Properties of Matter: Elasticity, stress and strain

Elasticity

1. Elasticity

When an external force acts on a solid body, its dimensions or shape may change. This change is called deformation. Depending on the direction and nature of the applied force, deformation may involve a change in length, volume, shape, or a combination of these.

For example:

• A wire undergoes elongation when a tensile force acts along its length.
• A solid body undergoes a change in volume when subjected to uniform external pressure.
• A block undergoes a change in shape when a tangential force acts on its surface.

In crystalline solids, deformation generally involves changes in interatomic distances and bond angles. In rubber and other polymers, deformation may also involve the uncoiling, orientation, and stretching of polymer chains.

Elasticity: Elasticity is the property of a material by virtue of which it tends to regain its original shape and dimensions after the removal of the external deforming force, provided that the deformation remains within its recoverable range.

1.1 Perfectly Elastic and Plastic Bodies

Perfectly elastic body: An ideal body that completely recovers its original shape and dimensions after the removal of the deforming force is called a perfectly elastic body.

Perfectly plastic body: An ideal body that retains its deformation after removal of the applied force, without elastic recovery, is called a perfectly plastic body.

Rigid body: An ideal body whose shape and dimensions remain unchanged under the action of external forces is called a rigid body.

Perfect elasticity, perfect plasticity, and absolute rigidity are idealizations. Real materials exhibit different degrees of elastic, plastic, and time-dependent deformation.

2. Stress

When an external force deforms a body, internal forces develop within the material. These forces arise from interactions between its constituent particles and resist the imposed deformation.

The internal force transmitted across a unit area of an imaginary surface within a material is called stress.

For a uniformly loaded specimen, the average stress is given by

\[ \boxed{\sigma=\frac{F}{A}} \]

where

• \(F\) = magnitude of the relevant force,
• \(A\) = area over which the force acts,
• \(\sigma\) = normal stress.

The SI unit of stress is newton per square metre (\(\mathrm{N\,m^{-2}}\)), also called the pascal (Pa).

\[ [\text{Stress}] = \frac{[F]}{[A]} = \frac{MLT^{-2}}{L^2} \] \[ \boxed{[\text{Stress}]=ML^{-1}T^{-2}} \]

Thus, stress has the same dimensional formula and SI unit as pressure.

2.1 Types of Stress

According to the direction of the applied force relative to the surface and the resulting deformation, stress is classified into the following types.

(i) Longitudinal Stress

When a force acts normally to the cross-sectional area of a body along its longitudinal axis, the resulting stress is called longitudinal stress.

\[ \boxed{\sigma_L=\frac{F}{A}} \]

Longitudinal stress may be tensile or compressive. A tensile force tends to increase the length of the body, whereas a compressive force tends to decrease it.

Fig. 2.1: Longitudinal tensile stress acting on a solid specimen.

(ii) Bulk or Volumetric Stress

When a body is subjected to uniform normal pressure acting on its entire surface, it undergoes a change in volume. The corresponding loading is described as bulk or hydrostatic stress.

For hydrostatic compression, the applied pressure is

\[ \boxed{P=\frac{F_n}{A}} \]

where \(F_n\) is the magnitude of the normal force acting on a surface of area \(A\).

An increase in external pressure generally produces a decrease in volume. A hydrostatic tensile stress, when physically sustainable, tends to increase the volume.

Fig. 2.2: Schematic representation of hydrostatic compression.

(iii) Shearing or Tangential Stress

When a force acts tangentially to a surface of a body, it produces a tendency for adjacent layers to slide relative to one another. The resulting stress is called shearing stress.

\[ \boxed{\tau=\frac{F_t}{A}} \]

where \(F_t\) is the tangential force and \(A\) is the area of the surface.

Fig. 2.3: Shear deformation caused by a tangential force.

3. Strain

The deformation of a body is expressed quantitatively in terms of strain. For simple deformation modes, strain represents the relative change in dimensions or the angular distortion produced by an applied stress.

Strain: Strain is a measure of the relative deformation of a body produced under the action of external forces.

For elementary elasticity, strain is classified into longitudinal strain, volumetric strain, and shearing strain.

3.1 Longitudinal Strain

Longitudinal strain is defined as the ratio of the change in length of a body to its original length.

Let the original length of a wire be \(L\). If the applied longitudinal force changes its length by \(\Delta L\), then

\[ \text{Longitudinal strain} = \frac{\text{Change in length}} {\text{Original length}} \] \[ \boxed{\varepsilon_L=\frac{\Delta L}{L}} \]

For tensile deformation, \(\Delta L>0\), and hence the longitudinal strain is positive. For compressive deformation, \(\Delta L<0\), and the strain is negative under the usual sign convention.

3.2 Volumetric Strain

Volumetric strain is defined as the ratio of the change in volume of a body to its original volume.

If the initial volume is \(V\) and the change in volume is \(\Delta V\), then

\[ \text{Volumetric strain} = \frac{\text{Change in volume}} {\text{Original volume}} \] \[ \boxed{\varepsilon_V=\frac{\Delta V}{V}} \]

Volumetric strain is positive for an increase in volume and negative for a decrease in volume.

3.3 Shearing Strain

Shearing strain describes the angular distortion produced when adjacent layers of a body undergo relative tangential displacement.

Consider a rectangular block of height \(h\), whose lower surface is fixed. When a tangential force is applied to its upper surface, the upper surface undergoes a horizontal displacement \(\Delta x\).

If \(\theta\) is the angle of shear, then geometrically

\[ \tan\theta=\frac{\Delta x}{h} \]

For small angular deformation, when \(\theta\) is measured in radians,

\[ \tan\theta\approx\theta \]

Therefore, the engineering shearing strain is

\[ \boxed{ \gamma=\frac{\Delta x}{h} =\tan\theta\approx\theta } \]

Thus, the shearing strain is approximately equal to the angle of shear for small deformations.

Important: Strain is dimensionless because it is expressed as a ratio of quantities having the same dimensions. Shearing strain is also dimensionless, although its corresponding angle may be expressed in radians.

4. Hooke's Law

Robert Hooke formulated the fundamental law of elastic deformation in the seventeenth century. According to Hooke's law, within the proportional limit of a material, stress is directly proportional to the corresponding strain.

\[ \text{Stress}\propto\text{Strain} \]

Therefore,

\[ \boxed{ \text{Stress}=E\times\text{Strain} } \]

or

\[ \boxed{ E=\frac{\text{Stress}}{\text{Strain}} } \]

Here, \(E\) represents the elastic modulus corresponding to the deformation mode. For longitudinal deformation, \(E\) specifically denotes Young's modulus.

The elastic modulus measures the stiffness of a material. A higher modulus indicates that a greater stress is required to produce a given small strain.

Since strain is dimensionless, the elastic modulus has the same dimensions and SI unit as stress.

\[ \boxed{[E]=ML^{-1}T^{-2}} \] \[ \boxed{\text{SI unit}=\mathrm{Pa}} \]

4.1 Types of Elastic Moduli

The elastic modulus depends on the nature of deformation. The three principal elastic moduli for an isotropic solid are Young's modulus, bulk modulus, and shear modulus.

(i) Young's Modulus

Young's modulus is defined as the ratio of longitudinal stress to longitudinal strain within the linear elastic region.

\[ Y=\frac{\sigma_L}{\varepsilon_L} \]

Substituting the expressions for longitudinal stress and strain,

\[ Y=\frac{F/A}{\Delta L/L} \] \[ \boxed{Y=\frac{FL}{A\Delta L}} \]

(ii) Bulk Modulus

Bulk modulus is defined as the ratio of an increase in hydrostatic pressure to the corresponding fractional decrease in volume within the linear elastic region.

\[ K=-\frac{\Delta P}{\Delta V/V} \]

Therefore,

\[ \boxed{K=-V\frac{\Delta P}{\Delta V}} \]

The negative sign ensures that the bulk modulus is positive for ordinary stable materials, because an increase in pressure produces a decrease in volume.

For infinitesimal changes, the isothermal bulk modulus is written as

\[ \boxed{ K_T=-V\left(\frac{\partial P}{\partial V}\right)_T } \]

(iii) Modulus of Rigidity

The modulus of rigidity, also called the shear modulus, is defined as the ratio of shearing stress to shearing strain within the linear elastic region.

\[ G=\frac{\tau}{\gamma} \]

Using

\[ \tau=\frac{F_t}{A}, \qquad \gamma=\frac{\Delta x}{h} \]

we obtain

\[ \boxed{ G=\frac{F_t h}{A\Delta x} } \]

4.2 Comparison of Elastic Moduli

Elastic modulus	Expression	Deformation
Young's modulus \(Y\)	\(\dfrac{FL}{A\Delta L}\)	Change in length
Bulk modulus \(K\)	\(-\dfrac{\Delta P}{\Delta V/V}\)	Change in volume
Shear modulus \(G\)	\(\dfrac{\tau}{\gamma}\)	Change in shape

5. Stress–Strain Curve and Elastic Limit

The mechanical behaviour of a material under loading can be studied using a stress–strain curve. It represents the variation of stress with strain during deformation.

For many ductile metals under uniaxial tensile loading, the engineering stress–strain curve exhibits an initial elastic region followed by yielding, plastic deformation, strain hardening, and finally fracture.

Fig. 5.1: Schematic engineering stress–strain curve for a ductile metal under tensile loading (not to scale).

5.1 Proportional Limit

The proportional limit is the maximum stress up to which stress is directly proportional to strain. In this region, Hooke's law is satisfied, and the stress–strain curve is linear.

\[ \boxed{\sigma=Y\varepsilon} \]

The slope of the linear portion of the stress–strain curve gives Young's modulus.

\[ \boxed{ Y=\frac{d\sigma}{d\varepsilon} } \]

5.2 Elastic Limit

The elastic limit is the maximum stress up to which a material can recover its original dimensions after unloading without permanent deformation.

The elastic limit need not coincide exactly with the proportional limit. A material may show recoverable nonlinear elastic deformation beyond its linear region.

5.3 Yield Point

The yield point marks the onset of appreciable plastic deformation. Beyond this stage, the material develops permanent strain that remains after the applied load is removed.

Some materials exhibit distinct upper and lower yield points, while others show a gradual transition from elastic to plastic behaviour.

For materials without a clearly defined yield point, the yield strength is commonly determined using a specified offset strain, such as 0.2%.

5.4 Plastic Region

Beyond yielding, deformation is no longer completely recoverable. The total strain contains both elastic and plastic contributions.

\[ \boxed{ \varepsilon_{\text{total}} = \varepsilon_{\text{elastic}} + \varepsilon_{\text{plastic}} } \]

During plastic deformation, many ductile metals undergo strain hardening, in which increasing stress is required to produce additional plastic strain.

5.5 Ultimate Tensile Strength

The ultimate tensile strength (UTS) is the maximum engineering stress attained during a tensile test.

\[ \boxed{ \sigma_{\mathrm{UTS}}= \frac{F_{\max}}{A_0} } \]

where \(F_{\max}\) is the maximum tensile load and \(A_0\) is the original cross-sectional area of the specimen.

After the maximum load, a ductile specimen may undergo localized reduction in cross-sectional area, called necking.

5.6 Breaking Point

The breaking point corresponds to the fracture of the specimen. The engineering stress at fracture is called the breaking or fracture stress.

\[ \boxed{ \sigma_b=\frac{F_b}{A_0} } \]

where \(F_b\) is the load at fracture. The corresponding strain is called the breaking strain.

For ductile materials, the engineering breaking stress may be lower than the ultimate tensile strength because of necking.

5.7 Factor of Safety

The factor of safety is a dimensionless quantity used in engineering design to provide a margin between the material's limiting strength and the allowable working stress.

When fracture strength is used as the limiting criterion,

\[ \boxed{ \text{Factor of safety} = \frac{\text{Breaking stress}} {\text{Allowable working stress}} } \]

For ductile materials, yield strength is often a more appropriate design criterion. In that case,

\[ \boxed{ n=\frac{\sigma_y}{\sigma_{\mathrm{allow}}} } \]

where \(\sigma_y\) is the yield strength and \(\sigma_{\mathrm{allow}}\) is the allowable working stress.

6. Elastic Recovery and Relaxation

When the external deforming force is removed, an elastic material tends to recover its original dimensions. For an ideal elastic solid, this recovery is immediate and complete.

However, some real materials exhibit time-dependent deformation and recovery because of viscoelastic effects.

Delayed elastic recovery: The gradual recovery of deformation after the removal of stress is called delayed elastic recovery or elastic aftereffect.

The characteristic time associated with this recovery is called a retardation or recovery time. Its value depends on the material, temperature, and deformation conditions.

Scientific distinction: Stress relaxation refers specifically to the decrease in stress with time when a viscoelastic material is maintained at constant strain. It is different from delayed recovery after unloading.

7. Elasticity and Stiffness of Materials

The elastic behaviour of a material should be distinguished from its stiffness and its ability to undergo large recoverable deformation.

The elastic modulus measures resistance to elastic deformation, whereas the elastic limit indicates the boundary beyond which permanent deformation begins.

Metals generally have relatively high Young's moduli and therefore undergo small elastic strains under moderate stress. Rubber-like elastomers have much lower elastic moduli and can undergo comparatively large recoverable strains.

Thus, rubber can be stretched much more than many metals, even though its Young's modulus is considerably smaller.

A material with a higher Young's modulus is described as stiffer, not necessarily more elastic in the sense of having greater recoverable strain.

8. Important Formulae

Quantity	Mathematical expression
Normal stress	\(\sigma=F/A\)
Shear stress	\(\tau=F_t/A\)
Longitudinal strain	\(\varepsilon_L=\Delta L/L\)
Volumetric strain	\(\varepsilon_V=\Delta V/V\)
Shearing strain	\(\gamma=\Delta x/h\)
Hooke's law	\(\sigma=Y\varepsilon_L\)
Young's modulus	\(Y=FL/(A\Delta L)\)
Bulk modulus	\(K=-\Delta P/(\Delta V/V)\)
Shear modulus	\(G=\tau/\gamma\)
Factor of safety	\(n=\sigma_{\mathrm{limiting}}/ \sigma_{\mathrm{allow}}\)