PSI - Issue 6

Anatolii Bochkarev / Procedia Structural Integrity 6 (2017) 174–181

177

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A.O. Bochkarev / Structural Integrity Procedia 00 (2017) 000–000

and the surface energy of the facial surfaces under z = ± h/ 2 Gurtin and Murdoch (1975, 1978) U ± = ∫ Ω ( τ 0 tr ε ± + λ s 0 2 tr 2 ε ± + µ s 0 ε ± ·· ε ± ) d Ω .

(10)

The last one also can be written as the sum of the energies of the deformed facial planes and their bending

1 2 ∫

(0) ·· ε (0) ) d Ω+

Ω (

(1) + U (2) =

(0) + 2 λ s

2 ε (0) + 4 µ s

4 τ 0 tr ε

0 ε

U + + U − = U

0 tr

2 ∫

Ω ((

0 ) (∆ w )

2 , 12 − w , 11 w , 22 ) ) d Ω . (11)

h 2

λ s 0 2

+ µ s

2 + 2 µ s

0 ( w

Using the effective elastic moduli (7) and (9), both pairs of summands of the potential energy of the deformed bulk phase and the surface energy (11) can be recombined into new pair of summands: the full energy of the deformed mid- and facial planes and the full energy of the plate’s bending

Ω (

(0) ·· ε (0) ) d Ω+

2 ∫

C ∗

4 τ 0 C ∗

tr ε (0) + ν ∗

2 ε (0) + (1 − ν ∗

W = W (1) + W (2) =

t tr

t ) ε

2 ∫

D ∗

Ω (

2 , 12 − w , 11 w , 22 ) ) d Ω . (12)

(∆ w ) 2 + 2(1 − ν ∗

f )( w

As can see, using the effective tangential and flexural elastic moduli allows keeping the classic structure of the potential energy of the deformed nanoplate with surface stresses. Finally, of practical interest is a simplified expression for the full energy of the deformed mid- and facial planes W (1) from (12) in the framework of the flexible plate theory, corresponding to the linearized von K´arma´n theory

2 ∫

Ω (

(0) ·· e (0) ) d Ω+

C ∗

4 τ 0 C ∗

tr e (0) + ν ∗

2 e (0) + (1 − ν ∗

W (1) = W (0) + ∆ W (1) =

t tr

t ) e

1 2 ∫

Ω (

2 , 2 + 2 T ∗ 12 w , 1 w , 2 ) d Ω . (13)

T ∗ 11 w

2 , 1 + T ∗ 22 w

The first summand in (13) is the same as one in (12), but calculated through the linear strains under the known stress field in the plane, and therefore is constant. And the second summand in (13) is its incremental gain under a non-zero deflection. Then the total energy (12) can be also expressed in terms of the energy increment gain under the non-zero deflection

W = W (0) + ∆ W = W (0) + ∆ W (1) + ∆ W (2) .

(14)