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Cracks in complex materials: varifold-based variational description

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Cracks in complex materials: varifold-based variational description

Paolo Maria Mariano

University of Florence - Italy

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

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

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

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

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:

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

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

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.

The energy

- Cases
- the latter being the n-vector containing 1 and all minors of the spatial derivative of n

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

Ingredients - 3

: a. e. approximately differentiable map

Assume

Graph

- n-current orientation over the graph

- Mass

Boundary current

Existence for extended weak diffeomorphisms

- Assumptions on the energy density

Sequel

: the space hosting minimizers

Existence for SBV-diffeomorphisms

Assumptions about the energy: H1-1 remains the same,

H2-1 changes in

: the space hosting minimizers

The relevant existence theorem follows

Another case

The interaction between deformation and microstructure depends on the whole set of minors of Du and Dn

: 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

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

- 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

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.

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