Multiscale analysis of dislocations Adriana Garroni Sapienza, Universit` a di Roma ”Mathematical challenges motivated by multi-phase materials: Analytic, stochastic and discrete aspects” Anogia, Crete June 22 - 26, 2009 Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 1 /21
Elastic vs Plastic deformations Single crystal Elastic deformation (reversible) Permanent deformation Elasto-plastic deformation The plastic deformation is due to slips on slip planes Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 2 /21
Elastic vs Plastic deformations Single crystal Elastic deformation (reversible) Permanent deformation Elasto-plastic deformation The plastic deformation is due to slips on slip planes In terms of the displacement u we can write Du = ∇ u L 3 + ([ u ] ⊗ n ) d H 2 Σ Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 2 /21
DISLOCATIONS NOTE: The slip in general is not uniform ⇐ ⇒ DEFECTS (dislocations) Dislocations are line defects in crystals (topological defects) At the microscopic level: Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
DISLOCATIONS NOTE: The slip in general is not uniform ⇐ ⇒ DEFECTS (dislocations) Dislocations are line defects in crystals (topological defects) At the microscopic level: Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
DISLOCATIONS NOTE: The slip in general is not uniform ⇐ ⇒ DEFECTS (dislocations) Dislocations are line defects in crystals (topological defects) At the microscopic level: Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
DISLOCATIONS NOTE: The slip in general is not uniform ⇐ ⇒ DEFECTS (dislocations) Dislocations are line defects in crystals (topological defects) At the microscopic level: Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
DISLOCATIONS NOTE: The slip in general is not uniform ⇐ ⇒ DEFECTS (dislocations) Dislocations are line defects in crystals (topological defects) At the microscopic level: Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
DISLOCATIONS NOTE: The slip in general is not uniform ⇐ ⇒ DEFECTS (dislocations) Dislocations are line defects in crystals (topological defects) At the microscopic level: Burgers vector Burgers circuit Dislocation core Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
TOPOLOGICAL SINGULARITIES OF THE STRAIN We can identify dislocations using the decomposition of the deformation gradient Du = ∇ u L 3 + ([ u ] ⊗ n ) d H 2 Σ = β e + β p - where [ u ] is the jump of the displacement along the slip plane Σ - ∇ u is the absolutely continuous part of the gradient In presence of dislocations Curl ∇ u = − ( ∇ τ [ u ] ∧ n ) d H 2 Σ = µ � = 0 µ is the dislocations density Then dislocations can be understood as Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 4 /21
TOPOLOGICAL SINGULARITIES OF THE STRAIN We can identify dislocations using the decomposition of the deformation gradient Du = ∇ u L 3 + ([ u ] ⊗ n ) d H 2 Σ = β e + β p - where [ u ] is the jump of the displacement along the slip plane Σ - ∇ u is the absolutely continuous part of the gradient In presence of dislocations Curl ∇ u = − ( ∇ τ [ u ] ∧ n ) d H 2 Σ = µ � = 0 µ is the dislocations density Then dislocations can be understood as ◮ singularities of the Curl of the elastic strain Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 4 /21
TOPOLOGICAL SINGULARITIES OF THE STRAIN We can identify dislocations using the decomposition of the deformation gradient Du = ∇ u L 3 + ([ u ] ⊗ n ) d H 2 Σ = β e + β p - where [ u ] is the jump of the displacement along the slip plane Σ - ∇ u is the absolutely continuous part of the gradient In presence of dislocations Curl ∇ u = − ( ∇ τ [ u ] ∧ n ) d H 2 Σ = µ � = 0 µ is the dislocations density Then dislocations can be understood as ◮ singularities of the Curl of the elastic strain ◮ regions where the slip is not uniform Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 4 /21
Why dislocations are important ⇒ Plastic behaviour Dislocations in crystals favor the slip = (Caterpillar, Lloyd, Molina-Aldareguia 2003) (Crease on a carpet, Cacace 2004) Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 5 /21
DIFFERENT SCALES ARE RELEVANT Microscopic - Atomistic description Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 6 /21
DIFFERENT SCALES ARE RELEVANT Microscopic - Atomistic description Mesoscopic - Lines carrying an energy - Interaction, LEDS, Motion... Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 6 /21
DIFFERENT SCALES ARE RELEVANT Microscopic - Atomistic description Mesoscopic - Lines carrying an energy - Interaction, LEDS, Motion... Macroscopic - Plastic effect - Dislocation density, Strain gradient theories... Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 6 /21
DIFFERENT SCALES ARE RELEVANT Microscopic - Atomistic description Mesoscopic - Lines carrying an energy - Interaction, LEDS, Motion... Macroscopic - Plastic effect - Dislocation density, Strain gradient theories... Objective: 3D DISCRETE − → CONTINUUM POSSIBLE DISCRETE MODELS Ariza - Ortiz, ARMA 2005 Luckhaus - Mugnai, preprint. Collaborations: S. Cacace, P. Cermelli, S. Conti, M. Focardi, C. Larsen, G. Leoni, S. M¨ uller, M. Ortiz, M. Ponsiglione. Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 6 /21
We have an almost complete analysis under different scales (mesoscopic and macroscopic) for special geometries. MESOSCOPIC ◮ Cilindrical geometry (dislocations are points) ◮ Screw dislocations - Burgers vector parallel to the dislocation line - (Ponsiglione, ’06) ◮ Edge dislocations - Burgers vector orthogonal to the dislocation line - (Cermelli and Leoni ’05) ◮ Only one slip plane (dislocations are lines on a given slip plane) ◮ A phase field approach for a generalized Nabarro-Peierls model (the phase is the jump along the slip plane and the energy is a Cahn-Hilliard type energy with non-local singular perturbation and infinitely many wells potential) (G.- Muller ’06, Cacace-G ’09, Conti-G.-Muller preprint) All the results above are based on the analysis of a ”semi-discrete” model. El Hajj, Ibrahim and Monneau for the 1D multiscale analysis for the dynamics. Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 7 /21
THE DISCRETE MODEL (Ariza-Ortiz, ARMA 2005) For simplicity we consider the cubic lattice. 3 1 E ( u , β p ) = X X 2 B ij ( l − l ′ )( du i ( l ) − β pi ( l ))( du j ( l ′ ) − β pj ( l ′ )) i , j =1 l , l ′ ∈ lattice bonds - u = displacements of the atoms; - du ( l ) = discrete gradient along the bond l ; - β p = eigen-deformation induced by dislocations (defined on bonds). β p = b ⊗ m where b ∈ Z 3 (Burgers vectors) and m ∈ Z 3 (normal to the slip plane) Four-point interaction energy with interaction coefficients B ij ( l − l ′ ) with finite range. Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 8 /21
PARTICULAR CASE: Anti-plane problem (screw dislocations) Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 9 /21
PARTICULAR CASE: Anti-plane problem (screw dislocations) Scalar (vertical) displacement u : Z 2 ∩ Ω → R . Take a two-point interaction discrete energy E discr ( u , β p ) = X | u ( i ) − u ( j ) − β p ( < i , j > ) | 2 < i , j > Dislocations are introduced through the plastic strain β p : { bonds } → Z . Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 9 /21
PARTICULAR CASE: Anti-plane problem (screw dislocations) Scalar (vertical) displacement u : Z 2 ∩ Ω → R . Take a two-point interaction discrete energy E discr ( u , β p ) = X | u ( i ) − u ( j ) − β p ( < i , j > ) | 2 < i , j > Dislocations are introduced through the plastic strain β p : { bonds } → Z . Minimizing w.r.t. β p β p E discr ( u , β p ) = E discr ( u ) = X dist 2 ( u ( i ) − u ( j ) , Z ) min < i , j > Note: β p corresponds to the projection of du on integers. Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 9 /21
PARTICULAR CASE: Anti-plane problem (screw dislocations) Scalar (vertical) displacement u : Z 2 ∩ Ω → R . Take a two-point interaction discrete energy E discr ( u , β p ) = X | u ( i ) − u ( j ) − β p ( < i , j > ) | 2 < i , j > Dislocations are introduced through the plastic strain β p : { bonds } → Z . Minimizing w.r.t. β p β p E discr ( u , β p ) = E discr ( u ) = X dist 2 ( u ( i ) − u ( j ) , Z ) min < i , j > Note: β p corresponds to the projection of du on integers. Remark: β p in general is not a discrete gradient. We can define a discrete Curl of β p , denoted by d β p , and α = d β p is the discrete dislocation density Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 9 /21
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