Necking

neck

1. NECKING CRITERIA & Notch tensile test

2. Instability in tension Ideal plastic materials Undergo necking after yielding with no strain hardening. Most metals Necking begins at maximum load with strain hardening increasing load-carrying capacity Instability occurs when > An increase in stress due to reduced cross-sectional area The increase in load-carrying ability due to strain hardening 2

3. The condition of instability, which leads to localized deformation is defined by dP =0 . From the constancy of volume relationship, From instability condition So that at a point of tensile instability 3

4. Therefore the point of necking can be obtained from the true stress-strain curve by Finding the point on the curve having a sub tangent of unity The point where the rate of strain hardening equals the stress 4

5. The maximum load can be determined from Considère’s construction when the stress-strain curve is plotted in terms of true stress σ and conventional strain e . Let point A represent a negative strain of 1. A line drawn from point ‘A’ which is tangent to the stress-strain curve will give maximum load with the slope of σ/ ( 1+ e). This strain is the true uniform strain ε u . Considère’s construction for the determination of the point maximum load 5

6. Flow instability (necking) in biaxial tension Necking in a uniaxial cylindrical tensile specimen is isotropic. However in a sheet specimen where the width of the specimen is much higher than the thickness, there are two types of flow instability: 1) Diffuse necking Provide a large extent of necking on the tensile specimen similar to necking from a cylindrical specimen. Diffuse necking might terminate in fracture but normally followed by localised necking. 2) Localised necking Localised necking is a narrow band with about equal to the sheet thickness and inclined at an angle to the specimen axis, across the width of the specimen. Give no change in width through the localised neck plain strain deformation . 6

7. With localized necking the decrease in specimen area with increasing strain is restricted to the thickness direction. From constancy of volume, d ε 2 = d ε 3= -d ε 1 /2, and d ε 3 = dt /t The increasing in load carrying ability due to strain hardening is given by Equating above two equation For a power law flow curve, ε u =2n for localized necking. 7

8. Stress distribution at the neck Necking introduces a complex triaxial state of stress in the necked region. The neck region is in effect a mild notch. The average true stress at necking , which is much higher than the stress would be required to cause a normal plastic flow due to stresses in width and thickness directions. (a) Geometry of necked region (b) Stress acting on element at point ‘ O’ 8

9. Bridgman made a mathematical analysis which provides a correction to the average axial stress to compensate for the introduction of transverse stresses. This analysis was based on the following assumptions: The contour of the neck is approximated by the arc of a circle. The cross section of the necked region remains circular throughout the test. The von Mises ’ criterion for yielding applies. The strains are constant over the cross section of the neck. According to Bridgman’s analysis, the uniaxial flow stress corresponding to that which would exist in the tension test if necking had not introduced triaxial stresses is where, ( σ x ) avg is the measured stress in the axial direction, R is the radius of curvature of the neck a is the linear distance. 9

10. Ductility measurement in tension test Measured elongation in tension specimen depends on the gauge length or cross-sectional area. The total extension consists of two components, the uniform extension up to necking and the localized extension once necking begins . The extent of uniform extension will depend on the Metallurgical condition of the material (through n) and the specimen size and shape on the development of necking. The shorter the gauge length the greater the effect of localized deformation at necking on total elongation Variation of local elongation with position along gauge length of tensile specimen 10

11. The extension of a specimen at fracture can be expressed by Where, α is the local necking extension e u L is the uniform extension The tensile elongation then is given by According to Barba’s law, and the elongation equation is 11

12. Dimensional relationships of tensile specimens for sheet and round specimens Elongation depends on the original gauge length L . The shorter gauge length the greater the percentage of elongation. 12

13. Difference between % elongation and reduction of area % Elongation: % Elongation is chiefly influenced by uniform elongation and thus it is dependent on the strain-hardening capacity of the material. Reduction of area: Reduction of area is more a measure of the deformation required to produce failure and its chief contribution results from the necking process. Because of the complicated stress state in the neck, values of reduction of area are dependent on specimen geometry and deformation behaviour, and they should not be taken as true material properties. Reduction of area is the most structure-sensitive ductility parameter and is useful in detecting quality changes in the materials. 13

14. Effect of strain rate on flow properties Strain rate is applied to the specimen can have an important influence on the stress. Strain rate unit is s -1 . 14 Flow stress dependence of strain rate and temperature

15. The crosshead velocity is v= dL / dt . The rate expressed in terms of conventional linear starin is The true strain rate is given by The above equation indicates that for a constant crosshead speed the true strain rate will decreases as the specimen elongates. The true strain rate is related to the conventional strain rate by the following equation: 15

16. To maintain a constant true strain rate using open loop control the deformation velocity must increases in proportion to the increases in the length of the specimen as For deformation occurring at constant volume a constant true strain rate is obtained if the specimen area changes as A general relationship between flow stress and strain rate, at constant strain and temperature as Where m is strain rate sensitivity. 16

17. The velocity dislocation motion was very strongly dependent on stress according to Strain rate is related to velocity of mobile dislocation by Therefore, If there is no change in the mobile dislocation density with increasing stress, m ’ =1/m . 17

18. Strain rate sensitivity (m) indicates any changes in deformation behaviour. Measurement of strain rate sensitivity can be linked to dislocation concept. High strain rate sensitivity is a characteristic of superplastic metals and alloys. From the definition of true strain rate Combining above two equations The above equation states that so long as m is less than 1 the smaller the cross sectional area, the more rapidly the area is reduced. 18 or

19. When m=1 the deformation is Newtonian viscous and dA / dt is independent of ‘ A’ and any neck is simply preserved during elongation and does not propagate inward. As ‘ m’ approaches unity, the rate of growth of incipient necks is drastically reduced. 19 Dependence of tensile elongation on strain-rate sensitivity Dependence of rate of decrease of area on cross sectional area for different values of ‘ m ’

20. Notch tensile test Notch tensile test is used to evaluate notch sensitivity (the tendency for reduced tensile ductility in the presence of a triaxial stress field and steep stress gradients). Notch tension specimens have been used for fracture mechanics measurements. Notch tensile specimen 60° notch with a root radius of 0.025 mm or less introduced into a round (circumferential notch) or a flat (double-edge notch) tensile specimen. The cross-sectional area under the notch root is one-half of the unnotched area. 20 Notch tensile specimen Stress distribution around tensile notches

21. Notch strength Notch strength is defined as the maximum load divided by the original cross-sectional area at the notch. Due to the constraint at the notch, the notch strength is higher than the tensile strength of the unnotched specimen. Notch-strength ratio (NSR) detects notch brittleness (high notch sensitivity) from If the NSR is < 1 , the material is notch brittle. 21

22. Thank you 22