Dynamic Brittle Fracture Research Articles

A three-dimensional coupled thermo-elastic-plastic phase field model for the brittle-ductile failure mode transition of metals

Dynamic brittle fracture and shear banding are two typical failure modes of metals, and the transformation of the brittle-ductile failure mode has been observed in the Kalthoff test. This paper establishes a thermo-elastic-plastic coupled three-dimensional phase field model to simulate brittle-ductile failure mode transition of metals. The expression for the variation of the Taylor-Quinney coefficient with stress triaxiality is adopted, and the critical energy release rate is automatically adjusted using the Taylor-Quinney coefficient. Then, the Kalthoff test is simulated using the proposed model. The brittle-ductile failure mode transformation phenomenon is reproduced, which agrees well with the experimental results. It can be well proved that impact velocity is crucial in determining the transition to failure mode. At low-velocity impact, the energy is insufficient to drive the plastic accumulation of the shear band, resulting in brittle tensile fracture. At high-velocity impact, the energy is sufficient to drive the formation of adiabatic shear bands, resulting in tensile shear failure. In addition, three-dimensional simulations show that the tip of the shear band exhibits a crescent-shaped non-two-dimensional extension state under finite thickness. This numerical framework provides a predictive tool to understand the evolution of the dynamic failure of metals under impact loading.

Multi-time-step coupling of peridynamics and classical continuum mechanics for dynamic brittle fracture

Peridynamics (PD), as a nonlocal theory, is well-suited for solving problems with discontinuities, such as cracks. However, the nonlocal effect of peridynamics makes it computationally expensive for dynamic fracture problems in large-scale engineering applications. As an alternative, this study proposes a multi-time-step (MTS) coupling model of PD and classical continuum mechanics (CCM) based on the Arlequin framework. Peridynamics is applied to the fracture domain of the structure, while continuum mechanics is applied to the rest of the structure. The MTS method enables the peridynamic model to be solved at a small time step and the continuum mechanical model is solved at a larger time step. Consequently, higher computational efficiency is achieved for the fracture domain of the structure while ensuring computational accuracy, and this coupling method can be easily applied to large-scale engineering fracture problems.

Modeling dynamic crack growth in quasicrystals: Unraveling the role of phonon–phason coupling

In this work, we employed the phase-field fracture (PFF) model for dynamic brittle fracture in quasicrystals (QCs). The dynamic PFF formulation is derived using both elastodynamics and elasto-hydrodynamic theory. The temporal discretization is achieved using an implicit generalized-α method, which is unconditionally stable for a particular choice of constants. The model incorporates two distinct crack driving forces: (a) the contribution of elastic strain energy from both phonon and phason fields and (b) the contribution solely from the elastic energy associated with the phonon field. To validate the implementation of QCs, we have conducted benchmarking against recent results in the literature, particularly in the context of quasi-static loading. We addressed various paradigmatic case studies to demonstrate the dynamic crack nucleation and propagation in planar 2D decagonal QCs, with a primary emphasis on understanding the interplay between phonon and phason fields. Phason walls, representing low-energy crack paths resulting from atomic rearrangements, play a significant role in crack propagation by introducing an additional energy component. Notably, increasing the phonon–phason coupling constant expedites both crack initiation and propagation. Furthermore, variations in the phonon–phason coupling parameter result in different crack trajectories, observed in both uniaxial and biaxial loading scenarios. The developed code can be downloaded from https://github.com/Hirshikesh/Quasicrystal.git.

Shack–Hartmann wavefront sensing: A new approach to time-resolved measurement of the stress intensity factor during dynamic fracture

The stress intensity factor describes the stress state around a crack tip in a solid material and is important for understanding crack initiation and propagation. Because stresses cannot be measured directly, the characterization of the stress intensity factor relies on the measurement of deformation around a crack tip. Such measurements are challenging for dynamic fracture of brittle materials where the deformation is small and the crack tip velocity can be high. Digital gradient sensing (DGS) is capable of full-field measurement of surface deformation with a sub-micrometer sensitivity and a sub-microsecond temporal resolution, but it has only been demonstrated on centimeter-scale specimens with a spatial resolution of ∼ 1 mm. This makes it challenging to measure deformations close to the crack tip. Here, we demonstrate the potential of Shack–Hartmann wavefront sensing (SHWFS), as an alternative to DGS, for measuring surface deformation during dynamic brittle fracture of millimeter-scale specimens. Using an opaque commercial glass ceramic as an example material, we demonstrate the capability of SHWFS to measure the surface slope evolution induced by a propagating crack with a micrometer spatial resolution and a sub-microsecond temporal resolution. The SHWFS apparatus has the additional advantage of being physically more compact than a typical DGS apparatus. We interpret our SHWFS measurements of the surface slope by comparing them with 2D analytical predictions and phase-field simulations as well as 3D linear-elastic FEM simulations, based on which we discuss the relevance of 3D effects around the crack tip. Then, we introduce our procedure for extracting the apparent stress intensity factor associated with the propagating crack tip using SHWFS measurements. We conclude by discussing potential future enhancements of this technique and how its compactness could enable the integration of SHWFS with other characterization techniques including x-ray phase-contrast imaging (XPCI) for multi-modal characterization of dynamic fracture.

Meso-scale size effects of material heterogeneities on crack propagation in brittle solids: Perspectives from phase-field simulations

Brittle solids are often toughened by adding a second-phase material. This practice often results in composites with material heterogeneities on the meso scale: large compared to the scale of the fracture process zone but small compared to that of the application. The specific configuration (both geometrical and mechanical) of this mesoscale heterogeneity is generally recognized as important in controlling crack propagation behavior and, subsequently, the (effective) toughness of the composite. Here, we systematically investigate how dynamic brittle fracture navigates through a linear array of mesoscale inclusions. Using a variational phase-field (PF) approach, we compute the apparent crack speed and fracture energy dissipation rate to compare crack propagation (and the resulting toughening) under Mode-I loading for various configurations of inclusions. We identify an interplay between the size of inclusion and that of the K-dominant zone in the presence of elastic heterogeneity: matching these two sizes gives rise to the best toughening outcome for a given area fraction of inclusions. We discuss mechanisms that rationalize this observation and the importance of the length scale parameter used in PF models in interpreting simulation results. Our work sheds physical insight into the interaction between size effects and material properties, thereby opening a venue for the rational design of functional (architected) composites for dynamic fracture applications.

Meshfree simulation of dynamic brittle and quasi-brittle fracture using a local damage model based on lattice particle method

In this paper, the particle size-dependent issue of Lattice Particle Method (LPM) for simulating dynamic brittle and quasi-brittle fracture is addressed by coupling LPM with a local damage model based on fracture energy. The proposed model is simple and more robust than the Stress Intensity Factor (SIF)-based approach as it can model mixed-mode fracture with multiple cracks without any application-dependent tuning parameter. Numerical procedures for estimating the dissipative energy and the crack tip velocity for LPM simulation are also proposed. A series of benchmark problems involving dynamic fracture and crack branching are simulated using the proposed model. Good agreements are found against existing experimental observation and solutions from other numerical methods. Although a Cartesian-like (structured) lattice configuration is employed in the current LPM method, physically meaningful and accurate crack patterns can still be captured without any special numerical treatment.

Supershear rupture with a two-scale damage model

An investigation of intersonic rupture is performed in a two-scale framework, with a damage model obtained by homogenization from microstructures with dynamically evolving mode II microcracks. The macroscopic fracture is analyzed in connection with the microcrack propagation. It is found that the supershear transition and the subsequent intersonic failure are achieved by collective effects of propagation and coalescence of sub-Rayleigh microcracks ahead the macroscopic rupture front. This is similar to the behavior experimentally observed for mode I dynamic brittle fracture (Guerra et al. 2012). Numerical simulations of a pre-cracked plate under shear loading are performed and characteristic features like the formation of shear shock waves, the mother-daughter or direct transition mechanisms, the specific evolutions of the rupture speed or the symmetry breaking near the crack tip are retrieved. Parametric studies allow to determine the influence of the size of the microstructure, the applied loading and the fracture properties of the solid. For the asymmetric impact of a Homalite plate problem (Rosakis et al., 1999), the numerical results obtained with the damage model are successfully confronted with the experimental observations.

An adaptive multi-patch isogeometric phase-field model for dynamic brittle fracture

This paper presents an adaptive multi-patch isogeometric phase-field model based on Nitsche's method for simulating dynamic brittle fracture. Complex structures are accurately represented with multiple patches, and Nitsche's method is used to link two adjacent patches to ensure the compatibility and continuity of physical quantities (such as displacements, stresses, phase-field variables) on the coupling edge. LR NURBS with the local refinement ability are used for spatial discretization, while a generalized-α method is used for temporal discretization. The staggered algorithm and hybrid PFM formulation are adopted to enhance the solution efficiency of multi-field coupled system. In addition, the local refinement is executed utilising the phase-field threshold. Several dynamic fracture examples are used to validate the proposed approach's accuracy and effectiveness. The proposed approach can effectively model dynamic fracture propagation, and numerical results demonstrate that the adaptive analysis may significantly lower the computing cost.

An effective correspondence-based peridynamics-FEM coupling model for brittle fracture

Correspondence-based peridynamics (CBPD) is attractive and promising because of its outstanding ability to employ the traditional constitutive relations for material modeling. However, application of CBPD to brittle fracture problems is somewhat limited due to the numerical instability issue and lack of a rational bond failure criterion. To enable CBPD to accurately and stably predict brittle fracture problems, an improved energy-based bond failure criterion is firstly established in this work, in which the expression of bond potential is re-derived, and its simplified expression for isotropic linear elastic solids is clearly defined. The critical bond potential is determined by assuming it is directly proportional to the bond length, and the discretization error for the critical energy release rate in the discretized model is corrected with a correction factor. Besides the zero-energy mode control method, additional stabilization technique to reduce the influence of ill-posed or singular reference shape tensor on brittle fracture simulations is also explored to avoid the unexpected irregular damages during the fracture process. In addition, a direct coupling model of CBPD and FEM is established to improve the computational efficiency, and the ghost forces at the interface are verified to be negligible. Finally, several numerical examples involving typical quasi-static and dynamic brittle fracture problems are investigated, and fracture behaviors including the crack paths, load-displacement curves, and crack propagation speeds are all well predicted, which sufficiently demonstrates the accuracy and stability of the proposed method.

A nonlocal gradient damage model with energy limiter for dynamic brittle fracture

A nonlocal gradient damage model with energy limiter for dynamic brittle fracture

Element‐based coupling modeling of peridynamics and classical continuum mechanics for dynamic brittle fracture

AbstractIn this work, the hybrid peridynamic and classical continuum mechanical model based on the element‐type numerical discretization is used to study the dynamic fracture problem of brittle materials. The weak form of the motion equation for the hybrid model is derived by applying Hamilton's principle. An element‐based spatial discretization scheme is set up using the peridynamics‐based finite element method (PeriFEM) (Han and Li). The ‐convergence and m‐convergence of the hybrid model are analyzed to verify the numerical convergence and the mesh dependency. The influence of the area of the peridynamic model subdomain and the hybrid model subdomain on the calculation efficiency and accuracy is discussed. And the relationship between the crack propagation speed and the crack branching phenomenon is also investigated. Good agreements are obtained by comparing the crack path and the crack speed profile predicted by the hybrid model with those in the literature. Numerical results verify the ability and efficiency of the hybrid model for dynamic fracture problem.

Interfaces in dynamic brittle fracture of PMMA: a peridynamic analysis

Interfaces in dynamic brittle fracture of PMMA: a peridynamic analysis

A comparison of numerical stability for ESPH and TLSPH for dynamic brittle fracture

Dynamic brittle fracture is a numerically challenging problem that involves crack nucleation, formation, propagation, and material fragmentation. In this work, we use two forms of Smoothed Particle Hydrodynamics (SPH), namely Eulerian SPH (ESPH) and Total Lagrangian SPH (TLSPH) augmented with the pseudo-spring or virtual-link analogy for seamless modelling of crack formation, subsequent propagation, and material fragmentation. Being particle-based in nature, SPH is naturally capable of capturing finite deformation in materials, and the pseudo-spring or virtual-link analogies provide modelling of multiple discrete cracks without any additional condition, such as visibility criteria. We simulate the crack branching and propagation in a brittle polymeric material subjected to biaxial tensile loading with a pre-existing central notch. The numerical results using ESPH and TLSPH agree with the previously published experimental and numerical results. We have also simulated the dynamic fragmentation of a cylinder and compared the results. This work shows the capability of both ESPH and TLSPH frameworks to model dynamic brittle fracture especially crack branching and curving. It is also observed that the ESPH and TLSPH frameworks present similar results for minor material deformation problems. However, the ESPH framework shows better stability and capability for finite material deformation problems.

Virtual modelling integrated phase field method for dynamic fracture analysis

A non-deterministic phase field (PF) virtual modelling framework is proposed for three-dimensional dynamic brittle fracture. The developed framework is based on experimental observations, accurate numerical modelling, and virtually foreseeable dynamic fracture prediction module through the machine learning algorithm. The uncertain system inputs, including variabilities of material properties, are incorporated into dynamic fracture analysis. Phase field method is implemented to simulate the dynamic fracture behaviours of 3D cracked structures with variabilities and then to create the training database for virtual damage model. The virtual damage model omits the physical finite element approximation process and reveals the virtual governing relationship between the variational system inputs and fracture responses. This advantage enables the virtual model to provide reliable crack propagation prediction based on either experiment-based or numerical simulation and greatly improves the computational efficiency of dynamic fracture analysis. To establish the accurate virtual model in the training process, a newly developed extended support vector regression (X-SVR) method with T-spline polynomial kernel functions is adopted for its outstanding performance in handling complex high-dimensional problems. Based on different real-world engineering scenarios, multiple fracture failure criteria, both strength-based and serviceability-based, are selected and demonstrated in numerical investigations to visualise the proposed framework's workflow. The effectiveness as well as accuracy are verified by these examples, and it is observed that the computational efficiency of dynamic fracture analysis is greatly improved. With the proposed framework, a continuously updating dynamic fracture surveillance system can be potentially built for practical applications.

A dissipation informed peridynamic model for dynamic brittle fracture

In this paper, we propose a dissipation informed peridynamic (PD) model for dynamic brittle fracture. The viscosity is introduced in for dissipation compensation to recover the physical mechanism of fracture. The general type of deformation in solids is considered and the bulk viscosity force state is nonlocally constructed from the bond level as a minimum variation of conventional PD. Rather than adjusting the artificial coefficient, the signal processing method, i.e. the wavelet method is employed for viscosity evolution. With the wavelet method, the spurious stress waves can be identified and the dissipation deficiency can be naturally quantified. The proposed model is then verified by a rigorous validation simulation against the experiment results, which is still missing in the PD literature. The real loading condition is modeled to reproduce the ‘quasi-steady-state crack propagation’ problem and twenty different cases are included to cover the whole fracture stage. The results demonstrate that the model is capable to recover the physical mechanism and capture the dynamic brittle fracture. New insights into the dynamic brittle fracture are also gained by investigating the viscosity effect and deriving the crack motion equation.

Numerical simulations of dynamic fracture and fragmentation problems by a novel diffusive damage model

In this paper, we report numerical simulations for dynamic fracture and fragmentation problems in brittle materials using a recently developed split strain energy diffusive mass-field damage model in terms of standard finite element method (FEM). The enhanced constitutive laws for mass source and mass flux by means of strain energy decomposition are adopted to overcome certain drawbacks of the original non-split model. We present theoretical formulations in a more general way that is suitable and valid for both small and finite deformation regimes. The coupled system of equations is derived in which the momentum is governed to describe the time-dependent deformation of brittle solids whereas the mass-diffusion balance equations are for the evolution of mass density. We also develop simple techniques for determining crack velocity and estimating amount of mass loss. Numerical examples are considered for both small and finite deformations. The accuracy and performance of the developed theory for dynamic brittle fracture are illustrated through comparison and verification, which are to compare the computed results with respect to experimental data and other numerical methods in terms of crack path trajectories, dynamic response, energy dissipation, and other relevant numerical aspects.

Prediction of the peak load and crack initiation energy of dynamic brittle fracture in X70 steel pipes using an improved artificial neural network and extended Finite Element Method

In this paper, a robust technique is presented to predict the peak load and crack initiation energy of dynamic brittle fracture in X70 steel pipes using an improved artificial neural network (IANN). The main objective is to investigate the behaviour of API X70 steel based on two experimental tests, namely Drop Weight Tear Test (DWTT) and the Charpy V-notch impact (CVN), for steel pipe specimens. The mechanical properties in the brittle fracture behaviour of API X70 steel pipes are predicted utilizing numerical approaches with different crack lengths. Next, to simulate the impact of API X70 steel pipes at lower temperatures through a numerical approach, a cohesive approach using the extended Finite Element Method (XFEM) is used. The data obtained are used as input for the proposed IANN using Balancing Composite Motion Optimization (BCMO), Particle Swarm Optimization (PSO) and Jaya optimization algorithms, to predict the peak load values and crack initiation energy of dynamic brittle fractures in API X70 steel with different crack lengths. The results show the effectiveness of ANN-PSO and ANN-BCMO based on the convergence of the results and the accuracy of the prediction of the peak load and crack initiation energy. Note that, the source codes are publicly available at https://github.com/Samir-Khatir/JAYA-ANN.git.

An asynchronous variational integrator for the phase field approach to dynamic fracture

AbstractThe phase field approach is widely used to model fracture behaviors due to the absence of the need to track the crack topology and the ability to predict crack nucleation and branching. In this work, the asynchronous variational integrator (AVI) is adapted for the phase field approach of dynamic brittle fracture. The AVI is derived from Hamilton's principle and allows each element in the mesh to have its own local time step that may be different from others'. While the displacement field is explicitly updated, the phase field is implicitly solved, with upper and lower bounds strictly and conveniently enforced. In particular, two important variants of the phase field approach, the AT1 and AT2 models, are equally easily implemented. Several benchmark problems are used to study the performances of both the AT1 and AT2 models, and the results show that the AVI for the phase field approach significantly speeds up the computational efficiency and successfully captures the complicated dynamic fracture behavior.

Modeling dynamic brittle fracture in functionally graded materials using hyperbolic phase field and smoothed particle hydrodynamics

We present a novel particle-based computational framework for the numerical simulation of dynamic crack propagation in functionally graded materials under highly dynamic loading conditions and large deformations. The approach is based on an innovative computational method that solves phase field of brittle fracture with smoothed particle hydrodynamics. The meshfree nature of the discretization technique allows for the simulation of scenarios involving extreme deformations and material separation, as opposed to conventional mesh-based computational techniques such as the finite element method. At the same time, the damage evolution is governed by a hyperbolic partial differential equation that allows for efficient explicit time integration and avoids the complexities of solving linear systems of equations. The framework is verified and validated against other computational approaches and experimental results. Finally, the proposed approach is applied to some challenging impact scenarios that involve fast dynamics or large deformations, and it is shown that it can be easily used for identifying material gradation profiles that manipulate crack propagation.

Adaptive modelling of dynamic brittle fracture - a combined phase field regularized cohesive zone model and scaled boundary finite element approach

Adaptive modelling of dynamic brittle fracture - a combined phase field regularized cohesive zone model and scaled boundary finite element approach