PSI - Issue 41

Fabrizio Greco et al. / Procedia Structural Integrity 41 (2022) 576–588 Author name / Structural Integrity Procedia 00 (2019) 000–000

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In computation mechanics, Lagrange approaches link the mesh grid to the continuum and follow it during the motion. On the other hand, the Eulerian approach adopts a fixed mesh so that the behavior of the continuum is analyzed while it passes through the computational grid. The ALE formulation introduces a third reference system that is, the mesh (or referential) coordinates system R  , in which the coordinate χ identifies the nodes of the computational mesh at time t =0 (Greco et al. (2020d)). Such a system can be moved liberally into the space according to specific conditions not necessarily associated with the mechanics of the problem under investigation. Besides, the governing equations of the problem can be formulated in the referential coordinate system. In the present work, the ALE formulation is employed to move the nodes of the computational mesh following the geometry evolution caused by the dynamic growth of material internal defects. Fig. 1 shows a schematic describing how the proposed modeling approach uses the moving mesh techniques. Specifically, it depicts the evolution of a pre-cracked continuum from an initial configuration (at time t =0) to a current configuration (at time t t  ). In addition, the figure illustrates the relationships between the spatial, material, and referential domains. As one can observe, passing from the initial to the current configuration, the material and the referential frames do not coincide, thus highlighting that the referential frame can be moved freely.

Fig. 1. The Arbitrary Lagrangian-Eulerian formulation: Relationships between Spatial ( R x ), Material ( R X ), and Referential ( R  ) domains

The links between the referential, material, and spatial frame are described by means of the following time dependent and bijective mapping functions:                   , , , ; , , , ; , , , t t t t t t t t t    X φ X x χ Φ χ x χ Ψ χ X    (1) In particular,   , t φ X represents the mapping function between the material and spatial domains,   , t Φ χ links the referential and spatial domains, and   , t Ψ χ connects the referential and material domains. According to the proposed approach, the governing equation of the solid mechanics must be solved at each time step concerning the referential frame ( χ ). Therefore, spatial and time derivatives of a generic vectorial field v can be expressed in the referential coordinates as follows:

M       v v χ X χ Ψ R

(2)

M M    v

R R  v J

1

(Spatial derivative)

 

Ψ



M R R R   v v  

1 v J X     Ψ

R





1   Ψ Ψ v J J X X   1 1    R R R R  

v v 

1 v J X v J X  1 R R R R R   

M R

R

R R R

2    

(Time derivatives)

(3)





Ψ

Ψ





1 v J X J X R R 

R

Ψ

Ψ