Exploring Piezoelectric Properties of Wood and Related Issues in Mathematical Description. Igor Dobovšek

Transcription

1 Exploring Piezoelectric Properties of Wood and Related Issues in Mathematical Description Igor Dobovšek University of Ljubljana Faculty of Mathematics and Physics Institute of Mathematics Physics and Mechanics

2 Contents Keywords: Wood, Elasticity, Piezoelectricity, Constitutive behaviour, Averaging of heterogeneous structure Introduction Discussion of experimental data Foundations: mathematical formulation Structure of Constitutive relation (EL & PE) Transition mechanisms between micro and macro scale: Averaging of EL and PE heterogeneous structure

3 Historical overview 1st publication describing phenomenon of PE (piezo-press) 1880, Curie bros. J & P, experiments on crystals: tourmaline, topaz, quartz, salt (mech. stress surface charge). Direct PE effect. Lippmann G. J. 1881, Converse PE effect (el. charge -> deformation) First applications: WW I, Langevin P., Ultrasonic submarine detector After: Power sonars, sensitive hydrophones, phono cartridges, microphones, accelerometers, ultrasonic transducers etc. Smart & intelligent material structures

4 Transducers: energy converters Mechanical energy Electrical energy Sensors Actuators Piezo electricity direct PE effect inverse PE effect

5 PE system El. field E V voltage Direct PE effect Apply, D E El. displacement Inverse PE effect Apply E D, PE voltage Mechanical behavior

6 Origins of PE Dielectrics: (nonconductors) D Crystals (mono): space lattice with periodic combination of differently charged ions. El. polarization induced by mechanical stress and deformation and vice versa PE textures: crystal aggregates formed by oriented crystals with PE properties Polarized dielectric: polarization-el. displacement + and el. charges at both ends of the element Crystal classes: 32 Centro-Symmetric: 11 FCC, BCC Non-Centro Symm. Exhibit PE behavior CS: unit cells with point symmetry (center of sym.) w.r.t. all features NCS: Ability to form el. dipole & el. dipole moment P polarization = Resultant of dipole moments P P 0 0 nonpolar crystal: no PE polar crystal: PE effect

7 Relation to wood: cellulose morphology WOOD: natural composite material Two phase: cellulose & matrix Crystallization: cellulose crystals in cell wall: stiff micro-fibrils Stress distribution (loading) in fiber directions Change in crystal lattice strain PE voltage: depends on mechanical deformation of crystalline sub-domains PE effect: deformation behavior of cellulose (skeleton of wood) PE effect

8 Nakai T. et al.: J Wood Sci (1998) 28, 255; (2004) 97; (2005) 193; (2006) 539; Wood Sci Technol (2005) 163; Japanese cypress (Chamaecyparis obtusa) T=20 o C, 60% humidity Density=0.33g/cm 3 Moisture=12% 150X36X7.5mm

9 Mathematical description Material body(continuum): defined on Material point: Position vector Orthonormal basis B [0, ), t [0, ) P( x) B x : F. C. C. S eˆˆ e eˆ eˆ e ˆ I ij eˆ i eˆ j { 1, 2, 3}, i j ij 3 B General frame: arbitrary c.s. orientation w.r.t. bulk element eˆ eˆ, (3) i ia a SO T I eˆ eˆ, (2) i ia a SO T cos sin 0 sin cos ( x L ) 3

10 eˆ i { eˆˆ, e, eˆ } Arbitrary coordinate system Principal coordinate system eˆ { eˆ, eˆ, eˆ } a R T L eˆ eˆ, (3) i ia a SO

11 General relation between stress and strain C :, ij C ijklkl S :, ij S ijklkl Voigt notation 11 1, 22 2, , , i, j, k, l 1 3 ij ,, ,,,,,,,,,,, Stress tensor-> vector C, C S, S

12 Mechanical properties The mechanical properties of wood are orthotropic: 3 orthogonal planes of symmetry: Radial (R), Tangential (T), Longitudinal-Axial (L), along which the coordinate axes of orthonormal basis are aligned x, y, z x, x, x R, T, L 1 2 3

13 S 11 S12 S S21 S22 S S 31 S32 S T S S = S S S66 S 1/ E, S / E, S / E, 11 R 12 RT T 13 RL L S / E, S 1/ E, S / E, 21 TR T 22 T 23 TL L S / E, S / E, S 1/ E, 31 LR R 32 LT T 33 L S 1/ G, S 1/ G, S 1/ G 44 TL 55 LR 66 RT E, E, E elastic Young s moduli R T L G, G, G shear moduli TL LR RT the set of Poisson s ratios

14 General theory of PE Linear relation between polarization vector and stress (strain) D D eˆ d : d eˆ, i, j, m 1 3; m m mij ij m Dm d mijij d d d d d d d d d d d d d d d d d d d , 22 2, , , m Dm d m d d d d d d d d d d d d d d d d d d d

15 WOOD: Piezoelectric properties Theoretical and Experimental verification: the PE properties for majority of wood species belong to the Schoenflies D2 or 222 international class of orthorhombic non-centro-symmetric systems. d d d d 36 d 2 d 2 d d 2 d 2 d d 2 d 2 d d d, d d 0?

16 Structure of constitutive equations S : E d, S E d D d : E, D d E S E d, S E d D d E, ij ijkl kl m m mij ij mk m mij k m D d E m m mk k stress vector tensor 2 ( )( N m ) strain vector ( tensor)( m m) Evector of applied el. field ( V m) permitivity( F m) dmatrix ( tensor) of piezoel. strain constants ( m V ) Smatrix tensor of compliance coefficients m N 2 ( ) ( ) 2 Dvector of electric polarization displacement ( C m )

17 Influence of structural heterogeneity on E and PE properties Micromechanical approach Problem of averaging of two phase material with different symmetry groups Anatomical structure of wood: two types of layered PE textures Prototype problem: Structural scheme of layered material two phase material: straight annual rings and no xylem rays Real problem: Structural scheme of wood as layered material: small curvature annual rings, xylem rays

18 Textures (A, B) - Elastic properties: Symmetry group : 2 11 S 11 S12 S S S S S 13 S13 S S S S S 2( S S ) S11, S12, S13, S33, S44 A, B 11 C 11 C12 C C C C C 13 C13 C C C C C ( C C ) / C11, C12, C13, C33, C44 A, B

19 Layered texture A+B: E symmetry group 2 : 2 11 S 11 S12 S S S S S 13 S23 S S S S S, S, S, S, S S33, S44, S55, S Coupling relations: EL compatibility equations A B AB A B AB A B AB A B AB

20 Rule of mixtures A B V V V A B VA VB A, B, A B 1 V V A B Homogeneous texture A B ij A ij B ij A B ij A ij B ij Combination of phases A & B requires more general texture morphology (symmetry group) GA( : 2) G B ( : 2) GA B (2 : 2) S, C G(2 : 2)

21 Textures (A, B) - PE properties: Symmetry group : 2 D d D d D A, B d d Coupling relations: PE compatibility equations Polarization-el. displacement D D, D D, D D 0 A A B B A B D D D A B 1 A 1 B 1 D D D A B 2 A 2 B 2 d G A B (2 : 2)

22 Final remarks Wood can be modelled as a two-phase material A two phase material structure can take into account the basic morphology of the specimen The model can accomodate the effect of the structural inhomogeneity by using structural layer theory, rule of mixtures and first order homogenization - averaging The symmetry structure of elastic and PE coupling with the corresponding set of material constants is known Can be perceived as a layered texture with changed morphological symmetry with xylem ray texture type of reinforcement in radial direction. The overall resulting symmetry of the texure is (2:2).