PSI - Issue 41

5

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

580

R X  is a

Ψ J is the Jacobian matrix of the mapping function ( , ) t Ψ χ , and

( ) ( ) M       X ,

( ) ( ) R       χ ,

where

convective term denoting the velocity of the mesh in the material domain.

2.2 Governing equations of the problem Fig. 2 depicts a two-dimensional homogeneous body

2 R  , whose external boundary  consists of two

t  , where prescribed displacements ( u ) and distributed tractions ( ( ) t p ) are

u  and

regions, referred to as

c  ) that departs from the external boundary

assigned, respectively. In addition, the body presents an internal crack (

T C . T C is the origin of a local coordinate system ( 1 2 ,

c c x x ), whose

and develops up to a crack tip denoted by

horizontal axis 1 c x is parallel to the internal crack faces. Under the hypothesis of homogeneous, isotropic, and linear elastic material behavior, the mechanics of the body is governed by the following weak form:   : p h d h d h dS h d                     C u u f u p u u u  (4) where, C is the fourth‐order constitutive tensor, h is the body thickness, f and p are the body force and surface traction vectors, respectively, and  is the mass density. Besides, u and  u represent the displacement field and a suitable set of virtual displacements. According to the proposed scheme, the governing equations of the problem (Eq.(4)) must be solved with reference to the Referential system χ . Therefore, the ALE formulation of Eq. (4) is necessary, which can be obtained by rearranging Eq.(4) using Eq.(2)-(3), as follows:         1 1 1 1 1 1 1 1 : 2 R R R p R R R R R R R R R R S R R R R R R R R R R R R R R R R R R R h J d h J d h J dS h J d                                                       Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ C u J u J f u p u u u J X u J X u J J X X u J X J X u         (5) respectively. In addition to the Eq.(5), the ALE formulation introduces further equations ensuring that the motion of the mesh nodes occurs with regularity, minimizing the distortions of the finite elements. Among the different ways to express the governing equations of the Moving Mesh problem, the proposed method adopts the Laplacian (or Poisson) smoothing approach, which consists of solving the following partial differential equation within the computational domain: 0 M M    x (6) being, 1 2 [ ] x x  x the position vector identifying the spatial coordinates of mesh nodes in the actual configuration of the body. Eq. (6) must be solved using a proper set of boundary conditions imposing that: ( i ) the displacement of the mesh node associated with the crack tip equals the incremental displacement prescribed by fracture criteria, and ( ii ) the external boundary of the computational domain are fixed: where, R  is the referential volume, whereas J  and S J are the Jacobian related to the volume and area,

 x n 

x

at T a C 

;

0 on







(7)

c

where,   x x X represents the nodal mesh displacement vector function, and a  is the crack tip velocity. Finally, the set of governing equations is completed by those associated with the fracture mechanics problem. In particular, such governing equations are two sets of fracture functions that serve to ( i ) identify the condition of crack initiation ( I F f ) and ( ii ) describing the kinematic of the crack tip ( P F f ). Such functions are defined as follows: