Issue 62

M. S. Shaari et alii, Frattura ed Integrità Strutturale, 62 (2022) 150-167; DOI: 10.3221/IGF-ESIS.62.11











, i j , ijkl k l u E u d  G G

, i j u E u  G

L

L u t d i i

L crack

L u b d

 





 



 





u t d

(5)

   G

, ijkl k l

i

i

i

i

L

L

crack

G









t

The input can be of any choice based on what is being computed for, and they are also not dependent on each other. Here, freedom is given for the crack to be modeled on the local mesh for analysis. If there is an equal and opposite magnitude of traction force applied at the faces of crack, the integral for crack face traction (the second term on the right side) of Eqn. 4 cancels out. The formulas derived for S-version FEM can be used along with the discrete quantities of  G and  L to set up a matrix as given in Eqn. 6.

 K K u GG GL G K K u LG LL L

0     G b                          G crack L L t t t b       

   

(6)

Where

   

   

       

   

G T B D B d

                G L G L

               





K

GG

G



G T B D B d





K

GL

L



L T B D B d





K

LG

L



(7)

L T B D B d





K

LL

L



    G L F F

 



G N b d L N b d i

G N f d L N f d i



 



G

G









 



i

i

G

G





From Eqn. 6, the nodal displacements of the local and global meshes are represented by the vectors   L u and   G u , respectively while   crack t ,   G t and   L t are the traction force vectors from Eqns. 4 and 5. Here,   GG K and   GL K are taken from the left side of Eqn. 4 while   LL K and   LG K are from the left side of Eqn. 5. From the equation,          GL LG K K proves that Eqn. 6 will be symmetrical. The magnitude of the vector displacement of the global and local mesh can be obtained simultaneously. A singular global matrix will be generated if the displacement field of the global and local mesh is similar, which may occur and be eliminated when discretization is carried out. The reader will notice the matrix   GL K and   LG K are symmetrical and describe the superimposed area of the stiffness matrix. Where   B is the strain-displacement matrix of the element and   D indicates the material’s stress-strain relationship [16]. The nodal forces for a global presence by   G F and   L F for local areas while, the i f represents the nodal forces on the crack front. Eqn. 6 can be used to compute global and local displacements. While the local mesh expands proportionately, the global mesh remains unchanged since the sole region of interest is contained within the crack front boundary. Additionally, the global mesh will not be re-meshed during the process. As a result, it accomplishes the superposition technique’s objective of reducing computation time. Using the energy release rate ( G ) formulation, the Stress Intensity Factor (SIF) is calculated concurrently with the expansion of the local mesh. By computing the stress condition at the crack tip, the SIF is primarily utilized in fracture mechanics to forecast crack growth. Thus, to determine the SIF, the energy release rate must be computed using the virtual crack closure method (VCCM), developed by Rybicki [14] in 1977. The following equation illustrates the energy release rate formula G :

1









1 U L G u u f 1  

 

2 2 U L u u f

(8)

1

2

h

2

153