MECHANICAL PROPERTIES OF SOLIDS

M ECHANICAL P ROPERTIES OF S OLIDS . INTRODUCTION In Chapter , we studied the rotation of the bodies and then realised that the motion of a body depends on how mass is distributed within the body. We restricted ourselves to simpler situations of rigid bodies. A rigid body generally means a hard solid object having a definite shape and size.

But in reality, bodies can be stretched, compressed and bent. Even the appreciably rigid steel bar can be deformed when a sufficiently large external force is applied on it. This means that solid bodies are not perfectly rigid. A solid has definite shape and size.

In order to change (or deform) the shape or size of a body, a force is required. If you stretch a helical spring by gently pulling its ends, the length of the spring increases slightly. When you leave the ends of the spring, it regains its original size and shape. The property of a body, by virtue of which it tends to regain its original size and shape when the applied force is removed, is known as elasticity and the deformation caused is known as elastic deformation.

However, if you apply force to a lump of putty or mud, they have no gross tendency to regain their previous shape, and they get permanently deformed. Such substances are called plastic and this property is called plasticity. Putty and mud are close to ideal plastics. The elastic behaviour of materials plays an important role in engineering design.

For example, while designing a building, knowledge of elastic properties of materials like steel, concrete etc. is essential. The same is true in the design of bridges, automobiles, ropeways etc. One could also ask — Can we design an aeroplane which is very light but sufficiently strong?

Can we design an artificial limb which is lighter but stronger? Why does a railway track have a particular shape like I ? Why is glass brittle while brass is not? Answers to such questions begin with the study of how relatively simple kinds of loads or forces act to deform different solids bodies.

In this chapter, we shall study the . Introduction . Stress and strain . Hooke’s law .

Stress-strain curve . Elastic moduli . Applications of elastic behaviour of materials Summary Points to ponder Exercises . STRESS AND STRAIN When forces are applied on a body in such a manner that the body is still in static equilibrium, it is deformed to a small or large extent depending upon the nature of the material of the body and the magnitude of the deforming force.

The deformation may not be noticeable visually in many materials but it is there. When a body is subjected to a deforming force, a restoring force is developed in the body. This restoring force is equal in magnitude but opposite in direction to the applied force. The restoring force per unit area is known as stress.

If F is the force applied normal to the cross–section and A is the area of cross section of the body, Magnitude of the stress = F / A ( . ) The SI unit of stress is N m – or pascal (Pa) and its dimensional formula is [ ML – T – ]. There are three ways in which a solid may change its dimensions when an external force acts on it. These are shown in Fig.

(a), a cylinder is stretched by two equal forces applied normal to its cross-sectional area. The restoring force per unit area in this case is called tensile stress . If the cylinder is compressed under the action of applied forces, the restoring force per unit area is known as compressive stress . Tensile or compressive stress can also be termed as longitudinal stress.

In both the cases, there is a change in the length of the cylinder. The change in the length ∆ L to the original length L of the body (cylinder in this case) is known as longitudinal strain . Longitudinal strain ∆ ( . ) However, if two equal and opposite deforming forces are applied parallel to the cross-sectional area of the cylinder, as shown in Fig.

. (b), there is relative displacement between the opposite faces of the cylinder. The restoring force per unit area developed due to the applied tangential force is known as tangential or shearing stress . As a result of applied tangential force, there is a relative displacement ∆ x between opposite faces of the cylinder as shown in the Fig.

. (b). The strain so produced is known as shearing strain and it is defined as the ratio of relative displacement of the faces ∆ x to the length of the cylinder L . Shearing strain ∆ x L = tan θ ( .

) where θ is the angular displacement of the cylinder from the vertical (original position of the cylinder). Usually θ is very small, tan θ is nearly equal to angle θ , (if θ = °, for example, there is only % difference between θ and tan θ ). It can also be visualised, when a book is pressed with the hand and pushed horizontally, as shown in Fig. .

(c). Thus, shearing strain = tan θ ≈ θ ( . ) In Fig. .

(d), a solid sphere placed in the fluid under high pressure is compressed uniformly on all sides. The force applied by the fluid acts in perpendicular direction at each point of the surface and the body is said to be under hydraulic compression. This leads to decrease (a) (b) (c) (d) Fig. .

(a) A cylindrical body under tensile stress elongates by ∆ L (b) Shearing stress on a cylinder deforming it by an angle θ (c) A body subjected to shearing stress (d) A solid body under a stress normal to the surface at every point (hydraulic stress). The volumetric strain is ∆ V/V, but there is no change in shape. elastic behaviour and mechanical properties of solids which would answer many such questions. in its volume without any change of its geometrical shape.

The body develops internal restoring forces that are equal and opposite to the forces applied by the fluid (the body restores its original shape and size when taken out from the fluid). The internal restoring force per unit area in this case is known as hydraulic stress and in magnitude is equal to the hydraulic pressure (applied force per unit area). The strain produced by a hydraulic pressure is called volume strain and is defined as the ratio of change in volume ( ∆ V ) to the original volume ( V ). Volume strain ∆ V V ( .

) Since the strain is a ratio of change in dimension to the original dimension, it has no units or dimensional formula. . HOOKE’S LAW Stress and strain take different forms in the situations depicted in the Fig. ( .

). For small deformations the stress and strain are proportional to each other. This is known as Hooke’s law. Thus, stress ∝ strain stress = k × strain ( .

) where k is the proportionality constant and is known as modulus of elasticity. Hooke’s law is an empirical law and is found to be valid for most materials. However, there are some materials which do not exhibit this linear relationship. .

STRESS-STRAIN CURVE The relation between the stress and the strain for a given material under tensile stress can be found experimentally. In a standard test of tensile properties, a test cylinder or a wire is stretched by an applied force. The fractional change in length (the strain) and the applied force needed to cause the strain are recorded. The applied force is gradually increased in steps and the change in length is noted.

A graph is plotted between the stress (which is equal in magnitude to the applied force per unit area) and the strain produced. A typical graph for a metal is shown in Fig. . .

Analogous graphs for compression and shear stress may also be obtained. The stress-strain curves vary from material to material. These curves help us to understand how a given material deforms with increasing loads. From the graph, we can see that in the region between O to A, the curve is linear.

In this region, Hooke’s law is obeyed. The body regains its original dimensions when the applied force is removed. In this region, the solid behaves as an elastic body. In the region from A to B, stress and strain are not proportional.

Nevertheless, the body still returns to its original dimension when the load is removed. The point B in the curve is known as yield point (also known as elastic limit ) and the corresponding stress is known as yield strength ( σ y ) of the material. If the load is increased further, the stress developed exceeds the yield strength and strain increases rapidly even for a small change in the stress. The portion of the curve between B and D shows this.

When the load is removed, say at some point C between B and D, the body does not regain its original dimension. In this case, even when the stress is zero, the strain is not zero. The material is said to have a permanent set . The deformation is said to be plastic deformation .

The point D on the graph is the ultimate tensile strength ( σ u ) of the material. Beyond this point, additional strain is produced even by a reduced applied force and fracture occurs at point E. If the ultimate strength and fracture points D and E are close, the material is said to be brittle . If they are far apart, the material is said to be ductile .

Fig. . A typical stress-strain curve for a metal. As stated earlier, the stress-strain behaviour varies from material to material.

For example, rubber can be pulled to several times its original length and still returns to its original shape. Fig. . shows stress-strain curve for the elastic tissue of aorta, present in the heart.

Note that although elastic region is very large, the material does not obey Hooke’s law over most of the region. Secondly, there is no well defined plastic region. Substances like tissue of aorta, rubber etc. which can be stretched to cause large strains are called elastomers .

. ELASTIC MODULI The proportional region within the elastic limit of the stress-strain curve (region OA in Fig. . ) is of great importance for structural and manufacturing engineering designs.

The ratio of stress and strain, called modulus of elasticity , is found to be a characteristic of the material. . . Young’s Modulus Experimental observation show that for a given material, the magnitude of the strain produced is same whether the stress is tensile or compressive.

The ratio of tensile (or compressive) stress ( σ ) to the longitudinal strain ( ε ) is defined as Young’s modulus and is denoted by the symbol Y . Y = σ ε ( . ) From Eqs. ( .

) and ( . ), we have Y = ( F/A )/( ∆ L/L ) = ( F × L ) /( A × ∆ L) ( . ) Since strain is a dimensionless quantity, the unit of Young’s modulus is the same as that of stress i.e. , N m – or Pascal (Pa).

Table . gives the values of Young’s moduli and yield strengths of some material. From the data given in Table . , it is noticed that for metals Young’s moduli are large.

Fig. . Stress-strain curve for the elastic tissue of Aorta, the large tube (vessel) carrying blood from the heart. Table .

Young’s moduli and yield strenghs of some material # Substance tested under compression u u Therefore, these materials require a large force to produce small change in length. To increase the length of a thin steel wire of . cm cross- sectional area by . %, a force of N is required.

The force required to produce the same strain in aluminium, brass and copper wires having the same cross-sectional area are N, N and N respectively. It means that steel is more elastic than copper, brass and aluminium. It is for this reason that steel is preferred in heavy-duty machines and in structural designs. Wood, bone, concrete and glass have rather small Young’s moduli.

Example . A structural steel rod has a radius of mm and a length of . m. A kN force stretches it along its length.

Calculate (a) stress, (b) elongation, and (c) strain on the rod. Young’s modulus, of structural steel is . × N m - . Answer We assume that the rod is held by a clamp at one end, and the force F is applied at the other end, parallel to the length of the rod.

Then the stress on the rod is given by Stress F A F r π