1.1. Stress-Strain relation

Once we workout the stress-strain relation using a rigorous mathematical derivation, we get (7)

The stress-strain relation describes how a material deforms (strain) in response to an applied force (stress). In this context, a mathematical equation has been derived that relates stress and strain for isotropic materials, which typically have uniform properties in all directions.

Think of a rubber band. When you pull it, the rubber band stretches (strain) proportionally to the force you apply (stress). The relationship between the amount you stretch it and the force you use can be described by a specific equation.

Here (λ,µ) are called Lame’s constants and are the two material constants for an isotropic material. Let us consider the component where i=j=1, i.e., (8)

Lame's constants, λ (lambda) and µ (mu), are parameters that describe the elastic properties of an isotropic material. They help define the relationship between stress and strain mathematically. When we focus on a specific component of stress and strain (where the indices are equal), we can derive constants related to the material's stiffness.

Imagine pulling on different sides of a piece of dough. The dough has certain properties (stiffness) and will respond in predictable ways based on the forces you apply — Lame's constants help quantify this behavior.

Comparing with equation (1), we can see that C_1111 (the constant relating σ and ε) will be given by the coefficient of ε_11, i.e., C_1111 = λ + 2µ (9)

From the mathematical derivation, we identify that one of the coefficients (C_1111) in the stress-strain equation can be expressed in terms of Lame's constants. This relationship is important as it helps us understand how stress and strain are interrelated for isotropic materials.

Consider a sponge. The way it compresses and expands when you squeeze it is dictated by its stiffness. The quantities λ and µ are like specific instructions on how the sponge should behave under various forces.

Likewise, we can also deduce C_1122 and C_1133 as C_1122 = C_1133 = λ (10)

For isotropic materials, certain stress components can be shown to have the same coefficient. This gives us a simplified structure in the equations we use for analysis. Recognizing these relationships allows us to reduce the complexity of material properties we need to experimentally determine.

If you think about a loaf of bread, applying even pressure on top causes it to compress equally across its surface, akin to how the constants in stress components behave when referencing an isotropic material.

These are additional constraints for the coefficients of the stiffness tensor for isotropic materials. They are not covered by either minor or major symmetry. In fact, a rigorous analysis proves that there are several other constraints in this case all of which finally lead to only two independent constants for the stiffness tensor of isotropic materials.

In isotropic materials, a deeper analysis reveals that despite the complexities of three-dimensional stress and strain, we can ultimately reduce the number of constants we need to consider down to just two. This is a significant simplification and streamlining in the analysis of material properties.

Imagine you're calibrating a complex instrument. Initially, there are many settings, but with more thorough understanding and testing, you find that you only need to adjust two main knobs for accurate results.

This also means that for isotropic materials, we just have to do two experiments to obtain its material constants and then generate the complete stress-strain relation (for a general material, we will accordingly need to do 21 experiments).

The practical result of having only two independent material constants means that determining the full stress-strain relationship of isotropic materials is much more straightforward than for general materials. This efficiency is crucial for engineers and material scientists.

If you have a set of scales that can weigh items with only two weights, it will be much faster and easier than if you had to test every possible combination to find the right weight for each item.

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