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The stress-strain data of Table 32.4 are plotted in Fig. 32.17 on cartesian coordinates. The most significant difference between the shape of this stress-strain curve and that of the load-deformation curve in Fig. 32.16 is the fact that the stress continues to rise until fracture occurs and does not reach a maximum value as the loaddeformation curve does. As can be seen in Table 32.4 and Fig. 32.17, the stress at the time of the maximum load is 86 kpsi, and it increases to 119 kpsi at the instant that fracture occurs. A smooth curve can be drawn through the experimental data, but it is not a straight line, and consequently many experimental points are necessary to accurately determine the shape and position of the curve. The stress-strain data obtained from the tensile test of the annealed A40 titanium listed in Tables 32.3 and 32.4 are plotted on logarithmic coordinates in Fig. 32.18.The elastic portion of the stress-strain curve is also a straight line on logarithmic coordinates as it is on cartesian coordinates. When plotted on cartesian coordinates, the slope of the elastic modulus is different for the different materials. However, when

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FIGURE 32.17 Stress-strain curve for annealed A40 titanium. The strain is the natural or logarithmic strain and the data of Tables 32.3 and 32.4 are plotted on cartesian coordinates.

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plotted on logarithmic coordinates, the slope of the elastic modulus is 1 (unity) for all materials it is only the height, or position, of the line that is different for different materials. In other words, the elastic moduli for all the materials are parallel lines making an angle of 45 with the ordinate axis. The experimental points in Fig. 32.18 for strains greater than 0.01 (1 percent plastic deformation) also fall on a straight line having a slope of 0.14. The slope of the stress-strain curve in logarithmic coordinates is called the strain-strengthening exponent because it indicates the increase in strength that results from plastic strain. It is sometimes referred to as the strain-hardening exponent, which is somewhat misleading because the real strain-hardening exponent is the Meyer exponent p, discussed previously under the subject of strain hardening. The strain-strengthening exponent is represented by the symbol m. The equation for the plastic stress-strain line is = 0 m (32.15)

and is known as the strain-strengthening equation because it is directly related to the yield strength. The proportionality constant 0 is called the strength coefficient. The strength coefficient 0 is related to the plastic behavior of a material in exactly

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FIGURE 32.18 Stress-strain curve for annealed A40 titanium plotted on logarithmic coordinates. The data are the same as in Fig. 32.17.

the same manner in which Young s modulus E is related to elastic behavior. Young s modulus E is the value of stress associated with an elastic strain of unity; the strength coefficient 0 is the value of stress associated with a plastic strain of unity. The amount of cold work necessary to give a strain of unity is determined from Eq. (32.14) to be 63.3 percent. For most materials there is an elastic-plastic region between the two straight lines of the fully elastic and fully plastic portions of the stress-strain curve. A material that has no elastic-plastic region may be considered an ideal material because the study and analysis of its tensile properties are simpler. Such a material has a complete stress-strain relationship that can be characterized by two intersecting straight lines, one for the elastic region and one for the plastic region. Such a material would have a stress-strain curve similar to the one labeled I in Fig. 32.19. A few real materials have a stress-strain curve that approximates the ideal curve. However, most engineering materials have a stress-strain curve that resembles curve O in Fig. 32.19. These materials appear to overyield ; that is, they have a higher yield strength than the ideal value, followed by a region of low or no strain strengthening before the fully plastic region begins.Among the materials that have this type of curve are steel, stainless steel, copper, brass alloys, nickel alloys, and cobalt alloys. Only a few materials have a stress-strain curve similar to that labeled U in Fig. 32.19. The characteristic feature of this type of material is that it appears to underyield ; that is, it has a yield strength that is lower than the ideal value. Some of the fully annealed aluminum alloys have this type of curve.

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