1.2.3. Shear Modulus (G)

From equation(16), we can see that shear modulus is given by

$G = \frac{\tau}{\gamma}$

So, if we can induce shear in a body and measure the corresponding shear stress, the ratio of stress to strain will give us the shear modulus.

Shear modulus (G) is a measure of a material's ability to resist shear deformation. It is calculated as the ratio of shear stress ($\tau$) to the shear strain ($\gamma$). When a shear force is applied to a material, it deforms, and the ability of that material to resist this deformation is captured by the shear modulus. The equation states that G is equal to the shear stress divided by the shear strain.

Think about a deck of cards. If you push the top card sideways while holding the bottom card in place, the top card experiences shear stress. How 'stiff' the cards are to this push reflects their shear modulus. If they easily slide, the shear modulus is low; if they resist sliding, the shear modulus is high.

Let us conduct another experiment with the rectangular bar. The bar is clamped at the bottom face and we apply shear force on the top face (face with normal along e2) as shown in Figure 5. The force is along e1 direction.

In this experiment, a rectangular bar is fixed at one end while a shear force is applied at the other end. By applying the shear force, we create a shear stress in the material. This setup allows us to measure how much the material deforms or shears due to that stress, which is essential to calculate the shear modulus.

Imagine trying to push the top of a stack of books sideways while keeping the bottom of the stack fixed. The top book will slide over the others, showing how the stack resists shear forces. Measuring how much the top book moves (the shear strain) in relation to how hard you push (the shear stress) helps us determine the shear modulus of the books.

If we draw the plot for τ vs γ, we get a curve as shown in Figure 6. The initial slope of this curve will give us the shear modulus, i.e.,

$G = \frac{d\tau}{d\gamma}$

The initial slope is measured because the linear relations are valid only for small shear strains.

The relationship between shear stress and shear strain can be visualized on a graph where shear stress (τ) is on the y-axis and shear strain (γ) is on the x-axis. The initial slope of this curve represents the shear modulus. At very small strains, materials tend to behave in a linear manner, meaning if you double the stress, the strain also doubles, which makes the calculation of G straightforward.

Consider a rubber band. If you slowly stretch it (small strains), you can see that it stretches proportionally to the force you apply. However, if you stretch it too much, it can snap and no longer obey this linear relationship. The initial slope of the stretch (when it's not too far stretched) correlates to its shear modulus.

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