Role of thermodynamics in modeling the behavior of complex solids

Transcription

1 IWNET Summer School 2015 Role of thermodynamics in modeling the behavior of complex solids Bob Svendsen Material Mechanics RWTH Aachen University Microstructure Physics and Alloy Design Max-Planck-Institut für Eisenforschung Thanks to Markus Hütter Role thermodynamics modeling solids July 11, / 62

2 outline introduction & preliminaries Eulerian fluids? Lagrangian solids? Continuum mechanics! "standard" non-equilibrium thermodynamics for solids GENERIC for solids Role thermodynamics modeling solids July 11, / 62

3 introduction & preliminaries outline introduction & preliminaries Eulerian fluids? Lagrangian solids? Continuum mechanics! "standard" non-equilibrium thermodynamics for solids GENERIC for solids Role thermodynamics modeling solids July 11, / 62

4 introduction & preliminaries a bit more detail comparison of Eulerian- & Lagrangian-based continuum mechanics configuration & fields derivatives (e.g., material time derivative) transport & balance relations elements of material theory for solids comparison of non-equilibrium thermodynamic approaches generalized Gibbs / entropy-based e.g. Meixner and Reik (1959), de Groot and Mazur (1962), Liu (1972), Müller (1985) General Equation for Non-Equilibrium Reversible-Irreversible Coupling GENERIC: e.g. Öttinger and Grmela (1997), Öttinger (2005), Grmela (2010) applications homogeneous thermoelastic solids (e.g., Hütter and Svendsen, 2011) homogeneous viscoplastic solids (e.g., Hütter and Svendsen, 2012) inhomogeneous thermoelastic solids (today) Role thermodynamics modeling solids July 11, / 62

5 introduction & preliminaries a little notation things Euclidean three-dimensional Euclidean space (E 3, V 3 ) first-order tensors a, b, c,... V 3 second-order tensors A, B, C,... Lin(V 3, V 3 ) fourth-order tensors A, B, C,... Lin(Lin(V 3, V 3 ), Lin(V 3, V 3 )) component form Cartesian basis vectors (i 1, i 2, i 3 ) = (i x, i y, i z ) arbitrary order A = 3 i,j,k,...=1 A ijk... i i i j i k scalar product arbitrary order A B = 3 i,j,k,...=1 A ijk... B ijk... Role thermodynamics modeling solids July 11, / 62

6 introduction & preliminaries a little notation partial derivative a f(a, b,...) f (a, b,...) f(a + ɛ, b,...) f(a, b,...) a := lim ɛ 0 b,... ɛ (total) time derivative f(t) := lim ɛ 0 f(t + ɛ) f(t) ɛ Role thermodynamics modeling solids July 11, / 62

7 outline Eulerian fluids? Lagrangian solids? Continuum mechanics! introduction & preliminaries Eulerian fluids? Lagrangian solids? Continuum mechanics! "standard" non-equilibrium thermodynamics for solids GENERIC for solids Role thermodynamics modeling solids July 11, / 62

8 Eulerian fluids? Lagrangian solids? Continuum mechanics! continuum mechanics & material behavior some basic aspects kinematics (body, placement, configuration, motion, velocity,...) balance relations (mass, momentum, energy,...) constitutive relations / material theory entropy / dissipation principle (second law) frame-indifference (Euclidean & material) material symmetry & constraints (solid, fluid,..., incompressible,...) material heterogeneity (chemical, structural,...)... environment / boundary conditions (variational) formulation of initial-boundary value problems & solution Role thermodynamics modeling solids July 11, / 62

9 Eulerian fluids? Lagrangian solids? Continuum mechanics! basics: body, placement, configuration, deformation kinematic space (classic spacetime): E 3 R material body: differentiable manifold M modeled on E 3 placement (chart) κ: M E 3 b x κ = κ(b) configuration (chart image) B κ = κ[m] E 3, x κ B κ deformation (of B κ into B γ ) χ γκ : B κ B γ x κ x γ = γ(κ 1 (x κ )) =: χ γκ (x κ ) Role thermodynamics modeling solids July 11, / 62

10 motion Eulerian fluids? Lagrangian solids? Continuum mechanics! motion: time-dependent (t R) sequence of deformations χ: R B r E 3 (t, x r ) x c = χ t (x r ) = χ(t, x r c x r 2 B r x c 2 B c special configurations reference (e.g., undeformed, initial) B r E 3 current (deformed, time t) B c = χ t [B r ] E 3 Role thermodynamics modeling solids July 11, / 62

11 Eulerian fluids? Lagrangian solids? Continuum mechanics! configurations & fields two configurations B r and B c = two representations of (time-dependent) physical fields f(t, x) referential / material / Lagrangian: field on B r f r (t, x r ) current / spatial / Eulerian: field on B c f c (t, x c ) same physical quantity (e.g., temperature) f r (t, x r ) = f c (t, x c ) = f c (t, χ(t, x r )) abstract form (partial composition notation) f r = f c χ Role thermodynamics modeling solids July 11, / 62

12 Eulerian fluids? Lagrangian solids? Continuum mechanics! derivatives of fields partial derivatives (notation) t f r f r t, t f c f c xr t, xc f r f r x, r t f c f c x c t material time derivative: with respect to / at fixed B r Lagrangian / referential field f r t f r Eulerian / spatial field f c := f c χ χ 1 spatial velocity field v := χ χ 1 = t χ χ 1 = t f c + f c χ χ 1 = t f c + f c v Role thermodynamics modeling solids July 11, / 62

13 Eulerian fluids? Lagrangian solids? Continuum mechanics! deformation & velocity gradients deformation gradient ("mixed" or "two-point" tensor field on B r ) F := χ velocity gradient ("Eulerian" tensor field on B c ) L := v or L := v χ connection (chain rule) F = χ = χ = (v χ) = ( v χ) χ = LF Role thermodynamics modeling solids July 11, / 62

14 Eulerian fluids? Lagrangian solids? Continuum mechanics! Eulerian & Lagrangian densities, fluxes area da = dx dy & volume dv = dx dy dz elements χ da = F dx F dy = (cof F ) da = (det F ) F T da χ dv = F dx F dy F dz = (det F ) dv relation between volume densities f r dv = B r f c dv = B c χ (f c dv) B r = f r dv = χ (f c dv) = f r = (det F ) f c χ relation between surface densities (fluxes) f r da = B r f c da = B c χ (f c da) B r = f r da = χ (f c da) = f r = (cof F ) T f c χ Role thermodynamics modeling solids July 11, / 62

15 Eulerian fluids? Lagrangian solids? Continuum mechanics! transport relations extensive quantity with density f f r dv = f c dv B r B c Lagrangian rate of change ( B r fixed) B r f r dv = t f r dv = B r B r f r dv Eulerian rate of change ( B c = χ t [ B r ] moving) f c dv = t f c dv + f c v n da B c B c B c Role thermodynamics modeling solids July 11, / 62

16 Eulerian fluids? Lagrangian solids? Continuum mechanics! Eulerian transport divergence theorem f c v n da = div f c v dv B c B c divergence form f c dv = ( t f c + div f c v) dv = ( f c + f c div v) dv B c B c B c pull-back form f c dv = B c B r (f c χ) det F dv = ( f c + f c div v) dv B c Role thermodynamics modeling solids July 11, / 62

17 Eulerian fluids? Lagrangian solids? Continuum mechanics! balance relation for extensive quantity with density f B f dv = p dv B }{{} production rate + f n da B } {{ } flux rate + s dv B }{{} supply rate differential forms in terms of material time derivative f r = p r + div f r + s r f c + f c div v = p c + div f c + s c in terms of partial time derivatives t f r t f c = p r + div f r + s r = p c + div(f c f c v) + s c Role thermodynamics modeling solids July 11, / 62

18 Eulerian fluids? Lagrangian solids? Continuum mechanics! specfic balance relations mass (no supply) ϱ r = m r div j r, ϱ c + ϱ c div v = m c div j c linear momentum (no production) ṁ r = div S r + b r, ṁ c + m c div v = div S c + b c total energy (no production) ė r = div h r + s r, ė c + e c div v = div h c + s c entropy η r = π r div φ r + σ r, η c + η c div v = π c div φ c + σ c From now on: only "Lagrangian" ("r" subscript will be dropped) Role thermodynamics modeling solids July 11, / 62

19 Eulerian fluids? Lagrangian solids? Continuum mechanics! a little material / constitutive theory balance relations contain constitutive quantities: j, m, S, h, s, φ,... momentum flux: first Piola-Kirchhoff stress, Cauchy stress P S r, T S c example: constitutive model for stress, free energy density ψ(f ) P (F ) = F ψ(f ) material frame-indifference (observer independence) ψ(f ) = ψ(qf ) Q Orth(V 3, V 3 ) material symmetry (e.g., anisotropy) ψ(f ) = ψ(f Q) Q G Orth(V 3, V 3 ) Role thermodynamics modeling solids July 11, / 62

20 "standard" non-equilibrium thermodynamics for solids outline introduction & preliminaries Eulerian fluids? Lagrangian solids? Continuum mechanics! "standard" non-equilibrium thermodynamics for solids GENERIC for solids Role thermodynamics modeling solids July 11, / 62

21 "standard" non-equilibrium thermodynamics for solids setting kinematics & balance relations material class: materially inhomogeneous solids selected applications of phase field modeling precipitate formation (phase separation) in multicomponent solids recrystallization in polycrystalline systems austenite martensite transformation in metallic systems dislocation dissociation / core-solute interaction in metallic nanocrystals polarization domain microstructure formation in ferroelectrics magnetization domain microstrcture formation in ferromagnetics... today: simple ideal case two-component, two-phase solid mixture for simplicity: thermoelastic phases with heat and mass transfer for simplicity: no mass production, no supplies Role thermodynamics modeling solids July 11, / 62

22 "standard" non-equilibrium thermodynamics for solids mass balance for two-component mixture component mass balance (no production) ϱ 1 = div j 1, ϱ 2 = div j 2 mixture mass density ρ = ϱ 1 + ϱ 2 for simplicity: assume mixture mass density constant ρ = 0 = ϱ 2 = ϱ 1 one independent component / mass balance relation ϱ ϱ 1, j j 1 = ϱ = div j Role thermodynamics modeling solids July 11, / 62

23 "standard" non-equilibrium thermodynamics for solids linear momentum & energy balance linear momentum, total energy balance for mixture (no supplies) ṁ = div P, ė = div h densities (linear momentum, total energy, kinetic energy) m = ρ χ, e = k + ε, k = ρ χ χ 2 = m m 2ρ total energy flux: mechanical + non-mechanical h = P T χ e mixture internal energy balance ε = P χ div e Role thermodynamics modeling solids July 11, / 62

24 "standard" non-equilibrium thermodynamics for solids entropy balance & generalized Gibbs mixture entropy balance η = π div φ (absolute) temperature, inverse temperature ("coldness") θ, ϑ = θ 1 generalized Gibbs relation η ϑ ε = π e ϑ ϑp χ div(φ ϑ e) = ˇψ + ε ϑ (negative) free entropy density, free energy density ˇψ := ϑ ε η = ϑ ψ, ψ := ε θ η Role thermodynamics modeling solids July 11, / 62

25 "standard" non-equilibrium thermodynamics for solids internal energy flux & entropy flux notation (energetic entropic) ˇf := ϑ f = θ 1 f for heat & mass transport entropy flux: purely thermal (Clausius-Duhem) e.g. Eckart (1940), Truesdell and Noll (1965), Gurtin and Vargas (1971), Šilhavý (1997) φ = ˇq, e = q + µ j (energetic) chemical potential µ internal energy flux: purely thermal e.g. Meixner and Reik (1959), de Groot and Mazur (1962), Müller (1968) e = q, φ = ˇq ˇµ j (entropic) chemical potential ˇµ = ϑ µ Role thermodynamics modeling solids July 11, / 62

26 "standard" non-equilibrium thermodynamics for solids entropy production rate & dissipation rate entropy production-rate density π = ˇP χ + ˇµ ϱ + ε ϑ ˇψ + q ϑ j ˇµ entropy principle: non-negative entropy production-rate (density) π 0 in terms of dissipation-rate density δ = θ π = P χ + µ ϱ η θ ψ φ θ j µ dissipation principle: non-negative dissipation-rate (density) θ 0, π 0 = δ = θ π 0 Role thermodynamics modeling solids July 11, / 62

27 "standard" non-equilibrium thermodynamics for solids a brief aside: Müller-Liu entropy principle entropy principle with balance relations as constraints e.g. Liu (1972), Müller (1985), Muschik et al. (2001) π = η + div φ λ ϱ ( ϱ + div j) λ m (ṁ div P ) λ e (ė div P T χ + div q) 0 Lagrange multipliers λ ϱ, λ m, λ e not today! Role thermodynamics modeling solids July 11, / 62

28 "standard" non-equilibrium thermodynamics for solids model class: isothermal phase field a (very) few landmarks van der Waals (1893): non-uniform density (see Rowlison, 1979) Landau (1937): order parameter PT (see Provatas and Elder, 2010) Cahn and Hilliard (1958): conservative (mass) Allen and Cahn (1979): non-conservative (structural) Khachaturyan (1983): defective solid mechanics (structural) Elder and Grant (2004): phase field crystal models for materially inhomogeneous free energy (density) weakly non-local (e.g., Cahn and Hilliard, 1958) ψ(..., x) ψ(..., ϱ(x), φ(x), ϱ(x), φ(x), ϱ(x), φ(x),...) more general: strongly non-local ψ(..., x) = w(x x ) ϕ(..., ϱ(x), φ(x), ϱ(x ), φ(x )) dv(x ) Role thermodynamics modeling solids July 11, / 62

29 "standard" non-equilibrium thermodynamics for solids isothermal formulation of Cahn and Hilliard (1958) isothermal, purely chemical case ψ(ϱ, ϱ, ϱ,...) expand ψ about the "homogeneous" state (ϱ, ϱ, ϱ,...) H = (ϱ, 0, 0,...) expansion up to second order ψ = ψ H + ϱ ψ H ϱ + ϱ ψ H ϱ ϱ ϱ ϱ ψ H ϱ + reduce order: partial integration, divergence theorem ϱ ψ H ϱ dv = ϱ ψ H n ϱ da div ϱ ψ H ϱ dv B B B Role thermodynamics modeling solids July 11, / 62

30 "standard" non-equilibrium thermodynamics for solids isothermal formulation of Cahn and Hilliard (1958) no-flux boundary conditions ϱ ψ H n B = 0 = ϱ ψ H ϱ div ϱ ψ H ϱ dependence of ϱ ψ H on ϱ div ϱ ψ H ϱ = ϱ ( ϱ ϱ ψ H ) ϱ reduced free energy model ψ(ϱ, ϱ) = ψ H (ϱ) + ϱ ψ H (ϱ) ϱ ϱ Λ H(ϱ) ϱ second-order coefficient Λ H = ϱ ψ H 2 ϱ ϱ ψ H Role thermodynamics modeling solids July 11, / 62

31 "standard" non-equilibrium thermodynamics for solids generalization: deformation & material inhomogeneity free energy density ψ( χ, ϱ, ϱ, φ, φ) elastic phases P = χ ψ inhomogeneous (local) dissipation rate density δ = (µ ϱ ψ) ϱ ϱ ψ ϱ φ ψ φ φ ψ φ j µ dissipation rate form δ dv = (µ δ ϱ ψ) ϱ dv (j µ + δ φ ψ φ) dv B B B ( ϱ ψ n ϱ + φ ψ n φ) da B Role thermodynamics modeling solids July 11, / 62

32 "standard" non-equilibrium thermodynamics for solids chemical potential & boundary conditions inhomogeneous chemical potential µ = δ ϱ ψ variational derivative δ x a := x a div x a boundary conditions ϱ ψ n ϱ B = 0, φ ψ n φ B = 0 "residual" dissipation-rate density δ = j µ φ δ φ ψ Role thermodynamics modeling solids July 11, / 62

33 "standard" non-equilibrium thermodynamics for solids kinetics thermodynamic fluxes & forces j := ( j, φ), f := ( µ, δ φ ψ) quasi-linear flux-force relation j = L(..., f)f dissipation principle δ = j f = f Lf 0 = D = sym L 0 (non-negative definite) simplest case: "diagonal" flux-force relations (no coupling) j = D(...) µ, φ = m(...) δ φ ψ dissipation-rate density δ = µ D µ + δ φ ψ m δ φ ψ 0 = D T = D, D 0, m 0 Role thermodynamics modeling solids July 11, / 62

34 "standard" non-equilibrium thermodynamics for solids the case of a dissipation potential potential-based flux-force relations j = µ d, φ = δφ ψ d dissipation potential d(..., µ, δ φ ψ) dissipation-rate density δ = µ µ d + δ φ ψ δφ ψ d d non-negative, convex in forces (sufficient for δ 0) µ µ d + δ φ ψ δφ ψ d d 0 treatment in the GENERIC framework: Hütter and Svendsen (2013) Role thermodynamics modeling solids July 11, / 62

35 "standard" non-equilibrium thermodynamics for solids summary of isothermal model relations Cahn-Hilliard mass balance µ = δ ϱ ψ = ϱ ψ div ϱ ψ, ϱ ψ n ϱ B = 0 ϱ = div j = div D µ, j B n = 0 reduced momentum balance ϱ χ = ṁ = div P = div χ ψ time-dependent Ginzburg-Landau, Allen-Cahn phase field φ = m δ φ ψ = m (div φ ψ φ ψ), φ ψ n φ B = 0 Role thermodynamics modeling solids July 11, / 62

36 "standard" non-equilibrium thermodynamics for solids model class: non-isothermal phase field free entropy density ˇψ(ϑ, χ, ϱ, ϱ, φ, φ) thermoelastic internal energy & entropy ε = ϑ ˇψ = η = ˇψ ϑ ϑ ˇψ thermoelastic stress, chemical potential P = θ χ ˇψ, µ = θ δϱ ˇψ boundary conditions ϱ ˇψ n ϱ B = 0, φ ˇψ n φ B = 0 "residual" entropy production rate density π = q ϑ j ˇµ φ δ φ ˇψ Role thermodynamics modeling solids July 11, / 62

37 "standard" non-equilibrium thermodynamics for solids non-isothermal kinetics for simplicity: "diagonal" flux-force relations q = θ 2 K(...) ϑ, j = θd(...) ˇµ, φ = θ m(...) δ φ ˇψ entropy production-rate density π = ϑ θ 2 K ϑ + ˇµ θd ˇµ + δ φ ˇψ θ m δφ ˇψ dissipation-rate density δ = ϑ θk ϑ + ˇµ D ˇµ + δ φ ˇψ m δφ ˇψ Role thermodynamics modeling solids July 11, / 62

38 "standard" non-equilibrium thermodynamics for solids summary of non-isothermal model relations Cahn-Hilliard mass balance ˇµ = δ ϱ ˇψ = ϱ ˇψ div ϱ ˇψ, ϱ ˇψ n ϱ B = 0 ϱ = div j = div θd ˇµ, j B n = 0 reduced momentum balance ϱ χ = ṁ = div P = div θ χ ˇψ time-dependent Ginzburg-Landau (TDGL), Allen-Cahn phase field φ = θ m δ φ ˇψ = m (div φ ˇψ φ ˇψ), ϑ φ ˇψ n φ B = 0 Role thermodynamics modeling solids July 11, / 62

39 "standard" non-equilibrium thermodynamics for solids temperature relation (sketch) from energy balance & free entropy ε = P χ div q, ε = ϑ ˇψ time derivative ε = ϑ ε ϑ + χ ε χ + ϱ ε ϱ + ϱ ε ϱ + φ ε φ + φ ε φ from entropy balance & free entropy η = π div φ, η = ˇψ ϑ ϑ ˇψ time derivative η = ϑ η ϑ + χ η χ + ϱ η ϱ + ϱ η ϱ + φ η φ + φ η φ Role thermodynamics modeling solids July 11, / 62

40 GENERIC for solids outline introduction & preliminaries Eulerian fluids? Lagrangian solids? Continuum mechanics! "standard" non-equilibrium thermodynamics for solids GENERIC for solids Role thermodynamics modeling solids July 11, / 62

41 GENERIC for solids GENERIC: brief review General Equation for Non-Equilibrium Reversible-Irreversible Coupling ẋ = L D x E + M D x S energy E, entropy S: functionals A[x] of x D x A functional derivative ("gradient") of A skew-symmetric Poisson operator L symmetric, non-negative definite friction operator M inner product, transpose of operator O D x B, O T D x A := D x A, O D x B skew-symmetry of L D x A, L T D x B = D x A, L D x B symmetry, non-negative definiteness of M D x A, M T D x B = D x A, M D x B, D x A, M D x A 0 Role thermodynamics modeling solids July 11, / 62

42 brackets & identities GENERIC for solids Poisson & dissipation brackets {A, B} := D x A, L D x B, [A, B] := D x A, M D x B bracket symmetry L T = L = {B, A} = {A, B} M T = M = [B, A] = [A, B] M NND = [A, A] 0 Jacobi identity {A, {B, C}} + {B, {C, A}} + {C, {A, B}} = 0 Role thermodynamics modeling solids July 11, / 62

43 GENERIC for solids functional evolution & orthogonality functional evolution A = D x A, ẋ = D x A, L D x E + D x A, M D x S = {A, E} + [A, S] orthogonality L D x S = 0, M D x E = 0 energy conservation Ė = {E, E} = 0 non-negative entropy production Ṡ = [S, S] 0 Role thermodynamics modeling solids July 11, / 62

44 GENERIC for solids functionals & gradient for phase field models volume density representation of functionals A[x] = a(x, x) dv B variation δa = ( x a δx + x a δ x) dv B = δ x a δx dv + ( x a)n δx da B B (bulk & boundary) functional derivative D x A B δ x a = x a div x a, D x A B ( x a) B n no-flux boundary conditions (energy, entropy) D x A B = ( x a) B n = 0 Role thermodynamics modeling solids July 11, / 62

45 GENERIC for solids choice of GENERIC variables GENERIC variables x = (χ, m, θ, ϱ, φ), m = ρ χ total energy and entropy E[x] = e( χ, m, θ, ϱ, ϱ, φ, φ) dv B S[x] = η( χ, θ, ϱ, ϱ, φ, φ) dv B total energy density e( χ, m, θ, ϱ, ϱ, φ, φ) = 1 m m + ε( χ, θ, ϱ, ϱ, φ, φ) 2ρ Role thermodynamics modeling solids July 11, / 62

46 GENERIC for solids gradients driving "forces" / gradients D χ E δ χ e D χ S δ χ η D m E δ m e D m S δ m η D x E = D θ E = δ θ e, D x S = D θ S = δ θ η D ϱ E δ ϱ e D ϱ S δ ϱ η D φ E δ φ e D φ S δ φ η particular energy gradient "components" δ χ e = δ χ ε, δ m e = m ρ = χ, δ θe = θ ε = c particular entropy gradient "components", temperature δ m η = 0, δ θ η = θ η, θ = θ ε θ η Role thermodynamics modeling solids July 11, / 62

47 GENERIC for solids reversible part GENERIC variable evolution momentum evolution reversible (stress thermoelastic) temperature evolution reversible & irreversible mass evolution irreversible (flux diffusive) order parameter evolution irreversible (relaxation) reversible-irreversible split x rev = (χ, m, θ rev, 0, 0), x irr = (0, 0, θ irr, ϱ, φ) reversible part ẋ rev = L D x E χ 0 L χm L χθ δ χ e L χm δ m e + L χθ δ θ e ṁ = L mχ 0 L mθ δ m e = L mχ δ χ e + L mθ δ θ e L θχ L θm 0 δ θ e L θχ δ χ e + L θm δ m e θ rev Role thermodynamics modeling solids July 11, / 62

48 GENERIC for solids consequences of orthogonality orthogonality L D x S = 0 L χm δ m η + L χθ δ θ η = 0 δ m η=0, δ θ η 0 = L χθ = 0 L mθ δ θ η + L mχ δ χ η = 0 = L mθ = L mχ δ χ η 1 θ η L θχ δ χ η + L θm δ m η = 0 δ m η=0, δ χ = η 0 L θχ = 0 reduced evolution relations (L χθ = 0, L θχ = 0, δ m e = χ, θ ε = θ θ η) χ = L χm δ m e + L χθ δ θ e = L χm χ ṁ = L mχ δ χ ε + L mθ θ ε = L mχ δ χ (ε θ η) θ rev = L θχ δ χ e + L θm δ m e = L θm χ Role thermodynamics modeling solids July 11, / 62

49 GENERIC for solids Poisson operator components operator component χ = L χm χ = L χm = I skew-symmetry D m A, L mχ D χ B = D χ B, L χm D m A = L mχ = I orthogonality, skew-symmetry L mθ = L mχ δ χ η 1 θ η = div χ η 1 θ η = D θ A, L θm D m B = D m B, L mθ D θ A = dv D m B div χ η D θa θ η B Role thermodynamics modeling solids July 11, / 62

50 GENERIC for solids Poisson operator components integration by parts dv D m B div χ η 1 B θ η D θa = da D θ A 1 θ η ( χη)n D m B B boundary condition ( χ η) n B = 0 D θ A, L θm D m B = dv D θ A 1 θ η χη D m B Poisson operator component L θm = 1 θ η χη B B dv D θ A 1 θ η χη D m B Role thermodynamics modeling solids July 11, / 62

51 GENERIC for solids summary: reversible part reversible part χ ṁ = θ rev = 0 I 0 I 0 div χ η 1 θ η 0 1 θ η χ η 0 m ρ div χ ψ 1 θ η χ η χ δ χ e δ m e δ θ e momentum balance, temperature evolution ṁ = δ χ ψ = div χ ψ = div P, θrev = L θm δ m e = 1 θ η χη χ Role thermodynamics modeling solids July 11, / 62

52 GENERIC for solids irreversible part GENERIC variable evolution temperature evolution reversible & irreversible mass evolution, order parameter irreversible (flux diffusive) heat conduction affects only θ irr mass diffusion, structural evolution affect θ irr, ϱ, φ reversible-irreversible split x rev = (χ, m, θ rev, 0, 0), x irr = (0, 0, θ irr, ϱ, φ) irreversible part ẋ irr = M D x S θ irr M θθ M θϱ M θφ δ θ η M θθ δ θ η + M θϱ δ ϱ η + M θφ δ φ η ϱ = M ϱθ M ϱϱ 0 δ ϱ η = M ϱθ δ θ η + M ϱϱ δ ϱ η φ M φθ 0 M φφ δ φ η M φθ δ θ η + M φφ δ φ η Role thermodynamics modeling solids July 11, / 62

53 GENERIC for solids consequences of orthogonality orthogonality M D x E = 0 M ϱθ δ θ ε + M ϱϱ δ ϱ ε = 0 = M ϱθ = M ϱϱ δ ϱ ε 1 θ ε M φθ δ θ ε + M φφ δ φ ε = 0 = M φθ = M φφ δ φ ε 1 θ ε reduced phase field relations ϱ = M ϱθ δ θ η + M ϱϱ δ ϱ η = M ϱϱ δ ϱ (η ϑ ε) = M ϱϱ δ ϱ ˇψ φ = M φθ δ θ η + M φφ δ φ η = M φφ δ φ (η ϑ ε) = M φφ δ φ ˇψ = same driving force as in "standard" formulation comparison with Ginzburg-Landau driving force θ δ x ˇψ δx ψ = θ 1 x ψ θ Role thermodynamics modeling solids July 11, / 62

54 GENERIC for solids mass diffusion & structural transition friction operator component for mass diffusion M ϱϱ = θd friction operator component for mobility M φφ = θ m reduced phase field relations ϱ = M ϱθ δ θ η + M ϱϱ δ ϱ η = div θd δ ϱ ˇψ φ = M φθ δ θ η + M φφ δ φ η = θ m δ φ ˇψ consistent with relations from standard NE thermodynamics Role thermodynamics modeling solids July 11, / 62

55 GENERIC for solids thermal friction operator split of M θθ (heat conduction affects only θ) M θθ = M Cθθ + M Pθθ, M Cθθ = 1 c θ2 K 1 c, c = θε conduction ( θ ε = θ θ η) M Cθθ δ θ η = 1 c div θ2 K θ η c M Cθθ δ θ ε = 1 c div θ2 K c c = 1 c = 0 div K θ orthogonality M D x E = 0 and M Cθθ δ θ ε = 0 M θθ δ θ ε + M θϱ δ ϱ ε + M θφ δ φ ε = 0 = M Pθθ = M θϱ δ ϱ ε 1 c M θφ δ φ ε 1 c Role thermodynamics modeling solids July 11, / 62

56 GENERIC for solids consequences of symmetry symmetry D θ A, M θϱ D ϱ B = D ϱ B, M ϱθ D θ A = D ϱ B div θd δ ϱ ε 1 c D θa dv integration by parts & boundary condition D ϱ B B = 0 D ϱ B div θd δ ϱ ε 1 c D θa dv = D θ A 1 c θd δ ϱε D ϱ B dv B B temperature / mass density coupling M θϱ = 1 c θd δ ϱε B likewise D θ A, M θφ D φ B = D φ B, M φθ D θ A yields θ-φ coupling M θφ = 1 c δ φε θ m Role thermodynamics modeling solids July 11, / 62

57 GENERIC for solids phase field part M Pθθ previous result from M D x E = 0 and M Cθθ δ θ ε = 0 M Pθθ = M θϱ δ ϱ ε 1 c M θφ δ φ ε 1 c θ-ϱ component M θϱ = 1 c θd δ ϱε = M θϱ δ ϱ ε 1 c = 1 c θd δ ϱε δ ϱ ε 1 c θ-φ component M θφ = 1 c δ φε θ m = M θφ δ φ ε 1 c = 1 c δ φε θ m δ φ ε 1 c result M Pθθ = 1 c θd δ ϱε δ ϱ ε 1 c + 1 c δ φε θ m δ φ ε 1 c Role thermodynamics modeling solids July 11, / 62

58 GENERIC for solids summary: irreversible part friction operator 1 c θ2 K 1 c + M Pθθ M = θd δ ϱ ε 1 c θ m δ φ ε 1 c 1 c θd δ ϱε 1 c δ φ ε θ m θd 0 0 θ m phase field part of M θθ M Pθθ = 1 c { δϱ ε θd δ ϱ ε + δ φ ε θ m δ φ ε } 1 c irreversible temperature evolution θ irr = 1 c div K θ + 1 c δ ϱε θd δ ϱ ˇψ + 1 c δ φ ε θ m δ φ ˇψ Role thermodynamics modeling solids July 11, / 62

59 summary & outlook GENERIC for solids comparison of Eulerian- & Lagrangian-based continuum mechanics comparison of non-equilibrium thermodynamic approaches "standard" non-equilibrium thermodynamics GENERIC applications / outlook homogeneous & inhomogeneous thermoelastic-viscoplastic solids strongly non-local GENERIC-based phase field (Hütter & S., in prep.) multiscale dislocation ensemble modeling (Koimann et al. 2015; in prog.) Role thermodynamics modeling solids July 11, / 62

60 GENERIC for solids references I Allen, S. M., Cahn, J. W., A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, Volume 27, pp , Cahn, J. W., Hilliard, J. E., Free energy of a non-uniform system. I. Interfacial energy., Journal of Chemical Physics, Volume 28, pp , de Groot, S., Mazur, P., Non-Equlibrium Thermodynamics, North Holland, Eckart, C., Thermodynamics of irreversible processes, II. Fluid mixtures, Physical Review, Volume 58, pp , Elder, K. R., Grant, M., Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Physical Review E, Volume 70, p , Grmela, M., Why GENERIC?, Journal of Non-Newtonian Fluid Mechanics, Volume 165, pp , Gurtin, M. E., Vargas, A. S., On the classical theory of reacting fluid mixtures, Archive for Rational Mechanics and Analysis, Volume 43, pp , Hütter, M., Svendsen, B., On the formulation of continuum thermodynamic models for solids as general equations for non equilibrium reversible irreversible coupling, Journal of Elasticity, Volume 104, pp , Role thermodynamics modeling solids July 11, / 62

61 GENERIC for solids references II Hütter, M., Svendsen, B., Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible-irreversible coupling, Continuum Mechanics and Thermodynamics, Volume 24, pp , Hütter, M., Svendsen, B., Quasi-linear versus potential-based formulations of force-flux relations and the GENERIC for irreversible processes: comparisons and examples, Continuum Mechanics and Thermodynamics, Volume 25, pp , Khachaturyan, A. G., Theory of Structural Transformations in Solids, Wiley, New York, Liu, I.-S., Method of Lagrange multipliers for exploitation of the entropy principle, Archive for Rational Mechanics and Analysis, Volume 46, pp , Meixner, J., Reik, H. G., Thermodynamik der irreversiblen Prozesse, in Flügge, S. (editor), Handbuch der Physik, Vol. III/2, Springer, Müller, I., A thermodynamic theory of fluid mixtures, Archive for Rational Mechanics and Analysis, Volume 28, pp. 1 39, Müller, I., Thermodynamics, Pitman, Muschik, W., Papenfuss, C., Ehrentraut, H., A sketch of continuum thermodynamics, Journal of Non-Newtonian Fluid Mechanics, Volume 96, pp , Öttinger, H. C., Beyond Equilibrium Thermodynamics, Wiley Interscience, Role thermodynamics modeling solids July 11, / 62

62 GENERIC for solids references III Öttinger, H. C., Grmela, M., Dynamics and thermodynamics of complex fluids, II. Illustrations of a general formalism, Physical Review E, Volume 56, pp , Provatas, N., Elder, K., Phase Field Methods in Material Science and Engineering, Wiley-VCH, Rowlison, J. S., Translation of J. D. van der Waals "The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density" (1893), Journal of Statistical Physics, Volume 20, pp , Truesdell, C., Noll, W., The Non-Linear Field Theories of Mechanics, first edition, Springer, Šilhavý, M., The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, Role thermodynamics modeling solids July 11, / 62