Quasi-static Crack Propagation Research Articles

Phase-field modeling of fracture with physics-informed deep learning

We explore the potential of the deep Ritz method to learn complex fracture processes such as quasistatic crack nucleation, propagation, kinking, branching, and coalescence within the unified variational framework of phase-field modeling of brittle fracture. We elucidate the challenges related to the neural-network-based approximation of the energy landscape, and the ability of an optimization approach to reach the correct energy minimum, and we discuss the choices in the construction and training of the neural network which prove to be critical to accurately and efficiently capture all the relevant fracture phenomena. The developed method is applied to several benchmark problems and the results are shown to be in qualitative and quantitative agreement with the finite element solution. The robustness of the approach is tested by using neural networks with different initializations.

Quasi-Static and Dynamic Crack Propagation by Phase Field Modeling: Comparison with Previous Results and Experimental Validation

In this paper, experimental tensile tests for pre-cracked high Carbon steel ‘C90’ specimens were performed for quasi-static and dynamic loading. High loading velocity affects the crack patterns by preventing deflection. On the other hand, an efficient numerical tool based on the phase field model was developed and validated to predict brittle fracture trajectories. A staggered numerical scheme was adopted to solve the displacement and damage fields separately. Implementation efficiency in initiating and propagating cracks, even from an undamaged microstructure, was proved. The effect of the critical fracture energy density Gc on the crack path was tested; with smaller Gc, the crack patterns become more complex. In addition, the impact of loading velocities was examined, and earlier and faster crack formation and greater crack branching is observed with higher impact velocity. In this study, bidimensional plane stress cases were treated. The phase field model with hybrid formulation was able to predict crack pattern and especially crack arrest and branching found in the literature. The developed model accurately determined the transition zone of the crack path topology that has been observed experimentally.

A peridynamic-informed deep learning model for brittle damage prediction

Physics-informed Neural Network (PINN) has been introduced recently to predict and understand complex physical phenomena by directly incorporating feedback from governing equations. While PINNs offer remarkable capabilities, challenges arise when addressing discontinuities and heterogeneity, potentially compromising accuracy and reliability in prediction outcomes. In this study, a novel approach that combines the principles of peridynamic (PD) theory with PINN is presented to predict quasi-static damage and crack propagation in brittle materials. To achieve high prediction accuracy and convergence rate, the linearized PD governing equation is enforced in the PINN’s residual-based loss function. The proposed PD-INN is able to learn and capture complicated displacement patterns associated with different geometrical parameters, such as pre-crack position and length. The computational method presented in this study can be implemented to any PD constitutive model like Bond-Based PD or State-Based PD as well as any horizon factor. Several enhancements like cyclical annealing schedule and deformation gradient aware optimization technique are proposed to ensure the model would not get stuck in its trivial solution. The model’s performance assessment is conducted by monitoring the behavior of loss function throughout the training process. The results obtained through PD-INN demonstrate a satisfactory level of agreement in 2D and 3D when compared with the ground truth deformation from high-fidelity techniques like PD direct numerical method and Extended finite element method (X-FEM). Our results show the ability of the nonlocal PD-INN to predict damage and crack propagation accurately and efficiently.

Quasistatic progressive crack propagation behavior of crystalline rocks based on SCB tests

To explore the influence of mineral grains on the mechanical strength and crack propagation behavior of rocks, this study investigates the quasi-static crack propagation characteristics in four types of granular structured rocks (three varieties of granite and one type of diabase rock) through a semi-circular bend three-point bending test. The implementation of a servo-controlled crack-opening displacement mechanism facilitated the realization of quasistatic crack propagation. The entire experimental procedure was documented on a camera. Subsequently, the quasistatic propagation and development of cracks in crystalline rocks were examined using digital image correlation, whereas the distribution of mineral grains and their relationship with the pattern of crack evolution were investigated under optical microscopy. The paths of the cracks exhibited noticeable deviation, which were further influenced by the mineral grains. Specifically, cracks tended to circumvent harder minerals while gravitating towards softer ones, highlighting the significant role of mineralogical composition in determining the fracture behavior of granite.

Continuum-molecular modeling of planar micropolar media: Anisotropy, chiral properties and length-scale effects

This paper presents a continuum-molecular formulation for bi-dimensional micropolar media within the mathematical formalism of a revised peridynamic theory with oriented material points. A variational procedure is considered to derive the fundamental governing equations of the model, which postulates that material points interact through pair potentials allowing generalized pairwise actions to be derived as energy conjugates to properly defined pairwise deformation measures. While continuity of mass is assumed, constitutive laws are produced for long-range micro-interactions to reproduce the effective behavior of the material at the macro-scale. The definition of proper micromoduli functions respecting material symmetries and invariance properties allows then to obtain a non-local micropolar continuum based on pseudo-discrete kinematics, while providing a mechanism-based description of material anisotropy and coupling behaviors uncovered by classical elasticity. An analytic micro-macro moduli correspondence procedure is also established, based on the formal analogy with the constitutive tensor of centrosymmetric planar micropolar continua. As an important result of this research, we show that distinctive chiral effects of micropolar elasticity can be reproduced by introducing a directional independent pseudo-scalar pair potential, which turns out to be analytically vanishing when the rotationally invariant part of the corresponding continuum elastic tensor is invariant to mirror reflections as well. Moreover, the proposed formulation demonstrates sensitivity to elastic bending size-effect as specific property of structured materials homogenized as micropolar continua, and related to the characteristic length of their underlying microstructure. The resulting constitutive non-locality, which plays an important role in fracture problems, is conceptually separated with respect to the intrinsic non-local character of the model related to the integral nature of its governing equations, and then does not vanish when the horizon reduces to zero. The theoretical findings and the effectiveness of the model are successfully verified through illustrative examples referred to representative cases of structured materials homogenized as micropolar continua, including length-scale dependent quasi-static crack propagation as well as the mechanism-based description of the coupling between bulk strain and micro-rotation in elastic bi-dimensional homogenized chiral lattices.

Experimental investigation of crack propagation mechanism in refill friction stir spot joints of AA6082-T6

Since many aluminum alloys preferred in structural engineering can be welded conventionally only with great effort and energy input interest in alternative joining techniques is growing, such as solid state joining processes. In this work, the effect of refill friction stir spot welding (refill FSSW) on the crack propagation behavior in AA6082-T6 is studied. To be able to identify the individual fracture mechanism, refill FSSW was performed as a blind weld, i.e. only in one sheet designed as C(T)100 specimens. The vertical distance between notch and spot weld was varied and tested in two phases. First, a cyclic pre-crack was induced and then the specimen was caused to fail in quasi-static conditions, resulting in two different fracture modes. The results showed that the cyclic crack is dominated by residual stresses but the microstructure mainly influences the quasi-static crack propagation. It was also found that a stress concentration occurs in the transition area even without a hook. Furthermore, it was found that the crack propagation is not exclusively driven by the local strength but also by the angle at which the crack hits the spot weld.

An explicit improved meshless numerical manifold method for dynamic crack propagation

Accurately simulating the dynamic propagation of cracks is critical to investigating the mechanisms of rock fracture under dynamic loading. The meshless numerical manifold method, characterized by its node-based interpolation approximation akin to the meshless method and the dual-cover of numerical manifold method, is particularly suitable for crack analysis and has been successfully applied to quasi-static crack propagation. This paper aims to extend its application to the simulation of dynamic crack propagation. Initially, the meshless numerical manifold method's nodal arrangement and numerical integration scheme are enhanced to address dynamic problems more effectively, referred to as the Improved Meshless Numerical Manifold Method (iMNMM). Subsequently, we introduce a novel mass lumping method grounded in rigorous mathematical principles, an energy-conserving extended degrees of freedom inheritance strategy and a crack propagation criterion based on dynamic stress intensity factors. Additionally, the viscous artificial boundaries and the “large mass” acceleration method are incorporated to impose the acceleration and zero-displacement boundary conditions. Further, the improved explicit meshless numerical manifold method based on the central difference method is established for dynamic crack propagation. Finally, for verification, iMNMM is tested on several numerical examples. Numerical results manifest that iMNMM can enable the simulation of dynamic crack propagation with larger, constant time steps.

Cohesive zone modelling to predict crack growth under fatigue loading

This manuscript presents the partial results in the development of numerical models based on Cohesive Zone Modelling to analyse quasi-static and fatigue crack propagation in adhesive interfaces. This work applied to adhesive interface between fibre laminates and structural epoxy adhesive of a wind turbine blade 3rd shear-web connection to the trailing edge panels. Fracture characterization experiments using the Double Cantilever Beam with Unequal Bending Moments (DCB-UBM) specimen are performed, and results are used to determine different modelling parameters, such as the mixed-mode cohesive laws and fatigue damage parameters for cohesive crack propagation. The developed numerical model is compared and validated against sub-component testing on two different scale levels, using a Modified Tear Test (MTT) specimen geometry with simplified geometry and loading, and a more complex geometry sub-component specimen with a simplified Shear Web Disbond Test (SBDT). The work is carried out as part of the CORTIR-II project, funded by the Danish Ministry of Energy.

Mixed Finite Element for Crack Analysis in Functionally Graded Material

This study presents a mixed finite element formulated using isoparametric techniques and extended to analyze quasi-static crack propagation in functionally graded materials. The mixed mode is analyzed for homogeneous and functionally graded beamssubjected to three-point bending and for homogeneous and functionally graded plates under uniform axial tension. The mixed finite element results are compared with previous experimental and numerical results published in the literature.

Correction to: Experimental investigation of the alternate recurrence of quasi-static and dynamic crack propagation in PMMA

Correction to: Experimental investigation of the alternate recurrence of quasi-static and dynamic crack propagation in PMMA

Experimental investigation of the alternate recurrence of quasi-static and dynamic crack propagation in PMMA

Experimental investigation of the alternate recurrence of quasi-static and dynamic crack propagation in PMMA

Mechanics of heterogeneous adhesive contacts

Contact of elastic solids with heterogeneous interface energy of adhesion is considered. The known results on the asymptotic analysis of the quasi-static crack propagation along with the Griffith-type approach (based on the energy balance) and the Irwin-type approach (based on the concept of the contact stress intensity factor) are applied to generalize the Johnson–Kendall–Roberts (JKR) theory of adhesive contact. Under the assumption of slightly heterogeneous adhesion, the effective interface energy is determined depending on the approach used. The asymptotic equivalence between the three derived models, including that based on the mixed Griffith–Irwin-type approach, has been established. The leading-order asymptotic approximation for the pull-off force is derived. The cases of the ray-shaped and periodically-patterned adhesion are considered in detail.

Quasi-static crack propagation in soft materials using the material-sink theory

In the present study, a finite element formulation for the material-sink approach aimed at modeling quasi-static crack propagation in hyperelastic solids is developed. Breakage of molecular bonds leads to material separation and appearance of two new surfaces of a crack. However, the bond breakage is diffusive, and the loss of local bonds leads to the localized material (molecular) loss. The latter notion triggers consideration of mass density as a variable that numerically decreases in the area where damage localizes into a crack. This physical notion requires mathematical consideration of mass balance as an additional and active law, which regularizes the computational model. From the numerical point of view, the developed finite element formulation has displacement and density degrees of freedom. Also, a monolithic approach was applied that ensures stable incrimination of the nonlinear problem. Numerical examples of the fracture of aneurysm material demonstrate the high robustness of the proposed approach.

Adaptive coupling peridynamic least-square minimization with finite element method for fracture analysis

This study presents an adaptive coupling peridynamic least-square minimization with the finite element method (PDLSM-FEM) for fracture analysis. The presented method utilizes the PDLSM modeling discontinuities while maximizing the FEM region for computational efficiency. Within the presented adaptive PDLSM-FEM, only elements intersecting with the crack path and their neighboring elements are defined as PD elements, whose stiffness matrices are derived based on PDLSM equations. The remaining elements are conventional finite elements. Numerical integration of interaction integral is proposed and implemented to evaluate the stress intensity factors (SIFs) for 2-D problems. The criterion of maximum hoop tensile stress is employed for failure prediction. New contributions of this work include the adaptive coupling of PDLSM with FEM for minimizing the PD region and the application of the adaptive PDLSM-FEM to quasi-static crack propagation analysis. Simulations of three 2-D plane stress plates and one 3-D block with static or quasi-static cracks propagation are performed. Results show the proposed method improves computational efficiency substantially and has reasonable accuracy and good capability of crack propagation prediction.

Dynamic Versus Quasi-Static Analysis of Crack Propagation in Soft Materials

Abstract Cracks usually propagate dynamically that makes them so dangerous. However, most crack simulations are based on quasi-static analyses because they are simpler than the dynamic ones. Is it correct to use quasi-static analyses instead of the dynamic ones? Will the quasi-static and dynamic simulations provide similar results? We try to answer these questions in the present work. We compare results of quasi-static and dynamic simulations of crack propagation in aneurysm material. We use the material-sink (MS) approach, which is based on the notion of the diffused bond breakage. The latter feature implies a local loss of material and, consequently, decrease of mass density, which, in its turn, means that both stiffness and inertia go down in the damaged zone. The cancellation of inertia is an important feature of the MS approach in contrast to more formal regularization theories as phase field, gradient damage, and other nonlocal formulations. The MS approach is implemented within commercial finite-element software abaqus. A reduced mixed finite-element formulation is adopted to circumvent the volumetric locking and an implicit staggered solution algorithm is developed via the user-defined element subroutine UEL. Considered examples show that the onset of crack instability under static loads is followed by the dynamic rather than quasi-static crack propagation. Moreover, dynamic and quasi-static simulations, generally, provide different results.

Seamless coupling of bond- and state-based peridynamic and finite element analyses

This study presents a variational approach to couple PeriDynamic (PD) and Finite Element (FE) analyses to take advantage of their salient features. The PD and FE domains share the same nodes along their interface without an overlap region and any constraint conditions. Therefore, this approach does not require a morphing or a blending function that facilitates coupling over a transition zone. Although this approach is general and not limited to any commercial FE software, the PD interactions for bond-based (BB), ordinary state-based (OSB) and non-ordinary state-based (NOSB) are modeled with MATRIX27 elements native to ANSYS, a commercial software, for the purpose of demonstration. It is a general framework to perform BB, OSB and NOSB PD coupled with traditional elements. It provides the ability to practicing engineers in industry for integrating PD into their FE based design tools. The PD domains can share a boundary with FE domains or be completely embedded in the FE domain. The resulting global system of equations includes the contributions arising from the PD nodes and FE nodes. These equations are solved simultaneously without requiring an iterative procedure through implicit methods. Therefore, it is a direct (seamless) coupling approach without an overlap zone. The results are free of displacement kinks; thus, unphysical stress concentrations do not appear near the interface between the PD and FE regions. Its accuracy is demonstrated by considering isotropic elastic plates subjected to various types of boundary conditions under quasi-static and dynamic loading conditions. Also, its accuracy is demonstrated for quasi-static crack propagation by considering double cantilever beam and three-point bend test specimens. The results from the coupled PD-FE approach agree well with those of FEM for all boundary and loading conditions.

On the role of bond-associated stabilization and discretization on deformation and fracture in non-ordinary state-based peridynamics

As it is well known, non-ordinary state-based peridynamic (NOSB PD) models suffer from inherent instability issues, which can severely affect deformation and fracture predictions. These issues have been identified and tackled through the years using stabilization methods that work by introducing additional terms in the formulation or by disturbing the radial symmetry of the horizon, leading to stabilized solutions. A popular and effective method is the bond-associated correspondence formulation. In this work, we make an in-depth investigation on the role of bond-associated stabilization and discretization on the deformation and fracture behaviour in NOSB PD by considering three different comprehensive and detailed numerical studies. Firstly, the convergence of the NOSB PD linear elastic formulation is investigated under both the L2 and H1 error norms. The role of the influence function is also considered and analysed. Different combinations of bond-associated horizon and point-associated horizon values are used to determine the role of these parameters in obtaining accurate solutions. Secondly, we consider a static analysis example where we compare PD with the finite element method (FEM), in order to verify the accuracy of the PD models for stress analysis. Lastly, a quasi-static crack propagation analysis benchmark example is presented to assess the accuracy of fracture predictions and the effect of varying the discretization parameters is studied for both load–displacement curves and crack paths.

Two-grid based sequential peridynamic analysis method for quasi-static crack propagation

We introduce two-grid based sequential analysis algorithm for implicit peridynamics to find the quasi-static solutions for crack propagation problems. The sequential analysis of implicit peridynamic formulation is capable of studying quasi-static fracturing of brittle materials. The failure model in peridynamics, however, usually allows one to break a single bond between two nodes under a certain damage criterion in tracing equilibrium points, which results in the whole simulation quite lengthy and inefficient. In order to circumvent the efficiency issue, we propose the two-grid based sequential peridynamic analysis algorithm, in which a coarse grid is used to find the equilibrium path with sequential analysis and the converged solution is properly prolongated to the fine grid. The failure model is also implemented in the coarse grid and the computed damage information is properly mapped into the fine grid. In this way, the proposed two-grid based scheme incorporates breaking multiple bonds into one step of sequential analyses. We document the superior behavior of the proposed scheme with respect to the conventional solution procedures of implicit peridynamics in both solution accuracy and computational efficiency for quasi-static crack propagation.

Improved concept for iterative crack propagation using configurational forces for targeted angle correction

In many applications, fracture mechanics is indispensable in predicting structural failure. In this paper, a concept for predicting discrete crack paths according to the criterion of maximum energy release rate, which uses the finite element method, is presented. Within existing approaches to determine the incremental crack propagation direction, on the one hand, the information of the current crack is used in explicit approaches, leading to inaccuracies. On the other hand, the information of introduced virtual cracks can be used in implicit approaches, with the required number of virtual cracks determining the computational effort. This work introduces a 2D concept for quasi-static crack propagation in elastic materials and that uses configurational forces to estimate an angle error of a virtual crack increment; the concept uses this angle error in an iterative crack correction. The concept is evaluated using a simplified model for one crack propagation increment and a three-point bending model that contains holes for predicting crack paths in combination with the incremental crack propagation method. The results are compared with those of existing explicit and implicit crack propagation direction concepts, as well as experimental results. It is shown that the presented concept fulfils the concept for maximum energy release rate as accurately as a computationally expensive implicit concept, while the computational effort of the proposed concept is close to fast explicit concepts.

Imposition of local boundary conditions in peridynamics without a fictitious layer and unphysical stress concentrations

This study introduces a general approach for the imposition of local boundary conditions in non-ordinary state-based peridynamics (NOSB PD) to eliminate the displacement kinks near the boundary without a fictitious layer under quasi-static loading conditions. It identifies the underlying reason for the unphysical displacement kinks. Under an imposed linear displacement field, the NOSB PD equilibrium equation is not satisfied near the boundary due to the unsymmetric horizon of material points. However, the equilibrium equation derived by using the PD differential operator is satisfied at such material points. Therefore, the material body is divided into three regions to satisfy the equilibrium equations and to impose displacement and tractions boundary conditions. This approach does not deviate from the original NOSB PD; however, it provides a simple solution to eliminate the displacement kink near the boundary, which leads to unphysical stress concentrations. Its efficacy is demonstrated by considering elastic rectangular and square plates subjected to various types of boundary conditions leading to homogeneous as well as nonhomogeneous deformations. The creep response of a rectangular plate further proves the robustness of the present approach. Also, a quasi-static crack propagation from a pre-existing crack in a square plate under mode-I, mode-II and mixed-mode loading conditions demonstrates its capability for failure prediction based on the critical stretch criteria. Finally, its applicability for 3D analysis is demonstrated by considering a rectangular prism under applied stretch and normal stress.