H. Singh and J. A. Hanna
The balance of pseudomomentum is discussed and applied to first gradient elasticity, ideal fluids, and the mechanics of inextensible rods and sheets. A general framework is presented in which the simultaneous variation of an action with respect to position, time, and material labels yields bulk balance laws and jump conditions for momentum, energy, and pseudomomentum. The example of first gradient elasticity of space-filling continua is treated at length. Then the pseudomomentum balance is employed to derive several results, beginning with the conservation of vorticity, circulation, and helicity in ideal fluids. A mathematical similarity is noted between the evaluation of circulation along a material loop and the J-integral of fracture mechanics. Integration of the pseudomomentum balance, making use of a prescription for singular sources derived by analogy with the continuous form of the balance, directly provides the propulsive force driving passive reconfiguration or locomotion of confined, inhomogeneous elastic rods. The conserved angular momentum and pseudomomentum are identified in the classification of conical sheets with rotational inertia or bending energy.