The Pressure/volume and Stress/strain Pair
As an example, consider the PV conjugate pair. The pressure acts as a generalized force - pressure differences force a change in volume, and their product is the energy lost by the system due to mechanical work. Pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables.
The above holds true only for non-viscous fluids. In the case of viscous fluids, plastic and elastic solids, the pressure force is generalized to the stress tensor, and changes in volume are generalized to the volume multiplied by the strain tensor (Landau & Lifshitz 1986). These then form a conjugate pair. If is the ij component of the stress tensor, and is the ij component of the strain tensor, then the mechanical work done as the result of a stress-induced infinitesimal strain is:
or, using Einstein notation for the tensors, in which repeated indices are assumed to be summed:
In the case of pure compression (i.e. no shearing forces), the stress tensor is simply the negative of the pressure times the unit tensor so that
The trace of the strain tensor is just the fractional change in volume so that the above reduces to as it should.
Read more about this topic: Conjugate Variables (thermodynamics)
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