PSI - Issue 26

Christos F. Markides et al. / Procedia Structural Integrity 26 (2020) 53–62 Ch. F. Markides et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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7

Fig. 4. The axial stress σ xx along y-axis. Red color corresponds to the notched beam and blue color to the intact one.

The locus along which ε xx and ε yy were analytically determined, was the C΄H΄ segment of the y-axis (not very close to the notch tip C and to the punch - for the case of the three-point bending test), for consistency with the experimental data (recall that strain gauges could not be attached very close to the above neighborhoods). Though not straightforward (three-point bending against four-point bending), the comparison reveals quite satisfactory similarities between theory and experiment, providing, at least, some qualitative evidence supporting the validity of the analytic solution.

2.4. Notched beam: The displacement field Having obtained Φ(ζ) and Ψ(ζ), φ(ζ) and ψ(ζ) are derived as (Muskhelishvili 1963): ( ) ( ) ( ) ( ) ( ) ( ) d , d     =        =       

(20)

In turn, the horizontal, u, and vertical, v, components of the displacement are calculated at every point of the notched beam through the well-known formulae ( μ is the shear modulus of the material ) (Muskhelishvili 1963): ( ) ( ) ( ) ( ) 1 u iv   + =   −     −     

2

(21)

E , plane strain 

(   +  −   ) ( 1 1 2

)

3 ,

 + 

 

 =

 = 

 + 

2 E , plane stress 1 

  −  