he analytical solutions of thick-walled cylinder of softening material and its stability

The constitutive law of elastoplastic material beingsimplified to three-line model (elastic-linear softening plastic-residualideal plastic model), and the material obeying Tresca yield criteria andassociated flow rule, the analytical solutions of thick-walled cylindersubject to internal pressure p were derived in the paper. The result showsthat the yield stress in the softening plastic region is the inverse squareof radial coordinate r.Firstly, the pressure p was taken as generalized force, the displacement utaken as generalized displacement, and the thick-walled cylinder taken as awhole system. On the basis of the solutions the stability problem ofthick-walled cylinder was then discussed. The p-u curve of balance path wasdrawn, on which each point denotes a balance state. The slope of the tangentline for each point can be considered the stiffness of thick-walledcylinder. The extreme value of generalized force is the critical point onthe curve, and the critical point separates the curve into two sections: thesection before the critical point is stable, and the stiffness is positive;the section after the point is unstable, and the stiffness is negative. Whenthe generalized force reaches the critical point, the displacement increasesquickly and the system loses its stability, while ideal plastic thick-walledcylinder loses its stability only when the plastic region penetrates throughthe whole cylinder. Therefore, the failure mechanism is completelydifferent: the former belongs to extreme value point destabilization, andthe latter belongs to strength failure. That is to say, the bearing capacityhas different mechanical meanings.