Brown University - Joint Materials/Solid Mechanics Seminar Series
Abstract: Cracks propagate in porous ductile solids through two mechanisms: coalescence of voids and formation of a shear band. The former mechanism prevails in mode I and the second one in mode II. In mixed-mode situations, both can occur, with a possible coupling between the two.
In recent years, void coalescence has been the subject of several theoretical studies, due notably to Gologanu, Leblond and Perrin. Their main idea was to schematize some representative volume element (RVE) in which coalescence was taking place, as a superposition of alternately sound and porous planar layers in which mechanical fields were considered as homogeneous. Gurson's classical "homogenized" model was used to describe the behavior of the porous layers. The onset of coalescence in this approach corresponded to sudden concentration of the strain rate in the porous layers, the sound ones becoming then rigid. The deformation mode of the RVE during coalescence was a pure extension in the direction perpendicular to the layers.
Concerning formation of shear bands, an old but seminal theoretical contribution was due to Drucker. He showed that in ductile materials loaded in shear, deformation tends to concentrate in thin planar layers linking voids, rather than to spread over the whole RVE. The effect is intimately tied to the presence of some porosity, in spite of the fact that void growth is limited under shear loading conditions.
We present a unified theoretical approach of void coalescence and formation of shear bands in ductile solids. It contains both Gologanu, Leblond and Perrin's treatment of void coalescence and Drucker's treatment of shear bands, and extends them in the sense that it also considers possible interactions between the two phenomena in mixed-mode situations. Its principle consists of extending the former authors' analysis, which was limited to axisymmetric macroscopic stress states, to superpositions of axisymmetric and shear stress states. The onset of void coalescence and/or formation of a shear band again corresponds to sudden concentration of the strain rate in the porous layer, but the deformation mode of the RVE is now a superposition of a pure extension and a shear deformation.
The theory is applied to the prediction of kink angles of cracks developing mixed-mode conditions in ductile solids. This is done by looking for the "most favorable" direction for the onset of void coalescence and/or formation of a shear band ahead of the crack tip. The predictions compare favorably with experimental results obtained at the French Commissariat a l'Energie Atomique. They reproduce in particular the experimental fact that such kink angles are much lower for ductile materials than for brittle ones.
Brown Analysis Seminar
Scientific Computing Seminar
Abstract: In a number of applications it is useful to compute angles between subspaces subject to linear constraints. In particular we will focus the statistical applications of computing linearly constrained canonical correlations and the corresponding canonical vectors. To do this we solve the more general problem of maximizing a bilinear form or its absolute value subject to linear constraints.
Brown University -
Graduate School, Dissertation Defense
PDE Seminar
Department of Mathematics Colloquium
Finished for This Semester!