Modelling of Strain Softening Materials Based on Equivalent Damage Force

Abstract

The main aim of the work presented in this paper was addressing localisation problem observed in the analysis of strain softening materials using finite element methods (FEM) combined with local continuum damage mechanics (CDM) approach. Strain softening is typically observed in damaged quasi brittle materials such as fibre reinforced composites and application of the CDM approach with the classic FEM features a number of anomalies, including mathematical (change of the type of partial differential equations leading to ill-posed boundary value problem), numerical (pronounced mesh dependency) and physical (infinitely small softening zone with the zero dissipated energy). These features of the classic FEM solutions have been already demonstrated in (Vignjevic, Djordjevic et al. 2014). The model proposed here is still based on the local CDM approach, but introduces an alternative definition of damage effects in the system of equilibrium equations. The constitutive equation in the model is defined in terms of effective stress, whilst the damage effects in the conservation of momentum equation are calculated as equivalent damage force (EDF), which contributes to the equilibrium on the right hand side of the momentum equation. The main advantages of this model are that the problem remains well posed, as the type of partial differential equations remains unchanged when the material enters softening, numerical stability, which is preserved without a need for regularisation measures, and significantly reduced mesh dependency. In addition, the EDF model can be combined with existing local CDM damage evolution functions and does not violate symmetry of the stiffness tensor. The EDF model was implemented in in-house developed coupled FEM - MCM code, where explicit FEM (Liu 2004) is coupled with a stable TotalLagrange form of SPH (Vignjevic, Reveles et al. 2006, Vignjevic, Campbell et al. 2009). Its performance is demonstrated in the analysis of a dynamic one dimensional stress wave propagation problem, which was analytically solved in (Bazant, Belytschko 1985). For a range of loading rates that correspond to the material softening regime, the numerical results shown nonlocal character with a finite size of the damaged zone, controlled with the damage characteristic length, which can be experimentally determined and is an input parameter independent of the discretisation density.