Thermal expansion test

Elements tested

B21

B22

B31

B32

PIPE21

PIPE31

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 1 for the elasticity models, Figure 2 for the inelasticity models, and Figure 3 for the viscoelasticity models. Material properties are listed in Table 1 for the elastic materials and in Table 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 4, Figure 5, and Figure 6, respectively, except for pipe elements, whose results are consistent with beam elements.

Input files

simple_expansion_one.inp: Mises plasticity test.
simple_expansion.inp: Other material models and elements.

Tables

Table 1. Material properties for elastic materials.
Material	Properties	Value
Isotropic elasticity (density=8032)	E	193.1 × 10⁹
Isotropic elasticity (density=8032)	$ν$	0.3
Orthotropic elasticity (density=7850) (ENGINEERING CONSTANTS)	$E_{1}$	2.0 × 10¹¹
	$E_{2}$	1.0 × 10¹¹
	$E_{3}$	1.0 × 10¹¹
	$ν_{12}$	0.3
	$ν_{13}$	0.23
	$ν_{23}$	0.34
	$G_{12}$	7.69 × 10¹⁰
	$G_{13}$	7.69 × 10¹⁰
	$G_{23}$	9.0 × 10⁹
Orthotropic elasticity (density=7850) (ORTHOTROPIC)	$D_{1111}$	2.24 × 10¹¹
	$D_{1122}$	1.23 × 10¹¹
	$D_{2222}$	4.79 × 10¹¹
	$D_{1133}$	4.21 × 10¹⁰
	$D_{2233}$	4.74 × 10¹⁰
	$D_{3333}$	1.21 × 10¹¹
	$D_{1212}$	7.69 × 10¹⁰
	$D_{1313}$	7.69 × 10¹⁰
	$D_{2323}$	9.00 × 10⁹
Lamina (density=7800)	$E_{1}$	2.0 × 10¹¹
	$E_{2}$	1.5 × 10¹¹
	$ν_{12}$	0.35
	$G_{12}$	2.00 × 10¹⁰
	$G_{13}$	9.00 × 10⁹
	$G_{23}$	8.50 × 10⁹
Foam hyperelasticity (density=0.001)	N	2
	$ν$	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)	$D_{1111}$	2.24 × 10¹¹
	$D_{1122}$	1.23 × 10¹¹
	$D_{2222}$	4.79 × 10¹¹
	$D_{1133}$	4.21 × 10¹⁰
	$D_{2233}$	4.74 × 10¹⁰
	$D_{3333}$	1.21 × 10¹¹
	$D_{1112}$	1.00 × 10⁶
	$D_{2212}$	2.00 × 10⁶
	$D_{3312}$	3.00 × 10⁶
	$D_{1212}$	7.69 × 10¹⁰
	$D_{1113}$	4.00 × 10⁶
	$D_{2213}$	5.00 × 10⁶
	$D_{3313}$	6.00 × 10⁶
	$D_{1213}$	7.00 × 10⁶
	$D_{1313}$	7.69 × 10¹⁰
	$D_{1123}$	8.00 × 10⁶
	$D_{2223}$	9.00 × 10⁶
	$D_{3323}$	1.00 × 10⁷
	$D_{1223}$	1.10 × 10⁶
	$D_{1323}$	1.20 × 10⁶
	$D_{2323}$	9.00 × 10⁹
Polynomial hyperelasticity (density=1000)	N	2
	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)	N	3
	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)	E	193.1 × 10⁹
	$ν$	0.3
	${\bar{g}}_{i}$	0.901001
	${\bar{k}}_{i}$	0.0
	$τ_{i}$	0.99
	$θ_{0}$	70
	$C_{1}$	4.92
	$C_{2}$	215
Visco-polynomial hyperelasticity (density=1000)	N	2
	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)
	${\bar{g}}_{i}$	0.901001
	${\bar{k}}_{i}$	0.0
	$τ_{i}$	0.99
	$θ_{0}$	70
	$C_{1}$	4.92
	$C_{2}$	215
Visco-Ogden hyperelasticity (density=1000)	N	3
	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)
	${\bar{g}}_{i}$	0.901001
	${\bar{k}}_{i}$	0.0
	$τ_{i}$	0.99
	$θ_{0}$	70
	$C_{1}$	4.92
	$C_{2}$	215
Visco-foam hyperelasticity (density=0.001)	N	2
	$ν$	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)
	${\bar{g}}_{i}$	0.901001
	${\bar{k}}_{i}$	0.0
	$τ_{i}$	0.99
	$θ_{0}$	70
	$C_{1}$	4.92
	$C_{2}$	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)
	${\bar{g}}_{i}$	0.901001
	${\bar{k}}_{i}$	0.0
	$τ_{i}$	0.99
	$θ_{0}$	70
	$C_{1}$	4.92
	$C_{2}$	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)
	${\bar{g}}_{i}$	0.901001
	${\bar{k}}_{i}$	0.0
	$τ_{i}$	0.99
	$θ_{0}$	70
	$C_{1}$	4.92
	$C_{2}$	215

Table 2. Material properties for inelastic materials.
Material	Properties	Value
Mises plasticity (density=8032)	E	193.1 × 10⁹
	$ν$	0.3
	$σ_{0}$	206893
	H	206893
Drucker plasticity (density=1000)	E	2.0 × 10⁷
	$ν$	0.3
	$σ_{0}$	40000
	H	40000
	$β$	40
	K	1.0
	$ψ$	20.0
Hill's plasticity (density=2500)	E	1.0 × 10⁹
	$ν$	0.3
	$σ_{0}$	1.0 × 10⁶
	H	4.0 × 10⁵
	$R_{11}$	1.5
	$R_{22}$	1.0
	$R_{33}$	1.0
	$R_{12}$	1.0
	$R_{13}$	1.0
	$R_{23}$	1.0
Crushable foam with volumetric hardening (density=500)	E	3.0 × 10⁶
	$ν$	0.0
	k	1.1
	$k_{t}$	0.1
	hardening	(2.2× 10⁵, 0.0)
		... ...
		(6.88× 10⁵, 10.0)
Crushable foam with isotropic hardening (density=500)	E	3.0 × 10⁶
	$ν$	0.0
	k	1.1
	$ν_{p}$	0.2983
	hardening	(2.2× 10⁵, 0.0)
		... ...
		(6.88× 10⁵, 10.0)
Ductile failure (density=5800)	E	2.0 × 10⁸
	$ν$	0.3
	$σ_{0}$	2.0 × 10⁵
	H	4.0 × 10⁵
	${\bar{ϵ}}_{f}^{p l}$	0.5
Mises plasticity (density=8032)(rate dependent)	E	193.1 × 10⁹
	$ν$	0.3
	$σ_{0}$	206893
	H	206893
	D	1000
	p	2.0
Hill's plasticity (density=2500)(rate dependent)	E	1.0 × 10⁹
	$ν$	0.3
	$σ_{0}$	1.0 × 10⁶
	H	4.0 × 10⁵
	$R_{11}$	1.5
	$R_{22}$	1.0
	$R_{33}$	1.0
	$R_{12}$	1.0
	$R_{13}$	1.0
	$R_{23}$	1.0
	D	4000
	p	6.0
Drucker-Prager/Cap plasticity(density=0.0024)	E	30000
	$ν$	0.3
	d	100
	$β$	37.67
	R	0.1
	$ϵ_{v o l}^{p l}$	0.0
	$α$	0.01
	hardening	(20.96, 0)
		... ...
		(655.6, 0.00249)
Porous metal plasticity(density=7.7 × 10⁷)	E	2.0 × 10¹¹
	$ν$	0.33
	$σ_{0}$	7.5 × 10⁸
	H	0.0
	$q_{1}$	1.0
	$q_{2}$	1.25
	$q_{3}$	1.0
	$ϵ_{N}$	0.1
	$s_{N}$	0.06
	$f_{N}$	0.04
	$f_{F}$	0.8
	$f_{c}$	0.5

Figures