Numerical integration of elasto-plasticity coupled to damage using a diagonal implicit Runge–Kutta integration scheme

Abstract

This article is concerned with the numerical integration of finite strain continuum damage models. The numerical sensitivity of two damage evolution laws and two numerical integration schemes are investigated. The two damage models differ in that one of the models includes a threshold such that the damage evolution is suppressed until a certain effective plastic strain is reached. The classical integration scheme based on the implicit Euler scheme is found to suffer from a severe step-length dependence. An alternative integration scheme based on a diagonal implicit Runge-Kutta scheme originally proposed by Ellsiepen (1999) is investigated. The diagonal implicit Runge-Kutta scheme is applied to the balance of momentum as well as the constitutive evolution equations. When applied to finite strain multiplicative plasticity, the diagonal implicit Runge-Kutta scheme destroys the plastic incompressibility of the underlying continuum evolution laws. Here, the evolution laws are modified such that the incompressibility of the plastic deformation is preserved approximately. The presented numerical examples reveal that a significant increase in accuracy can be obtained at virtually no cost using the diagonal implicit Runge-Kutta scheme. It is also shown that for the model including a discontinuous evolution law, the superiority of the diagonal implicit Runge-Kutta scheme over the implicit Euler scheme is reduced.

Topology optimization of finite strain viscoplastic systems under transient loads

“…In all cases, the implicit Newmark scheme is utilized, with the stability parameters γ = 0.5 and β = 0.25. An adaptive time‐stepping scheme is used such that Δ t n +1 = Δ t n ( f i t e r / N n e w t ) v , where N n e w t is the number of Newton iterations at the current time step t n (cf the work of Borgqvist and Wallin), and the parameters f i t e r and v are chosen as f i t e r = 4 and v = 0.8. Diverging steps are restarted with Δ t n +1 =Δ t n +1 /2.…”

A modified nonlinear fatigue damage accumulation model under multiaxial variable amplitude loading

“…The parameter η, as given in Eq. (15), can be easily calculated at the number of cycles N i equals to zero N i ¼ 0 ð Þand is therefore defined as the ultimate stress in torsion test τ u . Accordingly, it is clear, that η equals 475 MPa.…”

“…Various theories and models have been proposed to predict damage accumulation in materials. Fatigue damage cumulative theories can be classified into two categories: linear [1][2][3][4][5] and nonlinear [6][7][8][9][10][11][12][13][14][15][16] damage models. An overview of some predictive models was given by Fatemi and Yang [17] and Schijve [18], with interesting discussions on influencing factors.…”

A generic anisotropic continuum damage model integration scheme adaptable to both ductile damage and biological damage-like situations

“…A priori, any thermodynamically consistent damage evolution could be used with this anisotropic damage elasto-plastic model. Compared to other integration algorithms used for anisotropic damage coupled to elastoplasticity (Lemaitre and Desmorat, 2005;Borgqvist and Wallin, 2013;El khaoulani and Bouchard, 2013;Brünig et al, 2008;Menzel and Steinmann, 2001; Abu Al-Rub and Voyiadjis, 2003), this staggered procedure has the advantage that it makes no assumption on the damage evolution other than keeping a symmetric damage tensor. It can thus be used for different materials having different damage mechanisms.…”

“…The numerical integration algorithms of constitutive models incorporating anisotropic damage effects presented in the literature (Lemaitre and Desmorat, 2005;Borgqvist and Wallin, 2013;El khaoulani and Bouchard, 2013;Brünig et al, 2008;de Souza Neto et al, 2011) are usually limited to one given damage model and are not easily extended to other formulations or damage criteria. The numerical scheme in Simo and Ju (1987a,b) or in Jeunechamps and Ponthot (2013) is similar to the proposed approach in such a way that the integration can be considered as a triple operator split: elastic predictor, plastic corrector, damage corrector.…”