5. Hydrostatic and Deviatoric Parts in Stress Tensor

We can additively decompose the stress tensor into hydrostatic and deviatoric parts. Consider adding and subtracting $I (\sigma)[I]$ to a general stress tensor $\sigma$, i.e.,

The stress tensor can be decomposed into two components: the hydrostatic part and the deviatoric part. This is done by adding and subtracting a specific term from the stress tensor. The term being added and subtracted is related to the average stress, represented as $I (\sigma)[I]$, which is multiplied by the identity tensor. This transformation helps in simplifying the analysis of stress.

Think of a balloon filled with air. The pressure inside the balloon can cause it to expand uniformly (hydrostatic pressure), but if you poke the balloon with a stick, it will deform locally while the overall volume stays the same (deviatoric stress). The balloon's uniform internal pressure represents the hydrostatic component, while the localized deformation from poking represents the deviatoric component.

Let us denote the part within the parentheses as $\hat{Q}$. This is a special kind of decomposition because the first term is proportional to the identity tensor, whereas the second term is such that its first invariant is 0 which can be proved as follows:

In this decomposition, the part $\hat{Q}$ represents the deviatoric stress, which is characterized by having a first invariant of zero. The first invariant is a measure of the volume change of the stress matrix. Since $\hat{Q}$ leads to no volume change, we can say it has only shear stress effects, leading to deformations without a change in volume.

Imagine kneading a piece of dough. As you knead, the dough stretches and changes shape, but the amount of dough remains constant (no volume change), illustrating the concept of deviatoric stress.

Here, we used the fact that the first invariant of an identity tensor equals three. Due to this, $\hat{Q}$ is also called the deviatoric part of the stress tensor whereas the first term that is proportional to the identity tensor is called the hydrostatic part of the stress tensor.

The identity tensor inherently has the property that its first invariant equals three. This means that it contributes equally to all three dimensions of the stress matrix, leading to isotropic stress, which is the hydrostatic component. On the other hand, the deviatoric stress matrix is designed to have a zero first invariant, indicating it causes shear deformation without changing the volume.

Picture a cube of ice being pressed down on all sides uniformly by hands. The overall size of the cube does not change (hydrostatic part), but if you twist it with one hand while keeping the other hand still, the ice deforms (deviatoric part).

There always exists a coordinate system such that the matrix form of the deviatoric part has all its diagonal entries zero, i.e., $\hat{Q}$ has the following matrix form:

The representation of the deviatoric part in a specific coordinate system (often referred to as the principal coordinate system) allows it to have a matrix form with zeroes on the diagonal. This indicates that there is no normal stress on these planes, and the stress is purely shear. This is essential for understanding how materials deform under shear.

Imagine sliding a deck of cards across a table. The movement is shear, and there is no pressure pushing the cards upwards or downwards; you're just shearing them sideways, analogous to the pure shear state described by the deviatoric part.

The hydrostatic part has only got normal traction acting on the faces of the cuboid. This normal traction is the same on all the faces and so, it just tries to change the size of the cuboid without distorting its shape.

The hydrostatic part of the stress tensor is responsible for isotropic pressure, meaning it applies uniformly in all directions. This uniformity leads to a change in volume without causing any shear deformations in the material. It's a critical concept in understanding how fluids behave under pressure.

Think of a liquid-filled balloon. When you squeeze it, the pressure exerted is uniform across all sides, increasing the internal pressure without changing the balloon’s shape significantly. This represents the hydrostatic stress state.

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