5. Energy stored in the body due to an applied force

We now want to find the energy stored in a deformable body due to a force, say F. Would it simply be F · δ where δ is the corresponding displacement?

In this chunk, we start by pondering how to quantify the energy stored in a material when a force is applied. Specifically, we consider that the energy might just be the product of the force (F) applied and the resulting displacement (δ) it causes in the material. This set-up raises a key question about how to quantitate the energy stored in a material under stress.

Think of stretching a rubber band. When you apply a force to stretch it, the rubber band stores energy in the form of potential energy. The more you stretch it, the more energy it stores, similar to how we relate force and displacement in our formula.

To find an answer to this, consider a spring having spring constant k and clamped at one end. A force F is applied at the other end as shown in Figure 5.

We detail a specific example of energy storage by analyzing a spring. When a force is exerted on a spring, it stretches, which is its displacement. The energy stored in the spring due to this applied force is derived from a well-known physics formula that links force, displacement, and energy. This relationship is crucial because it provides a deeper understanding of the mechanics of deformation and energy storage in materials.

Imagine pulling on a slinky. If you pull it slowly, you can feel the tension building as it's stretched. The energy you put into stretching it gets stored as potential energy in the spring until you release it.

We know that the energy stored in the spring due to this force is E = (1/2)F · x.

Here, we provide the mathematical derivation for the energy stored in the spring. The formula shows that the energy stored is proportional to the force applied over the displacement. The factor of one-half arises because the force increases linearly from 0 to its maximum value as the spring is stretched, meaning the average force over the displacement is half the maximum force.

Visualize filling a balloon with air. At the start, it takes minimal effort to inflate it; as you blow more air in, it requires more effort. The energy used to inflate it doesn’t just disappear but is stored; similarly, in a stretching spring, energy is stored as it is progressively stretched.

However, the final energy stored is independent of the loading path taken in the process; it only depends on the final force/displacement.

This idea emphasizes that regardless of the manner in which a force is applied—as long as the final displacement is achieved—the energy stored remains the same. It indicates a fundamental property of linear elastic materials where energy conservation principles apply. Whether you apply the force slowly or suddenly finishing at the same displacement point, the stored energy remains unchanged.

For instance, if you squeeze a sponge under water, you can either do it slowly or quickly, but once you reach the same size, the energy spent compressing it remains constant. The key takeaway here is that the end state matters more than how you got there.

Now we can have multiple forces F1, F2,...Fn acting on the deformable body.

This segment introduces the concept of multiple forces acting on a body and how the total energy stored can be expressed as the sum of energies contributed by each force. It highlights the interactions between forces and how each force impacts the overall deformation and energy storage. Each respective displacement is linked to its corresponding force by influence coefficients.

Imagine a team of people pushing a car up a hill. Each person contributes their effort (force), and when combined, they all contribute to lifting the car. The total amount of work done (and energy stored) can be considered equal to the sum of the work done by each person.

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