Weak formulation of elastodynamics in domains with growing cracks

Abstract

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 for both the damped and undamped systems, while in the damped case we are also able to prove uniqueness and an energy balance.

An existence result for the fractional Kelvin–Voigt’s model on time-dependent cracked domains

“…For the term involvingF, we argue as we already did forḞ and by using two times Fatou's lemma we get For the fractional Kelvin-Voigt's model (2.9), we expect to have uniqueness of the solution, as it happens in [6,24] for the classic Kelvin-Voigt's one. Unfortunately, the technique used in the cited papers cannot be applied here, and we are able to prove it only when the crack is not moving (see Sect.…”

“…In the classical theory of linear viscoelasticity, the constitutive stress-strain relation of the so-called Kelvin-Voigt's model is given by σ (t) = Ceu(t) + Beu(t) in \ t , t ∈ (0, T ), (1.2) where C and B are two positive tensors acting on the space of symmetric matrices, and ev denotes the symmetric part of the gradient of a function v (which is defined as ev := 1 2 (∇v + ∇v T )). The local model associated with (1.2) has already been widely studied and we can find several existence results in the literature; we refer to [2,3,6,7,17,24] for existence and uniqueness results in the pure elastodynamics case (B = 0) and in the classic Kelvin-Voigt's one.…”

On the continuity of the trace operator inGSBV(Ω) andGSBD(Ω)

“…In our case the finitness of the jump sets is ensured by requiring the stronger condition J u ⊂ Γ, but on the other hand J u might have possible interaction with both Dirichlet and Neumann part. Minimization problems like (5.5), arise for example in the minimizing movements technique, for example in [5] or in [13], in order to solve respectively the wave equation or the equations of elastodynamics, in a prescribed arbitrary growing cracks domain.…”

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

“…where E is the elasticity tensor, given a sequence of subsets (C(t)) t≥0 of Ω satisfying some regularity requirements. In an attempt to better understand (1.8), in [14] for the scalar-valued case and [38] for the vector-valued case, the wave equation…”