PAMM · Proc. Appl. Math. Mech. 14, 151 – 152 (2014) / DOI 10.1002/pamm.201410063 Modeling and Simulation of Damage Processes based on a Gradient-Enhanced Free Energy Function Stephan Schwarz 1, ∗ , Philipp Junker 1 , and Klaus Hackl 1 1 Ruhr-University Bochum, Department of Civil and Environmental Engineering, Institute for Computational Engineering, Chair of Mechanics of Materials Taking into account softening effects in connection with conventional inelastic material models can cause ill-posed boundary value problems. These problems can be established by obtaining no unique solution for the resulting algebraic system or by having a strong mesh dependence of the numerical results. This is the consequence of losing ellipticity of the governing field equations. A possible approach to solve these problems is to introduce a non-local field function in the model which includes an internal material length scale. For this purpose a gradient-enhanced free energy function is used for the current continuum damage model from which two variational equations are resulting. Calculations with less effort can be achieved due to the enhancement of the free energy function in comparison to other approaches. The mentioned model is applied to a material with locally varying damage properties (yield limits). Furthermore, the model is able to describe crack propagation in cases of completely damaged material. Therewith, a matrix material including precip- itates, such as carbides, is modeled. This allows to investigate ship screws, which usually exhibit the mentioned composition, with regard to the influence of cavitation. Cavitation describes the implosion of risen vapor bubbles, whereby the impact on screws causes heavy damages which can lead to a complete destruction. c  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Material Model The following gradient-enhanced Helmholtz free energy function is used in the current material model, see also [1], [2] and [3]. Ψ = 1 2 f (d) ε : C : ε + α 2 (ϕ − d) 2 + β 2 |∇ϕ| 2 (1) Hereby, we chose a local damage function in the form of f (d) = exp(−d) and the local damage parameter can be identified by d. Besides the local parameter we also use a non-local damage parameter represented by the field function ϕ. The energy function considers two types of penalizations: The slip between the local and non-local field is penalized by (ϕ − d) 2 and gradients in the non-local field are penalized by |∇ϕ| 2 . The strength of penalization can be controlled by the model parameters α for the local effects and β for the non-local effects. Based on the minimization of the potential functional, that is the Gibbs free energy, the variational equations with respect to the displacements u and with respect to the field function ϕ can be derived. These variational equations allow us to define the Helmholtz equation α (ϕ − d) − β Δϕ =0 as well as the boundary condition ∇ϕ · n =0. Since the model contains the internal variable d an evolution equation is needed. This can be found with the principle of the minimum of the dissipation potential which takes into account the rate of the free energy function ˙ Ψ and the dissipation functional D. After minimization and solving for the rate of the local damage parameter an expression for the evolution equation is obtained. But previously, the driving force can be calculated and modified with use of the Helmholtz equation as follows p = − ∂ Ψ ∂d = − f ′ (d) 1 2 ε : C : ε + α (ϕ − d) = − f ′ (d) 1 2 ε : C : ε + β Δϕ (2) Thus, the evolution equation can be written in the form ˙ d = − | ˙ d| r ∂ Ψ ∂d = | ˙ d| r p = | ˙ d| r  −f ′ (d) 1 2 ε : C : ε + β Δϕ  (3) To update the local damage parameter the yield function φ = p 2 − r 2 and the Kuhn-Tucker conditions ˙  ≥ 0 ,φ ≤ 0 , ˙ φ =0 are used. The yield function as well as the consistency parameter ˙  = | ˙ d|/r can be found by performing the Legendre transformation D ∗ = sup ˙ d  p ˙ d −D  = sup ˙ d  p ˙ d − r| ˙ d|  = sup ˙ d  | ˙ d| r p 2 − r| ˙ d|  = sup ˙ d  | ˙ d| r ( p 2 − r 2 )  (4) ∗ Corresponding author: e-mail Stephan.Schwarz@rub.de, phone +49 234 32 21452, c  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim