# Elasticity and viscosity

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## Introduction

"The fundamental property of a fluid is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the common surface. This property is the basis of Hydrostatics and is verified by the complete agreement of the deductions of that science with experiment. Very slight observation is enough, however, to convince us that oblique stresses may exist in fluids in motion. Let us suppose for instance that a vessel in the form of a circular cylinder, containing water (or other liquid), is made to rotate about its axis, which is vertical. If the motion of the vessel be uniform, the liquid is soon found to be rotating with the vessel as one solid body. If the vessel be now brought to rest, the motion of the fluid continues for some time, but gradually subsides, and at length ceases altogether; and it is found that during this process the portions of fluid what are further from the axis lag behind those which are nearer, and have their motion more rapidly checked. These phenomena point to the existence of mutual actions between contiguous elements which are partly tangential to the common surface. For if the mutual action were everywhere wholly normal, it is obvious that the moment of momentum, about the axis of the vessel, of any portion of fluid bounded by a surface of revolution about this axis, would be constant. We infer, moreover, that these tangential stresses are not called into play so long as the fluid moves as a solid body, but only whilst a change of shape of some portion of the mass is going on, and that their tendency is to oppose this change of shape."

A treatise on the mathematical theory of the motion of fluids; Horace Lamb; Cambridge Press: Cambridge; 1879; pp 2-3.

"Long-range elasticity has been observed only in substances and materials known to contain flexible molecular chains. The chains may be crosslinked into a network structure, but this is not essential. Stretching produces molecular configurations in which the orientation about the single bonds of the structure is less random than in the unstretched specimen. This produces a decrease in the entropy of the system. Elastic recovery is a result of the tendency for the system to increase its entropy. Energy changes may also take place in the stretching and recovery processes, but they are not necessary for the phenomenon. It can readily be shown by thermodynamic reasoning that, if energy changes are negligible, the elastic modulus should be directly proportional to absolute temperature. Certain rubber samples have, in fact, been found to conform to this relationship."

Physical chemistry of high polymers; Maurice L. Huggins; John Wiley & Sons: New York; 1958; p 102.

"If portions of a mass of liquid are caused to move relatively to one another, the motion gradually subsides unless sustained by external forces; conversely, if a portion of a mass of liquid is kept moving, the motion gradually communicates itself to the rest of the liquid. These effects, which are matters of immediate observation, were ascribed by Newton to a 'defectus lubricitatis,' i.e., a 'lack of slipperiness' between the particles of the liquid, which may be fairly interpreted to mean a property resembling friction between solid surfaces, and Newton, in fact, uses the the term 'attritus,' i.e. friction, several times in the course of his deduction. These terms "internal friction" and "viscosity" have been used indifferently to describe this property of liquids; it is not clear when the latter work acquired a strict technical meaning, since etymologically (from viscum, the mistletoe) it would ween to have been applied originally to liquids possessing the property to an abnormal degree."

The viscosity of liquids; Emil Hatschek; G.Bell and Sons: London; 1928; p. 1.

## What is the meaning of elasticity?

Elasticity is a material property characterizing the compressibility of a fluid - how easy a unit of the fluid volume can be changed when changing the pressure working upon it.

 Strain in tension: $\varepsilon =\frac{\Delta l}{l}\,\!$ Strain in shear: $\gamma =\frac{\Delta x}{h}\,\!$ Stress: $\sigma =\frac{F}{A}\,\!$ \begin{align}  & \sigma =G\gamma \text{ for linear-shear elasticity} \\ & \sigma =E\varepsilon \text{ for linear-tensile elasticity} \\  \end{align}\,\!
\begin{align}  & G\text{ or }E=\frac{force}{area}\text{ } \\ & G\text{ or }E=\frac{energy}{volume} \\ & G\text{ or }E=\text{bond energy}\centerdot \text{bond density} \\  \end{align}\,\!

 When we stretch a solid, just a little; or when we move liquid, for a very short time, we expect the stress (force per area) to be proportional to the strain: $\sigma =G_{0}\gamma \,\!$ The stored energy per unit volume is the work done by the stress: $w=\int\limits_{0}^{\gamma }{\sigma d\gamma }=G_{0}\int\limits_{0}^{\gamma }{\gamma d\gamma }=\frac{1}{2}G_{0}\gamma ^{2}\,\!$ $G_{0}$ has units of energy/volume. To be “soft” matter the energy per molecule is about kT so that the energy per volume is: $G_{0}\approx \frac{kT}{a^{3}}\,\!$

See Witten, p. 29

## Viscosity and molecular properties

Newtonian liquids $\sigma =\eta \dot{\gamma }\,\!$ Viscosity is an empirical constant.
The work per time is force x velocity. \begin{align}  & \dot{W}=Force\cdot velocity \\ & \text{ }=\left( \sigma \Delta x\Delta y \right)\left( \dot{\gamma }\Delta z \right) \\ & \dot{w}=\sigma \dot{\gamma }=\eta \dot{\gamma }\cdot \dot{\gamma } \\ & \dot{w}=\eta \dot{\gamma }^{2} \\  \end{align}\,\!

First calculation of volume work rate.

Therefore viscosity is energy per volume times time.

Think of flow as repeated steps: Stretch then relax.
“Affine” transition – as if in rubber.
Stress per step: $\sigma =G_{0}\gamma \,\!$
The energy stored per unit volume is $w=\int\limits_{0}^{\gamma }{\sigma d\gamma }=G_{0}\int\limits_{0}^{\gamma }{\gamma d\gamma }=\frac{1}{2}G_{0}\gamma ^{2} \,\!$
The work rate is the work divided by a relaxation time: $\dot{w}=\frac{G_{0}}{2\tau }\gamma ^{2} \,\!$ The second calculation of volume work rate.

See Witten pp 28-29.

These two derivations give us two models for the volume work rate.

## What is the meaning of viscosity?

The definition of "Viscosity" is the resistance to flow of a liquid (normally).

A high viscosity means that the liquid will not flow easily - imagine treacle.

A low viscosity means that the liquid flows very easily - imagine water or gasoline.

Work (rate) done by viscous flow. \begin{align}  & \dot{w}=\eta \dot{\gamma }^{2} \\ & \text{or }\dot{w}=\frac{\eta }{\tau ^{2}}\gamma ^{2} \\  \end{align}\,\!

Work (rate) done by stretching and relaxation. $\dot{w}=\frac{G_{0}}{2\tau }\gamma ^{2}\,\!$
Combining the two gives: $\eta =\frac{G_{0}\tau }{2}\,\!$

Viscosity is an energy/volume x a characteristic time.

## Elasticity of Cell Membranes

The cell membrane is the boundary between a cell and its surroundings and is mostly made up of lipids (a dominant type is the phospholipid) and proteins. Phospholipids play key roles in cell membranes because they have hydrophilic head groups and hydrophobic hydrocarbon tails. This creates a bilayer on the cell membrane. The fluid mosaic model (which was proposed by Singer and Nicolson in 1972) is the most widely accepted model for cell membranes. In this model, the cell membrane is viewed as a lipid bilayer which has proteins embedded in it. Meanwhile, the lipids can move freely in the membrane surface, like a fluid. There is a membrane skeleton beneath the lipid membrane made up of a network of proteins that link to the proteins embedded in the membrane.

To study the elasticity of cell membranes one has to study lipid bilayers. One can think of the bilayer as being a neumatic liquid crystal at room temperature. Thus one can write down the curvature energy per unit area as:

f_c=k_c/2(2H+c_o)2+kK

where c_o is the spontaneous curvature of the membrane. H and K are the mean and Gaussian curvature of the membrane surface, respectively. k and k_c are the bending rigidities.

Similarly, one can write down formulas for the free energy of a closed (or open, for that matter), bilayer which is under some osmotic pressure, delta-p:

F_h = integral(f_c+lambda)dA+delta-p*integral(dV)

where dA and dV are the area element of the membrane and the volume element enclosed by the closed bilayer, respectively. λ is the surface tension of the bilayer. From this equation, oenc an also describe the equilibrium shape of the bilayer as:

Science 175 (1972) 720
Wikipedia
http://arxiv.org/html/physics/0411169

## Application: Running Shoes

Viscosity and elasticity are key variables to consider when designing a running shoe. An article by Nigg et al. in the Journal of Biomechanics attempts to scientifically determine the effect of the rheological properties of the running shoe on muscle activation (e.g. hamstrings and quadraceps) during running. When running, a foot can experience a force equal to 1.5 to 2 times the person's body weight. However, the muscle vibrations are minimal, suggesting that the muscles are activated in such a way to behave like a critically damped oscillator. It seems plausible that this behavior would be affected by the type of running shoe worn.

The authors of the paper conducted an experiment with twenty male runners to see if this is indeed the case. To measure the efficiency of their stride, their oxygen consumption was monitored over the course of 6 minutes. During identical measurements (doing them at the same time would skew the results), the electrical activity in the relevant muscles was measured. Two different sole materials were used in their running shoes:

• mainly elastic (shore C = 45)
• mainly viscous (shore C = 26)

The rheology of the running shoes was determined by compressed them at 300 mm/s for a maximum deformation of 14 mm, to simulate actual running conditions. The corresponding force-deformation curves are shown below:

Surprisingly, there was no general trend between oxygen consumption and shoe material: some groups did better or the same with viscous shoes, whereas another group did better with elastic shoes. The size of these differences was about 2% of the total oxygen consumption. Although no broad trend between muscle activation and shoe elasticity was seen, the researchers did observe how the rheology of the shoe affected the pattern of muscle activation, as shown below:

Although these results are thought-provoking, they offer new clear advice as I search for a new pair of running shoes. The authors only mention the possibility of future studies to determine the optimal tuning' of shoe properties to individual runners, which may produce even larger differences in oxygen consumption.

For more information see: B. M. Nigg, D. Stefanyshyn, G. Cole, P. Stergiou, J. Miller, The effect of material characteristics of shoe soles on muscle activation and energy aspects during running, Journal of Biomechanics, Volume 36, Issue 4, April 2003, Pages 569-575.