PSI - Issue 33

862 Domenico Ammendolea et al. / Procedia Structural Integrity 33 (2021) 858–870 Domenico Ammendolea et al./ Structural Integrity Procedia 00 (2019) 000 – 000 Notice that the bijection of  implicates that Jacobian matrix of the mapping function has a positive determinant:



( ) det 0 with J  

(5)

J

=



R

X



In the proposed approach, the governing equations of the problem ( i.e. , the fundamental equation of structural mechanics and heat transfer) must be solved at each step of the propagation process, then in the moved configuration. The Jacobian matrix (Eq. (5)) permits projecting the governing equations from the referential system to the moved one. To this aim, the spatial derivatives referring to the moving frame must be used. By considering a generic vectorial field ( ) M v X , its gradient is expressed in the moving frame as follows:

R

v

v X



 

(6)

1

M  = = v

R

− 

v J

= 

M X X 

R





The ALE formulation introduces further equations into the problem that must be solved together with those arising from structural mechanics and heat transfer. Such equations govern the motion of the mesh nodes. Particularity, they avoid that mesh elements undergo relevant distortions. Among the different approaches available in the framework of the ALE, the proposed scheme adopts a regularization strategy consistent with a Laplace approach, which involves solving the Laplace equation defined in terms of the mesh point displacement vector function ( i.e. M R X X X  = − ), as follows: being ( ) 2   the nabla operator. Eq. (7) must be solved with the following boundary conditions imposing that the mesh movement follows the crack advance while ensuring that the mesh nodes on the external boundary of the computational domain stand at rest: at and 0 on c F T X n C X    =  =  (8) where, c n is the unit versor that define the direction of crack propagation referring the local coordinate system at the crack tip ( x 1 , x 2 ). The interaction integral method (M-integral) The Interaction Integral method, also known as M -integral, is a widespread approach used to extract fracture variables at the crack front, such as Stress Intensity Factors (SIFs) and T-stress (Yau et al. (1980)). In particular, the SIFs are essential to identify crack onset conditions and the direction of propagation. As a result, such variables play a key role in the proposed scheme because they address the motion of the mesh points. Since the mesh frame moves during the crack advance, the M -integral is formulated in the ALE context. Hence, similarly to the governing equation of the problem, the M -integral is evaluated in the moved configuration. The general expression of the M -integral derives from the J -integral applied to the superimposed state resulting from the union of two admissible fields. The first is the actual solution, which includes the displacement, stress, and strain fields of the problem under investigation for which SIFs worth be evaluated. In particular, these fields are evaluated through FE analysis. The second is an auxiliary solution with known SIFs. Usually, for homogeneous materials, suitable auxiliary fields rely on Williams’s crack tip asymptotic solutions (Williams (1956)). For the analysis of the fracture problem under thermal loading, the M -integral expression is derived starting from the J -integral proposed by Wilson and Yu (Wilson and Yu (1979)), whose integral domain expression is the following: 2 0 on X   =  (7) 2.3