Cracks in complex materials: varifold-based variational description

1. Cracks in complex materials: varifold-based variational description Paolo Maria Mariano University of Florence - Italy

2. Some prominent cases • Y. Wei, J. W. Hutchinson, JMPS, 45, 1253-1273, 1997 (materials with strain-gradient plastic effects) • R. Mikulla, J. Stadler, F. Krul, K.-H. Trebin, P. Gumbsch, PRL, 81, 3163-3166, 1998 (quasicrystals) • C. C. Fulton, H. Gao, Acta Mater., 49, 2039-2054, 2001 (ferroelectrics) • C. M. Landis, JMPS, 51, 1347-1369, 2003 (ferroelectrics) • F. L. Stazi, ECCOMAS prize lecture, 2003 (microcracked bodies) Tentatives for a non-completely variational unified description • PMM, Proc. Royal Soc. London A, 461, 371-395, 2005 • PMM, JNLS, 18, 99-141, 2008

3. Point of view • I follow here a variational view on fracture mechanics • As in G. Francfort and J.-J. Marigo’s proposal, deformation and crack are distinct but connected entities • In contrast to that proposal, fractures are represented by special measures: curvature varifolds with boundary • Griffith’s energetic description of fracture is evolved up to a form including effects due to the curvature of the crack lateral margins, the tip, and possible corners • Material complexity is described in terms of the general model-building framework of the mechanics of complex materials • Even possible non-local interactions among microstructures could appear in the energy considered • Miminizers of that energy are lists of deformation, descriptors of the material morphology, families of varifolds: pertinent existence theorems are shown • The jump set of the minimizing deformation is contained in the support of the minimizing varifold

4. Minimality of the energy is required over a class of bodies parameterized by families of varifolds and classes of fields

5. Consequences • Crack nucleation can be described without additional failure criterion: it is intrinsic to the variational treatment • Partially open cracks can be described • The list of balance equations coming from the first variation is enriched: such equations include curvature-dependent terms • Nucleation of macroscopic line defects in front of the crack tip is naturally described • Interaction with the crack pattern of microstructure line defects, and microstructure domain patterns can be accounted for by appropriate choices of functional spaces • Energy can be attributed to the tip and corners Remark The choice of a function space as ambient for minimizers is a constitutive prescription which can be considered analogous to the explicit assignment of the energy

6. Ingredients - 1 • A reference macroscopic place • Standard deformations • Descriptor map of the inner material morphology belonging to a function space equipped with a functional • which is l.s.c. in L1 • is compact for the L1 convergence for every k • s. t. if • and in L1 • Example:

7. Examples of descriptors of the inner material morphology Polymer chain n n n  n First moment of the distribution of n Porous body Slip systems generating plastic flows

8. Ingredients - 2 • A fiber bundle with typical fiber the Grassmanian of k-planes over the reference place, k=1,…, n-1, • Non-negative Radon measures V over such a bundle: varifolds • A subclass defined over • Densities s.t. defining rectifiable varifolds • Special case. Densities with integer values: integer rectifiable varifolds • MassM(V) of a varifold: over the set where V is defined

9. Ingredients – 2 sequel Stratified families k = 2, … , n-1

10. Why stratified varifolds? Approximate tangent spaces describe locally the crack patterns.The star of directions in a point collects all possibilities for the possible nucleation of a crack. Stratified families of varifolds:V2-support is C,V2-support is the whole C,V1-support is the tip alone.

11. The energy • Cases • the latter being the n-vector containing 1 and all minors of the spatial derivative of n

12. Some reasons for the curvature • Rupture due to bending of material bonds induces related configurational effects measured by the curvature • Surface microstructural effects – a coarse account of them • Analytical regularization

13. Functional choices for the deformation - 1

14. A closure theorem

15. Ingredients - 3 : a. e. approximately differentiable map Assume Graph • n-current orientation over the graph • Mass Boundary current

16. Functional choices for the deformation - 2

17. Another closure theorem

18. Existence for extended weak diffeomorphisms • Assumptions on the energy density

19. Sequel : the space hosting minimizers

20. An existence theorem

21. Existence for SBV-diffeomorphisms Assumptions about the energy: H1-1 remains the same, H2-1 changes in

22. : the space hosting minimizers The relevant existence theorem follows

23. Another case The interaction between deformation and microstructure depends on the whole set of minors of Du and Dn

24. Assumptions about the energy

25. : the space hosting minimizers • The microstructure may create domains • The closure theorem for SBV-diff implies that the energy two slides ago is L1-lower semicontinuous on The relevant existence theorem follows

26. Remarks • The comparison varifold can be even null – there is then possible nucleation • Stratified families of varifolds allow us to distribute energy over submanifolds with different dimensions (the tip, its corners, etc) • No external failure criterion has to be assigned a priori: energy and boundary conditions determine the minimizing varifold, then the crack pattern

27. Details in • M. Giaquinta, P.M.M., G. Modica, DCDS-A, 28, 519-537, 2010 “Nirenberg’s issue” • See also (for the varifold-based description of fractures in simple bodies) • M. Giaquinta, P.M.M., G. Modica, D. Mucci, Physica D, 239, 1485-1502, 2010 • P.M.M., Rend. Lincei, 21, 215-233, 2010 • M. Giaquinta, P.M.M., G. Modica, D. Mucci, Tansl. AMS, 229, 97-117, 2010

28. A model is a ‘speech’ about the nature, a linguistic structure over empirical data. It is conditioned by them but, at the same time, it transcends them.