COMPUTATIONAL MODELING OF SHAPE MEMORY MATERIALS

Transcription

1 COMPUTATIONAL MODELING OF SHAPE MEMORY MATERIALS Jan Valdman Institute of Information Theory and Automation, Czech Academy of Sciences (Prague) based on joint works with Martin Kružík and Miroslav Frost August 6, 2018

2 1 Introduction to shape memory materials 2 Mathematical models Nonlinear elasticity Sharp interface Dissipation Energetic solutions 3 Numerical implementation (main contribution) Simplified model with 2 variants of martensite Integer minimization problems Computational benchmarks

3 Table of Contents 1 Introduction to shape memory materials 2 Mathematical models 3 Numerical implementation (main contribution)

4 Motivation: martensitic patterns Source: http: //

5 Motivation: martensitic patterns Source: http: //

6 Shape memory alloys Principle of shape memory: high temperature: atomic grid with high symmetry (usually cubic): the so-called austenite, higher heat capacity

7 Shape memory alloys Principle of shape memory: high temperature: atomic grid with high symmetry (usually cubic): the so-called austenite, higher heat capacity low temperature: atomic grid with lower symmetry: martensite, lower heat capacity, typically in many symmetry-related variants

8 Shape memory alloys Principle of shape memory: high temperature: atomic grid with high symmetry (usually cubic): the so-called austenite, higher heat capacity low temperature: atomic grid with lower symmetry: martensite, lower heat capacity, typically in many symmetry-related variants Our model - two variants of martensite only!

9 Shape memory alloys

10 Shape memory alloys More details later.

11 Table of Contents 1 Introduction to shape memory materials 2 Mathematical models Nonlinear elasticity Sharp interface Dissipation Energetic solutions 3 Numerical implementation (main contribution)

12 Elasticity Ω IR 3 reference configuration

13 Elasticity Ω IR 3 reference configuration y : Ω IR 3 deformation

14 Elasticity Ω IR 3 reference configuration y : Ω IR 3 deformation F := y deformation gradient, det F > 0

15 Elasticity Ω IR 3 reference configuration y : Ω IR 3 deformation F := y deformation gradient, det F > 0 T : Ω IR 3 3 1st Piola-Kirchhoff stress tensor

16 Elasticity Ω IR 3 reference configuration y : Ω IR 3 deformation F := y deformation gradient, det F > 0 T : Ω IR 3 3 g : Γ 1 IR 3 1st Piola-Kirchhoff stress tensor density of surface forces

17 Elasticity Ω IR 3 reference configuration y : Ω IR 3 deformation F := y deformation gradient, det F > 0 T : Ω IR 3 3 g : Γ 1 IR 3 1st Piola-Kirchhoff stress tensor density of surface forces T (x) := ˆT (x, y(x)) constitutive law

18 Elasticity Ω IR 3 reference configuration y : Ω IR 3 deformation F := y deformation gradient, det F > 0 T : Ω IR 3 3 g : Γ 1 IR 3 1st Piola-Kirchhoff stress tensor density of surface forces T (x) := ˆT (x, y(x)) constitutive law divt = 0 equilibrium equations y = y 0 on Γ 0 Ω, g=tn on Γ 1 Ω boundary conditions

19 Hyperelasticity Assumption: 1st Piola-Kirchhoff stress tensor T has a potential: T ij := W ( y) F ij

20 Hyperelasticity Assumption: 1st Piola-Kirchhoff stress tensor T has a potential: T ij := W ( y) F ij W : IR 3 3 IR {+ } stored energy density J(y) := W ( y(x)) dx f y ds. Ω Γ 1 Minimizers of J satisfy equilibrium equations.

21 Properties of W (i) W : IR IR is continuous (ii) W (F ) = W (RF ) for all R SO(3) and all F IR 3 3 (iii) W (F ) + if det F 0 +

22 Properties of W (i) W : IR IR is continuous (ii) W (F ) = W (RF ) for all R SO(3) and all F IR 3 3 (iii) W (F ) + if det F 0 +!!! (iii) excludes convexity of W!!!

23 A sharp interface model only the first gradient of y non-diffuse boundary between phases (phase indicator z) { Y = y W 1,p (Ω, IR 3 ) : det y > 0 a.e., } det y(x) dx L 3 (y(ω)), Ω

24 A sharp interface model only the first gradient of y non-diffuse boundary between phases (phase indicator z) { Y = y W 1,p (Ω, IR 3 ) : det y > 0 a.e., } det y(x) dx L 3 (y(ω)), Ω Z := { z BV(Ω, {0, 1} M+1 ) : z i z j = 0 for i j, M i=0 } z i = 1 a.e. in Ω.

25 Loading We assume that the body is exposed to possible body and surface loads, and that it is elastically supported on a part Γ 0 of its boundary. The part of the energy related to this loading is given by a functional L C 1 ([0, T ]; W 1,p (Ω; IR 3 )) in the form L(t, y) := b(t) y dx + s(t) y ds + K y y D (t) 2 ds. Ω Γ 1 2 Γ 0 Here, b(t, ) : Ω IR 3 represents the volume density of some given external body forces and s(t, ) : Γ 1 Ω IR 3 describes the density of surface forces applied on a part Γ 1 of the boundary. The last term in with y D (t, ) W 1,p (Ω; IR 3 ) represents energy of a spring with a spring stiffness constant K > 0.

26 Dissipation The total dissipation reads D(z 1, z 2 ) := D(z 1, z 2 ) := z 1 z 2 M+1. Ω D(z 1 (x), z 2 (x)) dx. The dissipation of a curve z : [0, T ] BV(Ω, {0, 1}) with [s, t] [0, T ] is correspondingly given by { N Diss D (z, [s, t]) := sup D(z(t i 1 ), z(t i )) j=1 : N N, s = t 0... t N = t }.

27 Total energy E(t, y, z) := E b (y, z) + E int (y, z) L(t, y).

28 Energetic solution (Mielke & Theil) We say that (y, z) Y Z is an energetic solution to (E, D) on the time interval [0, T ] if t t E(y(t), z(t)) L 1 ((0, T )) and if for all t [0, T ], the stability condition E(t, y(t), z(t)) E(t, ỹ, z) + D(z(t), z) (ỹ, z) Q. and the condition of energy balance E(t, y(t), z(t)) + Diss D (z; [0, t]) = E 0 + t 0 where E 0 = E(0, y(0), z(0)), are satisfied. E t (s, y(s), z(s)) ds

29 Existence A standard way how to prove the existence of an energetic solution is to construct time-discrete minimization problems and then to pass to the limit. For given N N and for 0 k N, we define the time increments t k := kt /N. Furthermore, we use the abbreviation q := (y, z) Q. Assume that at t = 0 there is given an initial distribution of phases z 0 Z and y 0 Y such that q 0 = (y 0, z 0 ) S(0). For k = 1,..., N, we define a sequence of minimization problems minimize E(t k, y, z) + D(z, z k 1 ), (y, z) Q. We denote a minimizer for a given k as (y k, z k ) Q.

30 Table of Contents 1 Introduction to shape memory materials 2 Mathematical models 3 Numerical implementation (main contribution) Simplified model with 2 variants of martensite Integer minimization problems Computational benchmarks

31 Model: a FEM mesh Figure: A FEM triangular mesh (left) and the corresponding rectangular mesh (right) for visualization. The FEM mesh consists of triangles T T and edges E E. The subset E I E denotes the set of internal edges.

32 Bulk energy with two variants of martensite The bulk energy E b is considered in the form ( ) E b (y, z) := z(x)ŵ 1 (F (y(x))) + (1 z(x))ŵ 2 (F (y(x))) dx, Ω where Ŵ 1, Ŵ 2 are densities in the form Ŵ 1 (F ) := W (FF 1 1 ), Ŵ 2 (F ) := W (FF 1 2 ).

33 Bulk energy with two variants of martensite Here, F 1, F 2 are given stretching matrices ( ) 1 ɛ F 1 :=, F := defined by a parameter ɛ > 0. ( ) 1 ɛ 0 1 Figure: Examples of mesh deformations corresponding to stretching matrices F 1 (left), F 2 (right) for ɛ = 0.3.

34 Two-dimensional compressible Mooney-Rivlin material model The form of W is given by W (F ) := α tr(f T F ) + δ 1 (det F ) 2 δ 2 ln(det F ). Parameters α, δ 1, δ 2 > 0 satisfy the relation δ 2 = 2α + 2δ 1 and it holds W (F ) for det F 0+ and I = argmin F W (F ).

35 Interfacial energy We consider the interfacial energy E i in the form E i (z) := α i z + (s) z (s) ds, E E I where z + and z denote restrictions of z to triangles T + and T and α i 0 is a parameter. T - n T + Ε Figure: An internal edge E E I shared by two elements T +, T T and its normal vector n.

36 Surface energy The surface energy E s is given as E s (F ) := α s cof F(s)n z + (s) z (s) ds. E E I It is based on the cofactor of the surface deformation gradient F := F (I n n), where I represents an identity matrix. A parameter α s 0 is given.

37 Model: a dissipation The dissipation is assumed in the form D(z, z k 1 ) := β z z k 1 dx, where β 0 is a given parameter. Ω

38 Discrete minimization problem The time sequence of incremental minimizations rewrites as: minimize ( ) zŵ 1 (F ) + (1 z)ŵ 2 (F ) + β z z k 1 T T T + ( z + z (α i + α s cof Fn ) ) E (1) E E I over y [P 1 (T )] 2, z P{0,1} 0 (T ). All terms are either constant on each triangle T T or constant on each edge E E I.

39 Discrete minimization problem in z only for given F (y) We introduce an incremental variable z := z z k 1 P{ z 0 k 1,1 z k 1 }(T ), (z-independent terms are dropped): minimize ( ) z[ŵ 1 (F ) Ŵ 2 (F )] + β z T T T + ( α z + z + z + k 1 z k 1 ) E (2) E E I over z P{ z 0 k 1,1 z k 1 }(T ). Further transformations lead to zero-one integer programming problem.

40 Discrete minimization problem in z only for given F (y) minimize ( ) zŵ 1,2 + β λ T + α σ E T T E E I over z P 0 [ z k 1,1 z k 1 ], λ P 0 [ 1,1] (T ), σ P0 [ 1,1] (E I) subject to constraints: z λ 0 z λ 0 on all triangles T T, z + z σ ( z + k 1 z k 1 ) ( z + z ) σ z + k 1 z k 1 on all edges E E I.

41 Time evolution: a full Dirichlet BC

42 Time evolution: a full Dirichlet BC

43 Time evolution: a full Dirichlet BC

44 Time evolution: a full Dirichlet BC

45 Time evolution: a full Dirichlet BC

46 Time evolution: a full Dirichlet BC

47 Time evolution: a full Dirichlet BC

48 Time evolution: a full Dirichlet BC

49 Time evolution: a full Dirichlet BC

50 Time evolution: a full Dirichlet BC

51 Time evolution: a full Dirichlet BC stored energy vs. time stored energy dissipated energy vs. time dissipation

52 Time evolution: a partial Dirichlet BC

53 Time evolution: a partial Dirichlet BC

54 Time evolution: a partial Dirichlet BC

55 Time evolution: a partial Dirichlet BC

56 Time evolution: a partial Dirichlet BC

57 Time evolution: a partial Dirichlet BC

58 Time evolution: a partial Dirichlet BC

59 Time evolution: a partial Dirichlet BC

60 Paper Miroslav Frost, Martin Kružík and Jan Valdman - Computational modeling of shape memory materials (in preparation)

61 Thank you for your attention!