On the jerky crack growth in elastoplastic materials
- PDF / 668,350 Bytes
- 40 Pages / 439.37 x 666.142 pts Page_size
- 53 Downloads / 281 Views
Calculus of Variations
On the jerky crack growth in elastoplastic materials Gianni Dal Maso1
· Rodica Toader2
Received: 14 October 2019 / Accepted: 31 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The purpose of this paper is to show that in elastoplastic materials cracks can grow only in an intermittent way. This result is rigorously proved in the framework of a simplified model. Mathematics Subject Classification 49K10 · 35J05 · 35J25 · 74A45 · 74C05
1 Introduction In this paper we give a contribution to the mathematical derivation of the properties of the quasistatic crack growth in elastoplastic materials. The study of this subject has a long history (see, e.g., [11,14,15]). Our aim is to obtain a precise mathematical result in a simplified model where perfect plasticity interacts with crack growth. In particular, under suitable assumptions we prove that cracks are piecewise constant in time. In our simplified model the reference configuration Ω is a bounded connected open subset of R2 with Lipschitz boundary. We consider only the antiplane case, so that the displacement u is a function from Ω into R. We assume that the cracks and the plastic slips may occur only on a prescribed segment Γ , whose interior is contained in Ω and whose end-points belong to ∂Ω. It is not restrictive to assume that Γ := {(x, 0) : a ≤ x ≤ b} for some a < b. Since there is no plastic part in Ω \ Γ , the displacement u belongs to H 1 (Ω \ Γ ) and the elastic energy is given by 1 2
Ω\Γ
|∇u|2 d xd y.
Communicated by A. Malchiodi.
B
Gianni Dal Maso [email protected] Rodica Toader [email protected]
1
SISSA, Via Bonomea 265, 34136 Trieste, Italy
2
DMIF, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy 0123456789().: V,-vol
123
107
Page 2 of 40
G. Dal Maso, R. Toader
We assume that at each time the crack has the form Γas := {(x, 0) : a ≤ x ≤ s} for some a ≤ s ≤ b and that the energy spent to produce it is equal to s − a. On Γsb := {(x, 0) : s ≤ x ≤ b} the plastic slip is determined by the jump of the displacement: [u] = u + − u − , where u + and u − are the traces of u on Γ from above and from below. The plastic dissipation distance between the current displacement u and a previous displacement u 0 is given by |[u] − [u 0 ]| d x. Γsb
The evolution is driven by a time-dependent Dirichlet boundary condition u = w(t) imposed on a prescribed Borel subset ∂ D Ω of ∂Ω. We first consider the incremental formulation. Given a subdivision 0 = t0 < t1 < · · · < tn−1 < tn = T of the interval [0, T ], for i = 1, . . . , n let (u i , si ) be a solution of the incremental minimum problem for the pair (u, s): 1 2 min |∇u| d xd y + s + |[u] − [u i−1 ]|d x . 2 Ω\Γ u∈H 1 (Ω\Γ ) Γsb u=w(ti ) on ∂ D Ω si−1 ≤s≤b
As in [6] we can prove (Theorem 2.5) that, passing to a subsequence, the piecewise constant interpolation of (u i , si ) converges, as the fineness of the subdivision tends to zero, to a quasistatic evolution, i.e., a pair (u, s) which satisfies the following conditions:
Data Loading...
