PSI - Issue 42

T. Fekete et al. / Procedia Structural Integrity 42 (2022) 1684–1691

1689

6

T. Fekete et al.: Extending reliability of FEM simulations… / Structural Integrity Procedia 00 (2019) 000–000

4. Corrections of the stress-strain curve

As mentioned earlier, the finel aim of this development is to achieve a level of accuracy of constitutive models, developed by full-field evaluation of the observed measurements, that –in extension to the global force and displacement fit– provides a sufficiently accurate match between the observed shape of the sample and the DT simulations over the full geometry. The method is still being developed and its first results are promising. However, experience shows that it is quite resource demanding, and therefore, at least for some time, is likely to remain an expensive approach. Therefore, based on the progression achieved so far, efforts have recently been extended to the development of an evaluation procedure, which is based on the literature, requires much less resources, but is expected to have satisfactory accuracy for many applications, such as the Bridgman correction for cylindrical tensile test specimens –Bridgman (1952)–. Choung and Cho developed a correction scheme for rectangular specimens that had previously been applied to structural steels used in the shipbuilding industry –see Choung and Cho (2008)–. The correction works the same way as the Bridgman correction for cylindrical specimens. It is known that the evaluation of tensile tests beyond the start of necking, i.e., in the ‘diffuse necking’ regime, is challenging. The reason is that the strain and stress fields become inhomogeneous, to a degree there, which is not followed by engineering models commonly used to date. However, correction-based methods are useful in this regime. Choung and Cho assume the flow curve to satisfy the Ludwik equation –see Ludwik (1909)–: ( ) n y pl K σ σ ε = + ⋅ (6)

pl ε stands for plastic strain, K denotes strength coefficient and

y σ means yield stress,

where σ is true flow stress,

n is the plastic hardening exponent. The corrective factor is expressed, as follows:

( ) pl σ ς ε σ = ⋅ eq

avax

where

1.4 1.4 ≤ ⋅ > ⋅

ε ε

 

1

for

n

( ) pl

= 

ζ ε

pl

2 α ε β ε γ ⋅ + ⋅ + pl pl

for

n

pl

(7)

with

= −

0.0704 0.0275 0.4550 0.2926 0.1592 1.024 n n n ⋅ − ⋅ − ⋅ +

α β γ

=

=

avax σ is the averaged axial true stress in the minimal cross section of the

eq σ means equivalent and

where

specimen, ς stands for stress correction factor, n and pl ε are as above, α , β and γ are coefficients of the polynomial. The onset of necking is checked using Considère’s criterion –see Considère (1885)–. The criterion was originally formulated in engineering strain – engineering stress system, but it has been transformed into the true strain – true stress system –see Tu, Ren, He and Zhang (2020)–. For the Ludwik equation, this means that necking approximately starts when the corresponding plastic strain is equal to the work hardening exponent n , that is n n ε = , where n ε is the plastic strain at the onset of necking. Due to the inhomogeneous, polycrystalline structure of the material, diffuse necking is initiated at multiple locations simultaneously, but finally a single necking becomes dominant. It is this dominant region where final rupture will appear.

5. Characteristics of the corrected stress-strain curve

The foregoing sections have presented the theoretical model implemented in the digital pair of tensile tests, the experimental setup, and a correction procedure that seems to offer promise for evaluation of the measurements.