A general kinematically admissible velocity field for axisymmetric forging and its application to hollow disk forging

Abstract

An approach for selecting general kinematically admissible velocity fields for axisymmetric forging problems is outlined. The approach accounts for the existence of a rigid zone at frictional interfaces and for the singular behavior of real velocity fields in the vicinity of maximum friction surfaces. The plastic work rate for a material obeying the von Mises yield criterion and its associated flow rule is expressed in terms of one arbitrary function of a single argument, its derivative, and anti-derivative. An upper bound solution for constrained forging is given. Comparison with an available upper bound solution is made and it is shown that the new kinematically admissible velocity field results in a more accurate solution at high friction. Other problems that can be treated are briefly discussed.

An Upper Bound Solution for Continued Compression of a Cylinder

“…The power dissipated in the process under consideration is composed of three components: the power dissipated within the plastic region, W V ; the power dissipated due to plastic shearing at the velocity discontinuity surface ac, W d ; and the power dissipated by friction at the sliding contact between points c and b, W f . All these components have been found in [18]. In particular:…”

“…A general upper bound solution for axisymmetric forging has been proposed in [18]. The corresponding solution for instantaneous compression of a solid cylinder can be derived from this general solution as a special case.…”

“…where ƞ c = (r c / R 0 ) 2 and ƞ b = (r b / R 0 ) 2 = ƞ bd (1). It has been shown in [18] that ƞ c = −2hf (1) independently of the function f(ζ). Therefore, using Eq.…”

“…It is worthy of note that these simple kinematically admissible velocity fields are in general capable of predicting rather accurate upper bounds on the load required to deform the specimen since mathematical properties of the extremum principles of plasticity allow close bounds to be obtained even from crude approximations for the actual stress and velocity fields (it is of course not true for flow pattern) [17]. A general kinematically admissible velocity for axisymmetric forging has been proposed in [18]. This field accounts for the existence of a rigid region and is of the same level of complexity as the fields used in Ebrahimi and Najafizadeh [5,9,10].…”

“…This field accounts for the existence of a rigid region and is of the same level of complexity as the fields used in Ebrahimi and Najafizadeh [5,9,10]. The objective of the present paper is to combine a particular case of the kinematically admissible velocity field proposed in [18] appropriate for the cylindrical compression test and the concept of sequential limit analysis.…”