Theory of Elasticity

Lecture -01

Introduction

Theory of Elasticity

The property of solid materials to deform under the application of an external force and to regain their original shape after the force is removed is referred to as its elasticity.

The external force applied on a specified area is known as stress, while the amount of deformation is called the strain.

Stress – It is the internal resistance offered by the body over the area on which the load is acting.

·  tensile stress - stress that tends to stretch or lengthen the material - acts normal to the stressed area

σ = Fn / A

where

σ = normal stress ((Pa) N/m2, psi)

Fn = normal component force (N, lbf (alt. kips))

A = area (m2, in2)

·  compressive stress - stress that tends to compress or shorten the material - acts normal to the stressed area

σ = Fn / A

where

σ = normal stress ((Pa) N/m2, psi)

Fn = normal component force (N, lbf (alt. kips))

A = area (m2, in2)

·  shearing stress - stress that tends to shear the material - acts in plane to the stressed area at right-angles to compressive or tensile stress

τ = Fp / A (2)

where

τ = shear stress ((Pa) N/m2, psi)

Fp = parallel component force (N, lbf)

A = area (m2, in2)

Force distribution

Internal force – the internal force are the reactive force which are setup due to external force.

External forces- The state of stress and strain in a body arise due to external influences. The external force can be divided into three types i.e

·  Body force

·  Traction force

·  Point load

Body force ( f) – It is the distributed force acting on every elemental volume of the body.

Unit = force /unit volume (N/mm3)

Body force vector f is given by

f=fxfyfz= fxfyfzT

Where fx,fyfz are the body force components in x, y & z direction respectively.

Work potential

f=vol UTf.dv

Where UT={ u v w}

u, v & w are the displacement components in x, y & z direction respectively.

Traction force - It is the distributed force acting on on the surface of the body.

Unit = force /unit area (N/mm2)

Traction force vector is given by

T=TxTyTz= TxTyTzT

Where TxTyTz are the traction components in x, y & z direction respectively.

Work potential

T=S UTT.ds

Where UT={ u v w}

u, v & w are the displacement components in x, y & z direction respectively.

Point load - it is the force acting at a particular point which causes displacement.

force vector is given by

P=PxPyPz= PxPyPzT

Work potential

P=i=0nUi TPi

Total Work potential

WP=-vol UTf.dv +S UTT.ds +i=0nUi TPi

Note - work potential is negative since energy is supplied externally.

Department of mechanical engineering