Long-Term Mechanical Properties of Bone

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Long-Term Mechanical Properties of Bone
David Taylor
Bone, like other structural materials, can fail
by long-term mechanisms such as fatigue, creep
and wear. Here we will discuss the fatigue and
creep properties of bone. Wear is an important
mechanism of failure in joints but it is really
the wear properties of cartilage which are
important there.
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Why study long-term failure?
Because bones fail by long-term mechanisms as a
result of strenuous exercise. Failures that occur
in this way are called Stress Fractures by
clinicians, to distinguish them from fractures
that occur due to trauma (e.g. a fall). Stress
fractures occur often in athletes, dancers and
sports people, and especially in military
recruits at the start of their training. Stress
fractures also occur in animals, e.g.
thoroughbred racehorses. They happen because
damage (especially cracks), builds up in the bone
faster than the bone can repair it or reduce
stress by remodelling and adaptation. Long-term
failures also occur during normal activities in
people with poor-quality bones (e.g. elderly
people or those with osteoporosis). The same
happens to chickens who are kept in battery
cages, because their bones dont develop full
strength. These failures are called Fragility
Fractures.
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Fatigue and Creep What are they?
You probably know this already but heres a brief
reminder. FATIGUE is long-term failure due to
cyclic (repetitive) loading. Failures occur by
the initiation and propagation of cracks, even
when the maximum stress is less than the UTS or
failure stress of the material. The stress range
Ds is defined as the difference between the
maximum and minimum stresses in the cycle the
mean stress smean is also important. Fatigue
occurs in all materials, though it is most
important in metals and in some polymers and
composites. CREEP is caused by high temperatures
(relative to the melting point of the material).
Failure occurs due to plastic deformation
essentially a gradual flowing of the material
due to diffusion and thermally-activated
dislocation motion.
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How is Fatigue Measured?
Either by recording the number of cycles to
failure of specimens as a function of stress
range (or strain range) or by measuring the rate
of crack growth da/dN (mm/cycle) as a function of
the applied range of stress intensity. Note the
logarithmic scales.
Crack Growth Rate da/dN (log.scale)
Stress Range (log.scale)
Number of cycles to failure, Nf (log.scale)
Stress Intensity Range DK (log.scale)
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How is Creep Measured?
One can record time to failure as a function of
stress and also the change in strain during a
test, which has a characteristic shape.
Stress (log.scale)
Strain
time
Time to failure, tf (log.scale)
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Fatigue in Bone
Creep and fatigue properties can be measured in
bone as in engineering materials. Here is a
typical strain/life curve for tests on cortical
bone.
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Fatigue in Bone
Here is some growth-rate data, also for cortical
bone.
Stress Intensity Range, DK (MPa.(m)1/2)
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Fatigue in Torsion
Bone is considerably weaker in torsion than in
tension or compression
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Frequency Effects
Frequency has a similar effect to strain rate.
Here is some data at 2Hz (a typical physiological
frequency) and 0.02Hz. The symbols O-C and O-T
refer to stress cycles that go from zero to
compression and zero to tension respectively.
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Other effects in fatigue behaviour
Quite a lot of things affect the fatigue
behaviour of bone, but there are some things
which, surprisingly, dont. These effects can be
expressed in terms of how much they increase or
decrease the fatigue strength, which is defined
as the stress range needed to get failure in a
given number of cycles (say 100,000). The table
below summarises various effects Parameter
Change Effect on Fatigue Strength (Multipl
ying Factor) Frequency From physiological
frequency (0.5-3Hz) x1.33 to high frequency
(30-125Hz) Age From old bone to young
(mature) bone x1.44 Temperature From
physiological temperature to x1.16 room
temperature Loading Type From zero-tension
to zero-compression x1.08 From zero-tension to
tension-compression x1.12 From zero-compression
to tension-compression x1.04
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Creep in Bone
Here is some typical creep data.
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Creep/Fatigue Interactions
This graph, due to Carter and Caler, shows
stress-life results for tension-compression
fatigue (no creep) and for zero-tension fatigue
(with creep), plus prediction lines using damage
mechanics (see below).
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Damage
What actually happens to bone when it is tested?
It does not show plasticity and yielding like
metals and some polymers. It is essentially
brittle the only response to stress is to become
damaged by cracking. However, like a good
engineering composite, the initial damage takes
the form of microscopic cracks, usually lying
between the secondary osteons. Increased static
loading (or increased number of fatigue cycles)
will increase the number of cracks. Eventually
one crack becomes large (greater than 1mm) and
this then grows rapidly to failure. Because of
the materials anisotropy, microcracks tend to be
alligned parallel to the long axis of the bone
even if the loading is also in this direction, as
occurs in some composite materials but never in
isotropic materials like metals. On the next page
is a typical microcrack seen on a transverse
section longitudinally it will be much longer.
Note how it has stopped when it reached the
osteons.
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Crack
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A schematic showing the 3D shape and position of
a typical crack in a bone, and a photograph of a
real crack obtained by laser scanning confocal
microscopy (thanks to Fergal OBrien)
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Damage Mechanics
Damage mechanics is a a type of theoretical
modelling that allows one to predict fatigue and
creep failures. It is used a lot in predicting
failure in engineering structures as well as in
bone. We assume that the total damage in a volume
of material can be represented by a quantity D,
which goes from 0 when there is no damage to 1
when failure occurs. If we apply a stress s to
the material, it will fail after a time tf. So
after some shorter time t has elapsed we assume
that the damage due to creep is Dc t/tf. This
assumption is known as the linear accumulation
of damage. Likewise for fatigue we can write Df
N/Nf (where Nf is a function of Ds) These
equations can be used to predict the amount of
damage, and the time (or the number of cycles)
needed for failure, in situations where the
stress (or stress range) is not constant. For
example consider a person who walks for a certain
number of steps and then runs for a certain
number how much damage has been done? We can
also use this approach to predict the combined
effect of creep and fatigue, in a situation (as
happens sometimes in bone) where both act
simultaneously. In this case we assume that the
total damage D Dc Df.
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Damage Mechanics - continued

• Carter and Caler carried out a damage mechanics
  analysis of bone the original paper is Carter
  DR and Caler WE (1985) A cumulative damage model
  for bone fracture. J Orthop Res 3 84-90. Their
  results are shown in the graph above.
• The important result from their analysis is that
  creep dominates at high stresses, but fatigue
  dominates for lower stresses, including all
  stresses in the physiological range (less than
  60MPa).

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Predicting Stress Fractures
Stress fractures are fatigue failures. An
interesting example is military recruits who
suffer a lot of stress fractures during initial
training. In theory we should be able to predict
the probability of getting a fracture using data
such as shown in this graph.
Weibull equation cumulative probability P1-exp-
(Nf/Nfo)m
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Predicting Stress Fractures
But the prediction doesnt work unless we include
repair and adaptation. Stress fractures occur
because fatigue cracks grow faster than they can
be repaired, and faster than the body can adapt.
This graph shows our prediction (from Taylor D
and Lee TC (2003) J.Anatomy 203 203-211). Repair
is especially important.
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Structural Design The Bone Tube

• Consider a simple example of a problem in
  structural design. Most bones are tubular,
  external radius R and thickness t. A question
  arises as to what is the optimum value for the
  ratio R/t for structural considerations, and is
  this value actually found in practice? Pauwels
  considered this question, assuming that the
  operative criterion was minimum weight.
• Consider a tube loaded in simple axial tension or
  compression in this case it is obvious that the
  stress for a given weight (and thus for a given
  cross sectional area) is not affected by the
  ratio R/t no optimisation occurs. If however the
  tube is loaded in bend, the stress will be
  reduced by increasing the radius, even if t is
  decreased to maintain constant weight. In this
  case the optimum tube has an infinitely large
  radius and infinitely small thickness. So neither
  of these simple models yields a realistic optimum
  value for R/t.

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• However, Pauwels pointed out that the tube is not
  empty it is filled with bone marrow, which has a
  finite density. This adds to the overall weight,
  leading to a finite optimum value for R/t.
• Currey and Alexander noted that this value will
  vary depending on the failure criterion used. The
  graph below shows their predictions using four
  different criteria impact strength ultimate
  strength yield strength (which they considered
  to give the same predictions as fatigue strength)
  and elastic stiffness.
• Measurement of this ratio in real bones showed
  that there was a lot of variation, between 1 and
  3.7, but with a clear preference for values
  around 2, which lead them to conclude that bone
  is optimised for impact or ultimate strength.

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It is worth noting that the total saving in
weight is quite small, the optimum condition
being about 90 of the weight of a solid bone of
the same strength. THE END!