Updated notes on nonlinear viscosity

I have just updated my notes on nonlinear viscosity.  The notes are written for a course on plasticity.  When I update the notes, I will post a link on my twitter account:  https://twitter.com/zhigangsuo.

Nonlinear viscosity.  A purely viscous fluid has no memory.  When the state of stress changes, the rate of deformation changes instantly, with no delay.  A model of viscosity specifies a relation between the state of stress and the rate of deformation.  Water is viscous: the state of stress is linear in the rate of deformation.

Ice is also viscous, but the relation between the state of stress and the rate of deformation is nonlinear. The flow of ice contributes to the dynamics of glaciers.  Nonlinear viscosity prevails in most materials.  Think of ice creams, skin creams, greases, toothpastes, chocolates, polymers, ceramics, and metals.

Why do we study nonlinear viscosity? We study nonlinear viscosity for its own sake, and for the insight into more complex rheological behavior.  Rheology is a study of unity, as well as diversity.  Viscosity, plasticity and viscoplasticity have the same microscopic origin:  atoms and molecules change neighbors.  The continuum theories of viscosity, plasticity and viscoplasticity have an identical structure (Prager 1961).  These theories must feed into the theories of more complex rheological behavior, such as elastoplasticity and viscoelasticity (Reiner 1945).

Nonlinear viscosity highlights, in purest forms, many salient features of rheology: deformation of arbitrary magnitude, dissipation of energy, use of invariants, use of convex functions, and concern over the uniqueness of solution.    

Homogeneous deformation of a small piece. Inhomogeneous deformation of a body.  In a test such as pure shear, a material flows by homogeneous deformation.  The test determines the relation between the shearing stress and the rate of shear, known as the flow curve.

The flow curve, together with the balance of forces and compatibility of deformation, governs the flow in a pipe, squeezing of a film, and lubrication between parts in a machine.  In these examples, the body of fluid flows by inhomogeneous deformation.  We regard the body as a sum of many small pieces.  Each small piece undergoes a homogeneous deformation.  In these examples, each small piece deforms by shear, just as that in the test of pure shear.  The shearing flows of the small pieces constitute the inhomogeneous flow of the body.

In a more complex flow, each small piece still undergoes a homogeneous deformation, but the rate of deformation is a tensor of all its components.  To analyze inhomogeneous deformation in general, we need a relation between the rate of deformation and arbitrary state of stress.

The second-invariant model of viscosity.  A particularly popular relation between the state of stress and the rate of deformation is the second-invariant model of viscosity, also known as the generalized Newton’s model of viscosity, or the J2 model.  The model assumes that the state of deviatoric stress equals a scalar times the rate of deformation, and that the scalar is a function of the second invariant of the rate of deformation.  The scalar function of a scalar is fixed by using a flow curve (i.e., a relation between the stress and the rate of deformation), measured by subject the material to a simple test, such as pure shear and uniaxial tension.  The model predicts all components of the rate of deformation under any state of stress, or the other way around. The second-invariant model readily accommodates viscoplasticity, i.e., the Bingham fluids.

Power-law creep.  Ilyushin theorem.  Very viscous fluids flow very slowly.  They creep.  Neglecting the effect of inertia, the balance of forces, as well as the compatibility of geometry, gives linear equations.  The only nonlinearity comes from the flow curve.  When the flow curve obeys the power law, creeping flows obey a scaling relation (Ilyushin 1946).

Theoretical properties of the second-invariant model.  The second-invariant model has several significant theoretical properties.  The model, introduced to plasticity by von Mises (1928), is a special case of the theory of dissipation function (Rayleigh 1871).  The model satisfies the thermodynamic inequality.  The model has a convex dissipation function and a convex flow potential.  (We briefly outline the mathematics of convex functions.)  The solution to the boundary-value problem of any creeping flow is unique (Hill 1956).  The boundary-value problems obey variational principles.