PSI - Issue 6

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

180

7

1.5

SSSS (1) SSSF (1) SCSF (1) SSSS (2) SCSF (2) SCSF (2)

SSSS (1) SSSF (1) SCSF (1) SSSS (2) SCSF (2) SCSF (2)

2

min / Λ min , τ 0 =0

1

cr / p cr

1.5

p s

0.5

Λ s

1

0

0

50

100

150

200

250

0

50

100

150

200

250

a , nm

a , nm

Fig. 2. Comparison of the behaviour of the relative eigenvalues (a) and end-critical load (b) in dependence on the length of the side of a square plate for two sets of surface constants in Problems SSSS, SSSF and SCSF

Taking A k as generalized Lagrange coordinates, the energy increments ∆ W (1) and ∆ W (2) are quadratic forms of these coordinates. In the equilibrium position, the generalized forces corresponding to these gener alized coordinates are equal to zero:

(1) kl − ph ∑ l =0

= 2 τ 0 ∑ l =0

D ∗ a 2 ∑ l =0

∂ ∆ W (1) ∂A k

∂ ∆ W (2) ∂A k

∂W ∂A k

(1) kl +

(2) kl ( k = 1 , 2 , 3 ... ) (25)

A l � W

0 =

=

+

A l W

A l W

where the coefficients � W (1) (2) kl are dimensionless. If K is the number of the terms in the series (24), this equation can be rewritten as a 1 / Λ-eigenvalue problem for the matrix W (1) with the weight � W (2) 1 Λ � W (2) A = W (1) A, � W (2) = W (2) + 2 τ 0 a 2 D ∗ � W (1) , Λ = pha 2 D ∗ . (26) 2.4. Numeric results Finally, we illustrate how the static boundary conditions SSSS, SSSF and SCSF affect the relative per formance of the compressive buckling nanoplate from aluminium with the elastic moduli λ = 58 . 2 GPa and µ = 26 . 1 GPa for the bulk phase. The corresponding properties of the surface are determined by the elastic constants (N/m), obtained by simulation methods Miller and Shenoy (2000): (1) λ s = 6 . 85, µ s = − 0 . 38, and τ 0 = 0 . 911; (2) λ s = 3 . 49, µ s = − 5 . 43, and τ 0 = 0 . 569. Our interest is primarily in the redistribution of the bending energy of the static boundary conditions due to the dependence on the effective Poisson’s ratio. This redistribution is characterized by the behaviour of the relative minimal eigenvalues Λ s min | τ 0 =0 / Λ min (with the mark s for a nanoplate and without any mark for a macroplate). For the sake of simplicity, let a = b (i.e., the plate is square). Fig. 2(a) shows that there is no redistribution of the bending energy in problem SSSS. But in problems SSSF and SCSF, a redistribution takes place. This leads to a reduction of the minimum eigenvalue compared with the macroplate for both Sets 1 and 2. The greatest redistribution of bending energy occurs in Problem SCSF, where there is the smallest relative eigenvalue. It is also interesting to compare the relative end-critical loads p s cr /p cr in all three problems that is shown in Fig. 2(b). Here, for the different Sets 1 and 2, there is a different picture. For Set 1, a growth in the critical load is observed in all three problems, and curiously, to a greater extent in problem SSSF, and less in problem SCSF. Problem SSSS occupies an intermediate position. For Set 2, the order of the problems is preserved. However, the behaviour is already different: in Problems SSSS and SSSF, the critical load increases, and in problem SCSF, it decreases. kl , W (1) kl , and W