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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