PSI - Issue 41

Jesús Toribio et al. / Procedia Structural Integrity 41 (2022) 736–743 Jesús Toribio / Procedia Structural Integrity 00 (2022) 000–000

738

3

L/2

x = a - r

R

r

a

A

D/2

Fig. 1. Axisymmetric notched geometries used in the calculations.

Local strain at the notch tip (  L ) is calculated over a local reference length B = 0.01 D that was selected after checking that it was small enough for numerical results to be independent of the chosen size and also much higher than relevant material microstructural parameters (the average size of the pearlite colony for the selected steel).

u L B =

u L 0.01 D

 L =

(2)





where u L is the relative displacement between the ends of the local reference length B (B = 0.01 D) Global strain is the global displacement u G (relative displacement between the sample ends) divided by a characteristic length of the geometry (the sample diameter D) to get a dimensionless variable: (3) It is convenient for the sample length to be long enough to have its ends under uniaxial tension. This is achieved    G = u G D

when the sample length is four times the diameter (L = 4D). Local and global strain rates are defined, accordingly, as:

u L i+1 - u L i 0.01 D ∆t u G i+1 - u G i D ∆t

d  L /dt =

(4)

d  G /dt = (5) where indexes i and i+1 are used to designate the value at instants t and t+∆t respectively. The time discretization interval ∆t was chosen to guarantee the convergence of the elastic-plastic finite element computation. Results of such calculations, for L = 4D, are shown in Fig. 2, i.e, the relationship between local and global strain rates for the four geometries, as a function of global strain. In this plot, local strain rate was computed exactly at the notch tip, which corresponds to the sample surface. The relationship between local and global strain rates changes with time, as the plastic zone spreads or the global strain (or more properly the dimensionless global displacement) increases. It should be pointed out the influence of the spreading of the plastic zone on the evolution of local strain rate, thus emphasizing the importance of the sample geometry and the constitutive equation of the material.