Force exerted on a body can cause a change in either the shape or the motion of the body. The unit of force is the newton, N. No solid body is perfectly rigid and when forces are applied to it, changes in dimensions occur. Such changes are not always perceptible to the human eye since they are so small. For example, the span of a bridge will sag under the weight of a vehicle and a spanner will bend slightly when tightening a nut. It is important for engineers and designers to appreciate the effects of forces on materials, together with their mechanical properties.
The three main types of mechanical force that can act on a body are:
(i) tensile,
(ii) compressive, and
(iii) shear 1.2. Tensile Force -
Tension is a force that tends to stretch a material, as shown in Figure 1.1. For example,
(i) the rope or cable of a crane carrying a load is in tension
(ii) rubber bands, when stretched, are in tension
(iii) when a nut is tightened, a bolt is under tension
A tensile force, i.e. one producing tension, increases the length of the material on which it acts.
1.3. Compressive Force -
When a body is subjected to two equal and opposite axial pushes F (also called compressive load) as shown in Figure. 1.2
(i) a pillar supporting a bridge is in compression
(ii) the sole of a shoe is in compression
(iii) the jib of a crane is in compression
A compressive force, i.e. one producing compression, will decrease the length of the material on which it acts.
1.4. Shear Force -
Shear is a force that tends to slide one face of the material over an adjacent face. For example,
(i) a rivet holding two plates together is in shear if a tensile force is applied between the plates—as shown in Figure 1.3
(ii) a guillotine cutting sheet metal, or garden shears, each provide a shear force
(iii) a horizontal beam is subject to shear force
(iv) transmission joints on cars are subject to shear forces A shear force can cause a material to bend, slide or twist.
1.5. Stress -
Forces acting on a material cause a change in dimensions and the material is said to be in a state of stress. Stress is the ratio of the applied force F to cross-sectional area A of the material. The symbol used for tensile and compressive stress is σ (Greek letter sigma). The unit of stress is the Pascal, Pa, where 1 Pa = 1 N/m2. Hence
1.6. Strain -
The fractional change in a dimension of a material produced by a force is called the strain. For a tensile or compressive force, strain is the ratio of the change of length to the original length. The symbol used for strain is ε (Greek epsilon). For a material of length L metres which changes in length by an amount x metres when subjected to stress,
percentage strain = ε × 100
For a shear force, strain is denoted by the symbol γ (Greek letter gamma) and, with reference to
Figure 1.5, is given by:
The fractional change in a dimension of a material produced by a force is called the strain. For a tensile or compressive force, strain is the ratio of the change of length to the original length. The symbol used for strain is ε (Greek epsilon). For a material of length L metres which changes in length by an amount x metres when subjected to stress,
Strain is dimension-less and is often expressed as a percentage, i.e.
percentage strain = ε × 100
For a shear force, strain is denoted by the symbol γ (Greek letter gamma) and, with reference to
Figure 1.5, is given by:
1.7. Elasticity, limit of proportionality and elastic limit -
Elasticity is the ability of a material to return to its original shape and size on the removal of external forces.
Plasticity is the property of a material of being permanently deformed by a force without breaking. Thus if a material does not return to the original shape, it is said to be plastic. Within certain load limits, mild steel, copper, polythene and rubber are examples of elastic materials; lead and plasticine are examples of plastic materials.
If a tensile force applied to a uniform bar of mild steel is gradually increased and the corresponding extension of the bar is measured, then provided the applied force is not too large, a graph depicting these results is likely to be as shown in Figure 1.7. Since the graph is a straight line, extension is directly proportional to the applied force.
plastic and no longer returns to its original length when the force is removed. The material is then said to have passed its elastic limit and the resulting graph of force/extension is no longer a straight line. Stress, σ = F/A, from Section 1.5, and since, for a particular bar, area A can be considered as a constant,
then F ∝ σ.
Strain ε = x/L, from Section 1.6, and since for a particular bar L is constant, then x ∝ ε. Hence for stress applied to a material below the limit of proportionality a graph of stress strain will be as shown in Figure 1.8, and is a similar shape to the force/extension graph of Figure 1.7.
1.8. Hooke’s law -
Hooke’s law states:
Within the limit of proportionality, the extension of a
material is proportional to the applied force
It follows, from Section 1.7, that:
Within the limit of proportionality of a material, the strain produced is directly proportional to the stress producing it
Young’s modulus of elasticity
Within the limit of proportionality, stress α strain, hence
stress = (a constant) × strain
This constant of proportionality is called Young’s modulus of elasticity and is given the symbol E. The value of E may be determined from the gradient of the straight line portion of the stress/strain graph. The dimensions of E are pascals (the same as for stress, since strain is dimension-less).
elasticity, E, include:
Aluminium alloy 70 GPa , brass 90 GPa, copper 96 GPa,titanium alloy 110 GPa, diamond 1200 GPa, mild steel 210 GPa, lead 18 GPa, tungsten 410 GPa, cast iron 110 GPa, zinc 85 GPa, glass fibre 72 GPa, carbon fibre 300 GPa.
Stiffness
A material having a large value of Young’s modulus is said to have a high value of material stiffness, where stiffness is defined as:
For example, mild steel is a much stiffer material than lead.
Since L and A for a particular specimen are constant, the greater Young’s modulus the greater the material stiffness.
Hooke’s law states that extension x is proportional to force F, provided that the limit of proportionality is not exceeded, i.e. x ∝ F or x = kF where k is a constant.
1.9 Ductility, brittleness and malleability -
Ductility is the ability of a material to be plastically deformed by elongation, without fracture. This is a property that enables a material to be drawn out into wires. For ductile materials such as mild steel, copper and gold, large extensions can result before fracture occurs with increasing tensile force. Ductile materials usually have a percentage elongation value of about 15% or more. Brittleness is the property of a material manifested by fracture without appreciable prior plastic deformation. Brittleness is a lack of ductility, and brittle materials such as cast iron, glass, concrete, brick and ceramics, have virtually no plastic stage, the elastic stage being followed by immediate fracture. Little or no‘waist’ occurs before fracture in a brittle material undergoing a tensile test. Malleability is the property of a material whereby it can be shaped when cold by hammering or rolling. A malleable material is capable of undergoing plastic deformation without fracture. Examples of malleable materials include lead, gold, putty and mild steel.
(b) A typical load/extension curve for a brittle material is shown in Figure 1.10(b), and an example of such a material is cast iron.
(c) A typical load/extension curve for a ductile material is shown in Figure 1.10(c), and an example of such a material is mild steel.
Experiments have shown that under pure torsion (see Chapter 10), up to the limit of proportionality, we have Hooke’s law in shear, where
1.10. Thermal strain -
If a bar of length L and coefficient of linear expansion α were subjected to a temperature rise of T , its length will increase by a distance αLT, as described in Chapter 20. Thus the new length of the bar will be:
L + αLT = L(1 + αT )
Now, as the original length of the bar was L, then the thermal strain due to a temperature rise will be:
