Stress and Strain in Plastics Machinery's Handbook, 31st Edition
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For metals, Young’s modulus is expressed in terms of 10 6 lb f /in nient (see Units of Pressure and Stress starting on page 2861.) Secant modulus is the ratio of stress to corresponding strain at any point on the stress- strain curve (see Modulus of Elasticity E ). 2 , N/m 2 , or Pa, as conve Proportional limit is the greatest stress at which a material is capable of sustaining the applied load without losing the proportionality of stress to strain. This limit is the point on the stress-strain curve where the slope begins to change, as shown at A on each of the curves in Fig. 5. Proportional limit is expressed in lb f /in 2 (MPa or GPa). Yield point is the first point on the stress-strain curve where an increase in strain occurs without an increase in stress, indicated by B on some of the curves in Fig. 5. The slope of the curve is zero at this point; however, some materials do not have a yield point. Ultimate strength is the maximum stress a material withstands when subjected to a load, and is indicated by C in Fig. 5. Ultimate strength is expressed in lb f /in 2 (MPa or GPa). Elastic limit is indicated by D on the stress-strain curve in Fig. 5 and is the level beyond which a material is permanently deformed when the load is removed. Although many materials can be loaded beyond their proportional limit and still return to zero strain when the load is removed, some plastics have no proportional limit in that no region exists where the stress is proportional to strain (i.e., where the material obeys Hooke’s law). Yield strength is the stress at which a material shows a specified deviation from stress to strain proportionality. Some materials do not show a yield strength clearly, and it may be desirable to choose an arbitrary stress level beyond the elastic limit, especially with plastics that have a very high strain at the yield point, to establish a realistic yield strength. Such a point is seen at F on some of the curves in Fig. 5 and is defined by constructing a line parallel to OA at a specified offset strain H . The stress at the intersection of the line with the stress-strain curve at F would be the yield strength at H offset. If H were at 2 percent strain, F would be described as the yield strength at a 2 percent strain offset. Poisson’s Ratio n is the ratio of the lateral contraction to longitudinal elongation. Under a tensile load, a rectangular bar of length L with sides of widths b and d lengthens by an amount D L , producing a longitudinal strain of (5) The bar is reduced in its lateral dimensions and the associated lateral strains will be opposite in sign, resulting in (6) If the deformation is within the elastic range, Poisson’s ratio of the lateral to the longitu dinal strains will be constant. The formula is: (7) When a bar like that in Fig. 4 is stretched in tension, mass must be conserved; if there is no change in diameter, the average density diminishes inversely as the strain increases. If the area decreases in percentage as much as length increases in percentage, the stretched volume would equal the original volume, and density would be preserved. In fact, most real materials do contract laterally when stretched, but not enough to preserve constant density; therefore, density usually does diminish with elongation. Values of v for most engineering materials lie between 0.20 and 0.40, and these values hold for unfilled rigid thermoplastics. Values of n for filled or reinforced rigid thermoplas tics fall between 0.10 and 0.40 and for structural foam between 0.30 and 0.40. Rigid ther moset plastics have Poisson’s ratios between 0.20 and 0.40, whether filled or unfilled, and elastomers can approach 0.5. Table 1 lists values of n determined at 73°F (23 ° C) for some plastics by measuring of lateral contraction and longitudinal elongation in tensile tests. L L ε = ∆ b d d ε =− ∆ =− ∆ b v Longitudinal Strain Lateral Strain = ∆ = d ⁄ d L ⁄ L ∆
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