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# Stress-strain graphs

**If you plot a stress against strain of a material with the (linear) elastic behaviour, you get a straight line.**

i.e. stress is proportional to strain. The gradient of the above straight line is the Young's modulus, E

and

E is constant and does not change for a given material, no matter what the size of the sample we test. It can be considered as a property of the material. The value of E reflects the stiffness of the material. **Stiffer materials have higher values of E.** Young's modulus values of different materials are often listed in the form of a table in reference books so scientists and engineers can look them up.

**Units of the Young modulus E:** Nm^{-2} or Pa.

* Note:* The value of E in Pa can turn out to be a very large number. It is for this reason that, some times the value of E may be given M

**Nm**.

^{-2}* Note:* Because 'stress' and 'strain' are (uniquely) related to force and extension, it is not surprising that the two graphs, stress v/s strain and force v extension, have similar shapes and characteristics.

We can experimentally determine the value of E by choosing a specimen of the material in a convenient shape and form. For example, it is easier to deal with a specimen in the form of a long, thin wire for determining the value of Young's modulus of a metal. In principle we can apply different forces to a wire by hanging different weights on it and measure the extension of the wire for the magnitudes of the force applied to draw a stress strain graph. We have already noted that strain is a small number so it needs to be measured more accurately. **We can do this by using :**

**Searle's apparatus**

**This is a schematic diagram of the apparatus.**

We actually use two wires of equal lengths attached to a rigid support. Although the support is rigid it to can 'give' slightly under the forces applied. This can affect results. By using two wires, spurious strain can be eliminated from the measurements. One wire acts as a control wire. We can accurately measure extension of the other (test) wire. Both control and test wires are attached to the other ends by a horizontal bar supporting a spirit level. The bar is hinged to the control wire so that when the test wire is extended due to the addition of weights on the side of the test wire, the spirit level is tilted by a small amount. We can remove any tilt of the spirit level and restore it to the horizontal position by turning the screw of a micrometer, which is positioned on the test wire side and making the bar mounted spirit level travel in the desired direction.

* Caution:* It is possible that a wire under tension can snap suddenly and damage eyes. Wear safety glasses. It is also possible that weights attached to the wires could fall down and land on your feet or other part of the body.

**Experimental determination consists of the following steps:**

* Step 1:* Attach equal weights both wires to make them equally taut.

* Step 2: *Measure the initial length of the wire several times to obtain the average value of

*lo*

* Step 3:* Measure the diameter of the wire at several points along the wire and the average value of the diameter (d) and then calculate the circular cross-sectional area

**From the formula:**

A = (πd^{2})

* Step 4:*Adjust the spirit level so that it is in the horizontal position by turning the micrometer. Record the micrometer reading to use it as the reference reading.

* Step 5:*Load the test wire with a further weight. Wait while the wire is being stretched to the equilibrium position and the spirit level is maximally tilted.

* Step 6:* Adjust the micrometer screw to restore the spirit level into the horizontal position.

* Step 7:* Subtract the first micrometer reading from the second micrometer reading to obtain the extension (e) of the test wire.

* Step 8:* Calculate stress and strain from the formulae
and

* Step 9:* Repeat steps 4,5,6 to obtain more values of stresses and strains

* Step 10:* Plot the above values on stress strain graph; it should be a straight line. Determine the value of the gradient E.

**Worked example**

**A wire of length 2m and diameter 0.4mm is hung from the ceiling. Find the extension caused in the wire when a weight of 100N is hung on it. Young Modulus (E) for the wire is 2.0 x 10 ^{11} Pa.**

**Answer:****e ~ 8 mm.**

In this 'Learn-it' so far, we have drawn stress-strain graphs for the elastic behaviour of a material. In the elastic region the stress-strain graph is a straight line. We can, however draw a stress strain graph beyond the elastic region. The graph, then becomes non-linear because Hooke's law is not obeyed and stress is not proportional to strain.

**Here are schematic stress-strain graphs of copper and glass.**

* Note:* that both graphs end at points marked X. These points are called

**breaking points**. A material physically breaks at its breaking point. The stress at the breaking point is called the

**breaking stress**of the material. Breaking stress of a material, in principle, is related to the energy required to break internal bonds between the atoms or the molecules of the material. It is very important for designers and engineers to know the value of the breaking stress for the materials they use.

**This diagram schematically shows the stress strain curve of rubber. It is different from the other two stress-strain graphs, above, in the following respects:**

- Within the range of the stress and strain of the graph, rubber undergoes high strains (extension) without breaking. For example, one kind of rubber (polyisoprene) can be stretched ~500% without breaking
- Although, rubber on loading returns to its original length (zero extension), the stress-strain graph has two branches (generated by loading up stress and unloading stress). The loop formed by the two branches is called
**hysteresis loop.**It actually represents the fact that rubber is not a very good material for storing energy. In one loading and unloading cycle the strain energy, represented by the area bound by the hysteresis loop is lost and eventually dissipated as heat.