{"title":"Steady-state mode I crack propagation in a brittle solid","authors":"Martin Kroon","doi":"10.1007/s10704-026-00919-w","DOIUrl":"10.1007/s10704-026-00919-w","url":null,"abstract":"<div><p>A steady-state, finite element analysis of brittle, dynamic crack propagation in PMMA is presented. The steady-state assumption means that all time derivatives are translated into a spatial derivative instead. The bulk PMMA is modelled as a linear elastic, isotropic material. The crack is modelled by use of a non-standard cohesive zone model that ensures that material stability is maintained. The cohesive zone contains two lengths, which allows for a regularisation of the crack problem. The boundary conditions were adjusted so that the results could be compared to experimental studies. The stress and strain fields at the crack tip resulting from the numerical analysis were shown, and the possible implications for damage evolution and crack branching were discussed. The study supports the idea that microcracks are initiated at some distance from the crack plane and then grow and join the main crack. The study suggests that the propagating crack goes from a ’simple crack’ to an unstable crack when the peak in the maximum principal strain – which appears at some distance from the crack plane – exceeds the dynamic fracture strain of the material.</p></div>","PeriodicalId":590,"journal":{"name":"International Journal of Fracture","volume":"250 2","pages":""},"PeriodicalIF":2.5,"publicationDate":"2026-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10704-026-00919-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Analysis of thermoelastic crack problems in two-dimensional orthotropic materials by PDDO method","authors":"Jun Lei, Yong Lu, Yanpeng Gong, Weihui Hu","doi":"10.1007/s10704-026-00917-y","DOIUrl":"10.1007/s10704-026-00917-y","url":null,"abstract":"<div><p>As a novel meshless method, the Peridynamic Differential Operator (PDDO) method has distinct advantages in naturally handling discontinuous problems. In this study, it is further extended to investigate thermoelastic fracture problems in two-dimensional (2D) orthotropic materials. The stress intensity factors (SIFs) at crack tips are evaluated by using an interaction integral. A modified maximum circumferential stress criterion is applied to determine the crack propagation direction. Then, the meshless PDDO is employed to simulate crack propagation paths in some 2D structures with various geometric configurations under coupled thermo-mechanical loadings. The present results are compared with the existing numerical results to validate its effectiveness and high accuracy in computing the SIFs and simulating crack propagation paths.</p></div>","PeriodicalId":590,"journal":{"name":"International Journal of Fracture","volume":"250 2","pages":""},"PeriodicalIF":2.5,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147607402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Virtual Element Method for dynamic crack analysis","authors":"Philipp Wappler, Kevin Schmitz, Andreas Ricoeur","doi":"10.1007/s10704-026-00913-2","DOIUrl":"10.1007/s10704-026-00913-2","url":null,"abstract":"<div><p>The Virtual Element Method (VEM) constitutes a generalization of the standard Finite Element Method (FEM), allowing for substantially more flexibility in the mesh design. This feature makes the method particularly suitable for problems with evolving discontinuities and singularities, and thus for crack problems. Dynamic crack analyses oftentimes incorporate impact loads, requiring sophisticated procedures of time integration, which are known from the FEM. Furthermore, the crack tip loading quantities are highly dependent on structural damping, while algorithmic damping is inevitable for numerical stability. Established approaches to quantify crack tip loading are basically applicable within the VEM framework, however, have to be adapted to characteristics like the absence of shape functions within elements. The Displacement Interpretation Method (DIM), the Modified Crack Closure Integral and the <i>J</i>-integral are elaborated in this context, restricting analyses to linear elastic fracture mechanics and stationary cracks. The accuracy of the approaches is verified comparing results of analytical and other numerical calculations. Numerical studies reveal the impact of physical parameters associated with structural damping and load rate, as well as of numerical factors such as approximation order, contour of the <i>J</i>-integral, extrapolation domain of the DIM and algorithmic damping. After all, the VEM appears to be a highly suitable numerical approach for dynamic fracture mechanics with a view to future crack growth simulations.</p></div>","PeriodicalId":590,"journal":{"name":"International Journal of Fracture","volume":"250 2","pages":""},"PeriodicalIF":2.5,"publicationDate":"2026-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10704-026-00913-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147606967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Adaptive phase-field modeling of three-dimensional thermally-induced brittle fracture in COMSOL","authors":"Jia-Nan He, Sundararajan Natarajan, Weihua Fang, Tiantang Yu","doi":"10.1007/s10704-026-00918-x","DOIUrl":"10.1007/s10704-026-00918-x","url":null,"abstract":"<div><p>This work proposes an adaptive thermomechanical phase-field framework for simulating three-dimensional (3D) thermally-induced fracture in brittle materials. Adaptive mesh refinement is performed via the COMSOL application programming interface (API), using a user-defined phase-field threshold as the refinement indicator. COMSOL’s built-in adaptive meshing produces a locally refined mesh without hanging nodes. A staggered iteration scheme is adopted to ensure computational robustness of the fracture phase-field framework. To further improve computational efficiency in 3D fracture problems, the restarted generalized minimum residual (GMRES) method, combined with smoothed aggregation algebraic multigrid (SAAMG), is adopted as an iterative linear system solver. Through the analysis of three numerical examples, the accuracy and efficiency of the present framework can be validated. The open-source model enables users without programming experience to simulate fractures in a multiphysics environment.</p></div>","PeriodicalId":590,"journal":{"name":"International Journal of Fracture","volume":"250 2","pages":""},"PeriodicalIF":2.5,"publicationDate":"2026-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147561990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Effect of environmental conditions on fracture of composite materials and thin films","authors":"Victor Birman","doi":"10.1007/s10704-026-00915-0","DOIUrl":"10.1007/s10704-026-00915-0","url":null,"abstract":"<div><p>Environmental conditions, i.e., temperature and moisture, affect mechanical properties of materials, including composites. In this paper, we concentrate on one of the aspects of the effect of environment on fracture in a composite lamina. The paper demonstrates an analytical approach to account for the effect of the changes in environment on the strain energy release rate of composite materials and thin films. These analytically determined strain energy release rates should be compared to fracture toughness to predict the susceptibility of the material to fracture. Numerical examples are presented for several polymeric and metal matrix composites using available experimental data for compliances affected by the environment. While a general conclusion about the effect of the environment on the strain energy release rate of composites cannot be achieved, the materials considered in the examples demonstrated an increase of the strain energy release rate with an elevated temperature or moisture indicating their potential vulnerability to fracture. Additionally, the effect of temperature on the strain energy release rate of thin films delaminated from the substrate is analyzed. The condition of the increase of the strain energy release rate for mode I fracture in a one-dimensional straight-faced blister has been derived. This condition depends on the ratio of the buckling and residual stresses and temperature-dependent moduli of elasticity of the film.</p></div>","PeriodicalId":590,"journal":{"name":"International Journal of Fracture","volume":"250 2","pages":""},"PeriodicalIF":2.5,"publicationDate":"2026-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10704-026-00915-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147559710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Microstructural mechanisms for enhancing energy dissipation and spall resistance in SiC","authors":"Tyler Ragan, Tengyuan Hao, Min Zhou","doi":"10.1007/s10704-026-00910-5","DOIUrl":"10.1007/s10704-026-00910-5","url":null,"abstract":"<div><p>Key microstructural features such as grain size, grain shape, crystallographic orientations, and grain boundary properties strongly influence fracture resistance of ceramics. Well-designed microstructural features contribute to the materials’ ability to withstand dynamic loading and deformation by facilitating mechanisms that dissipate energy and prevent catastrophic failure. In this research, a microstructure-explicit and fracture process-explicit computational framework is employed to better understand how these microstructural features impact the dynamic fracture response of SiC during high-rate compression loading and spallation. Utilizing the cohesive finite element method (CFEM), the model incorporates anisotropic bulk constitutive and fracture behaviors of grains and misorientation angle-dependent grain boundary properties to resolve complex crack paths and fracture patterns. The model also captures the effects of intergranular and transgranular fracture, and friction between crack faces. A key observation is that grain size gradient can be utilized to balance and optimize both compressional energy dissipation and spall strength. Additionally, it was shown that the effect of crystallographic texture on spall strength is strongly dependent on both the volume fraction and orientation of preferentially aligned grains. Lastly, the study finds that the anisotropic fracture behavior of the grain boundary has a more pronounced effect on both spall strength and energy dissipation than the degree to which anisotropy changes as a function of misorientation angle. The findings provide insight into microstructural features that optimize energy dissipation and spall strength under specific loading conditions. Furthermore, the model framework can be extended to guide the microstructural design of other ceramics and ceramic composites.</p></div>","PeriodicalId":590,"journal":{"name":"International Journal of Fracture","volume":"250 2","pages":""},"PeriodicalIF":2.5,"publicationDate":"2026-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147363028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Energy release rate governed stair cracking in brittle laminates and shale fracking","authors":"Xiaguang Zeng, Yujie Wei","doi":"10.1007/s10704-026-00911-4","DOIUrl":"10.1007/s10704-026-00911-4","url":null,"abstract":"<div><p>Stair-like wavy cracking commonly arises in brittle laminated materials due to the interplay between heterogeneous mechanical properties of layered structures and external boundary conditions. Based on the energy release rate (ERR) criterion and the straight-crack equivalence assumption, we derive an explicit solution for the continual kinking conditions under which crack propagation in brittle laminates through repeated kinking (stair cracking) governed by three parameterized factors. The theoretical solution yields phase diagrams mapping three fracture modes determined by (1) the ratio between the critical fracture toughness when cracking along the interface (<span>({G}_{text{dc}})</span>) and that during traversing fracture (<span>({G}_{text{mc}})</span>), (2) the angle between the initial crack and interfaces, and (3) the stress ratios of boundary conditions. Such an efficient engineering approach, with ERR playing the central role, achieves high accuracy in predicting crack path patterns, as validated through finite-element simulations.</p></div>","PeriodicalId":590,"journal":{"name":"International Journal of Fracture","volume":"250 2","pages":""},"PeriodicalIF":2.5,"publicationDate":"2026-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147342675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the mechanisms of ductile failure under dynamic loadings","authors":"Amine Benzerga, Alain Molinari","doi":"10.1007/s10704-026-00909-y","DOIUrl":"10.1007/s10704-026-00909-y","url":null,"abstract":"<div><p>The mechanisms of void-mediated dynamic ductile failure are discussed in the context of a recently proposed indicator of void coalescence. The indicator is developed based on unit cell calculations within the confines of a continuum description and inertial effects. Here, it is shown that this indicator may inevitably be satisfied in a loading program as a mere consequence of pressure-sensitive plastic flow. Furthermore, insofar as an indicator is not just meant to interpret unit cell results, but rather be used in structural simulations of dynamic failure, the proposed indicator cannot be used as basis to develop a void coalescence criterion. When interpreted in light of recent advances in micromechanical modeling of void growth and coalescence, the reported on unit cell calculations are found to confirm the conclusions of prior work in the literature, the most important of which being that dynamic loading may frustrate internal necking as a potent mode of void coalescence such that the latter would rather occur by mere impingement. A basic formulation of concurrent failure criteria (internal necking versus impingement) under dynamic loading conditions is presented with open questions highlighted for future research.</p></div>","PeriodicalId":590,"journal":{"name":"International Journal of Fracture","volume":"250 2","pages":""},"PeriodicalIF":2.5,"publicationDate":"2026-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147341454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Wrinkling–buckling interaction in a hyperelastic bilayer with interfacial delamination under compression: critical load analysis and comparison with approximate models","authors":"V. L. Bogdanov, V. M. Nazarenko, A. L. Kipnis","doi":"10.1007/s10704-026-00912-3","DOIUrl":"10.1007/s10704-026-00912-3","url":null,"abstract":"<div><p>The aim of this study is to develop and apply a semi-analytical approach to determine the critical parameters of stability loss under compression of a bilayer consisting of a thin film on a semi-infinite substrate in the presence of an interface delamination of a given length. Using the methods of three-dimensional linearized stability theory, the corresponding boundary value problem is reduced to an eigenvalue problem with respect to the parameters of the compressive loading for a system of Fredholm integral equations of the first kind, which is solved numerically. Since the components of the bilayer are considered as hyperelastic materials with an arbitrary form of elastic potential, the obtained results cover a wide range of mechanical characteristics – both in the small and finite (large) deformation modes. Based on examples involving specific types of elastic potentials, the influence of the constitutive model on the critical parameters of stability loss is investigated. For cases of relatively short and relatively long delaminations, a comparison of the results obtained using the proposed approach is made with the results obtained using known approximate approaches used to estimate critical wrinkling and buckling deformations, respectively. In the case of intermediate-length delaminations, it is shown that due to the interaction of wrinkling and buckling mechanisms, the critical values of relative compression in the bilayer are significantly lower than those predicted by the approximate models. At the same time, the obtained results are in good agreement with the results of numerical simulations reported in previous studies and, thus, can be used to predict the onset of interfacial failure in compressed bilayers with thin films.</p></div>","PeriodicalId":590,"journal":{"name":"International Journal of Fracture","volume":"250 2","pages":""},"PeriodicalIF":2.5,"publicationDate":"2026-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147340325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Maria Laura De Bellis, Giulio Alfano, Anna Pandolfi, Elio Sacco
{"title":"Multiscale modeling of interacting fracture networks","authors":"Maria Laura De Bellis, Giulio Alfano, Anna Pandolfi, Elio Sacco","doi":"10.1007/s10704-025-00904-9","DOIUrl":"10.1007/s10704-025-00904-9","url":null,"abstract":"<div><p>This study introduces a theoretical and numerical model for the spontaneous nucleation and propagation of distributed brittle fractures in initially intact materials. The formulation adopts a kinematic description of nested discontinuities characterized by a micromechanical interface law accounting for cohesion, unilateral contact and friction. The model allows for a unified treatment of the mixed-mode frictional fracture. The implementation of the model into a finite element framework is validated against benchmark tests, exhibiting independence of the discretization. The solution of selected quasi-static boundary value problems demonstrates the predictive capabilities of the framework in capturing key features observed in laboratory tests and in field problems involving confined quasi-brittle materials.</p></div>","PeriodicalId":590,"journal":{"name":"International Journal of Fracture","volume":"250 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146082588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}