20 2.Fundamental Mechanical Properties of Materials o O' 02 0y1 y 0y0 E0 81 E1 E2 g" E2E3 permanent permanent permanent deformation deformation deformation (a) (b) (c) FIGURE 2.8.Increase of yield strength (and reduction of ductility)by re- peated plastic deformation.(a)Sample is moderately stressed until some plastic deformation has occurred,and then it is unloaded, which yields permanent deformation.(b)The sample is subsequently additionally permanently deformed.Note that the coordinate system has shifted after unloading from eo to e1.(c)Limit of plastic deforma- tion is reached after renewed stressing. engineering strain,as defined in Equations(2.1)and (2.2),are essentially sufficient for most practical purposes.However,as mentioned above,the cross-sectional area of a tensile test spec- imen decreases continuously,particularly during necking.The latter causes a decrease of o beyond the tensile strength.A true stress and true strain diagram takes the varying areas into con- sideration(Figure 2.9(a)).One defines the true stress as: 0= (2.5) where A;is now the instantaneous cross-sectional area that varies during deformation.The true strain is then: (2.6) ISee Problems 8 and 9.engineering strain, as defined in Equations (2.1) and (2.2), are essentially sufficient for most practical purposes. However, as mentioned above, the cross-sectional area of a tensile test spec￾imen decreases continuously, particularly during necking. The latter causes a decrease of beyond the tensile strength. A true stress and true strain diagram takes the varying areas into con￾sideration (Figure 2.9(a)). One defines the true stress as: t
A F i , (2.5) where Ai is now the instantaneous cross-sectional area that varies during deformation. The true strain is then:1 t
 li l 0 d l l
ln l l 0 i 
ln A A 0 i  . (2.6) 20 2 • Fundamental Mechanical Properties of Materials 1See Problems 8 and 9. (a) (b) (c) 0 1 1 y2 y3 2 2 3 ' " y1 y1 ' ' " " y0 y2 permanent deformation permanent deformation permanent deformation FIGURE 2.8. Increase of yield strength (and reduction of ductility) by re￾peated plastic deformation. (a) Sample is moderately stressed until some plastic deformation has occurred, and then it is unloaded, which yields permanent deformation. (b) The sample is subsequently additionally permanently deformed. Note that the coordinate system has shifted after unloading from 0 to 1. (c) Limit of plastic deforma￾tion is reached after renewed stressing.