Multi-field Compliant Mechanisms of Adaptive Foldable Structures

Transcription

1 016 Multi-Scale Structural Mechanics Meeting Air Force Office of Scientific Research Dayton, OH, July 18-, 016 Multi-field Compliant Mechanisms of Adaptive Foldable Structures Project duration: September PI/co-PIs: Anastasia H. Muliana, Andrea Bonito, KR Rajagopal Mechanical Engineering and Mathematics Departments Texas A&M University 1

2 Outline Overview Project Goal: combined theoretical and computational approach for foldable polymeric based compliant structures triggered by non-mechanical stimuli (electrical, thermal, light, solvent) Highlight Research Findings: 1) Constitutive model of light-activated shape memory polymers (LASMP) and foldable structures by LASMP ) Time-dependent electro-active plate/shell structures 3) Computational tool for simulating folding of compliant systems Summary and Publications Research Personnel Acknowledgment

3 Foldable Compliant Systems DARPA and NASA morphing wing concept Folding of planar structures by electrical activation (Paik et al. 011) Folding of wings that are fabricated from Veritex shape memory composites (Havens et al., SPIE Proceeding vol. 576, 005) LASMP (Lendlein et al., Nature 005) Light activated shape memory polymers (LASMP) Electro-active fiber composites (smart-material.com) Inflatable active polymer (Jordi et al. 010) Electro-active part 3

4 Macro-scale compliant structures: slender beams and flat sheets (plates/shells) with embedded actuators a) folding line b) actuator actuator polymer Activation systems: 1) electric field using piezoelectric and active fiber composites ) ionic polymer composites and solvent diffusion 3) light activated shape memory polymers 4) thermally activated composites polymer Computational tool for simulating shape changes of compliant systems: compliant geometries and constitutive material models 4

5 Research goals: to develop computational model for simulating timedependent shape reconfigurations, activated by non-mechanical stimuli, and predict life performance of active compliant structures. Expected research outcomes: support the development of digital models of adaptive compliant systems which can be exploited prior to the design of real multifunctional structures Research tasks: 1) Formulate constitutive models for active constituents including time (rate)-dependent and inelastic responses ) Formulate governing equations (partial differential equations, PDEs) for deformations in compliant systems: slender beams, membrane, thin-shell/plate 3) Develop numerical methods for solving the nonlinear PDEs with non-mechanical activation 5

6 Light Activated Shape Memory Polymers (LASMPs) 6

7 Advantages of LASMPs: low cost, lightweight, undergo large deformations, easily processed, nontoxic, can be made biodegradable photosensitive molecules PMC LASMP Typical stress vs strain plot of LASMPs stress strain PNR LASMP 7

8 Due to their ability in retaining certain shapes by light irradiation, LASMPs are good candidates for active components in shape reconfiguration structures. Constitutive modeling of LASMPs (incompressible materials) Response of original network T p B Ka T is the Cauchy stress and B is the stretch tensor Response of two-networks mixture T pt1+ T (Assumption: two networks are uniformly distributed and are constrained to have the same displacement.) PMC: T T T PNR: T 1 T 1 T 1 T p B B Isotropic case: a Ka b Kb Anisotropic case: T p B B F J 1 n n F J T a Ka b Kb C1 Kb 1 Kb Kb Kb F K 1 m m F T C Kb 1 Kb Kb Kb n. C n ; K m. C m 1 Kb Kb Kb 1 Kb Kb Kb 8

9 Relaxation behaviors PNR, Long et al. (010) 9

10 Quasi-linear viscoelastic model for incompressible body dd( t s) T I F S S F d( t s) t e e T ( t) p ( t)( D( t) ( ) ) ( ) 0 D s ds t p : Lagrangian multiplier F( t) : deformation gradient D( t) : normalized relaxation modulus S e D ( t) : nd Piola-Kirchhoff stress of elastic part Response of two-networks mixture p T T T 1+ t e dda ( t s) e T T1 FKa ( t)( SDa ( t) S ( ) ) ( ) 0 Da s ds FKa t d( t s) t e ddb( t s) e T T FKb ( t)( SDb( t) SDb( s) ds) FKb ( t) t1 d( t s) F( t) F ( t) F ( t ) Kb Ka 1 10

11 Relaxation behaviors of PNR LASMPs i e c io 3.8 MPa; c 80 ao i~ i~ t/ i Di 1 e io io a~ 0.4; a 15sec ao 11

12 Implementation within finite element (FE) method W T T F F J C F F F 1 0 PMC: PNR: T T1 T T 1 T T 1 MZ J MZ J MZ J MZ J MZ J MZ J S We We = =4 = C CC C C F F F F c ijkl ip jq kr ls pqrs 1 = + ( + + )e e e e Z J c ik jl ik jl il jk il jk i j k l 1 J MZ J Z J 1

13 Validation of FE analyses with analytical solutions Uniaxial extension Radial inflation 13

14 Numerical simulations Geometry size (mm): 60 x 0 x x layers Material: Top: Hyperelastic material (neo-hookean), Bottom: LASMP, Tension: stretch ratio=3.0 Tension and radiation Removal of load 14

15 Geometry size (mm): 45 x30 x 4 x Material: Beam: Linear elastic material, Connector: LASMP Bending: Moment: -800 Nmm 15

16 Geometry size (mm): 50 x30 x 4 x Material: Beams: Linear elastic material, Link: LASMP, Bending: Moment: 400 Nmm; 000 Nmm

17 Geometry size (mm): 50 x30 x 4 x 4 Material: Bars: Linear elastic material, Links: LASMP, Bending: Moment: 000 N mm (loaded sequentially) 17

18 Electro-active thin-shell/plate 18

19 Piezoelectric and active fiber composite actuators for flexible and foldable structures Thin sheet with distributed actuators actuator actuator polymer polymer Typical response of polarized piezoelectric materials is nonlinear Problem descriptions and assumptions Thin flexible plates/shells with multiple patches and foldable capability without undergoing large strains and stretch Large deformation in thin and flexible sheet is mainly due to large rotations; thus the shell experiences small (linearized) strains Patches are made of a piezoelectric material and perfectly bonded to the substrate Piezoelectric materials exhibit nonlinear time-dependent electro-mechanical responses Polymeric sheets show viscoelastic behaviors Co-rotational Lagrangian approach is used for numerical analyses (finite element formulation) 19

20 Effect of piezoelectric patches on the shell substrate Under stimulating voltage (electric field) input, the top actuator patch undergoes positive strain and the bottom patch undergoes negative strain h h h h z dy dz x z dx dz y m dy x m dx y From continuity of strain at interface and bending moment equilibrium: m m h x pl 3 1 pl xi pl yi h E E y pl 3 1 pl yi pl xi where xi yi 1 p p 1 p pl p p 1 p p 1 p pl p p x pl p y y pl p x p x x p p y p y y p p x p x, y p p p E p 1 pl p k E pl 1 p : unconstrained strains of piezoelectric material due to electric field input g, k g 3th h t 0 h t 3ht 3 3

21 Normalized relaxation modulus Strain (%) Active fiber composites (AFCs) Test Model E(V/m) Test model E AFC =9.7 GPa; d 33 =110 pm/v; β 33 =.55 f (m/v) E c_afc =1.4MV/m Time (min) 1

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23 Constitutive model for AFC d t t E E j ME j i ( t) Sij j ( t) dik Ek ( t) Sij t s ds dij t s ds ds ds 0 0 t t EM j j Di ( t) dik k ( t) ij E j ( t) dij t s ds ij t s ds ds ds 0 0 d de de Nonlinear response: f d de t t E i j ME i j i () t Sij t s ds dij t s ds 0 j ds E 0 j ds g d t t EM i j i j Di () t dij t s ds ij t s ds 0 j ds E 0 j ds f g de 3

24 Co-rotational finite element (CRFE) method CRFE is based upon explicit separation of rigid body motions including translations and rotations from deformational motions. Benefit: existing linear finite element models can be taken advantage of for deformational part of motion while nonlinear analysis is excluded for the rigid body motions part C 0 : initial configuration C R : co-rotated configuration C D : deformed configuration T, T 0 : transformation matrices of the coordinate systems T T [ e 0 e 0 e 0 ], T [ e e e ] T x = T0 x, x = Tx 1 3 R : orthogonal rotation matrix R T : 0 T T to express rigid body 0 rotation u u u ; R R R, 1,,3 a ra da a da 0 a 4

25 Co-rotational finite element (CRFE) method (cont. d) Deformational (elastic) nodal displacement : Internal force in LCS: As for nonlinear approach, iterative incremental approach is used: Nodal displacement in GCS: P f n m n m n m p { u θ u θ u θ } T d d1 d1 d d d3 d3 f K d da u a R a, d { u T ω T } T, a a a : projector matrix to extract deformational translations and rotations from the total translations and rotations of the element ( a purely geometrical matrix) d d Pd T Internal force of an element : f T P H f T T T el T T T T T T T T T T T T f T P H f T P H f T P H f T P H f ( K K K K ) d el el el el GR GP GM M K G K M = Geometric stiffness : = Material stiffness : K K K K GR GP GM T P H K HPT T T T M el el transformation of local stiffness to the global frame 5

26 Flat shell triangular element Obtained by combining 9-DOF bending triangular element and 9-DOF membrane triangular element Plate bending element based on Kirchhoff plate theory: U u u u b z1 x1 y1 z x y z3 x3 y3 T Membrane element (optimal version of LST with drilling degree of freedom) U u u u u u u m x1 y1 z1 x y z x3 y3 z3 T zi 1 u x yi u y xi 6

27 Solution Procedure After obtaining structural internal forces and tangent stiffness by assembling internal force and tangent stiffness of each element. Newton- Raphson algorithm with load control is adopted to derive the displacement solution of the structure under applied loads. For each iteration (i) at the (n+1) th load step: 1 U K F F ( n1) ( n1) ( n1) ( i) St ( i) ext St ( i) ( n 1) : load factor i in ( n 1) max ( n) Convergence criterion: U U ( n1) ( i1) 3 10 ( n1) 7

28 Viscoelastic isotropic substrate Relaxation modulus: E( t) e t GPa Poisson s ratio is constant =0.3 E e = 3 KV/m It is necessary to incorporate proper material responses, e.g., nonlinear and time-dependent electro-mechanical models, for the constituents for precise controls of shape reconfigurations in adaptable structures. 8

29 T=0.1 sec T=1 sec E e = 0.5 MV/m, 9

30 Combined AFCs and LASMPs as active structures AFCs LASMPs Advantages Fast actuation, precise control, relatively low voltage input Shape retention after removal of stimuli Disadvantages No shape retention Slow actuation, need other stimuli to control shape AFCs LASMP 30

31 Flexible beams having LASMPs and AFCs E e = 0. MV/m, =0.5 E e = 0.6 MV/m, =0.5 E e = 0.6 MV/m, =0.5 31

32 Computational Simulations of Folding of Thin Plates 3

33 General numerical methods for solving the nonlinear PDEs of large deformation in compliant plates with non-mechanical activation Consider a bi-layer thin plate due to thermal activation Assuming linear heat conduction model (Fourier law): 0 t Hyperelastic materials are considered for the bi-layer with the following strain energy density Plate deformation Asymptotic when thickness orthogonal to deformed plate Reduction to two-dimensional geometric plate, gives the following strain energy: has deformation components effect of external stimuli, i.e., I 33

34 Perspectives: Temperature Model for the Spontaneous curvature E. Smela et al, Science (1995) Temperature controlled self opening / closing box 34

35 Effect of Tensor Spontaneous Curvature Left: Right: rotation by Corkscrew shape 35

36 Summary Constitutive material models for LASMPs and electro-active composites, incorporating time-dependent effect, have been formulated and integrated to finite elements in order to analyze responses of compliant structures. Shape changes in active foldable and flexible systems have been considered by formulating governing equations for thin beams and plates/shells capable of undergoing large deformations when subjected to non-mechanical stimuli (light, electric field, thermal, and diffusion of solvent). For precisely controlling shape reconfigurations in adaptable structures it is necessary to consider time-dependent and nonlinear coupling responses of the constituents, as they are crucial in determining magnitude and duration of external stimuli that have to be prescribed. A general computational tool for simulating folding of compliant systems due to thermal activation has been developed for elastic plates, which allows for analyzing various shape changes in foldable systems. Currently it is being modified to incorporate more complex constitutive material models and other stimuli. 36

37 Journal publications: 1. Muliana AH Large deformations of nonlinear viscoelastic and multi-responsive beams Int. J. Nonlinear Mechanics, 71, pp , Tajeddini V and Muliana A, Nonlinear deformations of beams with piezoelectric patches subjected to electric and mechanical actuations Composites Structures, 13, pp , Yuan Z, Muliana, A., and Rajagopal, KR, Modeling Responses of Light Activated Shape Memory Polymers Mathematics Mechanics of Solids, 19, pp , Parthasarathy S, Muliana, A., and Rajagopal, KR, A Fully Coupled Model for Diffusion-Induced Finite Deformations in Polymers Acta Mechanica, 7, pp , Bartels S, Bonito A, and Nochetto RH, Bilayer Plates: Model Reduction, G-Convergent Finite Element Approximation and Discrete Gradient Flow, Communication on Pure and Applied Mathematics, in press 016, DOI: /cpa Yuan Z, Muliana A, and Rajagopal KR, Quasi-linear viscoelastic of light-activated shape memory polymers in review 7. Tajeddini V and Muliana A, Deformations of Flexible and Foldable Electro-active Composite Structures, in review Conference abstracts and presentations: 1. Tajeddini V. and Muliana A., Nonlinear deformation of flexible plates with piezoelectric actuators ASME Mechanics of Materials Conference, San Diego June 30-July, Yuan Z, Muliana, A., and Rajagopal, KR, Modeling Responses of Light Activated Shape Memory Polymers ASME Mechanics of Materials Conference, San Diego June 30-July, Parthasarathy S, Muliana, A., and Rajagopal, KR, A Fully Coupled Model for Diffusion-Induced Finite Deformations in Polymers ASME Mechanics of Materials Conference, San Diego June 30-July, Muliana, A, Sohrabi A, Tajeddini V, Nonlinear Electro-mechanical Responses of Ferroelectric Ceramics and Active Composites International Conference of Computational Materials, Auckland NZ, July 13-17,

38 Graduate Students: Vahid Tajeddini (PhD, graduated 016) Sudharsan Parthasarathy (PhD) Zhi Yuan (PhD) Ruyue Song (PhD) Acknowledgement: This research is sponsored by the US Air Force Office of Scientific Research (AFOSR) under grant FA Contact: Anastasia Muliana 38