When we talk about the deformation of a material, we are talking about how much it changes shape when subjected to some force, stress or load. Elastic deformation, in particular, refers to deformation that is reversible, or in other words, where the material can return back to its original shape once the forces acting on it are removed. When we apply a force to a material, it changes shape and stores energy, and this stored energy is referred to as the elastic potential energy per unit volume of the material.
Linear Elastic Deformation
When a material deforms linearly, it means that the deformation is proportional to the force applied. In other words, if we double the force, the deformation will also double. To calculate the elastic potential energy per unit volume in this case, we need to calculate the area under the curve of either the stress-strain or strain-stress graph. This is because the force applied is directly proportional to the stress applied, whereas the deformation or change in length is proportional to the strain.
To illustrate this, let’s consider the example of a spring. When a spring is compressed or stretched, the deformation it undergoes is proportional to the force applied. If we plot the force applied against the deformation or change in length of the spring, we would get a straight line as shown below:
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F ----------|----------
|
|
0 x1, x2
Here, F represents the force applied, and x1 and x2 represent the initial and final lengths of the spring respectively. The slope of this graph gives us the stiffness of the spring, which is defined as the force required to produce unit deformation or change in length. The area under this curve represents the elastic potential energy per unit volume of the spring.
Non-Linear Elastic Deformation
Now, let us consider a material that deforms non-linearly, such as a rubber band. In this case, the deformation is not proportional to the force applied. If we plot the stress-strain curve or strain-stress curve for such a material, we would get a curve that looks something like this:
Here, sigma represents the stress applied, and epsilon represents the strain. The curve is not a straight line, but rather a curve that changes direction. In this case, the area under the stress-strain curve or strain-stress curve is not equal to the elastic potential energy per unit volume of the material.
The Role of Integration
The reason why we cannot simply calculate the area under the curve to find the elastic potential energy per unit volume in this case is due to the non-linear nature of the deformation. The area under the curve represents the work done in deforming the material, but for a non-linear deformation, this work cannot be represented by a single number. Instead, we need to find the work done for each infinitesimal change in deformation, and then integrate over the range of deformation to find the total work done or the elastic potential energy per unit volume.
To illustrate this, let’s consider the example of a rubber band stretched between two fixed points. When we apply a small amount of force to the rubber band, it changes length by a small amount, and stores some energy. If we repeat this process for small intervals of force, we can calculate the work done for each interval, and then integrate over the entire range of deformation to find the elastic potential energy per unit volume of the rubber band.
Conclusion
In conclusion, when we calculate the elastic potential energy per unit volume of a material that deforms linearly, we simply need to find the area under the stress-strain or strain-stress curve. However, for materials that deform non-linearly, we need to find the work done for each infinitesimal change in deformation, and then integrate over the range of deformation to find the total work done or elastic potential energy per unit volume.
Elastic Potential Energy Per Unit Volume,stress-strain And Strain-stress Curve
When we talk about the deformation of a material, we are talking about how much it changes shape when subjected to some force, stress or load. Elastic deformation, in particular, refers to deformation that is reversible, or in other words, where the material can return back to its original shape once the forces acting on it are removed. When we apply a force to a material, it changes shape and stores energy, and this stored energy is referred to as the elastic potential energy per unit volume of the material.
Linear Elastic Deformation
When a material deforms linearly, it means that the deformation is proportional to the force applied. In other words, if we double the force, the deformation will also double. To calculate the elastic potential energy per unit volume in this case, we need to calculate the area under the curve of either the stress-strain or strain-stress graph. This is because the force applied is directly proportional to the stress applied, whereas the deformation or change in length is proportional to the strain.
To illustrate this, let’s consider the example of a spring. When a spring is compressed or stretched, the deformation it undergoes is proportional to the force applied. If we plot the force applied against the deformation or change in length of the spring, we would get a straight line as shown below:
| F ----------|---------- | | 0 x1, x2Here, F represents the force applied, and x1 and x2 represent the initial and final lengths of the spring respectively. The slope of this graph gives us the stiffness of the spring, which is defined as the force required to produce unit deformation or change in length. The area under this curve represents the elastic potential energy per unit volume of the spring.
Non-Linear Elastic Deformation
Now, let us consider a material that deforms non-linearly, such as a rubber band. In this case, the deformation is not proportional to the force applied. If we plot the stress-strain curve or strain-stress curve for such a material, we would get a curve that looks something like this:
| sigma_max _|_ | \ | \ | \ sigma_min _|----\_\_----- | ` epsilon_max epsilon_minHere, sigma represents the stress applied, and epsilon represents the strain. The curve is not a straight line, but rather a curve that changes direction. In this case, the area under the stress-strain curve or strain-stress curve is not equal to the elastic potential energy per unit volume of the material.
The Role of Integration
The reason why we cannot simply calculate the area under the curve to find the elastic potential energy per unit volume in this case is due to the non-linear nature of the deformation. The area under the curve represents the work done in deforming the material, but for a non-linear deformation, this work cannot be represented by a single number. Instead, we need to find the work done for each infinitesimal change in deformation, and then integrate over the range of deformation to find the total work done or the elastic potential energy per unit volume.
To illustrate this, let’s consider the example of a rubber band stretched between two fixed points. When we apply a small amount of force to the rubber band, it changes length by a small amount, and stores some energy. If we repeat this process for small intervals of force, we can calculate the work done for each interval, and then integrate over the entire range of deformation to find the elastic potential energy per unit volume of the rubber band.
Conclusion
In conclusion, when we calculate the elastic potential energy per unit volume of a material that deforms linearly, we simply need to find the area under the stress-strain or strain-stress curve. However, for materials that deform non-linearly, we need to find the work done for each infinitesimal change in deformation, and then integrate over the range of deformation to find the total work done or elastic potential energy per unit volume.