Stresses and Strains - 2

Hooke's Law

According to this law, Stress in a material is directly proportional to Strain within elastic limit.

$\sigma =E\times\varepsilon $

           $\sigma=$ Stress
           $\varepsilon=$ Strain
           $E=$ Modulus of Elasticity

Hooke's law is valid for homogeneous isotropic and linearly elastic material.

Deformation of Load under Axial Load

Case 1. Bar of uniform section

$\delta =\frac{PL}{AE}$

Case 2. Stepped Bars

$\delta ={{\delta }_{AB}}+{{\delta }_{BC}}+{{\delta }_{CD}}$

$=\frac{{{N}_{1}}{{l}_{1}}}{{{A}_{1}}{{E}_{1}}}+\frac{{{N}_{2}}{{l}_{2}}}{{{A}_{2}}{{E}_{2}}}+\frac{{{N}_{3}}{{l}_{3}}}{{{A}_{3}}{{E}_{3}}}$

${{N}_{1}},{{N}_{2}},{{N}_{3}}=$ Resultant forces in the member 1, 2 and 3 respectively

Case 3. Tapered circular bar

$\delta =\frac{PL}{\frac{\pi }{4}E\times {{D}_{1}}{{D}_{2}}}$

Case 4. Tapered Rectangular bar

$\delta =\frac{PL}{E\times t(a-b)}{{\log }_{e}}\frac{a}{b}$

Case 5. Deformation due to self weight

$\delta =\frac{\gamma {{L}^{2}}}{2E}=\frac{WL}{2AE}$

$\gamma=$ unit weight of bar
$W=$ Total Weight of bar

Case 6. Deformation in a composite bar

${{\delta }_{1}}=\frac{{{P}_{1}}L}{{{A}_{1}}{{E}_{1}}}=\frac{{{P}_{2}}L}{{{A}_{2}}{{E}_{2}}}={{\delta }_{2}}$

Case 7. Deformation due to change in temperature

$\delta =L\times \alpha \times \Delta T$

       When both ends are fixed

${{\delta }_{R}}=\frac{RL}{AE}$

$Stress=E\times \alpha \times \Delta T$

Case 8. Deformation due to change in temperature in a fixed bar with one yielding support

$\delta =L\alpha \Delta T-\frac{RL}{AE}$

$Stress=\frac{E(L\alpha \Delta T-\delta )}{L}$

Poisson's Ratio

Whenever a homogeneous and isotropic material is under stress, the longitudinal side will elongate or contract and all the transverse sides will contract or elongate.
The ratio of lateral strain and longitudinal strain is known as Poisson's ratio.

${{\varepsilon }_{x}}=\frac{{{\sigma }_{x}}}{E}$        ${{\varepsilon }_{y}}=-\mu \frac{{{\sigma }_{x}}}{E}$        ${{\varepsilon }_{z}}=-\mu \frac{{{\sigma }_{x}}}{E}$

Possion's Ratio of Engineering Materials

Cork	$0$	Concrete	$0.1-0.2$
Aluminium	$0.33$	Steel	$0.27-0.3$
Perfectly Elastic Rubber	$0.5$	Metal	$0.25-0.4$
Cast Iron	$0.2-0.3$	Rubber	$0.45-0.5$

Modulus of Rigidity (G)

Modulus of rigidity or shearing modulus is the ratio of shear stress and shear strain.

Bulk Modulus (K)

Bulk Modulus is the ratio of Direct stress and Volumetric Strain.

Relation between E,G,K and $\mu$

$E=2G(1+\mu )$       $E=3K(1-2\mu )$

$\mu =\frac{3K-2G}{6K+2G}$       $E=\frac{9KG}{G+3K}$

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Stresses and Strains - 1

Whenever an external load is applied to a solid body, it deforms if the body is restrained from motion either fully or partially.
If the body is not restrained against the motion it may undergo displacements without a change in shape or size and these displacements are called rigid body displacements.

Stress is a resistance generated by the body to counteract the external forces applied to it. It has the same unit as pressure i.e., force/area.

Types of Stresses

1. Normal Stress
2. Shear Stress

Normal Stress can be further classified as

1. Axial Stress
2. Bearing Stress
3. Bending Stress

Axial Stress -  Load is applied along the axis of a body i.e., normal to the section.
                       It can be either tensile or compressive in nature.

Bearing Stress - It is a compressive stress generated when one body is supported on another.

Bending Stress - Bending tension and compression produces normal stress.

Shear Stress

It is the stress that acts parallel to the applied force. Normal stress acts perpendicular to the plane causing either tension or compression while shear stress acts tangentially, that's why it is also known as tangential stress.

                                          Source: Quora

It is of 2 types:

1. Direct Shear Stress
2. Indirect Shear Stress 

Direct Shear Stress - It is that shear stress which tries to cut the section.
Indirect Shear Stress - It arises due to either tension/ compression or torsion.

Shear Stress on opposite faces of a stressed element is equal in magnitude and opposite in nature.

                                                       Source: Javelin-Tech

Stress tensor - Stress is not a vector quantity because its results cannot be obtained by parallelogram law of vector addition. It is a mathematical quantity called tensor.

It is represented by,
                            $\sigma =\left( \begin{matrix} {{\sigma }_{xx}} & {{\tau }_{xy}} & {{\tau }_{xz}} \\ {{\tau }_{yx}} & {{\sigma }_{yy}} & {{\tau }_{yz}} \\ {{\tau }_{zx}} & {{\tau }_{zy}} & {{\sigma }_{zz}} \\ \end{matrix} \right)$

Only 6 components are required to define the condition of stress at a given point in a 3D system because of the complimentary shear stress concept.
And in a 2D system, only 3 stress components are required to define the condition of stress at a given point.

Normal Strain

It is the deformation per unit length.

                                                       $\varepsilon =\frac{\delta }{L}$

$\delta=$ Change in length
$L=$ Original Length

Stress-Strain Curve for Mild Steel

                                              Source: Quora

Failure Plane

Material	Tension Test	Torsion Test
Ductile	${{45}^{{}^\circ }}$	${{90}^{{}^\circ }}$
Brittle	${{90}^{{}^\circ }}$	${{45}^{{}^\circ }}$

Properties of Material

1. Elasticity: It is the property of a material by virtue of which, it returns to its original position after unloading.
   If the material is unloaded before the elastic limit has reached, it will follow the original curve otherwise it will have a residual strain and the unloading curve will be different from original curve.
   
   
   
   
   
2. Plasticity: It is a property of a material which describes the deformation of a solid material undergoing inelastic strain beyond the strain at the elastic limit.
3. Creep: It is a property by virtue of which a material undergoes additional deformation with the passage of time under sustained loading within elastic limit.
   
   
   
4. Relaxation: It is the decrease in stress under constant strain.
   
   
5. Fatigue: It is a deterioration of a material caused by repeatedly applied loads results in progressive and localised structural damage.
   
   
   
   
   Endurance limit is the stress level below which even large number of stress cycle cannot produce fatigue failure.
   For structural stress, it is half of the .ultimate strength and due to corrosion, endurance limit further reduced by 50%
6. Resilience: It is a property by which a material absorbs energy when it is deformed elastically and release that energy upon unloading.
   
   
                                       Source: NPTEL
   The area under the load-deformation curve within elastic limit is called resilience.
   
7. Toughness: It is the ability of a material to absorb mechanical energy upto failure.
   
   
   
                                       Source: NPTEL


8. Tenacity: It is a property of a metal to resist fracture under the action of tensile load.

Standard Stress-Strain Curves

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