PSI - Issue 28

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

1983

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2 Theoretical Framework This section presents an overview of the Arbitrary Lagrangian-Eulerian method (ALE) and the Interaction Integral approach (M-integral). In particular, since the proposed framework aims to simulate crack propagations in 2D linear elastic continuum media, the formulations of both methods are discussed for 2D problems. The Arbitrary Lagrangian-Eulerian formulation (ALE) Fig. 1 shows a two-dimensional structure affected by a crack propagation process under increasing loads ( p ). The crack propagation develops toward the inner of the structure, changing the shape of internal boundaries. To reproduce the geometry evolution caused by crack propagation, standard FE approaches require several remeshing processes that could increase computational times in case of complex configurations. In the present approach, a moving mesh strategy based on the Arbitrary Lagrangian-Eulerian method (ALE) is adopted to describe the evolution of pre-existing discontinuities, thereby limiting the recourse to remeshing events. 2.1

Fig. 1. A schematic of 2D structure affected by a crack propagation process under the action of increasing loads: Referential and Moving coordinate systems.

The ALE approach is based on the definition of two configurations for the mesh points, known as fixed and current ones (Funari and Lonetti (2017), Funari et al. (2019)). The positions of the mesh points for the fixed and current configurations are referred to as a Referential (R) and a Moving (M) coordinate system, respectively. The mapping between these coordinate systems, and then between the two configurations of nodes, are governed by the following expressions:     1 M R R M X X X X            (1) where,   T M M M X X Y   and   T R R R X X Y   are the actual and referential position vector functions, respectively, while : R M C C    with   X Y      is assumed to be invertible with continuous inverse. In addition to Eq. (1), the mapping between the actual and referential coordinate systems is defined in terms of the Jacobian matrix of the transformation ( J  ), which is expressed by means of the following relationships:

X                     Y R R X Y R R X X Y Y

   J     1 M

  

with

(2)

J 



R



The Jacobian of the transformation is used to express the governing equations of the structural problem in the current configuration of the system. In particular, since 2D fracture problems are considered, the governing equations