended finite element method. A 4-noded quadrilateral plate bending element based on field and edge consistency requirement with 20 degrees of freedom per element is used for this study. The natural frequencies and mode shapes of simply supported and clamped square and rectangular plates are computed as a function of gradient index, crack length, crack orientation and crack location. The effect of thickness and influence of multiple cracks is also studied.     

Analysis of cracked anisotropic plates using the hybrid finite element method

This paper presents a convenient and efficient method to obtain accurate stress intensity factors for cracked anisotropic plates. In this method, a complex variable formulation in conjunction with a hybrid displacement finite element scheme is used to carry out the stiffness and stress calculations of finite cracked plates subjected to general boundary and loading conditions. Unlike other numerical methods used for local analysis such as the boundary element method, the present method results in a symmetric stiffness matrix, which can be directly incorporated into the stiffness matrix representing other structural parts modeled by conventional finite elements. Therefore, the present method is ideally suited for modeling cracked plates in a large complex structure.     

The eXtended finite element method for cracked hyperelastic materials: A convergence study

The present work aims to look into the contribution of the extended finite element method for large deformation of cracked bodies in plane strain approximation. The unavailability of sufficient mathematical tools and proofs for such problem makes the study exploratory. First, the asymptotic solution is presented. Then, a numerical analysis is realized to verify the pertinence of solution given by the asymptotic procedure, because it serves as an eXtended finite element method enrichment basis. Finally, a convergence study is carried out to show the contribution of the exploitation of such method. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamic analysis of cracked linear viscoelastic solids by finite element method using singular element

The finite element method for the dynamic problem of cracked linear viscoelastic solids is developed using the singular element where the displacement function is taken from the analytical solution near the crack-tip. The time variation of the dynamic stress intensity factors is determined for a center crack and an oblique crack in standard linear viscoelastic rectangular plates subjected to dynamic loading.

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Résumé La méthode par éléments finis permettant d'aborder le problème dynamique des solides linéaires viscoélastiques fissurés est développée en recourant à un élément singulier pour lequel la fonction de déplacement est prise dans une solution analytique au voisinage del'extrémité de la fissure. La variation dans le temps des facteurs d'intensité de contrainte dynamique est déterminée pour une fissure centrale et pour une fissure oblique dans des plaques rectangulaires standard en matériau linéaire viscoélastique soumises à une sollicitation dynamique.

     

A numerical contact algorithm in saturated porous media with the extended finite element method

In this paper, a coupled hydro-mechanical formulation is developed for deformable porous media subjected to crack interfaces in the framework of extended finite element method. Governing equations of the porous medium consist of the momentum balance of the bulk together with the momentum balance and continuity equations of the fluid phase, known as formulation. The discontinuity in fractured porous medium is modeled for both opening and closing modes that results in the fluid flow within the fracture, and/or contact behavior at the crack edges. The fluid flow through the fracture is assumed to be viscous and is modeled by employing the Darcy law in which the permeability of fracture is obtained using the cubic law. The contact condition in fractured porous medium is handled by taking the advantage from two different algorithms of LATIN method and penalty algorithm. The effect of contact on fluid phase is employed by considering no leak-off from/into the porous medium. The nonlinearity of coupled equations produced due to opening and closing modes is carried out using an iterative algorithm in the Newton–Raphson procedure. Finally, several numerical examples are solved to illustrate the performance of proposed X-FEM method for hydro-mechanical behavior of fractured porous media with opening and closing modes.     

Delamination modelling of GLARE using the extended finite element method

In this paper, an application of the Extended Finite Element Method (XFEM) for simulation of delamination in fibre metal laminates is presented. The study consider a double cantilever beam made of fibre metal laminate in which crack opening in mode I and crack propagation were studied. Comparison with the solution by standard Finite Element Method (FEM) as well as with experimental tests is provided. To the authors’ knowledge, this is the first time that XFEM is used in the fracture analysis of fibre metal laminates such as GLARE. The results indicated that XFEM could be a promising technique for the failure analysis of composite structures.     

Fracture mechanics analysis of cracked 2-D anisotropic media with a new formulation of the boundary element method

A new formulation of the boundary element method (BEM) is proposed in this paper to calculate stress intensity factors for cracked 2-D anisotropic materials. The most outstanding feature of this new approach is that the displacement and traction integral equations are collocated on the outside boundary of the problem (no-crack boundary) only and on one side of the crack surfaces only, respectively. Since the new BEM formulation uses displacements or tractions as unknowns on the outside boundary and displacement differences as unknowns on the crack surfaces, the formulation combines the best attributes of the traditional displacement BEM as well as the displacement discontinuity method (DDM). Compared with the recently proposed dual BEM, the present approach doesn't require dua elements and nodes on the crack surfaces, and further, it can be used for anisotropic media with cracks of any geometric shapes. Numerical examples of calculation of stress intensity factors were conducted, and excellent agreement with previously published results was obtained. The authors believe that the new BEM formulation presented in this paper will provide an alternative and yet efficient numerical technique for the study of cracked 2-D anisotropic media, and for the simulation of quasi-static crack propagation.     

Efficient prediction of deterministic size effects using the scaled boundary finite element method

This paper develops an efficient numerical approach to predict deterministic size effects in structures made of quasi-brittle materials using the scaled boundary finite element method (SBFEM). Depending on the structure’s size, two different SBFEM-based crack propagation modelling methodologies are used for fracture analyses. When the length of the fracture process zone (FPZ) in a structure is of the order of its characteristic dimension, nonlinear fracture analyses are carried out using the finite element-SBFEM coupled method. In large-sized structures, a linear elastic fracture mechanics (LEFM)-based SBFEM is used to reduce computing time due to small crack propagation length required to represent the FPZ in an equivalent nonlinear analysis. Remeshing is used in both methods to model crack propagation with crack paths unknown a priori. The resulting peak loads are used to establish the size effect laws. Three concrete structures were modelled to validate the approach. The predicted size effect is in good agreement with experimental data. The developed approach was found more efficient than the finite element method, at least in modelling LEFM problems and is thus an attractive tool for predicting size effect.     

Dynamic crack propagation analysis of orthotropic media by the extended finite element method

Dynamic crack propagation of composites is investigated in this paper based on the recent advances and development of orthotropic enrichment functions within the framework of partition of unity and the extended finite element method (XFEM). The method allows for analysis of the whole crack propagation pattern on an unaltered finite element mesh, defined independent of the existence of any predefined crack or its propagation path. A relatively simple, though efficient formulation is implemented, which consists of using a dynamic crack initiation toughness, a crack orientation along the maximum circumferential stress, and a simple equation to presume the crack speed. Dynamic stress intensity factors (DSIFs) are evaluated by means of the domain separation integral method. The governing elastodynamics equation is first transformed into a standard weak formulation and is then discretized into an XFEM system of time dependent equations, to be solved by the unconditionally stable Newmark time integration scheme. A number of benchmark and test problems are simulated and the results are compared with available reference results.     

Simulations of piezoelectric Lamb wave delay lines using a finite element method

The method of piezoelectric finite elements was applied to the simulation of piezoelectric Lamb-wave delay lines with and without acoustical absorbers. In this finite-element analysis, free as well as electrically driven vibrations were computed. The shapes of the symmetric Lamb modes were determined by the solution of eigenproblems, and the transient mechanical build-up process was studied for a switched electrical sine wave excitation. The transfer function, the group delay time, and the impedance matrix of devices of different designs are calculated. The good agreement between simulation and experimental results indicates the suitability of the finite-element method for optimizing acoustic delay-line devices.     

Evolutionary topology optimization using the extended finite element method and isolines

This study presents a new algorithm for structural topological optimization of two-dimensional continuum structures by combining the extended finite element method (X-FEM) with an evolutionary optimization algorithm. Taking advantage of an isoline design approach for boundary representation in a fixed grid domain, X-FEM can be implemented to improve the accuracy of finite element solutions on the boundary during the optimization process. Although this approach does not use any remeshing or moving mesh algorithms, final topologies have smooth and clearly defined boundaries which need no further interpretation. Numerical comparisons of the converged solutions with standard bi-directional evolutionary structural optimization solutions show the efficiency of the proposed method, and comparison with the converged solutions using MSC NASTRAN confirms the high accuracy of this method.     

Fracture mechanics analysis using the wavelet Galerkin method and extended finite element method

This paper presents fracture mechanics analysis using the wavelet Galerkin method and extended finite element method. The wavelet Galerkin method is a new methodology to solve partial differential equations where scaling/wavelet functions are used as basis functions. In solid/structural analyses, the analysis domain is divided into equally spaced structured cells and scaling functions are periodically placed throughout the domain. To improve accuracy, wavelet functions are superposed on the scaling functions within a region having a high stress concentration, such as near a hole or notch. Thus, the method can be considered a refinement technique in fixed‐grid approaches. However, because the basis functions are assumed to be continuous in applications of the wavelet Galerkin method, there are difficulties in treating displacement discontinuities across the crack surface. In the present research, we introduce enrichment functions in the wavelet Galerkin formulation to take into account the discontinuous displacements and high stress concentration around the crack tip by applying the concept of the extended finite element method. This paper presents the mathematical formulation and numerical implementation of the proposed technique. As numerical examples, stress intensity factor evaluations and crack propagation analyses for two‐dimensional cracks are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

The calculation of stress intensity factors using the finite element method with cracked elements

The calculation of stress intensity factors for complicated crack configurations in finite plates usually presents substantial difficulty. A version of the finite element method solves such problems approximately by means of special cracked elements. A general procedure for evaluating the stiffness matrix of a cracked element is developed, and numerical results obtained by the simplest elements are compared with those provided by other methods.

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Zusammenfassung Die Berechnung von Spannungsintensitätsfaktoren für komplizierte Rißgefüge in endlichen Platten bereitet gewöhnlich erhebliche Schwierigkeiten. Fine Variante finite element method löst annähernd solche Probleme mit Hilfe von spezieller gerissenen Elementarteilen.Es wird ein allgemeines Verfahren zur Ermittlung der Steifheits-Matrix eines gerissenes Elementarteilchens aufgestellt. Die numerischen Ergebnisse welche mit den einfachsten Elementarteilen bestimmt wurden, werden mit den nach anderen Verfahren erzielten Ergebnissen verglichen.

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Résumé Pour une plaque finie et une configuration de fissures compliquée, le calcul des coefficients d'intensité de contraintes s'avère normalement difficile, voire pratiquement impossible.Toutefois, une variante de la méthode des éléments finis permet de résoudre ce genre de problèmes de façon approximative moyennant l'adoption d'un élément fissure.Dans cet article l'auteur développe une méthode générale permettant d'évaluer la matrice de raideur d'un élément fissuré.Ensuite il procède pour des éléments simples à une comparaison des résultats numériques obtenus respectivement par d'autres méthodes et par la sienne.

     

A combined space–time extended finite element method

The Newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering. Easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities. Zienkiewicz proposed an approach using finite element concept in time, which allows a new look at the Newmark method. The idea of this paper is to propose, thanks to this approach, the use of a time partition of the unity method denoted Time Extended Finite Element Method (TX‐FEM) for improved numerical simulations of time discontinuities. An enriched basis of shape functions in time is used to capture with a good accuracy the non‐polynomial part of the solution. This formulation allows a suitable form of the time‐stepping formulae to study stability and energy conservation. The case of an enrichment with the Heaviside function is developed and can be seen as an alternative approach to time discontinuous Galerkin method (T‐DGM), stability and accuracy properties of which can be derived from those of the TX‐FEM. Then Space and Time X‐FEM (STX‐FEM) are combined to obtain a unified space–time discretization. This combined STX‐FEM appears to be a suitable technique for space–time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities. Copyright © 2005 John Wiley & Sons, Ltd.     

New parametric study of nugget size in resistance spot welding process using finite element method

Resistance spot welding process (RSW) is one of important manufacturing processes in automotive industry for assembling bodies. Quality and strength of the welds and therefore body mainly are defined by quality of the weld nuggets. The most effective parameters in this process are: current intensity, welding time, sheet thickness and material, geometry of electrodes, electrode force, and current shunting. In present research, a mechanical–electrical–thermal coupled model in a finite element analysis environment is made using. Via simulating this process, the phenomenon of nugget formation and the effects of process parameters on this phenomenon are studied. Moreover, the effects of welding parameters on temperature of faying surface are studied. Using this analysis, shape and size of weld nuggets are computed and validated by comparing them with experimental results from published articles. The methodology developed in this paper provides prediction of quality and shape of the weld nuggets with variation of each process parameter. Utilizing this methodology assists in adjusting welding parameters so that costly experimental works can be avoided. In addition, the process can be economically optimized to manufacture quality automotive bodies.     

Strong tangential discontinuity modeling of shear bands using the extended finite element method

A method is developed for modeling of shear band with strong tangential discontinuity by means of cohesive surfaces within the extended finite element method (XFEM). A rate-independent non-associated plasticity model is incorporated along the strong discontinuity to consider the highly localized regions. Once the localization is occurred, tangential enrichment degrees of freedom are added to the localized elements, and the discontinuity is captured regardless of mesh resolution and alignment. By introducing the tangential enrichment function, the discontinuity is only imposed in the tangential direction, while the continuity across the shear band is automatically fulfilled. Adopting bilinear quadrilateral elements within the context of XFEM allows for the plastic deformation of shear band to be obtained with quadratic distribution within an enriched element. Since the strong discontinuity approach is employed, the singularity of strain field at the position of displacement jump is attained through a Dirac delta distribution. By means of this singularity, the cohesive shear traction is derived for the J2 plasticity model and is applied onto the band interfaces in order to reproduce the dissipative mechanism of the band. Several numerical examples are analyzed to assess the accuracy and robustness of the proposed approach.     

A modified extended finite element method approach for design sensitivity analysis

This paper describes a modified extended finite element method (XFEM) approach, which is designed to ease the challenge of an analytical design sensitivity analysis in the framework of structural optimisation. This novel formulation, furthermore labelled YFEM, combines the well‐known XFEM enhancement functions with a local sub‐meshing strategy using standard finite elements. It deviates slightly from the XFEM path only at one significant point but thus allows to use already derived residual vectors as well as stiffness and pseudo load matrices to assemble the desired information on cut elements without tedious and error‐prone re‐work of already performed derivations and implementations. The strategy is applied to sensitivity analysis of interface problems combining areas with different linear elastic material properties. Copyright © 2015 John Wiley & Sons, Ltd.     

The effective electroelastic property of cracked piezoelectric media

Summary This paper presents a study on the effective electroelastic property of piezoelectric media with parallel or randomly distributed cracks. The theoretical formulation is derived using the dilute model of distributed cracks and the solution of a single dielectric crack problem, in which the electric boundary condition along the crack surfaces is governed by the crack opening displacement. It is observed that the effective electroelastic property of such cracked piezoelectric media is nonlinear and sensitive to loading conditions. Numerical simulations are conducted to show the effects of crack distribution and electric boundary condition upon the effective electroelastic property. The transition between the commonly used electrically permeable and impermeable crack models is studied.