PSI - Issue 39

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

653 5

at

and

0 on

X n η ∆ ⋅ = 

C

X ∆ =

∂Ω

(5)

c

F

T







where, c n  crack tip ( x 1 , x 2 ).

is the unit versor that define the direction of crack propagation referring the local coordinate system at the

2.2. Governing equations of the problem

2 R Ω ⊂ made of a functionally graded material, whose gradation occurs

Figure 2 shows a two-dimensional domain

linearly along a generic design direction. The external boundary consists of two portions, named as u ∂Ω and t ∂Ω , where prescribed displacements ( u  ) and distributed tractions ( t  ) are imposed. Besides, the domain has an initial crack ( Γ C ) that departs from the external boundary and develops internally up to a crack tip denoted by T C . The crack tip is the origin of a local coordinate system ( x 1 , x 2 ), whose horizontal axis x 1 is parallel to the crack faces. This system is adopted in the proposed model as a reference for the crack front variables ( i.e. , Stress Intensity Factors), and the kinking angle ( c θ ). Because of the gradation, Young’s modulus, Poisson’s ratio, density, and fracture toughness vary spatially, i.e. , ( ) E E X =  , ( ) X ν ν =  , ( ) X ρ ρ =  , ( ) IC IC K K X =  . By assuming quasi-static loading conditions, small strains, and the absence of interaction forces between the crack faces, the governing equation of the solid mechanics can be expressed by means of the following weak form:

Ω ∫

Ω ∫

∫

(6)

: C u   − ∇ ⋅∇ Ω + ⋅ sym sym u d δ

0

f u d δ

t u dS δ  

Ω + ⋅

=





 







∂Ω

t

( ) C C X =   

and t 

being,

is the fourth-order constitutive tensor, X 

the position vector, f 

the body force and surface

traction vectors, respectively. Eq. (6) is expressed with reference to the Referential system. Since the proposed model depicts the crack advance as a sequence of moved configurations, the above weak expression must be formulated in moved coordinates using the ALE approach. Therefore, by introducing Eq. (3) in Eq. (6), one achieves: ( ) ( ) ( ) ( ) ( ) 1 1 : 0 R R t M R R R R M M R R C X u X J u X J J d f X uJ d t X n u J d δ δ δ − − Φ Φ Ω Ω Γ Ω Ω ∂Ω     − ∇ ∇ Ω + Ω + ⋅ Γ =     ∫ ∫ ∫                (7) configuration, whereas J Ω and J Γ are the Jacobian related to the volume and area, respectively. Eq. (7) must be solved together with the equations related to fracture mechanics. Because the crack path is unknown in advance, the fracture mechanics problem must be stated in an incremental form by using the classic Karush-Kuhn Tucker conditions, as follows: 0, 0, 0 F F F F f f η η = ≤ ≥   (8) where, R Ω is the referential volume, t ∂Ω is the boundary portions subjected to external tractions in the referential