I am familiar with the formula for energy density: [itex]\frac{1}{2}[/itex] * [itex]\frac{Force*Extension}{Area*length}[/itex] and also the formula for elastic potential energy: [itex]\frac{1}{2}[/itex] * [itex]Force*Extension[/itex]. I noticed that there is a 1/2 in both formulas because we are concerned with the average force in each case and that would be half of the maximum force 'F'. As for the stress formula, I was surprised to see it given in my textbook as [itex]\frac{Force}{Area}[/itex]. Several websites define stress as the 'average force per unit area', so why don't we multiply by 1/2 like the other two cases?
Your formulas for energy density seem to be approximations which are only exact as long as Force is proportional to Extension (Hookes law). In general the elastic potential energy is ##\int_0^x F(x') dx'## where x is extension. if ##F(x)=\alpha x## then this integral becomes ##1/2 \alpha x^2=1/2 F(x)X##.
Yup, these are introductory-level formulas for materials obeying Hooke's Law. Can you please explain why it's Force/Area and not 1/2 * Force/Area?
The elastic potential energy = the work done by when the force is applied. If you imagine applying the force slowly, so you can ignore the kinetic energy of the slowly moving object, the force will increase gradually from 0 to its maximum value. the work done = (average force) x distance, and the time-averaged value of the force is half the final force. That's where the factor of 1/2 comes from. On the other hand, the stress only depends on the current force, not on its time-average. Presumably, when the websites you mentioned said "average force" they were talking about averaging something over the area, not over time. But the words "average force over the area" don't sound right. "The average stress over the area = the total force applied to the area / the area" would be better. If you give us the links to some of the websites you mentioned, we might be able to explain what they really meant, or tell you if they are just wrong.