2.1. Taylor’s expansion

The Taylor’s expansion of σ with respect to strain components will be as follows:

(8)

All the derivatives have been evaluated at zero strain state, i.e., ϵ = 0.

Taylor's expansion is a mathematical method to approximate a function. In this case, we are using Taylor's expansion to express stress (σ) in terms of strain components (ϵ). The formula provided indicates that at the zero strain state, the derivatives used in the equation give us the rates of change of stress with respect to strain. This is foundational in material science as it allows us to relate how materials deform under stress.

Think of Taylor's expansion like estimating the height of a roller coaster at various points. If you start from a point where you know the height (zero strain state), you can use the slopes (derivatives) at that point to predict the height at nearby points, just like using the behavior of stress at a zero strain state to predict how it will behave at small strains.

The first term (denoted as σ0) is called the residual stress and gives the stress in the zero strain state or the reference configuration. The reference configuration is typically chosen such that the stress in the body is zero in this state. Accordingly, we will assume σ0 to be zero for this course. The second term is linear in strain while the third term is quadratic in strain. We had also said earlier that for this course, we will work with very small strains only. Thus, we can neglect the terms which are quadratic or higher order in strain components as they are much smaller than the linear term. Thus, we are left with:

(9) This is called the linear stress-strain relation.

In this chunk, we discuss the residual stress, which represents the state of stress when there is no deformation (zero strain). For this course, we simplify our analysis by assuming this residual stress is zero. The stress can then be approximated using only the linear term based on small strains, which gives us a solid foundation for working with the linear stress-strain relation. By ignoring higher-order terms, we make our calculations simpler and more practical while ensuring significant accuracy for small deformations.

Imagine stretching a rubber band. When you first stretch it slightly, the relationship between how much you stretch it and the tension you feel is straightforward (linear). However, if you stretch it too much, the behavior becomes complex and non-linear. For small stretches (small strains), we only care about that simple relationship, making it easier to predict how the rubber band will behave.

These derivatives have a special significance in solid mechanics: they denote the components of the stiffness tensor. It is a fourth-order tensor because it has four indices all running from 1 to 3 and is denoted as C:

(10)

Thus, we can write:

(11)

The derivatives obtained from the Taylor expansion are crucial as they correspond to the stiffness tensor components. This tensor characterizes how a material resists deformation under stress. As a fourth-order tensor, it encompasses a wide range of material behaviors, indicating that the relationship between stress and strain is influenced by material properties and the direction of applied forces. Knowing these components allows engineers to predict and design materials that will perform well under specified conditions.

Think of the stiffness tensor like a fabric that holds its shape when stretched. Different materials (like cotton versus steel) have different stiffness properties. When you pull on the fabric, the way each type responds differs based on its stiffness tensor. Understanding that tensor helps you know how that fabric will behave in a quilt or a tent.

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