ABAQUS Verification Manual  


2.2.32 Thermal expansion test

Product: ABAQUS/Explicit  

Elements tested

B21    B22    B31    B32    C3D8R    C3D10M    CPE4R    CPE6M    CPS4R    CPS6M    CAX4R    CAX6M    M3D4R    S4R    S4RS    S4RSW    SAX1    T2D2    T3D2   

Features tested

Thermal expansion defined by a predefined temperature field is tested for the following material models: isotropic elasticity, orthotropic elasticity, anisotropic elasticity, lamina, hyperelasticity with polynomial and Ogden forms, hyperelasticity with Arruda-Boyce and Van der Waals forms, hyperfoam, Mises plasticity, Drucker-Prager plasticity, Hill's potential plasticity, crushable foam plasticity with volumetric hardening, crushable foam plasticity with isotropic hardening, ductile failure plasticity, rate-dependent Hill's potential plasticity, rate-dependent Mises plasticity, Drucker-Prager/Cap plasticity, porous metal plasticity, visco-hyperelasticity with polynomial and Ogden forms, visco-hyperelasticity with Arruda-Boyce and Van der Waals forms, and visco-hyperfoam.

Problem description

The verification tests consist of a set of single element tests that include a combination of all the available elements with all the available materials. All elements are loaded by ramping up the temperature from an initial value of 0° to a final value of 100°. The undeformed meshes are shown in Figure 2.2.32–1 for the elasticity models, Figure 2.2.32–2 for the inelasticity models, and Figure 2.2.32–3 for the viscoelasticity models. Material properties are listed in Table 2.2.32–1 for the elastic materials and in Table 2.2.32–2 for the inelastic materials. The thermal expansion coefficient for all materials is 0.00005.

The degrees of freedom in the vertical direction are constrained for all the nodes, and deformation is allowed only in the horizontal direction. Nodes associated with elements C3D8R and C3D10M are constrained in the out-of-plane direction, which causes a plane strain condition to apply for these elements.

Results and discussion

The time history plots for isotropic elasticity, Mises plasticity, and viscoelasticity for all of the elements are shown in Figure 2.2.32–4, Figure 2.2.32–5, and Figure 2.2.32–6, respectively.

Input files

simple_expansion_one.inp

Mises plasticity test.

simple_expansion.inp

Other material models and elements.

Tables

Table 2.2.32–1 Material properties for elastic materials.

MaterialPropertiesValue
Isotropic elasticity (density=8032)E193.1 × 109
0.3
Orthotropic elasticity (density=7850) (ENGINEERING CONSTANTS)2.0 × 1011
1.0 × 1011
1.0 × 1011
0.3
0.23
0.34
7.69 × 1010
7.69 × 1010
9.0 × 109
Orthotropic elasticity (density=7850) (ORTHOTROPIC)2.24 × 1011
1.23 × 1011
4.79 × 1011
4.21 × 1010
4.74 × 1010
1.21 × 1011
7.69 × 1010
7.69 × 1010
9.00 × 109
Lamina (density=7800)2.0 × 1011
1.5 × 1011
0.35
2.00 × 1010
9.00 × 109
8.50 × 109
Foam hyperelasticity (density=0.001)N2
0.01
uniaxial test(–0.0217, –0.05)
 ... ...
 (–0.02896, –0.80)
simple shear test(0.0140, 0.08, 0.0046)
 ... ...
 (0.2987, 0.72, 0.1904)
Anisotropic elasticity (density=7850)2.24 × 1011
1.23 × 1011
4.79 × 1011
4.21 × 1010
4.74 × 1010
1.21 × 1011
1.00 × 106
2.00 × 106
3.00 × 106
7.69 × 1010
4.00 × 106
5.00 × 106
6.00 × 106
7.00 × 106
7.69 × 1010
8.00 × 106
9.00 × 106
1.00 × 107
1.10 × 106
1.20 × 106
9.00 × 109
Polynomial hyperelasticity (density=1000)N2
uniaxial test(155060, 0.1338)
 ... ...
 (6.424 × 106, 6.6433)
biaxial test(93840, 0.02)
 ... ...
 (2.465 × 106, 3.45)
planar test(60000, 0.0690)
 ... ...
 (1.82 × 106, 4.0621)
Ogden hyperelasticity (density=1000)N3
uniaxial test(155060, 0.1338)
 ... ...
 (6.424 × 106, 6.6433)
biaxial test(93840, 0.02)
 ... ...
 (2.465 × 106, 3.45)
planar test(60000, 0.0690)
 ... ...
 (1.82 × 106, 4.0621)
Arruda-Boyce hyperelasticity (density=1000)uniaxial test(155060, 0.1338)
 ... ...
 (6.424 × 106, 6.6433)
biaxial test(93840, 0.02)
 ... ...
 (2.465 × 106, 3.45)
planar test(60000, 0.0690)
 ... ...
 (1.82 × 106, 4.0621)
Van der Waals hyperelasticity (density=1000)uniaxial test(155060, 0.1338)
 ... ...
 (6.424 × 106, 6.6433)
biaxial test(93840, 0.02)
 ... ...
 (2.465 × 106, 3.45)
planar test(60000, 0.0690)
 ... ...
 (1.82 × 106, 4.0621)
Viscoelasticity (density=8032)E193.1 × 109
0.3
0.901001
0.0
0.99
70
4.92
215
Visco-polynomial hyperelasticity (density=1000)N2
uniaxial test(155060, 0.1338)
 ... ...
 (6.424 × 106, 6.6433)
biaxial test(93840, 0.02)
 ... ...
 (2.465 × 106, 3.45)
planar test(60000, 0.0690)
 ... ...
 (1.82 × 106, 4.0621)
0.901001
0.0
0.99
70
4.92
215
Visco-Ogden hyperelasticity (density=1000)N3
uniaxial test(155060, 0.1338)
 ... ...
 (6.424 × 106, 6.6433)
biaxial test(93840, 0.02)
 ... ...
 (2.465 × 106, 3.45)
planar test(60000, 0.0690)
 ... ...
 (1.82 × 106, 4.0621)
0.901001
0.0
0.99
70
4.92
215
Visco-foam hyperelasticity (density=0.001)N2
0.0
uniaxial test(–0.0217, –0.05)
 ... ...
 (–0.02896, –0.80)
simple shear test(0.0140, 0.08, 0.0046)
 ... ...
 (0.2987, 0.72, 0.1904)
0.901001
0.0
0.99
70
4.92
215
Visco-Arruda-Boyce hyperelasticity (density=1000)uniaxial test(155060, 0.1338)
 ... ...
 (6.424 × 106, 6.6433)
biaxial test(93840, 0.02)
 ... ...
 (2.465 × 106, 3.45)
planar test(60000, 0.0690)
 ... ...
 (1.82 × 106, 4.0621)
0.901001
0.0
0.99
70
4.92
215
Visco-Van der Waals hyperelasticity (density=1000)uniaxial test(155060, 0.1338)
 ... ...
 (6.424 × 106, 6.6433)
biaxial test(93840, 0.02)
 ... ...
 (2.465 × 106, 3.45)
planar test(60000, 0.0690)
 ... ...
 (1.82 × 106, 4.0621)
0.901001
0.0
0.99
70
4.92
215

Table 2.2.32–2 Material properties for inelastic materials.

MaterialPropertiesValue
Mises plasticity (density=8032)E193.1 × 109
0.3
206893
H206893
Drucker plasticity (density=1000)E2.0 × 107
0.3
40000
H40000
40
K1.0
20.0
Hill's plasticity (density=2500)E1.0 × 109
0.3
1.0 × 106
H4.0 × 105
1.5
1.0
1.0
1.0
1.0
1.0
Crushable foam with volumetric hardening (density=500)E3.0 × 106
0.0
k1.1
0.1
hardening(2.2× 105, 0.0)
 ... ...
 (6.88× 105, 10.0)
Crushable foam with isotropic hardening (density=500)E3.0 × 106
0.0
k1.1
0.2983
hardening(2.2× 105, 0.0)
 ... ...
 (6.88× 105, 10.0)
Ductile failure (density=5800)E2.0 × 108
0.3
2.0 × 105
H4.0 × 105
0.5
Mises plasticity (density=8032)(rate dependent)E193.1 × 109
0.3
206893
H206893
D1000
p 2.0
Hill's plasticity (density=2500)(rate dependent)E1.0 × 109
0.3
1.0 × 106
H4.0 × 105
1.5
1.0
1.0
1.0
1.0
1.0
D4000
p6.0
Drucker-Prager/Cap plasticity(density=0.0024)E30000
0.3
d100
37.67
R0.1
0.0
0.01
hardening(20.96, 0)
 ... ...
 (655.6, 0.00249)
Porous metal plasticity(density=7.7 × 107)E2.0 × 1011
0.33
7.5 × 108
H0.0
1.0
1.25
1.0
0.1
0.06
0.04
0.8
0.5


Figures

Figure 2.2.32–1 Simple expansion test for elastic materials.

Figure 2.2.32–2 Simple expansion test for inelastic materials.

Figure 2.2.32–3 Simple expansion test for viscoelastic materials.

Figure 2.2.32–4 Mises stress versus time for isotropic elasticity.

Figure 2.2.32–5 Mises stress versus time for Mises plasticity.

Figure 2.2.32–6 Mises stress versus time for viscoelasticity.