# Self-Strained Piezothermoelastic Compo- site Beam Analysis Using First-Order Shear Deformation Theory*

```Self-Strained Piezothermoelastic Composite Beam Analysis Using First-Order
Shear Deformation Theory*
G.E. Blandford, T.R. Tauchert and Y. Du
University of Kentucky
Lexington, KY
ICCE/5 Las Vegas
July 5-11, 1998
_________
*To appear: Composites Part B Engineering Journal
Assumptions
 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 beam depth
 Displacement assumptions consistent with Mindlin theory
Finite Element Equations
Coupled Equations
[K ]{a u }  [K c ]{a  }  {FP }  {Ft }  {F }
[K c ]T {a u }  [K  ]{a  }   {Fq }  {P }
Uncoupled Equations
[K ]{a u }  {FP }  {Ft }  {F }  [K c ]{a  }
[K] = Stiffness matrix
[K c ] = Coupling matrix
[ K  ] = Permittivity matrix
{FP } = Nodal concentrated load vector
{Ft } = Traction load vector
{F } = Thermal load vector
{Fq } = Electric charge load vector
{P } = Pyroelectric load vector
{a u } = Nodal displacement vector
{a  } = Nodal electric potential vector
Sample Problems
Material Property Data
PVDF
PZT
Graphite-Epoxy
k 1 = k2 = k3 = k0
k 1 = k2 = k3 = k0
k1 = 100k0, k2 = k3 = k0
E1 = E2 = E3 = E0
E1 = E2 = E3 = 30 E0
E1 = 90E0, E2 = E3 = 5E0
12 = 13 = 23 = 0
12 = 13 = 23 = 0
12 = 13 = 23 = 0
G12 = G13 = G23 = 0.375E0
G12 = G13 = G23 = 11.25E0
G12 = G13 = 4E0, G23 = 1.5E0
1 = 2 = 3 = 0
1 = 2 = 3 = 0.01 0
1= 0.00020, 2= 3 = 0.20
11 = 22 = 33 = 0
11 = 22 = 33 = 150 0
---
d31 = d32 = -d24 = -d15 = d0
d31 = d32 = -7d0, d24 = d15 =
---
d33 = -1.2d0
24d0, d33 = 14d0
p 3 = p0
p3 = -30p0
---
Problem 1
 Simply Supported Beam
 Orthotropic Layers 2 and 4 – Graphite Epoxy
 Piezoelectric Layers 1, 3, 5 – PVDF or PZT
Spacially Varying Electro-Magnetic Potential Gradient
1( x )   0 sin (x / L)
w *mid  plane
Layup
L/h
Ref. 1 - U
FEM - U FEM - C
*xx max
Ref. 1 - U
FEM - U FEM - C
Layer, Position
PVDF/90/
Isotropic/
90/PVDF
40
-0.3729 -0.3717 -0.3704
8.359
8.417
8.407 2, t (1, U &amp; C)
5
-0.3870 -0.3695 -0.3682
8.377
8.417
8.407 2, t (1, U &amp; C)
PZT/90/
Isotropic/
90/PZT
40
5.5460
5.4423 5.1769
-661.3
-661.3 -675.6 3, t (1, U &amp; C)
5
5.4609
5.2687 5.0103
-602.2
-661.3 -676.0 3, t (1, U &amp; C)
PVDF/90/
Isotropic/
0/PVDF
40
-0.2761 -0.2760 -0.2758
24.824
25.068 25.066 4, t (1, U &amp; C)
5
-0.2787 -0.2747 -0.2744
24.204
25.068 25.066 4, t (1, U &amp; C)
PZT/90/
Isotropic/
0/PZT
40
5.6510
5.5588 5.3181
-656.9
-658.6 -710.3 4, t (1, U &amp; C)
5
5.3630
5.3963 5.1660
-571.8
-658.6 -711.0 3, t (1); 4, t (U&amp;C)
Spacially Varying Temperature Gradient
1 ( x )  0 sin (x / L)
w *mid  plane
Layup
L/h Ref. 1 - U FEM - U FEM - C
*xx max
Ref. 1 - U FEM - U FEM - C
Layer, Position
PVDF/90/
Isotropic/
90/PVDF
40
-0.0732
-0.0721
-0.0659
1.503
1.503
1.350
2, t (1, U &amp; C)
5
-0.0739
-0.0721
-0.0659
1.480
1.503
1.352
2, t (1, U &amp; C)
PZT/90/
Isotropic/
90/PZT
40 -0.00206 -0.00204 0.00957
-0.698
-0.699
1.790 2,t (1 &amp; U); 1,b (C)
5 -0.00183 -0.00204 0.00915
-0.687
-0.699
1.768 2,t (1 &amp; U); 1,b (C)
PVDF/90/
Isotropic/
0/PVDF
40
-0.0742
-0.0738
-0.0689
7.461
7.525
7.064
4, t (1, U &amp; C)
5
-0.0675
-0.0738
-0.0689
6.150
7.525
7.070
4, t (1, U &amp; C)
PZT/90/
Isotropic/
0/PZT
40 -0.00294 -0.00291 0.00978
-0.709
-0.711
1.858 2,t (1 &amp; U); 1,b (C)
5 -0.00247 -0.00291 0.00938
-0.649
-0.711
1.834 2,t (1 &amp; U); 1,b (C)
Problem 2
 Simply Supported Beam, L/h = 20
 T = Total Thickness of PZT Layers
 1 ( x )  0 sin (x / L)
CONCLUSIONS
Five-Layer Hybrid Laminate
 Finite element displacement and lamina stress results compare
favorably with the uncoupled analytical elasticity solutions of
Tauchert for moderately thick laminates.
 Coupled/Uncoupled analyses nearly identical using PVDF
laminae for the electric flux load case.
 Coupled analyses predict a reduction in displacement and
stress magnitudes using PVDF laminae for the temperature
 Coupled analyses showed that the hybrid laminates with PZT
top and bottom surface laminae resulted in a slight reduction
in displacements but slightly increased maximum lamina
 Hybrid laminates with PZT laminae subjected to the temperature gradient loading produced self-strain deformations due to
the pyroelectric effect that exceeded the temperature load deformations, which resulted in a change in the maximum stress
location compared to the uncoupled analyses.
 Thick beams (length-to-depth ratios of five) analytical elasticity and FE displacement results did not agree, as should be
expected since first-order shear deformation theory is not applicable.
 Uncoupled analytical and finite element stress results through
the thickness of the beam do compare favorably provided the
same through thickness temperature distribution is used.
Three-Layer Hybrid Laminate
 Results show that the direct piezoelectric effect has an excess
capacity to eliminate the bending displacements caused by the
temperature gradient for this particular beam configuration.
 Excess capacity occurred for all PZT thickness to laminate
thickness ratios, but particularly for T/h = 0.15 – 0.75 with a
maximum at T/h = 0.40.
```