Fracture Problems Research Articles

Prevention of the Fracture Problem Occurring in Automotive Alternator Heatsink Blocks Using Artificial Intelligence

In this study, prevention of fracture in vibration fatigue testing of automotive alternator heatsink blocks was investigated using an artificial neural network. Automotive components such as alternator heatsink blocks are subjected to high cyclic vibration fatigue loads throughout their lifespan, which can lead to the formation and propagation of fatigue cracks and ultimately component failure. The basic parameters affecting the resonant frequency of the heatsink blocks, including geometry and loading conditions, are determined. Data-driven decision making provides advanced predictive insights to analyze data for prediction and decisions using artificial intelligence approaches. An efficient artificial neural network model was defined to predict the resonance frequency in the vibration fatigue test. While the artificial neural network was trained to establish a functional relationship between the parameters and the resonance frequency, regression analysis was used to develop a predictive model to detect the resonance frequency of the heatsinks. The proposed approach aims to provide a comprehensive framework for preventing fracture problems in vibration fatigue tests of automotive alternator heatsinks and ultimately contribute to the reliable design and performance of these critical components. While the artificial neural network approach achieved high classification accuracy in predicting the new natural frequency, the regression model was also able to make accurate predictions. The results of this study showed that the time spent on design and simulation can be significantly reduced in preventing breakage problems that may occur before dynamic tests such as vibration tests of alternator components.

Interfacial crack analysis in piezoelectric bi-material using coupled FE-XEFG approach

The current paper emphasizes the study of interface crack problems in piezoelectric bi-material having k-class singularity using coupled finite element (FE) and extended element free Galerkin (XEFG) technique. Heaviside and asymptotic crack-tip functions are incorporated to capture crack-tip singularity and discontinuities across crack surfaces for local enrichment within FE-XEFG framework. Few fracture problems are numerically simulated to extract mixed mode intensity factors (IFs) by modified form of domain-based interaction integral. Parametric studies based on geometrical parameters and loading conditions, such as mechanical loading effect, electrical loading effect, effect of crack length, and aspect ratio on fracture parameters have been simulated.

A Fast Time-Discontinuous Peridynamic Method for Stress Wave Propagation Problems in Saturated Porous Media

When employing numerical methods to simulate the behavior of saturated porous media under impact load, the Gibbs phenomenon often occurs, significantly compromising the accuracy of numerical calculations. To improve the accuracy of numerical solutions, this study introduces a fast time-discontinuous peridynamic (TDPD) method for simulating the propagation of fluid pressure and solid stress waves within saturated porous media. In this method, the classical u–p and u–U models are reformulated from spatial differential equations into integral equations to facilitate the simulation of fracture problems. Subsequently, the basic field variables are independently interpolated in the temporal domain, with the introduction of jump terms representing the discontinuities of variables between adjacent time steps. Then an integral weak form in the temporal domain of the spatially discrete governing equations is constructed and the basic formula of the TDPD is derived. Additionally, an efficient approximation method that directly uses the internal force to calculate its time derivative is further proposed. These characteristics ensure that the TDPD can effectively and efficiently capture the inherent sharp gradient features of wave propagation in solid phase and fluid while controlling spurious numerical oscillations. Several representative numerical examples demonstrate that the TDPD can obtain more accurate results compared with conventional peridynamic solution schemes. Moreover, the TDPD can also be viewed as a novel time integration technique, holding substantial potential for high-precision numerical solutions of hyperbolic equations in diverse physical contexts.

Mechanics-informed, model-free symbolic regression framework for solving fracture problems

Data-driven methods have recently been introduced to address complex mechanics problems. While model-based, data-driven approaches are predominantly used, they often fall short of providing generalizable solutions due to their inherent reliance on pre-selected models. Model-free approaches, such as symbolic regression, hold promise for overcoming this limitation by extracting solutions directly from datasets. However, these approaches remain unexplored when dealing with high-dimensional fracture mechanics problems and require significant customization to be effective. In this work, we propose a new symbolic regression framework that integrates mechanics knowledge to enhance the ability to generalize solutions. This framework also includes a model-free variable separation scheme to decouple high-dimensional problems into simpler sub-problems with manageable complexity while preserving data fidelity. We demonstrate the advantages of this framework through two fracture mechanics problems, showing that it can potentially provide generalizable, analytical solutions to novel, easy-to-use fracture testing configurations.

A computationally efficient Element Edge point numerical integration scheme in the meshless method framework for solving fracture problems

The fracture problem is modeled using the Radial Point Interpolation Meshless Method (RPIM) to solve for displacements. Further, the stresses and Stress Intensity Factor (SIF) are calculated using obtained displacements. An effective numerical integral quadrature called the Element Edge point(EE) scheme is used as an alternative to conventional Gauss Quadrature to improve computational efficiency. A comparative study based on two numerical integration schemes, the Element Edge point scheme and Gauss Quadrature, is conducted on four benchmark problems of thick, cracked plates owing to plane strain conditions. The study reveals that the proposed Element Edge point (EE) scheme is computationally efficient and works well in the meshless method framework.

Effect of porosity gradient on fracture mechanics of bi-directional FGM structures: Phase field approach

This research aims to develop a computational model that can accurately predict the fracture behavior of porous bi-directional Functionally Graded Materials (FGMs). The Voigt model for homogenization, is established to account the effects of porosity fraction and gradient distribution within the FGMs, providing valuable insights about the brittle crack propagation. The study employs the UMAT subroutine in ABAQUS software and establishes an analogy between the phase field evolution law and the heat transfer equation, enabling efficient analysis of complex fracture problems. To validate the model, 2D fracture benchmark cases are analyzed, demonstrating its ability to capture different failure modes and the intricate material behavior of porous FGMs under fracture conditions. Furthermore, newly parametric analyses, that highlights the impact of various values of porosity’s volume fraction and FGM’s power law indexes on the brittle fracture path, are conducted to further validate the effectiveness of the newly developed phase field model in predicting the fracture behavior of bi-directional porous FGMs.

Numerical Simulation of Hydraulic Fracture Propagation in Unconsolidated Sandstone Reservoirs

In order to comprehensively understand the complex fracture mechanisms in thick and loose sandstone formations, we have carefully developed a coupled finite element numerical model that captures the complex interactions between fluid flow and solid deformation. This model is the cornerstone of our future exploration. Based on this model, the crack propagation problem of hydraulic fracturing under different engineering and geological conditions was studied. In addition, we conducted in-depth research on the key factors that shape the geometry of hydraulic fractures, revealing their subtle differences and complexities. It is worth noting that the sharp contrast between the stress profile and mechanical properties between the production layer and the boundary layer often leads to fascinating phenomena, such as the vertical merging of hydraulic fracture propagation. The convergence of cracks originating from adjacent layers is a recurring theme in these strata. Sensitivity analysis clarified our understanding, revealing that increased elastic modulus promotes longer crack propagation paths. As the elastic modulus increases from 12 GPa to 18 GPa, overall, the maximum crack width slightly decreases, with a less than 10% reduction rate. The increased fluid leakage rate will significantly shorten the length and width of hydraulic fractures (with a maximum decrease of over 70% in fracture width). The increase in viscosity of fracturing fluid causes a change in fracture morphology, with a reduction in length of about 32% and an increase in fracture width of about 25%. It is worth noting that as the leakage rate of fracturing fluid increases, the importance of the viscosity of fracturing fluid decreases relatively. Strategies such as increasing fluid viscosity or adding anti-filtration agents can alleviate these challenges and improve the efficiency of fracturing fluids. In summary, our research findings provide valuable insights that can provide information and optimization for hydraulic fracturing filling and fracturing strategies in loose sandstone formations, promoting more efficient and influential oil and gas extraction work.

Study on the PDDO-based meshfree method in numerical simulation of shell ductile fracture considering a non-local GTN model

A meshfree method is developed based on the peridynamic differential operator (PDDO) for ductile damage and fracture problems in metal shell structures. The kinematic equations coupled to classical continuum mechanics (CCM) are derived with the motion variables discretized by the PDDO. The elastoplastic and fracture behavior of the material are described by applying the Gurson-Tvergaard-Needleman (GTN) model with shear modification, and the non-local form of the model improves the computational convergence for different modeling scales. The zero-energy model in numerical computations is effectively controlled by introducing an hourglass force based on the average displacement state. The particles contact algorithm and multi-crack visualization algorithm are developed to simulate the fracture of shell structures under collision loads. By comparing with experiments, it is verified that the proposed PDDO-based meshfree method can accurately predict the ductile fracture of shell structures subjected to in-plane and out-of-plane loads.

Implementation of the Zienkiewicz–Pande Model into a Four‐Dimensional Lattice Spring Model for Plasticity and Fracture

ABSTRACTPlasticity and fracture problems have always been hot topics in numerical methods. In this work, a universal implementation procedure for the elasto‐plastic constitutive model is developed in the four‐dimensional lattice spring model (4D‐LSM), in which the Jaumann stress rate is incorporated to exclude the influence of the rigid rotation in the particle stress, expanding the ability of 4D‐LSM to deal with large elastic deformation problems by its own to large plastic deformation problems. As an example, the Zienkiewicz–Pande (ZP) constitutive model is implemented. Several numerical examples are carried out to check the performance of the implemented model. Through a comparison with analytical solutions, available experimental data, and other numerical results, the stability of the developed plastic framework and the correctness of the stress calculation scheme are verified. Meanwhile, numerical results show that the developed code is capable of solving elasto‐plastic large deformation problems. With the advantage of 4D‐LSM in handling fracture problems, the ability of the embedded model to solve plastic fracture problems is verified with a simple maximum deformation failure criterion.

Accurate prediction of discontinuous crack paths in random porous media via a generative deep learning model

Pore structures provide extra freedoms for the design of porous media, leading to desirable properties, such as high catalytic rate, energy storage efficiency, and specific strength. This unfortunately makes the porous media susceptible to failure. Deep understanding of the failure mechanism in microstructures is a key to customizing high-performance crack-resistant porous media. However, solving the fracture problem of the porous materials is computationally intractable due to the highly complicated configurations of microstructures. To bridge the structural configurations and fracture responses of random porous media, a unique generative deep learning model is developed. A two-step strategy is proposed to deconstruct the fracture process, which sequentially corresponds to elastic deformation and crack propagation. The geometry of microstructure is translated into a scalar of elastic field as an intermediate variable, and then, the crack path is predicted. The neural network precisely characterizes the strong interactions among pore structures, the multiscale behaviors of fracture, and the discontinuous essence of crack propagation. Crack paths in random porous media are accurately predicted by simply inputting the images of targets, without inputting any additional input physical information. The prediction model enjoys an outstanding performance with a prediction accuracy of 90.25% and possesses a robust generalization capability. The accuracy of the present model is a record so far, and the prediction is accomplished within a second. This study opens an avenue to high-throughput evaluation of the fracture behaviors of heterogeneous materials with complex geometries.

An in-depth analysis of the Radial Point Interpolation Method’s parameters in relation to the accuracy of fracture problem

The fracture problem has been modeled using Radial Point Interpolation Method (RPIM) to examine the effects of support domain size and shape parameter value on the calculation of Stress Intensity Factor (SIF). The study focuses on an edge crack problem subjected to normal load and shear loads, determining that the support domain size of ‘ α ≥ 3.5 and shape parameter value between ‘ n = 1.9 -2.1’ yield results of accuracy with error % ≤ 3 % . Additionally, research extended by varying the problem domain size and crack location to verify consistency. The findings indicate that selecting the optimal shape parameter value will ensure better accuracy over the enrichment techniques with lesser computational cost.

Optimization of Supercritical CO2 Fracturing Based on Random Forest-Particle Swarm Optimization Model and Pre-existing Fracture Network

Summary On a global scale, shale oil/gas has become an important alternative energy source for conventional oil and gas. The potential advantages of supercritical CO2 (ScCO2) make it an ideal alternative to hydraulic fracturing, used for shale reservoir transformation and production increase while also promoting the geological storage of CO2, which is in-line with today’s carbon capture, utilization, and storage technology and helps to address the challenges of global climate change. To further study the fracture propagation and optimization of a complex fracture network (CFN) in ScCO2 fracturing under complex geological conditions using the cohesive module of ABAQUS to establish a fluid structure coupling model and completing indoor and field experimental verification, we introduce the global embedded cohesion zone model (CZM) combined with Python to generate two natural-fracture (NF) distribution models, conjugate and power law, to establish a dispersed mesh model. Based on this model, we studied the fracture propagation problem of ScCO2 fracturing under different engineering and geological conditions. The simulation results will be used as data-driven data to establish an optimization model of the random forest-particle swarm optimization algorithm (RF-PSO) and optimize the CFN. Research has shown that (1) ScCO2 is more inclined to pass through NFs and propagate in the rock matrix, and hydraulic fractures (HFs) combine better with NFs. Compared with hydraulic fracturing, ScCO2 fracturing has significant advantages (only the fracture width is lower than hydraulic fracturing, its initiation pressure and fracture length are much better than hydraulic fracturing, and there are more small fractures, making it easier to form a CFN). (2) During the process of fracture propagation, once dominant fractures form, the trend of the “Matthew effect” is inevitable. The process of fracture propagation is influenced by multiple factors, especially the distribution of NFs; the larger the reservoir filtration coefficient is, the more ScCO2 fracturing fluid that is lost, which is more unfavorable for fracturing construction. While maintaining the same amount of fracturing fluid injection, as the displacement increases, the fracture complexity increases, and the fracturing control range expands. Compared with other parameters, the effect of fracturing fluid temperature (FFT) on the expansion of ScCO2 fracturing fractures is not significant. (3) The established RF-PSO optimization model has an error of 2.89%, which can well adapt to CFN optimization problems under complex NF conditions and reduce uncertainty. We propose in this article a research method for fracture network optimization from fracture modeling, dynamic simulation, and optimization modeling. By combining numerical simulation and machine learning, the CFN optimization design of ScCO2 fracturing under CFN conditions is achieved, providing a research approach for the optimization of fracturing in fractured reservoirs.

Variational damage model: A novel consistent approach to fracture

The computational modeling of fractures in solids using damage mechanics faces challenge when dealing with complex crack topologies. One effective approach to address this challenge is by reformulating damage mechanics within a variational framework. In this paper, we present a novel variational damage model that incorporates a threshold value to prevent damage initiation at low energy levels. The proposed model defines fracture energy density (ϕ˜) and damage field (s) based on the energy density (ϕ), crack energy release rate (Gc), and crack length scale (ℓ). Specifically, if ϕ≤Gc2ℓ, then ϕ˜=ϕ and s=0; otherwise, ϕ˜=−Gc24ℓ21ϕ+Gcℓ and s=1−Gc2ℓ1ϕ. Furthermore, we extend the model with a threshold value to a higher-order version. Utilizing this functional, we derive the governing equation for fractures that evolve automatically with ease. The formulation can be seamlessly integrated into conventional finite element methods for elastic solids with minimal modifications. The proposed formulation offers sharper crack interfaces compared to phase field methods using the same mesh density. We demonstrate the capabilities of our approach through representative numerical examples in both 2D and 3D, including static fracture problems, cohesive fractures, and dynamic fractures. The open-source code is available on GitHub via the link https://github.com/hl-ren/vdm.

Adaptive PF-CZM for multiphysics fracture analysis in functionally graded materials

Functionally graded materials (FGM) hold significant relevance in engineering due to their tailored material property gradation, designed for specific engineering applications. The challenge in fracture analysis of FGM stems from the spatial variation of material properties, which complicates the prediction of crack topology. Local and global refinement strategies are impractical for fracture analysis in FGM due to the unpredictable nature of crack topology, which renders local refinement infeasible. Additionally, global refinement is not advisable as it leads to a significant increase in degrees of freedom, adversely affecting computational efficiency.The novelty of this research lies in the incorporation of spatial variation in both the length scale and material properties, enhancing the realism of FGM domain modeling. To preserve the required length scale, it is necessary to adopt a minimum mesh size, which consequently results in a substantial increase in the degrees of freedom and, thereby, escalates the computational cost. To address these challenges, the study employs an adaptive mesh refinement (AMR) algorithm integrated with a phase-field cohesive zone model (PF-CZM), providing a robust solution for accurate fracture analysis in FGM. Based on the ideas stemming from the need for efficient and realistic modeling, the AMR-PF-CZM framework refines the mesh efficiently in regions of crack growth based on crack-driving energy and phase field variable, thereby eliminating the need for pre-refinement. The findings demonstrate a 76%–85% increase in computational efficiency and accuracy of the AMR-PF-CZM approach compared to the non-adaptive PF-CZM. Furthermore, the developed algorithm’s applicability to dynamic fracture and multi-physics problems, specifically addressing mechanical and thermal fracture in FGM, underscores the importance of this approach in capturing complex fracture phenomena.

A phase field model with modified volumetric-deviatoric decomposition for the mixed-mode fracture of rock

Recent attempts of phase field models to simulate rock fracture problems have been regarded as fruitful, owing to their powerful crack characterization capability. However, existing phase field models still exhibit limitations in simulating quasi-coplanar shear cracks and related coalescence patterns that often occur in rock failure. In this study, we propose a novel phase field model to simulate the mixed-mode fracture, in which a modified volumetric-deviatoric decomposition that combines the advantages of the volumetric-deviatoric decomposition and the spectral decomposition is conducted to distinguish between tensile, tensile-shear, and compressive-shear fractures; to reduce unrealistic damage in the area outside the crack trajectory, a threshold parameter is introduced into the degradation function to control whether the related energy participates in damage evolution; and the hybrid formulation is employed to efficiently and robustly solve the displacement and phase fields alternately. The feasibility of the proposed phase field model is first validated by a benchmark example. Next, the application of this model to the simulations of different rock specimens under uniaxial compression reveals a better agreement with experimental observations than previous simulation results, demonstrating an advancement over existing phase field models.

An uncoupled ductile fracture criterion for a wide range of stress states in sheet metal forming failure prediction

In industrial production, sheet metal rupture will inevitably occur in the process of forming, especially in the shape of complex, relatively small thickness of the sheet. It is vital to accurately predict the failure of ductile metals under various working conditions. To deal with the problem, an uncoupled ductile fracture criterion (DFC) for a broad range of stress states is proposed based on the micro-mechanism of ductile fracture of metals, which redefines the relationship between the number of void nucleating and the equivalent plastic strain and takes into account the different deformation modes of the voids in the growth stage. Then, in order to verify the validity and advantages of the proposed DFC, the 3D fracture surface of AA 2024-T351 and AISI 1045 steel are constructed by the new criterion based on their fracture data under various stress states, and compared with the commonly used DF2016 criterion and Hu criterion. The prediction results show that the new criterion is better than them, both in terms of the maximum prediction error and the average prediction error. Furthermore, to further prove the effectiveness of the proposed model, five samples are designed for tension tests to calibrate the new DFC using a hybrid experimental–numerical approach, and the calibrated criterion is applied to cupping simulations of AA 6061 and compared with cupping experimental results. The results indicate that the Erichsen cupping number (IE) of the cupping simulation and experiment are 6.88 mm and 7.05 mm, respectively, and the error between them is only 2.411 %, and the location of the fracture is also basically the same. Therefore, all comparison results show that the proposed DFC can forecast the fracture problem in sheet forming more accurately under various stress states.

An adaptive peridynamics with correspondence material model for coupled creep-plastic fracture problems

This paper introduces an adaptive peridynamics with a correspondence material model (A-PD-CMM) to effectively simulate the coupled creep-plastic behavior and the creep crack propagation in metal materials. In this method, to describe the coupled creep-plastic deformation with considering the creep damage, the multiaxial coupled constitutive model that integrates the Wen-Tu creep damage model and the J2 plastic model is developed. Moreover, a damage associate criterion is incorporated into the peridynamics framework to effectively capture the influence of damage on the force state. To detail the rapid damage evolution during the tertiary creep stage, a time-adaptive solving strategy is proposed. Additionally, the constitutive integration algorithm of the multiaxial coupled model is efficiently realized by decoupling it into two parts, that is, an implicit computation of creep-plasticity and an explicit evolution of creep damage. The efficacy and engineering applicability of the proposed A-PD-CMM are validated through several representative numerical examples, showcasing its potential in addressing complex material behavior.

General particle dynamics: On the correlation of nonlocal and local theories for large deformation problems

This paper presents a detailed investigation on the General Particle Dynamics (GPD) method and its implications for addressing challenges in large deformation mechanics and fracture modeling. The GPD method integrates nonlocal interactions and nonlocal differential operators to overcome limitations of classical continuum mechanics in capturing long-range interactions and material behaviors under extreme deformation conditions. Through a systematic examination of the physical and mathematical foundations of GPD, including its governing equations and constitutive relations, we elucidate the robust and applicability of the proposed method in simulating complex deformation phenomena. Furthermore, the relationships among GPD, conventional mechanical theories (Smoothed Particle Hydrodynamics), nonlocal theory (Peridynamics) and Molecular Dynamic theories are discussed. Numerical examples related to large deformation and fracture problems are tested to verify the ability of the proposed method. The numerical results show that GPD inherits the advantages of the classical method and also has its unique properties in solid fracture mechanics and large deformation problems, making it a promising approach in computational mechanics.

A lattice model with a progressive damage applied to fracture problems of wood

A lattice model with a progressive damage applied to fracture problems of wood

Cohesive phase-field model for dynamic fractures in coal seams

A novel rate-dependent cohesive phase-field model that can simulate fluid-driven dynamic quasibrittle fracture in soft coal rocks is presented. The coal matrix and fractures are described by a unified fluid continuity equation, and the Poiseuille flow in the fractures is modeled by deformation-dependent permeability. Dynamic evolution equations for porosity, permeability and the Biot coefficient during phase-field evolution are developed. The phase-field evolution equation is derived from the total energy generalization based on the Francfort–Marigo variational principle. To simulate the rate-dependent dynamic fracture problem accurately, the effects of the phase-field evolution dissipation energy and inertia energy are considered. The above multifield coupled system is solved via a staggered approach within the finite element framework via the Newton–Raphson iterative algorithm. The independence and convergence of the proposed model were verified, and its accuracy was validated via an analytical solution for crack width and a benchmark test for dynamic crack propagation. The propagation law of single/multicluster hydraulic fractures in coal seams with interlayers was investigated via the proposed model. The simulations revealed that the number of clusters and the mechanical properties of the interlayers significantly influence the fracture morphology. Finally, hydraulic fracturing of fracture-developed coal seams was simulated, confirming the potential of the proposed model.