Phase-field Fracture Research Articles

Phase-field approach for precise fracture tracking in anisotropic rocks: Integrating orthotropy-based energy decomposition and two-fold symmetric fracture toughness

Rock formations are known to exhibit material anisotropy, both in terms of elastic and fracture properties. This means that the fracture path in such formations is not a priori known but rather a complex unknown that requires robust numerical techniques to predict accurately. In this context, the phase-field model is considered particularly effective, provided that certain physical considerations are carefully adjusted to align with the physics of the problem. While addressing elastic anisotropy is well-established, the tension–compression asymmetry necessary to inhibit crack interpenetration in phase-field fracture models needs to account for the specific material anisotropy. Additionally, to accurately capture crack propagation, it is critical to simultaneously account for orientation-dependent fracture toughness in such materials. To address this, the present study employs an anisotropic phase-field model that integrates the generalized spectral decomposition proposed in the literature for orthotropic materials with a two-fold symmetric fracture toughness, to predict the fracture trajectories in rock-type samples under fixed mixed-mode loading ratios. While each of the two aspects has primarily been applied to model orthotropic plates under simple tensile and shearing loading conditions in the literature, here we study their applicability in complex loading scenarios. To this end, the experimental data from notched semi-circular specimens of Grimsel Granite undergoing complex mixed-mode loading obtained in our previous work is considered. We focus on two given mode-mixity ratios and perform numerical studies. Our results emphasize the importance of considering this generalized decomposition for phase-field modeling of fracturing in rock-type materials, particularly under loading conditions where the crack might otherwise be unrealistically driven into the compressive region. Although certain features are well captured by considering anisotropy in elasticity alone, our findings demonstrate that incorporating a two-fold symmetric fracture toughness proves to be advantageous for more precise tracking of the fracture path.

Phase field numerical strategies for positive volumetric strain energy fractures

AbstractA reasonable crack driving force is the key to preventing compression cracks and eliminating unrealistic crack propagation in phase field fracture analysis. Thus, examining the roles of volumetric strain energy (VSE) and deviatoric strain energy (DSE) in influencing crack propagation is necessary. This paper presents a comprehensive analysis of the following four different approaches to energy decomposition‐driven crack propagation: the positive volumetric strain energy method (VSEM), deviatoric strain energy method (DSEM), volumetric‐deviatoric strain energy method (VDSEM), and no decomposition method (NM). All of the strain energy is involved in the development of the crack propagation. The following four examples are investigated: two benchmark single‐side notched examples, a complex crack containing a hole example, and a crack coalescence in an asymmetric double notch example. The VSEM's numerical findings align with literature and experimental observations, confirming that positive VSE rather than DSE promotes crack propagation.

The Determination of the Elastoplastic and Phase-Field Parameters for Monotonic and Fatigue Fracture of Sintered Steel Astaloy™ Mo+0.2C

This paper presents a procedure for determining the elastoplastic parameters of phase-field fracture of sintered material. The material considered was sintered steel Astaloy™ Mo+0.2C of three densities: 6.5, 6.8 and 7.1 g/cm3. The stress–strain curve and Wöhler curve, which are experimentally obtained, are utilized for validation of the numerical simulations. For modelling of damage evolution, a CCPF (Convergence check phase-field) algorithm was used as a numerical framework. During calibration of the numerical parameters, two-dimensional as well as three-dimensional modelling was used. A comparison of different fatigue degradation functions known from the literature is also made. To improve the efficiency of numerical simulations of fatigue behaviour, the cycle skip technique is also employed.

Fatigue crack growth in functionally graded materials using an adaptive phase field method with cycle jump scheme

Functionally graded materials provide versatility in adjusting the volume fractions of constituent materials to meet specific design requirements. However, this customization often introduces mode-mixity at the crack tip, posing challenges in predicting fracture under cyclic loading with discrete approaches and computationally expensive with conventional phase-field fracture models. To address these issues, this paper introduces an adaptive phase-field fracture formulation with cycle jump scheme to elegantly predict fatigue crack nucleation and growth in functionally graded materials. Within this framework, the effective properties at a point are estimated using the Mori–Tanaka homogenization scheme, while the crack growth due to cyclic load is captured by incorporating an additional fatigue degradation parameter. Moreover, the computational efficiency of the proposed framework is improved through an adaptive mesh refinement and explicit cycle jump scheme. The adaptive refinement scheme utilizes an error indicator derived from both the displacement solution and phase-field variable. The adaptive refinement scheme is integrated with efficient quadtree decomposition, which generates a hierarchical mesh structure. Hanging nodes resulting from the quadtree decomposition are efficiently handled using a polygonal finite element method. The proposed framework is validated against experimental and numerical results reported in the literature. Furthermore, we investigate the fatigue crack growth resistance across a broad range of material gradation directions, gaining valuable insights and identifying functionally graded materials with high fatigue resistance.

Orientation-dependent deformation and failure of micropillar shape memory ceramics: A 3D Phase-Field Study

Some microscopic samples of zirconia-based shape memory ceramics (SMCs) have shown full martensitic phase transformation (MPT) over multiple loading cycles without cracking. However, the occurrence of MPT is strongly influenced by grain orientation. Depending on the specific grain orientation relative to the loading direction, alternative mechanisms such as plastic slip and fracture may emerge. This study introduces a phase-field (PF) based framework that integrates a PF-MPT model, a PF fracture model, and a crystal viscoplasticity model to investigate the effects of grain orientation on MPT, plastic slip, and fracture mechanisms in SMC micropillars. Single crystal micropillars are created to distinguish the orientations that facilitate each mechanism. A wide range of grain orientations are found to predominantly exhibit MPT. Micropillars with grain orientations close to the (100) and (001) directions primarily experience fracture, with minimal plastic slip. Additionally, samples oriented along the (110) direction show a significant amount of plastic slip. A pole figure is constructed to elucidate the interplay between MPT, cracking, and slip under compressive loading conditions. This research provides valuable insights into the intricate behavior of SMCs under different loading scenarios, crucial for optimizing their performance in practical applications.

Fracture as an Emergent Phenomenon

The mechanics of fracture propagation provides essential knowledge for the risk tolerance design of devices, structures, and vehicles. Techniques of free energy minimization provide guidance, but have limited applicability to material systems evolving away from equilibrium. Experimental evidence shows that the material response depends on driving forces arising from mechanical fields. Recent years have witnessed the development of new methods for modeling complex dynamic and quasistatic fracture. New approaches may differ remarkably from previous ones, as they involve implicit coupling between damaged and undamaged states, allowing fracture to be modeled as emergent phenomena.The focus of this workshop is on the most advanced techniques for modeling fracture, represented by eigenerosion methods, variational approaches, phase field fracture models, and non-local approaches. Technical progress is contingent on the further development of the mathematical framework underlying these techniques. This is necessary for accurate and reliable computational modeling of fracture for multiple freely propagating cracks. The objective of this workshop is to mathematically identify and discuss open issues related to fracture modeling and to highlight recent advances. Addressing fundamental issues will foster exchange between the different communities, essential for advancing the field.

Effect of nitrogen-doped type on fracture toughness improvement and crack growth resistance of carbon nanotube/epoxy nanocomposites: Combined multiscale analysis approach

Recently, nitrogen-doped carbon nanotubes (N-doped CNTs) have received great attention in nanocomposite design. It has become highly necessary to develop predictive models to elucidate their toughning behavior. In this study, the effects of CNTs with three different types of N-doped functional groups (quaternary, pyrrolic, and pyridinic) on the fracture toughness (FT) and crack growth of polymer nanocomposites are predicted using a multiscale analysis approach. To scale up from the nanoscale to the macroscale, a multiscale analysis approach integrating molecular dynamics, micromechanics theory, linear fracture mechanics theory, and a phase-field fracture model (PFFM) is adopted. The toughness enhancement trends of the three different types of N-doped functional groups were quantified by considering four toughening mechanisms (CNT debonding, plastic nanovoid growth, CNT pull-out, and CNT rupture), and compared with experimental result. The results show that the excellent interphase and interfacial properties of quaternary and pyridinic functional groups significantly improve the FT and crack growth resistance of N-doped CNT/epoxy nanocomposites. Our study provides high-performance solutions for experimental studies pertaining to the FT and crack growth of N-doped CNT/epoxy nanocomposites.

Formation mechanism of radial and circular cracks promoted by delamination in drying silica colloidal deposits.

Cracks with radial and circular patterns are appealing in nature and industry. Although morphologies and propagation conditions of cracks are extensively studied, the formation mechanism of crack pattern by the interaction of channel fracture and interfacial delamination remains elusive. Here, we present the transition of radial to coexisting radial and circular crack patterns when the thickness of colloidal deposits on both hard and soft substrates exceeds a critical value, through the colloidal volume fraction dependence. In addition, a thickness-dependent phase diagram from radial crack to coexistence of radial and circular cracks was constructed with respect to the radius and the volume fractions of silica colloidal deposits. A phase-field fracture model is developed to elucidate how the formation of radial cracks is facilitated by simultaneous delamination. The warping-induced radial tensile stress at the bottom surface of the striped deposit is proportional to the thickness. It leads to subsequent nucleation and growth of circular cracks in thick deposits. This work provides insight into the formation mechanism of complex crack patterns in drying colloidal deposits and revolutionizes the design space of crack-based micro-nano structures.

ANSYS implementation of the phase field fracture approach

ANSYS implementation of the phase field fracture approach

A phase-field length scale insensitive mode-dependent fracture model for brittle failure

This article presents a consistent phase-field length scale insensitive mode-dependent fracture model for brittle failure by proposing mode-factor dependent degradation functions that incorporate the effect of two additional fracture parameters, namely the mode-II critical energy release rate and mode-II fracture strength, on the overall mechanical response. Using the proposed mode-factor dependent degradation functions and employing a recently proposed modified strain decomposition scheme, we provide analytical expressions for the mode-I and mode-II fracture strengths corresponding to the mode-dependent parts of the elastic energy. Adopting a modified Benzeggagh–Kenane (B–K) criteria, we propose mode-dependent driving forces by deriving expressions for the critical energy release rate corresponding to individual fracture modes. The proposed model provides a consistent coupling between the different fracture modes and can thus predict fracture for all possible mode-mixity ratios. A parametric study is carried out to unravel the effect of mixed-mode fracture parameters on the mechanical response of isotropic materials by considering a few representative numerical examples. The numerical results from the proposed model show an excellent agreement with experimental results reported in the literature.

Phase field modeling of underloads induced fatigue crack acceleration

This study proposes a novel methodology to model the fatigue crack growth acceleration under underloads using a phase field fracture framework. In this model, fatigue crack growth is characterized by the degradation of fracture toughness, with an emphasis on employing a representative loading strategy instead of explicit cyclic loading, thus accelerating simulations of high-cycle fatigue. The model integrates a zone-based crack acceleration approach that responds to single-cycle underload. Notably, the model adeptly captures the dynamics described by the Paris-Erdogan law. Post-underload crack growth acceleration is simulated by identifying an acceleration zone near the crack tip, inspired by existing models based on the plastic zone. This zone is defined by a strain energy density threshold, and the underload ratio governs the rate of fatigue damage accumulation attenuation within this area. The implementation leverages the UMAT user subroutine in Abaqus, utilizing coupled temperature-displacement elements where temperature analogously represents the phase field parameter. Experimental validation of the model confirms its ability to accurately reflect the loss of fatigue life and the acceleration of crack growth rates in compact tension and middle tension specimens. Additionally, the model shows promise for extension to periodic underloads, highlighting its potential for simulating real-world fatigue scenarios effectively.

A thermodynamic framework for ductile phase-field fracture and gradient-enhanced crystal plasticity

This study addresses ductile fracture of single grains in metals by modeling of the formation and propagation of transgranular cracks. A proposed model integrates gradient-extended hardening, phase-field modeling for fracture, and crystal plasticity. It is presented in a thermodynamical framework in large deformation kinematics and accounts for damage irreversibility. A micromorphic approach for variationally and thermodynamically consistent damage irreversibility is adopted. The main objective of this work is to analyze the capability of the proposed model to describe transgranular crack propagation. Further, the micromorphic approach for damage irreversibility is evaluated in the context of the presented ductile phase-field model. This is done by analyzing the impact of gradient-enhanced hardening considering micro-free and micro-hard boundary conditions, studying the effect of the micromorphic regularization parameter, evaluating the performance of the model in ratcheting loading and testing its capability to predict three-dimensional crack propagation. In order to solve the fully coupled global and local equation systems, a staggered solution scheme that extends to the local level is presented.

Phase Field Damage Framework for Modeling Thin-Film Silicon Anodes in All-Solid-State Batteries

Recent advancements in solid electrolyte research have revived interest in the concept of silicon anodes for All-Solid-State Batteries (aSSBs). Tan et al. have recently proposed stable cyclic performance of micro-grain size pure silicon anodes paired with sulfide-based solid electrolytes [1]. This development has opened pathways toward potential aSSBs without encountering dendrite growth issues. This rediscovery of the thin-film a-silicon anodes concept has sparked computational interest in simulating the fracture-chemo-mechanical silicon lithiation mechanism. Understanding chemical reactions during lithiation and crack evolution during de-lithiation processes are intriguing topics. Especially, elastic-plastic deformation coupled with phase-field continuous fracture is crucial in comprehending the underlying physics and enhancing performance of battery cell through reverse engineering approaches.In this work, we develop a thermodynamically consistent deformation-diffusion-damage coupled framework to model the evolution of dense, thin-film-like a-silicon anodes during lithiation and delithiation. During delithiation, tensile stress develops inside the film as the lithiated film overcomes the initial stack pressure and compressive stress prevalent during lithiation. As tensile stress and strain evolve, randomly distributed weak spots lead to crack nucleation. This fracture occurs throughout the silicon film, forming sharp vertical cracks, and depending on surface conditions along the solid-state electrolyte, some interfacial cracks may emerge, leading to permanent contact loss and resulting in capacity reduction.From the model, we observed “mud crack” like phenomena, where crack distribution is not controlled by particle’s property, but entire cluster’s thickness matters. While flat model suggests that one that nucleate the crack was weak spots where they make stress concentration over entire body, whether they can evolve into big vertical crack mostly governed by connectivity of such weak spot maps. However, from our interface roughness model, what governs the crack evolution most was not the randomly distributed weak spot, but its geometric relative thickness difference between one another. Due to manufacturing process, perfect interface is hard to be achieved and such geometric irregularity always exist, thus, additive and binder studies along the silicon anode and its interface between sse would be key to achieve long and stable life span for a-silicon based aSSBs. Reference [1] Tan, Darren HS, et al. "Carbon-free high-loading silicon anodes enabled by sulfide solid electrolytes." Science 373.6562 (2021): 1494-1499.

Mechanical Failure of Core-Shell Cathode Particles: The Effects of Concentration-Dependent Material Properties and Phase Field Fracture Modelling

The use of core-shell cathode particles in lithium-ion batteries is an attractive approach to enhancing energy density whilst retaining lifetime, through reducing undesired reactions at the electrode-electrolyte interface and limiting the volume change of the electrode particles. However, mechanical failure through the fracture and debonding of the core-shell interface is a major challenge. In this work, we employ a coupled finite-element model to predict and mitigate the mechanical failure of core-shell cathode structures, taking as example a particle of NMC811 (core) coated with NMC111 (shell). In particular, we focus on two aspects:The first one involves the assumptions of material properties as inputs of the model. The material properties are often considered constant by battery modelling researchers, yet these parameters can vary significantly during charge/discharge. For example, experiments have unveiled a three-orders-of-magnitude drop in the diffusion coefficient of NMC materials during discharge [1]. Here, we incorporate material properties obtained from experimental data, including concentration-dependent diffusion coefficient obtained from GITT measurement and partial molar volume derived from in situ X-ray diffraction data. Our results indicate that when assuming a concentration-dependent partial molar volume, the maximum values of tensile hoop stress in the shell are nearly three times lower than those predicted with constant average properties, diminishing the likelihood of fracture.When accounting for concentration-dependent diffusion coefficient, large concentration gradient is observed near the outer surface of the core due to reduced lithium mobility at high states of lithiation, hindering full electrode capacity utilisation. The significant concentration gradient and capacity underutilisation align with direct observations from experiments [2]. Notably, these phenomena cannot be captured in modelling if assuming constant diffusion coefficient. These findings offer new perspectives on the performance and design of core-shell electrode particles. Safe design maps in terms of core size and shell thickness to mitigate fracture and debonding are obtained.The second focus of this work is the utilisation of phase field fracture model in predicting cracking behaviours within the core-shell particle. Phase field modelling provides a continuous representation of cracks, enabling the tracking of complex cracking patterns [3]. We predict cracking behaviors caused by volume changes during charge/discharge, as well as inaccessible capacity and newly created active surface area caused by cracking. Our multiphyics model can hopefully act as a catalyst, accelerating the development process of the core-shell structured cathode particles and shortening the time to market of advanced batteries with extended lifetime.

Electro-Chemo-Mechanical Modeling of Composite Cathodes in All-Solid-State Li-Ion Batteries

A model-based understanding can assist and accelerate developing all-solid-state batteries (ASSB). In addition to chemo-mechanical influences within electrode particles (e.g., NMC) [1-2], a solid electrolyte (e.g., argyrodites) introduces additional interfacial interactions between electrode and electrolyte phases [3-5]. The present research derives and implements a coupled multi-physics finite-element model that captures electrochemical, transport, and structural behaviors of composite electrode structures. The models incorporate concentration-dependent and anisotropic material properties that are based on previously published combinations of experiment and density functional theory (DFT). These include stiffness and fracture toughness, porosity and crystallographic orientation, and operating conditions such as charge/discharge rates and external pressure.Figure 1 illustrates predicted stresses developed during electrode manufacturing. The relatively complex cathode microstructure is based on replicating scanning electron microscopy (SEM) images [6]. The composite electrode consists of electrode, electrolyte, and pore phases (Fig. 1a). As illustrated in Fig. 1b, the ASSB synthesis process involves applying and removing high compressive pressure, which causes plastic deformation and introduces residual stresses. Figure 1c shows residual von Mises stresses near electrode-electrolyte interfaces that can be on the order of a gigapascal. The synthesis-generated residual stresses serve as initial conditions for modeling the chemo-mechanics during battery cycling.During cell operation, spatially varying Li concentrations cause material deformation and associated stresses. Figure 1d shows predicted crack nucleation and growth during operation. Depending on the stress levels, crack nucleation and growth leads to cell degradation and capacity fade. The models predict interfacial fracture and phase separations using phase-field fracture theory. In phase-field formulation, the sharp cracks are approximated as diffuse cracks using a process-zone. High values of the phase parameter ξ (Fig. 1d) represent cracked surfaces. The structural disintegration and loss of active surface areas increase the tortuous path for Li/Li-ion transport, which eventually manifests as capacity-fade.The simulations are validated using published experimental work. The modeling approach, which combines phase-field and finite-element algorithms, is implemented using the COMSOL Multiphysics software. The models are expected to inform microstructure/manufacturing design and optimal operating conditions that improve cycling performance and limit/prevent mechanical damage.[1] K. Taghikhani, P.J. Weddle, J.R. Berger, and R.J. Kee. Modeling coupled chemo-mechanical behavior of randomly oriented NMC811 polycrystalline Li-ion battery cathodes. J. Electrochem. Soc., 168(8):080511, 2021.[2] R. Xu, Y. Yang, F. Yin, P. Liu, P. Cloetens, Y. Liu, F. Lin, and K. Zhao, Heterogeneous damage in Li-ion batteries: experimental analysis and theoretical modeling. J. Mech. Phys. Solids, 129, 160, 2019.[3] K. Taghikhani, P.J. Weddle, R.M. Hoffman, J.R. Berger, and R.J. Kee. Electro-chemo-mechanical finite-element model of single-crystal and polycrystalline NMC cathode particles embedded in an argyrodite solid electrolyte. Electrochim. Acta, 460:142585, 2023.[4] A. Bielefeld, D.A. Weber, R. Rueß, V. Glavas, and J. Janek. Influence of lithium ion kinetics, particle morphology and voids on the electrochemical performance of composite cathodes for all-solid-state batteries. J. Electrochem. Soc., 169(2):020539, 2022.[5] P. Minnmann, F. Strauss, A. Bielefeld, R. Ruess, P. Adelhelm, S. Burkhardt, S.L. Dreyer, E. Trevisanello, H. Ehrenberg, T. Brezesinski, F.H. Richter, and J. Janek. Designing cathodes and cathode active materials for solid-state batteries. Adv. Energy Mater., 12(35):2201425, 2022.[6] C. Doerrer, I. Capone, S. Narayanan, J. Liu, C.R.M. Grovenor, M. Pasta, and P.S. Grant. High energy density single-crystal NMC/Li6PS5Cl cathodes for all-solid-state lithium-metal batteries. ACS Appl. Mater. & interfaces, 13(31):37809–37815, 2021. Figure 1

Verification of strain energy splits of phase field fracture model using Westergaard’s problem under mixed-mode loading

One challenge in verifying strain energy splits in phase field fracture models is the lack of reference problems with exact solutions in mixed-mode loading that could serve as benchmark problems. In this work, different strain energy splits used in the variational phase field model have been applied to the classical Westergaard’s problem. In contrast to other benchmark problems used in the literature, the proposed problem not only includes the singular term as a boundary condition but also implicitly considers all terms. To the authors’ knowledge, this is the first time in the literature that this reference problem has been used to rigorously verify the accuracy of different energy splits of the phase field model. Several critical factors such as the pre-crack definition, mesh size, mixed mode ratio and the length scale parameter have been analysed. A good correlation between the numerical estimations and the analytical predictions was found for some pre-crack definition approaches under mode I. However, we demonstrated that the compatible regularisation length scale and mesh size are significantly lower under mode II loading (i.e., when the crack kinks) than under mode I.

A length-free phase field model for epoxy adhesive materials based on modified Drucker–Prager criterion

Characterizing the cracking behavior of epoxy adhesive material is of great importance for composite structures. Recently, the phase field model has emerged as the preferred method for predicting the crack propagation. This paper presents a length-free phase field model considering the characteristics of epoxy adhesive materials. The proposed model is formulated within a framework that decomposes the phase field fracture driving force, ensuring that it adequately considers the tension/compression asymmetry of the epoxy adhesive materials. The effects of hydrostatic pressure, deviational stress, and friction are incorporated into the model by employing a modified Drucker-Prager failure criterion. The failure criterion is included in the phase field model using the effective stress invariants. Furthermore, the internal length scale, previously considered to lack real physical meaning, is reformulated based on material strength. This model addresses the challenge of measuring the internal length of epoxy materials. It serves as a valuable reference for applying phase field models to epoxy materials. The accuracy of the model is validated through an L-shaped plate simulation, and an attempt using the model to predict the crack development in thick epoxy adhesive joints is undertaken. Several representative samples are simulated, including three double cantilever beams with different configurations. The numerical results demonstrate a good correlation with experimental data, highlighting the potential of the model in simulating the crack propagation in epoxy adhesive joints.

Anisotropic damage evolution in solid fractures: A novel phase field approach with multiple failure criteria and directional-dependent structural tensor

This study proposes a novel phase-field fracture model based on unified phase field theory, aiming to overcome current limitations in simulating material complex fracture behaviors. Through this model, analytical solutions for two-dimensional bars subjected to tensile or compressive stresses are provided, enabling the coupling of multiple failure criteria and further proficient simulation of mode-I, mode-II, and mixed mode-I/II fractures, effectively addressing challenges faced in modelling materials with different or complex failure modes under various loading conditions. Furthermore, to account for the strong anisotropic failure behavior of materials, a novel directional-dependent structural tensor is proposed. The tensor correlates fracture energy with crack surface orientation, facilitating precise characterization of material damage evolution with multiple potential crack orientations. This tensor ensures the consistency of phase-field fracture evolution with predefined fracture patterns. The effectiveness of the proposed model is validated through case studies, emphasizing its robustness and superior predictive capability in capturing fracture behavior under various conditions. This research provides a more accurate and universally applicable approach for simulating material failure, particularly for complex or multiple failure mode material failure simulations.

Study on fracture of hyperelastic Kirchhoff-Love plates and shells by phase field method

Thin walled structures such as plates and shells are widely used in many engineering fields. To Predict its fracture behavior is of great significance for integrity design and strength evaluation of engineering structures. Numerical simulation of the fracture behavior of hyperelastic plates and shells is a challenge due to complex kinematic description, hyperelastic constitutive relationship, geometric nonlinearity and the degradation on elastic parameter caused by fracture damage. Combining Kirchhoff Love (K-L) shell theory with the fracture phase field method, and numerically discretizing the first and second order partial derivatives of displacement field and phase field by using T-splines and meeting the requirements of K-L plate and shell theory for the C1 continuity of the shape function, a model for the isogeometric analysis numerical formulation of the phase field fracture in hyperelastic K-L plates and shells is established. The fracture failure behavior of hyperelastic K-L plates and shells under the uniform load and displacement load is simulated, and the effect of the Gaussian curvature on the fracture behavior of hyperelastic K-L shells is studied. The simulation results show that the present numerical scheme can effectively capture the complex crack propagation path of plates and shells under the uniform load, and the displacement field can effectively reflect the crack distribution of materials. The thin shell with negative Gaussian curvature shows the excellent fracture performance under the internal pressure, and can withstand the greater internal pressure.

Study of adhesive wear mechanisms in asperity junctions based on phase field fracture method

Adhesive wear that is mostly induced by adhesion between asperities of rough surfaces in contact is one of the most common forms of wear of materials. However, the adhesive wear mechanisms are still not fully understood. In this work, the phase field fracture method that does not require any assumption on the crack path is used to explore the debris formation mechanism during asperity separation in adhesive wear. It is found that for ideal materials without defects the size of wear particles depends on the contact length. A small contact length leads to fracture at the junction and the formation of small wear particles. On the contrary, a large contact length will cause fracture in the bulk, resulting in large wear particles. Moreover, the angle of crack propagation decreases with increasing base angle of asperity and the ratio of the asperity height to the base edge width. The variation of fracture angle is mainly related to the ratio of asperity height to the base edge width that will influence the angle of the interaction force between asperities in the wear process. For materials with initial hole defect, the formation mechanism of wear particles will be little affected when the hole is rather small or far away from the junction. However, as the hole diameter further increases, the wear particles will change from large particles to small ones. And for materials with initial crack defect, the crack propagation always starts from the initial crack during adhesive wear. Moreover, for adhesive wear of materials with an initial crack at the bottom of asperity, fracture angle in the bulk is significantly positively correlated with the contact length and large wear particles are easy to form. However, during adhesive wear of asperity with the initial crack inside the bulk of asperity, small particles are more likely to form. This work contributes to further understanding of adhesive wear process.