Calculus of Variations and Geometric Measure Theory

E. Tasso

Weak formulation of elastodynamics in domains with growing cracks

created by tasso on 27 Nov 2018
modified on 25 Nov 2019


Accepted Paper

Inserted: 27 nov 2018
Last Updated: 25 nov 2019

Journal: Annali di Matematica Pura ed Applicata
Year: 2019


In this paper we formulate and study the system of elastodynamics on domains with arbitrary growing cracks. This includes homogeneous Neumann conditions on the crack sets and mixed general Dirichlet-Neumann conditions on the boundary. The only assumptions on the crack sets are to be $(n-1)$-rectifiable with finite surface measure, and increasing in the sense of set inclusions. In particular they might be dense, hence the weak formulation must fall outside the usual context of Sobolev spaces and Korn's inequality. We prove existence of a solution both for the damped and undamped systems, while in the damped case we are also able to prove uniqueness and an energy balance.