The Hertz contact theory is a study of the local stress and strain distribution law of two objects due to contact with pressure. 1881 H.R. Hertz first studied the elastic deformation of glass lenses under the force of bringing them into contact with each other. He assumes that:
1) Small deformation occurs in the contact area.
2) The contact surface is elliptical.
3) The object in contact can be regarded as an elastic semi-space, and only the distributed normal pressure acts on the contact surface.
Any contact that satisfies the above assumptions is called a Hertz contact. When the contact surface is approximately a quadratic paraboloid, and the size of the contact surface is much smaller than the object size and the curvature radius of the surface, the Hertz theory can give the results consistent with the experimental. In the Hertz contact problem, since the deformation near the contact area is strongly restrained by the surrounding, the stress is highly concentrated. Besides, the contact stress is nonlinear with the applied pressure and is determined by the elastic modulus and Poisson’s ratio of the material.
It is clear that the type of structure in Hertz contact can be different. In this example, we only look at the cylinder-plate contact. The schematic diagram of the contact analysis is as follows.
Where the geometric parameters are given: a = 4 mm, b = 16 mm, r = 8 mm, and the force is F = 100 N.
Now let’s take a look at how to conduct Hertz contact analysis in WELSIM.
First set the material properties. Add a material object and name it “myMat”, setting Young’s modulus and Poisson’s ratio to 1100 and 0.3, respectively.
To simplify the analysis, a symmetric structure model is applied. Geometry models can be built from a CAD tool and imported into WELSIM. The previously created myMat material is assigned to these two parts.
In the mesh settings, set the maximum element size to 0.3 and click Mesh button. The generated mesh is as follows, with a total of 3153 nodes and 1042 Tet4 elements.
Set the contact pair. Scope surfaces of the cylinder and plate, and assign to a contact pair. Set the Contact Type to Frictionless, and the other properties remain the default values.
Set the boundary conditions, such as fixed supports and symmetric faces. (To save the length of this article, we only show one fixed support boundary condition here. The example project file in the installation directory contains all detailed settings)
A downward force in the Z direction is applied to the upper end of the structure, and the magnitude is 100.
Click the Compute button, and you will get the results quickly. Add deformation and stress result objects and evaluate these results. Since five substeps are defined, there are five result sets.
Z-direction deformation result contour.
von-Mises stress result contour.
Turn on the deformation and mesh display, zoom in onto the local view of the contact area, you can see the deformation of the structures, as well as the increased contact area.
This example is saved in examples/structural_contact/contact_Hertz_ex9.wsdb in the installation directory. If you have any questions about WELSIM’s contact analysis, please leave us a message or visit https://welsim.com.