Stress Strain Problem: Finding the Final Length

In summary, the material undergoes a total strain of .02 after being stressed. After the stress is removed, the material's new length is .6167 mm.
  • #1
Saladsamurai
3,020
7
HELP! Stress Strain Question

Homework Statement



In part (a) of the problem, we found that due to a certain stress, the amount of

elastic strain that a material undergoes is
[itex]\epsilon_E=.0087[/itex]

and the amount of plastic strain is
[itex]\epsilon_{pl}=.0113[/itex].

The total strain is therefore
[itex]\epsilon_T=.02[/itex]We are then told that a sample of this material with original length
[itex]l_o=610 \ mm[/itex] undergoes that same stress involved in part (a).

What is the new length [itex]l_f[/itex]after the stress is removed ?So I believe the idea behind this is that we gain the elastic portion of the strain back, but the plastic elongation should be added onto the original length.

I wrote this quantitatively as:

[tex]l_f=l_0+\epsilon_{pl}\Delta l[/tex] (1)

To find the change in length we have:

[tex]\epsilon_T=\frac{\Delta l}{l_0} \Rightarrow \Delta l=\epsilon_Tl_0[/tex] (2)

Therefore (1) becomes:

[tex]l_f=l_0+\epsilon_{pl}(\epsilon_Tl_0)[/tex]

[tex]\Rightarrow l_f=l_0(1+\epsilon_T\epsilon_{pl})[/tex]

Plugging in numbers we have lf=.6101 mm

but the correct answer is .6167 which is waayyy off.

What am I missing here?
 
Last edited:
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  • #2


First find the length when the total load is applied (i.e., the length incorporating elastic and plastic strain). Then find the length change associated with elastic recovery of this sample, and subtract this.
 
  • #3


I will try that. But how is that different? What concept am I missing here? I do not see the difference between adding the plastic elongation to the initial length and subtracting the elastic elongation from the final length.
 
  • #4


The mistake you made originally was taking [itex]\epsilon_{pl}[/itex] as the proportion of plastic strain, rather than the amount of plastic strain. The product [itex]\epsilon_T\epsilon_{pl}[/itex] doesn't have any meaning in this problem.

Personally, I think an answer of [itex]l_f=l_0(1+\epsilon_{pl})=0.6169[/tex] is good enough and simpler. But the approach I described gets you to .6167 (actually .616786), so perhaps it's the approach intended by the person who wrote the problem.
 
  • #5


Mapes said:
The mistake you made originally was taking [itex]\epsilon_{pl}[/itex] as the proportion of plastic strain, rather than the amount of plastic strain.

I don't understand this statement?

It was given that this was the plastic strain:confused:

Edit: hold on... let me think...
 
  • #6


Nope... I still don't understand. If by definition strain is [itex]\epsilon_T=\frac{\Delta l}{l_0} [/itex] then isn't it by definition a proportion?

i.e., percent change in length.
 
  • #7


In your equation (1) you've multiplied strain by change in length. This doesn't mean anything; strain is already change in length divided by original length. The term [itex]\epsilon\Delta L[/itex] just isn't useful or meaningful here.

However, you could calculate that

[tex]\epsilon_{pl}/ \epsilon_T=\epsilon_{pl}/(\epsilon_{pl}+\epsilon_{E})=56.5%[/tex]

of the total strain is plastic, and multiply this percentage by the total change in length to get the plastic change in length. That's what I meant by the proportion of plastic strain.
 
  • #8


Okay. I kind of follow you now. But what is the total change in length is it not [itex]\epsilon_{tot}*l_o[/itex] ?

So my original (1) should have been:

[tex]l_f=l_0(1+\frac{\epsilon_{pl}}{\epsilon_{tot}})[/tex]

arrgggg goddamn Latex
 
  • #9


Saladsamurai said:
Okay. I kind of follow you now. But what is the total change in length is it not [itex]\epsilon_{tot}*l_o[/itex] ?

Yes, that's the total change in length. Did I say something different?
 
  • #10


I don't know :rofl: Perhaps I did! I hate this class! Thanks for the help!
 
  • #11


Just keep at it and soon it'll seem easy. Good luck.
 

1. What is stress and strain?

Stress is the force applied per unit area on a material, while strain is the resulting deformation of the material. Stress is typically measured in units of force per unit area (such as N/m^2 or Pa), while strain is a unitless quantity.

2. How are stress and strain related?

Stress and strain are related by the material's Young's modulus, which is a measure of the material's stiffness. The relationship between stress and strain is linear for most materials within their elastic limit, meaning that the strain is directly proportional to the stress applied.

3. What is the difference between elastic and plastic deformation?

Elastic deformation is temporary and reversible, meaning that the material returns to its original shape after the stress is removed. Plastic deformation, on the other hand, is permanent and causes a permanent change in the material's shape.

4. What is the yield point and ultimate tensile strength?

The yield point is the stress at which the material starts to deform plastically, while the ultimate tensile strength is the maximum stress that a material can withstand before breaking. These are important measures of a material's strength and ductility.

5. How is stress and strain data used in materials testing?

Stress and strain data is used to determine a material's mechanical properties, such as its Young's modulus, yield strength, and ultimate tensile strength. This data is crucial in understanding a material's behavior under different types of stress and can help engineers design safe and efficient structures and products.

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