Elliptic and Parabolic Equations - Abstract

Naumann, Joachim

Variational problems in perfectly plastic fluid theory

The stationary motion of an incompressible fluid is governed by the system of PDEs beginequation nablacdotmathbfu = 0, qquad -nablacdot S+nabla p=mathbff, endequation where $mathbfu=(u_1,ldots,u_n)$ velocity, $S=S_ij$ deviatoric stress, $p$ pressure, $mathbff$ external force.
We consider the following constitutive law (power law model): beginequation S=S_varepsilon = g D ^varepsilon-1 Dqquad (varepsilon>0, ; g>0) endequation ($D=D(mathbfu)= D_ij(mathbfu)$, $D_ij(mathbfu) = displaystylefrac12(partial_iu_j+partial_ju_i)$ rate of strain). The limit case $varepsilon=0$
[0.2cm] characterizes the constitutive behavior of a perfectly plastic fluid (“von Mises solid”; R. von Mises (1913)): beginequation S_0