Issue Section:
Discussion
Keywords:
composite materials, laminates, shear deformation
Topics:
Composite materials, Plates (structures), Shear deformation, Laminates
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=u0−zw,x+3z2h−2z3h3 Qxλx,v=v0−zw,y+3z2h−2z3h3 Qyλy
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¨0−I1w¨,x+2h3 I8ψ¨x+w¨,x
(3)
Nxy,x+Ny,y=I0v¨0−I1w¨,y+2h3 I8ψ¨y+w¨,y
(4)
Qx,x−4h2 Rx,x+Qy,y−4h2 Ry,y+43h2 Px,xx+2Pxy,xy+Py,yy+p=I0w¨+43h2 I3u¨0,x+v¨0,y+43h2 I4−4I63h2ψ¨x,x+w¨,xx+ψ¨y,y+w¨,yy−4I43h2 w¨,xx+w¨,yy
(5)
Mx−43h2 Px,x+Mxy−43h2 Pxy,y−Qx−4h2 Rx=2h3 I8u¨0−I9w¨,x+2h3 I7ψ¨x+w¨,x.
(6)
My−43h2 Py,y+Mxy−43h2 Pxy,x−Qy−4h2 Ry=2h3 I8u¨0−I9w¨,y+2h3 I7ψ¨y+w¨,y.
(7)
For Ray’s theory, the equations of motion are
Nx,x+Nxy,y=I0u¨0−I1w¨,x+I8 Q¨xλx
(8)
Nxy,x+Ny,y=I0v¨0−I1w¨,y+I8 Q¨yλy
(9)
Mx,xx+2Mxy,xy+My,yy+p=I0w¨+I1u¨0,x+v¨0,y−I2w¨,xx+w¨,yy+I9Q¨x,xλx+Q¨y,yλy
(10)
Mx−43h2 Px,x+Mxy−43h2 Pxy,y−Qx−4h2 Rx=2h3 I7λx Q¨x+I8u¨0−I9w¨,x
(11)
My−43h2 Py,y+Mxy−43h2 Pxy,x−Qy−4h2 Ry=2h3 I7λy Q¨y+I8v¨0−I9w¨,y
(12)
with I7=9I2/4h2−6I4/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^nsδψs−43h2 Pnδw,n+Q^xnx+Q^yny+43h2 {Px,x+Pxy,ynx+Py,y+Pxy,xny+Pns,s}−4I33h2 u¨0+I4−4I63h2ψ¨x+w¨,x−I4w¨,xnx−4I33h2 v¨0+I4−4I63h2ψ¨y+w¨,y−I4w¨,ynyδwds−∑i 43h2 ΔPnssiδwsi
(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=Qα−4h2 Rαα=x,y,M^β=Mβ−43h2 Pββ=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+3δQx2hλx Mx−43h2 Pxnx+Mxy−43h2 Pxyny+3δQy2hλy My−43h2 Pyny+Mxy−43h2 Pxynx−Mnδw,n−Mnsδw,s+Mx,x+Mxy,ynx+Mxy,x+My,yny+−I1u¨0+I2w¨,x−Q¨xλx I9nx+−I1v¨0+I2w¨,y−Q¨yλy I9nyδwds.
(15)
Substituting
32hλx δQx=δψx+δw,x=δψn+δw,nnx+δψs+δw,ssx
32hλy δQy=δψy+δw,y=δψn+δw,nny+δψs+δw,ssy
(16)
in Eq. (15) reduces it to
∫Nnδun+Nnsδus+M^nδψn+M^nsδψs−43h2 Pnδw,n+Pn,sδw,s+Mx,x+Mxy,xnx+Mxy,y+My,yny+−I1u¨0+I2w¨,x−Q¨xλx I9nx+−I1v¨0+I2w¨,y−Q¨yλy I9nyδwds.
(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.