Heterogeneous Elasto-plasticity

Heterogeneous Elasto-plasticity πλάσσειν G. F. & Alessandro Giacomini

Small strain elastoplasticity

Small strain elasto-plasticity the rheology A model with brake and spring: ε p ε σ with σ σ c ε p 0 σ = σ c ε p =0 σ <σ c ε p 0 σ = σ c Response: σ σ c ε p ε ε

Small strain elasto-plasticity the formulation Du + Dut Eu := = e + p 2 p M N N dev σ = Ae; div σ =0 inω A : Hooke s law M N N dev := {τ symmetric : tr τ =0} τ = tr τ N i + τ D ε p p σ K := {τ :f (τ D ) 0} with K closed convex (f conv., f (0) < 0, f τ ) set of admissible stresses

Small strain elasto-plasticity the formulation M N N dev := {τ symmetric : tr τ =0} τ = tr τ N i + τ D Du + Dut Eu := = e + p 2 p M N N dev σ = Ae; div σ =0 inω A : Hooke s law Flow rule: ṗ(t) Aσ := ε p p σ K := {τ :f (τ D ) 0} with K closed convex (f conv., f (0) < 0, f τ ) set of admissible stresses (σ(t)) and λf (σ(t)) = 0 τ M N N dev : λ 0s.t.τ = λ f τ ṗ(t) N K (σ(t)), the normal cone to K at σ(t) K(t) b.c. : u(x, t) =w(x, t) AC ([0, T ]; H 1 2 ( d Ω; R 3 )) d Ω Dirichlet bdary: open / t Ω:= Ω \ d Ω : open, no forces

A remark about stress admissibility Lipschitz domain Ω From div σ =0 + σ L 2 (Ω; M N N sym K), we get: (σ D ν) τ (the tangential part of σν) (Kν) τ aprioriwelldefinedasanelementofh 1 2 00 ( dω; R N ) Since ṗ =[ẇ(t) u(t)] ν on d Ω, with [ẇ(t) u(t)] ν, we expect a boundary flow rule: [ẇ(t) u(t)] N (Kν)τ ((σ D ν) τ ) on d Ω in fact well defined as an element of L ( d Ω; R N ) maybe not unique unless bdary is C 2

The variational approach to elastoplasticity

Variational evolution in a nutshell Define: diss. pot. : H(p) :=sup{σ D p : σ K} q dissipation: H(q) := H Ω d Ω q (x) d q total diss.: D(0, t; p) :=sup part. of [0,t] i H(p(t i+1) p(t i )) total energy: E(t) :=1/2 Ω Ae(t) e(t)dx + D(0, t; p) At each time t, (u(t), e(t), σ(t) := Ae(t), p(t)) satisfies Global min.: 1/2 Ω Ae(t) e(t)dx 1/2 Ω Aη ηdx + H(q p(t)) (ve) de Energy cons.: dt (t) = σ(t) Eẇ(t)dx Proof through time discretisation: Find (u i, e i, p i )kin. compatible solving min 1/2 Ae edx + H(p p i 1 ) Ω Ω

Variational evolution & classical formulation 1 Global minimality H(q) Ω Ae η dx H( q) (v,η,q) kin. compat. with b.c. w =0 equilibrium + Neumann b.c.+ σ K

Variational evolution & classical formulation 1 Global minimality H(q) Ω Ae η dx H( q) (v,η,q) kin. compat. with b.c. w =0 equilibrium + Neumann b.c.+ σ K To go further, need to define the duality σ D, p. Not so clear because σ D not continuous!

Isues of duality Here σ and p are arbitrary provided that σ satisfies eqm. + Neumann b.c.+stress adm. & p is assd. to (u, e, p) kin.compatiblewithw as b.c. First define σ D, p as a distribution: σ D, p(ϕ) = ϕσ (e Ew) dx Ω Ω σ [(u w) ϕ] dx OK since σ L N Known result: If Ω isc 2 and Ω [ d Ω] is a C 2 (N 2)-hypersurface, then σ D, p is a finite Radon meas. on R N (Kohn-Temam 1983) Thm: ΩLipschitz.Thenσ D, p is a finite Radon meas. on R N \ Ω [ d Ω] and σ D, p σ D L p, σ D, p a = σ D p a

Variational evolution & classical formulation 2 Just using the definition of the duality: σ D, p d Ω =(σ D ν) τ (w u)h N 1 d Ω p (Ineq) H p σ D, p as measures p Here again σ and p are arbitrary provided that σ satisfies eqm. + Neumann b.c. + stress adm. & p is assd. to (u, e, p) kin.compatiblewithw as b.c. From energy equality + Reshetnyak s lower semi-continuity thm.: H(ṗ) Ḋ(0, t, p) = Ω σ(t) (ė Eẇ)(t) dx = σ D, ṗ(ω d Ω) l.s.c. en. eq. duality Hill s maximal plastic work principle σ D, ṗ(ω d Ω) = sup τd adm.τ D, ṗ(ω d Ω)

Heterogeneous elastoplasticity

A multiphase domain No ordering property of the K i s We will need C 1 interfaces

A multiphase domain No ordering property of the K i s We will need C 1 interfaces Define the dissipation : H(x, p) :=H i (p) =sup{σ D p : σ D K i } in each phase i. Since we expect p to be a measure, how do we define H on Ω i Ω j?

A multiphase domain No ordering property of the K i s We will need C 1 interfaces ////////////// inf {H i, H J } because destroys convexity Inf-convolution: H(x,ξ):= inf{h(a ν(x)) + H( b ν(x)); a b = c}, if ξ = c ν(x), else destroys l.s.c./ Need to re-establish l.s.c. of H: Thm: If (u n, e n, p n ) kin. compatible and the natural weak conv. hold (BD L 2 M b )thenh(p) lim inf n H(p n ) why C 1 -interfaces are necessary

Heterogeneous evolution Existence of a variational evolution We recover all results of homogeneous case + interfacial conditions:

Heterogeneous evolution Existence of a variational evolution We recover all results of homogeneous case + interfacial conditions: Stress adm.: (σ D ν) τ (K i ν) τ (K j ν) τ

Heterogeneous evolution Existence of a variational evolution We recover all results of homogeneous case + interfacial conditions: Stress adm.: (σ D ν) τ (K i ν) τ (K j ν) τ Flow rule: u i u j N (Ki ν) τ (K j ν) τ ((σ D ν) τ ) Exact expression of H on interfaces is used in deriving (Ineq) For now unable to find concrete example where the interfacial flow rule makes a difference!

Homogenization Rescaled heterogeneous variational evolution: x replaced by x/ε for multiphase torus Y with C 1 interfaces. Homogenization: with approp. i.c. s, ε n s.t., for all t [0, T ], u n (t) u(t) weakly* in BD(Ω ) e n (t) w-2 E(t) two-scale weakly in L 2 (Ω Y; M N N sym ) p n (t) w -2 P(t) two-scale weakly* in M b (Ω Y; M N D ). Here, E(x, y) L N x L N y + P Eu L N y = E y µ in Ω Y with µ M b (Ω Y; R N )), E y µ M b (Ω Y; M N N sym ),µ(f Y)= 0, F Borel Ω. Further (u(t), E(t), P(t)) is a two-scale quasistatic evolution: defined as before with explicit y-dependence; for example: dissipation H hom (Q) := Y Ω d Ω H y, Q (x, y) Q d Q