Tissue Mechanics
Dawn Mackey
Related reading:
Hamill and Knutzen: Chapter 2, pp. 36-63
Outline
• Part I: Introduction to the Mechanical
Behaviour of Biological Tissues
• Part II: Mechanisms of Tissue Injury and
Tolerance
Biewener, A. Overview of structural mechanics.
(http://www.sfu.ca/%7Estever/kin402/readings/kin402
_readings.htm)
pp 1-5, 10-11
The Musculoskeletal System
•
•
•
•
•
•
Outline
• Modes of loading
• Internal forces and moments
• Stiffness of a structure under uniaxial loading
• Structural versus material behaviour
• Linear versus nonlinear materials
• Elastic modulus of a material
• Stress-strain diagrams
• Viscoelastic properties of tissues
bone
tendons
ligaments
fascia
cartilage
muscle
Forces cause motions and deformations
force
Part I: Introduction to the Mechanical
Behaviour of Biological Materials
motion
deformation
• if the net external force or moment applied to a body is not zero,
it will undergo gross motion
• if a body is subjected to external forces or moments but remains
in static equilibrium, there will be local changes in the shape of
the body, or deformations
• in our previous treatment of static equilibrium, we assumed that
bodies were infinitely stiff and could not deform
• we now relax this assumption to examine the force-deformation
behavior of biological materials
There are five primary modes of loading
• compression and
tension are both “axial
loading”
• by convention,
compressive loads are
negative, and tensile
loads are positive
• in reality, we rarely
have pure axial
loading, pure bending
or pure torsion, but
instead a “combined
loading” situation
1
Each mode of loading creates a distinct pattern
of internal stress
tension
σ =P
compression
A
σ = − P shear
σ =P
A
A
Tensile loads cause a body to stretch
• Consider a structure of length L and cross-sectional
area A, that is loaded by two equal and opposite
forces of magnitude P
• we will use the Greek delta (δ) to represent the
change in length (or deflection) of the structure after
application of the forces
bending
σ=
My
I
T
σ= y
torsion
J
A
Stiffness is the slope of the load-deformation
curve
Biological structures have nonlinear stiffness
• the slope of the loaddeformation curve is the
stiffness k (in N/m)
• if a structure is made from a linear
material, the stiffness will be
constant (regardless of geometry)
• stiffness is affected by
the structure’s geometry
(length and crosssectional area)
A
• if the structure is made from a
nonlinear material, the stiffness
will vary with the load
load P
(N)
• a steel spring has a linear stiffness
• for this reason, stiffness
is referred to as a
structural property
P
linear
constant
stiffness k
δ
P
nonlinear
• biological tissues have nonlinear
stiffness
stiffness
k
varying
stiffness k
deflection δ (m)
Stiffness depends on length and cross-sectional
area
Axial stiffness increases
with increasing crosssectional area, and
decreases with
increasing length
P
δ
Relationship between structural and material
P
properties
A
• if we divide load P by crosssectional area A, and deflection δ by
original length L, the force-deflection
curves collapse onto one common
trace
• the slope of this trace is the modulus
of elasticity E (or Young’s modulus):
P
E=
δ
δ
PA
δ L
•the units of E are [N/m2], often called
pascals, Pa
Ε
δ
L
P
A
Ε
δ
L
2
Relationship between structural and material
properties (cont)
AXIAL
STRESS
P
σ=
A
[Pa]
F
A (x-sectional area)
E (modulus)
L
F
AXIAL STRAIN ε =
[m/m]
δ
L
M
structural
slope = stiffness k
(N/m)
ε=
• The relationship that defines modulus of
elasticity is referred to as Hooke’s Law:
PA
E=
δ L
Based on Hooke’s Law, the
stiffness of a structure under
axial loading (which, by
definition, is the the ratio of
load divided by deflection)
is:
δ=
PL
EA
δ
=
AE
L
[N/m]
("PLEA" or " FLEA" formula)
Example: stress and strain under uniaxial
loading
A cylindrical rod with radius r =
1.26 cm is tested in a uniaxial
tension test. Before applying any
load, the “gage length” (distance
between A and B) is 30 cm. After
applying a tensile load F of 1000 N,
the distance between A and B
increases to 31.5 cm. Determine the
tensile stress σ and tensile strain ε in
the rod, and the modulus of elasticity
E.
l
slope = modulus E
(N/m2 or Pa)
Stiffness under uniaxial loading
P
∆l
material
Hooke’s Law for uniaxial loading
k=
F
A
strain ε (percent)
deflection ∆L (m)
• Hooke’s Law is sometimes written as
follows:
σ=
l
stress σ (M/m2 or Pa)
• the ratio δ/L is the strain
ε under axial loading;
strain is dimensionless
although it is sometimes
referred to as having
units of [m/m] or %
The modulus of elasticity does not depend on geometry; it
is a material property rather than a structural property
force F (N)
• the ratio P/A is the stress
σ under axial loading;
the units of stress are
[N/m2] or pascals [Pa]
Modulus of elasticity is a material property
P
L+δ
L
structural
stiffness
under axial
loading
A
P
Example: stress and strain under uniaxial
loading (cont)
σ=
1000
1000
F
=
=
A π ( 0.0126) 2 0.0005
= 2 x 10 6 Pa = 2 MPa
ε=
∆L (0.315 - 0.30)
=
= 0.05 m/m
L
0.30
E=
σ 2 x 106
=
= 40 MPa
ε
0.05
3
Definitions related to the stress-strain
diagram
stress
energy to failure
6
yield strength
1
(kPa) 2
ultimate strength 2
1
3
ultimate strain
5
yield strain
4
elastic limit
1
toughness
6
σ
EL
AS
TIC
modulus
Viscoelastic Properties of Tissues
A
PL
I
ST
3
C
rupture
6
(area
under
curve)
4
5
strain ε (m/m)
• Tissues exhibit both
elastic and viscous
properties =
“viscoelastic”
• Like most polymers,
tendon and ligament
exhibit “viscoelastic”
characteristics:
– Stress relaxation
– Creep
– Hysteresis (energy
dissipation)
– Increased stiffness with
increased rate of
loading
deformation
Tendon and ligament exhibit viscoelastic
(rate-dependent) behaviour
elastic
viscoelastic
stress
time
In purely elastic materials (springs), a step
increase in load (stress) causes an instantaneous
deformation that remains constant over time and
disappears after the load is removed. In
viscoelastic materials (tendons and ligaments),
deformation is time-dependent.
Schematic representation of cyclic creep
in the MCL of the knee
There is a time dependant increase in elongation
when a viscoelastic material is subjected to a
repetitive constant stress (cyclic creep)
• Any deformation or residual deformation
alters the mechanical response of the tissue
reducing its stress bearing capacity.
• The tissues that frequently get injured due
to occupational biomechanical hazards are
ligaments, tendons, muscle and nerves
(cartilage and bones less so).
• All biological tissues are viscoelastic so we
will review the properties of viscoelastic
structures.
Stress Relaxation and Creep
• Stress Relaxation: A
step increase in
deformation (or strain)
causes an increase in
load (or stress), that
reduces in magnitude
with time.
• Creep: A step increase
in load (or stress)
causes a deformation
(or strain) that
increases in magnitude
with time.
(A) Schematic representation of stress-relaxation
(decreasing stress over time under a constant strain).
(B) Schematic representation of creep (increasing
deformation over time under a constant stress).
Hysteresis
• Loading and
unloading curves are
not the same.
Load
• Work done during
lengthening
(deformation) is
greater than the work
recovered during
shortening (return).
• Shaded area represents
lost energy (heat).
• Tissues have to absorb
the energy of
set
deformation which
can cause problems.
Hysteresis Loop
)
rm
fo
e
(d
ad
Lo
nl
U
d
oa
)
rn
tu
e
(r
Deformation
4
Hysteresis during cyclic loading of a
knee ligament
• As a ligament
undergoes cyclic
loading its relaxation
behaviour results in
continuously
decreasing stress.
• Repetitive stress
causes failure at a
lower load than that
required to cause
failure in a single
application.
The modulus and strength of bone depends on
rate of load
Bone is viscoelastic it’s modulus and
strength increase with
increasing rate of
loading
Part II: Mechanisms of Tissue Injury and
Tolerance
Outline:
• Definitions of injury
• Factor of risk
• Tissue tolerance
• Fatigue failure
• Chronic vs. acute injuries
• Review of tendons, ligaments, cartilage, and bone
Stiffness and strength increase with rate of
loading
Load
Fracture
Quick
Slow
Fracture
Deformation
Review questions
• what are the five modes of loading, and what patterns of
stress do each of these create?
• what is the difference between shear force and axial force?
between a twisting moment and a bending moment?
• what is the difference between a linear material and a
nonlinear material?
• for uniaxial loading, how do you convert force-deflection data
to stress-strain data?
• what parameters affect the stiffness of a structure under
uniaxial loading?
• what is the 'PLEA" formula for deflection of a structure under
uniaxial loading?
• what is definition of yield stress? of failure stress?
• what is the definition of viscoelastic?
• what are 4 viscoelastic characteristics of biological tissues?
WCB Definition of Musculoskeletal
Injury (MSI)
• 4.46 WCB Ergonomics (MSI) Regulations:
MSI “means an injury or disorder of the
muscles, tendons, ligaments, joints, nerves,
blood vessels or related soft tissue including
a sprain, strain and inflammation, that may
be caused or aggravated by work”
5
Biomechanics is well-suited to study the
causes and effects of human MSI
BIOMECHANICAL FACTORS
PSYCHOSOCIAL FACTORS
N
IO
AT
CO
CU RP
LT O R
UR AT
E EO
FA
CT RGA
OR N
S IZ
Tools, materials
Work station layout
Work / recovery cycles
Work rate & duration
&
Peak and cumulative forces on
tissues, postures of arms and torso,
forces on hands, perceived physical
demands
Design of:
Perceptions of:
• Reward adequacy
• Job satisfaction
• Control over work
• Psychological demands
• Work social environment
• Co-worker support
• Input to decisions
A
IZ
AN
RG S
O OR
TE C T
RA A
O EF
RP UR
CO ULT
C
PERSONAL FACTORS
O
TI
N
&
• Many different disciplines have a role to play
in a comprehensive understanding of injury.
• However, of all the scientific disciplines,
physics and its sub-discipline mechanics are
arguably most central to the study of injury.
• The fundamental relation between mechanical
energy and injury highlights biomechanics as
the logical discipline to study the causes and
effects of human musculoskeletal injury.
Whiting & Zernicke, 1998
CORPORATE ORGANIZATION
& CULTURE FACTORS
• Technique & motor skill
• CV & respiratory endurance
• Tissue tolerance/previous injury
• Muscular endurance & strength
• Coping ability/willingness
• Height, weight, age, sex
• Life style / Behaviour
• Knowledge
Biomechanical Factors
In Kin 380 we
focus on the
biomechanical
factors
A global theory of
musculoskeletal injury
precipitation. Kumar
1999, (modified from
Kumar 1991).
Risk for injury depends on the ratio of
applied load to failure load
• The applied load
depends largely on the
task characteristics
• The failure load is
affected by factors
such as the rate of
tissue loading, tissue
fatigue, frequency of
loading, and age
φ =" factor of risk"
applied load
failure load
φ ≥ 1 → failure
φ < 1 → no failure
φ=
• Kumar (1999) argues that overexertion can
be created by exceeding the normal
physical and physiological in any one of:
force (Fx), exposure time (Dy), range of
motion (Mz).
• Kumar symbolically represents
overexertion (OE) with the equation below.
OE = ∫ ( Fx, Dy, Mz )
The strength of bone declines with number of
cycles during cyclic loading
Like all materials,
bone is susceptible to
fatigue - when subject
to repetitive or
fluctuating stress, it
will fail at a stress
level much lower than
that required to cause
fracture on a single
application of load
ultimate
stress
stress
σ
(N/m2)
Fatigue (or
endurance) limit
100
106
log (number of cycles
to failure, N)
6
Tissue tolerance declines with repetitive
loading
Acute trauma
Tissue strength (conditioning) is also a
factor in risk of injury
Keep the big picture in
mind. If you have little
movement/ exercise
then the tissues become
more susceptible to
injury due to poor
conditioning.
Risk of
Injury
Load
Injury Threshold
Tolerance
Chronic
Repetitive
Repetition
Hippocrates (460-377 B.C.)
“All parts of the body which have a function, if
used in moderation and exercises in labours to
which each are accustomed, thereby become
healthy and well-developed: but if unused and left
idle, they become liable to disease, defective in
growth, and age quickly. This is especially the
case with joints and ligaments, if one does not use
them.”
LeVay 1990. p30.
Fatigue Characteristics
Sandover (1983; 1985) proposed a model of fatigue-induced
failure of the intervertebral joint using the data of Lafferty
(1978) and others:
logN = log(Su/Sp)x
N = (Su/Sp)x
where:
N = # of cycles to failure, Su = static failure stress,
Sp = applied repetitive stress, and x = constant.
too little
too much
Movement (repetition), force (lifting)
physical activity, sitting or standing
Fatigue Failure
• Compression fracture is the common failure
mode of the vertebra-disc complex in severe
axial loading. This mechanism does not apply
to repetitive loading within the linear portion
of the stress-strain curve.
• Low back pain and back disorders associated
with whole-body vibration and repeated shocks
point to a chronic degeneration of tissues,
rather than acute failure.
Fatigue life of
animate
tissues.
Straight lines
represent the
functions
N = (σu/σ)x
The value of exponent x varies between tissues and test method
from x = 5 for cortical bone (Carter et al. 1981) to 20 for
cartilage (Weightman, 1976).
The degree of fatigue damage is given by the summation of
ni/Ni,
where:
ni is the number of cycles at a particular stress level, and
Ni are the number of cycles to failure at that stress
7
Biomechanical Injury
Sudden
Force
Impact / Sudden
Trauma
Chronic
Volitional
Activity
Overuse
Trauma
Cumulative Loading
Fractures
Contusions
Concussion,
Sprains and
Strains etc
Pinched nerves
Tenosynovitis
Low back pain
Tendonitis, etc.
Ligament and Tendon Injury
Acute vs. Chronic Injuries
If you had a force vs
time graph the area
under the curve would
be an impulse (Ft => the
cumulative loading of
that tissue)
Acute
Force
• Assessing the effect of cumulative loading
is a difficult thing.
• If there is adequate recovery time then even
high cumulative loads may be safe.
• On the other hand a one time high peak
force over a very short period of time (low
cumulative load) may exceed the strength of
the tissue and cause injury (acute injury).
Chronic
Time
Ligaments are composed largely of water
and type I collagen
• These are the most frequently
injured connective tissues. Both
are made of collagen.
– Reversible deformation to 4%.
– Rupture around 8-10% deformation.
• Muscles are rarely affected by
unrecovered deformation, it is
generally the passive structures of
muscles (e.g. the sarcolemma) that
tear.
Ligaments get strength from collagen
and elasticity from elastin fibres
Collagen
• Deformation range
is small (6-8%)
• Strength
– 50% of that of
cortical bone
tested in
tension
Elastin
• Deformation range is
large (>100%, 150%
Fawcett 1986)
• Strength
– weak
8
Elastic Fibres (Elastin and microfibrils)
Progressive failure of the anterior cruciate ligaments
(cadaver knee tested in tension to failure at a
physiologic strain rate, Noyes 1977)
Deformation range
of collagen (6-8%)
Stress
Stress strain curve
of elastin
200
The quantity of healed tissue does not
determine the quality
•
•
•
•
Maximum load to 100%
failure for
primate anterior
cruciate
ligaments does not
return to pre50%
immobilization
status even after
12 months of
reconditioning
(Noyes 1977)
100%
91%
79%
61%
Reconditioned 12 months
• Rarely in an occupational setting does the
magnitude of the load exceed the tensile
strength of the collagenous fibres.
• More commonly repetitive motions with
inadequate recovery periods are the problem.
• In these cases the cross sectional area can be
reduced increasing the stress concentration.
• Prolonged static loading, which results in creep,
also renders the tissues vulnerable.
Ligament and tendon healing takes time!
Reconditioned 5 months
Mechanisms of Ligament and Tendon Injury
Immobilized 8 weeks
100
Strain (percent)
Control
0
Healed MCL exhibits inferior structural and
material properties after injury
Eighteen weeks of remobilization were necessary to
reverse the detrimental effects of a six-week
immobilization on the structural properties of ligaments
(Laros et al., 1971).
Structural properties (geometry) nearly normal but
mechanical properties of healed ligaments almost
always remain inferior when compared to normal tissue.
This is possible as the tissue accumulates mass to
compensate for inferior tissue quality.
Some areas of healed MCL were up to 2.5 times larger
than controls (Ohland et al., 1991)
9
healthy ligament
Articular Cartilage
0 weeks post injury
6 weeks post injury
Articluar cartilage, a dense
white connective tissue, coats
(1-7 mm thick) the ends of
bones articulating at synovial
joints. It serves two
purposes:
1:Spreads the load. Cartilage
can reduce the maximum
contact stress by 50% or
more.
2:Reduces friction during
movement.
14 weeks post injury
Prolonged Static Loading of Articular
Cartilage
Static Loading
• Static loading on ligaments and joints
therefore stress the tissue and reduce the
effectiveness of its mechanical response.
Articular cartilage has a combination of elastic and
viscoeleastic properties. As load is applied, deformation
increase with time, first in an elastic fashion, then with a slow
creep. With the removal of the load there is an elastic recoil
and the a slow recovery to the base line.
Bone Injury
• Work related injuries are commonly
associated with damage to soft
tissues or joints.
• However the prevalence of bone
fractures is still high.
• In 1995 factures accounted for 6.1%
of two million total injuries in
private industry in the States.
• Many solutions would revolve
around fall prevention issues which
are not the focus of Kin 380.
• This time dependant quality is due to the
fact that connective tissue has interstitial
fluid that gives it viscous properties as well
as elastic properties.
Stress/Strain Curves
Metal (ductile)
Stress
Glass (brittle)
Bone
Strain
10
Bone is stronger in compression than in
tension.
Stress to Fracture
Tension
Compression
Compression
Tension
Shear
Compression
Tension
Shear
The strength and modulus of bone vary
with the direction of loading
• Bone is stronger in
compression than in
tension
• Bone is anisotropic: its
modulus and strength (in
tension and compression)
depend on the orientation
of the tissue with respect
to the load
• For cortical bone:
(σ ult )tesnion
long
(σ ult )tesnion
trans
Elong ≈ 17 GPa
Etrans ≈ 11 GPa
In lifting, bending moments cause
tension and compression on the disc and
vertebrae.
≈ 135 MPa
comp
(σ ult )long
≈ -190 MPa
≈ 50 MPa
comp
(σ ult )trans
≈ -130 MPa
Bone fracture results from excessive
microdamage
• Bone failure occurs from
the formation and
propagation of bone
cracks, caused by
excessive monotonic or
repeated (fatigue) loading.
• In cortical bone, the cracks
form in transverse and
longitudinally-oriented
Haversian lamellae.
Bone fracture results from excessive
microdamage (cont)
In trabecular bone,
excessive loading causes
cumulative failure of
individual trabeculae.
Whole-bone fracture may
occur when the rate of
microfracture formation
exceeds the rate of
microfracture healing.
(a)
stress
(b)
strain
Mean and Range of Disc Compression
Failures by Age
10000
microcalluses
(healed trabeculae)
Compressive
Forces
Resulting in
Disc-Vertebrae
Failures at
L5/S1 Level
(Newtons)
Should job design
factor in age?
8000
6000
4000
2000
(Adapted from
Evans, 1959, and
Sonoda, 1962)
0
<40
40-50
AGE
50-60
>60
11
Review questions
• in addition to biomechanical factors, what other
factors affect risk for musculoskeletal injury?
• how is the “factor of risk” defined?
• how does the strength of a tissue change with
repeated loading?
• what is the most common way that ligaments and
tendons get injured in occupational settings?
• what does it mean that bone is anisotropic?
12