Phase-Field Approximation of a Vectorial, Geometrically Nonlinear Cohesive Fracture Energy
Sergio Conti, Matteo Focardi, Flaviana Iurlano
Abstract
Abstract We consider a family of vectorial models for cohesive fracture, which may incorporate $$\textrm{SO}(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SO</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -invariance. The deformation belongs to the space of generalized functions of bounded variation and the energy contains an (elastic) volume energy, an opening-dependent jump energy concentrated on the fractured surface, and a Cantor part representing diffuse damage. We show that this type of functional can be naturally obtained as $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> -limit of an appropriate phase-field model. The energy densities entering the limiting functional can be expressed, in a partially implicit way, in terms of those appearing in the phase-field approximation.