Validation Case: Bonded Contact on a Quarter Shaft

This bonded contact validation case belongs to solid mechanics. This test case aims to validate the following parameter:

The simulation results obtained with SimScale were compared to the analytical results presented in [Roark]$^1$.

Geometry

Two geometries are used for this bonded contact validation. The first one consists of a quarter shaft. Radius is 0.1 $m$ and length is 0.5 $m$:

The second geometry has the same dimensions, however, it is split exactly in half, between the ABC and A’B’C’ planes:

The 3D geometry is a 90$^0$ section of a cylinder with dimensions as tabulated below:

Table 1: Geometry dimensions in meters

Tool Type: Code Aster

Analysis Type: Linear static

Mesh and Element Types: The meshes for Cases A through D were created in SimScale. The standard algorithm was used.

Table 2: Mesh characteristics

Find below the mesh used for Case D. It is a standard mesh with second-order tetrahedral cells.

Material:

Steel (linear elastic)
- $E$ = 208 $GPa$, $v$ = 0.3
- Therefore, the shear modulus is 80 $GPa$.

Boundary Conditions:

Constraints
- Fixed support on face ABC
- Face A’B’C’ is rotated with an angle $(\theta)$ of 2e-4 $rad$
Contacts
- Single part shaft
  - Cyclic symmetry: face AA’B’B is tied to face AA’C’C. Rotation axis: z-axis. Sector angle: 90º
- Split shaft
  - Cyclic symmetry: face AA”B”B is tied to face AA”C”C. Rotation axis: z-axis. Sector angle: 90º
  - Cyclic symmetry: face A”A’B’B” is tied to face A”A’C’C”. Rotation axis: z-axis. Sector angle: 90º
  - Bonded contact: both parts are bonded at A”B”C”.

The analytical solution for maximum shear stress $\tau_{max}$ given below is based on Roark$^1$.

$$\large{\tau _{max}}=\frac {\theta.G.r}{l} = 3.2 \ [MPa]$$

The results obtained from SimScale for the maximum shear stress $\tau_{max}$ at point B’ are compared with the analytical solution by [Roark].

Case	Quantity	Roark	SimScale	Error (%)
(A)	Maximum shear stress $\tau_{max} [MPa]$	3.2	3.209	+0.289
(B)	Maximum shear stress $\tau_{max} [MPa]$	3.2	3.191	-0.279
(C)	Maximum shear stress $\tau_{max} [MPa]$	3.2	3.195	-0.152
(D)	Maximum shear stress $\tau_{max} [MPa]$	3.2	3.180	-0.639

Table 3: Comparison of results at point B’.

Inspecting the Cauchy stress magnitude for Case B in the post-processor:

References

W. C. YOUNG, R. G. BUDYNAS. Roark’s Formulas for Stress and Strain. Seventh Edition. McGraw-Hill. 2002. p. 415.

Last updated: November 7th, 2023

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