Identification of some nonsmooth evolution systems with illustration on adhesive contacts at small strains

Abstract

A class of evolution quasistatic systems which leads, after a suitable time discretization, to recursive nonlinear programs, is considered and optimal control or identification problems governed by such systems are investigated. The resulting problem is an evolutionary Mathematical Programs with Equilibrium Constraints (MPEC). A subgradient information of the (in general nonsmooth) composite objective function is evaluated and the problem is solved by the Implicit programming approach. The abstract theory is illustrated on an identification problem governed by delamination of a unilateral adhesive contact of elastic bodies discretized by finite-element method in space and a semi-implicit formula in time. Being motivated by practical tasks, an identification problem of the fracture toughness and of the elasticity moduli of the adhesive is computationally implemented and tested numerically on a two-dimensional example. Other applications including frictional contacts or bulk damage, plasticity, or phase transformations are outlined.

Normally Admissible Stratifications and Calculation of Normal Cones to a Finite Union of Polyhedral Sets

“…Due to the definition of normal cone, we see that (38) contains a hidden constraint 0 ≤ y n ≤ y n−1 , meaning that a glue cannot heal back to its original state y 0 . When considering optimal control or parameter identification in such model, it is advantageous to compute gph N , see [3].…”

“…First, we realize thats = (1,3,4,8,6), wheres ∈ is the unique index such that (ȳ,z) ∈ s . Employing (40), we realize thatI 1 (s 1 ) = {1}, I 2 (s 2 ) = {3}, I 3 (s 3 ) = {3, 4, 5}, I 4 (s 4 ) = {1, 8} and I 5 (s 5 ) = {5, 6}.…”

Cohesive zone-type delamination in visco-elasticity

“…Fig. 1, line 1, also upon renormalization, the cohesive zone surface energy density φ coh (without memory), then goes over to the adhesive contact energy φ adh from (1). An adhesive contact model involving φ adh thus can be seen as a cohesive zone model without the memory of the history of maximal separations.…”

“…[21,22], there is a large amount of analytical results on adhesive contact delamination models, see e.g. [28,25,7,8,46,36,31,44,45,37,52,43,27,47,48,33,1]. In adhesive contact models, the internal variable z : [0, T ]×Γ C → [0, 1] has the meaning of active bonds in the adhesive that glues the two parts of the body along Γ C .…”

“…on Γ C . For the limit κ → ∞ in the energy term (1) the brittle constraint could be re-obtained, also necessitating a sophisticated design of recovery sequences involving the proof of additional fine convergence properties of the semistable internal variables z k .…”