PSI - Issue 71

Arun K. Singh et al. / Procedia Structural Integrity 71 (2025) 90–94

92

and crack branching at higher crack velocity to result in increase of effective surface energy (Freund, 1998). Hence, effective dynamic surface energy d  may be expressed in terms of effective static surface energy s  , crack tip velocity v da dt = , limiting crack tip velocity l v and exponent n as (2) In Eq.2, l v is assumed to be the same as reference crack velocity in Eq.1. Further, Lagrangian could be expressed in the terms of total energy L of the system consisting surface energy, kinetic energy of crack tip as well as strain energy of crack having length a at a constant external stress 0  (Neal-Sturgess, 2012) yields 2 2 2 2 2 0 0 2 2 1 2 n s l l D a a v v L a v E v E           = + − −         (3) The Euler-Lagrange equation is given by (Neal-Sturgess, 2012) 0 1   = +         n    d s l v v   

d dL dL dt dv dv

      −   =    

(4)

2 IC s K E  = and 2

2

2

0 ID K a   = , following expression is obtained after neglecting

Also assuming

acceleration of crack tip

(

)

 

  

1 n v −

n

2

1  −

K

IC

n

v

l

2

K

=

ID

2 2   −    1 2 l v v D 

(5)

1 n = . Moreover, above equation further 0 D = . As a result, crack velocity

Noting that Eq.5 leads to the Mott’s model in Eq.1 for modifies upon neglecting inertia of the crack tip that is,

ID K reduces to ( ) 1 n v  −

dependent

  

n

2 K K =

2

1  −

ID IC

n

v



l (6) It may also be concluded from Eq.5, under quasistatic crack propagation, that is, ~0 v , results in ID IC K K = (Griffith’s model). Eq.5 is now used for nonlinear regression analysis of the experimental data related to DSIF vs. crack velocity of AISI 4340 steel from literature (Rosakis et al., 1984) for estimating the physical parameters namely n , l v and k . In order to determine the optimal value of the aforementioned parameters, solver “lsqcurvefit” of MATLAB → was used. In