Critical Plastic Strain

About the development of the FEM module/workbench.

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HarryvL
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Critical Plastic Strain

Post by HarryvL »

The issue of what to do with stress concentrations in FEA came up in several posts:

https://forum.freecadweb.org/viewtopic. ... 21#p215672

https://forum.freecadweb.org/viewtopic. ... 31#p220591

https://forum.freecadweb.org/viewtopic. ... 97#p266484

The paradox is that we appear to get punished in “FEM-assisted-design” for more accurate prediction of stress concentrations.

As explained in the above posts, the solution is to take account of plastic strains in the analysis and thereby redistribute peak stresses to lower loaded regions, as will happen in reality.

Then the design question becomes “what is an acceptable plastic strain?” or put differently “will the heavily loaded region crack before the ultimate limit load of the construction detail is reached?”

In my previous posts I recommended to limit the equivalent plastic strain (PEEQ in CCX and ABAQUS terms) to well below 0.05 or 5%.

Now we have fcFEM I can experiment with a more refined check for ductile fracture.

The specification of a critical plastic strain is the topic of active research (see https://www.vtt.fi/inf/julkaisut/muut/2 ... 177-17.pdf and https://www.vtt.fi/inf/julkaisut/muut/2 ... 741-16.pdf) and for the moment design codes make very conservative assumptions; typically eps_cr = 5%.

A better approach (that is both practical and widely accepted) is use of the Stress Modified Critical Strain (SMCS) criterion (see the above reports). It links critical plastic strain (eps_cr) to stress triaxiality (T):

eps_cr = alpha * exp(-beta * T)

where alpha is a material (ductility) parameter, beta=1.5 and T = pressure / sig_von_Mises.

This shows that steel under high isotropic tension (pressure>>0) is much more brittle than steel in pure shear (pressure=0). Alpha factors quoted in literature range for construction steel from 1-5, with the lower values applicable for higher strength steels.

I implemented the SMCS criterion in fcFEM and would like to demonstrate its application for the case of a plate with hole of previous posts.

The design question is: “will the material crack before the plastic limit load of the detail is reached?”. To answer this question I conservatively assume ductility at the lower end of the range (i.e. alpha=1.0) and then increase the load on the detail until the limit load is reached:

fcFEM_Plate_with_Hole_EpsP_Ratio.png
fcFEM_Plate_with_Hole_EpsP_Ratio.png (30.58 KiB) Viewed 5525 times

The acceptability of the induced plastic strain can now be reviewed by plotting the ratio of equivalent plastic strain to critical plastic strain in the last load step:

fcFEM_Plate_with_Hole_eps_critical_ratio.png
fcFEM_Plate_with_Hole_eps_critical_ratio.png (114.49 KiB) Viewed 5525 times

If the plastic strain ratio is low, so is the risk of ductile fracture. As can be seen from the plot, the maximum plastic strain ratio is of the order 0.02, which is far below the critical value of 1.0.
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HarryvL
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Re: Critical Plastic Strain

Post by HarryvL »

I added the critical plastic strain ratio as a post-processing step for CCX results. This is available from the VTK pipeline and for export to ParaView:

Spanner - Critical Strain Ratio from CCX (2).png
Spanner - Critical Strain Ratio from CCX (2).png (130.48 KiB) Viewed 4858 times
Spanner - Critical Strain Ratio from CCX.png
Spanner - Critical Strain Ratio from CCX.png (42.64 KiB) Viewed 4858 times
Plate with Hole - Critical Strain Ratio from CCX (2).png
Plate with Hole - Critical Strain Ratio from CCX (2).png (151.7 KiB) Viewed 4858 times
Plate with Hole - Critical Strain Ratio from CCX.png
Plate with Hole - Critical Strain Ratio from CCX.png (83.49 KiB) Viewed 4858 times

However, the 3 additional material parameters still need to be added to the material input objects
thschrader
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Re: Critical Plastic Strain

Post by thschrader »

HarryvL wrote: Mon Sep 05, 2022 11:43 am However, the 3 additional material parameters still need to be added to the material input objects
Do you mean the yield-points?
dnv208.JPG
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HarryvL
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Re: Critical Plastic Strain

Post by HarryvL »

thschrader wrote: Mon Sep 05, 2022 11:53 am
HarryvL wrote: Mon Sep 05, 2022 11:43 am However, the 3 additional material parameters still need to be added to the material input objects
Do you mean the yield-points?
dnv208.JPG
No, I mean alpha, beta and sig_y to compute the critical strain (see model at beginning of the post). I guess it is only 2, because sig_y is implied in the stress-strain curve.
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HarryvL
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Re: Critical Plastic Strain

Post by HarryvL »

PS: you give yield stress first and strain second in your list. I thought it was the other way around?
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Re: Critical Plastic Strain

Post by thschrader »

yield-stress first, than strain (and maybe temperature)
See ccx manual 2.20 page 533:
plastic_command_ccx.JPG
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HarryvL
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Re: Critical Plastic Strain

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HarryvL wrote: Mon Sep 05, 2022 11:43 am I added the critical plastic strain ratio as a post-processing step for CCX results. This is available from the VTK pipeline and for export to ParaView:
CCX predicts a critical strain ratio of 0.2 at full load (1.0 signifies rupture), but cannot push beyond that due to breakdown of the solver at collapse.

fcFEM happily continues at collapse and at the end of the following load-deflection curve computes a critica strain ratio of 0.56 ... getting closer to rupture.

fcFEM beyond max CCX load (F-u).png
fcFEM beyond max CCX load (F-u).png (20.5 KiB) Viewed 2867 times

fcFEM beyond max CCX load (csr).png
fcFEM beyond max CCX load (csr).png (162.93 KiB) Viewed 2867 times
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HarryvL
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Re: Critical Plastic Strain

Post by HarryvL »

I have now integrated the critical strain ratio calculation in FC such that no additional material parameters need to be entered.

Starting from the definition:

critical strain ratio = PEEQ / (alpha * exp(-beta * T)),

where:
PEEQ = equivalent plastic strain (produced by CCX),
T = stress triaxiality = pressure / sig_von_Mises (calculated by FC),
beta=1.5,
alpha is a material (ductility) parameter,

So alpha is the only unknown. However, this can be linked to the critical plastic strain in the stress-strain curve, for which the triaxiality T = 1/3.

With this it can be shown that alpha = 1.65 * eps_cr_uniaxial (the last strain entered by the user in the MaterialMechanicalNonlinear object)

So this is how I have now implemented the Critical Plastic Strain calculation in FC. It takes all required parameters from the user input.

I will give an example of how this works and then start to prepare a pull request.
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Re: Critical Plastic Strain

Post by Kunda1 »

sweet!
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thschrader
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Re: Critical Plastic Strain

Post by thschrader »

For testing:
DNV-RP-C208 manual, example B.1.2 page 44, cantilever beam with notch
https://rules.dnv.com/docs/pdf/DNVPM/co ... P-C208.pdf

zip contains FC-file, SMath-file, force definition pdf
DNV_T_section.zip
(125.85 KiB) Downloaded 68 times

Bonus file:
Frame buckling, example B2 page 46:
bucklingFrame.FCStd
(203.09 KiB) Downloaded 68 times
exampleDNV.JPG
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