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Flow Stress

Flow Stress

The flow stress is a fundamental parameter to determine toque and power of metal forming equipment. It is defined as a stress that results in the material flow in a one-dimensional stress state. When determine force and power requirement for a forming process, two factor groups are actually considered: (1) Metal to be deformed, (2) Tool type, or forming process (rolling, forging, extrusion, etc.). The part (2) is not covered here. The factor group related to metal itself is from flow stress point of view. This group of factors consists of strain, strain rate, temperature, initial grain size, and so on. We will discuss the detailed influence factors as follows.

Influence Factors for Flow Stress

Various material and process related parameters affect the flow stress, as is showed in Fig. 1. With increase of the second phase, the flow stress may increase in the case of some materials, or decrease for some other materials (a). With higher purity of material, the flow stress value also become higher (b). Crystal structure change resulted from the hardening and softening during cold and hot forming strongly affects the metal flow stress (c). With a change of phase composition, the flow stress changes d). In the chart (e), the cast structure of the entry material in the curve 1 is rougher than in the curve 2, so the flow stress denoted by curve 2 is higher than those by curve 1. The finer the grain size is, the higher the flow stress becomes (f).

Fig. 1: Influence factors on the flow stress [101]

a) Chemical composition; b) Purity; c) Cristal structure;
d) Phase constitution; e) Exit microstructure; f) Grain size
g) Heat treatment state (1: Heated to temperature δ
A and cool it to the forming temperature δU, 2: Heated to forming temperature δU directly); h) Sample state; I) Entry material (1: from smelting metallurgy, 2: from powder metallurgy)

Translation: Korngroβe grain size; Bedingungen - conditions

Heat treatment has a significant influence on the flow stress of cold forming. However, for hot forming, the heat treatment only has a slight change in its flow stress. In chart (g) curve 1, the material is heated to the temperature δA and cooled it to the forming temperature δU, while in the case of curve 2, the material is heated to forming temperature δU directly. The difference of flow curve is presented. During flow stress measurement, sample state also plays a roll. Material may have different mechanical property in different direction (h). In addition, how the material is produced, is also an important factor. Chart (i) shows a material produced with smelting metallurgy (curve 1) receives higher flow stress than those from power metallurgy.

Besides the material related factors discussed above, forming process parameters, such as strain, strain rate and temperature, also play an extremely important role. Such forming parameters even receive much high attention from metal forming engineers.

Fig. 2: Dependence of flow stress on the important forming technical influence factors [101]

Translation: Flieβspannung flow stress, Einphasenraum - One phase territory; Umformtemperatur deformation (forming) temperature; Umformgrade strain; Umformgeschwindigkeit strain rate

With the increased temperature, the material becomes softer, so the flow stress decreases. One exception to this is the temperature range within which a phase transformation occurs (see Fig. 3a). In this temperature range the flow stress could increase significantly with the increase of temperature.

The influence of strain can be described in a very rough way as, flow stress increase with a increase of strain. This statement is correct for cold forming (Fig. 3b, curve 1). For hot forming, sometimes it is not correct, especially when the strain is very high. In this case, the flow stress reach a certain peak value, then may decrease or keep constant with the increase of the strain (see Fig. 2b and 3b). The influence of strain dependents on two major factors: work hardening and softening (recovery and recrystallization). During cold rolling, recrystallization does not occur. Work hardening is a dominant factor. However in the case of hot rolling, recrystallization also play a very important role. The recrystallization also depends on the strain, strain rate and temperature. So it is a very complicated process. The peak stress is determined by both the stain and the temperature.

In general, with the increase of strain rate, the flow stress increase (Fig. 2c). The contribution to the cold forming is much smaller that to the hot forming. Hensel also suggested with his measurement, that when the strain rate is over 1000-2000/s (rolling speed 30-50m/s) range, flow strain almost keeps constant with the increase of strain rate. Before this range there is a peak value of the flow stress.

Different ways to measure flow stress normally receive different value. In Fig. 3c, the flow stress measured with upsetting (curve 1) is much higher than those taken with a torsion test (curve 2). In the strain range of 0.3 0.5, the difference of flow stress between those from upsetting and from torsion can be about 15%.




Fig. 3: Further influence factors on flow stress [101]
a: 1 one phase, 2 with phase change
b: 1 cold forming, 2 hot forming
c: 1 Upsetting test, 2 Torsion test

Flow Stress Model

Various studies have been done to model flow stress during hot and cold forming processes. In a straightforward way the influence from material, temperature, strain and strain rate can be considered (the grain size influence will be discussed elsewhere):


s F0 is material specific factors. The grain size influence is included in this factor.

K1 is the temperature factor, defined as below, in which A1 and m1 are constants.

K2 is the strain factor, showed below, in which A2, m2 and m4 are constants:

This factor is particularly designed for the hot forming process, where softening effect also play an important role.

K3 is the strain rate factor. A3, m3 are constants:

The valid range for the strain rate is 0.05-100/s for most steels.

For each material, there is a set of values for s F0, A1, m1, A2, m2, m4, A3 and m3. For example, for the steel AISI 15, we may use

Table 1: Flow stress for AISI 1015 and AISI 1045 for hot forming


s F0 (Mpa)








AISI 1015









AISI 1045









For cold forming, or in the case of small deformation, a simpler formula can be used:

This simplified form cannot describe the material softening factors (recovery, recrystallization, etc.) during stain hardening, so it is not suggested to use it for the hot forming (such as hot rolling), especially for a high reduction. If this equation is applied in cold forming, temperature factor can also be ignored in most cases.

There are more ways to model the flow stress. They maybe newer, or more accurate, and they are more complicated. We will contribute an article to specifically discuss the modeling of flow stress.

Mean Flow Stress

Since the metal deformation exists between the lower limit φ=0 and upper limit φ=φmax, so its necessary to use a mean flow stress s Fm to calculate to determine the driving torque, power and the deformation energy.

The physical meaning of the mean flow stress can be demonstrated in the Fig. 2, where the wid can be understood as the generated work per cubic unit due to the effect of strain and stress.

Fig. 2: Flow curve during cold forming [101]
(a) and hot forming (b) to determine mean flow stress s Fm.

For cold forming (i.e.forming temperature below the recrystallization temperature), mean flow stress can be determined as follows:

For the hot forming (i.e. forming temperature above the recrystallization temperature), due to the softening effect caused by recovery and recrystallization, the following equation is suggested to determine the mean flow stress:



[101] Arno Hensel: Technologie der Metallformung -Eisen- und Nichteisenwerkstoffe. Dt. Verl. Fϋr Grundstoffind., 1990. ISBN 3-342-00311-1

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