Elasticity & Safety: From Hooke’s Law to Smart Design
What would happen if we couldn't predict how much weight a bridge could support? We would either be unable to build it at all or, worse, it could collapse under heavy traffic. To prevent such disasters, we must be able to quantify and predict how materials react to external forces. This process is mathematically formulated through Hooke’s Law.
Modeling Normal Stress (Axial Loading)
Normal stress occurs when a force acts perpendicular to a material's cross-section, causing it to stretch or compress.
$$\sigma = E\epsilon$$
\(\sigma\) (Normal Stress): The intensity of internal force per unit area.
$E$ (Young’s Modulus): A fundamental material property representing its resistance to elastic deformation.
\(\epsilon\) (Normal Strain): The ratio of deformation relative to the original length.
Modeling Shear Stress (Torsional Loading)
While normal stress involves stretching, shear stress involves sliding or twisting forces acting parallel to the surface.
$$\tau = G\gamma$$
\(\tau\) (Shear Stress): The force per unit area acting parallel to the surface.
$G$ (Shear Modulus): The material's resistance to shape change, also known as rigidity.
\(\gamma\) (Shear Strain): A measure of the angular deformation (how much the object has been tilted or twisted).
Poisson's Ratio (\(\nu\))
Next is Poisson's ratio(\(\nu\)). It serves as a fundamental measure of volumetric change and is a key parameter in determining a material's stiffness. The following formulas define the strain ratios under tensile loading, where the material is elongated along its axis:
Sectional Strain (\(\epsilon_{area}\))
This represents the ratio of the change in cross-sectional area to the original area.
ϵarea=ΔAA≈−2νϵ (where ϵ>0 denotes the axial tensile strain)
Volumetric Strain (\(\epsilon_v\))
This is the ratio of the change in volume to the original volume. A Poisson's ratio of 0.5 implies that the material is incompressible, resulting in zero volumetric strain.
$$\epsilon_v = \frac{\Delta V}{V} = \epsilon (1 - 2\nu)$$
Shear Modulus ($G$)
The Shear Modulus relates the shear stress to the shear strain.
$$G = \frac{E}{2(1+\nu)}$$
Bulk Modulus ($K$)
The Bulk Modulus measures a substance's resistance to uniform compression.
$$K = \frac{E}{3(1-2\nu)}$$
The graph below illustrates the variation of G and K with respect to Poisson’s ratio using MATLAB.
Summary
In this post, we explored Hooke’s Law and Poisson’s ratio, specifically focusing on how $G$ and $K$ each relate to Poisson’s ratio.