by balasubramanian datchanamourty and george e. blandford university of kentucky lexington, ky

ByBalasubramanian Datchanamourty

and George E. Blandford

University of KentuckyLexington, KY

Assumptions

Finite Element Equations

Buckling Analysis

Numerical Results

Summary and Conclusions

Each lamina is generally orthotropic

Piecewise linear variation of electromagnetic potential through the depth of each piezoelectric lamina

Piezoelectric surface is grounded where it is in contact with structural composite material

Linear variation of temperature through the plate thickness

Displacement assumptions consistent with Mindlin theory

Nonlinear strains consistent with von Karman approximation

N N

N

uuuuu u uQ e u

Nuu Q e

Qu Q QQ eee

e

[K ] [K ] [0] ˆ[K ] [K ] [K ] {u } {f }

ˆ[K ] [K ] [K ] [K ] [0] [0] { } {0}

{0}ˆ[K ] [K ] [K ] {Q }[0] [0] [0]

u u

e e

{f } {f }

{f } {f }

{0} {0}

= ith element node displacement vector; five displacements per node: u, v, w, x, y

= ith element node electromagnetic potential

= ith element Gauss point transverse shear stressresultant vector; two per node: Qx and Qy

eiˆ{u }

eiˆ{ }

ei

ˆ{Q }

= mechanical load vector

= electrical load vector

= temperature-stress load vector

= pyroelectric load vector

= nonlinear temperature-stress load vector

ue{f }

e{f }

ue{f }

e{f }

uN e{f }

uu[K]

u[K]

uQ[K]

[K]

Q[K]

QQ[K]

= linear stiffness matrix for

= linear coupling matrix between and

eˆ{u }

eˆ{u }eˆ{ }

= linear coupling matrix between and eˆ{u } eˆ{Q }

= linear matrix for

= linear stiffness matrix for

eˆ{ }

eˆ{Q }

= linear coupling matrix between and eˆ{ } eˆ{Q }

uuN[K ]

uN[K ]

= nonlinear stiffness matrix consistent with the von Karman approximation

= nonlinear coupling matrix between displace-ments and electromagnetic potentials in the piezoelectric laminae

L[K ] [K ] { U} {0}

L[K ] uu u

u T

[K ] [K ]

[K ] [K ]

[K ] u[K ] [0]

[0] [0]

= geometric stiffness matrix

= linear coefficient matrix

= inplane stress magnification factor

N(U) [K(U)]{U} {F } {F} {0}

(U) = residual force vector

L N[K(U)] [K ] [K ]

N[K ] = nonlinear stiffness matrix consistent with a total Lagrangian formulation

N{F}, {F } = linear and nonlinear force vectors

Nonlinear Solution Schematic

Thermal Buckling of (0/90/0/90)s Graphite-Epoxy laminate plus top and bottom piezoelectric lamina – PVDF or PZT

Simply supported square plate

PVDF PZT Graphite-Epoxy

E1 = E2 = E3 =2 GPa E1 = E2 = E3 = 60 Gpa E1 =138 GPa, E2 = 8.28 GPa

12 = 13 = 23 = 0.333 12 = 13 = 23 = 0.333 12 = 0.33

G12 = G13 = G23 = 0.75 GPa G12 = G13 = G23 = 22.5 GPa G12 = G13 = G23 = 6.9 GPa

1 = 2 = 3 = 1.2x10-4 /0C 1 = 2 = 3 = 1x10-6 /0C1 = 0.18x10-6 /0C

2 = 27x10-6 /0C

11 = 22 = 33 = 1x10-10 F/m 11 = 22 = 33 = 1.5x10-8 F/m ---

d31 = d32 = -d24 = -d15

23x10-12 0C/N

d31 = d32 = -1.75x10-8 0C/N

d24 = d15 = 6x10-10 0C/N ---

p3 = -2.5x10-5 0C/K/m2 p3 = 7.5x10-4 0C/K/m2 ---

2

0a

Th

a/hAnalytical MF1

UC2 UC2 C3

5 1.457 1.457 1.50210 1.811 1.813 1.86915 1.898 1.899 1.95820 1.930 1.932 1.99225 1.946 1.947 2.00830 1.954 1.956 2.01635 1.960 1.961 2.02240 1.963 1.964 2.02560 1.969 1.970 2.03180 1.971 1.972 2.034

100 1.972 1.973 2.0351000 1.973 1.975 2.037

1MF FE Mixed Formulation2UC Uncoupled Piezoelectric Analysis3C Coupled Piezoelectric Analysis

T

a/hMF1

UC2 C3

5 4.208 -6.58410 5.475 -9.01015 5.799 -9.67520 5.922 -9.93125 5.981 -10.05530 6.013 -10.12335 6.033 -10.16540 6.045 -10.19260 6.069 -10.24280 6.077 -10.260

100 6.081 -10.2681000 6.088 -10.283

2

0a

Th

1MF FE Mixed Formulation2UC Uncoupled Piezoelectric Analysis3C Coupled Piezoelectric Analysis

T

Results have demonstrated the impact of piezoelectric coupling on the buckling load magnitudes by calculating the buckling loads that include the piezoelectric effect (coupled) and exclude the effects (uncoupled).

As would be expected, the relatively weak PVDF layers do not significantly alter the calculated results when considering piezoelectric coupling. The net increase is about 3% for the thermal loaded ten-layer laminate (PVDF/0/90/0/90)s.

However, adding the relatively stiff PZT as the top and bottom layers produces significant differences between the uncoupled and coupled results. A reversal of stress is required to cause buckling in the coupled analyses due to the sign on the pyroelectric constant for the PZT material. Neglecting the sign change, an increase of approximately 67% is observed in the absolute buckling load magnitude for the coupled analysis compared with the uncoupled analysis.

Looking into different stacking sequences – symmetric and anti-symmetric stacking

Looking into the effect of the piezoelectric thickness effect on buckling for the two cases above.

Six layer laminate: (PZT5/0/90)s

Simply supported

a = b = 0.2m

h = 0.001 m