Mechanical Properties of Solids: Exhaustive Study Guide

Introduction to Mechanical Properties of Solids

In the previous study of the rotation of bodies (Chapter 6), it was established that the motion of a body relies heavily on the distribution of mass within it. While those studies often simplified situations by assuming objects were rigid bodies—defined as hard, solid objects with a definite, unchanging shape and size—real-world physics reveals that no body is perfectly rigid. In reality, all solid bodies can be stretched, compressed, or bent. Even a steel bar, which appears appreciably rigid, can undergo deformation when subjected to a sufficiently large external force. A solid possesses a definite shape and size, but a force is required to deform or change these properties.

Two fundamental properties govern how materials respond to these deforming forces: elasticity and plasticity. Elasticity is the property of a body by virtue of which it tends to regain its original size and shape once the applied deforming force is removed. The accompanying change in shape is referred to as elastic deformation. A common example is a helical spring; when its ends are gently pulled, its length increases, but it resumes its original form upon release. Conversely, substances like putty or mud, which show no tendency to regain their previous shape and instead remain permanently deformed, are classified as plastic materials, a property known as plasticity. Putty and mud are considered close to ideal plastics.

Understanding the elastic behavior of materials is a cornerstone of engineering design. Knowledge of the elastic properties of steel and concrete is essential for designing buildings, bridges, automobiles, and ropeways. This study allows engineers to answer critical questions: such as whether an aeroplane can be designed to be both ultra-light and sufficiently strong, or why a railway track utilizes a specific I-shape. It also explains why certain materials, like glass, are brittle, while others, like brass, are not. These inquiries are addressed by examining how simple loads or forces act to deform different solid bodies.

Stress and Strain

When external forces are applied to a body maintaining static equilibrium, the body undergoes deformation. The magnitude of this deformation depends on the nature of the material and the strength of the applied force. Although the deformation might not be visually noticeable in all materials, it is consistently present. When a body is subjected to a deforming force, an internal restoring force is developed. This restoring force is equal in magnitude and opposite in direction to the applied force. The restoring force per unit area is defined as stress. If $F$ is the force applied normal to the cross-section and $A$ is the cross-sectional area, the magnitude of stress is expressed as:

$\text{Stress} = \frac{F}{A}$

The SI unit for stress is $N\,m^{-2}$ or pascal ($Pa$), and its dimensional formula is $[ML^{-1}T^{-2}]$. Solids can change their dimensions in three distinct ways depending on how the external force is applied, leading to different types of stress and strain.

Longitudinal stress occurs when a cylinder is stretched or compressed by forces applied normal to its cross-sectional area. If the cylinder is stretched, the restoring force per unit area is called tensile stress. If it is compressed, it is called compressive stress. This leads to a change in length; the ratio of the change in length ($\Delta L$) to the original length ($L$) is called longitudinal strain:

$\text{Longitudinal Strain} = \frac{\Delta L}{L}$

Tangential or shearing stress arises when two equal and opposite deforming forces are applied parallel to the cross-sectional area of a body, such as a cylinder. This causes a relative displacement between the opposite faces. The restoring force per unit area developed due to this tangential force is the shearing stress. The resulting shearing strain is defined as the ratio of the relative displacement of the faces ($\Delta x$) to the length of the cylinder ($L$), often expressed as:

$\text{Shearing Strain} = \frac{\Delta x}{L} = \tan(\theta)$

where $\theta$ is the angular displacement. For small deformations, $\tan(\theta) \approx \theta$. This effect can be visualized by pushing a book horizontally with your hand while it remains on a surface.

Hydraulic stress occurs when a solid (such as a sphere) is submerged in a fluid and compressed uniformly on all sides by pressure acting perpendicular to every point of its surface. This results in a decrease in volume without changing the geometrical shape. The internal restoring force per unit area developed here is the hydraulic stress, which is equal in magnitude to the hydraulic pressure ($p$). The corresponding strain is volume strain, defined as the ratio of the change in volume ($\Delta V$) to the original volume ($V$):

$\text{Volume Strain} = \frac{\Delta V}{V}$

Because strain is a ratio of similar dimensions, it is a dimensionless quantity and has no units.

Hooke's Law and the Stress-Strain Curve

Hooke's law states that for small deformations, the stress and strain in a material are proportional to each other. This empirical law is expressed as:

$\text{Stress} \propto \text{Strain}$$\text{Stress} = k \times \text{Strain}$

where $k$ is the proportionality constant known as the modulus of elasticity. While found valid for most materials, some substances do not exhibit this linear relationship. The specific relationship between stress and strain for a material is typically identified through experimental testing, such as a tensile test where a cylinder or wire is stretched and the resulting force and change in length are recorded.

A typical stress-strain curve for a metal reveals several regions. In the region from O to A, the curve is linear, and Hooke's law is obeyed; the material behaves elastically and returns to its original dimensions when the force is removed. From A to B, stress and strain are no longer proportional, but the body still regains its original shape upon unloading. Point B is called the yield point or elastic limit, and the stress at this point is the yield strength ($\sigma_y$).

If the load increases beyond B, the material enters the plastic deformation region (B to D). In this stage, even a small increase in stress causes a rapid increase in strain. If the load is removed at a point C within this region, the material does not return to zero strain; it retains a permanent set. Point D represents the ultimate tensile strength ($\sigma_u$). Beyond point D, additional strain is produced even if the force is reduced, leading eventually to the fracture point E. Materials where D and E are close together are called brittle, whereas those where D and E are far apart are described as ductile. Some materials, like rubber or the elastic tissue of the aorta (elastomers), can be stretched to several times their original length. While they have a large elastic region, they often do not obey Hooke's law and lack a well-defined plastic region.

Elastic Moduli

The modulus of elasticity is a characteristic property of a material defined as the ratio of stress to strain within the elastic limit.

Young's Modulus ($Y$) is the ratio of tensile (or compressive) stress ($\sigma$) to longitudinal strain ($\epsilon$):

$Y = \frac{\sigma}{\epsilon} = \frac{F/A}{\Delta L/L} = \frac{F \times L}{A \times \Delta L}$

The SI unit is $N\,m^{-2}$ or $Pa$. Metals typically have large Young’s moduli, meaning they require significant force for small deviations in length. For instance, to increase the length of a thin steel wire (area $0.1\,cm^2$) by $0.1\%$, a force of $2000\,N$ is required. For the same area and strain, aluminium requires $690\,N$, brass $900\,N$, and copper $1100\,N$. Consequently, steel is more elastic than copper, brass, or aluminium and is preferred for heavy-duty machinery.

Shear Modulus ($G$), or the modulus of rigidity, is the ratio of shearing stress ($\sigma_s$) to shearing strain ($\theta$):

$G = \frac{\sigma_s}{\text{shearing strain}} = \frac{F/A}{\Delta x/L} = \frac{F}{A \times \theta}$

For most materials, the shear modulus is approximately one-third of the Young's modulus ($G \approx Y/3$).

Bulk Modulus ($B$) defines the response to hydraulic stress and is the ratio of hydraulic stress ($p$) to volume strain ($\Delta V/V$):

$B = -\frac{p}{\Delta V/V}$

The negative sign indicates that as pressure increases, volume decreases. The reciprocal of the bulk modulus is called compressibility ($k$):

$k = \frac{1}{B} = -\frac{1}{\Delta p} \times \frac{\Delta V}{V}$

Solids have the highest bulk moduli and are the least compressible. Gases are approximately a million times more compressible than solids because their molecules are poorly coupled compared to the tight atomic coupling in solids.

Examples of Elastic Calculations

Example 8.1 involves a structural steel rod with a radius of $10\,mm$ ($0.01\,m$) and a length of $1.0\,m$, subjected to a $100\,kN$ ($10^5\,N$) force. The Young's modulus of structural steel is $2.0 \times 10^{11}\,N\,m^{-2}$. The stress is calculated as:

$\text{Stress} = \frac{10^5}{\pi \times (10^{-2})^2} = 3.18 \times 10^8\,N\,m^{-2}$

The elongation ($\Delta L$) is calculated by:

$\Delta L = \frac{(3.18 \times 10^8) \times (1)}{2.0 \times 10^{11}} = 1.59 \times 10^{-3}\,m = 1.59\,mm$

The resulting strain is $0.16\%$.

In Example 8.3, a human pyramid is supported by a performer. The total mass is $280\,kg$, but the performer being the support has a mass of $60\,kg$, meaning the legs support an extra $220\,kg$. The weight is $220 \times 9.8 = 2156\,N$. Each femur supports half this weight ($1078\,N$). With a Young's modulus for bone ($Y = 9.4 \times 10^9\,N\,m^{-2}$), length ($0.5\,m$), and radius ($2.0\,cm$), the compression in each thighbone is calculated:

$\Delta L = \frac{1078 \times 0.5}{9.4 \times 10^9 \times 1.26 \times 10^{-3}} = 4.55 \times 10^{-5}\,m$

Example 8.5 calculates the fractional compression of water at the bottom of the Indian Ocean (depth $3000\,m$). The pressure exerted is:

$p = h \rho g = 3000 \times 1000 \times 10 = 3 \times 10^7\,N\,m^{-2}$

Given the bulk modulus of water ($2.2 \times 10^9\,N\,m^{-2}$), the fractional compression ($\Delta V/V$) is:

$\frac{\Delta V}{V} = \frac{3 \times 10^7}{2.2 \times 10^9} = 1.36 \times 10^{-2} = 1.36\%$

Poisson's Ratio and Elastic Potential Energy

Poisson's Ratio describes the phenomenon where a longitudinal strain in one direction produces a lateral strain in a perpendicular direction. Simon Poisson noted that within the elastic limit, the lateral strain is proportional to the longitudinal strain. For a wire with original diameter $d$ and length $L$, undergoing contraction $\Delta d$ and elongation $\Delta L$, the Poisson's ratio is defined as:

$\text{Poisson's Ratio} = \frac{\Delta d/d}{\Delta L/L}$

This is a dimensionless number dependent only on the material; for steel, it is between $0.28$ and $0.30$.

When a wire is stretched, work is done against inter-atomic forces, which is stored as elastic potential energy ($U$). For an elongation $l$, the force is $F = \frac{YAl}{L}$. The work done ($W$) to increase the length from $0$ to $l$ is:

$W = \int_0^l \frac{YAl}{L} dl = \frac{YA l^2}{2L}$

This can be rewritten in terms of the total volume ($AL$):

$W = \frac{1}{2} \times \text{Young's modulus} \times \text{strain}^2 \times \text{volume}$$W = \frac{1}{2} \times \text{stress} \times \text{strain} \times \text{volume}$

The elastic potential energy per unit volume ($u$) is:

$u = \frac{1}{2} \sigma \epsilon$

Applications of Elastic Behaviour

The practical application of these principles is seen in structural engineering. For instance, in crane design, the radius of the steel rope must ensure the stress does not exceed the elastic limit. For a $10\text{-ton}$ load (Mg), and a yield strength of $300 \times 10^6\,N\,m^{-2}$, the area $A \ge Mg/\sigma_y$ results in a radius of roughly $1\,cm$. To provide a factor of safety (factor of $10$), a radius of $3\,cm$ is recommended. Because a single wire of that thickness would be too rigid, cranes use multiple thin wires braided together for flexibility.

In building and bridge design, beams must resist bending (sagging). A beam of length $l$, breadth $b$, and depth $d$ loaded at the center with weight $W$ sags by an amount $\delta$:

$\delta = \frac{W l^3}{4 b d^3 Y}$

To minimize sagging, materials with a high Young's modulus are used. Furthermore, increasing the depth ($d$) of the beam is more effective than increasing breadth ($b$) because $\delta \propto d^{-3}$. However, a very deep and thin beam might buckle. To prevent this, engineers use the I-shape cross-section, which provides a large load-bearing surface and sufficient depth while minimizing weight and cost. Pillars with distributed ends are also used to support larger loads than those with rounded ends.

The height of mountains on Earth is also limited by the elastic properties of rocks. At the mountain's base, the vertical force per unit area is $h \rho g$. While this is a vertical force, it creates a shear component because the sides of the mountain are free. If the shear stress exceeds the elastic limit ($30 \times 10^7\,N\,m^{-2}$ for rock), the rock will flow. Setting $h \rho g = 30 \times 10^7\,N\,m^{-2}$ with $\rho = 3 \times 10^3\,kg\,m^{-3}$ and $g = 10\,m/s^2$ gives a maximum height of approximately $10\,km$, which is slightly above the height of Mt. Everest.

Questions & Discussion

Discussion on Tension in Suspended Wires:

In a scenario where a wire is suspended from a ceiling and a weight $F$ is attached to the other end, the ceiling exerts an equal and opposite force. However, the tension at any cross-section of the wire is $F$, not $2F$. Consequently, the tensile stress is defined simply as $F/A$.

Validity and Relevance of Moduli:

It is important to remember that Hooke’s law is only valid in the linear portion of the stress-strain curve. Furthermore, Young’s modulus and shear modulus are only applicable to solids, as only solids have fixed lengths and shapes. In contrast, the bulk modulus is relevant for solids, liquids, and gases because it deals with uniform stress and volume change while maintaining shape.

Common Misconceptions about Elasticity:

In everyday language, we often assume that a material that stretches more (like rubber) is more elastic. In physics, this is a misnomer; a material is considered more elastic if it requires a greater force to produce a change in length (like steel). Additionally, stress is not a vector quantity; unlike force, it cannot be assigned a specific direction relative to a section in the body. While a force acting on a specific portion of a body has a direction, the state of stress within the body does not follow vector addition rules.

Summary of Concepts:

1. Stress is the restoring force per unit area; strain is fractional change in dimension.
2. Three main types of stress: Longitudinal (tensile/compressive), shearing, and hydraulic.
3. Hooke’s Law: Stress is proportional to strain for small deformations ($Stress = k \times Strain$).
4. Moduli: Young's ($Y = σ/ε$), Shear ($G = σ_s/θ$), Bulk ($B = -p/(ΔV/V)$).
5. Elastomers (like rubber/aorta) do not obey Hooke’s law despite large elastic ranges.
6. Applications: Crane cables (braiding for flexibility), I-beams (preventing bending/buckling), and mountain height limits.