PSI - Issue 25

Ch.F. Markides et al. / Procedia Structural Integrity 25 (2020) 214–225

223

10

Ch. F. Markides et al., Structural Integrity Procedia 00 (2019) 000 – 000

where (Muskhelishvili, 1963):

3   

E



,

3 4 , for plane strain   











  1 1 2 2  



 

  

(28)

* 3 3            * 1

E  

*

, * 

, for plane stress









2

2   

1





To this displacement field certain terms are added (do not affecting stresses) to comply with the actual laboratory test. These terms are the opposite horizontal and vertical components of the displacement of specific points of the CSR that are supposed to remain fixed during the test (e.g., for both CSRc and CSRt the lower jaw is fixed (Fig.7(a, b)). It is here mentioned that in previous studies for the CSR (Markides et al., 2018; Kourkoulis et al., 2018), a typing error in the expressions for the displacements concerning β _ dislocation appears. These errors are here corrected in the respective expressions of Eqs.(26, 27). The correct expressions for ψ ( z ) and Ψ ( z ) due to β _ dislocation (Muskhelishvili, 1963; Markides et al., 2018) read as (CSRc (upper sign), CSRt (lower sign)):     2 2 2 2 1 2 1 2 2 2 3 1 2 2 1 2 2 2 1 1 1 ( ) , ( ) ( )d log 2 ( 2 ) 2 ( 2 ) ( ) ( ) 2 R R R R z z z z z z R z z R R R                                          (29) To reveal the character of deformation, Fig.7 was drawn for an unrealistically large force P . In particular, regard ing the CSRc, two cases were considered: (i) one with static friction between the CSR and the indenter attaining values not exceeded by the tangential component P T of P , and (ii) one with smooth contact. In case of friction, it is assumed that point z T of the CSR remains constant (Fig.8a) and the CSR at the lower support swings almost as a rigid body about z T . For smooth contact it is assumed that point z O is fixed ( z O theoretically belongs to the CSR, on its outer peri phery; thus opposite displacements are realized at z O ). Here it is assumed that locally the CSR rotates about z O following the trace of the circular indentation of radius R O of CSR (Figs.8b, 9). Points, z T and z O are defined in Fig.9. With regard to these assumptions, smooth contact appears preferable. For example, without friction, the end point Q of the indentation is less possible to be pressed against the indenter (Fig.8(a, b)) avoiding possible local fracture. In any case, regarding the horizontal component of displacement u and the fact that the v displacement component is constant all along AB , the difference between the presence (blue color) and absence (red color) of friction, at least for the critical locus AB of interest, is restricted to a rigid body translation, thus not affecting stresses and in turn fracture (Fig.10). For Fig.10 to be drawn, use was made of the data adopted for drawing Figs.5. Similar conclusions are drawn for the CSRt, for which the deformed configuration is plotted only in the absence of friction in Fig.7b.

Fig. 7. (a) The deformed CSRc with or without friction at the CSR-indenter contact region; (b) The deformed CSRt for smooth contact. (a) (b)