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A Review of Development and Use of the Upper Bound Approach to Metal Forming Processes

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Grundlagen der Umformtechnik — Stand und Entwicklungstrends / Fundamentals of Metal Forming Technique — State and Trends

Summary

The development and use of the upper bound approach (UBA) are surveyed to indicate on what bases it stands, to what purposes it has been applied and how it has adapted itself to changing technological requirements and advancing computational facilities. The state of the art of and the problems set on UBA are then summarized. Finally the unique parts which UBA should play to continue useful in the future age of the finite element method are suggested.

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Abbreviations

f :

friction factor defined by f = τ f /k

k, k f :

flow stresses of workpiece material in simple shear and compression resp.

k fm :

mean of k f within workpiece

m :

strain-rate hardening exponent

SC :

work-tool interface

SD*:

assumed internal surface of velocity discontinuity

ST, SV :

workpiece surfaces where tractions and particle velocities are specified resp.

Ti :

vector component of specified traction

V :

workpiece volume

Vi :

vector component of specified tool velocity

V i , \({\dot V_i}\) :

vector components of actual velocity and acceleration of workpiece particle

V * i , V * t :

kinematically admissible velocity component and tangential velocity discontinuity resp.

V n , V n :

normal components to SC of tool velocity and workpiece particle velocity resp.

\({\dot W_C}*\) :

upper bound to total rate of work being done by tool over SC

\({\dot W_d},\,{\dot W_f}\) :

calculated rates of work due to plastic deformation and frictional sliding resp.

\({\dot W_s},\,{\dot W_t}\) :

calculated rates of work due to internal shearing and traction resp.

X i :

i-th axis of Eulerian co-ordinate system

\(\bar \dot \varepsilon *,\;\dot \varepsilon _{ij}^*\) :

kinematically admissible equivalent strain-rate and component of strain-rate tensor resp.

\(\int {d\bar \varepsilon}\) :

actual equivalent strain = \(\int {\bar \dot \varepsilon \,dt}\)

\(\bar \dot \varepsilon _m^*\) :

average of \(\bar \dot \varepsilon *\) within workpiece

θ:

current temperature of workpiece particle

μ:

coefficient of friction

ρ:

density of workpiece material

\(\bar \sigma\) :

equivalent stress due to von Mises

τ f :

frictional stress

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  1. Department of Mechanical Engineering, Yokohama National University, Japan

    H. Kudo

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Kudo, H. (1983). A Review of Development and Use of the Upper Bound Approach to Metal Forming Processes. In: Grundlagen der Umformtechnik — Stand und Entwicklungstrends / Fundamentals of Metal Forming Technique — State and Trends. Berichte aus dem Institut für Umformtechnik der Universität Stuttgart, vol 74. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82186-8_3

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