PSI - Issue 39

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

652

4

( ) R

( ) 1 M

M

R

(1)

X −

X

X X

= Φ

= Φ

  

  

Figure 1. The Arbitrary Lagrangian-Eulerian formulation: Referential (R) and Moved (M) configurations

[

]

[

]

and

are the moved and referential position vector functions of the mesh implicates that the Jacobian matrix of the mapping function has a

where,

T M M M X X Y = 

T R R R X X Y = 

nodes, respectively. The bijection condition of Φ  positive determinant, i.e. :

X ∂  ∂Φ

( ) det 0 with J Φ >

(2)

J 

=

Φ

R





The proposed approach involves the solution of the structural problem in the i -th moved configuration, i.e. , at each step of the crack advance. The Jacobian matrix J Φ  serves for projecting the governing equations of the problem ( i.e. , the fundamental equations of solids mechanics) from the referential to the moved configuration. To this aim, the spatial derivatives referring to the moving frame must be used. In particular, the gradient of a generic vectorial field ( ) M v X   in the moved configuration is expressed as follows:

R

v 

v X

∂

∂ ∂

1

M ∇ = =

R

(3)

− Φ

v 

v J  

= ∇





M X X ∂

R

∂Φ

∂







The use of the ALE formulation requires the solution of further equations devoted to ensuring the consistency of the motion of the mesh nodes. Such equations make sure that the motion of the mesh nodes takes place in such a way as to minimize the distortions of the finite elements. Among the various approaches in the context of the ALE formulation, the proposed model employs a regularization strategy consistent with a Laplace scheme. Such scheme consists of solving the Laplace equations of the mesh point displacement vector function ( i.e. M R X X X ∆ = −    ), i.e. : 2 0 on X ∇ ∆ = Ω  (4) where ( ) 2 ∇ ⋅ is the nabla operator. Eq. (4) must be solved together with proper boundary conditions, which impose that ( i ) the motion of the crack tip node equals the incremental displacement derived by the solution of the fracture problem and ( ii ) zero displacements on the external boundary of the computational domain: