# In continuum mechanics, what is work potential in the context of total potential energy?

I'm reading a book on the finite element method. Specifically I'm looking at the background material where they are discussing potential energy, equilibrium, and the Rayleigh–Ritz method.

The book claims that potential energy is equal to strain energy + work potential energy. I see that strain energy is like spring energy from a spring being stretched or compressed. However, I don't see what work potential represents. I feel that if a spring system is perturbed via forces, the work put into the system should be equal to the amount stored in the springs. (By integrating f(x) over x). The book seems to suggest otherwise since they are simply multiplying the force applied in the equilibrium position, times the displacement.

What's going on here, and especially what is the intuitive meaning behind work potential?

I know which book you are referring to. It is the book "Finite elements in Engineering" by Chandrupatla and Balegundu.I also have the same question. We learnt in Physics that the Work done by the force is stored as Potential Energy. There was no mention of Work Potential.

They probably mean external work (load * displacement) potential; summed with (inner) strain potential gives total potential energy. See e.g. here (just a quick search).

The total potential energy of a system is consisted from two terms:

1. Strain energy of the material
2. the potential lost due to work done by external force.

Let's assume we are pulling a spring from one end while the other end is fixed to the wall.

At the first, there is no stored energy so called strain energy in the spring and indeed all the energy is kept as the external force in our hand. By applying the external load, the kept energy decreases (due to -ive sign) and store as the strain energy into the material. In a nut shell, during the loading process, the kept energy of external force convert into the strain energy and store into the material.

Here is what I think. If a body under the action of several forces is in equilibrium, then, in totality, its potential energy must include both the stored elastic energy and the work done by all the forces applied on it; and it must be minimum.

In case of a stretched spring in equilibrium under the action of a force, the spring cannot potentially do any work because it is in equilibrium. Now, if the force acting on it is released, then the spring would possess elastic potential energy at that very instant in time. This is because the force has been removed from it and now it would be free to convert the stored energy into work.

I think the intention of the authors, Prof. Belegundu and Prof. Chandrupatla, was that they probably considered total potential energy of the body in the sense of capacity of the body to do work. If taken this way, the expression starts making some sense. I hope this helps.