Dynamic Crack Problems Research Articles

A stable numerical framework for long-time dynamic crack analysis

This paper introduces a stable numerical framework designed to address dynamic crack problems over long-time intervals. The initial step involves the temporal discretization of the governing dynamic equilibrium equations using the arbitrary order Krylov deferred correction method. To ensure precise boundary condition matching, a novel numerical implementation is incorporated into the Krylov deferred correction technique. Subsequently, the resulting system of spatial partial differential equations at each time node is solved using the meshless generalized finite difference method with 4th-order expansions in regions close to the crack-tips and 2nd-order expansions in areas far from the crack-tips. This combined approach capitalizes on the strengths of both the Krylov deferred correction technique and the generalized finite difference method, enabling stable simulations of dynamic cracks with the large time step and without the need for mesh generation. Notably, we refine the collocation nodes near crack tips to attain accurate numerical results for displacement and stress field. Several numerical experiments involving diverse impact loadings are conducted to validate the developed framework. Furthermore, a comparison is made between the dynamic stress intensity factors obtained using our approach and those from existing methods.

Non-Fourier thermoelastic interaction of two collinear cracks in a functionally graded layer

Designed with a predetermined composition profile, functionally graded materials receive much attention in serving as the thermal barrier coating to resist severe thermal loading. By means of the dual-phase-lag theory, the non-Fourier effect is considered in establishing the theoretical thermoelastic model of the cracked layer under sudden thermal shocks. The novelty of the present work lies in revealing the interaction effects of two collinear Griffith cracks embedded in a functionally graded layer on the transient temperature and thermal stress fields under harsh conditions. Via employing the Fourier sine and cosine transforms, coupled with the Laplace transform, the dynamic mixed mode crack problem is transformed into a group of Cauchy-type singular integral equations. Numerical calculations are implemented to evaluate the transient temperature and stress intensity factors. The influences of the crack spacing, the nonhomogeneous parameters, and the thermal lags on the thermal and stress concentrations are explored. The results indicate that both mode I and II thermal stress intensity factors of the outer crack tips are noticeably higher than those of the inner crack tips, even up to 71% and 128%, respectively. A larger fracture risk may occur with the closer two adjacent cracks.

A preconditioning method with auxiliary crack tip subproblems for dynamic crack propagation based on XFEM

In this paper, a domain decomposition based preconditioning method is developed to accelerate the Krylov subspace method for solving the linear system arising from the extended finite element discretization of the dynamic crack problem. Based on the observation that the crack tip area has a significant impact on the convergence of the iterative method while the other mesh points do not, the finite element mesh is partitioned into two types: the regular subdomains and the crack tip subdomains. To construct the additive Schwarz preconditioner, the global matrix is partitioned into submatrices which are solved exactly in the crack tip subdomains by the LU factorization and inexactly in the regular subdomains by an incomplete LU factorization. As the crack propagates, we develop a scheme to update the subdomain problems without resolving of them, in which, in order to save the computational cost, only the subdomains around the crack are updated, and all the other regular subdomains remain unchanged. To further speed up the Krylov subspace method, an auxiliary subproblem including crack tips of the previous and current time steps is constructed and solved to provide a better initial guess. We carefully studied the properties of the linear system and the performance of the proposed algorithm. The numerical experiments indicate that the proposed method works well for the simulation of dynamic crack propagation with multiple crack tips.

Numerical analysis of dynamic thermoelastic two-dimensional crack problem combined with non-Fourier heat transfer equation by finite-difference time-domain method

In recent years, a variety of microscopes using heat sources, such as photoacoustic microscopes, have been developed. In this observation principle, reflection and diffraction phenomena of two waves with different properties, namely thermal wave and elastic wave, are utilized. However, the thermal wave cannot be reproduced using the classical Fourier-type heat conduction equation. In order to overcome this mathematical difficulty, we had derived a coupled nonlinear partial differential equation consisting of the heat conduction with a delay time and the dynamic thermoelastic equation, and had simulated numerically the propagation process of thermal and elastic waves for one-dimensional problem. In this study, thermal and elastic waves were simulated for a two-dimensional plane problem with and without defect using the finite-difference time-domain method. The results showed that the reflection and interference phenomena of two waves can employ for detecting small internal defects existing near the surface of target sample.

Study on the Fracture Process Zone near the Mode I Dynamic Crack Tip

Evaluation of the shape and size of the fracture process zone near the mode I dynamic crack tip is still a problem unsolved completely at present. The research on the relationship between the fracture process zone and crack velocity near the mode I dynamic crack tip is quite limited, and some researchers have also developed experimental methods or numerical methods. In this research, based on the theory of elastodynamics and the complex stress function method, an approximate method for solving the mode I dynamic crack problem was proposed. The fracture process zone near the mode I dynamic crack tip was analyzed. The results showed that the areas of the fracture process zone determined based on the approximate method are nearly the same as the results obtained based on the well-known stress fields. The approximate method could provide a good reference for determining the fracture process zone near the mode I dynamic crack tip since no analytic methods had been found for evaluating the fracture process zone near the dynamic crack tip to the authors’ knowledge.

Dynamic fracture analysis of Kane–Mindlin plates using the dual boundary element method

In this paper, a new dual boundary element formulation is presented for dynamic crack problems in finite-thickness plates under in-plane loadings. The proposed formulation is based on a first-order plate theory (Kane–Mindlin theory) taking into account a coupled out-of-plane fracture mode due to in-plane shear loading and the effect of plate thickness on stress intensity factors, which cannot be considered within the framework of the two-dimensional elastic theories. The dynamic stress intensity factors are evaluated based on the crack opening displacements. Three benchmark examples are presented including the mixed-mode fracture of a finite plate. The effect of plate thickness on the dynamic stress intensity factors is investigated. The dynamic stress intensity factors obtained using the proposed formulation are shown to be in good agreement with the results from 3D finite element simulations.

Numerical simulation of dynamic fracture behavior in bulk superconductors with an electromagnetic-thermal model

Single-grain GdBCO bulk superconductors have significant applications potential due to their ability to trap stable and large magnetic fields. The internal fracture of these bulk superconductors caused by Lorentz force and thermal load is a key issue in their practical applications. The aim of this work presented here is to investigate the mechanical behavior of the bulk superconductor during pulsed field magnetization. The H-formulation and heat transfer equation are used to obtain the electromagnetic force and thermal load in the bulk with and without defects. Numerical simulations show strong local enhancement of the electromagnetic load at the crack tips. Moreover, dynamic stress intensity factors at the crack tips are presented based on the two-dimensional state-based peridynamic theory. In addition, the crack propagation path is predicted. Finally, different dynamic crack problems are considered to discuss the influence of crack, void, inclusion, and hole on the mechanical stability of the bulk. Results show that defects increase the risk of damage of superconducting bulks.

Static and dynamic fracture analysis in elastic solids using a multiscale extended isogeometric analysis

The main objective of this paper is to develop a novel multiscale approach for simulating static and stationary dynamic crack problems in two-dimensional elastic solids through an extended isogeometric analysis using the Non-Uniform Rational B-Splines (NURBS). This multiscale strategy consists of two techniques that the Nitsche’s method is applied for coupling different length-scales meshes on one hand, and on the other hand, a local enrichment technique is taken to describe the cracks independently of the mesh in the refined region by introducing the enrichment function into the isogeometric analysis displacement approximation. This method inherits the advantage of XIGA and simultaneously improves his limitation on the local mesh refinement. The proposed approach possesses several advantages in fracture modeling: (i) the singularities of the crack-tip fields can be captured precisely and conveniently by the enrichment techniques; (ii) higher-order convergence rates is achieved and (iii) stress smoothing results are achieved owing to higher continuity of NURBS basis functions. Numerical results of static and stationary dynamic crack problems validate its accuracy and efficiency.

Dynamic dual boundary element analyses for cracked Mindlin plates

In this paper, a new dual boundary element formulation is presented for dynamic crack problems in Mindlin plates. The displacement and traction boundary integral equations are derived in the Laplace frequency domain to allow for a boundary-only formulation. The cracked plate is modelled with the dual boundary element method and dynamic plate bending stress intensity factors are evaluated. Four benchmark examples are presented including mode I and mixed mode deformation. Such stress intensity factors obtained are shown to be in excellent agreement with finite element results as well as published results.

ACCURATE COMPUTATION ON DYNAMIC SIFS USING IMPROVED XFEM

Compared to the standard extended finite element method (XFEM), the improved XFEM overcomes, in theory, the linear dependence and the ill-conditioning issues, and improves in magnitude of orders the efficiency of linear system solve. In particular, in dynamic crack propagation problems, thanks to the exclusion of the extra dofs on crack tip enriched nodes, the new method eliminates the issue of energy inconsistency or the inconservitive energy transfer caused by dof dynamics, and provides more accurate dynamic stress intensity factors (DSIFs) with much less numerical oscillation. To the best of our knowledge, numerical solution on DSIFs for crack propagation under dynamic loading remains engineeringly unsatisfied. In this paper, the extra-dof free XFEM is extended to implicitly dynamic crack propagation problems-still a remaining difficulty for the current XFEMs. The implicit Newmark algorithm is used for time discretization. A dynamic interaction integral method is employed for DSIFs for both stationary and moving cracks under dynamic loading. Compared with the interaction integral method for static cracks, the dynamic method considers the effects of crack growth speed and inertia terms. The paper investigates in detail the influences of element size, mass matrix formulation, time step size, crack tip enriched zone size, inertia term, crack growth speed, and J-domain mesh/cell sizes of the interaction integral. Numerical tests show satisified accuracy and efficiency of the new method for dynamic crack problems. In particular on the challenging benchmark problem Mode I semi-infinite stationary and then moving crack, the new improved XFEM provides the best DSIFs up to the time of publication.

An adaptively refined XFEM with virtual node polygonal elements for dynamic crack problems

By introducing the shape functions of virtual node polygonal (VP) elements into the standard extended finite element method (XFEM), a conforming elemental mesh can be created for the cracking process. Moreover, an adaptively refined meshing with the quadtree structure only at a growing crack tip is proposed without inserting hanging nodes into the transition region. A novel dynamic crack growth method termed as VP-XFEM is thus formulated in the framework of fracture mechanics. To verify the newly proposed VP-XFEM, both quasi-static and dynamic cracked problems are investigated in terms of computational accuracy, convergence, and efficiency. The research results show that the present VP-XFEM can achieve good agreement in stress intensity factor and crack growth path with the exact solutions or experiments. Furthermore, better accuracy, convergence, and efficiency of different models can be acquired, in contrast to standard XFEM and mesh-free methods. Therefore, VP-XFEM provides a suitable alternative to XFEM for engineering applications.

The Effects of Newmark Method Parameters on Errors in Dynamic Extended Finite Element Method Using Response Surface Method

The Newmark method is an effective method for numerical time integration in dynamic problems. The results of Newmark method are function of its parameters (β, γ and ∆t). In this paper, a stationary mode I dynamic crack problem is coded in extended finite element method )XFEM( framework in Matlab software and results are verified with analytical solution. This paper focuses on effects of main parameters in Newmark method for dynamic XFEM problems. Also use of the response surface method (RSM) a regression model is presented for estimating error of dynamic stress intensity factors (DSIF) with high validity according to results of analysis of variance (ANOVA). This work enables one to understand the effect of Newmark parameters on error of DSIFs and to find optimum β and γ for a determined number of time steps (N). This procedure is highly effective in order to manage the computational cost and enhance the accuracy at the desired domain. The effect of the considered parameters on error, is investigated using RSM in Minitab software and optimum state for minimization of errors is illustrated.

A new symplectic analytical singular element for crack problems under dynamic loading condition

In this paper, the two-dimensional (2D) crack problem under dynamic loading condition is studied. Firstly, by the precise time domain expanding algorithm, the recursive formulation of the governing equation is derived, and the recursive equation can be solved by FEM. Then, a new Symplectic Analytical Singular Element (SASE) defined by the analytical symplectic eigen expanding terms is further developed for the investigated problem. The area near the crack tip is represented by using the SASE, while the other area is meshed by conventional elements. Taking advantage of the SASE, the crack problem can be solved with higher accuracy. Simultaneously, the dynamic stress intensity factors (DSIFs) can be obtained directly without any post processing through the relationship between the DSIFs and the eigen expanding coefficients. The numerical accuracy and stability is investigated carefully through the conducted numerical examples. It has been demonstrated that the proposed SASE for dynamic crack problem is effective and efficient.

Dynamic analysis of crack problems in functionally graded materials using a new graded singular finite element

In this research a recently developed graded singular element is used for studying dynamic crack problems in linear elastic isotropic functionally graded materials (FGMs) with spatially varying elastic parameters. The formulation for the graded singular element is obtained by analyzing the crack tip stress field using the Westergaard stress function method. Shape functions are extracted by presenting expressions for displacements near the crack tip in terms of the complex functions with unknown coefficients. The stiffness and consistent mass matrices for the singular element are then determined based on the principle of minimum potential energy and kinetic energy of the element, respectively. The global stiffness and mass matrices are then generated by assembling matrices for the singular and adjacent regular elements by considering the displacement continuity at their common boundary. The results from the current method are compared with existing solutions to evaluate the accuracy of the technique. A set of simulations are performed using this element for analyzing dynamic crack problems in FGMs whose elastic properties vary normal to the crack line. Compared with traditional elements, the proposed element includes higher order terms in addition to the singular term, which leads to accurate description of the stress field near the crack tip. It also significantly reduces the calculation time needed in comparison with using very fine mesh of traditional elements. With the use of the singular element, no post processing technique is required for calculating fracture parameters such as stress intensity factors, energy release rate, and mode mixity.

Determination of Dynamic Crack Initiation and Propagation Toughness of a Rock Using a Hybrid Experimental-Numerical Approach

AbstractIn order to determine the mode-I (opening mode) dynamic fracture toughness of a rock, cracked-straight-through Brazilian disc (CSTBD) specimens with strain gauges glued on the specimen’s surface were diametrically impacted by a split Hopkinson pressure bar. A hybrid experimental-numerical approach was used to determine the dynamic initiation toughness, and it was also used in conjunction with a universal function to obtain the dynamic propagation toughness. Precautions were taken for getting the time history of dynamic stress intensity factors. Numerical accuracy, stability, and convergence to the solution were guaranteed by analyzing beforehand a widely-recognized benchmark dynamic crack problem. Using the numerical technique the result of the conducted trial analysis corresponded almost exactly with the classical result. The dynamic initiation toughness and dynamic propagation toughness of the rock increase with increasing dynamic loading rate and crack propagation speed, respectively, demonstra...

Analysis of Dynamic Crack Propagation using MLSDifference Method

This paper presents a dynamic crack propagation algorithm based on the Moving Least Squares(MLS) difference method. The derivative approximation for the MLS difference method is derived by Taylor expansion and moving least squares procedure. The method can analyze dynamic crack problems using only node model, which is completely free from the constraint of grid or mesh structure. The dynamic equilibrium equation is integrated by the Newmark method. When a crack propagates, the MLS difference method does not need the reconstruction of mode model at every time step, instead, partial revision of nodal arrangement near the new crack tip is carried out. A crack is modeled by the visibility criterion and dynamic energy release rate is evaluated to decide the onset of crack growth together with the corresponding growth angle. Mode I and mixed mode crack propagation problems are numerically simulated and the accuracy and stability of the proposed algorithm are successfully verified through the comparison with the analytical solutions and the Element-Free Galerkin method results.

Dynamic energy release rate in couple-stress elasticity

This paper is concerned with energy release rate for dynamic steady state crack problems in elastic materials with microstructures. A Mode III semi-infinite crack subject to loading applied on the crack surfaces is considered. The micropolar behaviour of the material is described by the theory of couple-stress elasticity developed by Koiter. A general expression for the dynamic J-integral including both traslational and micro-rotational inertial contributions is derived, and the conservation of this integral on a path surrounding the crack tip is demonstrated.

Effects of different crack‐face boundary conditions on the dynamic intensity factors in linear magnetoelectroelastic solids

AbstractNew multifield materials for the development of smart devices and structures are receiving increasing attentions. Composite materials consisting of piezoelectric and piezomagnetic phases with an additional magnetoelectric coupling effect [1] offer advanced opportunities and may be used for broadband sensing, actuating devices and many other smart devices and structures which are required in engineering sciences. The dynamic crack analysis of magnetoelectroelastic solids is of special importance because such materials are often very brittle and the corresponding results have a direct relevance to the design and optimization of smart structures. The boundary element method (BEM) has been shown to be very attractive and efficient for solving dynamic crack problems in linear magnetoelectroelastic solids [3]. A critical issue in the numerical simulation of crack problems in linear magnetoelectroelastic solids is a realistic description of the mechanical and electromagnetical crack‐face boundary conditions, which is considered in this paper. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamic Crack Problems Using Meshless Method

A Meshless Approximation Based on the Radial Basis Function (RBF) Is Developed for Analysis of Dynamic Crack Problems. A Weak Form for a Set of Governing Equations with a Unit Test Function Is Transformed into Local Integral Equations. A Completed Set of Closed Forms of the Local Boundary Integrals Are Obtained. as the Closed Forms of the Local Boundary Integrals Are Obtained, there Are No any Domain or Boundary Integrals to Be Calculated Numerically in this Approach.

The singular edge-based smoothed finite element method for stationary dynamic crack problems in 2D elastic solids

In this paper, the recently developed singular edge-based smoothed finite element method (sES-FEM) is further developed for dynamic crack analysis in two-dimensional elastic solids. The objective of this work is to provide an efficient and accurate numerical simulation tool for the dynamic fracture behaviors of linear elastic solids in the framework of the strain smoothing approaches. Following this approach, the strains are smoothed and the system stiffness matrix is thus performed using the strain smoothing technique over the smoothing domains associated with the element edges. In order to accurately capture the singular fields at the crack-tip, a two-layer singular 5-node crack-tip element is employed. The governing dynamic equations are transformed into a weakened weak (W2) form, which is then discretized into a sES-FEM system of time-dependent equations to be solved by the unconditionally stable implicit Newmark time integration method. To analyze the fracture behaviors of linear elastic solids, mixed-mode dynamic stress intensity factors (DSIFs) are evaluated using the domain forms of the interaction integrals in terms of the smoothing technique. Four test examples including pure mode-I and mixed-modes are studied to validate the accuracy of the proposed method. The computed results for the normalized DSIFs are compared with analytical and other numerical reference solutions in a wide range of benchmark dynamic crack problems which shows high accuracy of the sES-FEM.