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”
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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
Great article, Ryan! What’s the name of this text book?
Biomechanics of Cartilage I think.
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
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.
Where can we find the article by Wolf?
It’s a two page paper included in the end of the 46 page pdf homework file.
Thanks Herberto!
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.
Amanda, I agree with you. I believe that this is a typo and that they mean force.
Amanda – I was thinking the same thing. I was wondering if it should be kg-f ?
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.
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.
Should be multiplication. See this: http://www.convertunits.com/from/kg/to/N
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.
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?
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
Does this equation look right?
P={ [ 2*pi*(3^2/3)*(Re^2/3) ] / [ 3*(4^2/3)*(Ee^2/3) ] } ^3
Were you able to find 3mm diameter steel ball working back from the result? I’m about 25% off
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?
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?
Erik I’m not sure where your formula came from. What is your thought process ?
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.
What did you all use for the radius of the cartilage since it is assumed flat?
A flat object would have an infinite radius.
I used the Re = Rsphere