12 the STRENGTH OF MATERIALS
12.4 THE ENERGY STORED IN STRETCHED MATERIALS In section 12.2 we saw that elastic strain energy is stored in extended (or compressed) springs. Stretched materials, such as a guitar string or a rubber band, also store elastic strain energy. The energy stored, E, is equal to the work done in stretching the material and (provided that the material follows Hooke’s law) is given by 1 E=2 F Δl
From this it can be shown that 1 × stress × strain E per unit volume = 2
In most circumstances, we can assume that all the work done in stretching the material is stored as elastic strain energy, and is then available to do work. But in some cases energy is transferred during the stretching as internal energy, raising the temperature of the material. One example of this is rubber. To demonstrate this to yourself, take a rubber band, stretch it and then let it return to its original size. Quickly repeat this a number of times and then hold it to your lips. You should notice how warm it is. More work is done in stretching a rubber cord than is released when the cord is unloaded. This type of behaviour is known as hysteresis (see Figure 20). Each time the rubber is stretched and released, some energy is transferred as internal energy in the rubber.
1 =2 × Young modulus × strain2
Energy per unit volume is measured in the unit of joule per metre cubed (J m–3). (For the derivation of this, see Assignment 1, which follows this section.)
Tensile stress / GPa
This expression is only valid for materials that follow Hooke’s law. However, for all materials, the work done in extending the material (which is equal to the energy stored per unit volume of the material) can be found by calculating the area below the stress–strain curve (Figure 19). If the material is tested to destruction, the area below the graph is a measure of the material’s toughness – that is, how much energy it can absorb before breaking. x breaking point
3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0
Kevlar
breaking point spider silk x toughness
0
0.1
0.2 0.3 Strain
0.4
Figure 19 The area below the curve is equal to the energy stored per unit volume, a measure of toughness.
Area A represents the energy transferred to internal energy in the rubber band, in each loading and unloading cycle. 50 Force / N
40
loading
30 20
area A unloading
10 0
0
0.01
0.02 0.03 0.04 Extension / m
0.05
Figure 20 Force–extension graph for loading and unloading a rubber band, showing hysteresis. The same force can cause two different extensions, depending on the history of the sample.
Most types of rubber obey Hooke’s law only over a limited range of extensions. Rubber tends to stretch easily at first and become much stiffer at high extensions, so the expression for strain energy ( 21 × Young modulus × strain2) will only be approximately correct over part of the cord’s extension. However, we can find the work done in stretching the cord by calculating the area below the force–extension graph (Figure 20), plotted as the rubber is loaded. (Alternatively, we can find the work done per unit volume by calculating the area under the stress–strain graph.) The elastic strain energy recovered is the area below the curve as the rubber is unloaded. The area between the two curves is the energy transferred to the rubber as internal energy. The larger the area of this ‘hysteresis loop’, the greater the rise in internal energy of the rubber, and the hotter the rubber will get. Rubber is said to be resilient if it has a hysteresis loop with a small area.
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