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

RemarkThe 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

•

•

Porous body Slip systems generating plastic flows

First moment of the distribution of

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

• Mass M(V) of a varifold: over the set where V is defined

Ingredients – 2 sequel

Stratified families

k = 2, … , n-1

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

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

Functional choices for the deformation - 1

A closure theorem

Ingredients - 3

• n-current orientation over the graph

Boundary current

: a. e. approximately differentiable map

Assume

Graph

• Mass

Functional choices for the deformation - 2

Another closure theorem

Existence for extended weak diffeomorphisms

• Assumptions on the energy density

Sequel

: the space hosting minimizers

An existence theorem

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 caseThe interaction between deformation and microstructure

depends on the whole set of minors of Du and D

Assumptions about the energy

: 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

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

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.