Equivalent theorem of Hellinger-Reissner and Hu-Washizu variational principles

Abstract

; e O. d s O. or = e/j (2d) do-O they satisfy the following energy identity A ~-B = %o-~ (2e) 2.3 Strain-displacement relations eo = l (lli,j + llJ, i) (in r) (3) £, 2.4 Boundary conditions For given surface displacement u i = i7, (in F,) (4) For given external force on boundary surface O-,:m.,.. = Pi (in F~) (5) Let F be the total boundary surface, then F = r,, + Fa (6) The known variational principles ate recapitulated at below: 1) The Minimum Potential Energy Principle J,,(ui)= III(A-f,,,,)d II ,,,,dS (7) where t6 is one kind of independent variable, the equations ~2) of strain-stress relations , the equations(3) of strain-displacement relations and the boundary conditions (4) for given surface displacement are its variational constraints. 2) Hellinger-Reissner Variational Principle JHR (O'O' lli) = ~(B + O-ij,./lli + .[)lli) d r-IIo-onjgi dS-IIui(o-on/-~i)dS (8) F, I'd where O-~/,uj are two kinds of independent variables, and the equations (2) of strain-stress relations are its variational constraints. 3)