Stochastic and deterministic analysis of models of defects in - PowerPoint PPT Presentation

Stochastic and deterministic analysis of models of defects in discrete systems Andrea Braides (Roma Tor Vergata ) Mathematical challenges motivated by multi-phase materials: analytic, stochastic and discrete aspects Anogia, June 26 2009

A prototypical model for defects A “non-defected” simple model: the discrete membrane: quadratic mass-spring systems. Ω ⊂ R d , u : ε Z d → R ε d � u i − u j � 2 � E ε ( u ) = ε NN (NN = nearest neighbours (in Ω) ) As ε → 0 E ε is approximated by the Dirichlet integral � |∇ u | 2 dx F 0 ( u ) = Ω

A prototypical ‘defected’ interaction: at a ‘defected spring’ � u i − u j � u i − u j � 2 � 2 substitute by ∧ C ε ε ε ( truncated quadratic potential ) The spring ‘breaks’ when u i − u j � = C ε ε

Note: Truncated quadratic potentials capture the main features of classes of discrete potentials. For example (asymmetric) truncated quadratic potentials can be used to derive limit energies for Lennard-Jones interactions by a comparison and scaling argument � α ′ z 2 , β ′ � � α ′′ z 2 , β ′′ � min ≤ J ( z ) ≤ min ( z > 0) NOTE: sup α ′ = inf α ′′ = 1 2 J ′′ (0) =: α (Taylor expansion at 0 ) sup β ′ = inf β ′′ = J (+ ∞ ) =: β (depth of the well) (B-Truskinovsky, B-Lew-Ortiz, etc.)

The Blake-Zisserman weak membrane The meaningful scaling for C ε is (of order) 1 ε , in which case the energy of a ‘broken’ spring scales as a surface: ε d · 1 ε = ε d − 1 . If only ‘defected’ springs are present the total energy ε d �� u i − u j � 2 ∧ 1 � � E ε ( u ) = ε ε NN is then approximated as ε → 0 by an (anisotropic) Griffith fracture energy (Chambolle 1995) � � |∇ u | 2 dx + � ν � 1 d H d − 1 F 1 ( u ) = Ω \ S ( u ) S ( u ) S ( u ) = discontinuity set of u (crack site in reference config.) ν = ( ν 1 , . . . , ν d ) normal to S ( u ) , � ν � 1 = � i | ν i | (lattice anisotr.) H d − 1 = surface measure; u ∈ SBV (Ω)

Models of defects in discrete systems Q: describe the overall effect of the presence of defects 1. ( Probabilistic setting ) Assume that the distribution of defects is random, and the probability of a defected interaction is p ∈ (0 , 1) . Is the limit deterministic? What is its form? How does it depend on p ? 2. ( “G-closure” approach ) Fix any family of distributions of defects W ε , and compute all the possible limits of the corresponding energies. What type of energies do we get? How does it depend on the local volume fraction of the defects? NOTE: a possible limit energy is always sandwiched between F 0 (Dirichlet, from above) and F 1 (Blake and Zisserman, from below); in particular it equals F 0 if no fracture occurs.

Random defects: a model for variational problems with percolation (We restrict to dimension d = 2 ) Let ω : { ( i, j ) NN in Z 2 } → { strong, defected } be a realization of an i.i.d. random variable such that � with probability p strong ω ( i, j ) = defected with probability 1 − p Define for i, j NN in ε Z 2  � � z 2 ε , j i if ω = strong  ε f ε ij ( z ) = � � z 2 ∧ 1 ε , j i if ω = defected  ε ε and the energy � u i − u j � � E ω ε d f ε ε ( u ) = ij ε NN

Tools for variational problems with percolation Clusters of strong/defected connections If p < 1 / 2 (resp., p > 1 / 2 ) almost surely there exists a (unique) infinite connected component ( cluster ) of strong (resp., defected) connections in Z 2 . paths of connections in the clusters defected strong

“Measure-theoretical” properties of clusters Each cluster is uniformly distributed : for all (large) cubes # disjoint paths connecting opposite sides is proportional to the area of the side N>>1 Consequence: if p < 1 / 2 then the functionals E ω ε are equi- coercive on H 1 (Ω) (use Poincar´ e’s inequality on strong paths).

Metric properties of clusters We define a distance on the cluster as d ω ( x, y ) = min { length of path in the cluster joining x and y } This distance can be homogenized : a.s. (in ω ) � x ε , y � d ω → ϕ ( x − y ) , ε with ϕ = ϕ p deterministic, convex and one-homogeneous ( asymptotic chemical distance ). Consequence: if p > 1 / 2 cracks will follow a minimal path in the defected cluster (the proof uses the property that long paths not in the defected cluster contain a proportion of strong links).

The Percolation Theorem (i) (subcritical regime) if p < 1 / 2 then defects are a.s. negligible and the energy is approximated by � |∇ u | 2 dx F p ( u ) = F 0 ( u ) = Ω defined in H 1 (Ω) ; (ii) (supercritical regime) if p > 1 / 2 then a.s. the discrete energy is approximated by a fracture energy governed by the chemical distance; i.e., � � |∇ u | 2 dx + ϕ p ( ν ) d H 1 F p ( u ) = Ω S ( u ) defined in SBV (Ω) . (B-Piatnitski 2008)

Notes • other types of distributions of random defects ⇒ different percolation thresholds • asymptotic expansion close to p = 1 / 2 not known • analysis limited to d = 2 for the supercritical case • similar variational formulation for other problems: dilute spin systems, “spin glass”, etc. • definition and asymptotic properties of distances d ω depend on the problem – little studied by the percolation community • i.i.d. random variables essential to have energies defined on surfaces

The deterministic case: design of weak membranes Contrary to the random case it is essential to handle particular concentrations of defects on a single surface. A side result: discrete transmission problems interfacial strong springs limit interface K voids � � u i − u j � 2 1 (strong spring) � ε d c ε c ε E ε ( u ) = ij = ij ε 0 (void) NN

Theorem (B-Sigalotti) Let p ε be the percentage of strong springs over voids at the (coordinate) interface K . If � c ε | log ε | if d = 2 p ε = c ε if d ≥ 3 then E ε can be approximated by a “transmission energy” � � |∇ u | 2 dx + b | u + − u − | 2 d H d − 1 , F ( u ) = Ω K defined on H 1 (Ω \ K ) , where � c π if d = 2 2 b = C d c if d ≥ 3 4+ C d and C d is the 2 -capacity of a “dipole” in Z d .

The Building Blocks for the design Same geometry with voids substituted by defects interfacial strong springs concentrated capacitary contribution limit interface K defects diffuse surface energy due to defects Proposition. The same p ε give � � |∇ u | 2 dx + H d − 1 ( { u + � = u − } ) + b | u + − u − | 2 d H d − 1 F ( u ) = Ω K for u ∈ H 1 (Ω \ K )

Note: (i) surface contribution of defects and capacitary contribution of strong springs can be decoupled as they live on different micro- scopic scales (ii) the construction is local, and is immediately generalized to K a locally finite union of coordinate hyperplanes (i.e., hyper- planes with normal in { e 1 , . . . , e n } ) (iii) the limit functional F can be interpreted as defined on SBV (Ω) and can be identified with F 1 ,b,K , where � � |∇ u | 2 dx + ( a + b | u + − u − | 2 ) d H d − 1 F a,b,K ( u ) = Ω S ( u ) with the constraint S ( u ) ⊂ K

Limits of energies F 1 ,b,K 1. Weak approximation of surface energies (on coordinate hyperplanes) Suitable K h s.t. H d − 1 K h ⇀ a H d − 1 K ( a ≥ 1 ) 1/h K h C/h K Then F 1 ,b,K h Γ -converges to F a,ab,K 2. Weak approximation of anisotropic surface energies. For non-coordinate hyperplanes K we find locally coordinate K h s.t. H d − 1 K h ⇀ � ν K � 1 H d − 1 K K h K 1/h Then F a,b,K h Γ -converges to F a � ν K � 1 ,b � ν K � 1 ,K

Summarizing 1 and 2: since all constructions are local, in this way we can approximate all energies � � |∇ u | 2 dx + ( a ( x )+ b ( x ) | u + − u − | 2 ) � ν � 1 d H d − 1 F a,b,K ( u ) := Ω S ( u ) with a ≥ 1 , b ≥ 0 , K locally finite union of hyperplanes, and u s.t. S ( u ) ⊂ K .

3. Homogenization of planar systems K h 1 /h -periodic of the form We can obtain all energies of the form � � |∇ u | 2 dx + ϕ ( ν ) d H d − 1 , F ϕ ( u ) = Ω S ( u ) with ϕ finite, convex, pos. 1-hom., ϕ ≥ � · � 1

Note: The condition ϕ ≥ � · � 1 is sharp since we have the lower bound F ϕ ≥ F 1 (= F �·� 1 ) . Proof: choose ( ν j ) dense in S d − 1 , Π j := {� x, ν j � = 0 } , h K h = 1 h Z d + � Π j , j =1 h Z d + Π j . Then F a h , 0 ,K h = F ϕ on b h = 0 and a h ( x ) = ϕ ( ν j ) on 1 its domain, and the lower bound follows. Use a direct construction if ν belongs to ( ν j ) H d − 1 a.e. on S ( u ) , and then use the density of ( ν j ) .

4. Accumulation of cracks (micro-cracking) K h locally of the form K h 1/h 1/h 2 K We can obtain all energies of the form � � ψ ( | u + − u − | ) d H d − 1 , |∇ u | 2 dx + F ψ ( u ) = Ω S ( u ) √ with ψ finite, concave, ψ ≥ d . √ Note: ψ ≥ d is sharp by the inequality F ψ ≥ F 1 and √ d = max {� ν � 1 : ν ∈ S d − 1 }

√ Proof. Choose a j ≥ d , b j ≥ 0 such that ψ ( z ) = inf { a j + b j z 2 } ψ Z 1) For a planar K with normal ν , choose K h = � h j =1 ( K + j h 2 ν ) and a ( x ) = a j , b ( x ) = b j on K + j h 2 ν ; 2) To eliminate the constraint S ( u ) ⊂ K use the homogenization procedure of Point 3.