thermo-viscoelasticity using

Size: px

Start display at page:

Download "thermo-viscoelasticity using"

Gwenda Black
6 years ago
Views:

1 thermo-viscoelasticity Chair of Computational Mechanics University of Siegen Chair of Applied Mechanics and Dynamics Chemnitz University of Technology ECCOMAS,

2 Thermoviscoelastic continuum framework enhanced TC algorithm projection of the test function s

3 Structure preserving time integrators great interest in the last two decades Energy balance longtime stability enhanced computational robustness exact reproduction of the physical structure format Matrix-Vector structure demand the structural properties of a thermodynamically system TC algorithm reflects the physical structure after the discretization in time 3

4 Kinematics The nonlinear mapping φ(x, t) describes the position of the reference configuration X in B after time t: : F = ϕ X = x X measure: C = F T F x = ϕ(x, t) X entropy: s B internal Variable: C i C e = C C 1 i F i B z ϕ F X B t F e x (isotropy) 4

5 5 Physical structure state vector: total energy: ] z = [ϕ, p, s, [C i ] vn H = T + E kinetic & total internal energy: 1 T = p p E = e (C, s,c i ) ρ B total entropy: B Lyapunov-function: S = B s V = H θ S

6 Evolution equations I equations of motion: ϕ = 1 ρ p ṗ = DIVP first & second Piola-Kirchhoff stress: P = F S S = e C thermal evolution equation: ṡ = 1 (DIV Q D int) θ Piola-Kirchhoff flux: with: θ = e s Q = K Grad θ K = κ J C 1 J = detc 6

7 7 Evolution equations II viscous evolution equation: compliance tensor: with: Ċ i = C i V 1 Σ vis V 1 = 1 I devt 1 + I vol V dev V vol n dim I devt = I T I vol I vol = 1 n dim I I stresses: Σ vis = C i Γ internal dissipation: Γ = ψvis C i D int = Σ vis : V 1 : Σ vis I T = I I

8 framework H.C. Öttinger (5) initial value problem: state vector: ż = (L + L c i ) δzh + (M + M c i ) δzs z(t ) = z ] z = [ϕ, p, s, [C i ] vn degeneracy conditions: δzs [L + L c i ] δzh = δzh M δzs = B B enhanced condition: δzh M c i δzs = δzh L c i δzh B B consequences for isolated systems: Ḣ = Ṡ V 8

9 Functional derivatives and matrices functional derivatives: DIVP 1 δzh = p ρ θ [Γ ] vn δzs = 1 matrices: 3,3 I 3,3 3,7 L = I 3,3 3,3 3,7 L ci = 7,7 7,6 6,7 [C 7,3 7,3 7,7 i ] vn 6,6 6,1 6,6 6,6 6,1 6,6 M = 1,6 1 DIV Q 1,6 θ Mci = 1,6 1 θ Dint 1,6 6,6 6,1 6,6 6,6 6,1 6,6 9

10 Weak evolution equations weak equations of motion: 1 wϕ q = wϕ p ρ B B wp ṗ = wp DIVP B B weak thermal evolution equation: w s ṡ = B B w s 1 θ [DIV Q D int] weak viscous evolution equation: w Ci : C i = w Ci : C i V 1 : Σ vis B B 1

11 in time I initial value problem: z n+1 z n wz = wz (L + L ci ) G H + (M + M ci ) G S h n B B vector of invariants: ] π = [[C] vn, p, s, [C i ] vn discrete conditions: G S [L + L c i ] G H = G H M G S = B B G H M c i G S = G H L c i G H B B consequences the enhanced TC integrator: H(z n+1 ) H(z n ) = S(z n+1 ) S(z n ) V (z n+1 ) V (z n ) 11

12 in time II G-equivariant functional derivatives: ( ) DIV F n+ 1 S 1 G D H = p T p (3,1) n+ 1 θ 1 [ ], (3,1) G S = D s s (6,1) Γ 1 vn with: S 1 θ 1 Γ 1 = D C e = D s e = D Ci e semidiscrete matrices: 3,3 I 3,3 3,7 L = I 3,3 3,3 3,7, L vis = 7,7 7,6 )] 7,3 7,3 7,7 6,7 [C i (C in+ 1 vn 6,6 6,1 6,6 M = 1,6 1 6,6 6,1 6,6 DIV Q 1 1,6 θ 1, Mvis = 1,6 1 D int 1 1,6 θ 1 6,6 6,1 6,6 6,6 6,1 6,6 1

13 Partitioned discrete gradients O. Gonzalez (1996) D p T = 1 D s s = 1 ρ D C e = 1 [ ( ) C e C n+ 1, s n+1, C in+1 + H sn+1,c in+1 C ( ) ] + C e C n+ 1, s n, C in + H sn,c in C D s e = e(c n, s n+1,c in+1 ) e(c n, s n,c in+1 ) s + e(c n+1, s n+1,c in ) e(c n+1, s n,c in ) s D Ci e = 1 [ ) Ci e (C n, s n, C in+ + J Cn,s n C i 1 ) ] + Ci e (C n+1, s n+1, C in+ + J Cn+1,s n+1 C i 1 with: [ ( H ( ) = C e(c n+1, ) e(c n, ) C e J ( ) = C i [ e(c in+1, ) e(c in, ) Ci e ) C n+ 1, ( C in+ 1, ] : C ) ] : C i 13

14 in space split of the reference configuration B in elements Ω e : η Ω e X e ξ n e B B h = J e ζ ζ Ω ϕ e F e η Ω e e=1 ξ j e gradient: ξ Ω t e ζ x e η test functions: w e ϕ, w e p, w e s, w e C i trial functions: x e, p e, s e, C e i geometry: X e approximation: ( ) e = n af A=1 F e = j e J e 1 with: j e = xe ξ, J e = Xe ξ N A ( ) ea 14

15 Discrete weak evolution equations I equations of motion: w e x e n+1 x e n ϕ = w e h q n Ω e Ω e w e p p e n+1 pe n = h n Ω e Ω e Ω e 1 ρ p e n+ 1 F e T n+ 1 Grad w e p : De C e + w e p T e 1 n+ Ω e thermal evolution equation: ws e sn+1 e sn e [ ( ) w e ( = Grad s K h n θ pe 1 θ pe) ( 1 Grad with: Ω e + we s θ pe D inte ] + D inte = 4C e i n+ 1 D e C i e : V 1 : C e i n+ 1 D e C i e Ω e θ pe ) w e s θ pe Q e n+ 1 15

16 Discrete weak evolution equations II test function of the thermal evolution equation: w e s = D e se problem: D e se is not defined in the test space projection of the test function in space (I. Romero 1): w e s = θpe = n af B=1 N A θ pea equation of the projection: w e θ p θpe = w e θ p De se with: w e θ p = sn+1 e se n h n Ω e Ω e viscous evolution equation: w e Ci e C i : n+1 C e i n = wc e h i : 4Ci e V 1 : C n n+ 1 i e D e n+ 1 C i e Ω e Ω e 16

17 Disk 17 y z t s x r r 1 dimensions: r 1 =.8 r =. t s =.4 initial angular velocity: Dirichlet-boundaries: Neumann-boundaries: T e n+ 1 Q e n+ 1 n af = B n af = B N B T eb n+ 1 N B Q eb n+ 1 θ temperature contour radius r ω z = 6 inner ring is fixed in position T eb n+ 1 Q eb n+ 1 = ( ) 1 π sin 8 t n+ 1 ( ) π = 1 sin 8 t n+ 1

18 18 Internal energy & parameters internal energy with: and: e(c, s, C i ) = ψ com (C) + ψ vis (C, C i ) + e the (C, s) ψ com = µ (trc n dim ln J) + ψ vol (J) ψ vis = µ e (trc e n dim ln J e ) + ψ vol (J e ) e the (C, s) = k [θ(c, s) θ ] + θ n dim β ψvol (J) J ψ vol (J ( ) ) = λ ( ) [ln J ( ) + (J ( ) 1) ] parameters: λ = 3 µ = 75 ρ = 8.93 λ e = 3 µ e = 75 V dev = 1 V vol = 5 κ = k = 15 θ = 3 β =.1

19 Dirichlet-boundaries & ω z ω z S H time time V time 3 19

20 Neumann-boundaries - T T H 3.4 S time T 1 V time 5 time 1 1

21 Neumann-boundaries - Q Q H time 1 S time 1 5 time 1 V 1

22 Conclusions format Implementation of the enhanced format partitioned discrete gradients projection of the test function Dirichlet and Neumann boundaries Outlook time integration for the enhanced system with a thermal constraint for the temperatures

23 Thank you for your attention! Questions?! 3 The authors thank the German Research Foundation (DFG) for financial support (GR 397/-1)

Energy-consistent time-integration for dynamic finite deformation. thermoviscoelasticity. GAMM Annual Meeting,

Energy-consistent time-integration for dynamic finite deformation. thermoviscoelasticity. GAMM Annual Meeting, for dynamic finite thermo-viscoelasticity Chair of Computational Mechanics University of Siegen GAMM Annual Meeting, 9.04.0 Thermo-viscoelastic double pendulum (closed system) Thermodynamically Consistent