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Bilinear1D

Uniaxial Bilinear Material Using J2 Plasticity

The Bilinear1D material implements a simple plasticity model that employs both linear isotropic and kinematic hardening rules. Further details can be found Section 1.4.4 of Computational Inelasticity . Alternatively, refer to the corresponding section in Constitutive Modelling Cookbook for implementation details.

It is suitable for simple hysteresis modelling. For more complex behaviour, one shall use other models.

Syntax

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material Bilinear1D (1) (2) (3) [4] [5] [6]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, initial yield stress
# [4] double, hardening ratio, default: 0.0
# [5] double, beta, default: 1.0
# [6] double, density, default: 0.0

History Variable Layout

location value
initialize_history(0) back_stress
initialize_history(1) equivalent_plastic_strain

Remarks

  1. The parameter beta is employed for mixed hardening rule: zero \(0.0\) for kinematic hardening and unity \(1.0\) for isotropic hardening, anything falls in the range indicates a mixed hardening response.
  2. Negative hardening ratio (softening) is supported. However, the yield surface will always be non-negative.

Usage

Isotropic Hardening

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material Bilinear1D 1 1000.0 10.0 0.2 1.0
materialTest1D 1 0.001 20 40 60 80 100 100
exit

Example 1

Isotropic Softening

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material Bilinear1D 1 1000.0 10.0 -0.02 1.0
materialTest1D 1 0.001 20 40 60 80 100 100
exit

Example 2

Kinematic Hardening

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material Bilinear1D 1 1000.0 10.0 0.1 0.0
materialTest1D 1 0.001 20 40 60 80 100 100
exit

Example 3

Combined Hardening

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material Bilinear1D 1 1000.0 10.0 0.1 0.5
materialTest1D 1 0.001 20 40 60 80 100 100
exit

Example 4

Accuracy

There is no local iteration required. The result is accurate.

Python
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from plugins import ErrorLine
# note: the dependency `ErrorLine` can be found in the following link
# https://github.com/TLCFEM/suanPan-manual/blob/dev/plugins/scripts/ErrorLine.py

young_modulus = 2e5
yield_stress = 4e2
hardening_ratio = 0.05

with ErrorLine(
    f"material Bilinear1D 1 {young_modulus} {yield_stress} {hardening_ratio} .5",
    ref_strain=yield_stress / young_modulus,
    ref_stress=yield_stress,
) as error_map:
    error_map.contour("bilinear1d", center=-5, size=10, type={"abs"})

accuracy analysis