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# Begins is the yield stress σ o if we continue the

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begins is the yield stress σ o . If we continue the loading, then it is typically observed that large increases in strain occur with relatively small increase in stress. The slope in this region is

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referred to as plastic hardening. Plastic deformation is irreversible and dissipates part of the mechanical energy supplied by the testing machine. The decrease in stress after the ultimate stress is rather puzzling. In fact, it is an artifact of the way in which the stress and strain have been defined. Let the initial length of the specimen be L o and the initial cross-sectional area be A o . We usually define the stress as the force per unit initial area, i.e. σ =F/A o where F is the force supported by the specimen. In the elastic region the axial strains and the change in the cross-sectional area are very small. However, once necking starts the cross-sectional area is considerably smaller than the initial area. If we denote the current cross-sectional area by A , then the true stress is defined as σ t = F/A . The true stress does not decrease, but instead continues to increase. True strain is defined as follows. Let the incremental longitudinal true strain be d ε t =dL/L . For a deformation from an initial length of L o to the current length of L , the true strain is obtained by integration as L L t t L L Ln L dL d 0 0 (1) Since the definition of the engineering strain is 1 0 0 0 0 L L L L L L L (2) it follows from equations (1) and (2) that
) 1 ( Ln t (3) It is typically observed in experiments that plastic deformation does not change the volume. This observation can be used to find the relation between true stress σ t and engineering stress σ . Since volume is conserved, A o L o = AL , which implies that 1 0 0 L L A A Note that I have used (2) to get the second equality. Now using the definitions of the true stress and eng. stress we get 1 0 0 A A F A A F t Thus the relation between the true stress and engineering stress is ) 1 ( t (4) A simple polynomial relation can be used to relate the true stress σ t to the true strain t is as follows n t t H ) ( (7) where n is called the strain hardening coefficient. The quantity H is called the strength coefficient and corresponds to the value of the true stress at the true strain value equal to 1 b. Ductile fracture process: As a round specimen is stretched beyond the stage of necking, the process of failure starts in the specimen. Typically every material contains some micro voids,

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pores, etc. The deformation of the voids in the neck region is schematically depicted in Figure 2. The voids are elongated more in the direction of the force than in the transverse direction (Figure 2b). This leads to a situation where the force is essentially carried by thin strips of material between the voids. Eventually these strips break and the cross-section resembles that of a hollow cylinder (Figure 2c). When examined under a microscope, the morphology has a distinct
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