PSI - Issue 28

Domenico Ammendolea et al. / Procedia Structural Integrity 28 (2020) 1981–1991 Author name / Structural Integrity Procedia 00 (2019) 000–000

1984

4

of the structural problem can be defined by using a classic formulation based on plane strain or plane stress assumptions. The shape variation of the structure produced by crack propagation is reproduced through specific boundary conditions and field equations. The crack tip mesh node is enforced to grow along the direction of crack propagation for a distance equal to the increment of the crack extension, through the following boundary conditions:

c T   

c T   

T   X 

T    Y

 

 

cos

sin

(3)

, ) T T X Y     are the displacement components of the crack tip mesh node 1 2 ( , ) X X , and T   is the incremental displacement of the tip, which is

where c  is the angle of crack propagation, ( along the crack tip local coordinate system

determined by solving the following constrained problem:

F F F F f G             F F F f



0 with 0  

f  

F

(4)

where F f is the fracture function, F   is the fracture multiplier, and G is the Energy Release Rate associated to the crack area extension. Furthermore, boundary conditions are imposed to constraint mesh motion to do not exceed the computational domain:

0, on

(5)

M R X X X        

S

u

where u S is the external boundary of the structure. Finally, rezoning or regularization equations are adopted to modify the position of the mesh nodes while reducing the distortion of the element mesh. According to Laplace’s regularization method, the following relationships hold:

2 0 X Y         2

0

(6)

2.2 The interaction integral method (M-integral) The interaction integral method, also known as M-integral, is an efficient strategy to extract mixed-mode Stress Intensity Factors (SIFs) in both homogenous and heterogeneous materials. In the following, the M-integral formulation for homogenous materials is discussed only. However, more details regarding the applicability of the method with heterogeneous materials can be found in (Kim and Paulino (2003), Yu et al. (2009)). The M-integral is based on the definition of two independent equilibrium states: the first is the actual state (1) of the fracture problem for which SIFs would be evaluated; the second is an auxiliary state (aux) with known SIFs. Generally, the auxiliary field consists of Williams’s crack tip asymptotic solutions ( i.e.   1 2 O r for the displacements and   1 2 O r  for the stresses and strains) (Kuna (2013)). The M-integral expression is derived from the J-integral of the superimposed state, ( i.e. actual plus auxiliary ( 1 aux J  )), which can be simply written in the equivalent integration domain form as follows:     1 1 ,1 ,1 1 , 1 with 2 aux aux aux aux aux aux ij j ij j ik ik ik ik j i A J J J M M u u q dA                       (7) fields, respectively, while the last term ( M ) is the interaction integral. In addition, A is the area of 1 2 ( , ) q X X is a scalar function which has the value of the unity on the inner contour of A ( 1 S ) and zero on the outer one ( 0 S ) (Fig. 2). According to the equivalence between the J -integral and the Energy Release Rate, the relation between the M-integral and SIFs can be expressed as follows: where ( , 1 J and aux J are the classic J-integral expression written for the actual   , , i ij ij u   and auxiliary , ) aux aux aux i ij ij u   integration, while