“Work done” my a muscle during weight-lifting

I'm looking for a simplistic way to describe the "work" done by muscles during compound weightlifting movements. Perhaps not work in the precise physics sense, but an overall idea of how much the muscle was "targeted".

For isolation movements, I feel like this is easy to describe. For example, take a simple standing bicep curl. The most effort from the biceps is at the point where the forearm is parallel to the ground (IE the bicep is in-line with the force of gravity), and the least effort from the bicep is at the very bottom of the rep. So, the effort on the bicep changes continuously and can seemingly be loosely modeled with a trigonometric function.

Say I am bicep curling 25 lbs, the angle $$\theta=0$$ is the beginning of the lift (arms hanging down), and we'll say a full range of motion is through an angle of $$\theta=\frac{3\pi}{4}$$. Then a loose function of "bicep effort" might be:

$$b(\theta) = 25 \gamma \sin(\theta)$$

And a measure of "total bicep effort" through the concentric portion of the rep would be:

$$\int_{0}^{\frac{3\pi}{4}} 25 \gamma \sin(\theta) d\theta$$.

Where $$\gamma$$ is a constant standing in for the forearm length, or whatever other constants may be involved. (apologies for the lack of precision here, I'm aiming at constructing an elementary integration introduction example).

The question is, what is a similar loose, intuitive model for compound movements? In particular, idealized compound movements with two "main movers".

If we take an idealized bench press that involves only the pectoralis and triceps with 0 involvement from other muscle groups, how can we model this in a similar way? Maybe this is simpler than I am thinking?

Thank you.

• The term "volume" describes the physical work done by your muscles during exercise because it is the total load times the total number of reps. Even though a rep isn't a precise measure of length, each repetition is the exertion of a force over a distance so a repetition of a given exercise is a unit of work. The fact that a particular exercise requires different levels of intensity at different points in the movement is just a case for using more than one exercise to target a muscle group. – John Douma Feb 22 at 0:41