Note on Elasticity, ductility, tensile strength

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Elastic deformation

What is elasticity?

Generally we assume that elastic deformation is time independent i.e. applied stress instantaneously produce strain. However in reality elastic deformation taken time i.e. there is finite rate of atomic or molecular separation in the material after the application of loads or amount of loads. The deformation is function of time dependent elastic behavior of material is known as elasticity. This behavior of material is known as an elasticity. This behavior or an elasticity is small for metals but it is significant for polymer.

What is poisson’s ratio? Write the value of theoretical poisson’s ratio for isotropic material.

Materials when subjected to tension or longitudinal stress there is decrease in lateral dimension (shrink). When compressive stress is applied to material, the lateral dimension of material increases (Bulge), whereas linear linear dimension decreases. To account this behavior we define a physical quantity which is the ratio of lateral strain to axial strain (longitudinal strain). It is called Poisson’s ratio. it is denoted by \(\gamma\). Here, \(\gamma\) is unit less and dimensionless quantity. The sign of lateral strain and axial strain are opposite.

fig: Poisson's ratio
fig: Poisson's ratio(en.wikipedia.org)

Lateral strain=\(\frac{d_i-d_\circ}{d_\circ}\)

Axial strain=\(\frac{l_i-l_\circ}{l_\circ}\)

Where,\(l_i\) & \(d_i\)=instantaneous length and diameter

\(l_\circ $ d_\circ\)=original length and diameter

Theoretical value of Poisson’s ratio for isotropic material is lies in between \(\frac{1}{4}\) to \(\frac{1}{2}\).

Experimental value of Poisson’s ratio lies in between 0.24 to 0.3. The relationship between Young’s modulus of elasticity, shear modulus of elasticity and Poisson’s ratio is given by,

$$E=2G(1+\gamma)\dotsm(1)$$

Define shear modulus and write the limiting value of shear modulus interms of Young’s modulus for isotropic material.

Shear modulus:

It is the ratio of shearing stress to shear strain. It is denoted by G.

\(\therefore G=\frac{shear stress}{shear strain}=\frac{\tau}{\gamma}\)

Where, \(\tau\)=shear stress=\(\frac{F_\parallel}{A_\circ}=\frac{tangential load}{original cross-section area}\)

Shear strain=tan\(\phi=\frac{\Delta L}{L}\simeq \phi\)

fig: shear modulus
fig: shear modulus(www.spaceflight.esa.int)

For isotropic material \(\gamma\)=0.25 $ E=2G(1+\(\gamma\))

$$\therefore G=\frac{0.5}{1+\gamma}E=\frac{0.5}{1+0.25}$$

$$G=0.4 E$$

Lower and upper yield point (atomic view point of stress-strain behavior)

fig: yield point
fig: yield point(physics.stackexchange.com)

In most of the structure the elastic deformation of materials will reverse when stress is applied. A structure or component that has plastically deform or experiences a permanent change in shape cannot regain its original shape. The value of stress at which plastic deformation starts is known as yield point. In some materials (low carbon steel) the strain-stress curves includes two yield points as shown in the diagram. The two yield point is due to atomic vibration during the transformation from elastic to plastic behavior. The yield strain is defined in this case as the average stress at the lower yield point.

What do you mean by tensile strength?

It is the value of stress after which the material will eventually break of this amount or stress is continuously applied.

fig: tensile strength
fig: tensile strength

For structural application the yield stress is usually more important property than tensile strength. Since, the yield stress has passed; the structure has deformed beyond structural limit or acceptable limits. In this case it cannot regain original shape and likely to be deform permanently.

What is ductility?

It is the measure of deformation at fracture. It is important mechanical property of material. It is measured in terms of percentage elongation upon fracture is called brittle. The material which has longer portion of strain-stress curve is ductile. Therefore,

\(\therefore\) ductility = % elongation fracture

=\(\frac{l_f-l_\circ}{l_\circ}\times\)100%

Where,

\(l_f\)=length of material of fracture

\(l_\circ\)=original length of material

It is also measured in terms of percentage reduction in area and given by,

% RA (reduction in area)=\(\frac{A_\circ-A_f}{A_\circ}\times\)100%

\(A_f\)=area of cross section ductility fraction

\(A_\circ\)=original area from section

The yield strength, tensile strength and modulus of elasticity, all decrease with increase in temperature whereas, ductility increase with increase in temperature.

References:

Callister, W.D and D.G Rethwisch. Material Science and Engineering. 2nd. New Delhi: Wiley India, 2014.

Lindsay, S.M. Introduction of Nanoscience . New York : Oxford University Press, 2010.

Patton, W.J. Materials in industry . New Delhi : Prentice hall of India, 1975.

Poole, C.P. and F.J. Owens. Introduction To Nanotechnology. New Delhi: Wiley India , 2006.

Raghavan, V. Material Science and Engineering. 4th . New Delhi: Pretence-Hall of India, 2003.

Tiley, R.J.D. Understanding solids: The science of Materials. Engalnd : John wiley & Sons , 2004.

1. important equations

Lateral strain=\(\frac{d_i-d_\circ}{d_\circ}\)

Axial strain=\(\frac{l_i-l_\circ}{l_\circ}\)

\(G=\frac{shear stress}{shear strain}=\frac{\tau}{\gamma}\)

\(ductility = % elongation fracture

=\(\frac{l_f-l_\circ}{l_\circ}\times\)100%

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