PSI - Issue 66

H.A.F.A. Santos et al. / Procedia Structural Integrity 66 (2024) 195–204

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4

H.A.F.A. Santos et al. / Structural Integrity Procedia 00 (2025) 000–000

The Neumann (or static) boundary conditions of the problem are

nN − ¯ N = 0 , nV − ¯ V = 0 , nM + ¯ M = 0 , on Γ N

(4)

with

n =

1 if x = L − 1 if x = 0

The constitutive relationships are taken as

22 χ, in

(5)

+ C

12 χ, V

= C

33 γ, M

= C

+ C

N = C

11 ε

12 ε

Ω

where

=

=

=

=

E ( z ) z 2 dA , C

f s G ( z ) dA

(6)

E ( z ) z dA , C 22

E ( z ) dA , C 12

C 11

33

A

A

A

A

where E and G are the Young’s and shear moduli, respectively, of the beam, such that

E 2(1 + ν )

(7)

G =

with ν the Poisson’s ratio. f s is the standard shear correction factor. Note the coupling between the axial force N and bending moment M inEq. (5). A transversely FGM is assumed, although axially or both transversely and axially FGM responses can be consid ered. Two types of material distribution functions are usually adopted for transversely FGMs: the exponential-law and the power-law. The former is based on the following relationship for the Young’s modulus

β z + h

2

E ( z ) = E

(8)

L e

with

1 h

E U E L

ln

(9)

β =

where E U and E L are the Young’s modulus of the upper and lower parts of the beam, respectively. The power-law assumes the following transverse distribution of the Young’s modulus

p

1 2

z h

U − E L )

E ( z ) = E

(10)

+ ( E

+

L