Manufacturing Processes - MFRG 315 - 3.2 Metal Forming Determining the Flow Stress - 1
In most mechanical engineering applications the change in dimensions under load is small, hence in calculating engineering stress and strain, no account is taken of any changes in dimension. In tension, stress is load / original area, engineering strain is change in length / original length.

In bulk deformation processes, the change in dimension will often be large, this must be accounted for in calculating stress and strain.

In elastic deformation, Poisson ratio (about 0.3) effects are due primarily to interatomic distance changes. In plastic deformation the volume remains essentially constant, atoms move with little change in their spacing, hence Poissons ratio is about 0.5.

In general the stress state in metal forming operations is triaxial.

Analysis is simplified if the co-ordinate system is orientated so that shear stresses disappear and only the three normal principal stresses act.

Plastic flow occurs when the stress state satisfies a yield criterion.

It is generally accepted that the yielding of ductile materials is not affected by hydrostatic stresses. Hence yielding is affected by the deviatoric stresses, that is the difference between the complex stress state occurring and the corresponding hydrostatic stress.

The two most widely used yield criteria are those developed by Tresca and by von Mises, see notes on failure criteria.

Yield criteria specify the onset of plastic deformation - we are interested in maintaining plastic deformation, the appropriate term is then the flow stress. It should be noted that the yield strength given in text books is of limited use as many metals work harden when cold worked and the stress to maintain flow rises with the deformation.

For certain stress states, von Mises predicts a flow stress that is 15% higher than the uniaxial flow stress. Not all metals obey the von Mises criterion, but for conservative analysis it is usual to use 1.15 sigmaf (frequently denoted: = 2 k) as the flow stress in plane strain.

In bulk deformation our interest starts with stresses above the yield stress, changes in dimensions must be considered and we use true stresses and strains.

The true, natural or logarithmic strain is defined as the natural logarithm of the ratio of the instantaneous length l, to the original length lo:

True strain, epsilon = ln ( l / lo ) = ln ( Ao / A )

However, to give ratios greater than one, when carrying out a compression test, the ratio of the larger to the smaller value is used, ie: ln ( lo / l) Cold Working
Here the flow stress, sigmaf is commonly a power function of the strain:

where K is the strength coefficient and n is the strain hardening exponent.

In a steady state process, such as wire drawing and rolling, the work strain hardens as it passes through the deformation zone. A mean flow stress can be used for these calculations, this is found by integrating the above equation between the limits of strain. For an annealed material this gives:

Alternatively a flow stress curve may be plotted and an average found by visual averaging.

Annealing- Cold work increases the yield strength (YS) of the material, the TS also rises, but less rapidly and the TS/YS ratio approaches unity. The ductility decreases. Hence only a limited amount of cold working is possible before annealing is required. In many cases cold working is used to produce high strength wire or sheets and the deformation - annealing sequence must be carefully planned to finish at the required section and mechanical properties. Very heavily cold worked metal can after annealing give a very fine grain structure having a high strength combined with reasonable ductility. Hot Working
Here the strength is substantially strain rate sensitive:

where C is a strength coefficient and m is the strain rate sensitivity exponent.

C is found at a strain rate of unity and m is the slope of the line. Both C and m vary with temperature. Increasing the temperature usually increases the strain rate sensitivity and hence m, but always decreases the flow stress and hence C.

When hot working, diffusion and softening effects occur during the deformation.

Typical values for n and m are:

  • cold working: n = -0.05 to 0.4
  • hot working: m= 0.05 to 0.3

More specific values can be found in texts, see reference 1.

Deformation in the temperature range 0.3 Tm to 0.5 Tm is usually referred to as 'warm working', and is characterised by effects intermediate between those of hot and cold working.

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David J Grieve, 24th October 2008.