Validation Case: 3D Triaxial Load Primary Creep (NAFEMS Test 11)

This triaxial load primary creep validation case belongs to solid mechanics. This test case aims to validate the following parameters:

• Creep material behavior.
• Standard and reduced integration elements.
• Automatic time stepping.

The simulation results of SimScale were compared to the analytical results derived from [NAFEMS_R27]\(^1\).

Geometry

The geometry consists of a cube with an edge length \(l\) = 0.1 \(m\).

The coordinates for the points in the cube geometry are as tabulated below:

	A	B	C	D	E	F	G	H
x	0	0.1	0.1	0	0	0.1	0.1	0
y	0	0	0.1	0.1	0	0	0.1	0.1
z	0	0	0	0	0.1	0.1	0.1	0.1

Analysis Type and Mesh

Tool Type: Code Aster

Analysis Type: Nonlinear static

Mesh and Element Types: The mesh used in cases A and B was created using the standard algorithm within SimScale. The same mesh is used in both cases – the only difference between the runs is the element technology integration. Table 2 shows more details about the cases.

Case	Mesh Type	Number of Nodes	Element Type	Element Technology
(A)	Standard	235	2nd order tetrahedral	Standard
(B)	Standard	235	2nd order tetrahedral	Reduced integration

Find below the mesh used for cases A and B. It’s a standard mesh with second-order tetrahedral cells.

Simulation Setup

Material:

• Steel (linear elastic)
  • \(E\) = 200 \(GPa\)
  • \(\nu\) = 0.3
  • \(\rho\) = 7870 \(kg/m³\)
  • Creep formulation: Time hardening
    • \(A\) = 2.6041667e-46 \(1/s\)
    • \(N\) = 5
    • \(M\) = -0.5

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:

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 target value: 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.

$$\epsilon_{xx}^c = – \epsilon_{zz}^c = \frac{0.004218}{60} \sqrt{t} \tag{1}$$

$$\epsilon_{eff}^c = \frac{0.004871}{60} \sqrt{t} \tag{2}$$

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

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)	0.133380	0.133107	-0.205
(B)	0.133380	0.133107	-0.205

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

\(\epsilon_{yy}^c\) and \(\epsilon_{zz}^c\) also show very good agreement with the analytical solution, having an error of 0% and -0.205%, respectively.

Last updated: November 29th, 2023