Why do Newtonian fluids have a single viscosity constant for both shear and normal stresses, while solids have different constants for each?

Question

I noticed that in Newtonian fluids, stresses are proportional to deformation, whether caused by normal stress or shear stress, with the constant ratio being the viscosity coefficient. I don’t understand how this constant ratio arises, given that the nature of the stresses is different.

For example, if we consider two components of the stress tensor:

The $\textbf{shear component}$ is given by the relation:

$\tau_{xy} = \mu \cdot \gamma_{xy}$

where the $\textbf{strain}$ $\gamma_{xy}$ (shear strain) is given by:

$\gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}$

The $\textbf{normal component}$ is given by the relation:

$\sigma_{xx} = \mu \cdot \epsilon_{xx}$

where the $\textbf{strain}$ $\epsilon_{xx}$ (normal strain) is given by:

$\epsilon_{xx} = 2 \frac{\partial u}{\partial x}$

From these two relations, we observe that despite the difference in the physical nature of the components (shear stress caused by transverse velocity gradients, and normal stress caused by stretching or compression), both are proportional to the rate of deformation with the same coefficient, which is the viscosity.

In contrast, in solids, the proportionality constants differ depending on the nature of the stress. For example:

Tensile stress is proportional to strain via $\textbf{Young’s modulus}$ $E$, as given by the relation: $\sigma = E \cdot \epsilon$

Shear stress is proportional to shear strain via the $\textbf{shear modulus}$ $G$, as given by the relation: $\tau = G \cdot \gamma$

Clearly, $E$ is not equal to $G$.

---

$\textbf{My question}$: What is different in fluids compared to solids that allows us to deal with a single proportionality constant regardless of the nature of the stress?

Answer 1

It seems like you are missing the distinction between strain and strain rate. Strain is the amount that the material is deformed, while strain rate is how fast the material is deforming or the change in velocity vs distance perpendicular to the velocity.

It is not true "that in Newtonian fluids, stresses are proportional to deformation, whether caused by normal stress or shear stress". In a Newtonian fluid shear stress is proportional to the shear strain rate, not shear strain. That coefficient of proportionality is the viscosity. This means that a fluid can undergo unlimited deformation under a fixed shear stress, given sufficient time.

In contrast, a normal stress could in principle be considered proportional to the normal strain, with a constant of proportionality called the compressibility. This means that a fluid would undergo only a fixed deformation under a fixed normal stress, even with sufficient time.

A fluid's compressibility and its viscosity are not the same constant, and to my knowledge there is not a formula that relates them in general. Viscosity very specifically describes a fluid's behavior under shear stress, not under normal stress.

Answer 2

For solids, there are two independent constants, the Young's modulus E and the Poisson ratio $\nu$. All other mechanical properties are expressible in terms of these. For. example, the shear modulus G which is related to E and $\nu$ by $$G=\frac{E}{2(1+\nu)}$$In the case of a Newtonian fluid, there are also two independent constants, the shear viscosity $\mu$ and the volumetric viscosity $\lambda$. See Transport Phenomena by Bird et al, Chapter 1.