1.2.1. Young’s Modulus (E)

We know that local longitudinal strain (ϵ_jj) is given by . If the elongation is uniform along the length of the bar, the local strain will be equal to the global strain. Thus:

longitudinal strain = (23)

Thus, we have:

(24)

The local longitudinal strain is a measure of how much a material stretches or compresses in the direction of the applied force. It is defined as the change in length of the material divided by its original length. If the elongation occurs uniformly, this local strain is the same as the global strain, meaning the entire length of the material stretches or compresses uniformly.

Imagine a rubber band. When you pull it at both ends, the length increases uniformly across the entire band. This change in length—how much it stretches in comparison to its original length—represents the longitudinal strain.

If we now draw the graph of σ vs ϵ for the above experiment by measuring the change in length, we will get a curve as shown in Figure 4. The initial slope of this graph (shown as the red dotted line) gives us the Young’s Modulus (E).

Thus, E is given by

(25)

The graph of stress (σ) versus strain (ϵ) shows how a material behaves under load. The slope of this graph at the point of origin (where both stress and strain are zero) is defined as Young's Modulus (E). It quantifies the stiffness of the material—higher values indicate a stiffer material that deforms less for the same amount of stress.

Think of a spring: if you pull a strong spring compared to a weak one, the strong spring stretches less than the weak one for the same amount of force. The Young’s modulus is like the spring constant, indicating how much the spring (or material) resists stretching.

While computing this derivative, we should not have σ₂₂ or σ₃₃ present. Essentially, the beam should be stretched in such a way that it can freely shrink in the lateral directions. We can also prove the above formula using equation (13). We can put the conditions of the experiment into this, i.e. σ₂₂ = 0 and σ₃₃ = 0. So, we will get:

(26)

When measuring Young's Modulus, it is crucial to ensure that no lateral (sideways) stresses are applied, as this could affect the measurement accuracy. Specifically, the material should only be stretched in one direction without any constraints that would inhibit its ability to contract in the perpendicular directions. The formula we derived earlier also supports this condition, confirming that those lateral stresses must be zero for accurate measurements.

Imagine stretching a soft dough. If you try to stretch it too much in one direction while pushing the sides, it will bulge outwards. To measure how much it stretches correctly, you must hold the sides firm, allowing it to only extend in the direction you are pulling.

On one thing to note here is that equation (13) gives us the slope of the σ vs ϵ plot to be a constant. But in reality, the slope is not constant as evident from Figure 4. This is because the linear stress-strain relation (13) works only for small strains. On the other hand, Figure 4 shows the curve even for larger strains, and this is the reason that we had computed the slope of this graph at ϵ = 0.

Theoretically, the relation between stress and strain is linear only for small deformations. However, as materials undergo larger strains, their behavior can become nonlinear, meaning that the same increase in stress results in different amounts of strain. Therefore, the slope of the stress-strain curve changes as the material is deformed beyond its elastic limit, making it essential to measure the slope at smaller deformations for calculating Young's Modulus.

Consider a rubber band again: when you stretch it a little, it returns to shape and behaves very predictably. But if you overstretch it, it could lose shape and no longer return to its original form, and it wouldn’t follow the same predictable path of behavior. This shows how materials can behave differently under different levels of strain.

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