### Concept of stress and strain

The mechanical properties of metals are directly related to the answer of the question how do metals responds to external loads. To understand how materials deform i.e. elongate, compress, twist or break due to applied load temperature and duration of load. There are four standard test method for the study of the mechanical properties of metal. The four different types of test are:

1. Tensile test
2. Compression
3. Shear
4. Torsion

The different types of test are related to following diagram

To compare specimen of different size the load is calculated per unit area, where is called stress. Now engineering stress is defined as load applied perpendicular to area of cross section per unit original area of cross section. It is denoted by $$\sigma$$.

Engineering stress=$$\sigma$$=$$\frac{F}{A_\circ}$$

Where,

F=load applied perpendicular to $$A_\circ$$

$$A_\circ$$=original area cross section

#### Engineering strain:-

It is defined as the change in length per unit original length due to stress. It is denoted by $$\epsilon$$.

Engineering strain=$$\epsilon$$=$$\frac{\Delta L}{L_\circ}$$

Sometimes engineering strain is also expressed in percentage(%) in this case, it is called percentage strain.

Percentage strain(5)=$$\frac{\Delta L}{L_\circ}\times$$100%

Strain and stress are taken as positive for tensile loads and taken as negative for compressive loads.

#### Shearing stress:

It is defined as the tangential load (parallel force to surface) per unit original area of cross section. It is denoted by $$\tau$$.

$$\therefore$$ shear stress=$$\tau$$=$$\frac{F}{A_\circ}$$

Where,

F= applied load parallel to surface

$$A_\circ$$= original cross section

#### Shear strain:-

It is defined as angular displacement of topmost layer due to shear stress with respect fixed bottom most layer of object. It is denoted by gamma ($$\gamma$$) and is given as,

$$\gamma$$=tan$$\phi$$=$$\frac{\Delta L}{L_\circ}$$

For small $$\phi$$, $$\gamma=\phi=\frac{\Delta L}{L_\circ}$$

Q.What is torsion?

Torsion is the variation of pure shear. The shear stress in this case is a function of applied torque. Shear strain is related to angle of twist.

Angle of shear =$$\phi=\frac{\Delta L}{L}\Rightarrow \Delta L=L. \phi \dotsm(1)$$

Angle of twist=$$\theta=\frac{\Delta L}{r}\dotsm(2)$$

From (1) and (2),

$$L. \phi =r.\theta$$

$$\theta=\frac{L}{r}.\phi \dotsm(3)$$

Twist is related to shear.

Unit of stress:

Stress is always measure in N/$$m^2$$ or pascal. And, strain is unitless.

#### Geometric Considerations of the Stress State

The computed stresses from the tensile, compressive, shear and torsional force states represented in above. In which their act either parallel or perpendicular to planar faces of the bodies. The stress state is a function of the orientations of the planes upon which the stresses are taken to act. For example, consider the cylindrical tensile specimen that is subjected to a tensile stress applied parallel to its axis. Furthermore, consider also the plane p-p’ that is oriented at some arbitrary angle relative to the plane of the specimen end-face. Upon this plane p-p’, the applied stress is no longer a pure tensile one. Rather, a more complex stress state is present that consists of a tensile (or normal) stress that acts normal to the p-p’ plane and, in addition, a shear stress that acts parallel to this plane; both of these stresses are represented in the figure. By using mechanics of materials principles, it is possible to introduce the equations for and in terms of$$\sigma’$$ and $$\tau’$$,as follows

$$\sigma’=\sigma cos^2 \theta=\sigma\biggl(\frac{1+cos2\theta}{2}\biggr)$$

$$\tau’=\sigma sin\theta cos\theta=\sigma \biggl(\frac{sin2\theta}{2}\biggr)$$

These same principles help to transform of stress components from one coordinate system to another coordinate system that has a different orientation.

#### 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. All mechanical properties:

torsion

shear

compression

tensile

2. Important equations

Engineering strain=$$\epsilon$$=$$\frac{\Delta L}{L_\circ}$$

Percentage strain(5)=$$\frac{\Delta L}{L_\circ}\times$$100%

$$\therefore$$ shear stress=$$\tau$$=$$\frac{F}{A_\circ}$$

$$\gamma$$=tan$$\phi$$=$$\frac{\Delta L}{L_\circ}$$

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