It is the contention of the authors that the “zeroth-order” shear deformation theory presented by Ray 1 is mathematically equivalent to Reddy’s third order theory 2. The notation of Ref. 1 is used herein. Ray’s approximations for the in-plane displacements  
u=u0zw,x+3z2h2z3h3Qxλx,v=v0zw,y+3z2h2z3h3Qyλy
are identical to those of the Reddy’s theory, 2, with  
ψx+w,x=3Qx2hλx,ψy+w,y=3Qy2hλy.
(2)
Hence the equations of motion and boundary conditions of Ray’s theory are mathematically equivalent to those of the dynamic version of Reddy’s theory. This is established explicitly by comparing the governing equations of the two theories.
For Reddy’s theory, the equations of motion are  
Nx,x+Nxy,y=I0u¨0I1w¨,x+2h3I8ψ¨x+w¨,x
(3)
 
Nxy,x+Ny,y=I0v¨0I1w¨,y+2h3I8ψ¨y+w¨,y
(4)
 
Qx,x4h2Rx,x+Qy,y4h2Ry,y+43h2Px,xx+2Pxy,xy+Py,yy+p=I0w¨+43h2I3u¨0,x+v¨0,y+43h2I44I63h2ψ¨x,x+w¨,xx+ψ¨y,y+w¨,yy4I43h2w¨,xx+w¨,yy
(5)
 
Mx43h2Px,x+Mxy43h2Pxy,yQx4h2Rx=2h3I8u¨0I9w¨,x+2h3I7ψ¨x+w¨,x.
(6)
 
My43h2Py,y+Mxy43h2Pxy,xQy4h2Ry=2h3I8u¨0I9w¨,y+2h3I7ψ¨y+w¨,y.
(7)
For Ray’s theory, the equations of motion are  
Nx,x+Nxy,y=I0u¨0I1w¨,x+I8Q¨xλx
(8)
 
Nxy,x+Ny,y=I0v¨0I1w¨,y+I8Q¨yλy
(9)
 
Mx,xx+2Mxy,xy+My,yy+p=I0w¨+I1u¨0,x+v¨0,yI2w¨,xx+w¨,yy+I9Q¨x,xλx+Q¨y,yλy
(10)
 
Mx43h2Px,x+Mxy43h2Pxy,yQx4h2Rx=2h3I7λxQ¨x+I8u¨0I9w¨,x
(11)
 
My43h2Py,y+Mxy43h2Pxy,xQy4h2Ry=2h3I7λyQ¨y+I8v¨0I9w¨,y
(12)
with I7=9I2/4h26I4/h4+4I6/h6. Using Eq. (2), it is observed that Eqs. (8), (9), (11), (12) are identical to Eqs. (3), (4), (6), (7). Forming the combination Eq. (10)–Eq. 11 yields Eq. (5).
For Reddy’s theory, the boundary conditions are obtained from the following boundary integral formed after using Green’s theorem in Hamilton’s principle  
Nnδun+Nnsδus+M^nδψn+M^nsδψs43h2Pnδw,n+Q^xnx+Q^yny+43h2{Px,x+Pxy,ynx+Py,y+Pxy,xny+Pns,s}4I33h2u¨0+I44I63h2ψ¨x+w¨,xI4w¨,xnx4I33h2v¨0+I44I63h2ψ¨y+w¨,yI4w¨,ynyδwdsi43h2ΔPnssiδwsi
(13)
where si are locations of plate corners and  
un=u0nx+v0ny,us=u0sx+v0sy
 
Nn=Nxnx2+Nyny2+2Nxynxny,
 
Nns=Nxnxsy+Nynysy+Nxynxsy+nysx
 
Q^α=Qα4h2Rαα=x,y,M^β=Mβ43h2Pββ=x,y,xy
(14)
with sx=ny,sy=nx. The expressions of M^n,M^ns;Mn,Mns;Pn,Pns are similar to those of Nn,Nns, and of ψn,ψs;w,n,w,s are similar to those of un,us. For Ray’s theory the corresponding boundary integral is  
Nnδun+Nns+δus+3δQx2hλxMx43h2Pxnx+Mxy43h2Pxyny+3δQy2hλyMy43h2Pyny+Mxy43h2PxynxMnδw,nMnsδw,s+Mx,x+Mxy,ynx+Mxy,x+My,yny+I1u¨0+I2w¨,xQ¨xλxI9nx+I1v¨0+I2w¨,yQ¨yλyI9nyδwds.
(15)
Substituting  
32hλxδQx=δψx+δw,x=δψn+δw,nnx+δψs+δw,ssx
 
32hλyδQy=δψy+δw,y=δψn+δw,nny+δψs+δw,ssy
(16)
in Eq. (15) reduces it to  
Nnδun+Nnsδus+M^nδψn+M^nsδψs43h2Pnδw,n+Pn,sδw,s+Mx,x+Mxy,xnx+Mxy,y+My,yny+I1u¨0+I2w¨,xQ¨xλxI9nx+I1v¨0+I2w¨,yQ¨yλyI9nyδwds.
(17)
Substituting the expressions of Mx,x+Mxy,y and My,y+Mxy,x from equations of motion (11) and (12) into Eq. (17) and using Eq. (2), reduces it to exactly the same expression as in Eq. (13). Hence Ray’s theory is not a new theory since its equations of motion and boundary conditions are mathematically equivalent to those of Reddy’s theory. The results of this theory for any boundary conditions will be identical to those of Reddy’s theory. The statics results of Ray’s theory in Table 1 agree with Reddy’s results, 2. The difference in Table 6 from Reddy’s results is due to neglect of some inertia terms by Ray while obtaining Navier’s solution. Ray’s theory is not a zeroth-order theory but Reddy’s third order theory in disguise. Moreover, the displacement approximation of Ray’s theory is valid only for the case of cross-ply and antisymmetric angle-ply laminates since for the general lay-up, the given expressions of λx,λy would not be valid.
1.
Ray
,
M. C.
,
2003
, “
Zeroth-Order Shear Deformation Theory for Laminated Composite Plates
,”
ASME J. Appl. Mech.
,
70
, pp.
374
380
.
2.
Reddy, J. N., 1997, Mechanics of Laminated Composite Plates Theory and Analysis, CRC Press, Boca Raton, FL.