# Dr.R.Narayanasamy - Mohr's circle and Formability

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19-Jan-2017Category

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Mohrs circle & FormabilityDr.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

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Mohrs circle representation of a three-dimensional state of stress.2

Mohrs circle (three dimensional) for the state of stress Uniaxial tension

- 1 max=1/ 2

2 = 3 = 0(a)

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https://www.youtube.com/watch?v=IIWSgApnIpM

Uniaxial tension (video)

Mohrs circle (three dimensional) for the state of stress Uniaxial compression

1 = 2 = 03 max

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https://www.youtube.com/watch?v=53QjIamBW3w

Uniaxial compression (video)

Uniaxial tension/CompressionTensile/Compressive stress is applied in one directionA maximum shear stress is achieved in the positive region for tensile stress and vice versa for the compressive stressThe maximum shear stress is uniform for tension & compression and which is larger than biaxial tension/compression.

Mohrs circle (three dimensional) for the state of stress biaxial tension

3 = 0 2 1 max = 2 1 3

31 2 3= 0

http://youtube.com/watch?v=FlOjVZis4SA

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Mohrs circle (three dimensional) for the state of stress biaxial tension2 - Tensile stress right angle to 1 (Tensile stress)Decrease in principal shear stress on two sets of planes out of threeMaximum shear stress is decreased from what would be for uniaxial tension

biaxial tension simulation (video)

Mohrs circle (three dimensional) for the state of stress biaxial compression

1 3 = 0 1 3

1 2 2 3 max = 2

3 = 0https://www.youtube.com/watch?v=h_bL40T5dl4

biaxial compression simulation (video)

Mohrs circle (three dimensional) for the state of stress triaxial tension (unequal)

1 2 = 3 max = 2= 3

1 3 2 1 = 2 2 = 23

Mohrs circle (three dimensional) for the state of stress triaxial tension (equal)1 2 = 3 max = 2 =3

1 3 2 1 = 2 = 3

1 = 2 = 3

Mohrs circle (three dimensional) for the state of stress triaxial tension (unequal)&(Equal)Tensile stress applied in the three directionsMaximum shear stress is reducedFor equal triaxial tension Mohrs circle will be a point and no shear stress acts on the body. This will reduce the ductility, because plastic deformation is produced by shear stressEqual triaxial state of stress will lead to brittle failure

Mohrs circle (three dimensional) for the state of stress triaxial compression (unequal)

2 = 3 1

max = 2 = 3

1 2 3

triaxial compression simulation (video)

Mohrs circle (three dimensional) for the state of stress combined tension and compression1 max = 2 = 3

3 2 1 1 = -2 2 = -23

2 = 3 1 max = 2 = 3 Example: Wire drawing

Mohrs circle (three dimensional) for the state of stress combined tension and compression (Wire drawing)Compressive stresses are applied to the lateral surface of tensile stress, which gives large maximum shear stress than uniaxial stress.Materials subjected to tension + compression will have good formability before fracture.It is applied in plastic working of metals. (Ex: More ductility is achieved in wire drawing through a die. Here applied load is tensile in nature whereas the reaction between the die and metal is compressive in nature).

Drawing operation (video)

Drawing operation (video)

Engineering Stress Strain CurveEngineering tension test provide basic design information on strength of material.In tension test uniaxial tensile force is applied on the specimen and its elongation was observed.Engineering stress strain curve was formed using load vs elongationThe stress used in this curve is average longitudinal stress = load/original c/s area

Engineering stress strain curve

Tensile strength & yield strength (strength parameters)

Percentage elongation and reduction of area (ductility parameters)

Engineering stress strain curveThe strain used for engineering stress strain is average linear strain. (elongation of gauge length/original length)

Both stress and strain are obtained by dividing the load and elongation by constant factors the (load-elongation) curve will have same shape as engineering stress strain curve.

Engineering stress strain curveThe shape and magnitude of stress strain curve depends on:1. chemical composition of material2. heat treatment3. prior history of plastic deformation4. strain rate5. temperature6. state of stress imposed during testing.

Engineering stress strain curveStress is proportional to strain (Elastic region)As load increases the value corresponding to yield strength the specimen undergoes plastic deformationThe stress to produce continued plastic deformation increases with increasing plastic strain (metal strain hardens)The volume of specimen remains constant during plastic deformation

Engineering stress strain curveAs the specimen elongates it decreases uniformly along the gauge length in cross sectional area.Initially, strain hardening compensates for decrease in area and the engineering stress in proportional to engineering strain (engineering stress is proportional to load)At certain point decrease in cross sectional area is > than increase in deformation load arising from strain hardening (this condition occurs in the specimen which is weaker than the rest portion). The specimen begins to neck/thin down locally.

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

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