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Mechanical Properties of Real Fluids and Solids

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Mechanical Properties of Real Fluids and Solids Powered By Docstoc
					Mechanical Properties of Real
     Fluids and Solids




          Presented by
    Ken Bole and Brian Sabino
                  Overview
•   Fluids
•   Viscosity
•   Plasticity of Metals
•   Nonlinear Elastic Materials
•   Linear and Quasi-linear Viscoelastic Bodies
•   Viscoplastic Materials
                   Fluids
Two types based on pressure-volume relationship
              Fluids (cont.)
Gaseous fluid equation of state


Relates pressure (p), volume (V) and absolute
 temperature (T), and


where N0 is Avogadro’s number and k is
 Boltzman’s constant
              Fluids (cont.)
For liquids, we have Van der Waals eqn.



where a/V2 is the attractive force between gas
particles, and b is the molecular volume of the
particles
               Fluids (cont.)
• Van der Waals
  isothermals
  – negative p is the
    liquid’s tensile
    strength
  – minima indicate
    ideal tensile
    strength of liquid
              Fluids (cont.)
• Differences between experimental and
  theoretical tensile strength values
  – Vapor-nucleating agents
  – Small bubbles
  – Liquid tearing away from container walls
                 Viscosity
Given by Newton in terms of shear flow with
uniform velocity gradient
       y            t


             x           u(y)


                    t
where u is systemic velocity of the fluid and t
is shear stress on surface normal to y-axis
            Viscosity (cont.)
For shear stress t, Newton proposed




Here m is the coefficient of viscosity
  – In liquids, diminishes rapidly as T increases
  – In gases, increases as T increases
          Plasticity of Metals
If a ductile metal is pulled, the load applied
may be plotted against elongation using:




where l0 is the original length and l is the
length under the load
Plasticity of Metals (cont.)
    Nonlinear Elastic Materials
Rubber cannot be described by Hooke’s law
 due to its nonlinear stress-strain curve




Rubber shows features of viscoelasticity
    Nonlinear Elastic Materials
              (cont.)
• Criteria for viscoelasticity:
   – Stress relaxation at constant strain
   – Creep at constant stress
   – Hysteresis in periodic oscillation
Nonlinear Elastic Materials
          (cont.)
    Nonlinear Elastic Materials
              (cont.)
If a pseudoelastic strain energy function
exists, then the stress-strain relationship can
be obtained by differentiation
    Nonlinear Elastic Materials
              (cont.)
For rubber,


But for biological tissues,
      Linear and Quasi-linear
        Viscoelastic Bodies
• 3 mechanical models of material behavior:
  (a) Maxwell model, (b) Voigt model,
  (c) Standard linear model
     Linear and Quasi-linear
    Viscoelastic Bodies (cont.)
Relationships between the load F and the
deflection u are:
      Linear and Quasi-linear
     Viscoelastic Bodies (cont.)
If we solve for u(t) when F(t) is a unit step
function, we get the creep function:
 Linear and Quasi-linear
Viscoelastic Bodies (cont.)
      Linear and Quasi-linear
     Viscoelastic Bodies (cont.)
If we solve for F(t) when u(t) is a unit step
function, we get the relaxation function:
 Linear and Quasi-linear
Viscoelastic Bodies (cont.)
     Linear and Quasi-linear
    Viscoelastic Bodies (cont.)
Generalized equation due to Boltzmann:
     Linear and Quasi-linear
    Viscoelastic Bodies (cont.)
Hysteresis of biological tissue variable,
though small, with strain rate
 Linear and Quasi-linear
Viscoelastic Bodies (cont.)
     Linear and Quasi-linear
    Viscoelastic Bodies (cont.)
Putting this into mathematical form, we get




or the stress at time t is the sum of all past
changes governed by the relaxation function
     Linear and Quasi-linear
    Viscoelastic Bodies (cont.)
Relaxation function G(t) of an infinite number
of Kelvin (standard linear) models in series
        Viscoplastic materials
While Newton’s viscosity law is good for
water, it’s not very good for most other fluids
   Viscoplastic Materials (cont.)
• Newtonian materials must flow under slight
  shear stress, but viscoplastic material can
  sustain stresses with slight shear in a state of
  rest
• Consider a viscoplastic body subjected to
  simple shear (s12 = s21 = t)
   – while t < K (yield stress), material is rigid
   – for t > K, material flows
Viscoplastic Materials (cont.)
   Viscoplastic Materials (cont.)
Bingham described the strain rate to stress
rate as




where the yield function F is
  Viscoplastic Materials (cont.)
These were later generalized

				
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posted:1/19/2014
language:English
pages:32