PSI - Issue 6

Anatolii Bochkarev / Procedia Structural Integrity 6 (2017) 174–181 A.O. Bochkarev / Structural Integrity Procedia 00 (2017) 000–000

178

5

Fig. 1. Compressive buckling of a rectangular nanoplate under the boundary conditions: (a) SSSS; (b) SSSF; (c) SCSF

2. A case study

To demonstrate the influence of surface stresses, the classical example of compressive buckling of a rect angular nanoplate will be studied under three cases of boundary conditions, investigated for macroplates in the first half of the twentieth century by Timoshenko (1907, 1910), Bubnov (1914), and Papkovich (1941): SSSS Fig 1(a); SSSF Fig 1(b); and SCSF Fig 1(c). In plane, the boundary conditions of the uniaxial compression p are imposed on the condition of the biaxial pre-tension 2 τ 0 due to structure of the state equation (6)

T ∗ 11 (0 , x 2 ) = T ∗ 11 ( a, x 2 ) = 2 τ 0 − ph, T ∗ 22 ( x 1 , 0) = T ∗ 22 ( x 1 , b ) = 2 τ 0 , T ∗ 12 (0 , x 2 ) = T ∗ 21 ( a, x 2 ) = 0 , T ∗ 21 ( x 1 , 0) = T ∗ 21 ( x 1 , b ) = 0 .

(15)

2.1. A case of the SSSS boundary conditions

In this simplest case, all four edges are S imply supported (SSSS):

w (0 , x 2 ) = 0 , w ( a, x 2 ) = 0 , w , 11 (0 , x 2 ) = 0 , w , 11 ( a, x 2 ) = 0; w ( x 1 , 0) = 0 , w ( x 1 , b ) = 0 , w , 22 ( x 1 , 0) = 0 , w , 22 ( x 1 , b ) = 0 .

(16)

As with biaxial compression buckling ( p 1 along the x 1 axis and p 1 along the x 2 axis), the problem was also investigated by Bubnov (1914). A solution has been sought for w ( x 1 , x 2 ) = A mn sin ( mπx 1 a ) sin ( nπx 2 b ) . (17)

It has been shown that the critical load satisfies the equation

a 2 h (

a 2 b 2 )

2

π 2 D ∗

2 + p

2 =

m 2 + n 2

p 1 m

2 n

.

(18)

The static boundary condition (15) corresponds to p 1 = p − 2 τ 0 /h and p 2 = − 2 τ 0 /h . Then the smallest critical load p s cr (with surface stresses taken into account) is reached at m = n = 1

a 2 h (

a 2 b 2 )

2

π 2 D ∗

4 τ 0 h

1 +

.

(19)

p cr =

+