Using a penalty term to deal with spurious oscillations in second gradient finite elements

Abstract

It is well known that it is necessary to introduce a length scale parameter in a continuum damage mechanics model to correctly simulate strain localization. The second gradient model, a special case of kinematically enriched continua, considers an internal length parameter by taking into account the second order derivatives of the displacements in the virtual power principle. In this paper, we show that the original second gradient finite element of Chambon and co-workers can present spurious oscillations, especially for mode I crack propagation problems. After providing the plane stress second gradient constitutive law, we propose to add a penalty term in the original formulation in order to improve numerical convergence and to avoid spurious oscillations in the local variables distributions. Two numerical examples using classical damage mechanics laws, a three points reinforced concrete beam and a trapezoidal notched specimen are used to test the performance of the formulation. Parametrical studies are also shown on the influence of the penalty parameter. The problem of unrealistic damage spreading for damage values close to one, occurring often in mode I crack propagation problems, is finally discussed.

A second gradient cohesive element for mode I crack propagation

“…In the second gradient model however it is very difficult to choose the appropriate value for the penalization parameter, which in fact has to be very large [24]. The combined use of Lagrange multiplier fields and penalization terms is a way out, as it improves the convergence performance and the sensitivity to the value of the penalization parameter [21] [24] [28]. In this latter case, Eq.13 becomes:…”

“…One particular type of generalized continua, the second gradient model developed by Chambon et al [14] [15] [16] [17], has demonstrated its ability to regularize strain localization in the framework of plasticity [15] [18] [19] and damage mechanics [2] [20] [21]. Shear banding problems [22] [23] [24] [25] and mode I crack propagation [2] can be reproduced up to a certain point.…”

“…On the other hand, as demonstrated by various authors [21] [26] [27] [28] [29], strain localization with a second gradient model can become problematic at the late stages of softening response, when the second gradient terms become significant compared to the first gradient terms. This is particularly true for sharp or narrow mode I fracture problems where high values of damage are expected as soon as the crack starts…”

“…Its performance is however less satisfying for mode I crack propagation problems (where damage increases rapidly and becomes close to 1). To illustrate this behavior, a notched trapezoidal beam with an imposed displacement at its lips [21] [5] is modeled hereafter, Figure 6. The damage evolution law is the same as in the 1D test (equation ( 19)).…”

A generalized Timoshenko beam with embedded rotation discontinuity

“…[22] who adopted a Cosserat type continuum, well suited for granular materials, to correctly model shear banding. This was soon followed with specific kinematically enriched models [23][24][25] and theoretical developments for classical and multiphase porous media and case studies for soils and concrete [26][27][28][29][30][31][32].…”