Mechanics -
Basic concepts
Elastic material
n Deforms upon
application
of force.
n Returns to original configuration
without energy dissipation when
force is released.
w All molecules return to
original configuration.
n Hooke’s experiment on metal
wires.
n Young extrapolated by
normalizing to area.
Plastic Deformation
Linearly elastic
SA: Yield stress
Assuming no creep.
Cardiovascular Biomechanics,
Spring 2004
1
Mechanics -
Basic concepts
Linearly elastic behavior follows Hooke’s law.
n
n
Stress (Force/Area) linearly proportional to strain (deformation (non-dimensional)).
Assumptions of:
w No creep.
n
Increase in strain over longer periods of time (slow flow).
w No stress relaxation.
n
Fall in initial stress value over longer periods of time.
w No plastic deformation.
n
Change in molecular structure.
Solid -vs- fluid: ability to withstand stress.
Viscoelastic material: elastic solid material that exhibits flow properties (creep,
stress relaxation).
Cardiovascular Biomechanics,
Spring 2004
2
Young’s Modulus & Poisson’s Ratio
Within linear range of behavior:
T = E l'
A
l
o
o
Proportionality constant is Young’s Modulus (E:
N/m2).
Stretching causes contraction in cross-sectional
area.
a' = -s l'
Circular cylinder:
a
l
s: Poisson’s ratio
o
n
=0.5 for incompressible material.
o
w No volume change.
Arterial walls can be assumed incompressible.
Cardiovascular Biomechanics,
Spring 2004
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Other components of stress
Tension/compression in longitudinal
direction has been considered.
Other stresses can be present.
Shearing Stress
Bending Stress
Cardiovascular Biomechanics,
Spring 2004
Torsional Stress
4
Properties of Blood Vessel Walls
Homogenous and isotropic assumptions.
n
E, s do not vary within the material.
Independent of direction in which stress is applied.
Is this true of blood vessel walls ?
n
n
n
n
~70% of walls consists of water (incompressible & inelastic).
Rest of material: elastic fibers.
Three types of fibers:
w Collagen
w Elastin
w Smooth muscle.
Cardiovascular Biomechanics,
Spring 2004
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Properties of Blood Vessel Walls
Elastin
Collagen
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Spring 2004
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Properties of Blood Vessel Walls
Non-linear behavior.
n
Measure effective Young’s modulus (Eeff) by measuring local
slope.
Peak Eeff for:
n
n
n
Elastin: 3 X 105 N/m2.
Collagen: ~108.
Smooth muscle: ~105-106 (??) - active structure.
Proportion of fibers varies depending on type
of artery or vein.
Cardiovascular Biomechanics,
Spring 2004
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Blood vessel walls
Endothelium: Single cell layer separating wall and blood
Elastic tissue: Elastin
Fibrous tissue: Collagen
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Spring 2004
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Measurement of Eeff
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Spring 2004
9
Changes in stress-strain characteristics
with time
3 consecutive pressurevolume curves for a
femoral artery
Note hysteresis.
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Spring 2004
10
Viscoelastic behavior
Purely elastic material:
n
No phase lag between strain and stress.
Viscoelastic material:
n
n
Strain lags stress.
Possible to represent major features
of viscoelastic behavior via a
Dynamic Young’s modulus (Edyn) and
effective viscosity (h)
Cardiovascular Biomechanics,
Spring 2004
11
Statics of an elastic tube
Use force balance to examine
relationship between radius,
pressure and wall tension.
Result:
pi ri - pe re = Sq h
Sq: Hoop stress
Cardiovascular Biomechanics,
Spring 2004
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