Chapter 1
Introduction to Terms and Test Procedures
Purpose:
1.
To define various terms used to describe the structural and material properties of
biological tissues.
2.
To illustrate the most common mechanical testing procedures used to study biological
tissues.
3.
To demonstrate how the structural and material properties of tissues are obtained from
the mechanical testing results.
Understanding the terminology used in mechanics is fundamental to understanding the
mechanics of biological materials. Like any field, there are specific terms used in mechanics to
describe specific characteristics of the material under study. The terms and definitions given
below are used throughout the rest of this text and therefore should be understood before
continuing.
Mechanics and Biomechanics:
Mechanics involves the analysis of dynamic systems. This is a broad field that includes
strength of materials, stress-strain relationships, mechanical design, fluid flow, heat and mass
transfer, control, dynamics, and motion of charged particles. Most organs and tissues within a
biological system are affected in some way by these various issues. Thus, ’Biomechanics’ is
‘mechanics’ applied to biological systems. The focus of the material presented in this book is on
strength of materials, stress-strain relationships, structural design, and dynamics.
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Force and Moments:
Biological tissues are designed to withstand a variety of forces and/or moments. A force
is a vector quantity and is defined as the action of one body on another. A force will cause a
body to deform and/or accelerate. If the body is constrained or acted upon by equal and opposite
forces, then the body deforms but does not accelerate. A moment is also a vector quantity and
represents the tendency of a force, acting at a distance from an axis of rotation of a body, to
cause that body to rotate. A moment can be expressed as the cross-product between the position
vector from the axis of rotation to the point of application of the force on the body and the force
vector. Any material subjected to a force or a moment deforms in some way. These
deformations are considered negligible in rigid body analyses, but play an important role in the
behavior of many biological tissues.
Stress:
When a force acts on an object, tending to produce a deformation in that object, then an
internal resistance to that deformation is produced in the material. The internal reaction to the
applied force is called stress. Stress is defined as the force per unit of cross-sectional area.
Forces can be applied to materials at any angle producing complex stress patterns within the
material. A material having a larger cross-sectional area can sustain a larger ultimate force
before failing compared to a smaller specimen of similar “material quality” (Figure 1). The
stress at failure would be the same for both specimens illustrated in (Figure 1). The size and
“quality” of the specimen affect the ultimate force that the specimen can sustain. Thus, the
concept of stress is important for comparing the behavior of different materials (e.g. ligament vs.
tendon) or the same material from different specimens (e.g. old vs. young or healthy vs. healing
tendon). Stress is a normalized force accounting for differences in a specimen’s size, but not the
“quality” of the material. If the cross-sectional area of a material is “A” and the force
transmitted through the material is “F”, then the stress (σ) in the material is the ratio F/A. Do
you see any problem with this definition? When testing a specimen its cross-sectional area does
not remain the same. During a tensile test the specimen will "neck down" or develop a smaller
cross-sectional area. Stress may be defined as the force per original cross-sectional area, or the
force per instantaneous cross-sectional area. It is technically more difficult to measure the
instantaneous cross-sectional area so most often the first definition is used. However, even the
determination of ‘original’ cross-sectional area can have problems because it can be affected by
the hydration level of the specimen and the extent of preload applied to the specimen.
Different tissues are exposed to different types of stresses based on their structure and
function. Ligaments and tendons can act only in tension. Whereas bones can sustain tensile
stresses, compressive stresses, and shear stresses. Biological tissues tend to fail at different
stress levels depending on the type of force being applied (e.g. tensile, compressive, shear). The
resistance to tensile force comes from the interatomic attractive forces of the material. The
resistance to compressive force comes from the interatomic repulsive forces of the material that
rise sharply at short interatomic distances (Simon, 1994). The ease that one portion of a
materials slides over another portion determines the material’s resistance to shear deformation.
Torsion produces shear stresses in a material.
Stress can not be measured directly, but the force applied to a specimen can. Therefore,
stress is determined experimentally by using a force transducer to quantify the force applied to
the specimen and dividing this force by the cross-sectional area of the specimen.
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Figure 1 - Size affects on ultimate failure load. Tissues with larger cross-sectional area will
have higher failure strengths and absorb more energy before failure compared to smaller tissues
having similar material properties. For the same force the larger tissue will deform less.
Strain:
The amount of deformation that a material experiences normalized with respect to some
reference length is defined as the strain. Strain can be represented in several ways. If L is the
absolute length of a specimen and Lo is some initial length, then the strain (ε) can be defined as:
ε = (L - Lo)/Lo
or
ε = (L - Lo)/L
The two definitions of strain are both dimensionless, but they yield different numerical results.
When reviewing the literature it is important to know how the strain was defined. Strain is often
reported in terms of microstrain (µε) (one microstrain is equal to a 1 micron change in length per
1 meter of original length). Stretch ratio, defined as L/Lo, is an additional term used to describe
tissue deformation. The longer the initial length of a tissue, the greater the elongation it can
withstand before failure (assuming similar material properties) (Figure 2).
When a material deforms in one direction it may also deform in other directions.
Poisson’s ratio (ν) is a term used to describe the relationship between transverse strain and
longitudinal strain. Poisson’s ratio is defined as the negative ratio of transverse and longitudinal
strains. A fully compressible material has ν=0. A completely incompressible material has
ν=0.5.
Strain is measured by quantifying the length and change in length of a specimen with a
ruler, caliper, or a variety of optical and electronic devices. It is usually better to have a noncontact method if possible so that the instrument itself does not effect the specimen being
studied.
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Figure 2 - Affects of tissue length on the deformation that can be withstood before failure.
Longer tissues can withstand greater deformation and absorb greater energy before failure
compared to shorter tissues having similar material properties. However, for a given elongation
of a tissue, the longer the tissue rest length the lower the force it will develop and the lower the
energy that it will absorb during the deformation.
Structural vs. Material Properties:
A biological tissue is often described in terms of its structural and material properties.
The structural properties characterize the gross tissue in its intact form and how it will behave.
Important structural properties are represented by a force-deformation curve as illustrated in
Figure 3. These structural properties characterize the relationship between force and
deformation and must be understood in order to predict how a tissue will behave in-vivo.
Material properties characterize the behavior of the material comprising the tissue and to
a first approximation are independent of the size of the tissue (Figure 3). The material properties
are expressed in terms of the stress-strain relationship of the material. Structural and material
property curves are similar in appearance, differing only by a scaling factor.
The stiffness of a material represents the materials ability to resist deformation. Stiffness
is commonly characterized by the slope of the linear region of a stress-strain curve, also referred
to as Young's Modulus. To describe the slope of other regions of the stress-strain curve a
Tangent Modulus is often defined. If a Tangent Modulus is defined it should have associated
with it a strain value or range of strains. Values for Young's Modulus for various materials as
reported by Frost (1971) are given in Table 1. A tissue can have a different modulus for
different loading conditions (e.g. shear modulus, compression modulus). The larger the stiffness,
the greater the force required to cause a given deformation. If the stress in a material is directly
proportional to the strain for all strains, the material is called a Hookean material.
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Figure 3 - Force-deformation and Stress-strain curves illustrating structural and material
properties respectively.
Table 1 - Examples of Young's Modulus for various materials.
Substance
Young's Modulus (GPa)
Aluminum
71
Titanium
120
Iron
140
Stainless Steel
200
cortical/trabecular Bone (wet)
17 / 1
Polymethylmethacrylate (pmma)
2.3
Nylon
3
Rubber
0.035
There are several additional terms of interest for describing the mechanical behavior of
tissues (see Figure 4). The stress in a material will increase linearly with increased strain for a
range of strains up to the proportional limit. If a force is applied to a material and then
removed, the material will return to its original shape as long as the force did not exceed the
elastic limit. If the material does not return to its original shape it is said to have experienced
plastic or a permanent deformation. A material has some ultimate stress and ultimate strain
that if exceeded, then the material will fail. That is not to say that a material does not experience
damage at stresses or strains below the ultimate values. The area under a force-deformation
curve represents the energy absorbed by the material. This can be an important parameter for
biological tissues. The area under the elastic portion of the force-deformation curve is a measure
of the material’s resilience. The area under the force-deformation curve of a material taken to
failure is a measure of the material’s toughness. A material that fractures before undergoing any
permanent deformation is said to be brittle. A material that reaches a yield point and then
undergoes further deformation before failing is said to be ductile (see Figure 5).
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Figure 4 - Illustration of a stress-strain curve with various quantities of interest identified.
Figure 5 – Force-deformation curves for materials having various combinations of structural
properties.
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Ideal materials are isotropic and homogeneous. A material is called isotropic when its
properties are the same in each of three coordinate axes (x,y,z). Tensile and compressive
properties may be different, but each respective property must be the same in three directions. A
material is said to be homogeneous if it is made of the same material throughout. Homogeneity
implies a well ‘mixed’ material. Depending on the scale, specimens from biological systems
may or may not be homogeneous. Typically biological tissues are considered anisotropic and
nonhomogeneous.
Viscoelastic Properties
Biological tissues are viscoelastic materials. This means that their behavior is time and
history dependent. A viscoelastic material possesses characteristics of stress relaxation, creep,
strain rate sensitivity, and hysteresis.
Force-relaxation (or stress-relaxation) is a phenomenon that occurs in a tissue stretched
and held at a fixed length (Figure 6). Over time the force present within the tissue continually
declines. Force-relaxation is strain rate sensitive. In general, the higher the strain rate, the larger
the peak force and subsequently the greater the magnitude of the force-relaxation.
In contrast to stress-relaxation that occurs when a tissue’s length is held fixed, creep
occurs when a constant force is applied across the tissue (Figure 7). If subjected to a constant
tensile force, then a tissue elongates with time. The general shape of the displacement-time
curve depends on the past loading history (e.g. peak force, loading rate).
Another time-dependent property is strain rate sensitivity (Figure 8). Different tissues
show different sensitivities to strain rate. For example, there may be little difference in the
stress-strain behavior of ligaments subjected to tensile tests varying in strain rate over three
decades while bone properties may change considerably (this concept is discussed in more detail
in later chapters).
The loading and unloading curves obtained from a force-deformation test of biological
tissues do not follow the same path (see Figure 9). The difference in the calculated area under
the loading and unloading curves is termed the area of hysteresis and represents the energy lost
due to internal friction in the material. The amount of energy liberated or absorbed during a
tensile test is defined as the integral of the force and the displacement. Hence the maximum
energy absorbed at failure equals the area under the force-displacement curve.
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Figure 6 - Example of a force or stress relaxation response. Such a response might occur in
tissues around the knee if the knee is moved to a specific knee position and maintained in that
position.
Figure 7 - Example of a creep response. Such a response might occur in a tendon subjected to a
constant muscle force.
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Figure 8 - Strain-rate affects on stress-strain properties. The faster a tissue is loaded the greater
the stiffness and the higher the ultimate failure load. Some tissues may exhibit higher ultimate
strains for higher strain rates (e.g. ligaments), others may have lower ultimate strains (e.g. bone).
Figure 9 - Loading and unloading curves during a tensile test of a biological tissue. The two
curves are not identical. The difference in area under the two curves is the area of hysteresis and
represents the energy lost due to internal friction within the material.
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Viscosity:
The viscosity of a fluid is a measure of the fluid's resistance to flow. Fluids with large
viscosity are used in slow moving bearings to reduce wear. Some viscosity values for various
fluids are listed below. Synovial fluid exists within certain joints and it may help to reduce
friction and wear of articulating surfaces.
Table 2 - Viscosities of some fluids. The viscosity of synovial fluid is within a range of many
common fluids.
Fluid
Viscosity (centipoise)
water
1
olive oil
84
synovial fluid
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Coefficient of Friction:
The coefficient of friction is that fraction of the force transmitted across two bearing
surfaces that must be used to initiate movement (µS -static friction) or keep the surfaces moving
(µD -dynamic friction). The static coefficient of friction between two surfaces is always greater
than the dynamic coefficient of friction. Joints of the human body are well designed to reduce
the coefficient of friction between articulating surfaces. Some coefficients of friction for various
surfaces are given in Table 3.
Table 3 - Coefficient of friction acting between various surfaces. The coefficient of friction
acting between articulating human joints is relatively low.
Surfaces
Coefficient of friction
µS
µD
rubber tire on dry concrete
1.0
0.7
aluminum on aluminum
0.8
0.5
steel on ice
0.03
teflon on teflon
0.05
ball bearings
0.1-0.001
graphite on graphite
0.1
hip implant materials
0.05
human joints
0.001
Testing Procedures:
Structural properties of biological tissues are usually determined through some form of
mechanical testing (e.g. tensile tests, compressive tests, bending and torsion tests). Customized
workstations utilizing force transducers, clamps, and an actuator to control the distance between
clamps are common place. Commercial systems are also available and vary in design depending
on the type of tissue being studied (e.g. macroscopic vs. microscopic, hard tissue vs. soft tissue
etc.) and the type of loading rates required. Instron and MTS are the two most common
suppliers of mechanical testing systems. Most systems allow either force control or length
control.
Mechanical testing of tissue in-vivo is very difficult and hence not commonly performed.
Some of the techniques that have been utilized include: 1) buckle transducers to monitor tendon
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and ligament forces, 2) telemetried pressure sensors to measure joint contact pressure, and 3)
strain gauges to quantify bone and ligament strain. Some non-invasive approaches have also
been employed. Ultrasound techniques have been used to detect changes in the speed of sound
in different tissues and these changes have been correlated with the tissue's elastic properties.
Various imaging techniques have also been used to quantify tissue geometry and deformation.
Summary:
There are several terms that the biomechanics students should understand in order to
describe the behavior of biological tissues. These include force, moment, stress, strain,
deformation, strength, energy, creep, relaxation, hysteresis, viscosity, and friction to name a few.
In addition to understanding the terminology the student should understand the difference and
relationship between structural properties and material properties as well as the methods
employed to quantify these properties.
References:
Basic Biomechanics of the Musculoskeletal System. 2nd Edition. Edited by Nordin, M. and V.H.
Frankel. Lea and Febiger, Philadelphia, 1989.
Frost, H.M. An Introduction to Biomechanics, Charles C. Thomas Publisher, Springfield, Ill.
1971.
Fung, Y.C. Biomechanics: Mechanical Properties of Living Tissues. Springer-Verlag
Publishing Company, New York, New York, 1981.
Handbook of Bioengineering. Edited by Skalak, R. and Chien, S. McGraw-Hill book Company,
New York, New York, 1987.
Orthopaedic Basic Science, Edited by Sheldon Simon. American Academy of Orthopaedic
Surgeons. Park Ridge, IL, 1994
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Sample Questions:
1. Calculate the magnitude and direction of the moment created on an object by a 30 N force
acting in the y-direction at a point 4 cm in the x-direction from axis of rotation. Assume a
standard x-y-z orthogonal coordinate system.
2. Determine the stress acting in a solid circular piece of bone having a 2 cm diameter subjected
to a 400 N compressive force.
3. If a material has Poisson’s ratio of 0.3, then calculate the transverse strain that occurs when
the material experiences a 2 microstrain in the longitudinal direction.
4. Determine the strain experienced by a Hookean material having an elastic modulus of 10 MPa
subjected to a 500 Pa stress.
5. Briefly describe the difference between a ductile and a brittle material.
6. Describe 4 characteristics of viscoelastic materials.
7. What would you expect to happen to the stress experienced by a muscle-tendon complex
subjected to a constant length stretch?
8. If the femur slides relative to the tibia during a portion of a gait cycle and if the dynamic
coefficient of friction between the articulating surfaces of a healthy tibia and femur is .001, then
what force is required to cause the femur to slide on the tibia if the normal force acting between
the surfaces is 600 N? How much would the force have to increase if a knee implant is present
having a dynamic coefficient of friction of 0.1?
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