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:

• Bonded contact

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:

	A	B	C	A’	B’	C’	A”	B”	C”
x	0	0.1	0	0	0.1	0	0	0.1	0
y	0	0	0.1	0	0	0.1	0	0	0.1
z	0	0	0	0.5	0.5	0.5	0.25	0.25	0.25

Analysis Type and Mesh

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.

Case	Geometry	Element Type	Number of Nodes	Element Technology
(A)	Quarter Shaft	1st Order Tetrahedral	8660	Standard
(B)	Quarter Shaft – Split	1st Order Tetrahedral	8846	Standard
(C)	Quarter Shaft	2nd Order Tetrahedral	62943	Reduced Integration
(D)	Quarter Shaft – Split	2nd Order Tetrahedral	63772	Reduced Integration

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

Simulation Setup

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”.

Reference Solution

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

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

Result Comparison

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

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

Last updated: September 24th, 2021