Mechanical Properties of Solids

Rigid Body Rigid Body: In physics, a rigid body is an idealization of a solid body in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it. But in reality, bodies can be stretched, compressed and bent. This means that solid bodies are not perfectly rigid.

Forces Concerned Deforming force:- A force acting on a body which produces change in its shape of body instead of its state of rest or uniform motion of the body. Body does not accelerate.  

Elasticity and Plasticity The property of a body, by virtue of which it tends to regain its original size and shape when the applied force is removed, is known as elasticity and the deformation caused is known as elastic deformation.eg Rubber Bands. If you apply force to a lump of putty or mud, they have no gross tendency to regain their previous shape, and they get permanently deformed. Such substances are called plastic and this property is called Plasticity. Putty and mud are close to ideal plastics. Ideal Elastic Materials are difficult (impossible) but mud and putty are very nearly ideal Plastic materials.  

Elastic Materials

Plastic materials

Reason for Elasticity and Plasticity Spring Ball Analogy For solids all the atoms are bonded to each other by inter-atomic (or inter-molecular) forces can be assumed as they are connected by springs. Normally each atom vibrates slowly about its equilibrium position. When a force is applied the on the solid these interatomic (intermolecular) distances change and exert a restoring force just like a spring. This restoring force drives the body to regain its original shape. This sums up for Elastic properties of matter. But as nothing is ideal there remains a small deformation and is the reason for plasticity.

Spring Ball Analogy for Solids

Stress Restoring force per unit area developed inside the body, when deforming force acts it is called Stress. S.I Unit of stress = N/m2 or Pascal (Pa)  Types of stress:- 1)Normal Stress and 2) Tangential Stress

Compressive Stress: when there is decrease in dimension Tangential Stress: When deforming force acts tangential to the surface of body. Causes Shear Strain.   Normal Stress: Compressive Stress: when there is decrease in dimension Tensile Stress: there is increase in dimension Compressive normal Stress Tensile Normal Stress C)Shear Stress

Strain Strain:- The ratio of change in dimension to the original dimension is called strain OR Fractional change in dimension is called Strain. Unit: Unitless Strain = Change in Dimension/Original Dimension

Types of Strain Longitudinal Strain: Change in Length/Original Length Shear Strain: Displacement of Face/ Height of face (It is the tangent of the angle by which the body gets distorted due to shear stress) Volumetric Strain: Change in Volume/Original volume

Hooke’s Law For small deformations the stress and strain are proportional to each other. This is known as Hooke’s law. Thus, Stress/strain = k where k is the proportionality constant and is known as modulus of elasticity. Hooke’s law is an empirical law and is found to be valid for most materials. However, there are some materials which do not exhibit this linear relationship

Modulus Of Elasticity: Young’s Modulus (Y) = Longitudinal Stress/Longitudinal Strain Modulus of rigidity or Shear Modulus (ƞ) = Tangential Stress/Shear Strain Bulk Modulus (K) = Normal Stress/ Volumetric Strain

Modulus Of Elasticity: Compressibility : The reciprocal of bulk modulus of a material is called its compressibility. Compressibility = 1/K

Stress-Strain Curve( for a metal)

Proportionality limit(A) – The stress at the limit of proportionality point P is known as proportionality limit. Elastic limit or Yield point - the point till which force can be applied to a wire so that on unloading it return to its original length is called the elastic limit. The corresponding Stress in called yield strength (σy) of the material.

Beyond this point any increase in stress will cause much greater increase in strain and also even if all forces are removed the body will not regain its original shape. There will always be a permanent deformation after Yield point and it is called Permanent Set.

Ultimate Tensile Strength(UTS) (Point D) is the Maximum Stress that the material can take without breaking. Beyond this point even a reduced stress can cause Fracture.

Point E is the Fracture point. If point D and point E are close the material is called Brittle and if they are close the material is called Ductile. If force is removed at point D it does not break.

FAQ: What if Force is greater than what Force is at UTS? The material snaps without warning. FAQ: Why does the material break at a stress lower than the Ultimate Tensile Stress? In this experiment we exert force and divide by initial area to find stress. But as the length increase and matter cant be created the area decrease.

So, actually the Stress at Breaking point is greater than Ultimate Tensile Stress but we plot graph using area of initial condition as finding area at each point is a tough job. So the original stress always increases till the breaking point.

Poisson’s ratio The ratio of lateral strain to longitudinal strain is called Poisson’s ratio. Poisons ratio = Fractional change in area per unit Fractional change in Length Poisson’s ration = (ΔArea/Area) / (Δlength/Length)

Applications of elasticity 1. Metallic part of machinery is never subjected to a stress beyond the elastic limit of material. 2. Metallic rope used in cranes to lift heavy weight are decided on the elastic limit of material. 3. In designing beam to support load (in construction of roofs and bridges) 4. Preference of hollow shaft than solid shaft 5. Calculating the maximum height of a mountain