Manfred Braun

The Balance of Material Momentum Applied to Water Waves

The balance of material momentum is applied to the motion of an ideal, incompressible fluid with special emphasis on water waves. To this end, the fluid flow is represented by its material or Lagrangian description. A variational approach using Hamilton’s principle is employed, with the incompressibility condition incorporated into the Lagrangian by means of a Lagrange multiplier. The balance of material momentum is obtained in its standard form known from nonlinear elasticity, however with the peculiarity that the dynamic Eshelby stress becomes hydrostatic and its divergence reduces to the (negative) gradient of an “Eshelby pressure”. The balance is applied to Gerstner’s nonlinear theory of water waves.

On Discontinuities of Material Momentum and Eshelby Stress in Hyperelasticity and Thermoelasticity

The balance of material momentum is applied to a propagating singular surface at which velocity and displacement gradient are discontinuous. The resulting jump condition contains an additional source term if the material is assumed purely hyperelastic. In a thermoelastic material, this imbalance is compensated by an ap-propriate jump of entropy such that, under adiabatic conditions, the strict balance is retained even on a singular surface.