PSI - Issue 66

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

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5

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

where 0 ≤ p ≤ + ∞ is known as the power-law (or volume fraction ) index . It is worth noting that, when p = 0, a single-material behaviour with modulus E U is obtained. On the other hand, for p → + ∞ this distribution renders a single-material behaviour with modulus E L . Following [10], a general crack j , with depth d j , located at x = ¯ x j , is herein modelled using a discrete spring approach in which Dirac’s delta generalized functions are considered in the beam’s flexural flexibility as

1 + n

− ¯ x j )

j = 1 β j δ ( x

1 C r

(11)

=

C 22

22

with δ the Dirac delta function and β j a coefficient related to the crack severity, given as

C 22 K j L

(12)

β j

=

where the crack parameter K j is defined, following the approach of [2], as

d j h

2

− 1 d j

h d j

C 22

= 0 . 9

(13)

K j

h

h

2 −

3. Complementary-Energy-Based Variational Principle

Following the methodology presented by Santos et al. [15, 19, 16], it can be shown that the total complementary energy of the boundary-value problem under study, Π c : U s ( Ω ) →R , comes out as

= − U

c ( N , M , V )

c ( N , M , V )

c , ext ( N , M , V )

(14)

+Π

Π

where U c is the complementary strain energy given as

N 2 − 2 C

2

C r

L 0

V 2 C 33

12 NM

+ C

11 M

1 2

22

(15)

dx

U c

+

=

− C 2

r 22

C 11 C

12

and Π

c , ext is the external complementary energy given as

+ [ nM ¯ φ ] Γ D

c , ext ( N , M , V )

= [ nN ¯ u ] Γ D

+ [ nV ¯ w ] Γ D

(16)

Π