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Elastic Properties
DOI 10.1615/hedhme.a.000522
5.4 PROPERTIES OF SOLIDS
5.4.5 Elastic properties
S.F. Pugh
A. Introduction
Solids deform when subjected to stresses. The strain is defined as the degree of distortion per unit length, e.g., the change in length per unit length of a wire when subjected to a tensile stress. The stress is defined as the applied force per unit area. A material is said to show perfectly elastic behavior when the stress-strain relationship is perfectly reversible. Furthermore if the strain is proportional to the applied stress then the behavior is known as linear elastic. Most engineering materials show approximately linear clastic behavior up to the onset of plastic deformation. Such behavior was originally described by Hooke and is therefore sometimes known as Hookean.
An important part of mechanical engineering design involves the computation of the distribution of strains in a structure when subject to the stresses imposed in service. Stresses may be imposed by fluid pressure or by fluid motion and also by nonuniform thermal expansion during changes in temperature. Elastic properties are traditionally regarded as being non-structure sensitive but there is one important aspect in which this is not true. The individual grains or crystals of metals are elastically anisotropic. Thus the elastic constants are a function of the orientation of the grain with respect to the orientation of the imposed stresses. The process of manufacture of components tends to introduce a certain degree of preferred orientation of the individual grains composing the structure and thus to introduce elastic anisotropy. It is probable that the existence of various degrees of preferred orientation in test specimens has led to the rather wide scatter in data for the clastic properties of metals and alloys. Because this scatter can introduce errors of as much as 20% in some cases in computing strains, the subject is dealt with in depth in this section. Table 531.3 should be regarded only as an example of the type of information in the literature. There is no reason to suppose, for example, that steels with 5-9% chromium should differ significantly in Young’s modulus from those containing slightly smaller or larger amounts of chromium as shown in that table.
Design codes require that stresses are less than the yield stress in a range in which structural materials are assumed to show linear elastic behavior. The behavior of real materials, however, is only approximately elastic so that on loading and unloading below the yield stress a narrow hysteresis loop is generated. For this reason materials have a nonzero damping capacity. The extent of the departure from elastic behavior becomes greater as the stress increases. Increases in the duration of loading and rise in temperature usually cause increased departure from elastic behavior. For many design purposes, perfect linear elastic behavior is assumed. The finer points of stress-strain behavior are included in this section since they could become very important. For example, the damping capacity of a heat exchanger tube might increase by an order of magnitude when the tube is pressurized. Similarly the elastic constants and damping capacity show significant changes when the temperature is increased in service, causing discrepancies between the behavior during testing cold and unpressurized and in service.
The present section therefore not only describes and defines the various ideal linear elastic moduli that are used in elementary engineering design but also indicates the major sources of error that can arise when the elastic behavior of real materials is not understood. The application of elastic properties in design and analytical procedures is discussed in Part 4 and includes calculation of stress-strain distributions in complex structures, linear elastic fracture mechanics, and vibration studies.
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