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Pre-Analysis & Start-Up

In the Pre-Analysis step, we will review the following:

Assumptions: Assumptions for classical Hertz contact mechanics are discussed.
Mathematical model: Governing equations and boundary conditions, as well as additional relations will be discussed.
FEM approach: We will discuss solution strategy used in solving a nonlinear problem in FEM.

Assumptions

This problem is a classic example of Hertz Contact Mechanics*, and hence, makes the following assumptions:

Surfaces are continuous and non-conforming, which means that initial contact is a point or a line. In our example of sphere-plate, the initial contact interface is in a form of a point.
Strains are small.
Solids are elastic. This means that the material response of stress and strain behaves linearly.
Surfaces are frictionless and cannot penetrate into each other.

For analytical solution, the following additional assumption is made.

5. Both objects (in our case, sphere and plate) are semi-infinitely large bodies (R1, R2 >> a, where a is contact radius).

Reference* S. Timoshenko and J.N. Goodier: "Theory of Elasticity" -- Chap. 13: Sect. 125, "Pressure between Two Spherical Bodies in Contact

Mathematical Model

As in any static analysis, the fundamental governing equations that we must keep in mind are the stress equilibrium equations (i.e. governing equation).

In the above set of equations, it can be shown that σ_ij = σ_ji due to moment balance! Furthermore, we begin by making valid assumptions with regards to our problem of interest. First, we assume that there is no body force (b = 0) anywhere in our model. In addition, we model our problem as a plane stress problem, which means that all of the out-of-plane stress components involving θ-direction, can be assumed to equal zero (σ_θ = τ_θr = τ_θz = 0). These assumptions lead to the following simplifications:

Next we list the relevant boundary conditions of our problem. The two types of boundary conditions, essential and natural, will be specified for all boundaries in our model. It must be noted that essential boundary conditions refer to displacement conditions and natural boundary conditions represent traction conditions. It is also important to observe that only one of these boundary conditions may be specified at a given boundary. In addition, only one of these boundary conditions is sufficient for a given boundary point.

Along the frictionless contact interface, we specify the following boundary conditions.

Here, r represents the radial position away from the axis of symmetry and a denotes the contact radius. We note that, due to the nonlinear nature of our problem, the contact radius a will change throughout the loading process. Even though the contact interface between the sphere and the surface is initially a single point, the contact interface will grow to become a surface as the sphere deforms.

Since the top of sphere is subjected to a point load, traction condition is specified at this location. We observe that since the load is being applied to a point, traction will be infinitely large.

Along the free surface of the sphere, the boundary condition may be specified as follows.

With symmetry condition, the following boundary condition is prescribed along the axis of symmetry.

By identifying the governing equations and defining the boundary conditions, we have set up the mathematical model. We will now establish several additional relationships, which are used in the postprocessing step for computing stress and strain fields using these nodal displacements. These relationships are commonly referred to as the constitutive equations. One of these equations is the strain-displacement relationship.

Second relationship is called Hooke's law. For our model, we assume isotropic material under plane stress, and so further simplifying the Hooke's law results in the following equations.