The “material properties” of the musculoskeletal tissues
The “material properties” of the musculoskeletal tissues depend on
the underlying micro-structures
They are typically inhomogeneous
The depend on age, disease, and other factors
Thus, we must consider
Tissue level (inhomogeneous, anisotropic)
Organ level (homogenized, isotropic or anisotropic)
Knowledge of the scale of interest is critical to our
material modeling approach
Challenges in biological material testing
Challenges in material testing include:
What is our scale of interest?
Isolation of homogeneous tissue
Measurement of deformation (porous tissue)
Control of environment (in-vivo vs ex-vivo)
Inter-specimen variability
We attempt generalization based on:
Density
Organic composition
Mineralization
…
Biological tissues adapt to their environment at the cellular,
structural, and molecular level
ie bone density and geometry adapt to load
Feedback loop (in time)
Mechanics respond to biology
Biology responds to mechanics
An important aspect of modern implant design is prediction of
in-vivo mechanobiologic response
How will the bone respond to the implant?
ie: will stress shielding be a problem?
Currently, we have little control and seek to understand
In future, we may also try to influence this response
Composition of bone
Bone composed of organics and inorganics
Loosely defined, organic compounds contain carbon
covalently bonded to hydrogen, oxygen or nitrogen, etc
By mass, bone is approximately:
60% inorganic material (calcium phosphate, etc)
30% organic
10% water
Collagen
Organic content of bone is mostly collagen
Collagen is the strongest and most abundant protein in the body
Rod-shaped molecules are about 300 nm long and 1.5 nm in diameter
Arranged in a quarter stagger pattern into fibrils which have
20-40nm diameter
“Starting with a single helical protein chain consisting of a variety
of amino acids connected by peptide bonds (top), three of these
polypeptide chains are combined to form the triple helix
tropocollagen molecule (second down), referred to most often as
simply the collagen molecule.”1
The different types of collagen have different types of polypeptide
chains. The collagen molecules are arranged in parallel in a regular
quarter-stagger arrangement to comprise the collagen fibril
(bottom).2
Bone is a hierarchical composite material
Bone is a hierarchical composite material
The four levels of bone microstructure, from the level of mineralized
collagen fibrils to cortical and trabecular bone. It is generally
assumed that at the former level, all bone is equal, although there
can be subtle differences in the nature of the lamellar architecture
and degree of mineralization between cortical and Trabecular
bone.3
Lowest hierarchical level
Collagen fibril (≈0.1 micron)
Sheet
lamellae
of uni-directional fibrils (most common) or blocks of pseudo
random “woven” fibrils (≈10 micron)
Diagram of a sector of the shaft of a long bone, showing the different
types of cortical bone, trabecular bone, and the various channels. The
osteons are
located between the outer and inner circumferential lamellae.4
Fluid resides around bone cells in holes (called lacunae)
These lacunae are interconnected by tiny channels (canaliculi) and
meet at gap junctions
Small molecules including ions pass between the cells and are
thought to help sense damage
Trabecular architecture
Differences between cortical and trabecular bone
The biggest difference is the cellular-like (spongy
structure of trabecular bone
Holes filled with marrow
The rods and plates tend to remodel more often, thus, trabecular
bone is less mineralized than the “older” cortical bone
The material properties are thus slightly worse on the tissue level
(even worse on the structural level due to cellular structures)
Elastic anisotropy
Thought experiment: isotropic material
Uniform rectangular block pulled on both ends
What strains do you expect for σxx≠0 (all others stresses =
0)?
We are all familiar with Hooke’s Law:
σxx=Eεxx
Rearranging:
εxx=σxxE
But what are the other strains?
But what are the other strains?εyy=−νεxxεzz=−νεxx=−νσxxEεij=0 for i≠j
Similarly, we can obtain similar equations in the other directions:
For σyy≠0, all others 0?
εyy=σyyEεxx=−νσyyEεzz=−νσyyEεij=0 for i≠j
For σzz≠0, all others 0?
εzz=σzzEεxx=−νσzzEεyy=−νσzzEεij=0 for i≠j
We’ve found a pattern for the normal stress-normal strain response
What about for shear?
For τxy≠0, all others 0?
γxy=τxyGγxz=0γyz=0εii=0Nosum
For τxz≠0, all others 0?
γxz=τxzGγxy=0γyz=0εii=0Nosum
For τyz≠0, all others 0?
γyz=τyzGγxy=0γxz=0εii=0Nosum
For multiple simultaneous stresses: use superposition
Isotropic constitutive behavior
{εxxεyyεzzεyzεxzεxy}=[1E−νE−νE000−νE1E−νE000−νE−νE1E00000012G00000012G00000012G]{σxxσyyσzzσyzσxzσxy} Note also: the shear modulus is: G=E2(1+ν) Thus there are only two constants that describe the
behavior of an isotropic material
In trabecular bone, rods and plates also align with loads
Bone properties have directionality (called anisotropy)
Principal material coordinate system
Principal material coordinate system for an orthotropic material. This
coordinate system (right) is aligned with the mutually orthogonal
“grain” axes of the material’s microstructure (left. As a class of
anisotropic materials, orthotropic materials have their grain along
three mutually perpendicular axes.
Spatial variations in the orientation of the principal material
coordinate system can occur. In many cases, a local coordinate system
in cylindrical coordinates can be used to describe such spatial
variations.
Anisotropic behavior
Anisotropic behavior
Fibrous composites exhibit a more complex constitutive response
Consider the following material description: {εxxεyyεzzεyzεxzεxy}=[S11S12S13S14S15S16S21S22S23S24S25S26S31S32S33S34S35S36S41S42S43S44S45S46S51S52S53S54S55S56S61S62S63S64S65S66]{σxxσyyσzzσyzσxzσxy}
This is generalized Hooke’s law (applicable to any linear elastic
material–called anisotropic).
For simplicity, we can write generalized Hooke’s law as:
{ε}=[S]{σ}
the values of S are called “elastic compliances”
Elastic constants
It is important to be able to consider the inversion of this system:
{ε}=[S]{σ}{σ}=[S]−1{ε}{σ}=[C]{ε}
The values of C are called Elastic constants
[C] and [S] are fully populated for an anisotropic
material
Finally, we can also use indicial notation to quickly write our
constitutive relationship:
εij=1E[(1+ν)σij−νδijσkk]
σij=2μεij+λδijεkkμ and
λ are called the Lamé constants. They can be found in standard
texts but will not be discussed further.
This method of expressing the equations is powerful and worthy of
study, however, we will not discuss it further
Other material descriptions
There are materials that fit between total anisotropy (21 constants)
and isotropic (2 constants).
In the aerospace world, a critical one is “orthotropic” (9
constants)
{εxxεyyεzzεyzεxzεxy}=[1Exx−νyxEyy−νzxEzz000−νxyExx1Eyy−νzyEzz000−νxzExx−νyzEyy1Ezz00000012Gyz00000012Gzx0000012Gxy]{σxxσyyσzzσyzσxzσxy}
Due to symmetry:
νyxEyy=νxyExxνzxEzz=νxzExxνzyEzz=νyzEyy
ET and EL are transverse (in plane) and longitudinal (out of
plane) modulus
νT and νL are transverse (in plane) and longitudinal (out
of plane) Poisson’s ratios
G is the out of plane shear modulus
Note: stresses and strains must be transformed into the local
coordinate system to apply the constitutive law
Alternatively, the stiffness or compliance matrix must be
transformed into the global coordinate system
See Bartel’s book or me for additional info
Have we covered it all?
Bone is anisotropic and inhomogeneous
Bone is rate dependent
Bone constitutive response depends on fatigue, age, damage, and
plasticity
In models of bone, it is often convenient to assume isotropic
properties
Difficult to obtain a “better” description
Transversely isotropic (and orthotropic) may be more appropriate
Mechanical properties of cortical bone
Asymmetric stiffness and strength
Asymmetric stiffness and strength
Stiffness and strength with age
Graph Showing Relationship Between Age and Bone Mass
Bone density peaks at about 30 years of age. Women lose bone mass more
rapidly than men.
(slide credit: @OpenStaxAnatomy2020 Ch. 6)
Volume fraction
Porosity is present in both cortical and trabecular bone
Define volume fraction (Vf) as the volume of actual bone tissue
to the bulk volume
Cortical 70-95%
Trabecular 5-60%
Extremes are the young adult and elderly
Bone density
Density is strongly dependent on porosity and volume fraction
It is also a primary indicator of bone strength and
stiffness
Apparent density is mass per bulk volume
Common measure of apparent density include:
Hydrated
De-hydrated
De-organified
Tissue density is mass per volume of actual bone tissue
(2.0g/cc)
Importantly, this volume excludes vascular pore spaces
Relationship between bone density and volume fraction
The volume fraction, tissue density, and apparent density are
related by ρapp=ρtissVf
Apparent densities
Cortical – ≈ 1.85 g/cc
Trabecular – 0.10-0.50 g/cc
Trabecular density decreases about 2% per decade after skeletal
maturity
Note also the cortical bone wall thickness decreases as you
age
Heterogeneity and variability
Strength
Mineral content
Mineral content is also important for mechanical properties
It is measured after heating bone to 700C for 24 hours
(de-organification and drying)
Content increases during skeletal growth and remains fairly constant
thereafter
Heterogeneity and variability
Stiffness
Density and strength
Average values
Regressions with age
σ=aρ+b
σ=aρb
Fatigue
Minor’s rule for fatigue
Hypothesis is that the fractional fatigue lives sum
together and predict failure
n∑i=1NiNFi=1
Where there are n different load levels
Works well in most brittle metals (implants!)
Not validated for bone!!!
In bone, common assumption to replace stress with strain
Account for various levels of porosity, etc (also reduces
variability in the test data)
What about bone remodeling?
Creep
De-vitalized bone exhibits creep
Resistance better in compression then tension
Difficult to test in-vivo response
Metals creep – may/may not be significant for ortho
Creep
Plasticity and micro-structural damage
Note the change in slope after yield – microdamage!
Strain rate sensitivity
Only a factor of two – mildly rate dependent
Probably not critical for most physiologic loads
Mechanical properties of trabecular bone
Trabecular bone behavior and large variability
Stress-strain behavior for compressive and tensile loading of bovine
(left) and human vertebral (right) trabecular bone, showing the wide
range in strength that is typical for trabecular bone.
Trabecular bone apparent density
Dependence of compressive on-axis strength on apparent density for
trabecular bone for two different sites bovine tibial (BPT) and human
vertebral (HVB) trabecular bone. The difference in slopes in the
linear relationship for each is due to the different architectures,
being mainly plate-like in the bovine bone and rodlike in the human
vertebral bone. When all the data are pooled, there is a strong
squared power law relationship, with r 0.94. (Bone Mechanics
Handbook. Editor SC Cowin. CRC Press Boca Raton, 2001)
Trabecular bone crush strength and age
Age-related reductions in compressive strength of human femoral and
vertebral trabecular bone.
Data from Mosekilde et
al. (1987) Bone 8:79-85 and McCalden et al. (1997) J Bone Jt Surg
79A:421-427.
Trabecular bone yield asymmetry
Compressive and tensile yield strains vs. human anatomic site for
trabecular bone specimens tested on-axis. Compressive yield strains
were always greater than tensile yield strains.
From Morgan and
Keaveny (2001) J Biomechanics 34: 569-57.
Trabecular bone yield anisotropy
Yield stress anisotropic – yield strain isotropic
Dependence of yield strain (left) and Young’s modulus (right) on
specimen orientation in tension and compression, for dense bovine
trabecular bone. For the off-axis orientation, the specimen axes were
offset 30-40 degrees from the principal trabecular (on-axis)
direction. Error bars show ±1 SD. In contrast to the yield strains,
which were isotropic, but asymmetric, elastic modulus and yield stress
(not shown) were clearly anisotropic. From Chang et al. (1999) J
Orthop Res 124582-585
Fatigue of trabecular bone
S-N fatigue curve for human vertebral trabecular bone
From Haddock et al. (2004) J Biomechanics 37:181-187.
Post-yield damage of trabecular bone
Typical compressive post-yield behavior of trabecular bone for
loading, unloading, and reloading along path 3-b-e as shown. The
initial modulus upon reloading (EINT REL) is statistically equal to
the Young’s modulus E. The reloading curve shows a sharp decrease in
modulus, denoted by ERESIDUAL- This modulus is similar to the
“perfect-damage” modulus (Epp, dashed line), which would occur if the
material behaved in a perfectly damaging manner with cracks occurring
at yield.5
Dependence of percent modulus reduction (between E and
ERESIDUAL) and strength reduction on the plastic strain that develops
with overloading.6
Failure prediction
Von Mises stress criterion is not good for bone
(particularly when shear stresses are high)
Tsai-Wu is a much better metric of strength, however,
requires significantly more experiments to establish the criterion
F1σ1+F2σ2+F3σ3+F11σ21+F22σ22+F33σ23+2F12σ1σ2+2F13σ1σ3+2F23σ2σ3+F44σ24+F55σ25+F66σ26=1 (Note vector representation of stress 1-6)
Requires tension, compression, and torsion tests in longitudinal
and transverse specimens
This is a big challenge in practical prediction of bone
failure
Most studies only compare stresses without
predicting failure