# What is the maximum work inequality?

In the Springer Handbook of Robotics, under the section dealing with manipulation, specifically friction limit surface, there's the following sentence.

When the part slips on the support, the contact wrench $$\omega$$ lies on the limit surface, and by the maximum-work inequality, the twist $$t$$ is normal to the limit surface at $$\omega$$

(here)

The closest I found was this wikipedia page on Principle of maximum work but it seems to relate to chemical reactions, and I couldn't quite relate the two.

So what is the maximum-work inequality (and how does that let us conclude that the twist is normal to the limit surface at the given contact wrench)?

From [1] Drucker's maximum work theorem states

[...] A material on which an external agency does positive work during an elastic-plastic stress cycle is considered strain hardening. As the inequality will not be satisfied if the material is strain softening, the postulate is often referred to as Drucker’s strain hardening postulate. To a close approximation the net plastic work done over the loading part of the cycle can be expressed as:

$$(\sigma_{ij}-\sigma_{ij}^{\circ})d\epsilon_{ij}^p + \frac{1}{2} d\sigma_{ij}d\epsilon_{ij}^p > 0 \tag{21}\label{1}$$

Drucker’s first and second postulate follows from this inequality. If it is assumed that the original state of stress is one of yielding, the first term cancels. In accordance with the inequality, the first postulate states that the plastic work done by an external agency during the application of additional stress is positive for a work hardening and zero for a non-hardening material. If the last part of the postulate is to be true, vectors representing the increments of stress and plastic strain have to be perpendicular. In a non-hardening material loading is assumed to be neutral, that is all load paths are tangential to the yield surface. Therefore the increment of strain is directed normally to it, thus resulting in the normality law. If the material is hardening, it must be assumed that the increment of stress is a linear combination of a plastic and an elastic part, and that the last is directed tangentially to the yield surface. As the elastic part of the stress increment does no irreversible work, the increment of plastic strain is also in this case directed normal to the yield surface.

Drucker’s second postulate is connected to the first part of the inequality above and states that the net work done by an external agency during a cycle of addition and removal of stresses is non-negative. If it is assumed that the original state of stress is not one of yielding, the last term in the inequality can be neglected. The result is:

$$(\sigma_{ij}-\sigma_{ij}^{\circ})d\epsilon_{ij}^p \ge 0 \tag{22}\label{2}$$

According to Callen [2]

4-5 THE MAXIMUM WORK THEOREM

The propensity of physical systems to increase their entropy can be channeled to deliver useful work. All such applications are governed by the maximum work theorem.

Consider a system that is to be taken from a specified initial state to a specified final state. Also available are two auxiliary systems, into one of which work can be transferred, and into the other of which heat can be transferred. Then the maximum work theorem states that for all processes leading from the specified initial state to the specified final state of the primary system, the delivery of work is maximum (and the delivery of heat is minimum) for a reversible process. Furthermore the delivery of work (and of heat) is identical for every reversible process.

[1]: Totten, Mackenzie: Handbook of Aluminum Volume 1 Physical Metallurgy and Processes, page 433-434

[2]: Callen: THERMODYNAMICS AND AN INTRODUCTION TO THERMOSTATISTICS 3rd ed, page 103