# Mechanical Properties of Solid

**Mechanical Properties of Solid**

**Stress:**

When the deforming force is applied to an object, the object deforms. In order to bring the object back to its original shape and size, there will be an opposing force generated inside the object.

This restoring force will be equal in magnitude and opposite in direction to the applied deforming force. The measure of this restoring force generated per unit area of the material is called stress.

Thus, Stress is defined as “The restoring force per unit area of the material”. It is a tensor quantity. Denoted by Greek letter σ. Measured using Pascal or N/m^{2}. Mathematically expressed as –

Where,

F is the restoring force measured in Newton or N.

A is the area of cross-section measured in m^{2}.

σ is the stress measured using N/m^{2} or Pa.

**Types of stress**

Therearemanytypesofstressinphysics,buttheyaremostlyclassifiedintotwoforms,whicharenormalstressandtangentialorshearstress.Sometypesofstressarediscussedinthefollowingsections.

Types of Stress

There are several types of stress in physics but it is mainly categorized into two forms that are Normal Stress and Tangential or Shearing Stress. Some stress types are discussed in the points below.

**Normal Stress:**

As the name suggests, Stress is said to be Normal stress when the direction of the deforming force is perpendicular to the cross-sectional area of the body. The length of the wire or the volume of the body changes stress will be at normal. Normal stress can be further classified into two types based on the dimension of force-

**Longitudinal stress**

Bulk Stress or Volumetric stress

**Longitudinal Stress:**

Consider a cylinder. When two cross-sectional areas of the cylinder are subjected to equal and opposite forces the stress experienced by the cylinder is called longitudinal stress.

Longitudinal Stress = Deforming Force / Area of cross-section = F/A

As the name suggests, when the body is under longitudinal stress-

The deforming force will be acting along the length of the body.

Longitudinal stress results in the change in the length of the body. Hence, thereby it affects slight change in diameter.

The Longitudinal Stress either stretches the object or compresses the object along its length. Thus, it can be further classified into two types based on the direction of deforming force-

- Tensile stress
- Compressive stress

**Tensile Stress:**

If the deforming force or applied force results in the increase in the object’s length then the resulting stress is termed as tensile stress. For example: When a rod or wire is stretched by pulling it with equal and opposite forces (outwards) at both ends.

**Compressive Stress**

If the deforming force or applied force results in the decrease in the object’s length then the resulting stress is termed as compressive stress. For example: When a rod or wire is compressed/squeezed by pushing it with equal and opposite forces (inwards) at both ends.

**Bulk Stress or Volume Stress**

When the deforming force or applied force acts from all dimensions resulting in the change of volume of the object then such stress in called volumetric stress or Bulk stress. In short, when the volume of body changes due to the deforming force it is termed as Volume stress.

**Shearing Stress or Tangential Stress**

When the direction of the deforming force or external force is parallel to the cross-sectional area, the stress experienced by the object is called shearing stress or tangential stress.

** Strain:**

Strain is the amount of deformation experienced by the body in the direction of force applied, divided by the initial dimensions of the body.

The following equation gives the relation for deformation in terms of the length of a solid:

where ε is the strain due to the stress applied, δl is the change in length and L is the original length of the material.

The strain is a dimensionless quantity as it just defines the relative change in shape.

**Types of Strain**

Strain experienced by a body can be of two types depending on stress application as follows:

**Tensile Strain**

The deformation or elongation of a solid body due to applying a tensile force or stress is known as Tensile strain. In other words, tensile strain is produced when a body increases in length as applied forces try to stretch it.

**Compressive Strain**

Compressive strain is the deformation in a solid due to the application of compressive stress. In other words, compressive strain is produced when a body decreases in length when equal and opposite forces try to compress it.

**Stress-strain curve**

When we study solids and their mechanical properties, knowing their elastic properties is most important. We can learn the elastic properties of materials by studying the stress-strain relationships of those materials under different loads. The stress-strain curve of a material gives its stress-strain relationship. On the stress-strain curve, stress and corresponding strain values are plotted. Below is an example of a stress-strain curve.

**Explaining Stress-Strain Graph**

The different regions in the stress-strain diagram are:

(i) Proportional Limit

It is the region in the stress-strain curve that obeys Hooke’s Law. In this limit, the stress-strain ratio gives us a proportionality constant known as Young’s modulus. The point OA in the graph represents the proportional limit.

(ii) Elastic Limit

It is the point in the graph up to which the material returns to its original position when the load acting on it is completely removed. Beyond this limit, the material doesn’t return to its original position, and a plastic deformation starts to appear in it.

(iii) Yield Point

The yield point is defined as the point at which the material starts to deform plastically. After the yield point is passed, permanent plastic deformation occurs. There are two yield points (i) upper yield point (ii) lower yield point.

(iv) Ultimate Stress Point

It is a point that represents the maximum stress that a material can endure before failure. Beyond this point, failure occurs.

(v) Fracture or Breaking Point

It is the point in the stress-strain curve at which the failure of the material takes place.

**Hooke's law**

In the 19th century, while studying springs and elasticity, the English scientist Robert Hooke noticed that many materials had a similar property when studying the stress-strain relationship. There was a linear region where the force required to stretch a material was proportional to the elongation of the material, known as Hooke's Law. Hooke's law states that the elongation of a material is proportional to the applied stress within the elastic limits of that material. Mathematically, Hooke's law is generally expressed as:

F = –k.x

Where F is the force, x is the longitudinal extension and k is the proportionality constant called the spring constant N/m.

**Elastic Constants**

Elastic constants are those constants which determine the deformation produced by a given stress system acting on the material.

Elastic constants are used to determine engineering strain theoretically.

They are used to obtain a relationship between engineering stress and engineering strain.

For a homogeneous and isotropic material, the number of elastic constants is 4.

**Types of Elastic Constants**

1.Young’s modulus or modulus of elasticity (E)

2.Shear modulus or modulus of rigidity (G)

3.Bulk modulus (K)

4.Poisson’s ratio (µ)

Relationship between Elastic Constants

« E = 2G (1+ µ)

« E = 3 K (1-2µ)

« –E = 9KG / G+3K

**Definition of modulus of elasticity **

According to Hooke's law, up to the limit of proportionality, "for small strains, the stress is directly proportional to the strain." Mathematically, Hooke's law is expressed as: Stress α Tension σ = E ε In the above formula, "E" is called the modulus of elasticity. σ is the stress and ε is the strain.

We can write the expression for modulus of elasticity using the above equation as:

E = (F*L) / (A * δL)

Thus, we can define the modulus of elasticity as the ratio of normal stress to longitudinal stress.

**Unit of Modulus of Elasticity :**

The unit of normal stress is Pascal, and longitudinal strain has no unit because it is the ratio of the change in length to the original length. Thus, the unit of modulus of elasticity is the same as the unit of stress and is Pascal (Pa). We most often use megapascals (MPa) and gigapascals (GPa) to measure the modulus of elasticity.

1 MPa = Pa

1 GPa = Pa

How to Measure Young’s Modulus or Modulus of Elasticity?

Let us take a rod of a ductile material that is mild steel. Now, do a tension test on the Universal testing machine. After the tension test, when we plot the stress-strain diagram, we get a curve like the one below.

From the curve, we see that from point O to B, the region is elastic. After that, the plastic deformation starts. Point A in the curve shows the limit of proportionality. For this curve, we can write the value of the modulus of elasticity (E) is equal to the slope of the stress-strain curve up to A.

Here are some values of E for the most commonly used materials.

Mild steel- E = 200 GPa

Cast iron E = 100 GPa

Aluminium E = 200/3 GPa

**Determination of the Young's modulus of a conductive material**

Make an experimental setup as shown in the figure to determine the value of Young's modulus of the wire material under tension.

Determination of the Young's modulus of a conductive material Take two identical straight wires (same length and equal radius), A and B. Now attach its end to a strong rigid support. Wire A is the reference wire and has a millimeter master scale M and a pan to determine weight. Thread B is an experimental thread. It also has a pan where known weights are placed. B connects a vernier weight V to the lower part of the thread. Now, when the weight is placed on the vessel attached to B, it exerts a downward force. This downward force (pull) causes wire B to stretch. This elongation (increase in length) of wire B is measured on the vernier scale. The reference wire A is used to compensate for the change in length due to the change in room temperature. At first, apply a small load to both wires A and B so that both are straight and take a Vernier reading. Now gradually increase the load on wire B and note the vernier reading. The difference between these two vernier readings indicates the change in the length of the thread.

Let the initial radius and length of wire B be r and L, respectively,

Then, the cross-sectional area of the wire would be pr^{2}

Let M be the mass that is responsible for an elongation DL in the wire B.

Then, the applied force is equal to Mg, where g is the acceleration due to gravity.

From the equation,

E= (F/A) / (DL/L)

= (F × L) / (A × DL)

The Young’s modulus of the material of the experimental wire B is given by,

Y = σ/ε

Y = (F/A)/(ΔL/L)

Y= (F × L) /(A × ΔL)

Y= (Mg × L) /(A × ΔL)

Interesting Fact about Modulus of Elasticity :

· The modulus of elasticity is standard. Robert Hooke (1635-1703) was an early scientist who worked on applied mechanics.

· Thomas Young explains this in Loongute's course on Natural Philosophy and Mechanical Art. He made a detailed study of the characterization of elasticity.

· According to Robert Hook, the value of E depends on both the geometry and the material.

· Thomas Young said that the value of E depends only on the material, not on its geometry.

· A small piece of rubber and a large piece of rubber have the same coefficient of elasticity. This is a fundamental property of any material that cannot be changed.

· The modulus of elasticity is a measure of the stiffness of a material.

Applications

· It is used in technology and medicine.

· The modulus of elasticity can be used to calculate how much a material stretches and how much potential energy is stored.

· The modulus of elasticity allows you to determine how a particular material reacts to stress.

· Elastic modulus is also used to characterize biological materials such as cartilage and bone.

**Shear Modulus:**

The shear Modulus of elasticity is one of the measures of the mechanical properties of solids. Other elastic moduli are Young’s modulus and bulk modulus. The shear modulus of material gives us the ratio of shear stress to shear strain in a body.

Measured using the SI unit pascal or Pa.

The dimensional formula of shear modulus is M1L-1T-2.

It is denoted by G.

It can be used to explain how a material resists transverse deformations but this is practical for small deformations only, following which they are able to return to the original state. This is because large shearing forces lead to permanent deformations (no longer elastic body).

**Modulus of Rigidity Formula:**

Where,

F is the force acting on the object.

A is the area on which the force is acting.

l is the initial length.

**Units:**

The modulus of rigidity is measured using the SI unit pascal or Pa.

Commonly it is expressed in terms of gigaPascal (GPa).

Alternatively, it is also expressed in pounds per square inch(PSI).

**Example of Modulus of Rigidity**

The following example will give you a clear understanding of how the shear modulus helps in defining the rigidity of any material.

Shear modulus of wood is 6.2×10^{8} Pa

Shear modulus of steel is 7.2×10^{10 }Pa

Thus, it implies that steel is a lot more (really a lot more) rigid than wood, around 127 times more.

**Bulk modulus:**

Bulk modulus is defined as the proportion of volumetric stress related to the volumetric strain of a specified material while the material deformation is within the elastic limit.

In simpler words, the bulk modulus is nothing but a numerical constant that is used to measure and describe the elastic properties of a solid or fluid when pressure is applied to all the surfaces.

The bulk modulus of elasticity is one of the measures of the mechanical properties of solids.

Other elastic modules include Young’s modulus and Shear modulus.

In any case, the bulk elastic properties of a material are used to determine how much it will compress under a given amount of external pressure.

Here, it is important to find and note the ratio of the change in pressure to the fractional volume compression.

The value is denoted with a symbol of K, and it has the dimension of force per unit area.

It is expressed in the units per square inch (psi) in the English system and newtons per square meter (N/m^{2}) in the metric system.

in the units per square inch (psi) in the English system and newtons per square meter (N/m2) in the metric system

**Bulk Modulus of Elasticity Formula**

It is given by the ratio of pressure applied to the corresponding relative decrease in the volume of the material.

Mathematically, it is represented as follows:

B = ΔP /(ΔV/V)

Where:

B: Bulk modulus

ΔP: change of the pressure or force applied per unit area on the material

ΔV: change of the volume of the material due to the compression

V: Initial volume of the material in the units of in the English system and N/m^{2 }in the metric system..

**Use **

· Bulk modulus measures the incompressibility of a solid.

· Moreover, the higher the K-value of a material, the higher its incompressibility.

For example, the K-value of steel is 1.6×10^{11}N/m^{2} and the K-value of glass is

4 ×10^{10}N/m^{2}. Here, the K-value of steel is more than three times the K-value of glass. This means that glass is more compressible than steel.

· Young's modulus is usually used for solids, but in gases the value of K is different because they are very compressible.

· The Bulk Modulus concept is also used for liquids.

· Liquid and entrained air temperatures are two factors that are heavily regulated by the bulk modulus.

**Elastic Potential Energy Formula**

Elastic potential energy is the potential energy stored by stretching or compressing an elastic object by an external force such as the stretching of a spring. It is equal to the work done to stretch the spring which depends on the spring constant k and the distance stretched.

Elastic Potential Energy imageIn other words,

Force required to stretch the spring is directly proportional to its displacement. It is given as

F =kx

Where in,

k = spring constant

x = displacement

The Elastic Potential Energy Formula of the spring stretched is given as

Elastic Potential Energy Formula

Where,

P.E = elastic potential energy and it’s expressed in Joule.