Elasto-plastic numerical models ⛓

Computational Mechanics of Tissues and Biomaterials

Biomedical Engineering - University of Rome Tor Vergata

Abstract

In this analysis some computational models are introduced for the finite element analysis of isotropic and kinematic hardening behaviors. The response to a cyclic load is analyzed for some stainless steels. These simulations exploit a linear quadratic finite element with elastic-plastic behavior with isotropic and kinematic hardening. These models are fundamental for analyzing the material response following a repeated cyclic load.

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Read more website

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Energy redefine:

\[CapitalPsi]e \[DoubleRightTee] \[Lambda]e/
    2 (Tr[\[DoubleStruckCapitalD]e])^2 + \[Mu]e Tr[\
\[DoubleStruckCapitalD]e . \[DoubleStruckCapitalD]e] + 
   1/2 H Tr[Transpose[\[Alpha]] . \[Alpha]];

Redefine yelding function:

fg \[DoubleRightTee] 
SMSSqrt[
 3/2 Tr[
   Transpose[\[Sigma]dev - q] . (\[Sigma]dev - q)]] - (\[Sigma]YO);

Does vector derivative require special definition ? Like $\mathbf{q}=-\frac{\partial \Psi}{\partial \boldsymbol{\alpha}}$ or $\mathbf{Q}{\alpha}=\boldsymbol{\alpha}-\boldsymbol{\alpha}{n}-\left(\lambda-\lambda_{n}\right) \frac{\partial f}{\partial \mathbf{q}}$ ?

Also update:

\[Alpha]n = {{\[DoubleStruckH]gnIO[[4]], \[DoubleStruckH]gnIO[[
   5]]}, {\[DoubleStruckH]gnIO[[5]], \[DoubleStruckH]gnIO[[6]]}};
\[Lambda]n = \[DoubleStruckH]gnIO[[7]];

Must update also q with symmetric flag:

SMSFreeze[q, -SMSD[\[CapitalPsi]e, \[Alpha], "Symmetric" -> True], 
  "Symmetric" -> True];

Output for the procedure:

\[DoubleStruckCapitalQ] = {\[DoubleStruckCapitalQ]\[Epsilon][[1, 
    1]], \[DoubleStruckCapitalQ]\[Epsilon][[2, 
    2]], \[DoubleStruckCapitalQ]\[Epsilon][[1, 2]], Qq[[1, 1]], 
   Qq[[2, 2]], Qq[[1, 2]], Q\[Lambda]};

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Notes

• 21 dec - Testing for the cinematic hardening introduction.
• 22 dec - Kinematic hardening works. Talked with prof. See first_test README.md

• Before going on create the two different elements for FEM

• 24 dec - Creation of two different elements: Q1EPS and Q1EPS2 and done patch test.

• Before going one transform the notebook to easly select Q1EPS or Q1EPS2. Use procedure with isotropic/cinematic.

• 1 gen - Created Q1EPS2. Implemented stress factor calculation.
• 26 gen - Other test and implmented AnimationOfResponse[]to export mp4 video of deformation
• 27 gen - Fi notebook.
• 30 gen - Test
• 31 gen - Test
• 3 feb - Edited K, stress intensity factor