Convergence of a gradient damage model toward a cohesive zone model

Abstract

The study starts from a specific gradient damage model which admits a closed-form solution in the case of uniaxial tension. It enables to separate the parameters of the model between a length scale, characteristic of nonlocal effects, and macroscopic parameters which retain their meaning in a cohesive crack setting. A convergence analysis is performed: the response of a cohesive zone model is retrieved when the length scale goes to zero while keeping the macroscopic parameters constant.

Comparison between thick level set (TLS) and cohesive zone models

“…Note that similar analysis concerning cohesive and gradient damage models has been performed by Lorentz in [6]. Thus, based on this equivalence, a method to derive a local damage behavior to use in TLS from any cohesive behavior will be exhibited.…”

Numerical investigation of dynamic brittle fracture via gradient damage models

“…The implicit method 0 < β ≤ 1 2 may be suitable for intermediate situations between a quasi-static and an explicit dynamic calculation. From (12), the determination of the the new accelerationü n+1 requires the knowledge of the new deformation state u n+1 which itself determines the new damage field at time t = t n+1 via (16). For the implicit Newmark method β = 0, (18) can thus be regarded as a nonlinear equation in u n+1 , where nonlinearity results from the irreversibility condition when minimizing the total energy (16).…”

“…From (12), the determination of the the new accelerationü n+1 requires the knowledge of the new deformation state u n+1 which itself determines the new damage field at time t = t n+1 via (16). For the implicit Newmark method β = 0, (18) can thus be regarded as a nonlinear equation in u n+1 , where nonlinearity results from the irreversibility condition when minimizing the total energy (16). To decouple the (u n+1 , α n+1 ) problem, we use a staggered time-stepping procedure as used in [13,14,25] among others.…”

“…The gradient projection conjugate gradient algorithm initially proposed in [23] is used to solve (16) in an iterative fashion. It is designed for quadratic bound-constrained minimization problems.…”

“…These models settle down a unified and coherent numerical framework covering the onset and the space-time propagation of cracks with possible complex topologies and have been successfully applied to study various real-world dynamic fracture problems. Meanwhile, in the quasi-static setting, further physical insights into the gradient damage model seen as a phase-field model of fracture are given among others in [16,17] where a comparison with the cohesive zone model and the Griffith's linear elastic theory is conducted. In dynamics, more well-designed numerical experiments should be performed to carry out such verification, see [18] for instance for an investigation of the phase-field crack speeds for plane problems.…”