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Visco elasticity and Non Newtonian fluids

Discussion in 'Biomechanics, Sports and Foot orthoses' started by David Smith, Jun 20, 2008.

  1. David Smith

    David Smith Well-Known Member

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    Recent threads have brought up queries about modelling plantar fascia or any other biological tissue that has viscoelastic properties. Viscoelastic materials are referred to as non Newtonian - This brings up two questions:

    1) What are non Newtonian fluid materials?
    2) Can they be modelled reasonably, in terms of body mechanics, using Newtonian mechanics?

    1) Simply put - Non Newtonian fluids are those that do not conform to Newton’s definition of fluid viscosity (Newton’s equation). IE Newton’s equation says that shear stress is directly relative and proportional to the force applied and that it is inversely proportional to the viscosity.

    Non Newtonian fluids change characteristics relative to applied force. Therefore the equation requires definition of, and is a function of, some variable of shear stress or time

    There are two types of Non Newtonian fluid - viscoelastic and viscoplastic.

    Viscoplastic fluid reduces its shear with force and becomes thinner e.g. non drip paint and viscoelastic increases its resistance to shear and becomes thicker.

    Shear stress is the result of different velocities of layers of fluid within the whole fluid. To imagine shear stress imagine a stone falling thru oil. The oil that is in direct contact with the stones surface will be going at the same velocity (speed and direction) as the stone, but a small distance away the oil will be at a slightly slower velocity until at a given distance the oil velocity will be zero. This curve of velocity from Vo - VI is known as a velocity gradient.
    The velocity gradient equation is the change in velocity divided by the change in distance between layers. dv/dx
    The shear stress is the force divided by the area of the layer and so - s = F/A

    These ultimately can be used to give a coefficient of viscosity (CoV) which is as:

    CoV = n n = Fx/Av and the units are Pascal seconds where a Pascal is 1N/m^2. Commonly the unit used to indicate viscosity is the milli-Pascal second or mP-s. Water has a viscosity coefficient of approximately 1 dependent on temperature (I shall ignore the relationship of temperature to viscosity here)

    Therefore by Newton’s equation of n=Fx/Av where F = 10N x = 0.02 A = 0.05m^2 and v = 0.2m/s then 10 * 0.02 / 0.05 * 0.2 = 200 P-s.

    However if the velocity is higher say 2m/s then 10*0.02/0.05*2 = 2 P-s. Which is what you might expect i.e. the force is the same but the velocity is higher therefore the viscosity must be lower. (Very similar to coefficient of friction i.e. lower CoF = higher acceleration of mass for a constant force.) However try increasing the force value and the viscosity coefficient will stay the same because the velocity will increase proportionally. (Similar to F=MA - increase the force for a constant mass and the acceleration increases)

    The maths is unimportant but the concept of Newtonian fluids is that they obey Newton’s equation described above.

    Newton did not account for viscoelasticity /plasticity and so those fluids are now known as non Newtonian.

    Therefore, considering viscoelastic materials, as force is applied viscosity increases and so velocity of shear between layers becomes slower. As the velocity approaches zero the fluid becomes like a solid. The faster the force is applied, i.e. the force impulse curve is short and high, the more rapidly the fluid becomes solid like.

    In terms of strain i.e. the change in length of a biological tissue, like plantar fascia for instance, this would result in a reduction of strain coefficient relative to the force applied. Therefore the higher the force the lower the strain coefficient, relatively speaking. This takes me to point 2.

    2) To explain further, Normally strain is directly proportional to force and has no time element (except for the time it takes to apply the force). EG 1000N = 5mm change in length - therefore for a initial length of 120mm the coefficient of change in length (strain) is 5/120 = 0.0417mm/mm and stiffness coefficient = 1000/5 = 200N/mm. This is a constant regardless of the impulse characteristics. (except for creep or hysteresis but I shall ignore those for now).
    In the case of a viscoelastic material - The characteristic of the force impulse will directly change the characteristic of strain. IE 1000N applied in 2milliseconds (ms) will cause a strain of say 1mm, whereas as force impulse of peak 1000N applied over 1 second will give rise to a change of length of say 6mm.

    The strain rate cannot be defined by a simple equation of force/change in length or strain = stress /constant - (Young's modulus).

    A much more sophisticated equation must be used and it must be tailored to suit the material to be modelled and the parameters may be gathered from relevant experimental data. Simple models start as things such as the spring and dashpot evaluation. These can be very sophisticated also. (this is only for very brainy people who got gold stars for their maths work at school, which unfortunately was not me :confused:)

    Once this data is acquired and a suitable equation defined then Newton’s laws can still be applied.
    E.G. F/A = K (dv/dx)^n is one used for less complex non Newtonian fluids.
    However for the purposes of biomechanical evaluation using Newtons Laws the forces applied at one end of the tissue of interest will be equal and opposite to the force at the other end of the tissue, whatever it's stress strain characteristics.

    Data gathered by experimentation rather than modelling can give a precise knowledge of the force - deformation characteristics of a certain tissue of interest if the experiment can sufficiently isolate the tissue from other confounding variables, which can be and is difficult to achieve especially in vivo. Force- deformation characteristics are not strictly the same as stress- strain characteristics but may be useful. Defining stress-strain properties of tissue in vivo is made even more difficult since the stiffness is complicated by variables that change other than by purely mechanical and the mechanical ones change with time and physiological situation.

    This is a simplified version of non Newtonian fluid principles as I understand them and you might want to read up more than this to get a better picture.

    All the best Dave
  2. Last edited: Jun 21, 2008
  3. David Smith

    David Smith Well-Known Member


    Yeah, like the video:cool:

    Yes synovial fluid is a dilatant non Newtonian fluid ie its shear velocity decreases as the applied force increases and the change is proprtional to the shear stress and not time.

    Studies show that in diseased joints the viscosity coefficient is much lower than in normal joints.

    OSTEOARTHRITIS SEVERITY? Conrad BP et al (2003 Summer Bioengineering Conference, June 25-29)

    On the viscosity and Ph of synovial fluid and the Ph of the blood, Jebens EH et al (Journal Bone and Joint surgery Vol 41 2 1959)

    BTW Simon did you know that a Fibula is a small lie? You learn something every day eh!:D
  4. So the question then becomes: how "solid" is synovial fluid during function?
    That's got to be the worst joke of all time, go out and learn about comedy rather than all this philosophy you've been coming out with lately.
  5. drsarbes

    drsarbes Well-Known Member

    OK David:
    I got a terrible frontal headache from reading your post.
    I'll have to check the viscosity differences in a '99 Turkey Old Vine Zin at 61 degrees F as compared to 57 degrees.


  6. David Smith

    David Smith Well-Known Member


    Well if my understanding of science philosophy is a ggod as my understanding of comedy, then for a Philosopher I would make a good plumber :dizzy:

    Dr Sarbes
    99 Turkey Old Vin Zin, well I'll have a pint of that whatever it is. :drinks

    My next book to read is Zen and the art of comedy

    All the best Dave
  7. Jeremy Long

    Jeremy Long Active Member

    In addition to being just a bit dizzy after reading the initial post, I can relate positive conservative use of viscoelastic materials. In patients with acute metatarsal head pain stemming from lack of necessary plantar fat padding, these gels have been extremely effective for me in forefoot orthotic extensions. One tip that I learned through the course of my experience doing this is to be sure to encapsulate the gel. Ultraleather has worked very well in this fashion, since it's thin, durable, and maintains some flexibility in contact with the wearer.

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