# Material Evolution from Plasticity to Morphogenesis (17frg674)

Arriving in Banff, Alberta Sunday, June 11 and departing Sunday June 18, 2017

## Organizers

Manuel de León (Consejo Superior de Investigaciones Científicos)

Marcelo Epstein (University of Calgary)

(Instituto de Ciencias Matematicas)

(Ben-Gurion University of the Negev)

## Objectives

The primary objective of our stay in Banff is bringing together researchers from five different institutions, that come from different branches of science, engineering and mathematics, but with a same objective: to put their knowledge in common for a research project entitled Material evolution: from plasticity to morphogenesis''.

This gathering is part of an ongoing effort to overlap purely mathematical grounds and applications of engineering to material evolution. People in the proposed team are good representatives of these fields of research. The spanish team is an expert in differential geometry and geometrical mechanics, whilst members from Canada, Israel and US have a solid background of applications of differential geometry in biomechanics and materials.

From this meeting we expect some publications in reputable journals concerning the applications of geometric mechanics to the understanding of evolution and composition of organic and inorganic materials: from muscle tissue to crystals. We also plan to write a state of the art indicating potential ways to explore in the next future.

Apart from these generic objectives, we plan to discuss the following items:

• A generalization of the algebroid and groupoid theory to different kind of materials, as for instance, functionally graded materials and Cosserat media.
• Derivation of a Hamilton--Jacobi theory that is invariant under the MS setting.
• A further step consists of a geometric Hamilton--Jacobi equation with the implementation of stochastic variables that still preserves the MS setting.

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#### Bibliography

1. [bridges] T.J. Bridges, Canonical multisymplectic structure on the total exterior algebra bundle, Proc. Royal Soc. A: 462, 1531--1554 (2006).

2. [demours} F. Demours, Stochastic multisymplectic variational integrators for SE(3) strand, Imperial College, ESI Workshop, Vienna, Februrary 2015. \bibitem{MEE] M. Epstein, M. Elzanowski, \textsl{Material Inhomogeneities and their Evolution}, Springer-Verlag 2007.

3. [MEPMDL] M. Epstein, M. De Le\'on, Geometrical theory of uniform Cosserat media, J. Geom. Phys. 26, 127--170 (1998).

4. [MDLME] M. Epstein, M. De Le\'on, Homogeneity without Uniformity: towards a Mathematical Theory of Functionally Graded Materials, Int. J. Solids and Structures 37, 7577-7591 (2000).

5. [MEPMDLSEG] M. Epstein, M. De Le\'on, Unified geometric formulation of material uniformity and evolution, Math. Mech. Complex Syst. 4, 17--29 (2016).

6. [MEMDLVMJ] M. Epstein, M. De Le\'on, V. M. Jiménez, Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies, Preprint. Arxiv: 1607.04043 (2016).

7. [LeonSardon1] M. de Le\'on, C. Sard\'on, Hamilton--Jacobi theory on Nambu--Poisson manifolds, Sent to Journal of Mathematical Physics, Arxiv: 1604.08904 (2016).

8. [LeonSardon2] M. de Le\'on, C. Sard\'on, A Geometric approach to solve time dependent and dissipative Hamiltonian systems, Sent to Journal of Mathematical Physics, Arxiv: 1607.01239 (2016).

9. [MPS} J.E. Marsden, G.W. Patrick, S. Shkoller, {\sl Multisymplectic Geometry, variational integrators and nonlinear PDEs,] ArXiv:9807080, 1998.

10. [WNOLL] W. Noll, Materially uniform simple bodies with inhomogeneities, Arch. Rational Mech. Anal. 27, 1--32 (1967).

11. [SanzSerna] J.M. Sanz-Serna, M.P. Calvo, Numerical Hamiltonian problems, Appl. Math. and Math. Comput. 7, Springer Science-Business Media BV, 1994.

12. [TU] A.M. Turing, The Chemical Basis of Morphogenesis, Phil. Trans. Royal Society London B237, 37--72 (1952).

13. [WABL] C.C. Wang, F. Bloom, Material Uniformity and Inhomogeneity in Anelastic Bodies, Arch. Rational Mech. Anal. 53, 246--276 (1974).

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