# Strain Energy Functions and Springs

• Trying2Learn
In summary, the conversation discusses the relationship between potential energy, strain energy function, and force in a spring. The strain energy function is derived from Fung's Solid Mechanics and contains a negative sign. However, this negative sign may seem counterintuitive when considering the force on a point mass in a 1d bar. Ultimately, the force required to stretch the spring is the negative of the force on a point mass.
Trying2Learn
TL;DR Summary
Strain energy and elastic constants: why is the strain energy formed this way
(If this is in the wrong forum, please move it)

Here is the potential energy of a spring

Here is the strain energy function in elasticity

The look alike -- I like that.

If we want the force in the spring, we take the derivative of V with respect to the displacement and make the result NEGATIVE

However, we note the following property of the strain energy function (from Fung's Solid Mechancis, page 128)

So, real stupid question: what happened to the negative?

I mean, I am trying to work my way through how elasticity, strain energy, etc. lead to the number of constants relating stress and strain. I will work this through on my own. I am trying to put all of this in some sort of contextual balance.

I can intuit the need for a strain energy function without the negative. But part of me wants to see the negative to make it harmonious with a spring.

Can someone get me out of this?

Try replacing the spring with a 1d bar. Upon extension, the bar experiences a tensile stress which produces a negative force on a point mass at the end of the bar/string.

The first equation looks at the force on a point mass. The force required to stretch the spring the relevant distance is the negative of this force.

Last edited:
Trying2Learn
Ah, of course. Thank you!

## 1. What is strain energy and how is it related to springs?

Strain energy is the potential energy stored in a material when it is deformed or stretched. This energy is directly related to springs because springs store strain energy when they are compressed or extended.

## 2. What are strain energy functions and how are they used in relation to springs?

Strain energy functions are mathematical equations that describe the relationship between the strain energy stored in a material and the amount of deformation it undergoes. In relation to springs, these functions are used to calculate the amount of strain energy stored in a spring based on its material properties and the amount of deformation it experiences.

## 3. How do different materials affect the strain energy of a spring?

The material properties of a spring, such as its stiffness and strength, directly affect the amount of strain energy it can store. A stiffer material will have a higher strain energy, while a weaker material will have a lower strain energy.

## 4. Can strain energy functions be used to predict the behavior of springs under different loads?

Yes, strain energy functions can be used to predict the behavior of springs under different loads. By inputting the material properties and the amount of deformation, the function can calculate the strain energy stored in the spring and how it will behave under different loads.

## 5. Are there any limitations to using strain energy functions for springs?

While strain energy functions are useful for predicting the behavior of springs, they have limitations. These functions assume that the material is linearly elastic and that the deformation is small. In reality, materials may exhibit non-linear behavior and large deformations, which can affect the accuracy of the predictions made by the function.

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