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)
Famous quotes containing the words pressure, volume, stress, strain and/or pair:
“Today’s pressures on middle-class children to grow up fast begin in early childhood. Chief among them is the pressure for early intellectual attainment, deriving from a changed perception of precocity. Several decades ago precocity was looked upon with great suspicion. The child prodigy, it was thought, turned out to be a neurotic adult; thus the phrase “early ripe, early rot!””
—David Elkind (20th century)
“There is a note in the front of the volume saying that no public reading ... may be given without first getting the author’s permission. It ought to be made much more difficult to do than that.”
—Robert Benchley (1889–1945)
“It is not stressful circumstances, as such, that do harm to children. Rather, it is the quality of their interpersonal relationships and their transactions with the wider social and material environment that lead to behavioral, emotional, and physical health problems. If stress matters, it is in terms of how it influences the relationships that are important to the child.”
—Felton Earls (20th century)
“Finding everything deep—that is an inconvenient trait: it causes a person constantly to strain his eyes and eventually to find out more than he might have wished.”
—Friedrich Nietzsche (1844–1900)
“When strength is yoked with justice, where is a mightier pair than they?”
—Aeschylus (525–456 B.C.)