Nonlinear dynamic fracture problems with polynomial and strain-limiting constitutive relations

2021/08/09 by Victoria Patel, Patel, Victoria
Engineering · Mathematics · #35K99 #35M13 #74D10 #74H20 #74R99 #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2108.03896

openalex publication_date 2021/08/09 · openalex created_date 2021/08/16 · openalex updated_date 2026/07/28

Abstract

We extend the framework of dynamic fracture problems with a phase-field approximation to the case of a nonlinear constitutive relation between the Cauchy stress tensor \mathbbT , linearised strain \boldsymbolε(u) and strain rate \boldsymbolε(u_t ) . The relationship takes the form \boldsymbolε(u_t) + α\boldsymbolε(u) = F(\mathbbT) where F satisfies certain p-growth conditions. We take particular care to study the case p=1 of a `strain-limiting' solid, that is, one in which the strain is bounded \it a priori. We prove the existence of long-time, large-data weak solutions of a balance law coupled with a minimisation problem for the phase-field function and an energy-dissipation inequality, in any number d of spatial dimensions. In the case of Dirichlet boundary conditions, we also prove the satisfaction of an energy-dissipation equality.

Citations

• Optimal approximations by piecewise smooth functions and associated variational problems

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