Phase field approximation of cohesive fracture models

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• 作者：S. Conti^a ; ^{sergio.conti@uni-bonn.de" class="auth_mail" title="E-mail the corresponding author} ; M. Focardi^c ; ^{focardi@math.unifi.it" class="auth_mail" title="E-mail the corresponding author} ; F. Iurlano^a ; ^b ; ^{iurlano@iam.uni-bonn.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 49J45 ; secondary ; 26B30 ; 74R10 ; 35A35
• 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• 出版年：2016
• 出版时间：July-August 2016
• 年：2016
• 卷：33
• 期：4
• 页码：1033-1067
• 全文大小：616 K

文摘

We obtain a cohesive fracture model as Γ-limit, as 4915000360&_mathId=si1.gif&_user=111111111&_pii=S0294144915000360&_rdoc=1&_issn=02941449&md5=b65f16ee384160ad4519a8778a986db4" title="Click to view the MathML source">ε→0$\epsilon \to 0$, of scalar damage models in which the elastic coefficient is computed from the damage variable v   through a function 4915000360&_mathId=si2.gif&_user=111111111&_pii=S0294144915000360&_rdoc=1&_issn=02941449&md5=5594e62deaf47e0d432fc2c78ba413ad" title="Click to view the MathML source">f_ε$f_{\epsilon }$ of the form 4915000360&_mathId=si3.gif&_user=111111111&_pii=S0294144915000360&_rdoc=1&_issn=02941449&md5=35d435f38bf38d88e670554315bf20e9">4915000360-si3.gif">$f_{\epsilon }(v)=\min \{1,{\epsilon }^{\frac{1}{2}}f(v)\}$, with f diverging for v close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening s at small values of s   and has a finite limit as 4915000360&_mathId=si4.gif&_user=111111111&_pii=S0294144915000360&_rdoc=1&_issn=02941449&md5=85fe607013c8bf0ad1cbdfe191a318a1" title="Click to view the MathML source">s→∞$s\to \infty$. If in addition the function f is allowed to depend on the parameter ε, for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings.