Modeling and Simulation of Magnetic Shape Memory Polymer Composites

Transcription

1 Modeling and Simulation of Magnetic Shape Memory Polymer Composites Martin Lenz 1 with Sergio Conti 2 and Martin Rumpf 1 1) Institute for Numerical Simulation University Bonn 2) Department of Mathematics University Duisburg Essen Supported by DFG Priority Program 1095 Analysis, Modeling and Simulation of Multiscale Problems GAMM Annual Meeting 2006 on March 27th 31st 2006 in Berlin Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

2 Overview Introduction Table of Contents Physical Problem Magnetic Shape Memory Materials Crystal Polymer Composites Model Micromagnetic Elastic Model Rigid Particles and Linear Elasticity Simulation Energy Descent Boundary Element Method Results Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

3 Physical Problem Magnetic Shape Memory Effect Magnetic Shape Memory Materials Magnetic Shape Memory Material E.g. Nickel Manganese Gallium Discovered in 1996 by Ullakko et al. Crystal lattice aligns to magnetization H Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

4 Physical Problem Magnetic Shape Memory Effect Magnetic Shape Memory Materials Magnetic Shape Memory Material E.g. Nickel Manganese Gallium Discovered in 1996 by Ullakko et al. Crystal lattice aligns to magnetization H Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

5 Physical Problem Magnetic Shape Memory Effect Magnetic Shape Memory Materials Magnetic Shape Memory Material E.g. Nickel Manganese Gallium Discovered in 1996 by Ullakko et al. Crystal lattice aligns to magnetization H Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

6 Physical Problem Magnetic Shape Memory Effect Magnetic Shape Memory Materials Magnetic Shape Memory Material E.g. Nickel Manganese Gallium Discovered in 1996 by Ullakko et al. Crystal lattice aligns to magnetization H Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

7 Physical Problem Magnetic Shape Memory Effect Magnetic Shape Memory Materials Magnetic Shape Memory Material E.g. Nickel Manganese Gallium Discovered in 1996 by Ullakko et al. Crystal lattice aligns to magnetization In Detail External magnetic field Magnetization aligns to field Magnetic easy axis is short lattice axis Movement of domain boundaries Macroscopic deformation Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

8 Physical Problem Magnetic Shape Memory Effect Magnetic Shape Memory Materials Magnetic Shape Memory Material E.g. Nickel Manganese Gallium Discovered in 1996 by Ullakko et al. Crystal lattice aligns to magnetization In Detail External magnetic field Magnetization aligns to field Magnetic easy axis is short lattice axis Movement of domain boundaries Macroscopic deformation H Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

9 Physical Problem Magnetic Shape Memory Effect Magnetic Shape Memory Materials Magnetic Shape Memory Material E.g. Nickel Manganese Gallium Discovered in 1996 by Ullakko et al. Crystal lattice aligns to magnetization In Detail External magnetic field Magnetization aligns to field Magnetic easy axis is short lattice axis Movement of domain boundaries Macroscopic deformation H Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

10 Physical Problem Magnetic Shape Memory Effect Magnetic Shape Memory Materials Magnetic Shape Memory Material E.g. Nickel Manganese Gallium Discovered in 1996 by Ullakko et al. Crystal lattice aligns to magnetization In Detail External magnetic field Magnetization aligns to field Magnetic easy axis is short lattice axis Movement of domain boundaries Macroscopic deformation H Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

11 Physical Problem Magnetic Shape Memory Effect Magnetic Shape Memory Materials Magnetic Shape Memory Material E.g. Nickel Manganese Gallium Discovered in 1996 by Ullakko et al. Crystal lattice aligns to magnetization In Detail External magnetic field Magnetization aligns to field Magnetic easy axis is short lattice axis Movement of domain boundaries Macroscopic deformation H Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

12 Physical Problem Magnetic Shape Memory Materials Properties of Magnetic Shape Memory Materials Large deformations ( 10%) Compared to 0.2% for magnetostrictive, 0.1% for piezo ceramics, up to 10% for thermic shape memory materials Moderate magnetic fields ( 1 T) Same order of magnitude as for magnetostrictive materials, less than for piezo ceramics High operating frequency ( 10 3 Hz) Faster than thermic shape-memory materials (which are limited by heat conduction) High work output ( 10 5 Pa) Larger than piezo ceramics and magnetostrictive materials, but less than for thermic shape memory materials Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

13 Physical Problem Magnetic Shape Memory Materials Applications of Magnetic Shape Memory Materials Applications as Actuators, Dampers and Sensors Promises to exhibit significant advantages over other active materials in different applications Research in fabrication, characterization, application concepts, modeling and simulation DFG Priority Program 1239 Änderung von Mikrostruktur und Form fester Werkstoffe durch äußere Magnetfelder Experiment by O Handley et al. Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

14 Physical Problem Deformation of Polycrystals Crystal Polymer Composites Deformations of 10% can be achieved only for Single Crystals, but growing single crystals of the size necessary for applications is difficult Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

15 Physical Problem Deformation of Polycrystals Crystal Polymer Composites Deformations of 10% can be achieved only for Single Crystals, but growing single crystals of the size necessary for applications is difficult In Polycrystals the incompatibilities at grain boundaries may lead to significantly smaller deformations, increase of the necessary external field, or even inhibit switching altogether. Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

16 Physical Problem Embedding Crystals in a Polymer Crystal Polymer Composites Alternative Approach Embedding single crystals in a polymer bulk - Feuchtwanger et al. (2003) - Gutfleisch, Weidenfeller Similar to an approach used with magnetostrictive Terfenol-D (Sandlund, Fahlander et al. 1994, McKnight and Carman 1999) Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

17 Physical Problem Embedding Crystals in a Polymer Crystal Polymer Composites Alternative Approach Embedding single crystals in a polymer bulk - Feuchtwanger et al. (2003) - Gutfleisch, Weidenfeller Configuration Small single crystal particles in a polymer matrix Particles deform and possibly align due to applied magnetic field Deformation of polymer matrix, yellow to red encodes elastic energy density Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

18 Physical Problem Embedding Crystals in a Polymer Crystal Polymer Composites Alternative Approach Embedding single crystals in a polymer bulk - Feuchtwanger et al. (2003) - Gutfleisch, Weidenfeller Configuration Small single crystal particles in a polymer matrix Particles deform and possibly align due to applied magnetic field Deformation of polymer matrix, yellow to red encodes elastic energy density Here: 4.8% deformation of composite (11.6% in single crystal) with 30% volume fraction of active material Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

19 Model Micromagnetic Elastic Model Full Model for Micromagnetism and Elasticity Ω R d ω Ω ω p=1 Ω Q p=0 area occupied by composite area occupied by particles E[v, m, p] = Ematr elast = + Epart elast + + E ext + E demag + + E anis + + E exch + Ω\ω ω v(ω) v(ω) v(ω) v(ω) W matr( v) W part(( v)q, p) H ext m 1 2 H d 2 ϕ p((rq) T m) 1 2 m 2 Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

20 Model Micromagnetic Elastic Model Full Model for Micromagnetism and Elasticity Ω R d ω Ω ω Q Ω p=1 p=0 area occupied by composite area occupied by particles E[v, m, p] = Matrix Elasticity W matr stored energy density of polymer bulk v : Ω R d deformation Ematr elast = + Epart elast + + E ext + E demag + + E anis + + E exch + Ω\ω ω v(ω) v(ω) v(ω) v(ω) W matr( v) W part(( v)q, p) H ext m 1 2 H d 2 ϕ p((rq) T m) 1 2 m 2 Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

21 Model Micromagnetic Elastic Model Full Model for Micromagnetism and Elasticity Ω R d ω Ω ω Particle Elasticity W part Q Ω p=1 p=0 area occupied by composite area occupied by particles E[v, m, p] = Ematr elast = + Epart elast + + E ext + E demag + + E anis + + E exch + stored energy density v(ω) of particles, i.e. quadratic distance from strain to eigenstrain with respect to crystal lattice orientation p : ω {1,... d} phase parameter in particles Q : ω SO(d) lattice orientation in particles v : Ω R d deformation Ω\ω ω v(ω) v(ω) v(ω) W matr( v) W part(( v)q, p) H ext m 1 2 H d 2 ϕ p((rq) T m) 1 2 m 2 Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

22 Model Micromagnetic Elastic Model Full Model for Micromagnetism and Elasticity Ω R d ω Ω ω Q Ω p=1 p=0 area occupied by composite area occupied by particles E[v, m, p] = Interaction with External Field m : ω R d magnetization H ext R d external magnetic field Ematr elast = + Epart elast + + E ext + E demag + + E anis + + E exch + Ω\ω ω v(ω) v(ω) v(ω) v(ω) W matr( v) W part(( v)q, p) H ext m 1 2 H d 2 ϕ p((rq) T m) 1 2 m 2 Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

23 Model Micromagnetic Elastic Model Full Model for Micromagnetism and Elasticity Ω R d ω Ω ω Q Ω p=1 p=0 area occupied by composite area occupied by particles Demagnetization m : ω R d magnetization H d : Ω R d demagnetization field E[v, m, p] = Ematr elast = + Epart elast + + E ext + E demag + + E anis + + E exch + Ω\ω ω v(ω) v(ω) v(ω) v(ω) W matr( v) W part(( v)q, p) H ext m 1 2 H d 2 ϕ p((rq) T m) 1 2 m 2 H d = ψ ψ = div m distributionally Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

24 Model Micromagnetic Elastic Model Full Model for Micromagnetism and Elasticity Ω R d ω Ω ω Q Ω p=1 p=0 area occupied by composite area occupied by particles E[v, m, p] = Ematr elast = + Epart elast + + E ext + E demag + + E anis + Anisotropy + E exch + m : ω R d magnetization p : ω {1,... d} phase parameter in particles ϕ p : R d R anisotropy in phase p applied to magnetization in deformed lattice R SO(d) rotational part of deformation u = RU Q : ω SO(d) lattice orientation in particles Ω\ω ω v(ω) v(ω) v(ω) v(ω) W matr( v) W part(( v)q, p) H ext m 1 2 H d 2 ϕ p((rq) T m) 1 2 m 2 Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

25 Model Micromagnetic Elastic Model Full Model for Micromagnetism and Elasticity Ω R d ω Ω ω Q Ω p=1 p=0 area occupied by composite area occupied by particles Magnetic Exchange m : ω R d magnetization E[v, m, p] = Ematr elast = + Epart elast + + E ext + E demag + + E anis + + E exch + Ω\ω ω v(ω) v(ω) v(ω) v(ω) W matr( v) W part(( v)q, p) H ext m 1 2 H d 2 ϕ p((rq) T m) 1 2 m 2 Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

26 Model Reduction to Small, Rigid Particles Rigid Particles and Linear Elasticity Particles are small and hard particle deformations are affine Particles are single crystals lattice orientation Q constant on each particle Particles are single domain phase p and magnetization m constant on particles E exch = 0 Deformations are small linearized elasticity Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

27 Reduction to Cell Problem Model Homogenization Large numbers of small particles: Fully resolved simulation not feasible Homogenization: Study periodic configurations Consider unit square with some particles, periodic boundary conditions for magnetic field, affine periodic for deformation Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

28 Model Homogenization Periodic Configuration Keep in mind: Computational cell is part of periodic configuration Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

29 Model Homogenization Periodic Configuration Keep in mind: Computational cell is part of periodic configuration Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

30 Model Homogenization Periodic Configuration Keep in mind: Computational cell is part of periodic configuration Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

31 Model Homogenization Periodic Configuration Keep in mind: Computational cell is part of periodic configuration Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

32 Model Homogenization Periodic Configuration Keep in mind: Computational cell is part of periodic configuration Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

33 Model Homogenization Periodic Configuration Keep in mind: Computational cell is part of periodic configuration Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

34 Model Homogenization Periodic Configuration Keep in mind: Computational cell is part of periodic configuration Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

35 Numerical Relaxation Simulation Energy Descent Minimize over internal variables Particle deformation particle magnetization particle phase Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

36 Numerical Relaxation Simulation Energy Descent Minimize over internal variables Particle deformation particle magnetization particle phase degrees of freedom per particle Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

37 Numerical Relaxation Simulation Energy Descent Minimize over internal variables Particle deformation particle magnetization particle phase degrees of freedom per particle For a given particle configuration, the energy now is a function of the external magnetic field and the macroscopic deformation. Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

38 Computation of Energy Simulation Boundary Element Method Energy minimization: Gradient Descent Approximate gradient by finite differences Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

39 Computation of Energy Simulation Boundary Element Method Energy minimization: Gradient Descent Approximate gradient by finite differences Energy evaluation: Boundary element method Elasticity in polymer matrix Affine-periodic cell boundary Dirichlet particle boundary Affine periodic boundary Dirichlet boundary Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

40 Computation of Energy Simulation Boundary Element Method Energy minimization: Gradient Descent Approximate gradient by finite differences Periodic boundary Energy evaluation: Boundary element method Elasticity in polymer matrix Affine-periodic cell boundary Dirichlet particle boundary Neumann values jumping Demagnetization Computation in deformed unit cell H d jumps on particle boundaries Actual energy computed by partial integration Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

41 Results Exploring Configurations Regular versus Random Configuration Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

42 Results Exploring Configurations Regular versus Random Configuration 5% Strain 2% Strain Careful optimization necessary Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

43 Results Exploring Configurations Exploring Configurations: Particle Shape Simple lattice, one particle in the periodic unit cell External field horizontal, compress lattice horizontally Plot energy over stretch for different particle shapes Energy Density in MPa :1 1:1 1: Horizontal Stretch Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

44 Results Exploring Configurations Exploring Configurations: Particle Shape Simple lattice, one particle in the periodic unit cell External field horizontal, compress lattice horizontally Elongated particles better, but not very significant try softer polymer plot optimal stretch and work output for different aspect ratios Stretch E = 6 GPa E = 1.2 GPa Work Output in MPa E = 6 GPa E = 1.2 GPa :1 2:1 1:1 Aspect Ratio 1:2 1: :1 2:1 1:1 Aspect Ratio 1:2 1:4 Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

45 Results Exploring Configurations Exploring Configurations: Polymer Elasticity Plot strain and work output for different polymer elastic moduli Longer particles, (somewhat) softer polymer (In comparison: For particles is E M Pa) Strain :1 2:1 3:1 4:1 Work Output in MPa :1 2:1 3:1 4: Polymer Elastic Modulus in MPa Polymer Elastic Modulus in MPa Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

46 Results Exploring Configurations Exploring Configurations: Particle Alignment Do the same computation for particles that do not form chains 1 Work Output in MPa % Shift 25 % Shift 50 % Shift 75 % Shift 100 % Shift Polymer Elastic Modulus in MPa Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

47 Results Exploring Configurations Exploring Configurations: Particle Orientation Consider two particles that are not oriented in the direction of the external field, examine effect of misorientation 1.6 Work Output in MPa Rotation Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

48 Conclusion Results Conclusion Continuous Model Efficient Simulation Results identify important factors in composite design Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

49 Conclusion Results Conclusion Continuous Model Efficient Simulation Results identify important factors in composite design Elongated particles Softer polymer Orientation Alignment not necessary Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

50 Outlook Results Conclusion Simulation of Polycrystals Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20

51 Outlook Results Conclusion Simulation of Polycrystals (Preliminary numerics) Thank you for your attention! Contact: Martin Lenz (INS Bonn) Magnetic Shape Memory Composites GAMM / 20