PSI - Issue 6

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

176

3

Here λ s and µ s are the surface elastic moduli (the analogues of the surface Lam´e constants), and τ 0 is the residual surface stress (all three are assumed to be equal on both sides of the face); A is the unit tensor, and ∇ is the nabla operator (both two-dimensional in the plate plane), and w is a plate deflection. Taking into account the surface equilibrium condition Gurtin and Murdoch (1975, 1978) n ± · σ = ∇· τ ± , the state equation of the bulk phase ( − h/ 2 < z < + h/ 2) can be written in the form

E 1 − ν 2

νσ zz 1 − ν

( A · σ ) =

((1 − ν )( A · ε ) + ν A tr( A · ε )) +

A

(3)

where σ zz̸ = 0 Lu, et al. (2006); Huang (2008), E is Young’s modulus and ν is Poisson’s ratio. According to Kirchhoff’s hypothesis, we assume the displacement field to be linear in the thickness. This allows us to get the distribution law of the stresses acting in the plate plane through the thickness of the bulk phase

E 1 − ν 2

z ((1 − ν ) ∇∇ w + ν A tr( ∇∇ w )) + 2 τ 0 z h ν 1 − ν

( A · σ ) = σ (0) −

A ∆ w,

(4)

E 1 − ν 2 (

(1 − ν ) ε (0) + ν A tr ε (0) ) .

σ (0) =

and on the facial surfaces

h 2

(0) ∓

(2 µ s

0 ∇∇ w + λ s

(0) = τ

s 0 ε

(0) + λ s

(0) .

0 A tr( ∇∇ w )) , τ

τ ± = τ

0 A + 2 µ

0 A tr ε

(5)

Here ε (0) are the nonlinear von K´arm´an-type strains in the midplane. To pass to the two-dimensional equations of plate theory, we introduce a membrane forces tensor as the tensor of the resultant through thickness forces of the bulk phase stresses σ (4) and the facial surfaces stresses τ ± (5) and the moments tensor of these stresses using the effective elastic moduli

+ h/ 2 ∫

( A · σ ) dz = 2 τ 0 A + C ∗ ( (1 − ν ∗ t ) ε

(0) ) ,

T ∗ = τ − + τ + +

(0) + ν ∗

(6)

t A tr ε

− h/ 2

Eh 1 − ν 2

1 C ∗

C ∗ = C + 4 µ s

s 0 , ν ∗ t =

( νC + 2 λ s

0 + 2 λ

0 ) , C =

;

(7)

+ h/ 2 ∫ − h/ 2 ( A · σ ) z dz ) = − D ∗ ( (1 − ν ∗ f ) ∇∇ w + ν ∗ f A tr( ∇∇ w ) ) ,

M ∗ = n × (

h 2

( τ + − τ − ) +

(8)

1 D ∗ (

τ 0 ) , D =

h 2 2

h 2 6

h 2 2

h 2 6

Eh 3 12(1 − ν 2 ) .

ν 1 − ν

ν 1 − ν

D ∗ = D + h 2 µ s

τ 0 , ν ∗ f =

λ s

λ s

0 −

0 −

(9)

νD +

0 +

1.2. Potential energy of a plate with surface stresses

The potential energy of a deformed plate with surface stresses taken into account consists of the energy of the deformed bulk phase (which consists of the energies of the deformed midplane and the plate’s bending)