Introduction the rate at which at which shear


As a child, I was always amazed when playing with a solution composed of cornstarch and water. While cornstarch and water is a simple mixture, its properties are fascinating.  Upon mixing cornstarch with water, the solution is a liquid.  However, once the solution is rolled around in the palm of the hand, the solution hardens and becomes semi-solid. After the pressure from the palm of the hands is released, the solution returns to a liquid state. This seemingly magical change of state is characteristic of non-Newtonian fluids. When I first learned about non-Newtonian fluids through a Ted-Ed video, I remembered my fascination with the cornstarch and water mixture, and decided to structure my research paper around Non-Newtonian fluids.  Because I wanted to do a paper with a real-life application, I specifically found the use of Non-Newtonian fluids in armor interesting.

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The viscosity of a fluid is defined as the ratio of shear stress to shear rate.  In Rheology, the study of flow and non-Newtonian fluid, shear is a strain in a substance produced by pressure.  Shear stress is the pressure applied to a fluid; it is defined as a force caused shear that causes friction between fluid particles.  Shear rate is the rate at which at which shear stress is applied and appears.  Viscosity can ultimately be thought of as a fluid’s thickness and resistance to flow. 
Because viscosity is the shear stress (pressure) over shear rate (which is rate of change), viscosity (?) represents viscosity can be expressed by the formula
?= (F?/A)/(?vx/?z)

Newton’s law of Viscosity

Newton’s law of Viscosity states that the relationship between shear stress and shear rate is a constant, meaning that the viscosity of fluids is constant regardless of the amount of pressure applied.  Fluids that obey this law are Newtonian fluids.  Conversely, fluids in which viscosity changes as an applied pressure changes are known as non-Newtonian fluids.  
The formula for viscosity can be rearranged to produce the formula 
,which more closely resembles Newton’s second law.   

Non-Newtonian Fluids

Non-Newtonian Fluids are fluids whose viscosities are not constant.  Technically, all fluids are non-Newtonian since there is a small time interval in which shear stress (applied pressure) affects the viscosity of the fluid.  However, for most fluids, the time horizon in which this happens is insignificant and disregarded.  For non-Newtonian fluids, there is a significant difference between the time it takes a fluid to adapt to shear stress and the period in which the pressure is applied.  This relationship between time differences is proportional to the relationship between the viscous forces and relation forces, which is known as the Weissenberg number and is important in determining which fluids are non-Newtonian.  If the time it takes the fluid to adapt to stress is greater than the time in which the pressure is applied, the fluid becomes less viscous.  Fluids in which the adaptation time is less than the period of applied pressure become more viscous and are known as shear-thickening fluids.  Shear-thickening fluids can be time-dependent or time-independent.  Time-dependent fluids are fluids which remain more viscous after the applied pressure has been released; they are known as rheopectics. An example of a rheopectic is whipping cream.  Time-independent fluids are fluids which return to a fluid-like state after the applied pressure has been released; they are known as dilatants.


A dilatant is a shear-thickening fluid that is time-independent.  Dilatants are able to become a solid under pressure because of the suspension of particles.  At rest and at low levels of shear stress, there is no contact between suspended particles because the liquid between the particles acts as a lubricant. These particles are typically 1 nm to 1 ?m wide.
 However, once a large amount of shear stress is applied, the particles dilate, hence the name dilatant, until the particles come into contact.  The dilation magnifies the shear stress to a point where it is greater than the shear rate; this causes the fluid to become more viscous to the point where it is solid.  Common dilatants are the cornstarch and water solution and quicksand.

Rewriting Newton’s Law to Fit Dilatants

Newton’s law can be written into the Ostwald-de Waele equation, more commonly known as a power law equation. 
(F/A) = k (?vx/?z)n
While it is meant for Newtonian fluids, it can handle time-independent non-Newtonian fluids, which includes dilatants. In this equation, k is the flow consistency index, the viscosity of a fluid at a shear rate of 1s-1; k gives an idea as to how viscous a fluid can become.  n represents the flow behavior index, which dictates if a fluid is shear-thickening, shear-thinning, or Newtonian.  If n is greater than one, the fluid is a dilatant fluid.  From this equation, the viscosity (?) of a non-Newtonian fluid can be derived as being.  
? = k (?vx/?z)n-1
This equation can be graphed with viscosity as the dependent variable, and the shear stress rate as the independent variable.  k and n, are of course constants relative to each fluid.

Dilatants as a Liquid Body Armor

As dilatants instantly solidify under large amounts of pressure and then quickly liquify once the pressure has been released, they are the perfect candidate for liquid body armor.  Although research in rheology is slim, several body armors have been created with the use of dilatant.  However, these body armors combine standard fibrous-based armor with dilatant fluids.  Currently, there is no purely dilatant-based body armor.  Theoretically, a purely dilatant-based body armor would provide a lightweight and flexible full-body armor.

Selection of Dilatants

Assuming that a ballistic impact is kept consistent, and the force applied by the ballistic impact is constant; how well a dilatant can stop a ballistic impact depends on the viscosity of the dilatant and the shear stress caused by a ballistic impact. Because the shear stress caused by a ballistic impact is large, fluids that have large viscosities with large shear stress are ideal for armor.  Recalling that dilatants are composed of suspended particles, factors such as particle shape and size affect how viscous a dilatant becomes.  Since data concerning dilatants as armor is limited, I chose to analyze data based off of the variables of particle concentration and the size of particles.  

Research Question

In which way can the properties of dilatants be manipulated in order to determine which fluids are optimal for use against ballistic impact. 


I hypothesize that dilatants that have a higher particle concentration and a large size of particles will be optimal for use against ballistic impact.  My reasoning is that as the particles cause the dilatant to solidify, the more particles and the larger they are will allow for a more solid substance. 


Collection Method

Because the testing of non-Newtonian fluids for this purpose is beyond my scope of capabilities as a student, the data I am using comes from other sources, which are as follows.

Shear- thickening behavior of polymethylmethacrylate particles suspensions in glycerine–water mixtures, Jiang, W., Sun, Y., Xu, Y., Pen, C. (Figure 1, 2)
The effects of interparticle interactions and particle size on reversible shear thickening: Hard-sphere colloidal dispersions, Maranzano, B.J., Wagner, N.J. (Figure 3)

The data collected is as follows   

Figure 1

The graph represents the relationship between viscosity and shear rate for a fluid consisting of polymethyl- methacrylate (PMMA) particles in a glycerine and water mixture.  The legend on the upper-lefthand corner indicates the percent concentration (also known as volume fraction) of PMMA particles. 

Figure 2

Based off of the data from Figure 1, Figure 2 models the relationship between viscosity and the percent concentration. By using both graphs, it is obvious that viscosity increases as the percent concentration of particles increases.

Figure 3

The graph represents shear stress as a function of particle concentration by weight for a solution of silica particles dispersed in tetrahydrofurfuryl alcohol, abbreviated on the graph as HS.  The numbers proceeding HS on the legend indicate the size of the particles in nano-meters.  Because higher shear stress indicates a higher viscosity, and particles of smaller sizes cause higher levels of shear stress, smaller particles create a higher viscosity.  

Conclusion and Application

My hypothesis was correct in that a larger concentration of particles would be optimal for body armor as the viscosity is greater and therefore ideal. However, my hypothesis was incorrect in that larger particles would be less effective than smaller particles. Although smaller particles leave more room for fluid to act as a lubricant, the stress felt by the dilatant increases as the area of each particle decreases and the particles expand more. The larger expansion of particles causes the dilatant to become more viscous and act as a stronger solid. By taking these results into consideration, they can be applied to the development of liquid body armor.  The dilatant used in liquid body armor should combine smaller particles with a large concentration in order to maximize viscosity and optimally prevent injury from ballistic impact.