ISSN: 2277-9655
Impact Factor: 4.116
CODEN: IJESS7
[Yagnasri* et al., 6(3): March, 2017]
IC™ Value: 3.00
IJESRT
INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH
TECHNOLOGY
ZERO DELAMINATION OF SMART COMPOSITE LAMINATED ANGLE PLY
USING HSDT WITH ZIG ZAG FUNCTION
*
P.Yagnasri*
Department of mechanical engineering, Assistant Professor Gurunanak institutions technical
campus, Ibrahimpatnam
DOI: 10.5281/zenodo.376540
ABSTRACT
Thermal analysis of smart composite laminated plate using higher order theory with zig-zag functions. To develop
the analytical procedure and to investigate the thermal characteristics, the material is considered to be orthotropic
under thermal load based on higher order displacement model with zig-zag function, without enforcing zero
transverse shear stress on the top and bottom faces of the laminated plates. The related functions and derivations
for equation of motion are obtained using the dynamic version of the principle of virtual work or Hamilton’s
principle. The solutions are obtained by using Navier’s and numerical methods for anti-symmetric angle-ply with
specific type of simply supported boundary conditions SS. Computer programs have been developed to find the
stresses and deflections for various aspect ratios, side thickness ratios (a/h) and voltages.
KEYWORDS: C++, Hamilton’s principle, HSDT-higher order shear deformation theory, Navier’s Strokes
equation, SS- simply supported.
INTRODUCTION
In this work actuator is coupled to the top of the laminated composite material plates to achieve its thermal
characteristics, which are tabulated in non-dimensional form of various aspect ratios, side thickness ratios
(a/h) and voltages, and also to evaluate electric potential function by solving higher order differential equation
satisfying electric boundary conditions along the thickness direction of piezoelectric layer. These solutions
are plotted as a function of aspect ratio, thickness ratio (z/h) and voltages etc. The effects of shear deformation,
coupling, degree of orthotropy and voltage on the response of smart composite laminated plates are
investigated.
FORMULATION OF HSDT WITH ZIG- ZAG FUNCTION
A laminated plate of 0  x  a; 0  y  b and -h/2 z h/2 is considered.
The displacement components u (x, y, z, t), v (x, y, z, t) and w (x, y, z, t) at any point in the plate are expanded in
terms of the thickness coordinate. In this work the in plane displacements are expanded as cubic functions of the
thickness coordinate. The displacement field which assumes w (x, y, z, and t) constant through the plate thickness
is expressed as:
u ( x, y, z )  u o ( x, y) z x ( x, y) z 2 u o* ( x, y) z 3 x* ( x, y)   k s1 ( x, y)
v( x, y, z )  vo ( x, y) z y ( x, y) z 2 vo* ( x, y) z 3 y* ( x, y)   k s 2 ( x, y)
w( x, y, z )  wo ( x, y)





……….. 1
Where
uo, vo, wo , s1 and s2 denote the displacements of a point (x, y) at the midplane.
x, y are rotations about y and x–axes at midplane.
uo*, vo*, x*, y* are the corresponding higher–order deformation terms defined at the midplane.
θk is the Zig-Zag function, defined as:
http: // www.ijesrt.com
© International Journal of Engineering Sciences & Research Technology
[149]
ISSN: 2277-9655
Impact Factor: 4.116
CODEN: IJESS7
[Yagnasri* et al., 6(3): March, 2017]
IC™ Value: 3.00
Zk
hk
zk is the local transverse coordinate with its origin at the centre of the kth layer and hk is the corresponding layer
thickness.
 k  2(1)k
The Zig-Zag function is piecewise linear with values of –1 and 1 alternately at the different interfaces.
The strain components are:
3 *
 x   x 0  zk sx  z 2 *
x0  z k x
3 *
 y   y 0  zk sy  z 2 *
y0  z k y

z0
 xy  
xy0
 zk sxy  z 2  *xy0  z 3 k *
xy
 yz   sy  z yz 0  z 2 *y
 xz   sx  z xz0  z 2 x*
…….2
LAMINATE CONSTITUTIVE EQUATIONS
   C    eE
…… 3
As per equation 3
 (x, y, z) is the electro static potential
http: // www.ijesrt.com
© International Journal of Engineering Sciences & Research Technology
[150]
ISSN: 2277-9655
Impact Factor: 4.116
CODEN: IJESS7
[Yagnasri* et al., 6(3): March, 2017]
IC™ Value: 3.00
The superscript t denotes the transpose of a matrix.
Q11 = C11 C4 + 2 (C12 + 2 C33) S2 C2 + C22 S
Q12 = C12 (C4 + S4) + (C11 + C22 - 4 C33) S2 C2

Q13 = (C11 - C12 - 2 C33) S C3 + (C12 - C22 + 2 C33) C S3

Q22 = C11 S4 + C22 C4 + (2 C12 + 4 C33) S2 C2

Q23 = (C11 – C12 – 2 C33) S3C + (C12 – C22 +2 C33)C3 S

Q33 = (C11 – 2 C12 + C22 – 2 C33) S2 C2 + C33 (C4+S4)

Q44 = C44 C2 + C55 S2

Q45 = (C55 – C44) C S

Q55 = (C44 S2 + C55 C2)
x = 1 C2 + 2 S2, y = 2 C2 + 1 S2, xy= (1 - 2) SC
…… 5
The stress strain relationship in global X – Y – Z coordinate is written as
 x  Q11 Q12 Q13 0 0   x
  Q Q Q
 
0
0
y
21
22
23
  
  y
 xy   Q31 Q32 Q33 0 0   xy
   0 0 0 Q Q  
44
45 
 yz  
 yz
 xz   0 0 0 Q54 Q55  
 xz









……. 6
The governing equations of displacement model are derived using the principle of virtual work or Hamilton’s
principle.
T

(U  V  K ) dt  0
0
……
7
The virtual work statement shown in Eq (7), integrating through the thickness of laminate, the in-plane and
transverse force and moment resultant relations in the form of matrix obtained as:
http: // www.ijesrt.com
© International Journal of Engineering Sciences & Research Technology
[151]
ISSN: 2277-9655
Impact Factor: 4.116
CODEN: IJESS7
[Yagnasri* et al., 6(3): March, 2017]
IC™ Value: 3.00
T
PZ
 0   N   N 
N 
*


 N PZ 
 * 
 *
*T

N
N




0
 
 
       PZ 
  
   A | B | 0       M*PZ 
T
M   t
 K s  M  M 
 *    B | Db | 0   *      

M   0 | 0 | D  K  M *T    
s
   
    Q PZ 
 
     

Q 
s   0  Q*PZ 
Q* 
 *    

 
   0  

……. 8
Using Hamilton’s principle for total potential energy and by Equating the coefficients of each of virtual
displacements u0 ,v0, w0, x, y, u0*, v0*, x*, y*, s1, s2 to zero, the following equations of motion are
obtained:
N x N xy
 0  I 2 ( x  Rs
1 )  I 3 u
 0 *  I 4  x *

 I1u
x
y
N y N xy
v 0 :

 I1v 0  I 2 ( y  Rs 2 )  I 3 v 0 *  I 4  y *
y
x
Q x Q y
 0
w 0 :

 q  I1w
x
y
M xy
M
 0  I 3 ( x  Rs1 )  I 4 u
 0 *  I5  x *
x : x 
 Q x  I2u
x
y
M y M xy
y :

 Q y  I 2 v 0  I 3 ( y  Rs 2 )  I 4 v 0 *  I5  y *
y
x
u0 :
*
N x * N xy
 0  I 4 ( x  Rs1 )  I5 u
 0 *  I 6  x *
u0 :

 I3 u
x
y
*
*
v 0 :
x * :
N y *
y

N xy *
x
 I 3 v 0  I 4 ( y  Rs2 )  I5 v 0 *  I 6  y *
*
M x * M xy
 0  I5 ( x  Rs1 )  I6 u
 0 *  I7  x *

 Qx *  I4u
x
y
M y * M xy *
y :

 Q y *  I 4 v 0  I5 ( y  Rs2 )  I6 v 0 *  I7  y *
y
x
RMx RMxy
 0  I3 R( x  Rs1 )  I 4 Ru
 0 *  I5 R  x *
s1 :

 R Q x  I2Ru
x
y
RM y RMxy
s 2 :

 R Q y  I2Rv 0  I3 R( x  Rs2 )  I 4 Rv 0 *  I5 R  y *
y
x
*
…….9
The Navier’s solutions of simply supported anti symmetric angle ply laminated plates.
The SS boundary conditions for the anti-symmetric angle ply laminated plates are:
http: // www.ijesrt.com
© International Journal of Engineering Sciences & Research Technology
[152]
ISSN: 2277-9655
Impact Factor: 4.116
CODEN: IJESS7
[Yagnasri* et al., 6(3): March, 2017]
IC™ Value: 3.00
At edges x = 0 and x = a
u0=0,wo=0, y=0, Nxy=0, Mx=0, u0*=0, y*=0, Mx*=0, Nxy*=0, S1 =0
… 10.1
At edges y = 0 and y = b
v0=0,wo=0, x=0, Nxy=0, My =0, v0*=0, x*=0, My*=0, Nxy* =0, S2 =0
10.2
....
The displacements at the mid plane will be defined to satisfy the boundary conditions in Eq. (10). These
displacements will be substituted in governing equations to obtain the equations in terms of A, B, D parameters.
The obtained equations will be solved to find the behavior of the laminated composite plates.
RESULTS AND DISCUSSIONS
The material properties used for each orthotropic layer are
Graphite Epoxy:
E1/E2= 2.5
G12/E2= 0.5
G23/E2= 0.2 E2=E3= 102 N/ m2 μ12= μ23=μ13=0.25
𝛼1 =1 × 10-6 0𝐶 −1 , 𝛼2 =1.125 × 10-3 0𝐶 −1 , 𝛼3 =1.125 × 10-3 0𝐶 −1
PFRC Layer:
C11 = 32.6Gpa, C12 = C21 = 4.3Gpa, C13 = C31 = 4.76Gpa, C22 = C23 =7.2Gpa, C23 = 3.85Gpa, C44 =1.05Gpa, C55 =
C66 =1.29Gpa, e31= -6.76C/m2, g11=g22 = 0.037E-9C/V m, g33=10.64E - 9 C/V m
The center deflections and stresses are presented here in non-dimensional form using the following multipliers:
𝑚6 =𝑒2 𝛼1 𝑡0 , 𝑚7 =
𝑎12 𝑡 𝛼1 𝑡0
𝑡2
The variation of non-dimensionalized transverse deflection (w) against side thickness ratios as a function of
number of layers at voltage 100V for a simply supported angle ply smart composite laminated plates is showed in
figure1. From the figure 1 it is noted that as the side to thickness ratio is increasing the transverse deflection is
decreasing and is observed maximum for 2 layers with an error of 1.2 percent. If the coupling coefficients increase
in magnitude, hence the effect of coupling increases with the increase in modulus ratio for transverse deflections
(w). Figure 2 shows the variation of non-dimensionalized normal stress (𝜎𝑥 ) against thickness coordinates as a
function of number of layers at voltage 100V for a simply supported angle ply smart composite laminated plates.
From plot it is noted that, normal stress is observed maximum for 2 layers with 0.6 pecent error and the stresses
are tending to be maximum for smaller thickness coordinate ratios. Figure 3 shows the variation of nondimensionalized transverse shear stress (𝜏𝑥𝑧 ) against side thickness ratios as a function of number of layers at
voltage 100V for a simply supported angle ply smart composite laminated plates. From plot 3 it is observed,
transverse shear stress is maximum for 2 layers with 1.2 percent as error. As side to thickness ratio is increasing
transverse shear stress is gradually decreasing.
http: // www.ijesrt.com
© International Journal of Engineering Sciences & Research Technology
[153]
ISSN: 2277-9655
Impact Factor: 4.116
CODEN: IJESS7
[Yagnasri* et al., 6(3): March, 2017]
IC™ Value: 3.00
CONCLUSION
Analytical procedure is developed in this paper for thermal analysis of smart composite laminated plates subjected
to electromechanical loading. The non-dimensional transverse displacement and shear stresses are obtained for
various voltages, aspect ratios and thickness coordinates. It is concluded from the results that, the transverse
http: // www.ijesrt.com
© International Journal of Engineering Sciences & Research Technology
[154]
ISSN: 2277-9655
Impact Factor: 4.116
CODEN: IJESS7
[Yagnasri* et al., 6(3): March, 2017]
IC™ Value: 3.00
displacement varies through the thickness non linearly for plate subjected to temperature gradient than for those
subjected to mechanical loads. The zig-zag function is a valuable tool to enhance the performance of both classical
and advanced theories.
REFERENCES
[1] SurotThangjitham and Hyung Jip Choi [1991] Multi-layered anisotropic medium under the state of
generalized plane deformation for thermal stress analysis. J. Appl. Mech...(58)1021-1027.
[2] Theodore R Tauchert [1991] Response of flat plates under thermal loading.J. Appl. Mech. (44)347-360.
[3] Tungikar V.B and Koganti M.Rao [1994] Three dimensional exact solutions for temperature distribution
and thermal stresses in simply supported finite rectangular orthotropic laminates .Composite Structures
(27)419-430.
[4] Savoia. M and J.N.Reddy [1995] Stress analysis of multi layered plates subjected to thermal and
mechanical loading.Int.J.of Solid Structures (32) 592-608.
[5] Klaus Rohwer, RaimundRolfes and Holger Sparr [2001] Higher-order theories for thermal stresses in
layered plates Int J Solids and Structures. (38) 3673-3687.
[6] T.Kant and S.Swaminathan [2002] Analytical solution for static analysis of laminated composite and
sandwich plates based on higher order refined theory. Composite structures (56) 329-344.
[7] HemendraArya, R.P.Shimpi, and N.K.Naik [2002] A zig-zag model for laminated composite beam.
Composite Structures (56) 21-24.
http: // www.ijesrt.com
© International Journal of Engineering Sciences & Research Technology
[155]