Tip Elements Research Articles

DragonFlex – Smart Steerable Laparoscopic Instrument

DragonFlex – Smart Steerable Laparoscopic Instrument

Numerical simulation of crack propagation in layered formations

In the present study, a boundary element method based on the higher order displacement discontinuity formulation is presented to solve the general problem of hydraulic fracture propagation in layered formations. Displacement collocation technique is employed to model the higher order displacement variation along the crack and the special crack tip element near its ends. The hydraulic fracture propagation and its interaction with the layer interface in non-homogenous rock materials are studied by the proposed semi-analytical (hybridized boundary element-boundary collocation) method. The maximum tangential stress criterion (or σ-criterion) of fracture mechanics considering different elastic constants (Young modulus and Poisson’s ratio) is used to obtain the fracture path. The fracture propagation from stiff to soft and soft to stiff media for cracks having different inclination angles is modeled, and the effects of elastic constants on the hydraulic fracture propagation is studied. The results show that if the hydraulic fracture originates in the stiffer layer, its capability to cross the layer increases and is vice versa for the softer material. The comparison of the results gained from the numerical method with those in the literature show a good performance of the method in the case of propagation of hydraulic fracture in layered formations.

Numerical Crack Analysis of Blunt Rock Indenters by an Indirect Boundary Element Method

Linear elastic fracture mechanics principles are widely applied for the analysis of crack problems in rock fracture mechanics. Rock indentation is an important and complicated problem among rock engineering issues. In this paper, in addition to the fracture criterion of maximum tangential stress adjacent to crack tip, the higher order displacement discontinuity method (which is a version of the indirect boundary element method) has been used for modeling the crack propagation mechanism under blunt indenters. In order to achieve more accurate results, higher order boundary elements i.e. quadratic elements, has been used to calculate displacement discontinuities and also to reduce the singularities of stress and displacement fields near the crack tip, the special crack tip elements has been used to calculate the stress intensity factors (SIF) at the crack tips. In this modeling, the effect of crack angle on stress intensity factors has been investigated. The numerical results of stress intensity factors obtained from some example problems were compared to the theoretical and experimental results cited in the literature which always show a percentage error less than one percent. The simulated results may pave the way for increasing the efficiency of mining and drilling by improving the design of tools and indentation equipments.

On the use of power series solution method in the crack analysis of brittle materials by indirect boundary element method

A hybridized boundary element/power series solution method is proposed for evaluating displacements and stresses near the crack tips. In the indirect boundary element method, the special crack tip elements can be used to increase the accuracy of the stress and displacement fields near the crack ends. The special crack tip elements are usually used by assuming the crack mouth opening and crack mouth displacement variations in form of an infinite power series which can end up with the special singular integrals. The aim of this paper is to solve these singular integrals by using an infinite power series. The power series solution method has been used to solve some example problems in fracture mechanics and the corresponding semi-analytical results are compared with those obtained by using the system of partial differential integral equations. This comparison demonstrates the high accuracy of the proposed method.

Thermo-Mechanical Stress near Apex of a Bi-Material Wedge by a Novel Finite Element Analysis

A super wedge tip element for application to a bi-material wedge is develop utilizing the thermo-mechanical stress and displacement field solutions in which the singular parts are numerical solutions. Singular stresses near apex of an arbitrary bi-material wedge under mechanical and thermal loading can be obtained from the coupling between the super wedge tip element and conventional finite elements. The validity of this novel finite element method is established through existing asymptotic solutions and conventional detailed finite element analysis.

Multiscale numerical modeling of propagation and coalescence of multiple cracks in rock masses

A new multiscale numerical model in which the coordination between the internal boundary (such as cracks and holes) and meshes is not necessary for simulating the damage evolution of crack-weakened rock masses is proposed based on the extended finite element method and global–local analysis. The connection between different sizes of meshes behaves smoothly through combining the displacement projecting method and displacement loading method. The contact constraint on crack surfaces is embedded within the total stiffness matrix using the penalty method. The path of crack propagation and stress fields are determined through iterative computations. Because additional discontinuous functions and enriched tip elements are added into the displacement field, the geometry of cracks is independent of the finite element mesh. As a result, re-meshing is not necessary to model crack propagation using the present new model. As the frictional contact of crack surfaces is taken into account, the present method is suitable for modeling the growth and coalescence of multiple cracks in geomaterials under compressive loads as well as tensile loads. Finally, the present method is employed to simulate multiple cracks growth when frictional contact exists on the crack surfaces. Comparison between the numerical and experimental results shows that the present numerical results are in good agreement with the experimental ones.

An explicit residual‐type error estimator for ‐quadrilateral extended finite element method in two‐dimensional linear elastic fracture mechanics

SUMMARYThis paper deals with explicit residual a posteriori error estimation analysis for ‐quadrilateral extended finite element method (XFEM) discretizations applied to the two‐dimensional problem of linear elastic fracture mechanics. The result is twofold. First, to enable estimation procedures with application to XFEM, a specific quasi‐interpolation operator of averaging type is constructed. The main challenge here arises from the different types of enrichments implemented, and hence, to impose the constant‐preserving property of the interpolation operator on an element, we use the idea of an extension operator. An upper bound on the discretization error measured in the energy norm and associated local error indicators are then constructed and analyzed. The second result follows from the error analysis and concerns an alternative choice of branch functions used in XFEM applications. In particular, the branch functions have to be chosen to fulfill the divergence‐free conditions within the crack tip element and traction‐free boundary conditions on the crack faces. Then, the corresponding XFEM solution gains a better accuracy with less degrees of freedom. Finally, numerical examples are provided with comparative results. Copyright © 2012 John Wiley & Sons, Ltd.

Singular stress analysis of an anisotropic elastic medium containing polygonal holes using a novel hybrid finite element method

A novel hybrid finite element method based on a numerical procedure is proposed to compute singular field near V-shaped notch corners in an anisotropic material containing polygonal holes. The finite element method is established by the following three steps: (1) an ad hoc one-dimensional finite element formulation is employed to determined numerical eigensolutions of the singular field near an V-shaped notch corner; (2) a super corner tip element is constructed to determine the strength of the singular field, in which the independent assumed stress fields are extracted from the eigensolutions; (3) a novel hybrid finite element equation is obtained by coupling the super corner tip element with the conventional hybrid stress elements. In numerical examples, generalized stress intensity factors for interactions between two polygonal holes with various geometry, space position and material property are mainly discussed. All the numerical results show that present method yields satisfactory singular stress field solutions with fewer elements. Compared with the conventional finite element methods and integral equation methods, the present method is more suitable for dealing with micromechanics of anisotropic materials.

On the crack propagation modeling of hydraulic fracturing by a hybridized displacement discontinuity/boundary collocation method

Numerical methods such as boundary element and finite element methods are widely used for the stress analysis in solid mechanics. This study presents boundary element method based on the displacement discontinuity formulation to solve general problems of interaction between hydraulic fracturing and discontinuities. The crack tip element and a higher order boundary displacement collocation technique are used to study the hydraulic fracture propagation and its interaction with the pre-existing cracks and discontinuities in an elastic rock mass. The maximum tangential stress criterion (or -criterion) and the strain energy density criterion (SED) are used to obtain the fracture path and the results of both criteria are compared with each other. The comparison of numerical method with the results brought in the literature shows a good performance of the method in the case of interacting cracks.

Two-Dimensional Crack Analyses by XFEM Using Crack Tip Elements

The extended finite element method (XFEM) using a crack tip element (TIP element), which is enriched through only the Heaviside function, is applied to crack and its propagation analysis in two-dimensional elastic problems. In the proposed method, two-kind of signed distance functions are utilized in order to express crack geometry implicitly and finite elements, which has interaction with crack, are appropriately partitioned according to the level set values and then integrated numerically for derivation of stiffness matrix. The results by XFEM using TIP elements were compared with those by the conventional XFEM using both the asymptotic bases and the Heaviside function. It was shown that the TIP element provides appropriate stress intensity factors and crack propagation path.

Study of Crack Growth Based on Extended Finite Element Method

A special method based on the extended finite element method is developed for the simulation of dynamic crack growth. It shows great advantages in the simulations of moving crack and mixed mode crack. The extended finite element method for two-dimensional crack is described in this paper. The crack form of the extended finite element in the homogeneous medium is studied in detail, and the internal detail in crack tip element and crack penetration element is analyzed. At last, the displacement mode is generated.

Study of Crack Growth Based on Extended Finite Element Method

A special method based on the extended finite element method is developed for the simulation of dynamic crack growth. It shows great advantages in the simulations of moving crack and mixed mode crack. The extended finite element method for two-dimensional crack is described in this paper. The crack form of the extended finite element in the homogeneous medium is studied in detail, and the internal detail in crack tip element and crack penetration element is analyzed. At last, the displacement mode is generated.

Application of the Θ-Method to 3D BEM Analysis of Cracks in Piezoelectric Components

A general purpose direct BEM code has been developed for three-dimensional crack problems in piezoelectric structures. Special 3D non-continuous crack tip elements and several techniques for determining crack tip parameters were implemented. To calculate the electromechanical energy release rate for a virtual crack extension, the θ-method is employed, which was originally suggested by BONNET for linear elastic materials. The paper presents the generalization and numerical realization of the θ-method to 3D piezoelectric cracks. The great advantage of the θ-method is the direct computation of energy release rate, whereas the way via K-factors and the IRWIN matrix is more complicated. The efficiency and accuracy of the technique are shown for various example problems by comparing with analytical solutions.

Composite Grid Method on Computation of Stress Intensity Factors of Biomaterial Interface Crack

The stress intensity factors (SIFs) of biomaterial interface crack were analyzed using the Composite Grid Method (CGM). Two different scale grids were adopted: a coarse mesh for the whole area without considering interface crack and a fine mesh for the local singular area considering the impact of interface crack. It solved the global coarse mesh problem and the local fine mesh problem iteratively and obtained the final results. The calculation was carried out by CGM but without the demand of dense mesh near the crack tip and special crack tip elements compared with other methods. The numerical extrapolation method of determining the SIFs to avoid the oscillation singularity of asymptotic fields of elastic interface crack. Reasonable agreement was achieved and efficiency of CGM for such kind of problem was displayed.

Fracture mechanics based predictions of initiation and growth of multi-level delaminations in a composite specimen

Design and manufacture of composite test specimens with two embedded delaminations subjected to transverse tension is described in this paper. The behaviour of the specimens has been investigated through experimental testing and finite element analyses. Fracture mechanics approaches were applied to predict the initiation and growth of delaminations and the degradation of the load carrying capacity of the specimens. Models were generated in the FE software programs MSC.Nastran and Abaqus. The MSC.Nastran model was used to calculate strain energy release rates employing a crack tip element methodology. The Abaqus model was evaluated using the virtual crack closure technique. The accuracy of numerical solutions was established by qualitative and quantitative comparisons with the experimental measurements, the results obtained from the ultrasonic non-destructive inspections and microscopic examination. It was shown that fracture mechanics can be used to accurately predict the failure initiation locations and failure loads of composite specimens containing multi-level delaminations.

Displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space

This paper describes a displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space or full space. The formulation is based on hypersingular integral equations that relate displacement jumps and tractions along the crack. The integral kernels, which represent stress influence functions for ring dislocation dipoles, are derived from available axisymmetric dislocation solutions. The crack is discretized into constant-strength displacement discontinuity elements, where each element represents a slice of a cone. The influence integrals are evaluated using a combination of numerical integration and a recursive procedure that allows for explicit integration of hyper- and Cauchy singularities. The accuracy of the solution at the crack tip is ensured by adding corrective stresses across the tip element. The method is validated by a comparison with analytical and numerical reference solutions.

EFG Virtual Crack Closure Technique for the Determination of Stress Intensity Factor

The aim of this paper is to introduce a virtual crack closure technique based on EFG method for thread-shape crack. The cracked component is discretized and the displacement field is determined using a coupled FE/EFG method, by which EFG nodes are arranged in the vicinity of crack tip and FE elements in the remain part in order to improve computational efficiency. Two typical parameters, nodal force and crack opening displacement attached to crack tip are calculated by means of setting up an auxiliary FE zone around crack tip. Strain energy release rate (SERR), further stress intensity factor (SIF) are determined by the two parameters. The method to calculate SIF is named as virtual crack closure technique based on EFG method. It is showed by several numerical examples that using the method presented in this paper, SIF on the crack tip can be obtained accurately.

Effects of residual thermal stresses on the debond characterization of sandwich beams

The crack tip element method is applied to the formulation of energy release rates associated with face sheet debonding of sandwich beams with residual thermal stresses using the bi-layer Timoshenko beam model. Special attention is paid to the energy release rates of DCB and MMB tests for debond characterization. Mode-I and mode-II energy release rates are formulated including residual thermal stresses. The derived analytical results are verified by comparison with finite element analysis. The effect of residual thermal stresses on the evaluated apparent fracture toughness, resulting from the neglect of residual thermal stresses in the standard evaluation methods, is discussed.

Simulation of three-dimensional fatigue crack propagation using enriched finite elements

A three-dimensional methodology for simulation of fatigue crack propagation is presented. The method is leveraged by the use of enriched crack tip elements to compute the mixed-mode stress intensity factors. The crack growth model used and the crack propagation life calculation are also described. As examples, fatigue crack propagation of a mode-I surface crack and crack advancements of mixed-mode surface cracks are simulated. The predicted results showed excellent agreement with experimental data from the literature. Thus, it is concluded that the crack propagation method developed allows efficient and accurate simulation of three-dimensional fatigue crack propagation problems.

Delamination of layered structures on elastic foundation

This study presents an interface fracture mechanics analysis of delamination of a layered beam resting on a Winkler elastic foundation subject to general mechanical loads. A crack tip element on elastic foundation model is established first, through which, two concentrated forces existing at the crack tip are determined in closed-form. Then total energy release rate of the crack can be expressed in term of these two forces. By using available numerical results in the literature, the phase angle of the total energy release rate is also obtained. To verify the validity and accuracy of the solutions, debonding of a bonded overlay from the base structure resting on a Winkler elastic foundation is analyzed using the present solution. Comparisons with the baseline results obtained by finite element analysis suggest that the present analytical solution provides an excellent estimation of the total energy release rate and its phase angle for interface cracks in layered structure on elastic foundation. This study provides an approximated analysis of the debonding of a thin overlay debonding from the concrete pavement, where the effect of the base structure is simplified by a Winkler elastic foundation. This solution can also be used to analyze other similar delamination problems, such as local delamination in laminated composites, and face sheet delimitation in sandwich beams.