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Materials Testing for the Metal Forming Industry pp 10–85Cite as

Determination of Flow Curves for Bulk Metal Forming

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Abstract

In the simplest case the flow curve is obtained by measuring the force F and the elongation A L in the tensile test. In the range of strain below uniform elongation it is assumed that the stress is constant over the cross-section of the specimen. The ratio F/A where A is the actual cross-section is called flow stress σf:

$$ {\sigma _f} = \frac{F}{A}$$
(2.1)

For the plastic deformation of most metals the condition of volume conservation is fulfilled with good accuracy. Therefore the actual cross-section A can be calculated from the measured elongation:

$$A = \frac{{\pi r_0^2{L_0}}}{{{L_0} + \Delta L}} $$
(2.2)

where L0 is the initial gage length of the test piece. Flow stress is plotted vs. strain

$$ \phi = In\frac{{{L_0} + \Delta L}}{{{L_0}}}$$
(2.3)

The function

$$ {\sigma _t} = {\sigma _f}\left(\phi\right) $$
(2.4)

is called the flow curve. It indicates the stress required for plastic deformation to occur under uniaxial stress. The relation between an uniaxial and a multiaxial state of stress is described by the concepts of equivalent stress and equivalent strain. So in the general case of a triaxial state of stress in Eq. (2.4) φ must be replaced by φ̄.

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Abbreviations

A0 :

initial cross-section of tensile specimen (Figs. 2.5 and 3.1)

A:

actual cross-section of tensile specimen

Amin :

cross-section in the neck of a tensile specimen (beyond uniform elongation)

a:

width of a plane strain upsetting die (Fig. 2.20)

α:

angle of twist in the torsion test

α̇:

time derivate of α

B:

diameter of the head of a torsion test piece (Fig. 2.24)

b:

length of a plane strain upsetting die (Fig. 2.20)

b:

constant in Eq. (2.69)

C:

diameter of the head of a tensile test piece

C:

constant for the absolute magnitude of flow stress in Eq. (2.12) and (2.87)

C:

constant in Eq. (2.74)

C1 :

constant in Eq. (2.56)

c:

specific heat

D1 :

constant in Eq. (2.58)

d:

average grain diameter

d0 :

initial diameter of a tensile test piece (Fig. 2.5)

δ:

factor in Eq. (2.77)

ε:

engineering strain (e.g. relative increase of length in tension test)

et5 :

relative elongation at fracture (total elongation) of a tensile specimen with L0/d0 = 5

εt10 :

relative elongation at fracture (total elongation) of a tensile specimen with L0/d0 = 10

εu :

uniform elongation of a tensile test piece

F:

force

F’:

upsetting force divided through the initial cross-section (Table 2.2)

f(γ, γ̇):

“correction function” for the shape of the flow curve in Eq. (2.57)

f(γr, γ̇r):

averaged “correction function” (Eq. (2.62))

φ:

natural or true strain (log. deformation ratio)

φ0 :

natural strain at the beginning of test

φ1 :

natural strain after the first step of an interrupted upsetting test

Δφ:

increase of natural strain

φ̄d :

equivalent strain determined from diameter measurement in upsetting test

φ̄h :

equivalent strain determined from height measurement in upsetting test

φ̇:

strain rate (time derivate of φ)

φ̇1 :

reference value of φ̇

φ̄̇:

equivalent strain rate

φu :

strain at uniform elongation in tensile test

γ:

(infinitesimal) shear strain

γ̇:

shear strain rate (time derivate of γ)

γu :

shear strain at the radial distance u in a torsion test piece

γ̇u :

shear strain rate at radial distance u in a torsion test piece

γr :

shear strain at the circumference of a torsion test piece

γ̇r :

shear strain rate at the circumference of a torsion test piece

γp :

shear strain at the radial distance up in a torsion test piece

γ̇p :

shear strain rate at the radial distance up in a torsion test piece

G:

limit of error

h0 :

initial height of an upsetting test piece

h:

actual height of an upsetting test piece

h(φ̄):

function, defined by Eq. (2.87)

k = 3,4:

jk

k = 3,4:

coefficients, defined by Eq. (2.63)

k:

constant in the Hall-Petch equation (Eq. (2.7))

kR :

resistance to deformation

L0 :

initial gage length of tensile specimen (Figs. 2.5 and 3.1)

Lc :

length of reduced section of tensile specimen (Figs. 2.5 and 3.1)

Δ L:

increase of length of a tensile test piece

l0 :

length of reduced section of a torsion test piece

lp :

“effective length” of a torsion test piece

M:

torque

M0 :

“zero approximation” of torque

m:

strain rate sensitivity index

μ:

(Coulomb’s) coefficient of friction

ν:

constant in Eq. (2.74)

n:

strain hardening coefficient

n1 :

estimaed value of n

n’:

constant in Eq. (2.74)

Q:

activation energy

ϙ:

densitiy

ϙ:

contour radius of the neck of a deformed tensile test piece

ϙ:

contour radius of an upsetting test piece after bulging

R:

contour radius at the ends of the gage length of a tensile or torsion specimen, see Figs. 2.5, 2.24 and 3.1

R:

gas constant

r0 :

initial radius of a tensile or upsetting test piece

r:

actual radius of a tensile or upsetting test piece

r:

outer radius of a solid or tubular torsion test piece

r↑ :

inner radius of a tubular torsion test piece (r↑ = 0 for solid specimen)

rmax :

maximum radius of an upsetting test piece after bulging

Syl :

lower yield point

Sy∞ :

yield strength for infinite grain size

S:

decrease of height of an upset specimen

ṡ:

time derivate of s

ṡ0 :

initial value of s

σ:

normalized “measure of deformation” (Table 2.6)

σik :

components of the stress tensor

T:

temperature

Tm :

melting temperature

t0 :

initial value of the ring height of a Rastegaev test piece (Fig. 2.9)

t:

actual height of the ring

t:

time

τ:

shear stress

u0 :

initial ring width of a Rastegaev test piece (Fig. 2.9)

u:

actual width of the ring

u:

distance from the axis in a torsion test piece

up :

“critical” distance from the axis in a torsion test piece

χ:

coefficient of friction (Coulomb) under the lubricant in the Rastegaev test

X:

“normalized variation of dimensions” (Tables 2.2 and 2.6)

Z:

Zener-Hollomon parameter

Z:

axial coordinate

ω:

relative error

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Further Reading

  1. Barrett, CS.; Massalski, T.B.: Structure of Metals, 3rd Ed., New York, McGraw-Hill, 1966.

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  2. McLean, D.: Mechanical Properties of Metals, Huntington, N.Y., Robert E. Krieger Publ. Comp., 1977.

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  4. ASTM E 4–83a: Standard Practices for Load Verification of Testing Machines, 1983.

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  5. ASTM B 312–82: Standard Test Method for Green Strength for Compacted Metal Powder Specimens, 1982.

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  6. ASTM B 528–83a: Standard Test Method for Transverse Rupture Strength of Sintered Metal Powder Specimens, 1983.

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  7. ISO 4492–1985: Metallic Powders, Excluding Powders for Hardmetals — Determination of Dimensional Changes Associated with Compacting and Sintering, 1985.

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  1. Institut für Umformtechnik, Universität Stuttgart, Holzgartenstraße 17, 7000, Stuttgart 1, Germany

    Dr.-Ing. habil. Klaus Pöhlandt

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© 1989 Springer-Verlag Berlin Heidelberg

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Pöhlandt, K. (1989). Determination of Flow Curves for Bulk Metal Forming. In: Materials Testing for the Metal Forming Industry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-50241-5_2

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