Machinery's Handbook, 31st Edition
Basic Theory of Metal Working 1403 (8) where σ = true stress (lb/in 2 ); F = applied load in the test (lb); A = actual area (instanta neous) area resisting the load (in 2 ). True strain in a tensile test can be defined by dividing the total elongation into small increments of actual change in length. Then, using calculus, it can be shown that true strain is defined by the equation (9) where e = true strain (in./in.); and l = instantaneous length at any moment during elonga- tion (in.). If the true stress, based on the actual (instantaneous) cross section area of the specimen, is used, it is found that the stress-strain curve increases continuously up to fracture. If the strain measurement is also based on instantaneous measurements, the curve obtained is known as a true stress-strain curve (Fig. 2). The stress-strain curve in Fig. 2(a) can be represented by the equation (10) where K = the strength coefficient (lb/in 2 ), and n = strain-hardening (work-hardening) exponent. This equation is called the flow curve , and it represents the behavior of metals in the plas tic zone, including their capacity for cold strain hardening. When the curve shown in Fig. 2(a) is plotted on a logarithmic graph as in Fig. 2(b), it is found that the curve is a straight line, and the slope of the line is equal to the exponent n . The value of constant K equals the value of true stress at a true strain value to 1. The strain-hardening exponent may have a value from n = 0 (perfectly plastic solid) to n = 1 (elastic solid). For most metals, n has values between 0.10 and 0.50. A F σ = ln l l 0 ε = c m K n σ ε =
a) Nonlogarithmic b) Logarithmic Fig. 2. True Stress-Strain Curve for Medium Steel
Ductility is most commonly defined as the ability of a metal to plastically deform easily upon application of a tensile force without breaking or fracturing. Ductility may be expressed as either percentage of elongation or percentage of area reduction in the specimen. Elongation can be defined as (11) 100 f 0 # δ = −
l l l 0
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