# Predicting elastic and plastic wheel deformation

• RayRoc
In summary, the conversation discusses the problem of predicting elastic wheel deformation and possible approaches to solving it. The participants mention using finite layers and applying shear modulus to calculate stress, as well as considering the contact pressure and contact area. They also mention the limitations of a simplistic view and the complexity of the problem. Finally, they discuss using equations for calculating contact stresses and strain in order to find conditions for plastic deformation at the contact area.
RayRoc
I am trying to predict elastic wheel deformation. I am not quite sure how to do it. How might one approach the problem?

Take a steel wheel: 6 in. diameter, 2 in. thick, with a contact force of 1000lbs

E = 30,000,000 psi
Poisson ratio: .27

Would one calculate this in finite layers? Considering sections at a time and applying shear modulus through the layers to find the finite stress of the steel?

I have seen the stress distribution of a steel wheel against a track, and it looked like a strait coulomb from the contact area to the axial. Can it be as simple as:

Contact pressure = (volume of wheel displaced/ volume of .5 the wheel cross section) * Poisson ratio * E

There are of course many effects not realized by such a simplistic view especially when one wants to find the conditions that will cause plastic deformation at the contact area; which is ultimately what I want to know.

Contact stresses for a cylinder on a cylinder are as follows. The area of contact is a narrow rectangle of width 2b, and length l, where l is the length of the cylinders, and
$$b = \sqrt{ \frac{2F}{\pi l} \frac{ (1-\nu^2_1)/E_1 + (1- \nu^2_2)/E_2}{1/d_1 + 1/d_2} }$$
With the maximum pressure being
$$p_{max} = \frac{2F}{\pi b l}$$
You can apply this to a cylinder in contact with a plane by making $$d_2 = \infty$$

edit: Just seen that you're looking for deformation, not stress. Let's see...stress in the z-direction, perpendicular to the contact line is:
$$\sigma_z = \frac{-p_{max}}{ \sqrt{1+ z^2/b^2}}$$
Perhaps you could integrate that from z = 0, to z = r to calculate the summation of strain?

As a scientist, there are several potential approaches to predicting elastic and plastic wheel deformation. One approach could be to use mathematical modeling, where you would input the known parameters of the steel wheel (diameter, thickness, contact force, elastic modulus, Poisson ratio) and use equations and simulations to predict the deformation under different conditions. This could involve breaking the wheel into finite layers and calculating the stress distribution through the layers to determine the potential for plastic deformation.

Another approach could be experimental testing, where you would physically apply different forces and measure the resulting deformation of the wheel. This could help validate the predictions from the mathematical model and provide more accurate data for real-world conditions.

Additionally, considering the complexities and variables involved in predicting elastic and plastic deformation, it may be beneficial to collaborate with other experts in the field, such as engineers or materials scientists, to gather different perspectives and approaches.

Ultimately, the best approach may depend on the specific goals and constraints of the project, but a combination of mathematical modeling and experimental testing could provide the most comprehensive understanding of the problem. It is also important to constantly evaluate and refine the approach as new information and data is collected.

## 1. What is the difference between elastic and plastic wheel deformation?

Elastic deformation is when a material changes shape temporarily under stress and returns to its original shape when the stress is removed. Plastic deformation is when a material changes shape permanently under stress. In terms of wheels, elastic deformation can occur when driving over small bumps or uneven surfaces, while plastic deformation can occur when hitting larger obstacles or overloading the wheel.

## 2. How is elastic and plastic deformation predicted in wheels?

Elastic deformation can be predicted using Hooke's law, which states that the amount of deformation is directly proportional to the applied stress. Plastic deformation can be predicted using the yield strength of the material, which is the amount of stress that will cause permanent deformation.

## 3. What factors affect the amount of elastic and plastic deformation in wheels?

The material of the wheel, the shape and size of the wheel, the weight of the load, the speed and direction of motion, and the surface conditions all affect the amount of elastic and plastic deformation in wheels.

## 4. How accurate are predictions of elastic and plastic deformation in wheels?

The accuracy of predictions depends on the accuracy of the data used and the assumptions made. In general, predictions of elastic deformation tend to be more accurate than predictions of plastic deformation since plastic deformation is affected by a wider range of factors.

## 5. How can predictions of elastic and plastic deformation be used in wheel design?

Predictions of elastic and plastic deformation can be used to determine the appropriate material, size, and shape of a wheel for a specific application. This can help ensure that the wheel can withstand the expected amount of stress and load without experiencing excessive deformation or failure.

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