Outline
Introduction to
Mechanical Behavior of
Biological Materials
• Modes of loading
• Internal forces and moments
• Stiffness of a structure under uniaxial loading
• Stress and strain due to axial loading
• Elastic modulus of a material
• Definitions from the stress-strain diagram
• Shear stress and strain
• Stresses due to bending and torsion
• Composite structures: springs in parallel and in
series
Ozkaya and Nordin
Chapter 7, pages 127-151
Chapter 8, pages 173-194
KIN 201
2007-1
Stephen Robinovitch, Ph.D.
1
Forces cause motions and deformations
2
There are five primary modes of loading
motion
• compression and tension
are both “axial loading”
force
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
3
• 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 “combined
loading”
4
Each mode of loading creates a
distinct pattern of internal stress
tension
compression
P
!=
A
P
A
shear
P
!=
A
Internal forces and moments
-P
A
!=
P
P
A
P’
P’
P’
bending
!=
My
I
torsion
T
M
!=
Ty
J
T’
M’
5
Internal forces and moments (cont.)
• Fx: axial force (tends to
elongate the structure); we
shall simply use F
• Fxy, Fxz : shear force (tends to
shear the structure apart); we
shall simply use V
• Mx: twisting moment or
torque (tends to twist the
structure about its long axis);
we shall simply use T
• My, Mz : bending moments
(about the structure’s short
axes); we shall simply use M
• for a structure in
equilibrium, external
forces are balanced by
internal forces and
moments
• we usually resolve
internal forces and
moments into
components parallel and
orthogonal to the long
axis of the structure
6
Example: internal forces and moments
During single
legged stance, the
proximal femur is
loaded as shown.
y
V
M
F
x
7
Determine the
forces and moments
acting at the femoral
neck cross-section
A-A.
!
FJ
b
Section A-A:
e
c
a
c
D
a
b
8
Example: internal forces and moments
(cont.)
! Fx = 0 :
Tensile loads cause a body to stretch
• Consider a structure of original length L and crosssectional area A, that is loaded by two equal and
opposite forces of magnitude P
• we will use the Greek letter delta (") to represent the
change in length (or deflection) of the structure after
application of the forces
F = FJ cos"
! Fy = 0 :
V = FJ sin"
L+"
P
! Ma = 0 :
M = " FJsin# $ e
P
A
L
9
Biological structures have nonlinear
stiffness
P
linear
Stiffness under axial loading
• stiffness is the slope of
the load-deformation
load P
curve
(N)
• we shall denote stiffness
with the letter k
• units for stiffness are
N/m
• stiffness is affected by
the structure’s geometry
(length and crosssectional area), and is
therefore a structural
property
10
• if a structure is made from a
linear material, the stiffness will
be constant (regardless of
geometry)
• if the structure is made from a
nonlinear material, the stiffness
will vary with the load
stiffness
k
deflection " (m)
• a steel spring has a linear
stiffness
• biological tissues have nonlinear
stiffness
11
constant
stiffness k
"
P
nonlinear
varying
stiffness k
"12
Stiffness depends on geometry and material
Axial stiffness increases with
increasing cross-sectional area,
and decreases with increasing
length
Modulus does not depend on geometry
P
A
• if we divide P by the crosssectional area A, and " by the
original length L, the forcedeflection curves collapse onto one
common trace
%
• the slope of this trace is the
modulus of elasticity E (or Young’s
modulus)
• Young’s modulus does not depend
on geometry. It is a material property
rather than a structural property
13
often
!
L 14
The modulus of elasticity does not depend on geometry; it
is a material property rather than a structural property
F
A (x-sectional area)
E (modulus)
L
M
F
AXIAL STRAIN ! =
[m/m]
"
L
15
!=
l
%
structural
slope = stiffness k
(N/m)
deflection !L (m)
stress # (M/m2 or Pa)
AXIAL
STRESS
P
!=
A
[Pa]
force F (N)
• the ratio "/L is the
strain # under axial
loading; strain is
dimensionless although
it is sometimes referred
to as having units of
[m/m]
%
Modulus of elasticity is a material property
Definition of axial stress and strain
• the ratio P/A is the
stress " under axial
loading; the units of
stress are [N/m2] or
pascals [Pa]
• the units of E are
called Pascals, Pa
[N/m2],
!
L
P
A
!=
F
A
"l
l
material
slope = modulus E
(N/m2 or Pa)
strain $ (percent)
16
Hooke’s Law for axial loading
Stiffness under uniaxial loading
• The “constitutive relationship” that defines a
material’s stress-strain behaviour (in the elastic
region) is referred to as Hooke’s Law. For axial
loading, Hooke’s law is:
By definition, stiffness is the
ratio of load divided by
deflection. Based on Hooke’s
Law, the stiffness of a
structure under axial loading
is:
! = E"
• This can be expressed as:
PL
!=
EA
P AE
k= =
!
L
(" PLEA" or " FLEA" formula)
[N/m]
P
L+"
L
structural
stiffness
under axial
loading
A
P
17
Example: stress and strain under
axial loading
18
Example: stress and strain under
axial loading (cont.)
A section of tendon having a
cross-sectional area of 0.5 cm2 is
subjected to a tension test. Before
applying any load, the “gage
length” (distance between A and
B) is 30 mm. After applying a
tensile load F of 1000 N, the
distance between A and B
increases to 31.5 mm. Determine
the tensile stress #, the tensile
strain $, and the modulus of
elasticity E.
F
1000
=
A 0.00005
= 2 x 10 7 Pa = 20 MPa
!=
"=
#L (0.0315- 0.030)
=
= 0.05 m/m
L
0.030
! 2 x 107
E= =
= 400 MPa
"
0.05
19
20
Young’s modulus of various
orthopaedic biomaterials
Definitions related to the stressstrain diagram
ultimate strength 2
modulus
3
ultimate strain
5
yield strain
4
elastic limit
1
toughness
6
rupture
PL
1
T IC
1
stress
#
(kPa) 2
AS
yield strength
6
EL
energy to failure
3
AS
C
TI
6
(area
under
curve)
4
5
strain $ (m/m)
21
Stresses due to bending
Asymmetric structures have a directional
dependency to stiffness under bending.
the axial stress due to bending is given by :
M y
!b = b
Ia
where Mb = F " x is the bending moment; y is the distance
2
from the neutral axis; and I a = # y dA is the area moment
of inertia (different than the mass moment of inertia
tension
y
compression
“neutral axis”
(#b = 0)
2
# r dm)
F
For the same applied
bending moment, in
what orientation is the
beam “stiffer”(i.e., will
deform less)? What is
the reason for the
difference?
x
23
24
Area moment of inertia, a purely
geometrical parameter, determines axial
stress due to bending
Area moment of inertia (MOI) for
common cross-sections
For a rectangular
cross section, area
MOI varies with
the cube of
height, but only
the first power of
width. For a
circular section,
area MOI varies
with radius to the
4th power.
At what location are
the stresses greatest
in beam (1)? Is the
same true for beam
(2)? What theoretical
advantages are
provided by the
beam (2) design?
(1)
25
Flexural rigidity is the product EI
(2)
26
Shear Stress and Strain
Stress does not depend on modulus :
M y
!x = b .
Izz
But strain depends on modulus :
Mb y
"x =
.
EI z z
The product EIz z is called the
"flexural rigidity."
Shear stress causes angular
deformation of a material,
according to Hooke’s Law for
shear stress and strain:
! x y = G" x y
where G is the shear modulus
(in Pa), &xy is the shear stress in
the xy plane (in Pa), and 'xy is
the shear strain in the xy plane
(in radians).
Can you understand the differences
in flexural rigidity associated with
the different fracture fixation plate
designs on the right?
27
28
Tension and compression exist in a body
subject to pure shear loading
Shear stresses and
strains due to torsion
Planes oriented 45
deg to the direction
of loading will have
tensile or
compressive stress.
These normal
stresses will be
equal in magnitude
to the shear stress in
planes parallel to
the loading.
Shear stresses due to torsion are :
Tr
!=
J
where J is the polar moment
of inertia
2
" r dA.
A
Corresponding shear strain are :
Tr
# =
.
JG
The product JG is called the
"torsional rigidity."
29
Shear stresses due to
torsion
30
Mechanics of the spiral fracture
A
B
Why are shear stresses
in the tibia due to
torsion lower in section
A than section B, even
though the thickness of
the cortex is much
greater at B?
31
As we shall discuss
later in the course, bone
is weaker in tension
than in compression or
shear. “Spiral fractures”
due to torsional loading
therefore initiate and
propagate 45 deg to the
long axis of the bone,
where tensile stress is
maximum.
32
Combined loading: stresses due to
axial and bending loads
Composite structures: springs in parallel
(load sharing)
What is the distribution of stress at section A-A due to the
force F?
a
F
• each spring element has
the same deflection
F
A
Mb=F·a
A
x
tension
x
y
tension
compression
Mx
!bending =
I
compression
compression
F
!axial = "
A
ax 1
!total = F # " %
$ I A& 33
Composite structures: springs in series
(load transfer)
F
F
• each spring element
supports the same force
ka
• the deflection in each
spring element is not
the same
• the total effective
stiffness (ktotal), and the
total deflection of the
system is dominated by
the element with the
lowest stiffness
kb
F
ktotal
k k
= a b
ka + kb
F
F
• the force in each spring
element is not the same
• the total effective
stiffness (ktotal), and the
total deflection of the
system, is dominated by
the element with the
highest stiffness
kb
x
ka
F
ktotal = ka + kb
In series or parallel?
• running shoe and ground
SERIES
F
• right and left upper extremity
during push up
• ankle, knee, and hip joint during
single-legged stance
• sarcomeres in muscle
• articular cartilage and
subchondral bone in knee joint
ka
kb
PARALLEL
kb
F
ka
x
36
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 the formula for stress due to an applied bending moment?
• how do we define the effective stiffness of springs in parallel? of
springs in series?
37
• what is definition of yield stress? of failure stress?