198 Modern Physical Metallurgy and Materials Engineering Upper Unirradiated td point re (19x20° neutrons Luders strain per cm2) 100 Uniform elongatIon 60 Figure 7.1 Stress-elongation curves for (a)impure iron, (b)copper, (c) ductile-brittle transition in mild steel(after Churchman, Morford and Cottrell, 1957 to the elastic portion of the stress-strain curve from than the strain to fracture measured along the gauge For control purposes the tensile test gives valuable length the point of 0. 1 strain True stress-true strain curves are often plotted to information on the tensile strength(TS= maximum show the work hardening and strain behaviour at large ad/original area)and ductility(percentage reduction strains. The true stress o is the load P divided by the in area or percentage elongation) of the material. When area A of the specimen at that particular stage of strain it is used as a research technique, however, the exact and the total true strain in deforming from initial length ape and fine details of the curve, in addition to the lo to length Ii is e=Ji(dl/I)=In(11/lo). The true way in which the yield stress and fracture stress vary stress-strain curves often fit the Ludwig relation a with temperature, alloying additions and grain size, are ke" where n is a work-hardening coefficient 0. 1-0 robably of greater significance and k the strength coefficient. Plastic instability, or The increase in stress from the initial yield up to the necking, occurs when an increase in strain produces TS indicates that the specimen hardens during defor- no increase in load supported by the specimen, i.e mation (i.e. work-hardens). On straining beyond the dP=0, and hence since P= aA. then TS the metal still continues to work-harden, bi ll to compensate for the reduction in ross-sectional area of the test piece. The deforma- defines the instability condition. During deformation, tion then becomes unstable, such that as a localized the specimen volume is essentially constant (i.e. dv= region of the gauge length strains more than the rest, 0)and from it cannot harden sufficiently to raise the stress for fur- ther deformation in this region above that to cause dv =d(IA)= AdI+ldA=0 further strain elsewhere. A neck then forms in the we obtain lis region until fracture. Under these conditions. the da d/ eduction in area(Ao- At)/Ao where Ao and A, are the initial and final areas of the neck gives a mea- Thus, necking at a strain at which the slope sure of the localized strain, and is a better indication of the true stress-true strain curve equals the true198 Modern Physical Metallurgy and Materials Engineering "~ 200 Fracture Z ~- Lower y~etd point i ' --<.. I00- Luders strain C .... 1 t 1 1 1 ..... __ (a) Max tmum stress 200 I00 Totat etonga ion 1 . L- 0 in -1 10 Etongation ~ % (b) 6050 ~'f~ radiated 40 i Irradiated --, S 30 2O ,o ~ ....... '40 0 40 8'o Temperature ~ ~ (c) Figure 7.1 Stress-elongation curves for (a) impure iron, (b) copper, (c) ductile-brittle transition in mild steel (after Churchman, Mogford and Cottrell, 195 7). to the elastic portion of the stress-strain curve from the point of 0.1% strain. For control purposes the tensile test gives valuable information on the tensile strength (TS- maximum load/original area) and ductility (percentage reduction in area or percentage elongation) of the material. When it is used as a research technique, however, the exact shape and fine details of the curve, in addition to the way in which the yield stress and fracture stress vary with temperature, alloying additions and grain size, are probably of greater significance. The increase in stress from the initial yield up to the TS indicates that the specimen hardens during defor￾mation (i.e. work-hardens). On straining beyond the TS the metal still continues to work-harden, but at a rate too small to compensate for the reduction in cross-sectional area of the test piece. The deforma￾tion then becomes unstable, such that as a localized region of the gauge length strains more than the rest, it cannot harden sufficiently to raise the stress for fur￾ther deformation in this region above that to cause further strain elsewhere. A neck then forms in the gauge length, and further deformation is confined to this region until fracture. Under these conditions, the reduction in area (A0- A l)/Ao where A0 and A l are the initial and final areas of the neck gives a mea￾sure of the localized strain, and is a better indication than the strain to fracture measured along the gauge length. True stress-true strain curves are often plotted to show the work hardening and strain behaviour at large strains. The true stress o is the load P divided by the area A of the specimen at that particular stage of strain and the total true strain in deforming from initial length Io to length ll is e---- f/o' (dl/l)= ln(l~/lo). The true stress-strain curves often fit the Ludwig relation a = ke" where n is a work-hardening coefficient ~0.1-0.5 and k the strength coefficient. Plastic instability, or necking, occurs when an increase in strain produces no increase m load supported by the specimen, i.e. dP = 0, and hence since P -- oA, then dP = Ado + odA = 0 defines the instability condition. During deformation, the specimen volume is essentially constant (i.e. dV = 0) and from dV = d(/a) = Adl + ldA -- 0 we obtain do dA dl a -- A -- 1 --dE (7.2) Thus, necking occurs at a strain at which the slope of the true stress-true strain curve equals the true