# Homework 2 – Problem 3

Tuesday, February 2nd, 2016

The article by Wolf describes using steel balls to induce failure in cartilage. Calculate the peak pressure and peak tensile stress in the cartilage due to loading by the small steel ball using Hertz contact theory. Do these data predict the failure stress of the large ball? You may assume that the cartilage is flat.

## 118 thoughts on “Homework 2 – Problem 3”

1. Ryan Wesley Hoover says:

since this is static loading we should use deformation = (0.563*P^2/(Re*Ee^2))^0.333 ?

pressure po = 3*P/(2*3.14159*r^2)
tau max = 0.31*po
sigma = 0.333*po*(1-2*poisson)

We know the force, the steel ball radius, radius of cartilage = infinity. In order to get remaining data we need modulus of the articular cartilage. Are we to research on our own to find this? 0.45 – 0.8 MPa per http://www.cartilagehealth.com/images/artcartbiomech.pdf

1. Herberto Dutra says:

Great article, Ryan! What’s the name of this text book?

1. Ryan Wesley Hoover says:

Biomechanics of Cartilage I think.

2. Herberto Dutra says:

Good afternoon Dr Nauman,
In order to calculate the peak pressure, I was trying to calculate the Contact Radius (a). The formula requires the Ee, which then requires the E value of the cartilage. Can you advise on the value to use?
Thanks

1. Myra Fabro says:

Hi Herberto. I think it’s ok if you look it up and use the value you find. I looked at a few different sources for consistency in range and picked a value.

3. Alison Ling says:

Where can we find the article by Wolf?

1. Herberto Dutra says:

It’s a two page paper included in the end of the 46 page pdf homework file.

4. Alison Ling says:

Thanks Herberto!

5. Amanda Taylor says:

Alright, I feel like I am over thinking something, or need a nap. In the article Wolf saysit took a pressure of 4-5 kg for the 1.5 mm ball, what is this value actually a reference to, because pressure is not measured in kg.

1. Abraham William Hamilton says:

Amanda, I agree with you. I believe that this is a typo and that they mean force.

1. Myra Fabro says:

Amanda – I was thinking the same thing. I was wondering if it should be kg-f ?

6. Amanda Taylor says:

I am glad it wasn’t just me, best I could figure if you convert from kg-f to N it has a simple 1 kg-f to 9.81 N, and away you go.

1. Abe Hamilton says:

Yes, be sure to convert to N!

Multiplying the given ‘force/pressure of 4-5kg’ by 9.81 to get the force in Newtons resulted in me getting a value for the contact radius which was more than the diameter of the steel ball itself, which seems incorrect. Did others have this issue?

Oh you divide it by 9.81. That makes more sense.

1. Anthony N Corsten says:

Should be multiplication. See this: http://www.convertunits.com/from/kg/to/N

2. Alycia G Berman says:

I also have a value for contact radius that is larger than the diameter of the steel ball. Not sure what to make of it.

7. Abe Hamilton says:

I am a little confused what they are referring to when they say, “Do these data predict the failure stress of the large ball?”.

These calculations are for the stresses in the cartilage, not the steel ball… Am I missing something?

1. Amanda Taylor says:

My assumption was that the problem was asking you to use the small ball information to solve for the max shear stress which would be for the Cartilage, then use that info and work backwards inserting the values for the big ball (larger radius) to see if you come up with a “pressure” similar to the one that Wolf gave in his article of 9-10 Kg

1. Erik Hansen says:

Does this equation look right?
P={ [ 2*pi*(3^2/3)*(Re^2/3) ] / [ 3*(4^2/3)*(Ee^2/3) ] } ^3

2. Herberto Dutra says:

Were you able to find 3mm diameter steel ball working back from the result? I’m about 25% off

3. Erik Hansen says:

I performed the calculation two different ways. I determined the load applied and then I determined the resultant diameter, based on the tensile stress calculated using the small ball. I not have two different percent differences. Which should I be using? What is considered acceptable percent difference in the industry?

2. Nathan Shaw says:

I am confused at the wording too. I can see the case for both Amanda’s and Herberto’s interpretation. Dr. Nauman, could you please shed some light on this?

8. Herberto Dutra says:

Erik I’m not sure where your formula came from. What is your thought process ?

1. Erik Hansen says:

I combined the peak contact radius equation and peak pressure equation and rearranged so that P was on one side and all other variables were on the other. What was yours?

Also, found http://www.amesweb.info/HertzianContact/HertzianContact.aspx to be useful for verifying calculations.

10. Joshua J Liddy says:

What did you all use for the radius of the cartilage since it is assumed flat?

1. Erik Hansen says:

A flat object would have an infinite radius.

2. Herberto Dutra says:

I used the Re = Rsphere

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