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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)

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Cracks in complex materials: varifold-based variational description

Paolo Maria Mariano

University of Florence - Italy

some prominent cases
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
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
  • 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


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
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

First moment of the distribution of n

Porous body

Slip systems generating plastic flows

ingredients 2
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
ingredients 2 sequel
Ingredients – 2 sequel

Stratified families

k = 2, … , n-1

why stratified varifolds
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
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
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
Ingredients - 3

: a. e. approximately differentiable map



  • n-current orientation over the graph
  • Mass

Boundary current

existence for extended weak diffeomorphisms
Existence for extended weak diffeomorphisms
  • Assumptions on the energy density

: the space hosting minimizers

existence for sbv diffeomorphisms
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
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

  • 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.