Chapter 6
Figure 6.3: Distributions ¯v, ¯σy and ¯τyz for the first modes of square sandwich plate for C-C and C-F boundary conditions.
PZT-5A h
face 90º face 0º
face 90º face 0º core 0º PFRC 90º
PFRC 90º
Hybrid sandwich plate (b) Piezoelectric plate (a)
0.04h 0.04h
0.04h 0.04h 0.64h
0.1h 0.1h
Fig 6.2: Configurations of piezoelectric and hybrid plates
Figure 6.4: Configurations of piezoelectric plate (a) and hybrid sandwich plate (b).
The plate is subjected to the following load cases, 1. Pressure p2 =p0sin (πy/b) acting on top surface.
2. Actuation potentials ofϕ1=ϕ2 =ϕ0sin (πy/b) applied on both top and bottom surfaces.
The results are nondimensionalized as follows:
Load case 1.
(¯v,w,¯ ϕ) = 100(Sv, w,¯ 104ϕd0S2)Y0/p0hS4; (¯σx,σ¯y,τ¯yz,¯τzx) = (σx, σy, Sτyz, Sτzx)h/p0S2 ( ¯Dx,D¯y) = (Dx, Dy)/d0p0S
Load case 2.
(¯v,w) = (Sv, w)Y¯ 0/S2d0ϕ0; (¯σx,σ¯y,τ¯yz,τ¯zx) = (σx, σy, Sτyz, Sτzx)h/Y0d0ϕ0
where S = a/h, d0 = 374.0×10−12mV−1, Y0 = 10.3GP a for plate (a) and for plate (b) d0 = 100.0×10−12mV−1. The plates are designated in terms of their mechanical boundary conditions at the edges at y =∓b/2. For example, a plate which is clamped (C) at y/b =−0.5 and free (F) at y/b= 0.5, is called a C-F plate. The response of the piezoelectric plate (a) is obtained for the pressure load case, while the hybrid sandwich plate (b) is analyzed for both pressure and potential load cases.
The longitudinal variations of displacements, stresses, sensory potential and electrical dis- placements atz-locations where they are large, are plotted in Fig. 6.5 for the square thick (S = 5) piezoelectric plate under clamped-clamped (C-C) boundary conditions alongy-axis. The 3D FE (ABAQUS) results are also plotted for verification of the results obtained by 3D EKM. It is
Chapter 6
observed that the two-term EKM solution is in excellent agreement with 3D FE solution for all the response entities. The displacements are accurately predicted by the IZIGT over the range and are in good agreement with 3D EKM. The error in the value of ¯σy is more than that of the
¯
σx near the clamped edge. Similar trend is observed for shear stresses, in which ¯τyz is the least accurate near the edge. Electric displacements ¯Dx and ¯Dy and electric potential ϕ is also not accurately predicted by IZIGT near the clamped edge. It has been found that for the simply supported case, all the entities are accurately predicted by IZIGT and ITOT.
Fig.4
CC OC
OC OC
posin(m 2)
-
Figure 6.5: Longitudinal variations of displacements, stresses, electric potential and electric displacements for square piezoelectric plate (a) under pressure loading.
Figure 6.6 shows the longitudinal variation of ¯w,σ¯y,τ¯yz and electric displacement ¯Dzunder pressure loading for sandwich plate (b) with clamped-free (C-F) and clamped-simply supported (C-S) boundary conditions. Transverse deflection ¯wis accurately predicted by IZIGT for both C-F and C-S cases whereas ITOT under predicts the deflection with large error. In-plane normal stress ¯σy is predicted very accurately by the IZIGT as compared to the ITOT in the interior
part of the plate but near the clamped support, prediction is poor by both the theories. Both the theories give inaccurate results for the transverse shear stress and electric displacement near the clamped-supports.
Longitudinal variations of deflection, stresses and electric displacement under potential loading for moderately thick sandwich plate with C-F and C-S boundary conditions are pre- sented in Fig. 6.6. For C-F case, IZIGT produces accurate results for whole range except at the boundaries (clamped and free supports) whereas ITOT gives erroneous prediction of stresses even in the interior part of the plate. For C-S case, IZIGT over predict the deflection whereas ITOT under predict it. A mismatch in normal stress value is observed by both the theories in the interior as well as at boundaries of the plates. Both the theories fail to satisfy the boundary conditions at the edge (i.e. zero stress at free and simply supported edge).
Longitudinal variations of ¯v,w,¯ σ¯x,σ¯y,τ¯zx and ¯τyz under potential loading with free-free (F-F) support conditions is plotted in Fig. 6.7 for moderately thick sandwich plate. Transverse displacement is very accurately predicted by both IZIGT and ITOT whereas a slight mismatch is observed for in-plane displacement near the free support. The results of IZIGT are in close agreement with 3D EKM for both in-plane normal and transverse shear stresses. However, ITOT results are erroneous even in the interior part of the plate specifically for ¯σx. Transverse shear stresses are accurately predicted both by IZIGT and ITOT except at the edge. For transverse shear ¯τzxat the free edge, both IZIGT and ITOT under predict almost four times less than the exact value predicted by 3D EKM. This high value of free edge stresses which cause delamination in the structures are not predicted by these two-dimensional theories.
Through-thickness variations of transverse shear stresses τzx and τyz at three location of y/b(0.025,0.05, and 0.1) are plotted in Fig. 6.8. It is observed that IZIGT produces more close predictions of the stresses as compared to ITOT. However, both the theories produce erroneous results at the piezo-interface and through the core in the sandwich.
Chapter 6
Fig.8
Figure 6.6: Longitudinal variation of deflection, stresses and electric displacement under pres- sure loading for sandwich plate (b) under top and bottom close circuit conditions
Fig.9
Figure 6.7: Longitudinal variation of deflection, stresses and electric displacements under potential loading for sandwich plate (b) under top and bottom close circuit conditions
Chapter 6
Fig.10
Figure 6.8: Longitudinal variation of displacements and stresses under potential loading for sandwich plate (b) with F-F boundary conditions under top and bottom close circuit conditions
Fig.12
IZIGT
ITOT
IZIGT
ITOT
Figure 6.9: Through-thickness variation of transverse shear stresses under potential loading for sandwich plate (b) with free-free boundary conditions under top and bottom close circuit conditions
Chapter 6