C H A P T E R 6
(6.1) a profit has a different, more comfortable connotation: to bear fruit, to be useful. The yield strength, when speaking of a material, is the stress beyond which it becomes plastic. The term is well chosen: yield and the plasticity that follows can be profitable—it allows metals to be shaped and it allows structures to tolerate impact and absorb energy. But the unplanned yield of the span of a bridge or of the wing spar of an aircraft or of the forks of your bicycle spells disaster.
This chapter is about yield and plasticity. For that reason it is mainly (but not wholly) about metals: it is the plasticity of iron and steel that made them the structural materials on which the Industrial Revolution was built, enabling the engineering achievements of the likes of Telford1 and Brunel2. The dominance of metals in engineering, even today, derives from their ability to be rolled, forged, drawn and stamped.
6.2 Strength, plastic work and ductility: definition and measurement
Yield properties and ductility are measured using the standard tensile tests introduced in Chapter 4, with the materials taken to failure. Figures 6.1–6.3 show the types of stress–strain behavior observed in different material classes. The yield strength σy (or elastic limit σel)—units: MPa or MN/m2—requires careful definition. For metals, the onset of plasticity is not always distinct so we identify σy with the 0.2% proof stress—that is, the stress at which the stress–strain curve for axial loading deviates by a strain of 0.2% from the linear elastic line as shown in Figure 6.1. It is the same in tension and compression. When strained beyond the yield point, most metals work harden, causing the rising part of the curve, until a maximum, the tensile strength, is reached. This is followed in tension by non-uniform deformation (necking) and fracture.
FIGURE 6.1 Stress–strain curve for a metal.
For polymers, σy is identified as the stress at which the stress–strain curve becomes markedly non-linear: typically, a strain of 1% (Figure 6.2). The behavior beyond yield depends on the temperature relative to the glass temperature Tg. Well below Tg most polymers are brittle. As Tg is approached, plasticity becomes possible until, at about Tg, thermoplastics exhibit cold drawing: large plastic extension at almost constant stress during which the molecules are pulled into alignment with the direction of straining, followed by hardening and fracture when alignment is complete. At still higher temperatures, thermoplastics become viscous and can be moulded; thermosets become rubbery and finally decompose.
FIGURE 6.2 Stress–strain curve for a polymer.
The yield strength σy of a polymer–matrix composite is best defined by a set deviation from linear elastic behavior, typically 0.5%. Composites that contain fibres (and this includes natural composites like wood) are a little weaker (up to 30%) in compression than tension because the fibres buckle on a small scale.
Plastic strain, εpl, is the permanent strain resulting from plasticity; thus it is the total strain εtot minus the recoverable, elastic, part:
The ductility is a measure of how much plastic strain a material can tolerate. It is measured in standard tensile tests by the elongation εf (the tensile strain at break) expressed as a percentage (Figures 6.1 and 6.2). Strictly speaking, εf is not a material property because it depends on the sample dimensions—the values that are listed in handbooks and in the CES
(6.2)
(6.3) software are for a standard test geometry—but it remains useful as an indicator of the ability of a material to be deformed.
In Chapter 4, the area under the elastic part of the stress–strain curve was identified as the elastic energy stored per unit volume ( ). Beyond the elastic limit plastic work is done in deforming a material permanently by yield or crushing.
The increment of plastic work done for a small permanent extension or compression dL under a force F, per unit volume V
= ALo, is
Thus, the plastic work per unit volume at fracture, important in energy-absorbing applications, is
which is just the area under the stress–strain curve.
E x a m p l e 6 . 1
A metal rod with a yield strength σy = 500 MPa and a density ρ = 8000 kg/m3 is uniformly stretched to a strain of 0.1 (10%). How much plastic work is absorbed per unit volume by the rod in this process? Assume that the material is elastic perfectly-plastic, that is, it deforms plastically at a constant stress σy. Compare this plastic work with the elastic energy calculated in Example 4.4.
Answer
Assuming that the material is ‘elastic perfectly-plastic’, that is, the stress equals the yield strength up to failure, the plastic energy per unit volume at failure is:
This plastic energy is 80 times the elastic strain energy of 625 kJ/m3 in Example 4.4, but is still only 0.015%
of the chemical energy stored in the same mass of gasoline.
Ceramics and glasses are brittle at room temperature (Figure 6.3). They do have yield strengths, but these are so enormously high that, in tension, they are never reached: the materials fracture first. Even in compression ceramics and glasses crush before they yield. To measure their yield strengths, special tests that suppress fracture are needed. We will return to the fracture-controlled strength of ceramics in Chapter 8. For now, it is useful to have a practical measure of the strength of ceramics to allow their comparison with other materials. The measure used here is the compressive crushing strength, and since it is not true yield even though it is the end of the elastic part of the stress strain curve, we call it the elastic limit and give it the symbol .
FIGURE 6.3 Stress–strain curve for a ceramic.
Tensile and compression tests are not always convenient: you need a large sample and the test destroys it. The hardness test (Figure 6.4) avoids these problems, although it has problems of its own. In it, a pyramidal diamond or a hardened steel ball is pressed into the surface of the material, leaving a tiny permanent indent, the size of which is measured with a microscope. The indent means that plasticity has occurred, and the resistance to it—a measure of strength—is the load F divided by the area A of the indent projected onto a plane perpendicular to the load:
(6.4)
(6.5)
FIGURE 6.4 The hardness test. The Vickers test uses a diamond pyramid; the Rockwell and Brinell tests use a steel sphere.
The indented region is surrounded by material that has not deformed, and this constrains it so that H is larger than the yield strength σy; in practice it is about 3σy. Strength, as we have seen, is measured in units of MPa, and since H is a strength it would be logical and proper to measure it in MPa too. But things are not always logical and proper, and hardness scales are among those that are not. A commonly used scale, that of Vickers, symbol Hv, uses units of kg/mm2, with the result that
Figure 6.5 shows conversions to other scales.
FIGURE 6.5 Common hardness scales compared with the yield strength.
The hardness test has the advantage of being non-destructive, so strength can be measured without destroying the component, and it requires only a tiny volume of material. But the information it provides is less accurate and less complete than the tensile test, so it is not used to provide critical design data.