Dynamic Crack Propagation Problems Research Articles

The fully coupled thermo-mechanical dual-horizon peridynamic correspondence damage model for homogeneous and heterogeneous materials

To accurately address the thermally induced dynamic and steady-state crack propagation problems for homogeneous and heterogeneous materials involving crack branching, interfacial de-bonding and crack kinking, we propose the fully coupled thermo-mechanical dual-horizon peridynamic correspondence damage model (TM-DHPD). To this end, the integral coupled equations for TM-DHPD are firstly derived within the framework of thermodynamics. And then, the alternative dual-horizon peridynamic correspondence principle is used to derive the constitutive bond force state, heat flow state and their general linearizations. Moreover, the unified criterion for bond damage is proposed to characterize the internal bond damage in a single material and the interface bond damage in dissimilar materials. To ensure convergence and accuracy, the coupled equations are solved using the standard implicit method without the use of artificial damping. In both homogeneous and heterogeneous materials, some representative and challenging numerical cases are examined, such as dynamic crack branching in a centrally heated disk and multiple interface failure of thermal barrier coating. The numerical results are in good agreement with the available experimental results or the previous predictions, which shows the great potential of the proposed TM-DHPD in addressing the physics of numerous thermally induced fractures in the real-world engineering problems.

Multivariate uncertainty analysis of fracture problems through model order reduction accelerated SBFEM

An efficient Monte Carlo simulation (MCS) method that analyses the influences of multivariate uncertain parameters on the response of cracked structures is proposed. The scaled boundary finite element method (SBFEM) is used to solve the static and dynamic crack propagation problems and obtain a full-order snapshot model. The combination of a proper orthogonal decomposition with model order reduction and a radial basis function is used to achieve accelerated stochastic analysis based on MC simulations. The proper orthogonal decomposition-radial basis function (POD-RBF) method is more difficult to use directly for multivariate problems. Thus, we proposed an efficient operation suitable for multiple input variables. In crack propagation simulations, an optimized polygon meshing algorithm is used. Numerical results show that the proposed method can stochastically analyse multidimensional input variables and effectively obtain the responses of cracked structures, which verifies the effectiveness of the POD-RBF method.

Towards a unified nonlocal, peridynamics framework for the coarse-graining of molecular dynamics data with fractures

Molecular dynamics (MD) has served as a powerful tool for designing materials with reduced reliance on laboratory testing. However, the use of MD directly to treat the deformation and failure of materials at the mesoscale is still largely beyond reach. In this work, we propose a learning framework to extract a peridynamics model as a mesoscale continuum surrogate from MD simulated material fracture data sets. Firstly, we develop a novel coarse-graining method, to automatically handle the material fracture and its corresponding discontinuities in the MD displacement data sets. Inspired by the weighted essentially non-oscillatory (WENO) scheme, the key idea lies at an adaptive procedure to automatically choose the locally smoothest stencil, then reconstruct the coarse-grained material displacement field as the piecewise smooth solutions containing discontinuities. Then, based on the coarse-grained MD data, a two-phase optimization-based learning approach is proposed to infer the optimal peridynamics model with damage criterion. In the first phase, we identify the optimal nonlocal kernel function from the data sets without material damage to capture the material stiffness properties. Then, in the second phase, the material damage criterion is learnt as a smoothed step function from the data with fractures. As a result, a peridynamics surrogate is obtained. As a continuum model, our peridynamics surrogate model can be employed in further prediction tasks with different grid resolutions from training, and hence allows for substantial reductions in computational cost compared with MD. We illustrate the efficacy of the proposed approach with several numerical tests for the dynamic crack propagation problem in a single-layer graphene. Our tests show that the proposed data-driven model is robust and generalizable, in the sense that it is capable of modeling the initialization and growth of fractures under discretization and loading settings that are different from the ones used during training.

S-version finite element strategy for accurately evaluating local stress in the vicinity of dynamically propagating crack front in 3D solid

Developing a numerical method to accurately simulate the brittle crack propagation phenomenon is crucial for ensuring the safety of large-scale steel structures. Although recent studies have demonstrated that using the local fracture stress criterion as the fracture condition is effective, the most critical issue is that the methods for evaluating local stress in the vicinity of the dynamically propagating crack front have not been established. To address this, this paper proposes a strategy for analysing a dynamically propagating crack in 3D solids on the basis of the s-version finite element method (s-method). In the proposed strategy, the local mesh is defined in the vicinity of the crack front as an ideal structured mesh aligned with both the crack front direction and crack propagation direction to accurately simulate the local stress field. The dynamically propagating crack front was modelled by a combination of the nodal force release method and local mesh update. The proposed strategy was verified by evaluating the accuracies of the local stress in stationary and dynamically propagating circular crack problems as well as those of the stress intensity factor. The results demonstrate that the proposed strategy provides unprecedented accuracy and efficiency of local stress evaluation in problems of dynamic crack propagation in a 3D solid without requiring any complicated remeshing procedures. Therefore, the proposed strategy has potential as the basis of a numeral framework for analysing dynamic brittle crack propagation problems dominated by the local fracture stress criterion.

Analysis of dynamic crack propagation in two-dimensional elastic bodies by coupling the boundary element method and the bond-based peridynamics

A novel method for predicting dynamic crack propagation based on coupling the boundary element method (BEM) and bond-based peridynamics (BBPD) is developed in this work. The special feature of this method is that it can take full advantages of both the BEM and PD to achieve a higher level of accuracy and efficiency. Based on the scale of the structure and the location of cracks, the considered domain can be divided into a non-cracked region and a cracked region. For the non-cracked region, a meshfree boundary-domain integral equation method (meshfree-BDIEM) is employed in the analysis to reduce the dimension by one and to increase the computational efficiency. The bond-based PD is applied to simulate the cracked region, which can model the initiation and propagation of the cracks automatically. The boundary nodes from the BEM on the interfaces can interact with the material points from the PD directly. By using the displacement continuity and force equilibrium conditions on the interfaces, a combined model is obtained by merging the mass, stiffness, and force matrices from each region of the domain. Both the implicit and explicit BEM-PD coupled solution methods can deliver accurate results without inducing the ghost forces. Several benchmark problems of dynamic crack propagation have been modeled by using the method, which demonstrate that the developed BEM-PD coupled approach can be an efficient numerical tool to model the dynamic crack propagation problems.

Dynamic crack face contact and propagation simulation based on the scaled boundary finite element method

This paper extends the static contact problem of crack faces to include dynamic contact and crack propagation based on the relevant literature (Zhang et al., 2018). The scaled boundary finite element method is used to solve dynamic crack face contact and crack propagation problems. Based on the scaled boundary finite element shape function considering side-face loads, the system dynamic equilibrium equation and related characteristic matrix are presented. The Lagrange multiplier method is used to establish the contact model between the crack faces. The meshing and remeshing algorithm is carried out by using arbitrary polygonal meshes. When a crack propagates, the remeshing operation involves only a small region of polygons around the propagating crack tip. Mesh mapping technology is used to transfer the response parameters to newly generated nodes. The proposed method is validated by analysing the impact of two identical bars and the dynamic contact of crack faces in the Koyna Dam. The results show that the dynamic contact model can effectively simulate the opening and closing of crack faces, and the longer the crack is, the greater the influence of the non-linearity of the contact along a crack face on the dynamic stress intensity factor and dynamic response. Finally, a dynamic crack propagation problem is simulated. Compared with the dynamic crack response without considering the contact constraint, when considering the contact constraint, the crack initiation angle is closer to the corresponding theoretical solution, and the crack propagation path is closer to the corresponding experimental result. Thus, the dynamic contact constraint condition of crack faces has an important influence on crack propagation.

A time‐discontinuous peridynamic method for transient problems involving crack propagation

AbstractA time‐discontinuous peridynamic method (TDPD) for modeling transient problems of wave and crack propagation in solids is presented in this article. The main features of the TDPD are that the displacement field and the velocity field are independently interpolated in the temporal domain by using cubic and linear functions, respectively. Jump terms representing discontinuities of the variables are introduced between the adjacent time steps. The weak form and a new time integration scheme are derived by combining the peridynamic governing equations with the time‐discontinuous method. The displacement field is continuous while discontinuous jumps are introduced into the velocity field. Those characteristics enable the TDPD to accurately capture the sharp gradient in stress and effectively control the spurious numerical oscillation. The effectiveness of the TDPD is first demonstrated through two representative numerical examples of wave propagation. Three additional examples are presented to show the advantages of the TDPD in the simulation of dynamic crack propagation problems. It is shown that TDPD provides more accurate and satisfactory results when compared with the traditional time integration scheme.

A coupling model of XFEM/peridynamics for 2D dynamic crack propagation and branching problems

Peridynamics (PD) and extended finite element method (XFEM) are coupled in the explicit integration scheme to study the 2D dynamic fracture problems. The advantages of PD for the crack propagation and XFEM for the strong displacement discontinuity description of the crack are combined together. The proposed method can improve the computation efficiency in comparison of the pure PD, and can avoid using macroscopic criteria to judge the crack propagation orientation of XFEM. The coupling scheme for the stiffness and mass matrices of PD and XFEM is given in the coupling region. The horizon length δ is examined to study their influences on the convergence of the proposed method. The accuracy of the dynamic fracture simulation using the proposed method is also validated in comparison of experimental and other numerical methods.

Application of PDS–FEM to simulate dynamic crack propagation and supershear rupture

Dynamic crack propagation problems, including intersonic rupture, are simulated with two Hamiltonian based formulations of particle discretization scheme (PDS) FEM: the traditional displacement momentum form and the strain momentum form, for which consistent momentum conserving and symplectic integration schemes are derived. Numerical results are verified, and validated by comparing with photoelastic observations of a dynamic mode-I crack captured with a 1 Mfps camera. Both methods are successful in accurately reproducing the crack patterns observed in classical 2D and 3D dynamic rupture scenarios, as well as the near crack tip stress field during the propagation. The two methods are found to be numerically indifferentiable, although the displacement based method offers a significantly better computational performance. As a demonstrative application, we simulate the super-shear rupture in earthquakes, modeling the contact at the fault surface by a linear slip weakening friction law. The Burridge–Andrews mechanism naturally appears in the simulations, making the crack front jump from the sub-Rayleigh regime to the intersonic regime and propagate while producing shear Mach cones.

Discrete solid element model applied to plasticity and dynamic crack propagation in continuous medium

In this work, the discrete solid element model (DSEM) applied to plasticity and dynamic crack propagation problems in the continuous medium is developed and implemented. First, the Drucker’s postulate and the consistency condition are used to establish the plastic flow rule in the DSEM. The plastic scale factor which characterizes the plastic displacement increment of spherical elements is derived by the classical plastic mechanics. The elastoplastic contact constitutive equations of the strain-hardening material are established, which can be used for elasticity, unloading and loading. Second, to accurately simulate the mechanical behavior of continuity, the spring stiffness of spherical elements on the boundary of the DSEM is strictly deduced using the principle of conservation of energy, and the relationship between the spring stiffness and the continuous constitutive parameters is established. Third, the bilinear contact softening model is adopted to simulate the crack propagation in the continuous medium. The criteria for the crack propagation based on the fracture energy of the material are developed. The numerical examples are presented to show the capability and effectiveness of the DSEM in simulating the dynamic buckling and crack propagation problems. The obtained results are in agreement with experimental and numerical results published by other researchers.

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.

Stable discontinuous space–time analysis of dynamic interface crack growth in orthotropic bi-materials using oscillatory crack tip enrichment functions

ABSTRACT The problem of dynamic interface crack propagation in layered composites is addressed in this paper. Recently developed dynamic orthotropic bi-material enrichment functions are utilized to allow for accurate capturing of the oscillatory stress and displacement fields near the interface crack tip and proper evaluating of the crack tip velocity. The time discontinuous finite element and balance of recovery methods are employed during the crack propagation process in order to increase the stability and accuracy of the solution in dynamic crack propagation problems. Also, a suitable crack propagation criterion and an iterative algorithm are used to determine the interface crack tip velocity in each time step. Fracture mechanics parameters, such as dynamic stress intensity factor, phase angle and energy release rate are accurately predicted using the interaction integral method and the crack velocity dependent auxiliary fields. Numerical simulations of dynamic crack propagation in a homogeneous isotropic material, dynamic interface crack propagation in an isotropic bi-material, and dynamic interface crack propagation in an orthotropic bi-material are performed and the predictions are compared with the reference experimental and analytical results and the numerical data available in the literature.

Coupling of 2D discretized Peridynamics with a meshless method based on classical elasticity using switching of nodal behaviour

PurposeThe paper aims to use a switching technique which allows to couple a nonlocal bond-based Peridynamic approach to the Meshless Local Exponential Basis Functions (MLEBF) method, based on classical continuum mechanics, to solve planar problems.Design/methodology/approachThe coupling has been achieved in a completely meshless scheme. The domain is divided in three zones: one in which only Peridynamics is applied, one in which only the meshless method is applied and a transition zone where a gradual transition between the two approaches takes place.FindingsThe new coupling technique generates overall grids that are not affected by ghost forces. Moreover, the use of the meshless approach can be limited to a narrow boundary region of the domain, and in this way, it can be used to remove the “surface effect” from the Peridynamic solution applied to all internal points.Originality/valueThe current study paves the road for future studies on dynamic and static crack propagation problems.

Numerical modelling of two-phase ceramic composite response under uniaxial loading

Two-phase ceramic composites (e.g. Al2O3+ZrO2 – Zirconia-toughened Alumina (Birman and Byrd, 2000; Sadowski, 2012, 2011)), due to their extremely high strength and increasing toughness are widely used nowadays in various industries, e.g. (Sadowski and Golewski, 2012; Sadowski, 2008; Sadowski and Golewski, 2011). At the macroscale, the main dissipative mechanism is brittle cracking, that can be faithfully modelled within the framework of classical Linear Elastic Fracture Mechanics (LFEM). However, many essential properties, such as fracture toughness, depend on microstructural phenomena, related to grain size, orientation, residual stresses and various toughening mechanisms, that are intentionally introduced into the microstructure. Thus, a good understanding of these processes is the most important tool for a fully conscious design of these advanced materials, from manufacturing to the end of structural lifetime.In this paper, a basic framework with a numerical example for two-dimensional micromechanical analysis of multiphase brittle composites is described, using two radically different theoretical and numerical approaches: continuum mechanics with cohesive-type cracks, completely discrete (peridynamical) theory.One can conclude that simplified FE models can’t reflect mechanical response of complex systems with cracks in a quantitative way. Peridynamics, despite its numerical inefficiency, seems to be a powerful tool for both static and dynamic crack propagation problems in heterogenous materials under thermomechanical loading.

A 2D discrete model with a bilinear softening constitutive law applied to dynamic crack propagation problems

In the present work, we propose the incorporation of a bilinear softening constitutive law in a purely linear elastic discrete model previously developed by the authors. The new model includes information of the fracture energy for the constitutive law. Three numerical examples of dynamic crack propagation are provided to demonstrate the effectiveness and robustness of the modification proposed. The obtained results are presented in terms of the crack path, the crack speed ,and the dissipated energy. These results are compared with those obtained with a previous discrete model developed by the authors and other numerical results published by other researchers. Final comments on the performance of the new model are carried out.

Structural Design Optimization of Dynamic Crack Propagation Problems Using Peridynamics

Based on a bond-based peridynamics theory for dynamic crack propagation problems, this paper presents a design sensitivity analysis and optimization method. Peridynamics has a peculiar advantage over the existing continuum theory in the mathematical modelling of problems where discontinuities arise. For the design optimization of the crack propagation problems, a non-shape design sensitivity is derived using the adjoint variable method. The obtained adjoint sensitivity of displacement and strain energy turns out to be very accurate and efficient compared to the finite different sensitivity. The obtained design sensitivities are futher utilized to optimally control the position of bifurcation point in the design optimization of crack propagation in a plate under tension. A numerical experiment demonstrates that the optimal distribution of material density could delay the position of bifurcation.

Particle difference method for dynamic crack propagation

This paper presents a novel particle difference method (PDM) for solving dynamic crack propagation problems. The PDM is based on the strong formulation that directly discretizes the governing partial differential equations for space and takes advantage of fast computation speed owing to the exemption of numerical integration and the full differentiation of the approximation by using the Taylor polynomial expanded by the moving least squares method. It also differentiates the discrete equations for time through both explicit and implicit algorithms; specifically, the Newmark method (NM) is effectively modified for complete explicit and implicit time integrations. Nodal topology change due to the crack propagation modeling is easily conducted by simple addition and deletion of nodes. Dynamic fracture simulation is successfully performed by the help of the visibility criterion and dynamic energy release rate evaluation. Robustness and effectiveness of the PDM are thoroughly verified through various numerical experiments.

Adjoint design sensitivity analysis of dynamic crack propagation using peridynamic theory

Based on the peridynamics of the reformulated continuum theory, an adjoint design sensitivity analysis (DSA) method is developed for the solution of dynamic crack propagation problems using the explicit scheme of time integration. Non-shape DSA problems are considered for the dynamic crack propagation including the successive branching of cracks. The adjoint variable method is generally suitable for path-independent problems but employed in this bond-based peridynamics since its path is readily available. Since both original and adjoint systems possess time-reversal symmetry, the trajectories of systems are symmetric about the u-axis. We take advantage of the time-reversal symmetry for the efficient and concurrent computation of original and adjoint systems. Also, to improve the numerical efficiency of large scale problems, a parallel computation scheme is employed using a binary space decomposition method. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It is demonstrated that the peridynamic adjoint sensitivity involving history-dependent variables can be accurate only if the path of the adjoint response analysis is identical to that of the original response.

Shape Design Sensitivity Analysis of Dynamic Crack Propagation Problems using Peridynamics and Parallel Computation

Using the bond-based peridynamics and the parallel computation with binary decomposition, an adjoint shape design sensitivity analysis(DSA) method is developed for the dynamic crack propagation problems. The peridynamics includes the successive branching of cracks and employs the explicit scheme of time integration. The adjoint variable method is generally not suitable for path-dependent problems but employed since the path of response analysis is readily available. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It turns out that C1-continuous volume fraction is necessary for the accurate evaluation of shape design sensitivity in peridynamic discretization.

Finite element simulation of dynamic crack propagation process using an arbitrary Lagrangian Eulerian formulation

ABSTRACTIn this paper, the arbitrary Lagrangian Eulerian formulation is employed for finite element modelling of dynamic crack propagation problem. The application phase simulation of computational dynamic fracture is applied to model by which the crack propagation history and variation of crack velocity are predicted using the material dynamic fracture toughness. The dynamic solution of problem is accomplished using the implicit time integration method. The convective terms due to mesh‐material motion are taken into account via the convection equation. A robust and efficient mesh motion technique, that its equations need not to be solved at every time step, is employed in Eulerian phase. The mesh connectivity is preserved during the analysis. So, the successive remeshing of model is eliminated. When the dynamic fracture criterion is satisfied for crack growth, the presented algorithm allows the crack to advance by splitting the material particle at the crack tip. The dynamic energy release rate is calculated at each time step to determine dynamic stress intensity factor. The predicted results are compared with those obtained through the experimental study and remeshing technique cited in the literature. The proposed computational algorithm leads to an accurate and efficient simulation of dynamic crack propagation process.