PSI - Issue 47

Domenico Ammendolea et al. / Procedia Structural Integrity 47 (2023) 488–502 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Figure 1. The Arbitrary Lagrangian-Eulerian formulation: relationship between the Referential ( R  ) and Material ( X R ) frames. Such a mapping function permits expressing the material gradient of a generic vectorial field v in the Referential frame as follows:

M M    v v χ

M M    v

R R  v J

1

(2)



Ψ

X

χ Ψ



 

in which, ( ) R   represent the gradient operator in the Material and Referential frames, respectively. Further, Ψ J is the Jacobian matrix of ( ) Ψ χ . In particular, the bijection condition of ( ) Ψ χ implicates that Ψ J has a positive determinant ( i.e. , det( ) 0  Ψ J ). 2.2. Governing equations of the problem in the Moving Mesh setting The governing equations of the proposed model can be gathered in three groups. The first group accounts for the fundamental equations of solid mechanics; the second group comprises the equations governing the kinematics of the moving mesh problem; the third group entails the equations associated with fracture mechanics, devoted to defining the crack onset conditions and the direction of crack propagation. The governing equations of solid mechanics are written with reference to a two-dimensional domain 2 R  bounded by a contour  .  consists of two portions, referred to as t  and u  , where are imposed external tractions ( t ) and prescribed displacements ( i.e. , u=u ), respectively. Further, the domain  possesses a pre-crack ( C  ) that develops inside the material from the external boundary up to a crack tip C . The crack tip serves as the origin of a local system of coordinates ( 1 2 , C C X X ), whose horizontal axis 1 C X is parallel to the crack faces. Under the hypotheses of quasi-static loading conditions and small strains, the variational weak form of the governing equations of solid mechanics assumes the following expression with reference to the Material frame: : 0 M M M M M M d d d                      C u u f u t u (3) being C the fourth-order elastic tensor, u and  u the displacement field and an appropriate set of virtual displacements. Noting that the superscript M indicates that displacements and virtual displacements are expressed with reference to the material frame. Using Eq. (2) to express the material gradient of the displacement field in the Referential frame, the ALE formulation of Eq. (3) can be written as follows:       1 1 : 0 R R R R R R R R J d J d J d                              Ψ Ψ C u J u J f u t u (4) where, R  represents the referential volume, whereas J  and J  are the Jacobian corresponding to the area and the boundary of R  . ( ) M   and