Machinery's Handbook, 31st Edition
Stress and Strain in Plastics 569 For isotropic elastic materials it has been shown that bulk modulus is linked to Young’s modulus and Poisson’s ratio by the equation (14) Thus, all three moduli, E , G , and B, are linked through Poisson’s ratio. True stress and true strain are terms not in frequent use. In Fig. 5 the stress, sometimes called the engineering stress , is calculated from an increasing load F , acting over a constant area A . Because the cross-sectional area is reduced with most materials, use of that smaller cross-sectional area in the calculation yields what is called the true stress . In addition, the direct strain referred to earlier, that is, the total change in length divided by the original length, is often called the engineering strain . The true strain would be the instantaneous deformation divided by the instantaneous length. Therefore, the shape of such a stress- strain curve would not be the same as a simple stress-strain curve. Modulus values and stress-strain curves are almost universally based on engineering stress and strain. Other Measures of Strength and Modulus.— Tensile and compression properties of many engineering materials, which are treated as linearly elastic, homogeneous, and iso tropic, are often considered to be identical so as to eliminate the need to measure prop- erties in compression. Also, if tension and compression properties are identical, under standard beam bending theory, there is no need to measure the properties in bending. In a concession to the nonlinear, anisotropic nature of most plastics, these properties, par- ticularly flexural properties, are often reported on manufacturers’ marketing data sheets. Compression Strength and Modulus: Because of the relative simplicity of testing in ten sion, the elastic modulus of a material is usually measured and reported as a tension value. For design purposes it often is necessary to know the stress-strain relationship for com pression loading. With most elastic materials at low stress levels, the tensile and compres sive stress-strain curves are nearly equivalent. At higher stress levels, the compressive strain is less than the tensile strain. Unlike tensile loading, which usually results in a clear-cut failure, stressing in compression produces a slow and indefinite yielding that seldom leads to failure. Because of this phenomenon, compressive strength is customarily expressed as the amount of stress in lb f /in 2 (or Pa) required to deform a standard plastics test specimen to a certain strain. Compression modulus is not always reported because defining a stress at a given strain is equivalent to reporting a secant modulus. If a compres sion modulus is given, it is usually an initial modulus. Bending Strength and Modulus: When a beam of rectangular cross section is bent under a vertical load midway between the beam supports, the bottom surface is stretched in tension while the top surface is compressed. Extending from end to end along the vertical center of the beam is a plane of zero stress called the neutral axis . Whatever the shape of the cross section— constant along the beam—the neutral axis is located at the center of gravity (centroid) of that cross-sectional area. The theory of bending for simple beams makes the following assump- tions: the beam is initially straight, unstressed, and symmetric; the material is linearly elastic, homogeneous, and isotropic; the proportional limit is not exceeded; Young’s modulus for the material is the same in tension and compression; and all deflections are small so that cross sections remain planar during bending. For a beam of length L with a rectangular section of width b and depth h carrying a central load F, the formula for the maximum bending stress s at the bottom surface is (15) and the formula for the maximum deflection at the center of the beam is (16) Note that, if the deflection equation is solved for E , Young’s modulus, the modulus can be estimated by measuring the deflection of a beam under a known central force: bh FL 2 3 2 σ= y Ebh FL 4 3 3 = B E 3 1 2 ν = – ^ h
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