Barlat anisotropic plasticity

This problem contains basic test cases for one or more Abaqus elements and features.

This page discusses:

Elements tested
Features tested
Problem description
Results and discussion
Input files
References
Tables
Figures

Elements tested

C3D8R

Features tested

Classical metal plasticity model with Barlat yield surfaces.

Problem description

This verification test contains a set of single-element models subject to uniaxial tension for various off-axis angles between the material directions and the loading direction. The test is designed to verify the yield stress ratios and strain ratios (Lankford's r-values) obtained in the simulations against the analytical solutions.

Model

Figure 1 shows the elements used in the analysis. All elements are subject to uniaxial tensile loading in the global x-direction. Each element has a unique material orientation. The material orientation is assigned such that for the three rows of elements (from bottom to top), the loading direction lies in the material 1-2 plane, 2-3 plane, 3-1 plane, respectively; and for the seven columns of elements (from left to right), the off-axis angle of loading is 0°, 15°, 30°, 45°, 60°, 75°, 90°, respectively.

Material

Elastic

Young's modulus, $E = 70.0 \times 10^{3}$

Poisson's ratio, $υ = 0.33$

Plastic

Hardening:


Yield stress	Plastic strain
150.	0.000
265.	0.247

(The units are not important.)

Four materials with Barlat anisotropy coefficients are tested. Coefficients of the Barlat Yld2004-18p yield surface for material 2090-T3 and 6111-T4 are shown in Table 1. The yield surfaces of the material 2090-T3 are illustrated in Figure 2, Figure 3, and Figure 4 for three possible plane stress conditions, respectively. The other two materials, 2024-T3 and 2008-T4, use the Barlat Yld91 yield surface with anisotropy coefficients as shown in Table 2.

In addition, the Mises isotropic plasticity is recovered by using an equivalent definition for the Barlat yield surfaces.

Results and discussion

The yield stress ratio is defined as

R = \frac{σ_{x}}{σ^{0}},

where $σ_{x}$ is the stress in the loading direction that can be obtained as output variable S11 in the global frame and $σ^{0}$ is the reference yield stress that can be obtained as output variable YIELDPOT.

The strain ratio (Lankford's r-value) is defined as

r = \frac{ε_{y}}{ε_{z}},

where $ε_{y}$ and $ε_{z}$ are the plastic strains in the width and thickness directions, respectively, corresponding to output variables PE22 and PE33 in the global frame.

The results of the yield stress and strain ratios agree well with the analytical solutions for the four materials. Comparisons of the simulation results and analytical solutions for the material 2090-T3 with loading applied on three material planes are shown in Figure 5, Figure 6, and Figure 7.

The yield stress ratio and strain ratio of the Mises plasticity are always one. The models with equivalent definitions of the Mises yield surface also match the analytical solutions.

Input files

barlat_c3d8r_2090t3.inp: Analysis using the Barlat Yld2004-18p yield surface for the material 2090-T3.
barlat_c3d8r_6111t4.inp: Analysis using the Barlat Yld2004-18p yield surface for the material 6111-T4.
barlat_c3d8r_mises.inp: Analysis using the Barlat Yld2004-18p yield surface with an equivalent definition of the Mises yield surface.
barlat91_c3d8r_2024t3.inp: Analysis using the Barlat Yld91 yield surface for the material 2024-T3.
barlat91_c3d8r_2008t4.inp: Analysis using the Barlat Yld91 yield surface for the material 2008-T4.
barlat91_c3d8r_mises.inp: Analysis using the Barlat Yld91 yield surface with an equivalent definition of the Mises yield surface.

References

Barlat, F., H. Aretz, J. W. Yoon, M. E. Karabin, J. C. Brem , and R. E. Dick, “Linear Transformation-based Anisotropic Yield Functions,” International Journal of Plasticity, vol. 21, pp. 1009–1039, 2005.
Barlat, F., D. J. Lege , and J. C. Brem, “A Six-Component Yield Function for Anisotropic Materials,” International Journal of Plasticity, vol. 7, pp. 693–712, 1991.

Tables

Table 1. Anisotropic coefficients of the Barlat Yld2004-18p yield surface for material 6111-T4 and 2090-T3 (Barlat et al., 2005).
Abaqus notation	Barlat's notation	Material 2090-T3	Material 6111-T4
$α$	$a$	8.0	8.0
$c_{1122}^{'}$	$c_{12}^{'}$	-0.069888	1.078271
$c_{1133}^{'}$	$c_{13}^{'}$	0.936408	1.078271
$c_{2211}^{'}$	$c_{21}^{'}$	0.079143	1.216463
$c_{2233}^{'}$	$c_{23}^{'}$	1.003060	1.223867
$c_{3311}^{'}$	$c_{31}^{'}$	0.524741	1.093105
$c_{3322}^{'}$	$c_{32}^{'}$	1.363180	0.889161
$c_{1212}^{'}$	$\frac{1}{2} c_{66}^{'}$	0.477161	0.674547
$c_{1313}^{'}$	${\frac{1}{2} c}_{55}^{'}$	0.534530	0.278587
$c_{2323}^{'}$	$\frac{1}{2} c_{44}^{'}$	0.511885	0.250955
$c_{1122}^{"}$	$c_{12}^{"}$	0.981171	0.775366
$c_{1133}^{"}$	$c_{13}^{"}$	0.476741	0.922743
$c_{2211}^{"}$	$c_{21}^{"}$	0.575316	0.765487
$c_{2233}^{"}$	$c_{23}^{"}$	0.866827	0.793356
$c_{3311}^{"}$	$c_{31}^{"}$	1.145010	0.918689
$c_{3322}^{"}$	$c_{32}^{"}$	-0.079294	1.027625
$c_{1212}^{"}$	$\frac{1}{2} c_{66}^{"}$	0.702310	0.294894
$c_{1313}^{"}$	${\frac{1}{2} c}_{55}^{"}$	0.573550	0.556137
$c_{2323}^{"}$	$\frac{1}{2} c_{44}^{"}$	0.525830	0.557917

Table 2. Anisotropic coefficients of the Barlat Yld91 yield surface for material 2008-T4 and 2024-T3 (Barlat et al., 1991).
Coefficients	Material 2024-T3	Material 2008-T4
$α$	8.0	11.0
$\bar{a}$	1.378	1.222
$\bar{b}$	1.044	1.013
$\bar{c}$	0.955	0.985
$\bar{f}$	1.0	1.0
$\bar{g}$	1.0	1.0
$\bar{h}$	1.210	1.0

Figures