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Mathematical Modelling

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Process Modelling of Metal Forming and Thermomechanical Treatment

Part of the book series: MRE Materials Research and Engineering ((MATERIALS))

Abstract

At the very heart of metal forming analysis is the theory of plasticity. Initially proposed last century by Tresca and Mohr, it reached a stabilized form in the twenties as a result of work done by von Mises and Hencky. In the early sixties it was completed for infinitesimal analysis [2.1]. A brief description of this theory follows in section 2.2. The theory for large deformations is still being developed today.

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Abbreviations

cp :

specific heat

E:

Young’s modulus

fi :

surface traction

Fx,Fy,FZ :

forces

h:

heat transfer coefficient or height of element

H:

instantaneous height

J1,J2’J3 :

deviatoric stress invariants

k:

shear flow stress

kijkl :

elasticity constants

L:

length of bar element

m:

friction factor 0 ≤ m ≤ 1

nj :

normal unit vector

N1 ,N2 :

shape function or interpolations function

p:

hydrostatic pressure or applied pressure

pu,pl :

applied pressure in upper and lower dies

P:

total load

qi :

body loads or heat flux

t:

time

Ti :

surface loads or temperatures

ui,j :

displacement

dui,j :

displacement increments

vo :

axial velocity of die

Δv :

velocity discontinuity

\({v_{{t_1}}},{v_{{t_2}}}\) :

tangential velocities

α:

thermal expansion coefficient (or angle)

\(\bar \varepsilon\) :

effective strain

\(\dot \bar \varepsilon\) :

effective strain rate

εx,εy,εz,γxy,γzx,γyz :

generalized state of strain

εij.:

infinitesimal deformation

dε *ij :

admissable strain increment tensor

μ:

friction factor 0 ≤ μ ≤ 0.577

ρ:

density

\(\bar \sigma\) :

equivalent or effective stress

σx,σy,σz,τxy,τxz,τyz :

generalized state of stress

σz,σr,σt :

axisymmetric state of stress σz axial stress σr radial stress σt hoop stress

σf :

flow stress in a uniaxial test

σm :

average stress

σi = σl,σ2,σ3 :

principal stresses (σ1 > σ12 > σ3)

\({\sigma _i}^\prime = {\sigma _1}^\prime ,{\sigma _2}^\prime ,{\sigma _3}^\prime \left( {or{s_1},{s_2},{s_3}} \right)\) :

deviatoric stresses

σ *ij :

admissible stress tensor

σy :

yield stress

τ:

friction shear stress or tangential shear stress

τmax :

maximum shear stress

τy :

shear yield stress

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Authors and Affiliations

  1. Research Center, BBC Brown, Bovery & Company, Limited, CH-5401, Baden, Switzerland

    Claudio R. Böer Ph. D., Hans A. B. Rydstad (Tekn. Dr.) & Günther Schröder (Dr.-Ing.)

  2. MARC Analysis Research Corp., Palo Alto, Calif., USA

    Nuno M. R. S. Rebelo

Authors
  1. Claudio R. Böer Ph. D.
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  2. Nuno M. R. S. Rebelo
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  3. Hans A. B. Rydstad
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  4. Günther Schröder
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© 1986 Springer-Verlag Berlin, Heidelberg

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Böer, C.R., Rebelo, N.M.R.S., Rydstad, H.A.B., Schröder, G. (1986). Mathematical Modelling. In: Process Modelling of Metal Forming and Thermomechanical Treatment. MRE Materials Research and Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82788-4_2

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  • DOI: https://doi.org/10.1007/978-3-642-82788-4_2

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  • Print ISBN: 978-3-642-82790-7

  • Online ISBN: 978-3-642-82788-4

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