Mesh and Element Types: The mesh for cases A through F is a second-order hexahedral one-element mesh. It was created locally and imported to SimScale. For cases G through J, the standard algorithm was used to generate a second-order tetrahedral mesh.

The secondary creep formulation and element technology vary from case to case. The table below contains an overview of the configurations:

Case

Mesh Type

Nodes

Creep Formulation

Element Type

(A)

2nd order hexahedral

20

Norton

Standard

(B)

2nd order hexahedral

20

Norton

Reduced integration

(C)

2nd order hexahedral

20

Time hardening

Standard

(D)

2nd order hexahedral

20

Time hardening

Reduced integration

(E)

2nd order hexahedral

20

Strain hardening

Standard

(F)

2nd order hexahedral

20

Strain hardening

Reduced integration

(G)

2nd order standard

235

Norton

Standard

(H)

2nd order standard

235

Norton

Reduced integration

(I)

2nd order standard

235

Time hardening

Standard

(J)

2nd order standard

235

Time hardening

Reduced integration

Table 2: Overview of the mesh, creep formulation, and element technology used for each case.

Find below the mesh used for case G. It’s a standard mesh with second-order tetrahedral cells.

Simulation Setup

Material:

Steel (linear elastic)

\(E\) = 200 \(GPa\)

\(\nu\) = 0.3

\(\rho\) = 7870 \(kg/m³\)

Three creep formulations are used. Find here the parameters for each of them.

Norton:

A = 8.6805556e-48 \(1/s\)

N = 5

Time hardening:

A = 8.6805556e-48 \(1/s\)

N = 5

M = 0

Strain hardening:

A = 8.6805556e-48 \(1/s\)

N = 5

M = 0

Boundary Conditions:

Constraints

\(d_x\) = 0 on face ADHE;

\(d_y\) = 0 on face ABFE;

\(d_z\) = 0 on face ABCD.

Surface loads

\(t_x\) = 300 \(MPa\) on face BCGF;

\(t_y\) = 200 \(MPa\) on face CDHG;

\(t_z\) = 100 \(MPa\) on face EFGH.

Advanced Automatic Time Stepping

For all cases, the following advanced automatic time stepping settings were defined under simulation control:

Retime event: field change;

Target field component: internal variable V1 (accumulated unelastic strain);

Threshold value: 0.0001;

Time step calculation type: mixed;

Field change targetvalue: 0.00008.

Reference Solution

The equations used to solve the problem are derived in [NAFEMS_R27]\(^1\). As SimScale uses SI units, the reference solution was adopted to a time unit of seconds instead of hours.

$$\epsilon_{xx}^c = – \epsilon_{zz}^c = \frac{0.004218}{3600} t \tag{1}$$

$$\epsilon_{eff}^c = \frac{0.004871}{3600} t \tag{2}$$

$$\epsilon_{yy}^c = 0.0 \tag{3}$$

Result Comparison

Find below a comparison between SimScale’s results and the analytical solution presented in [NAFEMS_R27]\(^1\) for the average creep strain \(\epsilon_{xx}^c\) of the cube. The creep time is 3.6e6 \(s\) (equivalent to 1000 hours):

Case

[NAFEMS_R27]

SimScale

Error (%)

(A)

4.218

4.21875

0.0178

(B)

4.218

4.21875

0.0178

(C)

4.218

4.21875

0.0178

(D)

4.218

4.21875

0.0178

(E)

4.218

4.21875

0.0178

(F)

4.218

4.21875

0.0178

(G)

4.218

4.21875

0.0178

(H)

4.218

4.21875

0.0178

(I)

4.218

4.21875

0.0178

(J)

4.218

4.21875

0.0178

Table 3: Comparison of SimScale’s results against an analytical solution.

In Figure 3, we can see how \(\epsilon_{xx}^c\), \(\epsilon_{yy}^c\), and \(\epsilon_{zz}^c\) are evolving for case D.

\(\epsilon_{yy}^c\) and \(\epsilon_{zz}^c\) also show very good agreement with the analytical solution, having an error of 0% and 0.0178%, respectively.

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