Localization in random media and its effect on the homogenized - PowerPoint PPT Presentation

Localization in random media and its effect on the homogenized behavior of materials Fran¸ cois Willot Center for Mathematical Morphology & Center for Materials P. M. Fourt Soutenance d’Habilitation ` a Diriger des Recherches Mines ParisTech, October 8 2019

Banding patterns in elasticity : cracked polycrystal Onset of a low percolation threshold in polycrystals with hexa- gonal symmetry, as crystal anisotropy increases. Bulk modulus 0.5 Interpreted as the develop- 0.4 ment of weakly-loaded re- 0.3 gions around cracks. FFT 0.2 Self-consistent 0.1 0 0 0.2 0.4 0.6 Crack density To which extent may homogenization theories be used to predict not only the self-consistent but the entire elastostatic probability distribu- tion field ? Refs : Barthelemy & Orland (1997), Cule & Tor- quato (1998), Idiart et al (2006), Giordano (2007)

Elastostatic probability field distributions Homogeneous cracked body under plane strain & biaxial stress loading ~ (t) [GPa -1 ] P yy crack density FFT 2.5 2 =0.035 (N/S)a 2 FFT parallel randomly-oriented ~ 1.5 P yy y _ 1 P (t) ij _ 6 0.5 Isolated crack P yy 5 0 4 0 0.5 1 1.5 σ yy [GPa] � + ∞ � t � d s q ( s ) 3 _ � P P ij ( t ) = | s | P ij xy 2 _ s � t n � q = � t n � � / � t n � P ij P −∞ � �� � xx 1 P ij � �� � � �� � Eshelby 0 -0.5 0 0.5 1 σ ij [GPa] inclusion FFT/self- Eshelby Eshelby inclusion p.d.f inclusion consistent Modeling of the stress intensity factor around each crack as a Gaussian probability distribution q . Van Hove singularities smoothed out, except at σ ij = 0 ± and ±∞ .

Elastostatic probability field distributions Van Hove singularities for the Eshelby inclusion problem computed making use of asymptotic expansions of the local stress fields in regions of interest. Allows one to derive estimates for the corresponding singularities in the p.d.f. for populations of interacting cracks � + ∞ � t � d s q ( s ) � P ij ( t ) = | s | P ij y s −∞ Eshelby parallel randomly-oriented σ xx H ( t ) − log t t = 0 ± = { [ a + bH ( t )] | t | − 1 / 3 H ( t ) | t | − 2 / 3 [ a + bH ( t )] | t | − 2 / 3 σ yy log 2 | t | σ xy | t | − 1 / 2 | t | − 1 / 2 H ( t ) | t | − 5 [ a + bH ( t )] | t | − 5 σ yy , xx t = ±∞ = { σ xy | t | − 5 H ( t ) : Heaviside function. Powerlaw decay with exponent − 5 for the distribution of the stress field.

Elastostatic probability field distributions Comparison with Fourier numerical results. Crack density η = Na 2 / S = 0 . 035 Probability distribution of σ yy 10 t → ±∞ : aligned -2/3 ~ σ yy -1/3 ~ σ yy 1 self. cons. & random Eshelby incl. -1 10 � �� � � ∞ -2 P yy ( t ) ∼ ωπη � d s s 4 q (sign( t ) s ) 10 -5 ~ σ yy | t | 5 0 -3 10 σ yy <0 � 1089 / 512 (parallel) -4 10 FFT ω = 585 / 512 (random) Reconstruction -5 10 -2 -1 1 | σ yy | [GPa] 10 10 FFT data consistent with the scaling law ∼ t − 5 when t → ±∞ with much lower prefactor when t → −∞ . Analysis restricted to a low crack density. Local loadings in arbitrary directions, making use of a multivariate distribution q , not considered here.

Nonlinear random media : context and motivation Emergence of special flow paths in model of nonlinear varistors (Roux et al, 1987 ; Donev et al, 2002). Shortest path and minimum cut problems, or minimal manifolds. Strain localization. Related problems in mechanics (e.g. damage induced by cracks). A simple example : network of nonlinear conductors J i (x) J 0 J 0 0 -J 0 -J 0 0 E i (x) � Equivalence between effective current thre- J ( i ) χ 0 = min 0 shold and minimal path in the dual lattice path p i ∈ p Problem : nonlinear conducting material in the continuum with randomly-distributed insulating (or highly-conducting) inclusions, in 2D. Anti-plane perfect plasticity. Refs : Drucker (1966), Roux & Fran¸ cois (1991), Roux & Hansen (1992), Ponte Casta˜ neda & Suquet (1997), Donev et al (2002), Duxbury et al (2006), Jeulin & Ostoja-Starzewski (2007), Sillamoni & Idiart (2016), Furer & Ponte Casta˜ neda (2018)

Boolean model of disks : shortest path Equisized disks of radius D , homogeneous Poisson point process of intensity θ . Join points ❆ to ❇ by a path passing through disk with centers ❈ i Corresponding upper-bound : �� � � N ℓ 2 i + m 2 i − D + Z i =1 ξ ≤ � N i =1 ℓ i Idea : choose disk ❈ i +1 in the region : � C i+1 | C i +1 − C i | C 1 − C i 1 | = inf 1 | ; ❈ a disk center ; 1 � � C i C 1 > C i 1 + D , | C 2 − C i D | C 1 − C i 2 | ≤ b 1 | . b to be optimized on

Geodesics in the Boolean model of disks C i+1 Upper bound obtained by : C i P { ℓ i > ℓ } = exp ( − θµ 2 ( K )) with K =domain enclosed by the two √ curves and the line ℓ = ℓ i . � Sharpest bound obtained with the choice b = 3 / 2 : � 3 f � 2 / 3 3 + O ( f 4 / 3 ) ≈ 1 − 1 . 3534 f 2 / 3 , � 2 � ξ ≤ 1 − f → 0 2 π Γ 3 Boolean model of arbitrary grain and “cost” 0 < p < 1 in the grains : � � 2 / 3 w 2 1 − (1 − p ) 4 / 3 3 5 / 3 g f ξ ≤ � 2 � 4Γ A g 3 ξ ≤ 1 − (1 − p ) f A g Consistent with numerical computations & “second-order” nonlinear homogenization theory w g in the anti-plane problem (up to numerical pre- factor)

Two-scale Cox-Boolean model Boolean model of disks aggregated into clusters f = f clus f in , f clus , f in ≪ 1 Dilute limit expansion : ξ ≤ 1 − α (1 − p ) 4 / 3 f 2 / 3 clus , α = 1 . 35 Numerical computations : ξ ≈ 1 − α (1 − p ) 4 / 3 f 2 / 3 clus , α = 1 . 85 Scale separation : p ≈ 1 − α f 2 / 3 in Dilute expansion for the two-scale model : � � α 7 / 3 f 8 / 9 f 2 / 3 clus , α f 2 / 3 ξ ≈ 1 − max f clus in in

Two-scale Cox-Boolean model Bounds prediction in the dilute limit 1- ξ in = f f -3 clus 10 ~f 7/9 f in = cste -4 10 ~f 2/3 f clus = cste -5 10 ~f 8/9 -6 10 ~f 2/3 -7 10 -10 -9 -8 -7 -6 -5 f 10 10 10 10 10 10 Regime change when f clus = cste and f in varies. Related to the presence and absence of rugosity at the macroscopic scale.

Two-scale Cox-Boolean model Comparison with numerical results in the case f clus = cste . Varying values of f clus . 1- ξ ~f 8/9 f = 0.02 clus -3 10 f = 0.01 clus f = 0.003 -4 clus 10 f = 0.001 clus -5 10 ~f 4/5 -6 10 ~f 2/3 -7 10 -10 -9 -8 -7 -6 -5 f 10 10 10 10 10 10 Lowest effect of the voids in two-scale media observed at the regime change ( f 2 / 3 ∼ f clus ) in which case 1 − ξ ∼ f 4 / 5 . in

N -scale Cox-Boolean model Pores embedded in clusters of pores embedded in super clusters, etc. Total porosity : f = f 1 ... f N . Let f i = f β i ( � i β i = 1). ν N = 2 β N ν i = ν i +1 +2 3 β i +1 1 − ξ ∼ f ν 1 , 3 , 3 min { β i ; ν i +1 } , 1 ≤ i < N . Properties : 2 / 3 ≤ ν 1 ≤ (1 + 2 − N ) − 1 → 1 ( N → ∞ ). • Lowest exponent (i.e. highest effect of the pores) obtained when β 1 = 1 (clusters fraction decrease very slowly except at the highest scale) or β N = 1 (clusters fraction decrease very slowly except at the lowest scale). • Highest exponent (i.e. lowest effect of the pore) obtained when β i = ν i +1 . However, as N → ∞ “almost” any choice of β 1 leads to ν 1 ≈ 1, i.e. a linear correction .

′ Model of rigid grains m i+1 Rigid grains (minimal paths avoid grains) Boolean model ξ ≤ 1 − (log f ) f 3 , f → 0 . ′ Random sequential ad- m sorption model of squares. i+1 ξ ≈ 1 + f 2 / 32 , f → 0 (non-rigorous analysis) ξ ≤ 1 − 3 8 (log f ) f 2 , f → 0 . 4 ξ ≈ 1 + ( w ⊥ g ) g f 2 , f → 0 w ⊥ A g 32 A 2 g (grains with moderate aspect ratio)

RSA model of squares n 4/3 ( ξ−1) _ ξ−1=(5/ n) √ f Numerical data collapse ( n the number ξ−1=(1/32) f 2 3 10 of squares in the numerical simulation) f=0.1 f=0.03 Consistent with the scaling law 2 f=0.01 10 ξ ≈ 1 + (1 / 32) f 2 , f → 0 1 10 rigid squares _ 1 1 1 n 1/3 √ f 10 For squares with cost p > 1 � � (1 / 32) f 2 , ( p − 1) f ξ ≈ 1 + min i , � For a N -scale model of rigid squares ( f = f 1 ... f N , f i = f β i β i = 1) ξ − 1 ∼ f ν , ν = β 1 +max( β 1 , β 2 +max( β 2 , β 3 + ... +max( β N − 1 , 2 β N ) ... ) with 1 ≤ ν ≤ 2. Minimal value of ν (maximal effect of the inclusions) obtained when β i = 2 β i +1 . As N → ∞ , ν → 1 for almost all choices of β i . Consistent with the bound Y 0 / Y ≤ 1 + (7 / 2) f (Goldsztein, 2011)

Stokes flow in porous media Boolean model of oblate cylinders with high aspect ratio, and high volume fraction Pores/ cylinders Obstacles (vol. frac. q ) q = 14 % q = 2 . 7 % Stokes flow (viscosity µ , pressure p , velocity field � u ) u � = − k u = � µ � � µ ∆ � ∇ p , � � ∇ p �