2. Linear Stress-Strain Relation

Our goal is to express stress in terms of strain, i.e. σ = σ(ϵ). As stress and strain tensors are both symmetric, they have six independent components each. In general, each of the stress components will depend on each of the strain components.

In this section, we aim to understand how stress relates to strain mathematically. We denote stress by σ and strain by ϵ. Both stress and strain are represented as tensors, which are mathematical objects that can encompass multiple dimensions. Each tensor has components that describe different directions and magnitudes of stress and strain. Since both tensors are symmetric in nature, there are six unique components for each, meaning that not all stress components are independent of one another. This interdependence forms the basis of how we study materials under stress.

Imagine applying pressure to a sponge. When you push on one side, the entire sponge changes shape, and the amount of compression you apply relates directly to how much it will deform. The stress (the force you apply) influences the strain (how much the sponge squishes). Each small change in pressure results in a measurable change in size, similar to how the different components of stress influence the different components of strain in materials.

The functional form of this dependence can be anything (e.g., linear, exponential, logarithmic, etc.) which will vary from material to material. There is no physical law to derive the exact functional form, and therefore mechanicians resort to experiments on real material specimens and data fitting techniques to obtain these relations.

The relationship between stress and strain is not fixed – it can take various mathematical forms depending on the material being tested. For example, metals often exhibit linear relationships (as in Hooke's Law), while rubber might show a more complex relationship that isn't linear at all. Because these relationships can differ widely and can't be deduced purely from theory, researchers rely on experimental methods to observe how materials behave under various loads and to determine the precise nature of the stress-strain relationship.

Think about different types of elastic bands. Some are very stretchy, while others snap back quickly. To understand how much stretch occurs under a certain load, you would need to perform tests on each band. By measuring how much you pull and how much they stretch, you create a custom equation for each type – much like engineers do with materials to derive their stress-strain relationships.

The Taylor’s expansion of σ with respect to strain components will be as follows: All the derivatives have been evaluated at zero strain state, i.e., ϵ = 0. Here, all the indices run from 1 to 3. The first term (denoted as σ0) is called the residual stress and gives the stress in the zero strain state or the reference configuration.

Taylor's expansion allows us to express a function (in this case, the stress σ) as a series that approximates the value near a point – in this context, the point where strain is zero. The first term of this expansion represents the residual stress, which is the stress present when there is no deformation. The remaining terms quantify how much stress will change as strain increases. However, because we usually consider only very small strains in engineering problems, we primarily focus on the linear part of this expansion and neglect the higher-order terms that would have minimal impact.

Consider a rubber band again. When not stretched, it has some internal stress (residual stress). As you begin to stretch it, the amount of force you need to apply increases in a predictable way, which we can describe with a simple formula. This approximation simplifies calculations for engineers as they design products, ensuring they only consider relevant stress changes for tiny amounts of stretch.

Thus we are left with σ = C : ϵ. This is called linear stress-strain relation. We again remark that the derivatives in the equation are to be evaluated at the reference configuration. These derivatives have a special significance in solid mechanics: they denote the components of the stiffness tensor.

The final linear stress-strain relation is represented by σ = C : ϵ, meaning stress can be defined as a linear function of strain through a stiffness tensor C. This tensor represents the material's intrinsic response to deformation, encapsulating how stress and strain are related. Essentially, the equation states that stress is proportional to strain, with the stiffness tensor acting as the proportionality factor. Understanding this relationship is fundamental in solid mechanics because it allows engineers to predict how materials will respond under various loads.

Think of a fitted spring in a mattress. The stiffness of the spring determines how much weight it can support without deforming beyond a certain point. Similarly, in the stress-strain equation, the 'stiffness' of the material defines its resistance to deformation when a force is applied. A stronger spring (or a material with a higher stiffness value) reacts less to the same weight than a weaker one, analogous to how different materials behave under pressure.

We can observe that C has 34 = 81 terms since all the four indices go from 1 to 3. These 81 constants are called material constants which vary from material to material. However, it turns out that the 81 constants are not all independent.

The stiffness tensor C is a complex entity as it comprises 81 components due to its four indices each ranging from 1 to 3. This indicates a high level of interdependence due to the material's symmetric behavior in three dimensions. However, not all these 81 components are independent; symmetries reduce the number of unique equations we must solve when applying this tensor to practical problems. This streamlined approach is essential for simplifying calculations in materials science.

Visualize a solid cube made of rubber. While you might think of the cube having countless individual stresses at different points, in reality, many are correlated due to the material's nature. Just like echoing voices in a room, not every voice is separate; instead, they harmonize. Similarly, applying load to the rubber cube generates stress that uniformly interacts throughout, allowing you to reduce complex calculations into more manageable ones.

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