9.1 Introduction

9.2 Linear and Nonlinear Elasticity

Figure 9.1 shows the stress–strain curve of a material exhibiting perfectly linear elastic behavior. This is the behavior that is characterized by Hooke’s law (Chapter 3). All solids are linear elastic at small strains—by which we usually mean less than 0.001, or 0.1%. The slope of the stress–strain line, which is the same in compression as in tension, is Young’s modulus, E. The area (shaded) is the elastic energy stored, per unit volume: we can get it all back if we unload the solid, which behaves like a spring.

Figure 9.2 shows a nonlinear elastic solid. Rubbers have a stress–strain curve like this, extending to very large strains (of order 5 or even 8). The material is still elastic: if unloaded much of the energy stored during loading is recovered—that is why catapults can be as lethal as they are.

9.3 Load–Extension Curves for Nonelastic (Plastic) Behavior

Rubbers are exceptional in behaving elastically to high strains; as we said, almost all materials, when strained by more than about 0.001 (0.1%), do something irreversible: and most engineering materials deform plastically to change their shape permanently. If we load a piece of ductile metal (e.g., copper), in tension, we get the following relationship between the load and the extension (Figure 9.3). This can be demonstrated nicely by pulling a piece of plasticine modelling clay (a ductile non-metallic material). Initially, the plasticine deforms elastically, but at a small strain begins to deform plastically, so if the load is removed, the piece of plasticine is permanently longer than it was at the beginning of the test: it has undergone plastic deformation (Figure 9.4).

If you continue to pull, it continues to get longer, at the same time getting thinner because in plastic deformation volume is conserved (matter is just flowing from place to place). Eventually, the plasticine becomes unstable and begins to neck at the maximum load point in the force-extension curve (see Figure 9.3). Necking is an instability which we shall look at in more detail in Chapter 12. The neck grows rapidly, and the load that the specimen can carry through the neck decreases until breakage takes place. The two pieces produced after breakage have a total length that is slightly less than the length just before breakage by the amount of the elastic extension produced by the terminal load.

If we load a material in compression, the force-displacement curve is simply the reverse of that for tension at small strains, but it becomes different at larger strains. As the specimen squashes down, becoming shorter and fatter to conserve volume, the load needed to keep it flowing rises (Figure 9.5). No instability such as necking appears, and the specimen can be squashed almost indefinitely, only being limited by severe cracking in the specimen or plastic flow of the compression plates.

9.4 True Stress–Strain Curves for Plastic Flow

The apparent difference between the curves for tension and compression is due solely to the geometry of testing. If, instead of plotting load, we plot load divided by the actual cross-sectional area of the specimen, A, at any particular elongation or compression, the two curves become much more like one another. In other words, we simply plot true stress (see Chapter 3) as our vertical coordinate (Figure 9.6). This method of plotting allows for the thinning of the material when pulled in tension, or the fattening of the material when compressed.

Figure 9.6

But the two curves still do not exactly match, as Figure 9.6 shows. The reason is that a displacement of (for example) u = L₀/2 in tension and in compression gives different strains; it represents a drawing out of the tensile specimen from L₀ to 1.5L₀, but a squashing down of the compressive specimen from L₀ to 0.5L₀. The material of the compressive specimen has thus undergone much more plastic deformation than the material in the tensile specimen, and can hardly be expected to be in the same state, or to show the same resistance to plastic deformation. The two conditions can be compared properly by taking small strain increments

δɛ=δuL=δLL

about which the state of the material is the same for either tension or compression (Figure 9.7).

Figure 9.7

This is the same as saying that a decrease in length from 100 mm (L₀) to 99 mm (L), or an increase in length from 100 mm (L₀) to 101 mm (L) both represent a 1% change in the state of the material. Actually, they do not give exactly the same state in both cases, but they do in the limit

dɛ=dLL

the two curves exactly mirror one another (Figure 9.8). The quantity ɛ is called the true strain (to be contrasted with the nominal strain u/L₀ defined in Chapter 3) and the matching curves are true stress/true strain (σ/ɛ) curves.

Figure 9.8

Now, a final catch. From our original load-extension or load-compression curves we can easily calculate ɛ, simply by knowing L₀ and taking natural logs. But how do we calculate σ? Because volume is conserved during plastic deformation we can write, at any strain,

A0L0=AL

9.5 Plastic Work

When metals are rolled or forged, or drawn to wire, or when polymers are injection molded or pressed or drawn, energy is absorbed. The work done on a material to change its shape permanently is called the plastic work; its value, per unit volume, is the area of the crosshatched region shown in Figure 9.8; it may easily be found (if the stress–strain curve is known) for any amount of permanent plastic deformation, ɛ′. Plastic work is important in metal and polymer forming operations because it determines the forces that the rolls, or press or molding machine must exert on the material.

9.6 Tensile Testing

The plastic behavior of a material is usually measured by conducting a tensile test. Tensile testing equipment is standard in engineering laboratories. Such equipment produces a load/displacement (F/u) curve for the material, which is then converted to a nominal stress/nominal strain, or σ_n/ɛ_n, curve (see Figure 9.9), where

σn=FA0

Figure 9.9

Naturally, because A₀ and L₀ are constant, the shape of the σ_n/ɛ_n curve is identical to that of the load-extension curve. But the σ_n/ɛ_n plotting method allows us to compare data for specimens having different (though now standardized) A₀ and L₀, and thus to examine the properties of material, unaffected by specimen size. The advantage of keeping the stress in nominal units and not converting to true stress (as shown before) is that the onset of necking can clearly be seen on the σ_n/ɛ_n curve.

Now, we define the quantities usually listed as the results of a tensile test. The easiest way to do this is to show them on the σ_n/ɛ_n curve itself (Figure 9.10). They are:

• ▪ σ_y = yield strength (F/A₀ at onset of plastic flow)
  
• ▪ σ_0.1% = 0.1% proof stress (F/A₀ at a permanent strain of 0.1%) (0.2% proof stress is often quoted instead; proof stress is useful for characterizing yield of a material that yields gradually, and does not show a distinct yield point)
  
• ▪ σ_TS = tensile strength (F/A₀ at onset of necking)
  
• ▪ ɛ_f = (plastic) strain after fracture, or tensile ductility; the broken pieces are put together and measured, and ɛ_f calculated from (L – L₀)/L₀, where L is the length of the assembled pieces

Figure 9.10