Adoption of non-local computational approaches in analyzing fracture propagation is a wellknown necessity required in order to overcome mesh-dependent results and to guarantee computational accuracy [1]. Among all non-local philosophies available in the literature, procedures characterizing the kinematics of the analyzed continuum by the second-gradient of the displacement (i.e. the strain gradient) proved to be particularly effective. Among them, recently a variational formulation involving the strain gradient total deformation energy functional was developed [2]. It proved to be effective in reproducing brittle crack propagation, by means of the damage field, in two-dimensional domains, although it can be easily extended to threedimensional cases.Such an approach was implemented in a computational framework based on the Galerkin finiteelement method. The original implementation permitted to compute quasi-static responses by an explicit integration scheme in which the response relevant to each pseudo-time step was computed by as function of the damage field value at the previous step. As for most of the explicit integration procedures, the accuracy of the adopted scheme is strongly dependent on the amplitude of the pseudo-time steps. Hence, it was necessary to adopt dense discretizations of load paths thus increasing the computational burden.The present contribution exploits the possibility of using an iterative procedure, based on the fixed-point theorem, thus formulating an implicit integration scheme. It adopts a sequence of trial equilibrium states and aims to minimize the amplitude of a residual. To exploit such an approach to the strain gradient formulation presented in [2], different typologies of residuals, defined by means of the damage field, the elastic energy and the static response, have been investigated. Numerical analyses, providing a comparison with the original explicit scheme, prove the robustness of the investigated implicit integration algorithm, its accuracy, and its benefits in reducing the computational burden.
Implicit integration scheme in strain-gradient modelling of brittle fracture
Placidi L
2022-01-01
Abstract
Adoption of non-local computational approaches in analyzing fracture propagation is a wellknown necessity required in order to overcome mesh-dependent results and to guarantee computational accuracy [1]. Among all non-local philosophies available in the literature, procedures characterizing the kinematics of the analyzed continuum by the second-gradient of the displacement (i.e. the strain gradient) proved to be particularly effective. Among them, recently a variational formulation involving the strain gradient total deformation energy functional was developed [2]. It proved to be effective in reproducing brittle crack propagation, by means of the damage field, in two-dimensional domains, although it can be easily extended to threedimensional cases.Such an approach was implemented in a computational framework based on the Galerkin finiteelement method. The original implementation permitted to compute quasi-static responses by an explicit integration scheme in which the response relevant to each pseudo-time step was computed by as function of the damage field value at the previous step. As for most of the explicit integration procedures, the accuracy of the adopted scheme is strongly dependent on the amplitude of the pseudo-time steps. Hence, it was necessary to adopt dense discretizations of load paths thus increasing the computational burden.The present contribution exploits the possibility of using an iterative procedure, based on the fixed-point theorem, thus formulating an implicit integration scheme. It adopts a sequence of trial equilibrium states and aims to minimize the amplitude of a residual. To exploit such an approach to the strain gradient formulation presented in [2], different typologies of residuals, defined by means of the damage field, the elastic energy and the static response, have been investigated. Numerical analyses, providing a comparison with the original explicit scheme, prove the robustness of the investigated implicit integration algorithm, its accuracy, and its benefits in reducing the computational burden.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
