# Finding increased length after deformation

• Dell
In summary, the strain, as a function of the angle, is K*sin2(x). The change in length is the integral of the strain, which can be simplified to K*pi. However, this does not take into account the fact that the material is connected in a hoop, so the correct equation should include the differential element for segment length, R\,d\theta. This can be seen by looking at the equation for the change in angle, which is not equivalent to the change in length. This issue is also discussed in further detail in the provided thread.
Dell

the strain, as a function of the angle is K*sin2(x)

now i know that the change in length is the integral of the strain

=$$\int$$K*sin2(x)dx from 0->2pi

=K/2*$$\int$$1-cos(2x)dx

=K/2*(2pi - 0.5*sin(4pi) )

where does the R come from? i realize that the change in length should be dependant on the radius, but mathematically how do i come to that?

You're working with $\epsilon_\theta$, an angular strain. When you integrate it, you get an increase in angle, not length. Know what I mean?

thats what i thought, but what does that mean, are there not still 360 degrees?? does it mean that each radian is now longer than 1/2pi of the circumference of the original circle, sort of like the length of an arc??

Right. The equation doesn't know that the material is connected in a hoop. To fix this, use the differential element for segment length, $R\,d\theta$.

## 1. How does deformation affect the length of an object?

Deformation can cause an object to either increase or decrease in length, depending on the type of deformation and the material properties of the object.

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

Elastic deformation is temporary and reversible, meaning the object will return to its original length once the force is removed. Plastic deformation is permanent and irreversible, causing a permanent change in the object's length.

## 3. What factors influence the amount of length increase after deformation?

The type of material, the amount of force applied, and the duration of the force all play a role in the amount of length increase after deformation.

## 4. How do scientists measure the amount of length increase after deformation?

Scientists use instruments such as strain gauges or extensometers to accurately measure the change in length of an object after deformation.

## 5. Can objects return to their original length after plastic deformation?

No, plastic deformation causes a permanent change in an object's length and cannot be reversed. The object would need to undergo a process such as annealing to regain its original length.

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