Past OxMOS Workshop/Meeting/Lecture

Fracture mechanics is a significant scientific field of great practical importance. Recently the subject has been invigorated by a number of important accomplishments. From the viewpoint of fundamental science there have been interesting new developments aimed at understanding fracture at the atomic scale; simultaneously, active research programmes have focussed on mathematical modelling, experimentation and computation at macroscopic scales. The workshop aims to examine various different approaches to the modelling, analysis and computation of fracture. The programme will allow time for discussion.

Invited speakers include:

Andrea Braides (Università di Roma II, Italy)

Adriana Garroni (Università di Roma, “La Sapienza”, Italy)

Christopher Larsen (Worcester Polytechnic Institute, USA)

Matteo Negri (Università di Pavia, Italy)

Robert Rudd (Lawrence Livermore National Laboratory, USA)

We are concerned with the derivation of the $\Gamma$-limit to a three-dimensional geometrically exact

Cosserat model as the relative thickness $h>0$ of a flat domain tends to zero. The Cosserat bulk model involves

already exact rotations as a second independent field and this model is meant to describe defective elastic crystals liable to fracture under shear.

It is shown that the $\Gamma$-limit based on a natural scaling assumption

consists of a membrane like energy contribution and a homogenized transverse shear energy both scaling with $h$,

augmented by an additional curvature stiffness due to the underlying Cosserat bulk formulation, also scaling with $h$.

No specific bending term appears in the dimensional homogenization process. The formulation

exhibits an internal length scale $L_c$ which survives the homogenization process.

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A major technical difficulty, which we encounter in applying the $\Gamma$-convergence arguments,

is to establish equi-coercivity of the sequence of

functionals as the relative thickness $h$ tends to zero. Usually, equi-coercivity follows from a local coerciveness assumption.

While the three-dimensional problem is well-posed for the Cosserat couple modulus $\mu_c\ge 0$, equi-coercivity forces us

to assume a strictly positive Cosserat couple modulus $\mu_c>0$. The $\Gamma$-limit model determines the

midsurface deformation $m\in H^{1,2}(\omega,\R^3)$. For the case of zero Cosserat couple modulus $\mu_c=0$

we obtain an estimate of the $\Gamma-\liminf$ and $\Gamma-\limsup$, without equi-coercivity which is then strenghtened to a $\Gamma$-convergence result for zero Cosserat couple modulus. The classical linear

Reissner-Mindlin model is "almost" the linearization of the $\Gamma$-limit for $\mu_c=0$

apart from a stabilizing shear energy term.

Computational Mechanics (CM) has become

a scientific discipline in itself, being High Perfomance Computational

Mechanics (HPCM) a key sub-discipline. The effort for the most efficient use of

distributed memory machines provides a different perspective to CM scientists

relative to a wide range of topics, from the very physics of the problem to

solve to the numerical method used. Marenostrum supercomputer is the largest

facility in Europe and the 5th in the world (top500.org - Spring 2007). This

talk describes the research lines in the CASE Dpt. of the BSC applied to

Aerospace, Bio-mechanics, Geophysics or Environment, through the development of

Alya, the in-house HPCM code for complex coupled problems capable of running

efficiently in large distributed memory facilities.

Professor Qiang Du will go over some work on modelling interface/microstructures with curvature dependent energies and also the effect of elasticity on critical nuclei morphology.

There will be three discussion meetings based on aspects of the

programme open to all internal project members. Others interested in

attending should contact Carlos Mora-Corral.

Team meetings, held roughly every four weeks, are open to anyone who is

interested. OxMOS post docs and Dphil students will give updates on the

research.

There will be three discussion meetings based on aspects of the

programme open to all internal project members. Others interested in

attending should contact Carlos Mora-Corral.

Team meetings, held roughly every four weeks, are open to anyone who is

interested. OxMOS post docs and Dphil students will give updates on the

research.

There will be three discussion meetings based on aspects of the

programme open to all internal project members. Others interested in

attending should contact Carlos Mora-Corral.