Issue 64

M. A. Kenanda et alii, Frattura ed Integrità Strutturale, 64 (2023) 266-282; DOI: 10.3221/IGF-ESIS.64.18

                    x y z yz xz xy σ σ σ τ τ τ

                    x y z yz xz xy ε ε ε γ γ γ

11 C C C 0 0 0 C C C 0 0 0 C C C 0 0 0 0 0 0 C 0 0 0 0 0 0 C 0 0 0 0 0 0 C 12 13 12 22 23 13 23 33 44 55

         

         

=

(5)

66

where ( x y z yz xz xy σ , σ , σ , τ , τ , τ ) and ( x y z yz xz xy ε , ε , ε , γ , γ , γ ) are the stress and strain components, respectively, ij C indicating the stiffness coefficients are defined as:

        

E(z)(1- ν )

C =C =C =

11

22

33

(1-2 ν )(1+ ν ) ν E(z) (1-2 ν )(1+ ν ) E(z)

C =C =C =

(6)

12

13

23

C =C =C =G(z)=

44

55

66

2(1+ ν )

where ν represents Poisson’s ratio, E(z) and G(z) are the Young’s and shear modulus of the porous FG plate, respectively.

E QUATIONS OF MOTION

B

ased on Hamilton's principle, the equations of motion can be expressed by the following variation form:    0 t s p k δ E + δ E - δ E dt=0 (7) k δ E are the variation of strain energy, the variation of the potential energy due to the applied loads, and the variation of kinetic energy, respectively. The variation of strain energy of the porous FG plate takes the form below: where δ represents the variation operator, s δ E , p δ E , and

   V



δ E =

x x y y σ δε + σ δε + σ δε + τ δγ + τ δγ + τ δγ d zz xyxy yzyz xzxz

(8)

 V

s

The variation of potential energy due to the applied loads is given by:    A p A δ E =- q δ w d

(9)

in which q represents the transverse distributed load, A is the top surface. The variation of kinetic energy of the porous FG plate is deduced by the following integral as:





               u δ u v δ v w δ w + + t t t t t t

ρ

(10)

δ E =

(z)d

 

V

k V

Eqns. (8), (9), and (10) are substituted into Eqn. (7), and the linear strain field expressions (4a, 4b), and constitutive law expressions (5) are used in Eqn. (7). The equations of motion are obtained as:

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