Constitutive Relation

For a viscous material, the Cauchy stress tensor depends not only on the deformation gradient but also on its time derivative. That is

$$\mathbf{T} = \mathbf{T}(\mathbf{F},\dot\mathbf{F}).$$ (6.1.1)

Most fluids are isotropic and homogeneous; an exception being a liquid crystal which is anisotropic. Here we will study isotropic fluids only. For an isotropic fluid, equation (6.1.1) reduces to

$$\mathbf{T} = \mathbf{T}(\rho, \mathbf{D}),$$ (6.1.2)

and for a linear viscous fluid, we have

$$\mathbf{T} = -p(\rho )\mathbf{1} + \lambda (tr\, \mathbf{D})\mathbf{1} + 2\mu\mathbf{D}.$$ (6.1.3)

Here $p$ is the hydrostatic pressure, $\lambda$ the bulk viscosity and $\mu$ the shear viscosity. The word linear in linear viscous fluid signifies that the viscous part of the stress (the last two terms on the right-hand side of (6.1.3)) depends linearly upon the strain-rate tensor $\mathbf{D}$. A perfect fluid or an ideal gas has no viscosity, and therefore can be described by the constitutive relation

$$\mathbf{T} = - p(\rho )\mathbf{1}.$$ (6.1.4)

Thus a perfect fluid is a nonlinear elastic material in the sense that the Cauchy stress for it depends only on the present value of the deformation gradient. For an ideal gas,

$$p = RT\rho$$ (6.1.5)

where $R$ is the universal gas constant and $T$ is the temperature in degrees Kelvin. Thus for deformations of an ideal gas at a constant temperature

$$\mathbf{T} =\ \, \rho \mathbf{1}$$ (6.1.6)

and the value of the const. depends upon the gas and its temperature.

Equation (6.1.3) for $\mathbf{D} = \mathbf{0}$ gives

$$\mathbf{T} = - p(\rho )\mathbf{1}.$$ (6.1.7)

Hence in a fluid at rest, the state of stress is a hydrostatic pressure. Note that the state of stress in a perfect fluid or an ideal gas is always that of hydrostatic pressure whether or not it is being deformed. However, in a viscous fluid, no shear stress exists if and only if it is at rest. This is sometimes taken as the definition of a viscous fluid; viz. a viscous fluid at rest can not support any shear stresses. On the other hand, a solid body when subjected to shear or tangential tractions can stay stationary.

For a homogeneous fluid the viscosities $\lambda$ and $\mu$ are constants. Equation (6.1.3) implies that the principal axes of the stretching tensor or the strain-rate tensor $\mathbf{D}$ coincide with the principal axes of the stress tensor $\mathbf{T}$.

Whereas the constitutive relation (5.2.1) or (5.2.2) for a linear elastic material describes well its infinitesimal deformations, no such restriction is imposed on (6.1.3). Said differently, equation (6.1.3) describes deformations of a viscous fluid for all values of the stretching tensor $\mathbf{D}$. The material characterized by equation (6.1.3) is called a Navier-Stokes fluid. Usually fluids and gases are assumed to be incompressible. The constitutive relations for an incompressible ideal gas and an incompressible Navier-Stokes fluid are respectively

$$\mathbf{T} =\ -p\mathbf{1},$$ (6.1.8)

$$\mathbf{T} =\ -p\mathbf{1} + 2\mu \mathbf{D},$$ (6.1.9)

where the hydrostatic pressure $p$ is not determined by the deformation of the fluid. It is, however, determined by the boundary conditions. Even for a homogeneous fluid, the pressure $p$ is a function of the spatial coordinate $\mathbf{x}$ and time $t$. In (6.1.9) we have used the continuity condition, $tr\, \mathbf{D} = 0$, for an incompressible fluid.