Invertibility and Weak Continuity of the Determinant for the Modelling of Cavitation and Fracture in Nonlinear Elasticity

Abstract

In this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture. The main idea to unify the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of creation of new surface. Accordingly, we define a functional that measures the area of the created surface. This functional has relationships with the theory of Cartesian currents. We show that the boundedness of that functional implies the sequential weak continuity of the determinant of the deformation gradient, and that the weak limit of one-to-one a.e. deformations is also one-to-one a.e. We then use these results to obtain existence of minimizers of variational models that incorporate the elastic energy and this created surface energy, taking into account the orientation-preserving and the non-interpenetration conditions.

Γ-convergence Approximation of Fracture and Cavitation in Nonlinear Elasticity

“…Therefore, there exists a minimizing sequence det Dū j > 0 almost everywhere,ū j is one-to-one almost everywhere and E (ū j ) = 786 0 for all j ∈ N, the same proof of [8,Th. 4] shows that there existsū…”

“…To be precise, Henao and Mora-Corral 60 [8][9][10] showed that when the functional setting allows for cavitation and fracture,…”

Orientation-Preserving Young Measures

“…[BM84] for such examples, [HMC10] for examples in the context of cavitation and the work in [KKK12] on the weak continuity of null Lagrangians at the boundary. In particular, let p < d, q ≥ p and u ∈ W 1,q (Ω; R d ) such that det ∇u(x) < 0 a.e.…”

A micromechanical damage and fracture model for polymers based on fractional strain-gradient elasticity

“…A number of micromechanical and computational models, ranging from atomistic to continuum, have been put forth (cf., e. g., Leonov and Brown (1991); Krupenkin and Fredrickson (1999a,b); Tijssens et al (2000a,b); Estevez et al (2000a,b); Baljon and Robbins (2001); Socrate et al (2001); Drozdov (2001); Tijssens and van der Giessen (2002); Robbins (2003, 2004); Basu et al (2005); Saad-Gouider et al (2006); Zairi et al (2008); Seelig and Van der Giessen (2009) ;Reina et al (2013)), including consideration of nucleation and growth Figure 2: Crazing process in a steel/polyurea/steel sandwich specimen under opening mode fracture (Yong et al, 2009). of voids, craze nucleation, network hardening and disentanglement, chain strength, surface energy and other, that account, to varying degrees, for the observational evidence and relate macroscopic properties to material structure and behavior at the microscale. In parallel a large mathematical literature has evolved, discussing the possibility of cavitation in local models and possible nonlocal extensions which may ensure existence of minimizers, see for example Ball (1982); James and Spector (1991); Müller and Spector (1995); Conti and DeLellis (2003); Henao and Mora-Corral (2010). These advances notwithstanding, the connection between micromechanical properties and polymer fracture, and specifically any scaling laws thereof, has defied rigorous analytical treatment and characterization.…”