Assessment of non-local regularization techniques for the μ damage model

Purpose The µ damage model provides good results under bi-axial compression loads and enables stiffness recovery due to crack closing under cyclic loading. However, this model suffers from numerical damage localization and is unable to preserve mesh objectivity. This work regularizes five different internal variables in the µ damage model and presents two analytical cases which are used to evaluate the non-localized µ damage model for its effectiveness in preserving mesh objectivity. Design/methodology/approach The following work enriches the µ damage model by rewriting the constitutive variables in the non-local form. Five different options of f(x) are herein considered, the first invariant I, the second deviatoric invariant J, both the first and the second invariants, I and J, two scalar indicators of the local strains, “εμt” and “εμc” and the equivalent strain Yµ. These non-localization techniques are evaluated in terms of the mesh convergence and the damage distribution pattern. Findings Numerical examples demonstrate the superiority of the non-local damage model in tackling the pathological mesh dependency of the local model, provided suitable internal variables are taken to be non-local. Non-localizing the first invariant (I) or the second invariant (J) of the strain tensor alone results in the marginal improvement of the mesh sensitivity. Non-localizing both the first and the second invariants (I and J) can achieve mesh convergence on the magnitude prediction of the deflection curve, but cannot overcome the mesh sensitivity in the softening region modeling. In contrast, the µ damage models based on non-localizing either two scalar indicators of the local strains (“εμt and εμc”) or the equivalent strain Yµ effectively preserve the mesh objectivity in capturing the magnitude of the deflection and provide decent performances in the softening responses. Research limitations/implications For the future work, it is interesting to investigate the computational cost difference between non-localizing two internal variables (such as “µt and µc”) and only one variable (such as Yµ). To compute the non-local forms of two internal variables, it is anticipated that increasing the number of non-localized internal variables would result in higher computational costs. Additionally, non-localizing two variables may introduce convergence challenges, further increasing the computational costs. With an aid of appropriate experiment data, such as a case in which multiple variables dominate the damage evolution simultaneously, it would be valuable in evaluating the computational cost associated with different numbers of non-localized internal variables. Practical implications This paper demonstrates how to effectively eliminate the pathological mesh sensitivity in the µ damage model using appropriate non-local internal variables. Social implications This work contributes on demonstrating the method to eliminate the mesh sensitivity using the µ damage model. It also provides a deep understanding on how the non-localization technique achieves the mesh objectivity. Originality/value This is the first work to focus on performing analysis of the non-local regularization techniques for the recently proposed µ damage model. For the µ damage model users in future, they can directly use the conclusion of the work and save lot of efforts to figure out how to non-localize the µ damage model.