Graph of brittle and ductile material
Graph of brittle and ductile material
The area of stress strain curve is more for ductile material then brittle material as shown in figure.
What do you mean by toughness of material?
It is the property of material which is measured in terms of energy absorbed by material up to frature which is equal to total area under the stressstrain curve up to fracture. Let \(\epsilon_f\) be strain up to fracture.
Toughness for brottle material
(Toughness)\(_{brittle}\)=\(\int^\epsilon_0f \sigma d\epsilon\)=area of OAB
(Toughness)\(_ {ductile}\)=\(\int^\epsilon_0f \sigma d\epsilon\)=Area of OMFD
The unit of toughness for particular material is J/\(m^3\).
Define true stressstrain.
True stress is defined as load per unit actual area in the neck down region. It is denoted by \(\sigma_T\) and given by,
$$\sigma_T=\frac{F}{A_i}\dotsm(1)$$
Where,
F=load applied of the cross section
\(A_i\)=instantaneous area in the neck down region
True stress continuous to rise to point of fracture in constract to engineering stress. True strain is defined as,
$$\epsilon_T=ln\biggl(\frac{l_i}{l_\circ}\biggr)$$Where,
\(l_i\)=instantaneous length during the fracture
\(l_\circ\)=original length
If the volume of the material remain unchanged during fracture, then,
$$A_il_i=A_\circ l_\circ$$
The relation between engineering stress and true stress is given by,
$$\sigma=\frac{F_i}{A_\circ}$$
$$\sigma=\frac{l_il_\circ}{l_\circ}$$
$$\sigma_T=\frac{F}{A_i}=\frac{F}{A_\circ l_\circ}$$
$$\sigma_T=\sigma(1+\epsilon)$$
$$\sigma_T=\sigma(1+\epsilon)\dotsm(2)$$
Relation between true strain and engineering strain,
$$\epsilon_T=ln(1+\epsilon)\dotsm(3)$$
Hardness of material
What do you mean by Hardness of material and how is it measure explain.
Hardness is the measure of material resistance to localized plastic deformed such as dent or scratch. A quantitative scale (Moh’s scale) is determined by ability of material to scratch another material. The value of hardness in this scale is 1 for test material and 10 for hardness material i.e. for diamond, 1 for talc.
Different type of quantitative hardness test has been designed:
 Rockwell hardness tests
 Brinell hardness tests
 Knoop and vicker’s micro indentation hardness
To measure the hardness of material a small indenter in the shape of sphere, cone or pyramid is forced into the surface of material under the controlled magnitude and rate of loading. The depth or size of indentation is measured as shown in diagram. The test is somewhat approximately but popular because they are easy not destructive except small hole and indentation is formed
Both tensile strength and hardness may be regarded or degree of resistance to plastic deformation. Hardness is proportional to tensile strain but the proportionality constant is different for different for different material.
Test:
Formula to measure hardness by Brinnel test,
$$H_B=\frac{2P}{\pi D(D\sqrt(D^2d^2)}$$
Hardness for vicker’s microhardness test:
Intender: diamond pyramid
Depth of indentation free diagonal view of pyramid=\(d_1\)
Load=P
$$H_v=\frac{1.854 P}{d_1^2}$$
what are the limits of shape deformation?
For practical engineering design yield strength is usually the important parameter.
In this case,
Design stress,\(\sigma_d=N\sigma_c\)
Where ,
\(\sigma_c\)=anticipated maximum stress
And , N=designed factor
For the safety of material design, the design stress should be less than yield strain so that it will not overcome the stress during working.
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: PretenceHall of India, 2003.
Tiley, R.J.D. Understanding solids: The science of Materials. Engalnd : John wiley & Sons , 2004.
1.(Toughness)\(_{brittle}\)=\(\int^\epsilon_0f \sigma d\epsilon\)=area of OAB
(Toughness)\(_ {ductile}\)=\(\int^\epsilon_0f \sigma d\epsilon\)=Area of OMFD
. 2.Other important equation
$$\sigma_T=\frac{F}{A_i}$$
$$\epsilon_T=ln(1+\epsilon)$$

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