[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81912-en":3,"doc-seo-81912-105":31,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81912,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Modeling Fatigue-Induced Anisotropic Quasi-Brittle Damage Based on the Endurance Surface Concept","This work proposes a novel continuum damage framework for fatigue built on the endurance-surface concept, using the energy-release rate as the driving force. Damage evolution is governed by the distance of the thermodynamic driving force from the endurance surface, enabling accumulation over many cycles under constant-amplitude loading. Endurance thus directly determines the material’s physical endurance limit. Mesh-objective regularization is achieved via micromorphic gradient enhancement, extended to anisotropic evolution and microcrack closure–reopening effects for multiaxial, loading-path-dependent degradation.","arXiv :2607 .048 18v 1 [ cs .CE] 6 Jul 2026  \nModeling Fatigue-Induced Anisotropic Quasi-Brittle Damage Based  \non the Endurance Surface Concept  \nK. Feike P. Kurzeja K. Langenfeld J. Mosler∗  \nTU Dortmund University, Institute of Mechanics,  \nLeonhard-Euler-Str. 5, D-44227 Dortmund, Germany  \nAbstract  \nThis work proposes a novel continuum damage framework for fatigue that is based on the endurance-surface concept and uses the energy-release rate as the driving force. Damage evolution is specifically governed by the distance of the thermodynamic driving force from the endurance surface. In contrast to classic failure surfaces, this allows damage to accumulate over many cycles even under constant-amplitude loading. The endurance surface therefore directly dictates the physical endurance limit of the material. To obtain mesh-objective results, the formulation is regularized by a micromorphic gradient enhancement. The incorporation of anisotropic damage evolution and the microcrack-closure-reopening effect extends the framework to multiaxial fatigue and loading-path-dependent degradation. The chosen prototype damage evolution fulfills three requirements: reasonable physics, computational robustness, and calibration flexibility.  \nThe model is successfully calibrated to both the monotonic response of plain concrete and to the high-cycle fatigue behavior of low-alloy steel. The numerical examples cover monotonic failure of an L-shaped concrete specimen, stress–life behavior under cyclic loading, and combined axial–torsional fatigue. These cases demonstrate how the proposed formulation applies to practical scenarios ranging from standard quasi-brittle fracture benchmarks to classical fatigue characterization and complex multiaxial damage evolution. The examples demonstrate that the formulation captures progressive degradation over many cycles and reproduces characteristic stress–life behavior. The influence of anisotropic degradation becomes especially relevant under multiaxial loading conditions during the near-failure phase. Overall, the approach provides a thermodynamically consistent, gradient-enhanced, and computationally robust framework for simulating fatigue-driven damage in the high-cycle regime.  \nKeywords: continuum damage, high-cycle fatigue, endurance surface, anisotropy, gradient regularization  \n1 Introduction  \nFatigue plays a central role in the assessment of structural reliability since it accounts for over 50 % of mechanical failure cases [1] . Typical examples include mechanical components in rotating machinery, structural elements in transportation systems, and load-bearing parts in energy infrastructure. In many of these applications, the applied stress amplitudes remain well below the macroscopic yield limit of the material. Consequently, degradation evolves gradually over many cycles while the mechanical response is largely governed by elasticity at the macro-scale. This regime is commonly referred to as high-cycle fatigue, and the damage behavior is quasi-brittle, in contrast to low-cycle fatigue with plasticity-driven ductile behavior.  \nAnalyzing high-cycle fatigue requires a constitutive framework that can distinguish nondamaging cyclic states from states that induce progressive distributed degradation in a predominantly elastic regime. This challenge becomes even more pronounced for multiaxial high-cycle fatigue, because the endurance limit must be generalized from a scalar stress amplitude to a surface separating nondamaging and damaging cyclic states. Several strategies have been proposed to address this challenge. Crack-growth-based approaches, such as Paris-law formulations, typically focus on the propagation of preexisting cracks by means of sharp interfaces. However, these formulations are primarily based on heuristics rather than thermodynamic principles and distributed damage accumulation as well as crack initiation are not captured [2, 3, 4, 5] . Alternatively, diffuse representations of cracks, as i","cbCaioiHjK7igOuC","https://ap.wps.com/l/cbCaioiHjK7igOuC","pdf",2095445,2,1,19,"English","en",105,"# Introduction\n## Background and motivation for high-cycle fatigue modeling\n## Endurance surface concept and its role in damage accumulation","[{\"question\":\"What drives damage evolution in the proposed fatigue framework?\",\"answer\":\"The framework uses the energy-release rate as the driving force, with damage evolution governed by the distance of the thermodynamic driving force from the endurance surface.\"},{\"question\":\"How does the endurance surface enable damage accumulation under constant-amplitude loading?\",\"answer\":\"Unlike classic failure surfaces, the endurance surface allows damage to accumulate over many cycles even when the applied loading amplitude remains constant.\"},{\"question\":\"What model features extend the framework to multiaxial fatigue and complex loading paths?\",\"answer\":\"Anisotropic damage evolution and the microcrack-closure–reopening effect are incorporated, enabling loading-path-dependent degradation under multiaxial conditions.\"}]","Modeling Fatigue-Induced Anisotropic Quasi-Brittle Damage Based on the Endurance Surface Concept | 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drives damage evolution in the proposed fatigue framework?","Question",{"text":76,"@type":77},"The framework uses the energy-release rate as the driving force, with damage evolution governed by the distance of the thermodynamic driving force from the endurance surface.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the endurance surface enable damage accumulation under constant-amplitude loading?",{"text":81,"@type":77},"Unlike classic failure surfaces, the endurance surface allows damage to accumulate over many cycles even when the applied loading amplitude remains constant.",{"name":83,"@type":74,"acceptedAnswer":84},"What model features extend the framework to multiaxial fatigue and complex loading paths?",{"text":85,"@type":77},"Anisotropic damage evolution and the microcrack-closure–reopening effect are incorporated, enabling loading-path-dependent degradation under multiaxial 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