# Finding the components of stress in bone cement

#### nerdeagle24

Problem Statement
The volume shown below is a piece of bone cement, a project you have been working on
for 2 years. It is finally ready for mechanical analysis. To mimic one possible in vivo
condition, the material was loaded with the stresses shown. You must perform the
following analysis in order to prepare for the next round of tests.
a) What are the values of σxx, σxy, σyx, and σyy in the 2D state of stress?
b) Given values of E = 19 GPa and ν = 0.36, which are reasonable values for bone, find the
values of strain for the figure below and assume that the material is linear, elastic,
homogeneous, and isotropic.
c) In the next phase of testing the loading conditions are getting more complex. For that test,
understand the effect of the current loading conditions at that angle. For the conditions
shown, find the values of σxx’, σxy’, and σyy’ for CCW rotation about the z axis of θ = 28°.

Note, equal stress force on arrows across from each other.
Relevant Equations
Strain x = (1/E)(sigma +v sigma y)
Shear Stress = G*shear strains
I am really stuck on part a.

For part b:
strain x = (1/19*10^9)(137000+.36*95000)
strain y =(1/19*10^9)(-95000-.36*137000)
Is this right?

For part c:
sigma xx'=(sigma x+sigma y)/2+(sigma x -sigma y)cos(2*28)/2+sigma xy sin(s*28)
sigma yy'=(sigma x+sigma y)/2-(sigma x -sigma y)cos(2*28)/2-sigma xy sin(2*28)
sigma xy'=-(sigma x-sigma y)sin(2*28)/2+sigma xy cos(2*28)
Are these equations correct?

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#### nerdeagle24

I also think sigma xy = sigma yx

#### Chestermiller

Mentor
Before getting to part c, your equation for the strains in part b both have sign errors in them.

#### nerdeagle24

Before getting to part c, your equation for the strains in part b both have sign errors in them.
I gotta start with part A. I will look at the signs as well for part b. How do I even start part A?

#### Chestermiller

Mentor
The values of the stresses are shown in the figure. $\sigma_{xx}=137 kPa$, $\sigma_{yy}=-95 kPa$

#### nerdeagle24

The values of the stresses are shown in the figure. $\sigma_{xx}=137 kPa$, $\sigma_{yy}=-95 kPa$
Yes. But what about sigma xy and sigma yx? I know they are equal but how do you calculate them?

#### Chestermiller

Mentor
Yes. But what about sigma xy and sigma yx? I know they are equal but how do you calculate them?
There's nothing to calculate. There are no shear (tangential) loads applied in the figure. Those components of the stress tensor are zero.

"Finding the components of stress in bone cement"

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