Coupled temperature-displacement model change: steady state

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

The following topics are discussed:

Elements tested
Features tested
Problem description
Reference solution
Results and discussion
Input files

ProductsAbaqus/Standard

Elements tested

C3D8HT

C3D8RT

C3D8RHT

C3D8T

C3D10MT

C3D10MHT

CAX4RHT

CAX4RT

CAX6MHT

CAX6MT

CGAX3HT

CGAX3T

CGAX4HT

CGAX4RT

CGAX4RHT

CGAX4T

CGAX6MHT

CGAX6MT

CGAX8HT

CGAX8RT

CGAX8RHT

CGAX8T

CPE4RT

CPE4RHT

CPE4T

CPE6MHT

CPE6MT

CPEG3HT

CPEG3T

CPEG4HT

CPEG4RT

CPEG4RHT

CPEG4T

CPEG6MHT

CPEG6MT

CPEG8HT

CPEG8RHT

CPEG8T

CPS4RT

CPS4T

CPS6MT

CPS8RT

CPS8T

Features tested

Continuum coupled temperature-displacement elements are removed and added during a steady-state analysis.

Problem description

Model:

The models have dimensions 5.0 × 2.0 in the x–y plane with an out-of-plane dimension of 1.0. In the axisymmetric case the models have dimensions 2.0 × 5.0 in the r–z plane, and the inner radius, $R_{i}$ , equals 10⁵. The inner radius is large to ensure that the strains in the circumferential direction are approximately uniform, which allows a comparison of the results obtained in this analysis with those obtained analytically.

Material:

Conductivity	7.872 × 10⁻⁴
Density	0.2829
Thermal expansion	1.0 × 10⁻⁶
Elastic modulus	100 × 10⁴
Poisson's ratio	0.25

Loading and boundary conditions

The left side of the model is held at $θ$ =0.0. There is a film condition on the right side of the model. The sink temperature is $θ^{0}$ =100.0, and the film coefficient, h, is 1.0. After a steady-state solution is obtained, some of the elements in the model are removed. The temperatures along the new external boundary are held fixed. The removed elements are added back into the model in the last step, and a new film condition is applied on the right side. The new sink temperature is $θ^{0}$ =200.0, and the same h is used.

During all three steps the following mechanical boundary conditions are maintained: $u_{y}$ =0.0 at all points along y=0; $u_{x}$ =0.0 at the point (0,0).

Reference solution

The solution for the one-dimensional steady-state heat transfer problem is given in Heat transfer model change: steady state. The solution for the mechanical response of the model is

ϵ_{x} = \frac{\partial u}{\partial x} = ϵ_{x}^{t h e} = α θ (x) = α \frac{x}{L + k / h} θ^{0} .

The expression for $ϵ_{x}$ is integrated to give

u = \frac{1}{2} α \frac{x^{2}}{L + k / h} θ^{0} + f (y) .

The y-component of strain is given as

ϵ_{y} = \frac{\partial v}{\partial y} = α \frac{x}{L + k / h} θ^{0} .

Integrating for v gives

v = α \frac{x y}{L + k / h} θ^{0},

where the boundary condition that v=0 at y=0 is used to eliminate the terms that are only functions of x. The condition that

γ_{x y} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} = 0

is used to find $f (y)$ , and the x-displacement is given as

u = \frac{α}{2} \frac{(x^{2} - y^{2})}{L + k / h} θ^{0} .

These expressions are used to calculate the displacements in the model. The temperature distribution can be calculated with the expression from Heat transfer model change: steady state. The results for the axisymmetric case are obtained by replacing x with z and y with ( $r - R_{i}$ ) in the relations for temperature and displacements. In addition, the displacements are multiplied by a factor of ( $1 + ν$ ), where $ν$ is the Poisson's ratio. This takes into account the contribution from the approximately constant strain in the circumferential direction.

Results and discussion

The model produces the theoretical results in both the first and third steps for the element temperatures and for the quadratic element displacements. The displacements obtained using the model with linear elements do not match the theoretical results but are still reasonable.

Input files

pmce_c3d8ht_ctd.inp: General test of C3D8HT elements in steady-state analysis.
pmce_c3d8rht_ctd.inp: General test of C3D8RHT elements in steady-state analysis.
pmce_c3d8rt_ctd.inp: General test of C3D8RT elements in steady-state analysis.
pmce_c3d8t_ctd.inp: General test of C3D8T elements in steady-state analysis.
pmce_c3d10mht_ctd.inp: General test of C3D10MHT elements in steady-state analysis.
pmce_c3d10mt_ctd.inp: General test of C3D10MT elements in steady-state analysis.
pmce_cax4rht_ctd.inp: General test of CAX4RHT elements in steady-state analysis.
pmce_cax4rt_ctd.inp: General test of CAX4RT elements in steady-state analysis.
pmce_cax6mht_ctd.inp: General test of CAX6MHT elements in steady-state analysis.
pmce_cax6mt_ctd.inp: General test of CAX6MT elements in steady-state analysis.
pmce_cgax3ht_ctd.inp: General test of CGAX3HT elements in steady-state analysis.
pmce_cgax3t_ctd.inp: General test of CGAX3T elements in steady-state analysis.
pmce_cgax4ht_ctd.inp: General test of CGAX4HT elements in steady-state analysis.
pmce_cgax4rht_ctd.inp: General test of CGAX4RHT elements in steady-state analysis.
pmce_cgax4rt_ctd.inp: General test of CGAX4RT elements in steady-state analysis.
pmce_cgax4t_ctd.inp: General test of CGAX4T elements in steady-state analysis.
pmce_cgax6mht_ctd.inp: General test of CGAX6MHT elements in steady-state analysis.
pmce_cgax6mt_ctd.inp: General test of CGAX6MT elements in steady-state analysis.
pmce_cgax8ht_ctd.inp: General test of CGAX8HT elements in steady-state analysis.
pmce_cgax8rht_ctd.inp: General test of CGAX8RHT elements in steady-state analysis.
pmce_cgax8rt_ctd.inp: General test of CGAX8RT elements in steady-state analysis.
pmce_cgax8t_ctd.inp: General test of CGAX8T elements in steady-state analysis.
pmce_cpe4rht_ctd.inp: General test of CPE4RHT elements in steady-state analysis.
pmce_cpe4rt_ctd.inp: General test of CPE4RT elements in steady-state analysis.
pmce_cpe4t_ctd.inp: General test of CPE4T elements in steady-state analysis.
pmce_cpe6mht_ctd.inp: General test of CPE6MHT elements in steady-state analysis.
pmce_cpe6mt_ctd.inp: General test of CPE6MT elements in steady-state analysis.
pmce_cpeg3ht_ctd.inp: General test of CPEG3HT elements in steady-state analysis.
pmce_cpeg3t_ctd.inp: General test of CPEG3T elements in steady-state analysis.
pmce_cpeg4ht_ctd.inp: General test of CPEG4HT elements in steady-state analysis.
pmce_cpeg4rht_ctd.inp: General test of CPEG4RHT elements in steady-state analysis.
pmce_cpeg4rt_ctd.inp: General test of CPEG4RT elements in steady-state analysis.
pmce_cpeg4t_ctd.inp: General test of CPEG4T elements in steady-state analysis.
pmce_cpeg6mht_ctd.inp: General test of CPEG6MHT elements in steady-state analysis.
pmce_cpeg6mt_ctd.inp: General test of CPEG6MT elements in steady-state analysis.
pmce_cpeg8ht_ctd.inp: General test of CPEG8HT elements in steady-state analysis.
pmce_cpeg8rht_ctd.inp: General test of CPEG8RHT elements in steady-state analysis.
pmce_cpeg8t_ctd.inp: General test of CPEG8T elements in steady-state analysis.
pmce_cps4rt_ctd.inp: General test of CPS4RT elements in steady-state analysis.
pmce_cps4t_ctd.inp: General test of CPS4T elements in steady-state analysis.
pmce_cps6mt_ctd.inp: General test of CPS6MT elements in steady-state analysis.
pmce_cps8rt_ctd.inp: General test of CPS8RT elements in steady-state analysis.
pmce_cps8t_ctd.inp: General test of CPS8T elements in steady-state analysis.