PSI - Issue 39

6

Arturo Pascuzzo et al. / Procedia Structural Integrity 39 (2022) 649–662 Author name / Structural Integrity Procedia 00 (2019) 000–000

654

Figure 2. A two-dimensional cracked FGM domain

being F η  the incremental displacement of the crack tip. As expressed, f F depends on the stress intensity factors at the crack front ( i.e. , , I II K K ), the kinking angle ( θ c ), and the fracture toughness of the material ( ( ) IC K X  ). Generally, the fracture function and the kinking angle are defined by means of one of the classic fracture criteria, such as the Maximum Hoop Stress (Erdogan and Sih (1963))) or the Maximum Strain Energy Release Rate (Hussain et al. (1974)) one. Eqs. (7) and (8) are solved together to find the incremental crack tip displacement under the condition f F = 0. 2.3. The interaction integral method (M-integral) for the analysis of FGMs in moving coordinates The interaction integral method proposed by Yau et al. (Yau et al. (1980)) represents an effective and reliable approach to extract fracture variables at the crack front. The M -integral expression derives from the J -integral applied to a superimposed state formed by the actual solution of the problem under investigation (denoted as , , act act act u ε σ    ) and an auxiliary one consisting of the fundamental analytic solution of the fracture problem of a straight crack in infinity and remotely loaded plate (next denoted as , , aux aux aux u ε σ    ). For FGMs, there are different available formulations to define such auxiliary fields. According to Kim and Paulino (Kim and Paulino (2004a)), in the proposed model, an incompatibility formulation is adopted. Specifically, the incompatibility formulation consists of assuming the fundamental analytic asymptotic solutions for homogeneous materials proposed by Williams (Williams (1956)) (with material properties sampled at the crack tip) under the assumption that equilibrium and constitutive relationships are satisfied, but compatibility is violated: ( ) f K K K X θ =      , , , F F I II c IC f a fracture function that defines crack nucleation conditions and



1

1

r

r

(

)

(

)

aux u K =

aux

I

Tip

aux

II

Tip

, θ κ

, θ κ

g

K

g

+

i     =   =    ≠ aux ij aux ij aux σ ε ε

I

i

II

i

2

2

tip

tip

2

2

π

π

µ

µ

aux

aux

K

K

( ) θ

( ) θ

I

II

f

f

+

I

II

with ,

1, 2

i j

(9)

=

ij

ij

2

2

r

r

π

π

aux

( )

ijkl S X

σ

kl



1 2

(

)

, i j u u + aux

aux

, j i

ij



In Eqs. (9), [ ( ), ( )] I II ij ij f

f θ θ and [ ( , I g

)] are angular functions whose expression can be found in Tip κ is equal to (3 ) (1 ) Tip Tip ν ν − + for plane stress and 3 4 Tip ν − Tip

Tip

II

),

( ,

g

θ κ

θ κ

i

i

Tip µ is the elastic shear modulus, and

(Kuna (2012)),