Elastic Crack Problems Research Articles

Application of Papkovich–Neuber General Solution for Crack Problems in Strain Gradient Elasticity

Application of Papkovich–Neuber General Solution for Crack Problems in Strain Gradient Elasticity

A data‐driven bond‐based peridynamic model derived from group method of data handling neural network with genetic algorithm

AbstractIn the governing equation of motion of bond‐based peridynamics, the acceleration of a material point can be considered as the response function of all the displacements of material points in the horizon and a micro stiffness containing Young's modulus and length scale. A group method of data handling (GMDH) neural network is first developed to explicitly derive the discrete bond‐based peridynamic equation of motion based on measured data in this study rather than traditional complicated mathematical derivation. In order to discover the optimal structure more efficiently and to avoid exhaustive search, genetic algorithm is incorporated into GMDH structure. It is found that the prediction results obtained by GMDH model agree well with measured values both for training and testing data. Moreover, the derived equation of motion is expressed as the product of parameter composed of Young's modulus and length scale and linear combination of displacements of material points in the horizon, which is in accordance with the original bond‐based peridynamic formulation. Furthermore, numerical benchmarks associated with elastic deformation and crack problems are performed and compared with analytical solution or finite element analysis result to verify the validity and feasibility of the proposed model.

A parallel scalable domain decomposition preconditioner for elastic crack simulation using XFEM

AbstractIn this article, a parallel overlapping domain decomposition preconditioner is proposed to solve the linear system of equations arising from the extended finite element discretization of elastic crack problems. The algorithm partitions the computational mesh into two types of subdomains: the regular subdomains and the crack tip subdomains based on the observation that the crack tips have a significant impact on the convergence of the iterative method while the impact of the crack lines is not that different from those of regular mesh points. The tip subdomains consist of mesh points at crack tips and all neighboring points where the branch enrichment functions are applied. The regular subdomains consist of all other mesh points, including those on the crack lines. To overcome the mismatch between the number of subdomains and the number of processor cores, the proposed method is divided into two steps: solve the crack tip problem and then the regular subdomain problem during each iteration. The proposed method was used to develop a parallel XFEM package which is able to test different types of iterative methods. To achieve good parallel efficiency, additional methods were introduced to reduce communication and to maintain the load balance between processors. Numerical experiments indicate that the proposed method significantly reduces the number of iterations and the total computation time compared to the classical methods. In addition, the method scales up to 8192 processor cores with over 70% parallel efficiency to solve problems with more than degrees of freedom.

Stability and convergence analysis of singular integral equations for unequal arms branch crack problems in plane elasticity

In this paper, an unequal arms branch crack problem in a plane elasticity is treated. Using distribution dislocation function and complex variable potential method, the problem is formulated into a singular integral equation. The appropriate integration scheme, in which a point dislocation is set at the origin and the distribution dislocation, is applied through all arms of the branch crack to solve the obtained singular integral equations numerically. Stability, convergence, the order of convergence, and the error term of the solution are analyzed. Some numerical examples are examined to describe the behavior of stress intensity factors at the endpoints of each branch crack.

Alternative Integration Approaches in the Weight Function Method for Crack Problems

This study proposes two alternative approaches to complement existing integration strategies used in the weight function method for linear elastic crack problems. The first approach is based on an interpolation type integration scheme and the second approach is based on Gauss quadrature. The proposed approaches enable a computationally efficient numerical integration for computing stress intensity factors in 2D fracture problems. The efficiency is gained through a comparatively low number of integration points facilitated by higher-order approximation. The integration weights only need to be computed once for a given crack length-to-width ratio and can be applied to arbitrary continuous and smooth stress distributions. The proposed approaches show excellent accuracy. In particular, the Gauss quadrature approach exhibits several orders of magnitude more accuracy compared to the most commonly used trapezoidal integration.

Modified duality methods for solving an elastic crack problem with Coulomb friction on the crack faces

AbstractWe consider an equilibrium problem for an elastic body with a crack, on the faces of which unilateral non-penetration conditions and Coulomb friction are realized. This problem can be formulated as quasi-variational inequality. To solve it, the successive approximation method is applied. On each outer step of this method, we solve an auxiliary problem with given friction. We solve the auxiliary problem by using modified Lagrange functionals. Numerical results are presented.

An improved singular curved boundary integral evaluation method and its application in dual BEM analysis of two- and three-dimensional crack problems

In this paper, an improvement is made to an efficient direct method for numerical evaluation of high order singular curved boundary integrals. Then this improved singular integral evaluation method is employed to solve the strongly and hypersingular integrals involved in the dual boundary element method (Dual BEM), which combine the use of displacement and traction boundary integral equations to solve crack problems in a single domain formulation. The singular integral evaluation method is carried out based on a parameter plane expansion and radial integral approach, this paper proposed a new strategy for treating the singular radial integral, which plays a vital role in this method. In isoparametric coordinate system, the singular curved boundary integral is mapped into a singular square plane integral in intrinsic coordinates, then the radial integration method (RIM) is employed to transform the singular square plane integral into a regular line integral over the contour of intrinsic square plane and a singular radial integral over the path from source point to the contour of intrinsic square plane. A singularity isolation technique is utilized to divide the singular radial integral into two parts, the regular radial integral can be evaluated normally using Gauss quadrature and the singular radial that can be evaluated analytically by expanding the non-singular part of the integrand function into a power series. Compared with conventional local interpolation approach to deal with the singular radial integral, the newly proposed method has a more rigorous mathematical derivation, and can achieve more stable and precise results. Based on the successful implementation of direct evaluation of singular boundary integrals, Dual BEM is successfully applied to solve two- and three-dimensional elastic crack problems including straight and curved crack paths with continuous or discontinuous elements. Two different approaches, geometrical extrapolation method and J-integral method are used in the evaluation of stress intensity factors. Several numerical examples are given to validate effectiveness of the presented method.

A new crack-tip singular element for cracks in three-dimensional elastic bodies

Symplectic analytical singular element (SASE) is a special crack-tip element in the framework of finite element method (FEM) for modelling cracks. A group of SASEs, which have been developed by the authors, for various crack problems almost in two-dimensional (2D), but not touched yet more complex ones, i.e., in three-dimensional (3D) domain. The underlying reason lies in the fact that analytical symplectic eigen solution for 3D elastic crack problem is not yet available, a 3D SASE thus cannot be constructed through a simple generalization from the 2D SASEs. This study aims to fill out this 3D gap, we thus use a trail displacement field to construct a 3D SASE. The derived strain and stress fields still possess singularity in the vicinity of crack-tip. Finite element formulation is derived based on the minimum total potential energy principle. It is found that some of key features of a 2D SASE still exist in the developed 3D one, e.g., the SIFs can be calculated accurately without any post-processing. Numerical examples are considered to show the accuracy and performance of the proposed element.

Fracture analysis for a cusp-type crack problem

This paper provides an exact solution for a cusp-type crack problem in plane elasticity. The solution of the studied problem depends on the conformal mapping technique and complex potentials. The stress intensity factors at the cusp-type crack tip can be evaluated accordingly. Some particular characters for the SIF (stress intensity factor) solution have been found. Particularly, if a compression loading is applied along the direction of the cusp-type crack, the relevant SIF will take a positive value. This phenomenon is useful in the rupture of the rock with the cusp-type crack.

Stable generalized finite element methods for elasticity crack problems

SummaryGeneralized or eXtended finite element methods (GFEM/XFEM) for crack problems have been studied extensively. The GFEM/XFEM are called extrinsic if additional functions are enriched at every node in certain domains, while they are called degree of freedom (DOF)‐gathering if the singular enriched functions are gathered using cutoff functions. The DOF‐gathering GFEM/XFEM save the additional DOFs compared with the extrinsic approach. Both extrinsic and DOF‐gathering GFEM/XFEM suffer from difficulties of stabilities in a sense that their scaled condition numbers (SCN) of stiffness matrices could be much larger than those of the standard FEM. A GFEM/XFEM is referred to as stable GFEM (SGFEM) if it reaches optimal convergence orders, and its SCN is of same order as that of FEM. An extrinsic SGFEM was established in Zhang et al for the Poisson crack problems. Objective of this article is to propose the SGFEM for elasticity crack problems; both extrinsic and DOF‐gathering schemes are addressed. The main idea is to modify the enriched functions by subtracting their FE interpolants, which was developed by Babuška and Banerjee. To remove local almost linear dependence introduced by multifold enrichments at one node, we propose a local principal component analysis technique to identify and analyze “contributions” of multifold enrichments at one node. Numerical studies demonstrate that the proposed SGFEM and DOF‐gathering SGFEM are of optimal convergence and have the SCNs of same order as in the FEM.

Effective Two-Level Domain Decomposition Preconditioners for Elastic Crack Problems Modeled by Extended Finite Element Method

Effective Two-Level Domain Decomposition Preconditioners for Elastic Crack Problems Modeled by Extended Finite Element Method

A structured grid based B-Spline finite elements method combining local isogeometry analysis technique for crack problems

In this paper, a structured mesh based B-Spline finite elements (BSFE) method combining the isogeometry analysis technique is developed for analyzing crack problems. The BSFE is used for the global approximation and the isogeometry analysis technique is performed on the local crack tip region which is covered by several elements around the crack tip. The geometry of the selected crack tip region is described by the B-Spline basis functions in parameter space. Meanwhile, a specific function with relevant parameters is introduced into geometry description for catching the characters of the solution around the crack tip. In addition, the B-Spline basis functions are also used as basis functions to approximate the displacement field in the local region under the parameter space. The relevant B spline basis functions used for approximation in local region are replaced by the corresponding BSFE basis functions used in the interface elements adjacent to the local region and the seamless connection between the global approximation and the local isogeometry analysis can be easily achieved. For crack problems, no enrichment functions and remeshing are needed in this new method. This newly developed method is applied to the stress analysis of 2D linear elasticity crack problems in order to investigate its performance and study parameters. Numerical results show that the present method is highly accurate and stable. The new method has a promising potential for practical applications.

Stress intensity factor for an elastic half plane weakened by multiple curved cracks

• The multiple curved cracks problem in an elastic half plane is formulated by using the complex potential methods. • The integral equations are solved numerically using quadrature rule for the stress intensity factor. • The closer the cracks to the boundary the higher the value of SIF. Modified complex potential with free traction boundary condition is used to formulate the curved crack problem in a half plane elasticity into a singular integral equation. The singular integral equation is solved numerically for the unknown distribution dislocation function. Numerical examples exhibit the stress intensity factor increases as the cracks getting close to each other, and close to the boundary of the half plane.

A Finite Element Formulation for Crack Problem in Couple-Stress Elasticity

In this research, we investigated the crack behavior in the Mode I fracture in a plane strain state, considering the size effects and under the couple-stress theory using finite element method (FEM). First, using the eigenfunction expansion technique, we provided the stress, couple-stress, displacement, strain, and curvature fields at the crack-tip. Then, using the FEM, fundamental parameters that determine the stress and couple-stress fields at the crack-tip (i.e., stress intensity factor and couple-stress intensity factor) were obtained. In the formulation of FEM, we used a mixed variational method, in which displacement and rotation are considered as independent variables, and their kinematic constraints are applied using Lagrange multipliers. Based on this formulation, it was possible to implement the classical quarter-point elements for the crack-tip. The results show that under the framework of the couple-stress theory, energy release rate can still be considered as the fundamental measure and determinant parameter for the crack behavior, but the stress intensity factor alone cannot appropriately describe the crack behavior.

Mapped Displacement Discontinuity Method: Numerical Implementation and Analysis for Crack Problems

The displacement discontinuity method (DDM) is a kind of boundary element method aiming at modeling two-dimensional linear elastic crack problems. The singularity around the crack tip prevents the DDM from optimally converging when the basis functions are polynomials of first order or higher. To overcome this issue, enlightened by the mapped finite element method (FEM) proposed in Ref. [13], we present an optimally convergent mapped DDM in this work, called the mapped DDM (MDDM). It is essentially based on approximating a much smoother function obtained by reformulating the problem with an appropriate auxiliary map. Two numerical examples of crack problems are presented in comparison with the conventional DDM. The results show that the proposed method improves the accuracy of the DDM; moreover, it yields an optimal convergence rate for quadratic interpolating polynomials.

A continuous–discontinuous hybrid boundary node method for solving stress intensity factor

A novel boundary type meshless method called continuous–discontinuous hybrid boundary node method is proposed in this paper, in which the enriched discontinuous shape function is developed to solve linear elastic crack problems. Firstly, the whole boundary is divided into several individual segments, and variables on each one of those segments are interpolated, respectively. For continuous segments, radial point interpolation method is employed. In regard to discontinuous segments, the enriched discontinuous basis functions combining with radial point interpolation method are developed for simulating the discontinuity of displacement and stress field on surfaces of crack, and the near tip asymptotic field functions are employed for simulating the high gradient of stress field around crack tip, so that high accuracy and discontinuity property of a crack can be easily described. Stress intensity factors are calculated directly using displacement extrapolation by displacement field near crack tip. Some numerical examples are shown that the present method is effective and can be widely applied in some practical engineering.

Modeling complex crack problems using the three-node triangular element fitted to numerical manifold method with continuous nodal stress

A three-node triangular element fitted to numerical manifold method with continuous nodal stress, called Trig3-CNS (NMM) element, was recently proposed for linear elastic continuous problems and linear elastic simple crack problems. The Trig3-CNS (NMM) element can be considered as a development of both the Trig3-CNS element and the numerical manifold method (NMM). Inheriting all the advantages of Trig3-CNS element, calculations using Trig3-CNS (NMM) element can obtain higher accuracy than Trig3 element without extra degrees of freedom (DOFs) and yield continuous nodal stress without stress smoothing. Inheriting all the advantages of NMM, Trig3-CNS (NMM) element can conveniently treat crack problems without deploying conforming mathematical mesh. In this paper, complex problems such as a crucifix crack and a star-shaped crack with many branches are studied to exhibit the advantageous features of the Trig3-CNS (NMM) element. Numerical results show that the Trig3-CNS (NMM) element is prominent in modeling complex crack problems.

A modified dual scheme for solving an elastic crack problem

A dual scheme for solving a crack problem in terms of displacements is considered. The dual solution method is based on a modified Lagrange functional. The convergence of the method is investigated under a natural assumption of H1-regularity of the solution to the crack problem. A duality relation for the primal and dual problems is proposed.

A modified dual scheme for solving an elastic crack problem

A modified dual scheme for solving an elastic crack problem

Novel approach to FE solution of crack problems in the Laplacian-based gradient elasticity

Stress/strain concentration around a crack tip in ceramic foam-like structures, where the characteristic size of the foam cell is comparable with a typical length over which field quantities change significantly, is analysed. It is convenient to replace the foam structure by an effective continuum which retains all necessary characteristic features of the foam. To this end a homogenized cracked open cell ceramic foam is analysed using the strain/stress model based upon an implicit dependence of the non-local stress and strain on the local stress and strain in the form of an inhomogeneous Helmholtz equation which is solved together with the equilibrium equations. To preserve the advantage of the uncoupled system of equations, which means that one set of equations is solved prior the other one and that two sets of shape functions can be chosen independently, a recurrent sequence of well posed boundary value problems is constructed which can be solved using a classical 8-nodes isoparametric element for 2-D problems. The key idea behind the suggested procedure consists in replacing the scale parameter l by a parameter increment Δl chosen arbitrarily small. The given small disturbance causes a redistribution of the local stress/displacement field. After the new local stress/displacement field is known, another disturbance is introduced and so on. The procedure is stopped when the solution, exhibiting boundary layer behaviour, leads to the cohesive-like zone with length reaching a value uniquely related to the scale parameter l.