Discussion: “Zeroth-Order Shear Deformation Theory for Laminated Composite Plates” (Ray, M. C., 2003 ASME J. Appl. Mech., 70, pp. 374–380)

Kapuria , S.; Dumir , P. C.

doi:10.1115/1.1769415

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 = u_{0} - {zw}_{, x} + (\frac{3 z}{2 h} - \frac{2 z^{3}}{h^{3}}) \frac{Q_{x}}{λ_{x}}, v = v_{0} - {zw}_{, y} + (\frac{3 z}{2 h} - \frac{2 z^{3}}{h^{3}}) \frac{Q_{y}}{λ_{y}}

are identical to those of the Reddy’s theory, 2, with

ψ_{x} + w_{, x} = \frac{3 Q_{x}}{2 h λ_{x}}, ψ_{y} + w_{, y} = \frac{3 Q_{y}}{2 h λ_{y}} .

(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

N_{x, x} + N_{xy, y} = I_{0} u ¨_{0} - I_{1} w ¨_{, x} + \frac{2 h}{3} I_{8} (ψ ¨_{x} + w ¨_{, x})

(3)

N_{xy, x} + N_{y, y} = I_{0} v ¨_{0} - I_{1} w ¨_{, y} + \frac{2 h}{3} I_{8} (ψ ¨_{y} + w ¨_{, y})

(4)

Q_{x, x} - \frac{4}{h^{2}} R_{x, x} + Q_{y, y} - \frac{4}{h^{2}} R_{y, y} + \frac{4}{3 h^{2}} (P_{x, xx} + 2 P_{xy, xy} + P_{y, yy}) + p = I_{0} w ¨ + \frac{4}{3 h^{2}} I_{3} (u ¨_{0, x} + v ¨_{0, y}) + \frac{4}{3 h^{2}} (I_{4} - \frac{4 I_{6}}{3 h^{2}}) (ψ ¨_{x, x} + w ¨_{, xx} + ψ ¨_{y, y} + w ¨_{, yy}) - \frac{4 I_{4}}{3 h^{2}} (w ¨_{, xx} + w ¨_{, yy})

(5)

{(M_{x} - \frac{4}{3 h^{2}} P_{x})}_{, x} + {(M_{xy} - \frac{4}{3 h^{2}} P_{xy})}_{, y} - (Q_{x} - \frac{4}{h^{2}} R_{x}) = \frac{2 h}{3} [I_{8} u ¨_{0} - I_{9} w ¨_{, x} + \frac{2 h}{3} I_{7} (ψ ¨_{x} + w ¨_{, x})] .

(6)

{(M_{y} - \frac{4}{3 h^{2}} P_{y})}_{, y} + {(M_{xy} - \frac{4}{3 h^{2}} P_{xy})}_{, x} - (Q_{y} - \frac{4}{h^{2}} R_{y}) = \frac{2 h}{3} [I_{8} u ¨_{0} - I_{9} w ¨_{, y} + \frac{2 h}{3} I_{7} (ψ ¨_{y} + w ¨_{, y})] .

(7)

For Ray’s theory, the equations of motion are

N_{x, x} + N_{xy, y} = I_{0} u ¨_{0} - I_{1} w ¨_{, x} + I_{8} \frac{Q ¨_{x}}{λ_{x}}

(8)

N_{xy, x} + N_{y, y} = I_{0} v ¨_{0} - I_{1} w ¨_{, y} + I_{8} \frac{Q ¨_{y}}{λ_{y}}

(9)

M_{x, xx} + 2 M_{xy, xy} + M_{y, yy} + p = I_{0} w ¨ + I_{1} (u ¨_{0, x} + v ¨_{0, y}) - I_{2} (w ¨_{, xx} + w ¨_{, yy}) + I_{9} (\frac{Q ¨_{x, x}}{λ_{x}} + \frac{Q ¨_{y, y}}{λ_{y}})

(10)

{(M_{x} - \frac{4}{3 h^{2}} P_{x})}_{, x} + {(M_{xy} - \frac{4}{3 h^{2}} P_{xy})}_{, y} - (Q_{x} - \frac{4}{h^{2}} R_{x}) = \frac{2 h}{3} [\frac{I_{7}}{λ_{x}} Q ¨_{x} + I_{8} u ¨_{0} - I_{9} w ¨_{, x}]

(11)

{(M_{y} - \frac{4}{3 h^{2}} P_{y})}_{, y} + {(M_{xy} - \frac{4}{3 h^{2}} P_{xy})}_{, x} - (Q_{y} - \frac{4}{h^{2}} R_{y}) = \frac{2 h}{3} [\frac{I_{7}}{λ_{y}} Q ¨_{y} + I_{8} v ¨_{0} - I_{9} w ¨_{, y}]

(12)

with

I_{7} = 9 I_{2} / 4 h^{2} - 6 I_{4} / h^{4} + 4 I_{6} / h^{6} .

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

\int [N_{n} δ u_{n} + N_{ns} δ u_{s} + M^_{n} δ ψ_{n} + M^_{ns} δ ψ_{s} - \frac{4}{3 h^{2}} P_{n} δ w_{, n} + [Q^_{x} n_{x} + Q^_{y} n_{y} + \frac{4}{3 h^{2}} {(P_{x, x} + P_{xy, y}) n_{x} + (P_{y, y} + P_{xy, x}) n_{y} + P_{ns, s}} - \frac{4 I_{3}}{3 h^{2}} \{u ¨_{0} + (I_{4} - \frac{4 I_{6}}{3 h^{2}}) (ψ ¨_{x} + w ¨_{, x}) - I_{4} w ¨_{, x}\} n_{x} - \frac{4 I_{3}}{3 h^{2}} \{v ¨_{0} + (I_{4} - \frac{4 I_{6}}{3 h^{2}}) (ψ ¨_{y} + w ¨_{, y}) - I_{4} w ¨_{, y}\} n_{y}] δ w] ds - \sum_{i} \frac{4}{3 h^{2}} Δ P_{ns} (s_{i}) δ w (s_{i})

(13)

where

s_{i}

are locations of plate corners and

u_{n} = u_{0} n_{x} + v_{0} n_{y}, u_{s} = u_{0} s_{x} + v_{0} s_{y}

N_{n} = N_{x} {n_{x}}^{2} + N_{y} {n_{y}}^{2} + 2 N_{xy} n_{x} n_{y},

N_{ns} = N_{x} n_{x} s_{y} + N_{y} n_{y} s_{y} + N_{xy} (n_{x} s_{y} + n_{y} s_{x})

Q^_{α} = Q_{α} - \frac{4}{h^{2}} R_{α} (α = x, y), M^_{β} = M_{β} - \frac{4}{3 h^{2}} P_{β} (β = x, y, xy)

(14)

with

s_{x} = - n_{y}, s_{y} = n_{x} .

The expressions of

M^_{n}, M^_{ns};

M_{n}, M_{ns};

P_{n}, P_{ns}

are similar to those of

N_{n},

N_{ns},

and of

ψ_{n}, ψ_{s};

w_{, n}, w_{, s}

are similar to those of

u_{n}, u_{s} .

For Ray’s theory the corresponding boundary integral is

\int [N_{n} δ u_{n} + N_{ns} + δ u_{s} + \frac{3 δ Q_{x}}{2 h λ_{x}} \{(M_{x} - \frac{4}{3 h^{2}} P_{x}) n_{x} + (M_{xy} - \frac{4}{3 h^{2}} P_{xy}) n_{y}\} + \frac{3 δ Q_{y}}{2 h λ_{y}} \{(M_{y} - \frac{4}{3 h^{2}} P_{y}) n_{y} + (M_{xy} - \frac{4}{3 h^{2}} P_{xy}) n_{x}\} - M_{n} δ w_{, n} - M_{ns} δ w_{, s} + \{(M_{x, x} + M_{xy, y}) n_{x} + (M_{xy, x} + M_{y, y}) n_{y} + (- I_{1} u ¨_{0} + I_{2} w ¨_{, x} - \frac{Q ¨_{x}}{λ_{x}} I_{9}) n_{x} + (- I_{1} v ¨_{0} + I_{2} w ¨_{, y} - \frac{Q ¨_{y}}{λ_{y}} I_{9}) n_{y}\} δ w] ds .

(15)

Substituting

\frac{3}{2 h λ_{x}} δ Q_{x} = δ ψ_{x} + δ w_{, x} = (δ ψ_{n} + δ w_{, n}) n_{x} + (δ ψ_{s} + δ w_{, s}) s_{x}

\frac{3}{2 h λ_{y}} δ Q_{y} = δ ψ_{y} + δ w_{, y} = (δ ψ_{n} + δ w_{, n}) n_{y} + (δ ψ_{s} + δ w_{, s}) s_{y}

(16)

in Eq. (¹⁵) reduces it to

\int [N_{n} δ u_{n} + N_{ns} δ u_{s} + M^_{n} δ ψ_{n} + M^_{ns} δ ψ_{s} - \frac{4}{3 h^{2}} (P_{n} δ w_{, n} + P_{n, s} δ w_{, s}) + \{(M_{x, x} + M_{xy, x}) n_{x} + (M_{xy, y} + M_{y, y}) n_{y} + (- I_{1} u ¨_{0} + I_{2} w ¨_{, x} - \frac{Q ¨_{x}}{λ_{x}} I_{9}) n_{x} + (- I_{1} v ¨_{0} + I_{2} w ¨_{, y} - \frac{Q ¨_{y}}{λ_{y}} I_{9}) n_{y}\} δ w] ds .

(17)

Substituting the expressions of

M_{x, x} + M_{xy, y}

and

M_{y, y} + M_{xy, 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

λ_{x},

λ_{y}

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.

Discussion: “Zeroth-Order Shear Deformation Theory for Laminated Composite Plates” (Ray, M. C., 2003 ASME J. Appl. Mech., 70, pp. 374–380)

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Discussion: “Zeroth-Order Shear Deformation Theory for Laminated Composite Plates” (Ray, M. C., 2003 ASME J. Appl. Mech., 70, pp. 374–380)

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