Crack Discontinuities Research Articles

Phase-field modeling of 3D fracture in elasto-plastic solids based on the modified GTN theory

An appropriate failure model of ductile materials needs to consider complex phenomena at the microscopic level, such as the nucleation, growth, and coalescence of microvoids and the final fracture at the macroscopic level. A recent popular continuous phase-field method based on the regularization of sharp crack discontinuities can easily model macroscopic cracks. In this work, we developed a coupled model to predict the crack law when materials undergo ductile fracture. The coupling model combined the ductile phase-field fracture model of the material with the shear-modified GTN model to illustrate the microscopic damage accumulation caused by the growth of voids and the macroscopic crack initiation and propagation. In addition, a modified phase-field degradation function was established, combining phase field, void volume fraction, and shear damage. This paper also considers the crack propagation law when the materials undergo ductile fracture under different stress triaxialities. In this study, we used the subroutines UEL and UMAT in the commercial finite element software Abaqus to give the staggering solution algorithm for ductile phase-field fracture of the materials. The shear-modified GTN model was written by the subroutine UMAT, while the ductile phase-field fracture model was written by the subroutine UEL. The results of the numerical simulations showed good agreement with the experimental damage failure mechanisms and the load–displacement curves.

Extended virtual element method for two-dimensional linear elastic fracture

In this paper, we propose an eXtended Virtual Element Method (X-VEM) for two-dimensional linear elastic fracture. This approach, which is an extension of the standard Virtual Element Method (VEM), facilitates mesh-independent modeling of crack discontinuities and elastic crack-tip singularities on general polygonal meshes. For elastic fracture in the X-VEM, the standard virtual element space is augmented by additional basis functions that are constructed by multiplying standard virtual basis functions by suitable enrichment fields, such as asymptotic mixed-mode crack-tip solutions. The design of the X-VEM requires an extended projector that maps functions lying in the extended virtual element space onto a set spanned by linear polynomials and the enrichment fields. An efficient scheme to compute the mixed-mode stress intensity factors using the domain form of the interaction integral is described. The formulation permits integration of weakly singular functions to be performed over the boundary edges of the element. Numerical experiments are conducted on benchmark mixed-mode linear elastic fracture problems that demonstrate the sound accuracy and optimal convergence in energy of the proposed formulation.

A deep learning-based approach for refined crack evaluation from shield tunnel lining images

This paper develops a deep learning-based approach that extends the PANet model by adding a semantic branch which refines the process of crack evaluation to reduce inaccuracies associated with crack discontinuities and image skeletonization. The PANet model is used to segment cracks from shield tunnel lining images, and an A* algorithm incorporated into the semantic branch computes the width and the shortest length of each crack from the segmented binary images. A comparison of experimental results shows that the performance of the proposed approach is better than those of Mask R-CNN, U-Net, and DeepCrack. The proposed approach also demonstrates its superiority at mitigating crack disjoint problems and skeletonization error. The error rates of length and width quantification for the A* algorithm are lower than those for the medial-axis-skeletonizing algorithm. The evaluation metrics indicate that the proposed model is an alternative approach to segment and quantify cracks in the field.

Three dimensional cells with embedded strong discontinuity for material failure analysis by the boundary element method

The Boundary Element Method (BEM) associated with the Continuum Strong Discontinuity Approach (CSDA) can be used as a non-geometric alternative for the modelling of crack growth in physically non-linear three-dimensional problems of solid mechanics. This work makes use of the refereed approach through the implementation of volumetric hexahedral cells with embedded discontinuity that enables the evaluation of discontinuous jumps in the displacement field along a crack discontinuity surface. This is made with an internal equilibrium verification, using a regularized isotropic damage constitutive model. Although this formulation has been extensively studied for two-dimensional problems over the past few years, this is the first time that a three-dimensional extension is presented. The strong discontinuity regime in the cells were imposed directly after the end of the elastic regime, as a typical behaviour of the fracture process of brittle materials. These cells are needed only at the specific region of the solid domain where the cracks propagation is supposed to occur, while the remaining non-discretized regions of domains are considered to work in elastic regime. Some simple numerical examples shows the feasibility of such methodology.

Life enhancement of cracked structure by piezoelectric patching underneath thermo-mechanical loading environment

This article presents crack repair in the structure by piezoelectric material and predict the fatigue life of the structure. The extended finite element method is applied to model the crack discontinuity. Heaviside function is used for modeling the crack surface while branch functions are used for modeling the crack front. Effects of active and passive repair on structural integrity are investigated under a thermo-mechanical loading environment at zero R-ratio. Paris law is applied to predict life in terms of fatigue cycles. The efficacy of the repair is measured in terms of fatigue life and stress intensity factor.

Mapped phase field method for brittle fracture

Phase field method has proven capable of producing complex crack patterns in solids. It introduces a continuous phase field to regularize the sharp crack discontinuities. However, the applicability of this method to engineering problems is hindered by its computational costs. In this work, we proposed a mapped phase field method as a possible route to resolve this issue. The core of this method is a map that connects the physical domain to a parametric domain, which is essentially a local reparametrization of the physical domain where large gradients are expected. By the use of this map the strongly varying fields can be approximated by a much smoother function. The reparametrized solution is solved via standard finite element in the parametric domain and then mapped back to the physical domain. A simple analysis shows that, with a properly defined map, such method consumes much less computational resources compared with conventional phase field method, without loss in accuracy. The map can also be easily manipulated to adapt to the evolution of cracks, and thus providing a flexible framework to simulate complex crack patterns without any knowledge of the crack path in advance. The advantages of our proposed method are further shown by four different numerical examples: single edge crack under symmetric and anti-symmetric loading, three-point bending, and L-shape panel under cyclic loading.

Numerical estimation of a Mode III fracture Mechanics parameter for three-dimensional elliptic boundary value problems with a crack singularity

In this study two spatial problems of solid bodies with a surface crack singularity (V-notch) are examined. These are problems in the domain of linear elastic fracture Mechanics which are reduced to Laplace equation problems by considering a Lamé potential. The boundary singularity is numerically treated by first determining the general solution of the governing equation, in the vicinity of the surface crack and by expressing it as an asymptotic expansion, the coefficients of which are approximated by polynomials. The remaining numerical steps are followed according to the singular function boundary integral method (SFBIM), which in the literature is known as one of the so-called Trefftz methods. Very fast convergence and very high accuracy are observed in implementing the method to solve a general 3-D model problem of a solid body having a cross section in the form of an ellipse and a specific spatial model problem of a steel rivet, which connects metal members of a structure. In fact, the CPU time and the numerical error recorded with this numerical technique are significantly smaller than those achieved with the finite element method (FEM) which was used to solve the same problems. The calculated value of Mode III fracture Mechanics parameter (FMP) indicates that there is no danger of crack propagation. Thus, the extension of the method to this category of problems is quite interesting since it involves a novel application of this algorithm in fracture Mechanics, a domain in which the method demonstrates its capability to treat problems with crack discontinuities and provide results more efficiently (less computing effort and greater accuracy) than the classical FEM with grid refinement. This advantage of the SFBIM can be exploited for research or design purposes, in several fields of applied Mechanics, by implementing its algorithm with subroutines embedded in engineering packages and aiming to make them more efficient in solving specific problems.

Fracture analysis of plastically graded material with thermo-mechanical J-integral

In this paper, the influence of plasticity graded property and thermal boundary conditions have been investigated on the fracture parameter, i.e. J-integral using the extended finite element method. A complete computational methodology has been presented to model elasto-plastic fracture problems with geometrical and material nonlinearities. For crack discontinuity modeling, a partition of unity enrichment concept was employed with additional mathematical functions like Heaviside and branch enrichment for crack discontinuity and stress field gradient, respectively. The modeling of the stress–strain relationship of the material is implemented using the Ramberg–Osgood material model and geometric nonlinearity is modeled using an updated Lagrangian approach. The isotropic hardening and von-Mises yield criteria are considered to check the plasticity condition. The elastic predictor–plastic corrector algorithm is employed to capture elasto-plastic stress in a cracked domain. The variation in plasticity properties for plastically graded material is modeled by exponential law. Furthermore, the nonlinear discrete equations are numerically solved using a Newton–Raphson iterative scheme. Various cracked problem geometries subjected to thermal (adiabatic and isothermal conditions) and thermo-mechanical loads are simulated for stress contours and J-integrals using the elasto-plastic fracture mechanics approach. A comparison of the results obtained using extended finite element method with literature and the finite element analysis (FEA) package shows the accuracy and effectiveness of the presented computational approach. A component-based problem, i.e. a Brazilian disc subjected to thermo-mechanical loading, has been solved to show the adaptability of this work.

Fracture of thermo-elastic solids: Phase-field modeling and new results with an efficient monolithic solver

Thermally induced cracking occurs in many engineering problems such as drying shrinkage cracking of concrete, thermal shock induced fracture, micro cracking of two-phase composite materials etc. The computational simulation of such a fracture is complicated, but the use of phase-field models (PFMs) is promising as they can seamlessly model complex crack patterns like branching, merging, and fragmentation by treating the crack discontinuity as thin band of diffuse damage. Despite the success of phase-field models there are two major issues in previous PFMs of thermally induced fracture. Firstly, these models, which are mostly based on a PFM using a simple quadratic degradation function without any user-defined parameters, provide solutions that are sensitive to a length scale. Secondly, they are limited to brittle fracture only. As a solution, we extend the phase-field regularized cohesive zone model (PF-CZM) of Wu [JMPS, 103 (2017)], with a rational degradation function dependent on elasticity and fracture related material parameters, to thermoelastic solids. Furthermore, we present a monolithic BFGS algorithm to solve the three-field (displacements, phase-field and temperature) coupling equations altogether. Multiple thermally induced fracture problems, simulated within the framework of the finite element method, are presented with predictions in good agreement with previous findings and experiments. The proposed model is shown to give better performance in capturing the physics of thermally induced fracture: it provides length scale insensitive responses. And the monolithic solver is 4∼5 times faster than the conventional alternating minimization solver.

A phase-field modeling method for the mixed-mode fracture of brittle materials based on spectral decomposition

The phase-field method is artful to handle sharp crack discontinuities by regularizing the sharp crack surface topology with a function. In the phase-field theory, the phase field characterizing the crack is driven by the strain energy. The strain energy of the crack-driving part is the key to simulate the realistic crack surface. In this study, a new framework of the phase-field method is proposed based on the unified tensile fracture criterion, in which the crack-driving strain energy is evaluated by the spectral representation with a thermodynamic consistent projection operator to ensure the thermodynamic consistency. The mixed-mode fracture of brittle materials can be well simulated. Furthermore, a new degradation function is proposed to simulate the brittle materials with different degradation rates. The proposed phase-field modeling method is implemented by a staggered algorithm, in which the phase field and displacement field are successively solved and updated by the subroutines UEL and UMAT of the commercial finite element software ABAQUS. The comparison with experimental data shows that the proposed phase-field modeling method can quantitatively and qualitatively simulate both the cracking and mechanical behavior of mixed-mode fracture.

ABAQUS Implementation to Analyze the Stress Intensity Factor for Laminated Composite Plate

The objective of this work is Abaqus implementation to analyze the effect of fibre orientation angle on stress intensity factor for Single Edge Notched (SEN) 3D laminate composite plate under tensile loading by various cases. The Abaqus software is used to analyze the various problems through the extended finite element method (XFEM). The effects of various fibre orientation angles on Sress Intensity Factor (SIF), Von Misses stress and displacement are also investigated. The presented work would be useful for enhancing the use of Abaqus software to analyze the composite structure with crack and other discontinuities under various conditions.

XFEM with partial Heaviside function enrichment for fracture analysis

In this work, an extended finite element method with partial Heaviside function enrichment (XFEM-P) in the crack tip element is proposed for fracture analysis for the first time. In this method, the crack tip element is divided into cracked and uncracked domains naturally by the crack geometry. In the cracked domain, Heaviside function enrichment is employed to simulate the crack discontinuity across the crack surface. In the uncracked domain, the displacement from standard finite element method is employed. Ramp function is used to ensure the displacement continuity in the method. Through this way, the crack tip can be arbitrarily located inside the crack tip element, which is a great advantage over traditional counterpart, i.e., XFEM with only Heaviside function enrichment (XFEM-H). Thus the mesh work for the XFEM-P is much easier than that of the XFEM-H, resulting in great conveniences for crack propagation. The new method is applied to several examples. From the computed results, it is observed that the XFEM-P offers more accurate results than the XFEM-H in terms of strain energy and SIFs. The XFEM-P performs better than XFEM-H in numerical properties such as convergence rate and condition number.

New Methodological Approach towards a Complete Characterization of Structural Fiber Reinforced Concrete by Means of Mechanical Testing Procedures

This work proposes a novel methodology for the complete characterization of fiber reinforced concrete (FRC). The method includes bending tests of prismatic notched specimens, based on the Standards for FRC, tensile and pure shear tests. The values adopted by the standards for designing FRC are the obtained from bending tests, typically fR3, even for shear and pure tension loading. This paper shows that the remaining strength of FRC, supplied by the fibers, depends on the type of loading. In the case of shear and tensile loading the prescriptions of the standards may be unsafe. In this work, the remaining halves of specimens subjected to bending test are prepared and used for shear and tension tests. This means significant savings in specimen preparation and a greater amount of information for structural use of FRC. The results provide relevant information for the design of structural elements of FRC compared with the only use of data supplied by bending tests. In the case of tensile tests, fLOP values are 42% of the strength of the equivalent bending results, being 31% the average reduction in remaining resistance in comparison with the bending test. Pure shear tests showed, for 0.5 mm shear displacement, that the shear resistance is greater than 160% of that expressed according to bending tests. In addition, a video-extensometry system was used to analyze the crack generation and cracking patterns. The video-extensometry applied to shear tests allowed the assessment of the sliding values and crack opening values at the crack discontinuity. These values may be quite relevant for the study of the FRC behavior when subjected to shear according to the shear-friction model theories.

A 3D meshfree crack propagation algorithm for the dynamic fracture in arbitrary curved shell

In this work, we proposed a meshfree crack propagation algorithm for dynamic fracture in arbitrary curved shell structures. In the numerical simulation of shell fracture, the treatment of crack discontinuity and simulation of fracture fragments are involved. In meshfree methods, since the connection between particles can be changed easily, which makes it suitable for dealing with such problems. In this study, firstly, the crack initiation algorithm for both ductile and brittle material is proposed based on the damage evolution. Besides, in the simulation, we assume that the crack tip coincides with the stress point, which means that the crack tip moves from one stress point to another, and by connecting the crack tip, the crack path is generated. Then a 3D visibility criterion for arbitrary curved shell structures is established, based on which the support domain of stress points is cut by the crack path to model the crack discontinuity. Finally, the crack propagation algorithm for dynamic fracture in curved shells is established; several numerical examples are studied to verify the reliability of the proposed algorithm.

Elasto-Plastic Fracture Modeling for Crack Interaction with XFEM

Multiple cracks/voids have been extensively seen in structural components. The stress field gets affected due to the crack interaction, so the effect of multiple cracks should be considered in design analysis. Further, due to service load, the stress concentrations are induced at the crack tips, which is the source of failure of the component. Hence, the current work has been proposed to investigate the crack interaction phenomenon in structural components under mechanical loading conditions. A mesh-independent computational approach, namely “Extended Finite Element Method (XFEM),” has been implemented for crack discontinuities modeling. Ramberg–Osgood material model has been used for the nonlinear stress–strain relationship of material. Isotropic hardening has been used with von Mises yield criteria to decide the plasticity condition. Further, nonlinear discrete equations have been solved by the Newton–Raphson iterative scheme. Few crack interaction problems are numerically solved by the presented EPFM approach, and results are presented in the form of fracture parameters.

Optical method to quantify the evolution of whole-field stress in fractured coal subjected to uniaxial compressive loads

Understanding and quantitatively characterizing the hidden distribution and evolution of the whole-field stress in fractured coal is crucial for accurately predicting crack propagation and connection that affect the transport capability of coalbed methane in reservoirs during fracture stimulation. However, it is challenging to directly reveal and quantify the hidden stress field using traditional methods because of the difficulties associated with extracting and identifying the stress distribution around randomly distributed, discontinuous, irregular-geometry cracks as well as with fabricating a model with embedded cracks. This study presents an optical method to reveal and quantify the continuous evolutions of the whole-field principal stress difference and shear stress in fractured coal by combining three-dimensional (3D) printing technology and digital photoelasticity. 3D printing technology is used to replicate the transparent model of fractured coal. The ten-step phase-shifting method and quality-guided phase unwrapping algorithm in photoelasticity are used to estimate the whole field photoelastic parameters (isoclinic and isochromatic) in the fractured coal model. The comparison between experimental and numerical results is provided to further prove the efficacy of this method in handling the natural fractured coal structure. The effects of crack irregularity and discontinuity on the stress evolution in fractured coal are evaluated as well.

Water‐induced failure mechanics for concrete

AbstractThis work outlines a micro‐mechanical framework for modeling water‐induced damage mechanism of concrete. Concrete has a highly heterogeneous micro‐structure and its composite behavior is very complex. Due to that various effects must be considered for analyzing failure response at micro scale, e.g. modeling the solid skeleton, fluid bulk phases and their interaction. For obtaining a deeper understanding of the influence of water on the concrete at the micro‐level, a micro‐computed‐tomography (micro‐CT) scan has been conducted at IfB to illustrate the micro‐structure geometry and concrete content. These data are required to build the constitutive model and the design of the numerical simulations, in line with [1]. Then a micro‐mechanical model is developed for the coupled problem of fluid‐saturated heterogeneous porous media at fracture. The modeling of microscopic cracks in porous heterogeneous media can be achieved in a convenient way by a continuum phase‐field approach, which is based on the regularization of sharp crack discontinuities [2,3]. This avoids the use of complex discretization methods for crack discontinuities, and can account for complex crack patterns. The numerical examples proposed in this contribution stem out from the DFG Priority Program SPP 2020 “Cyclic Damage Processes in High‐Performance Concretes in the Experimental Virtual Lab”.

Extended virtual element method for the Laplace problem with singularities and discontinuities

In this paper, we propose the extended virtual element method (X-VEM) to treat singularities and crack discontinuities that arise in the Laplace problem. The virtual element method (VEM) is a stabilized Galerkin formulation on arbitrary polytopal meshes, wherein the basis functions are implicit (virtual)—they are not known explicitly nor do they need to be computed within the problem domain. Suitable projection operators are used to decompose the bilinear form on each element into two parts: a consistent term that reproduces the first-order polynomial space and a correction term that ensures stability. A similar approach is pursued in the X-VEM with a few notable extensions. To capture singularities and discontinuities in the discrete space, we augment the standard virtual element space with an additional contribution that consists of the product of virtual nodal basis (partition-of-unity) functions with enrichment functions. For discontinuities, basis functions are discontinuous across the crack and for singularities a weakly singular enrichment function that satisfies the Laplace equation is chosen. For the Laplace problem with a singularity, we devise an extended projector that maps functions that lie in the extended virtual element space onto linear polynomials and the enrichment function, whereas for the discontinuous problem, the consistent element stiffness matrix has a block-structure that is readily computed. An adaptive homogeneous numerical integration method is used to accurately and efficiently (no element-partitioning is required) compute integrals with integrands that are weakly singular. Once the element projection matrix is computed, the same steps as in the standard VEM are followed to compute the element stabilization matrix. Numerical experiments are performed on quadrilateral and polygonal (convex and nonconvex elements) meshes for the problem of an L-shaped domain with a corner singularity and the problem of a cracked membrane under mode III loading, and results are presented that affirm the sound accuracy and demonstrate the optimal rates of convergence in the L2 norm and energy of the proposed method.

Thermoelastic fracture modelling in 2D by an adaptive cracking particle method without enrichment functions

Thermoelastic fracture of brittle materials is a key concern for many types of engineering structures such as aerospace components and pressure vessels. These problems present difficulties in modelling as they involve coupling of the effects of thermal and mechanical loadings, and usually material behaviours are strongly discontinuous at crack locations, and highly nonlinear during crack propagation. Most current numerical methods applied to these problems use enrichment functions to model discontinuities of temperature and stresses at cracks, however, these enrichments bring extra unknowns into the model and can lead to numerical difficulties in attempting solutions. In this paper, an adaptive cracking particle method is developed for thermoelastic fracture. The method is meshless, so crack discontinuities can be introduced by modifying the influence domains of particles, rather than through external enrichments. Another benefit is an easy implementation of h-adaptivity, since no transition is required for different densities of particles, and it is easy to build dense groups of particles around crack tips. The results demonstrate that this new method can provide the same level of accuracy as others that employ enrichment functions, but using fewer degrees of freedom. A number of examples show the flexibility of the new method to model a range of thermoelastic problems in 2D including multiple cracks and crack propagation.

A finite-strain phase-field approach to ductile failure of frictional materials

This work presents a modeling framework for the ductile failure of frictional materials undergoing large deformations with a focus on soil mechanics. Crack formation and propagation in soil can be modeled in a convenient way by the recently developed continuum phase-field approach to fracture. Within this approach sharp crack discontinuities are regularized. It allows the use of standard discretization methods for crack discontinuities and is able to account for complex crack paths. In the present contribution, we develop a computational modeling framework for the phase-field approach to ductile fracture in frictional materials. It combines a non-associative Drucker–Prager-type elastic-plastic constitutive model with an evolution equation for the crack phase field in terms of an elastic-plastic energy density. An important aspect is the development of an isotropic hardening mechanism that accounts for both friction and cohesion hardening. In order to guarantee a locking- and hourglass-free response, a modified enhanced element formulation, namely the consistent-gradient formulation, is employed as a key feature of the finite-element implementation. The performance of the formulation is demonstrated by means of representative numerical examples that describe soil crack formation rooted in elastic-plastic fracture mechanics.