Mechanical Properties of Solids: Class 11 Physics · Chapter 8

8.1 Introduction

In Chapter 6 we treated a wheel or a rod as a rigid body, whose shape never changes however hard we push. That was a useful model, not the truth. Stand on a steel bench and it sags by a fraction of a millimetre; pluck a sitar string and it lengthens; press your thumb into an eraser and it dents. Every real solid changes its size or shape when forces act on it, even when the change is far too small to see.

What happens when the forces are taken away? Here solids fall into two broad families. A steel spring, a rubber band or a diving board springs back to its original size and shape: it shows elasticity (प्रत्यास्थता), and the deformation it suffered was elastic. A ball of wet clay keeps whatever shape you squeeze it into. Such materials show plasticity (सुघट्यता); kneaded atta dough and modelling clay come close to being perfectly plastic (Fig. 8.1).

Engineers depend on this distinction. Before the steel for the Chenab rail bridge in Jammu and Kashmir, whose deck stands 359 m above the riverbed, could be ordered, its designers had to know how much each grade of steel stretches, bends and twists under load, and at what load it would stop springing back. The same knowledge decides how thick a lift cable must be, why a railway rail has a narrow waist and wide flanges, and why a glass tumbler shatters when dropped while a steel one only dents. This chapter builds the ideas needed to answer such questions.

8.2 Stress and Strain

Hang a bucket of water from a steel wire. The wire lengthens a little and then stops: the bucket is in equilibrium. Something inside the wire must now be pulling up on the bucket with a force equal to its weight. That something is the set of forces between neighbouring atoms. Stretching drags the atoms slightly further apart than their natural spacing, and the bonds between them pull back like tiny stretched springs (Fig. 8.2). Squeeze a solid instead, and the atoms are pushed closer than they like, so the bonds push back.

These internal forces are restoring forces. In equilibrium, the restoring force across any cross-section balances the external force on it. What matters to the material is not the force alone but how widely it is spread: 100 N hanging from a cotton thread is very different from 100 N on a steel bar. So we define stress (प्रतिबल) as the restoring force per unit area of the surface across which it acts:

Stress has the unit of pressure, the pascal: $1\mathrm{P}\mathrm{a}=1\mathrm{N}{\mathrm{m}}^{-2}\mathrm{.}$ Its dimensional formula is $[\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-2}]\mathrm{.}$ Engineering stresses run to millions or even hundreds of millions of pascals, so the megapascal ($1\mathrm{M}\mathrm{P}\mathrm{a}={10}^6\mathrm{P}\mathrm{a}$) and gigapascal (${10}^9\mathrm{P}\mathrm{a}$) are handy units.

Depending on how the forces are applied, a solid is stretched or squashed, sheared, or squeezed from every side. Each case has its own stress and its own measure of deformation, called strain (विकृति): the change in some dimension divided by its original value. Fig. 8.3 shows all of them on the same block.

Pulling or pushing along a length. Grip a rod by its ends and pull. The forces act along its axis, at right angles to the end faces, and the stress inside is tensile. Turn the forces round so that they push the ends together and the stress becomes compressive. Either way it acts along the length, so the two are grouped as longitudinal stress. The length changes from $L$ to $L\pm \mathrm{\Delta }L$, and the deformation is measured by the

Sliding one face past another. Now let the two forces act along two opposite faces instead of across them, like your palm pushing the top of an eraser whose base is glued to the desk (Fig. 8.4). The restoring force per unit area of that face is a shearing stress, also called a tangential stress. The top face moves sideways by $\mathrm{\Delta }x$ relative to the bottom face, a distance $L$ below it, so every vertical edge tilts through an angle $\theta \mathrm{.}$ The shearing strain is

In practice $\theta$ is tiny, so we can use the small-angle approximation, with $\theta$ in radians:

How good is this? At $\theta =5^{\circ }=0.0873\mathrm{r}\mathrm{a}\mathrm{d}$ we get $\tan \theta =0.0875$: the two differ by less than 0.3 %. Shear strains in loaded metal parts are usually thousands of times smaller still.

Squeezing from all sides. Drop a steel ball to the bottom of a deep lake. The water presses on every bit of its surface, always at right angles and equally hard everywhere, so the ball becomes very slightly smaller but stays perfectly round. The restoring force per unit area, the hydraulic stress, simply equals the extra pressure $p$ of the water, and the ball regains its size when it is brought back up. The strain here is the

Each strain is a ratio of two lengths or two volumes (or, for shear, an angle in radians), so strain is a pure number with no unit and no dimensions.

8.3 Hooke's Law

Load a wire, a spring or a rod gently and measure how far it deforms. Double the load and the deformation doubles; triple it and it triples (Fig. 8.5). As long as the deformation stays small, stress and strain rise and fall together in a fixed ratio, a rule known as Hooke's law.

The constant $k$ is the modulus of elasticity (प्रत्यास्थता गुणांक). Because strain has no unit, $k$ has the unit of stress, the pascal. A large modulus means a stiff material, one that needs a large stress for even a small strain. Each kind of stress has its own modulus; we meet them in Section 8.5.

Nothing deeper forces Hooke's law to be true: it is empirical, a pattern found by measurement. Metals follow it well, but only up to a certain stress, while rubber and many body tissues curve away from a straight line almost from the start. The next section shows where the law gives out.

8.4 Stress–Strain Curve

How does a material behave all the way from a gentle tug to breaking point? Clamp a sample, say a wire, in a testing machine and stretch it slowly while the machine logs two numbers: the pulling force and the extension. This is a tensile test. Turning each pair into a stress (force ÷ original area) and a strain (extension ÷ original length) and plotting them gives the material's stress–strain curve, a fingerprint of how it deforms. Squeezing or shearing the sample instead gives other curves of the same kind.

Fig. 8.6 sketches the curve for a ductile metal such as mild steel. The strains in its first part are tiny, well under 1 %, so the drawing stretches that part out to make it visible. Follow the curve from the origin O:

• O to P: the proportional region. A straight line: Hooke's law holds. Take the load off anywhere along it and the sample is exactly as long as before. P marks the proportional limit.
• P to Y: elastic but not proportional. The graph bends away from the straight line, but no harm is done yet: unload and the sample retraces its path to O. The end of this stretch, Y, is the yield point or elastic limit, and the stress there is the material's yield strength ${\sigma }_y$.
• Beyond Y: plastic flow. The curve flattens: layers of atoms slip past one another, and the sample stretches a lot for little extra stress. Let the load off at K and the sample comes back along a line parallel to OP, not along the curve, ending up longer than it started. That leftover strain is a permanent set, and the deformation is plastic.
• U: the ultimate tensile strength. The curve peaks at the ultimate tensile strength ${\sigma }_u$. From here on the sample thins at one spot, forming a neck, and that spot keeps stretching under a falling force.
• F: fracture. The sample breaks at the narrowest part of the neck.

The gap between U and F tells you how a material fails. If F lies far beyond U, the material can be drawn into wires or bent into shape before it breaks: it is ductile (तन्य), like copper, gold or mild steel. If U and F almost coincide, the material breaks soon after it stops being elastic, with little warning: it is brittle (भंगुर), like glass, cast iron or a china cup (Fig. 8.7).

Rubber plays by different rules. Pull a rubber band to three or four times its length and it still returns to its first length, yet its graph curves almost from the first moment, and it never passes through a plastic stage before it snaps. A material whose elastic stretch is huge but far from proportional is an elastomer. Your body has some: the wall of the aorta, the large artery leaving the heart, balloons out at every heartbeat. Its curve rises gently and then sharply, so the artery gives easily at first and then resists.

8.5 Elastic Moduli

Designers work hard to keep structures in the straight-line part of the curve, below the proportional limit, where deformations are small and fully reversible. There the ratio of stress to strain is constant. This modulus of elasticity is a property of the material (it varies a little with temperature), not of the size or shape of the sample. Because there are different kinds of stress and strain, there are several moduli; Table 8.5 collects them.

8.5.1 Young's Modulus

Stretch a short steel rod, or squeeze it with a stress of the same size, and its length changes by the same amount either way. So a single constant serves for both, the Young's modulus $Y$ (यंग गुणांक): longitudinal stress $\sigma$ divided by the longitudinal strain $\epsilon$ it produces.

For a rod of length $L$ and cross-sectional area $A$ that a force $F$ stretches by $\mathrm{\Delta }L$, (8.1) and (8.2) give

Its unit is the pascal, like that of stress. Turned round, the same equation predicts the extension of any wire, $\mathrm{\Delta }L=FL\mathrm{/}(AY)$: a long, thin wire of a material with a small modulus stretches most.

Why are crane hooks, rails and building frames made of steel? Hang the same 100 N load from wires 1 m long and $1\mathrm{m}{\mathrm{m}}^2$ in cross-section: a steel wire stretches by 0.50 mm, a copper one by 0.83 mm and an aluminium one by 1.4 mm. Steel's modulus is among the largest of common materials, so it deforms least. Glass is about as stiff as aluminium; concrete, bone and wood are softer, at roughly 10–30 GPa; rubber's modulus is thousands of times smaller.

8.5.2 Shear Modulus

For shearing, the stress is the tangential force per unit area of the face along which it acts, and the strain is the angle of tilt. Their ratio is the shear modulus $G$ (अपरूपण गुणांक), also called the modulus of rigidity:

By (8.4) the strain is simply $\theta$, so

The unit of $G$ is again the pascal. The $G$ column of Table 8.3 shows that, for a given material, the shear modulus is smaller than Young's modulus, typically between a third and two-fifths of it: it is easier to make layers of atoms slide over each other than to pull them apart. Only solids have a shear modulus. A liquid has no shape of its own, so even the gentlest sideways push makes it flow instead of settling at a fixed angle.

Shear turns up all over engineering: in a bolt that stops two plates sliding past each other, in a twisted drive shaft, and in the wire of a coil spring, which is twisted rather than stretched as the spring extends.

8.5.3 Bulk Modulus

A body in a fluid whose pressure rises by $p$ is squeezed equally from all sides, as in Section 8.2: the hydraulic stress is $p$ and the volume strain is $\mathrm{\Delta }V\mathrm{/}V\mathrm{.}$ Their ratio defines the bulk modulus $B$ (आयतन गुणांक):

Squeezing always shrinks a stable body, so $\mathrm{\Delta }V$ and $p$ have opposite signs, and the minus sign makes $B$ a positive number. Its reciprocal, the compressibility $k$ (संपीड्यता), answers the opposite question: by what fraction does the volume shrink for each pascal of extra pressure?

Unlike $Y$ and $G$, the bulk modulus applies to liquids and gases as well as to solids. A fluid has no shape of its own, but it does have a volume, and squeezing it reduces that volume. Table 8.4 and Fig. 8.9 compare the three states of matter.

For the same extra pressure, water shrinks about 70 times as much as steel, and air at constant temperature over 20 000 times as much as water. Why such a range? Go back to the spring model of Fig. 8.2. In a solid every atom is already pressed against stiff bonds on all sides, so moving the atoms even slightly closer takes an enormous pressure. A liquid's molecules also touch, but they are held more loosely. A gas is mostly empty space, and squeezing it only crowds the molecules into less of that space; this is also why a gas's compressibility depends strongly on its pressure and temperature.

Looking ahead: in Chapter 14 you will find that these moduli also set the speed of sound. A stiff, light medium carries sound fast, which is why sound travels about four times faster in water than in air, and faster still in steel.

8.5.4 Poisson's Ratio

Stretch a broad rubber band and watch its width: it narrows as it lengthens (Fig. 8.10). A steel wire does the same, far less visibly. The strain along the force, the longitudinal strain, comes with a strain at right angles to it, the lateral strain. If a wire of diameter $d$ and length $L$ lengthens by $\mathrm{\Delta }L$ while its diameter shrinks by $\mathrm{\Delta }d$, the lateral strain is $\mathrm{\Delta }d\mathrm{/}d\mathrm{.}$ Below the elastic limit the sideways strain keeps step with the lengthwise one, so their ratio is a fixed property of the material, named after the French mathematician Siméon Denis Poisson (1781–1840): Poisson's ratio (पॉयसन अनुपात):

No units survive the division, so Poisson's ratio is just a number. For most metals it lies between about 0.25 and 0.35: close to 0.29 for steel and 0.33 for aluminium (Table 8.3). For rubber it is almost 0.5, the value at which the volume stays the same as the material stretches. The lesson is general: a force in one direction produces strains in other directions too, so a single modulus cannot describe everything a solid does. A material that behaves the same in every direction needs two independent constants, for example $Y$ and Poisson's ratio, to describe its elastic behaviour fully.

8.5.5 Elastic Potential Energy in a Stretched Wire

Pulling a wire longer means dragging its atoms apart against their bonds, and that takes work. Below the elastic limit none of it is wasted: the wire holds it as elastic potential energy and pays it back as it shortens. That is how a bow launches an arrow and a gulel (catapult) flings a pebble. How much energy does a wire hold?

Since $\sigma =Y\epsilon$, the energy density can also be written as $u=\frac{1}{2}Y{\epsilon }^2$ or $u={\sigma }^2\mathrm{/}2Y\mathrm{.}$ At a given stress, a material with a smaller Young's modulus stores more energy per unit volume. Good spring materials combine a high yield strength with a moderate modulus, so they can store a lot of energy without being permanently bent.

8.6 Applications of Elastic Behaviour of Materials

Every bridge, building, crane and aircraft is designed with the ideas of this chapter. The designer must know how much each part will deform under the loads it will meet, and must make sure no part is ever stressed beyond its elastic limit, with a generous margin for surprises. Four examples show the reasoning.

Lifting ropes. A crane hook or a lift cage hangs from a steel rope. The rope must never be stretched into its plastic region, or it would stay permanently longer and weaker. So the stress $Mg\mathrm{/}A$ caused by the heaviest load $M$ must stay below the yield strength ${\sigma }_y$, which sets a minimum cross-sectional area:

Beams. A beam carries loads across a gap: a floor joist, a bridge girder, a plank laid over a drain. Its job is to bend as little as possible. For a bar of rectangular cross-section with breadth $b$ and depth $d$, resting on supports a distance $l$ apart and loaded by a weight $W$ at the middle, the sag at the centre is

Deriving this needs more calculus than we have yet, but we can read it. The sag grows rapidly with span, as $l^3$, so long spans need extra support. A stiffer material (larger $Y$) helps. Most striking is the role of shape: the sag falls as $1\mathrm{/}b$ but as $1\mathrm{/}{d}^3\mathrm{.}$ Doubling the breadth halves the sag; doubling the depth cuts it to one-eighth. So beams are made deep rather than wide (Fig. 8.13).

Why do steel girders have an I-shaped cross-section? When a beam sags, its top layers are squeezed and its bottom layers stretched, while the layers near the middle are hardly strained at all, so metal far from the middle does most of the work. An I-section puts most of its steel into two wide flanges, top and bottom, joined by a thin web: deep and stiff, yet light and cheap. The wide flanges also keep it from rolling over and folding sideways, the failure called buckling that threatens a tall, thin plank whenever the load is a little off-centre, as with vehicles crossing a bridge. Railway rails use the same idea in a modified form.

Pillars. Columns and pillars carry loads in compression, and their ends matter. A pillar with rounded ends touches what it supports over a small area, so the stress there is large, and the pillar can tilt and buckle easily. A pillar whose ends spread into a broad base and a broad top shares the load over a larger area and is held firmly in line, so it can carry much more. The wide bases and bracketed tops of the stone pillars in many old buildings do exactly this job.

How high can a mountain be? Elasticity even limits the height of mountains. Picture a vertical column of rock from the summit of a mountain of height $h$ down to its base (Fig. 8.14). Its weight, spread over its cross-section, presses on the rock at the base with a stress of about $h\rho g$, where $\rho$ is the density of the rock. The base is not squeezed equally from all sides, since nothing presses on the slopes from outside, so this stress has a large shearing part. If it exceeds the stress at which hot, deep rock begins to flow, the base slowly squeezes out sideways and the mountain sinks until the stress falls back below that limit.