Shear Strain,Shear Stress, Shear Rate,Viscosity

Shear Strain:-

• To define the term STRAIN we will consider a cube of material with its base fixed to a surface as shown below in figure-1.

• If we now apply a constant 'pushing' force, F, to the upper part of the cube, assuming the material behaves as an ideal solid, it will obey Hooke's law of elastic deformation and will deform to a new position as shown in figure-1.
• This type of deformation (lower fixed upper moving) is defined as a SHEAR DEFORMATION.
• The deformation δ u and h are used to define the SHEAR STRAIN as :

                             Shear Strain =  δ u/h

• The shear strain is simply a ratio of two lengths (displacement / gap) and so has no units. It is important since it enables us to quote pre-defined deformations without having to specify sizes of sample etc

Shear Stress:-

• The SHEAR STRESS is defined as F/A (A is the area of the upper surface of the cube l x w) Since the units of force are Newtons and the units of area are m^2 it follows that the units of Shear Stress are N/M2 This is referred to as the PASCAL (i.e. 1 N/m^2 = 1 Pascal) and is denoted by the symbol σ (in older textbooks you may see it denoted as τ)

Shear Rate:-

• Consider the case of a cube of material that behaves as an ideal fluid. When we apply a shear stress (force) the material will continually deform at a constant rate as illustrated in figure-2.

• The rate of change of strain is referred to as the SHEAR STRAIN RATE often abbreviated to SHEAR RATE and is found by the rate of change of strain as a function of time i.e. the differential d.

       SHEAR STRAIN / d.TIME

Viscosity

• The Shear Rate obtained from an applied Shear Stress will be dependant upon the materials resistance to flow i.e. its VISCOSITY
• Since the flow resistance ≡ force / displacement it follows that ;

               VISCOSITY = SHEAR STRESS / SHEAR RATE

• The units of viscosity are Nm^2s   Which are better known as Pascal Seconds (Pa.s)
• If a material has a viscosity which is independent of shear stress then it is referred to as an ideal or NEWTONIAN fluid. The mechanical analogue of a Newtonian fluid is a viscous dashpot which moves at a constant rate when a load is applied as seen in figure-3.

Viscosity

The resistance of a fluid to flow

1 Ns/m^2 = 1 Pa.s = 10 Poise
Viscosity in Poise / Density = Kinematic viscosity in Stokes

Shear Stress

1 Pa = 1 N/m^2 = 10 dyn cm^-2

Kinematic Viscosity

The dynamic viscosity divided by density

Shear Rate

The rate of change of shear stress.  The velocity gradient perpendicular to the direction of shear flow (dv/dx).  Units 1/s or s-1

Shear Stress

The shear force per unit area

Shear Strain

A unit-less quantity, the relative displacement of the faces of a sheared body (for example a layer of fluid) divided by the distance between them.

Zero-shear Viscosity

The viscosity at the limit of low shear rate. The viscosity a product will ultimately attain when at rest and undisturbed.

Shear Stress

• The shear stress is part of the pressure tensor.
• However, here it will be treated as a separate issue.
•  In solid mechanics, the shear stress is considered as the ratio of the force acting on area in the direction of the forces perpendicular to area. 
• Different from solid, fluid cannot pull directly but through a solid surface. 
• Consider liquid that undergoes a shear stress between a short distance of two plates as shown in Figure (1.1).
• The upper plate velocity generally will be

                         U = f(A, F, h)...............................................(1.2)

• Where A is the area, the F denotes the force, h is the distance between the plates.
• From solid mechanics study, it was shown that when the force per area increases, the velocity of the plate increases also. 
• Experiments show that the increase of height will increase the velocity up to a certain range. 
• Consider moving the plate with a zero lubricant (h » 0) (results in large force) or a large amount of lubricant (smaller force).
• In this discussion, the aim is to develop differential equation, thus the small distance analysis is applicable.

• The viscosity coefficient is always positive. When n, is above one, the liquid is dilettante. 
• When n is below one, the fluid is pseudoplastic. 
• The liquids which satisfy equation (1.13) are referred to as purely viscous fluids. 
• Many fluids satisfy the above equation. Fluids that show increase in the viscosity (with increase of the shear) referred to as thixotropic and those that show decrease are called reopectic fluids (see Figure 1.5).
• Materials which behave up to a certain shear stress as a solid and above it as a liquid are referred as Bingham liquids. In the simple case, the “liquid side” is like Newtonian fluid for large shear stress.