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SOLID MATERIALS 32.38
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32.11 STRENGTH, STRESS, AND STRAIN RELATIONS
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The following relationships between strength, stress, and strain are very helpful to a complete understanding of tensile properties and also to an understanding of their use in specifying the optimum material for a structural part. These relationships also help in solving manufacturing problems where difficulty is encountered in the fabrication of a given part because they enable one to have a better concept of what can be expected of a material during a manufacturing process. A further advantage of these relations is that they enable an engineer to more readily determine the mechanical properties of a fabricated part on the basis of the original properties of the material and the mechanisms involved with the particular process used.
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32.11.1 Natural and Nominal Strain The relationship between these two strains is determined from their definitions. The expression for the natural strain is = ln ( f / 0). The expression for the nominal strain can be rewritten as f / 0 = n + 1. When the latter is substituted into the former, the relationship between the two strains can be expressed in the two forms = ln (n + 1) exp ( ) = n + 1 (32.19)
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32.11.2 True and Nominal Stress The definition of true stress is = L/Ai . From constancy of volume it is found that Ai = A0( 0 / i), so that = which is the same as = S(n + 1) S exp ( ) (32.20) L A0
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32.11.3 Strain-Strengthening Exponent and Maximum-Load Strain One of the more useful of the strength-stress-strain relationships is the one between the strain-strengthening exponent and the strain at maximum load. It is also the simplest, since the two are numerically equal; that is, m = u. This relation is derived on the basis of the load-deformation curve shown in Fig. 32.20. The load at any point along this curve is equal to the product of the true stress on the specimen and the corresponding area. Thus L = A
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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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SOLID MATERIALS 32.39
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FIGURE 32.20 A typical load-deformation curve showing unloading and reloading cycles.
Now, since = 0 m and = ln A0 A or A= A0 exp ( )
the load-strain relationship can be written as L = 0A0 mexp ( ) The load-deformation curve shown in Fig. 32.20 has a maximum, or zero-slope, point on it. Differentiating the last equation and equating the result to zero gives the simple expression = m. Since this is the strain at the ultimate load, the expression can be written as u = m (32.21)
32.11.4 Yield Strength and Percent Cold Work The stress-strain characteristics of a material obtained from a tensile test are shown in Fig. 32.18. In the region of plastic deformation, the relationship between stress and strain for most materials can be approximated by the equation = 0 m. When a load is applied to a tensile specimen that causes a given amount of cold work W (which is a plastic strain of W), the stress on the specimen at the time is W and is defined as
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
SOLID MATERIALS 32.40
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W = 0( W)m
(32.22)
Of course, W is also equal to the applied load LW divided by the actual crosssectional area of the specimen AW. If the preceding tensile specimen were immediately unloaded after reading LW, the cross-sectional area would increase to AW from AW because of the elastic recov ery or springback that occurs when the load is removed. This elastic recovery is insignificant for engineering calculations with regard to the strength or stresses on a part. If the tensile specimen that has been stretched to a cross-sectional area of AW is now reloaded, it will deform elastically until the load LW is approached. As the load is increased above LW, the specimen will again deform plastically. This unloadingreloading cycle is shown graphically in Fig. 32.20. The yield load for this previously cold-worked specimen before the reloading is AW . Therefore, the yield strength of the previously cold-worked (stretched) specimen is approximately (Sy)W = But since AW = AW, then (Sy)W = LW AW LW AW
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