Plastic Instability in Uniaxial Tension

Plastic Instability in Uniaxial TensionByDr. R. Narayanasamy, B.E.,M.Tech.,M.Engg.,Ph.D.,(D.Sc.),Professor, Department of Production Engineering,National Institute of Technology, Tiruchirappalli- 620 015 , Tamil Nadu, India.

True Stress vs Engineering StressEngineering Stress(s)=Load(P)/A0 A0 is the initial area of cross section of tensile sample.True Stress() =Load(P)/Ai Ai is the instantaneous area of cross sectionIn the Load-Extension graph, volume constancy principle can be applied upto the maximum load point.

Engineering Strain vs True Strain

Plastic Instability in Uniaxial Tension

Plastic Instability in Uniaxial Tension cont..For Constant Volume : A l = Constant .Differentiating on both sides, we get dA l +A dl =0 This can be written as: -(dA/A) = (dl/l) = dTherefore, the above equation becomes: (d/) = d This can be written as: (d/d) = This is the condition for plastic instability.

True Stress vs True Strain Plot

Plastic Instability in Uniaxial Tension

True Stress vs True StrainTrue Stress() and True Strain() can be related using Power Law equation according to Ludwik as follows: where is True Stress, is True Strain, n is strain hardening exponent and K is the strength coefficient.Unit for True Stress and Strength coefficient is MPa (metric unit).K and n are to be determined by curve fitting.

Determination of K and n- values

Determination of K and n- valuesStrain hardening exponent (n) is the ratio of the physical distance (mm) of a by the physical distance (mm) of b.

n value has no unit.n value represents the strain hardening ability of the material.n value varies from 0.1 to 0.5 for conventional metals.

The effect of Strain hardening exponent on True stress -True strain

Typical K and n values for various metals

Plastic Instability in uniaxial tensionWe know that: Differentiating on both sides, we get: (Or)

This can be written as: (Or)

Plastic Instability in uniaxial tension contAt maximum load, (Or) Hence, the above expression becomes:

Therefore, n = Where is the true uniform strain, which is denoted by the symbol .

Plastic Instability in uniaxial tension contIt is important to note that rate of strain hardening is not identical with strain hardening exponent (n)value.

Plastic Instability in uniaxial tensionThe rate of strain hardening can be written as follows:

Plastic Instability in uniaxial tension

Consideres construction for the determination of the point of maximum load

Necking in uniaxial tension test

illustration of diffuse necking and localized necking in case of sheet metal tensile specimen

Diffused necking & Localized neckingNecking in cylindrical specimen is symmetrical around tensile axis in case of isotropicFor sheet tensile specimen width is greater than thickness and two type of tensile instability occursDiffused necking: Its extension is greater than thickness. It will end in fracture and also it is followed by second instability (localized necking)Localized necking: Neck is narrow width is approximately equal to thickness inclined at an angle to the specimen axis.Localized necking corresponds to plane-strain deformation

Diffused necking & Localized necking cont

Diffused necking & Localized necking cont

ReferenceMechanical Metallurgy by George E.Dieter, McGraw Hill Publication, London,1988.

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