ABSTRACT

Hooke (1635-1703) was the first to observe a linear relationship between the extension of a wire due to a force applied along its axis and the proportionate scaling with the length. Mathematically, from this simple relationship it is possible to define the two parameters stress, a, and strain, s, given by

(2.1)

In practice, equations (2.1) apply only to thin wires in tension under relatively light loads. It is also necessary to have a more rigorous definition of the quantities being measured. In the first of the above equations, the true stress a,,. should be considered as the ratio of tensile force in the thin wire to that of the instantaneous cross section of the wire. Inherent to this equation is the assumption that the stress is uniformly distributed across the wire (a reasonable assumption in this instance). Under these circumstances the true stress is given by

Applied force F a = = -

'"'• Instantaneous area A, (2.2)

Unfortunately, even when measuring something as simple as a wire, it is often a lot more convenient to measure the dimensions of the wire before testing and use the original area for the calculation of stress. Being a more

(2.4)

Equations (2.1) to (2.4) above impose no limits on the possible values of stress or strain. In reality, all materials have limits, the values of which define the strength of a given structure. To evaluate these limits, it is common to apply a tensile or compressive load of known value to a specimen of uniform cross section and known length and measure subsequent extension. From such measurements it is relatively straightforward to compute the engineering stress/ strain curves. In general it is found that there are two quite distinct types of behavior. For almost all metals there is an initial linear-elastic (note that elastic does not necessarily imply linear) region in which the specimen will return to its original length if the load is released. This is followed by a rapid extension with increasing load as the stress is increased beyond a certain value. In this latter stage of deformation the material has undergone an irreversible (or plastic) flow. The ability to plastically deform without fracture is called ductility and is characteristic of many metallic solids. Other materials that might be used for flexure applications, such as many ceramics, exhibit brittle failure due to the growth of cracks in the plane of the maximum tensile stress. Because of these differences, the responses to simple uni-axial loads of metals and non-metals are discussed separately below.