[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83616-en":3,"doc-seo-83616-105":30,"detail-sidebar-cat-0-en-105":91},{"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":13,"seo_description":14,"update_tm":28,"read_time":29},83616,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","A Variational Nonlocal Phase-Field Model for Dynamic Fracture in Elastic Solids","A variational nonlocal phase-field formulation for dynamic fracture in elastic solids is developed, combining nonlocal kinematics with kernel-dependent function spaces to relax regularity while recovering the classical local theory as the nonlocal interaction domain shrinks. A nonlocal crack-surface functional is introduced as an integral counterpart of Ambrosio–Tortorelli regularization, making the diffusive crack length implicit. The model couples a nonlocal momentum balance with an irreversible nonlocal gradient-flow evolution, solved via SAV and staggered alternating schemes with finite elements. Benchmarks include Mode-I fracture, dynamic branching, shear (Kalthoff–Winkler-type), and fragmentation.","arXiv :2607 .01881v1 [math .NA] 2 Jul 2026  \nA Variational Nonlocal Phase-Field Model for Dynamic  \nFracture in Elastic Solids  \nQing Chenga , Yuqi Suna , Xuejun Xua  \na School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai, 200092, China  \nAbstract  \nWe develop a variational nonlocal phase-field model for dynamic fracture in elastic solids. The proposed formulation is distinguished by three main features. First, the model is formulated through nonlocal kinematics and kernel-dependent function spaces, allowing weaker regularity requirements while recovering the classical local theory as the nonlocal interaction domain vanishes. Second, a nonlocal crack-surface functional is introduced asan integral counterpart of the Ambrosio–Tortorelli regularization, so that the characteristic length of the diffusive crack is implicitly determined by the nonlocal interaction domain rather than by a prescribed length scale. Third, the degraded nonlocal elastic energy and the nonlocal crack-surface functional are combined into a variationally consistent dynamic fracture system, consisting of a nonlocal momentum balance and an irreversible nonlocal gradient-flow evolution law for the phase field. The coupled system is solved using two temporal discretization strategies: a structure-preserving scalar auxiliary-variable scheme and a staggered alternating scheme, both combined with finite element discretization in space. Numerical examples involving Mode-I fracture, dynamic crack branching, Kalthoff–Winkler-typeshear fracture, and fragmentation show that the proposed model captures complex crack initiation, propagation, branching, and interaction without explicit crack tracking. Quantitatively, the predicted crack-tip velocities remain below 0 .6cR in the dynamic branching and shear-loading tests, and the shear-loading benchmark gives an inclined crack path of approximately 48◦ , consistent with the characteristic Kalthoff–Winkler fracture pattern.  \nEmail addresses: [qingcheng@tongji.edu.cn](qingcheng@tongji.edu.cn) (Qing Cheng), [yuqisun@tongji.edu.cn](yuqisun@tongji.edu.cn) (Yuqi Sun), [xuxj@tongji.edu.cn](xuxj@tongji.edu.cn) (Xuejun Xu)  \nKeywords: Nonlocal phase-field, Dynamic fracture, Structure-preserving SAV scheme, Staggered alternating scheme,  \n\n| Nomenclature |  |  |  |  |\n| --- | --- | --- | --- | --- |\n| Symbol |  | Description |  |  |\n| Domains, boundaries, and geometric quantities |  |  |  |  |\n| Ω\u003Cbr>Bδ (x) | Material domain in Rd.\u003Cbr>Nonlocal interaction domain defined\u003Cbr>centered at x with radius δ . |  |  | as the open ball |\n| Fields | and kinematic quantities |  |  |  |\n| u | Displacement field. |  |  |  |\n| ψ | Phase-field variable, with ψ = 0 for and ψ = 1 for the cracked region. |  |  | intact material |\n| εδ | Nonlocal infinitesimal strain tensor. |  |  |  |\n| σδ | Nonlocal Cauchy stress tensor. |  |  |  |\n| Nonlocal kernels and operators |  |  |  |  |\n| Gδ (u) | Nonlocal |  | displacement gradient. |  |\n| Dδ (σδ) | Nonlocal tensor. |  | divergence operator acting on the stress |  |\n| Nδ (σδ) | Nonlocal |  | boundary traction operator. |  |\n| Lδ ψ | Nonlocal\u003Cbr>variable. |  | diffusion-type operator for the phase-field |  |\n| Crack-surface functional and phase-field evolution |  |  |  |  |\n| γδ (ψ, Lδ ψ) |  | Nonlocal crack-surface density functional. |  |  |\n| Aδ (ψ, Lδ ψ) |  | Nonlocal crack-surface functional. |  |  |\n| Y (ψ, D) |  | Nonlocal crack-surface driving force. |  |  |\n| Function spaces |  |  |  |  |\n| Vω (Ω) | Kernel-dependent nonlocal function space for displacement fields. |  |  |  |\n| Mω (Ω) | Kernel-dependent nonlocal function space for the phase-field variable. |  |  |  |\n\n1. Introduction  \nFracture of materials is a longstanding and fundamental problem in solid mechanics and continues to pose significant challenges for computational modeling. A central difficulty arises from the inherently multiscale and discontinu","cbCaih4HAttqLkcT","https://ap.wps.com/l/cbCaih4HAttqLkcT","pdf",2740046,2,1,51,"English","en",105,"# Introduction\n## Discrete/algorithmic fracture modeling\n## Continuous field-equation fracture modeling\n## Phase-field fracture models","[{\"question\":\"What are the main features of the proposed variational nonlocal phase-field model?\",\"answer\":\"The model uses nonlocal kinematics with kernel-dependent function spaces, introduces a nonlocal crack-surface functional as an Ambrosio–Tortorelli counterpart, and forms a variationally consistent dynamic fracture system coupling momentum balance with irreversible phase-field evolution.\"},{\"question\":\"How does the model determine the characteristic length of the diffusive crack?\",\"answer\":\"The diffusive crack length is determined implicitly by the nonlocal interaction domain, not by a prescribed length scale.\"},{\"question\":\"Which numerical schemes and physical tests are used to validate the model?\",\"answer\":\"The coupled system is solved with a structure-preserving scalar auxiliary-variable (SAV) scheme and a staggered alternating scheme, both using finite element discretization. Tests include Mode-I fracture, dynamic crack branching, Kalthoff–Winkler-type shear fracture, and fragmentation.\"}]",1784189289,129,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-variational-nonlocal-phase-field-model-for-dynamic-fracture-in-elastic-solids","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-variational-nonlocal-phase-field-model-for-dynamic-fracture-in-elastic-solids/83616/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What are the main features of the proposed variational nonlocal phase-field model?","Question",{"text":75,"@type":76},"The model uses nonlocal kinematics with kernel-dependent function spaces, introduces a nonlocal crack-surface functional as an Ambrosio–Tortorelli counterpart, and forms a variationally consistent dynamic fracture system coupling momentum balance with irreversible phase-field evolution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the model determine the characteristic length of the diffusive crack?",{"text":80,"@type":76},"The diffusive crack length is determined implicitly by the nonlocal interaction domain, not by a prescribed length scale.",{"name":82,"@type":73,"acceptedAnswer":83},"Which numerical schemes and physical tests are used to validate the model?",{"text":84,"@type":76},"The coupled system is solved with a structure-preserving scalar auxiliary-variable (SAV) scheme and a staggered alternating scheme, both using finite element discretization. Tests include Mode-I fracture, dynamic crack branching, Kalthoff–Winkler-type shear fracture, and fragmentation.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]