An extension of one of the extremum principles for a Bingham solid

Summary An extension of the extremum principle concerned with velocity fields for boundary value problems of an incompressible rigid visco-plastic (Bingham) solid is derived. This extension can be used to obtain close overestimates for the rate of work of the unknown surface tractions in certain problems of visco-plastic flow.Nomenclature k yield stress in pure shear - coefficient of viscosity - _ij stress tensor referred to rectangular cartesian axes Ox _i - s _ij stress deviator tensor - T _i surface traction - J (1/2s _ijs_ij ^1/2 - F _i body force per unit volume - v _i velocity vector - _ij = rate of deformation tensor - I (2 _ijij ^1/2 - v] magnitude of a velocity discontinuity in flow of a rigid perfectly plastic solid - S surface - V volume - S _t part of surface upon which T _iis prescribed - S _v part of surface upon which v _iis prescribed - S _d surface of velocity discontinuity in flow of a rigid plastic solid - x, y, z rectangular cartesian coordinates - u, v, w velocity components in the x, y and z directions respectively - rate of twist per unit length - T torque