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Overview
Strain is the result of the application of forces to solid objects. The forces are defined in a special way described by the general term, stress. For those readers needing a review of force principles, Appendix 4 discusses elementary mechanical principles, including force. In this section we will define stress and the resulting strain.
FIGURE 5.12 Tensile and compressional stress can be defined in terms of forces applied to a uniform rod.
Definition: A special case exists for the relation between force applied to a solid object and the resulting deformation of that object. Solids are assemblages of atoms in which the atomic spacing has been adjusted to render the solid in equilibrium with all external forces acting on the object. This spacing determines the physical dimensions of the solid. If the applied forces are changed, the object atoms rearrange themselves again to come into equilibrium with the new set of forces. This rearrangement results in a change in physical dimensions that is referred to as a deformation of the solid.
The study of this phenomenon has evolved into an exact technology. The effect of applied force is referred to as a stress and the resulting deformation as a strain. To facilitate a proper analytical treatment of the subject, stress and strain are carefully denned to emphasize the physical properties of the material being stressed and the specific type of stress applied. We delineate here the three most common types of stress-strain relationships.
Tensile Stress-Strain
In Figure 5.12a, the nature of a tensile force is shown as a force applied to a sample of material so as to elongate or pull apart the sample. In this case, the stress is defined as
where F = applied force in N
A = cross-sectional area of the sample in m2
We see that the units of stress are N/m2 in the SI units (or Ib/in2 in the English units) and they are like a pressure.
The strain in this case is defined as the fractional change in length of the sample:
where Dl = change in length in m (in)
l = original length in m (in)
Strain is thus a unitless quantity.
Compressional Stress-Strain
The only differences between compressional and tensile stress are the direction of the applied force and the polarity of the change in length. Thus, in a compressional stress, the force presses in on the sample, as shown in Figure 5.12b. The compressional stress is defined as in Equation (5.2).
The resulting strain is also defined as the fractional change in length as in Equation (5.3), but the sample will now decrease in length.
Figure 5.13a shows the nature of the shear stress. In this case, the force is applied as a couple (that is, not along the same line), tending to shear off the solid object that separates the force arms. In this case, the stress is again
where F = force in N
A = cross-sectional area of sheared member in m2
The strain in this case is denned as the fractional change in dimension of the sheared member. This is shown in the cross-sectional view of Figure 5.13b.
FIGURE 5.13 Shear stress is defined in terms of a couple that tends to deform a joining member as shown in this figure.
FIGURE 5.14 A typical stress-strain curve showing the linear region, necking and eventual break.
where Dx = deformation in m (as shown in Figure 5.13b)
l = width of a sample in m
Stress-Strain Curve
If a specific sample is exposed to a range of applied stress and the resulting strain is measured, a graph similar to Figure 5.14 resulls. This graph shows that the relationship between stress and strain is linear over some range of stress. If the stress is kept within the linear region, the material is essentially elastic in that if the stress is removed, the deformation is also gone. But if the elastic limit is exceeded, permanent deformation results. The material may begin to "neck" at some location and finally break. Within the linear region, a specific type of material will always follow the same curves despite different physical dimensions. Thus, we can say that the linearity and slope are a constant of the type of material only. In tensile and compressional stress, this constant is called the modulus of elasticity or Young's modulus, as given by
where stress = F/A in N/m2 (or Ib/in2)
strain = Dl/l unitless
E = Modulus of elasticity in N/m2
The modulus of elasticity has units of stress, that is, N/m2. Table 5.1 gives the modulus of elasticity for several materials. In an exactly similar fashion, the shear modulus is defined for shear stress-strain as
where Dx is defined in Figure 5.13b and all other units have been denned in Equation (5.7).
| Material | Modulus (N/m2) |
| Aluminum Copper Steel Polyethylene (plastic) |
6.89 x 1010 11.73 X 10'° 20.70 X 1010 3.45 x 108 |
EXAMPLE 5.5
Find the strain that results from a tensile force of 1000 N applied to a 10-m aluminum beam having a 4 X 10-4 m2 cross-sectional area.
Solution
The modulus of elasticity of aluminum is found from Table 5.1 to be E = 6.89 X 1010 N/m2. Now we have, from Equation (5.7),
so that
Although strain is a unitless quantity, it is common practice to express the strain as the ratio of two length units, for example, as m/m or in/in; also, because the strain is usually a very small number, a micro(m) prefix is often included. In this sense, a strain of 0.001 would be expressed as 1000 min/in, or 1000 mm/m. In the previous example, the solution is stated as 36.3 mm/m. In general, the smallest value of strain encountered in most applications is 1 mm/m. Because strain is a unitless quantity, it is not necessary to do unit conversions. A strain of 153 mm/m also could be written in the form of 153 min/in or even 153 mfurlongs/ furlong. Modern usage often just gives strain in "micros."
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