Title: BIOL 47506750: Biology of the ECM: Mechanics I: Stress, Strain and the 1D Mechanics of Biological Ti
1BIOL 4750/6750 Biology of the ECMMechanics
I Stress, Strain and the 1-D Mechanics of
Biological Tissues
- January 23, 2009
- Prof. David T. Corr
2Preliminary Considerations
- Equilibrium - balance of forces and moments
- Loading conditions
- Moment torque (distance X force)
- Torsion axial torque
- Compatibility - relations between displacements
and strains or deformations - (normal strains, shear strains, strain tensors)
- Constitutive Laws
- Stress-strain load-displacement
- Material Properties
- e.g. Youngs modulus (E, elastic modulus),
Poissons ratio, shear modulus -
- Isotropic, orthotropic, anisotropic elastic
materials
3Mechanics of Deformable Solids
STATICS (SF 0 SM 0) assumes perfect
rigidity
forces moments on structure
non-rigid (deformable)
imbalance
DYNAMICS (SF ma SM Ia) movement
MECHANICS OF MATERIALS internal forces in
material deformations
4Mechanics of Deformable Solids
- CONCEPTS
- STRESS intensity of internal forces
- how a force is felt by the body
- ( Force/Area)
- STRAIN deformations per unit length that occur
under load - ( deformation / undeformed length )
-
- Normalization to remove influence of specimen
geometry
5Internal Forces
Deformable solid (arbitrary shape)
subjected to external loads (F1, F2, F3,
F4) pass section through center view internal
resultant forces and moment
6Internal Forces
- Distribution of actual internal forces
internal force distribution
7Internal Forces
- Resultant
- Internal Force (FR)
- Moment (Mro)
- Represent the resultant effect of
- actual force distribution
8Internal Forces
- Differential area element ?A
- ?F internal force vector acting on ?A
- Resolve force into
- components
- ?Fn normal force
- ?Ft tangential force
9Normal Stress
- NORMAL STRESS (? )
- Intensity of force, or force per unit area,
acting normal to area ?A - Mathematically,
Tensile Stress (positive) force pulls on
?A, stress is Compressive Stress
(negative) force pushes on ?A, stress is
-
10Shear Stress
- SHEAR STRESS (? )
- Intensity of force, or force per unit area,
acting tangent to area ?A - Mathematically,
11Cartesian Stress Components
- General State of Stress
- cut out cubic volume element area (?V
?x?y?z) - each of the 6 faces has 3 stress components
acting on it - one normal
- two shears
12Cartesian Stress Components
- General State of 3-D Stress
- Described by six components ( ?x , ?y , ?z ,
?xy , ?yz , ?xz ) - Typical Units of Stress
- Force per unit area
- SI units MPa (N/m2 x 106)
- U.S. units ksi (lbs./in2 x 103)
3 normal stresses
3 shear stresses
13Average Normal Stress
- Axially Loaded Body
- (eg. tendon, collagen fiber, ligament, metal rod,
) - Centric load
- - load through centroid
- - no moments
- - no transverse loads
- - load is strictly axial
- Evaluate in area of uniform deformation
- away from point of load application
14Average Normal Stress
- Subjected to
- constant uniform deformation
- constant normal stress, ?
- each ?A subjected to force ?F ??A
- Summing forces over entire cross-section
- must equal internal resultant force
15Average Normal Stress
External force
- Average normal stress eqn. valid in uniform
- deformation region
- Localized distortions near ends
- Points of external load application
-
External force
16Effect of Localized Load
17Average Shear Stress
- loaded in simple shear
- (shear caused by direct action of applied load P)
- P external applied load
- V internal resultant shear force ( P/2)
- A area of section
- ?avg average shear stress
- assumed same at every point on section
- zero at free surface
- ?max much larger ?avg
- only an approximation
A
18Examples Single Shear
19Strain in Deformable Bodies
- Deformations due to loading
- Normalized to specimen geometry
20Strain
Deformable solid (arbitrary shape) subjected
to loading
21Strain
Normal Strain (e) elongation (or
contraction) of a line segment per unit length
Normal direction (AB) change in length Ds -
Ds
22Strain
Deformable solid (arbitrary shape) subjected
to loading
23Strain
Shear Strain (g) change in angle between
two, originally perpendicular, line segments
Change in angle CAB undeformed p/2
deformed q
24General State of Strain
25General State of Strain
- Described by six components ( ex , ey , ez ,
gxy , gyz , gxz ) - 3 normal strains
- 3 shearing strains
- Units of Strain
- dimensionless length/length (in./in. , m/m)
26Part II 1-D Mechanical Behavior of Materials
27Mechanics of Deformable Solids
- Plotting Stress vs. Strain
- Characteristic curve for the material
- material properties
- does not depend on specimen geometry
- Allows properties of different materials to be
compared - injured vs. unijured tissue
- graft material vs. native tissue
-
- Material properties (not structural) used in
design - structures (eg. bridges, aircraft, )
- tissue engineering applications
28Applied Mechanics
- Experimental methods frequently used to
- Characterize response to load.
- Characterize resistance to deformation.
- Determine material properties structural
properties - Stiffness
- Modulus of Elasticity
- Strength (yield, ultimate, fatigue, )
- Toughness
- Brittleness (ductile vs. brittle)
- Fatigue properties (S-N diagram run-out tests)
- Hardness
- etc.
29Mechanical Testing Modes
30Mechanics Brittle Materials
Youngs modulus failure stress failure
strain plastic strain (permanent) elastic
strain (recovered) toughness (area)
stored energy recovered energy
X
Stress (?)
E
?p
?e
Strain (?)
31Mechanics Ductile Materials
32Stress-Strain vs. Load-Displacement
- Stress-Strain
- remove the influence of specimen geometry
- used to
- compare behavior of different materials
- design / work problems at different scales
- material properties
- Load-Displacement
- consider geometric properties
- used to
- evaluate behavior of a structure
- analyze on same scale as design
- structural properties
33Common simplifying assumptions
- Linear
- Elastic modulus is constant, independent of
strain level - Isotropic
- exhibits the same elastic behavior in all
directions - Homogeneous
- material is spatially uniform (tensor is
symmetric) - Time-Independent
- stresses and strains are uniquely related,
independent of the rate of strain
34Putting the Bio in Biomechanics
- Biological materials and structures require
analyses beyond most typical material
characterization. - Biomaterials often display complex behavior
- Non-linear Elasticity
- Strain stiffening (toe-in)
- Viscoelasticity
- Stress relaxation
- Creep
- Strain Rate dependence
- Poroelasticity (Biot Theory - biphasic)
- Permeability aggregate modulus
- Activation Contractile Behavior
- Active tissues only
35Non-linear Elasticity
- exhibits a nonlinear response to applied load
- material properties are different at different
loads (or displacements) - stress-strain relation depends on level of strain
(non-unique) - possibly due to recruitment and/or orientation of
different structures. - e.g. strain stiffening
- /- linear portions.
Stress
Strain
36Non-linear Elasticity Collagenous Tissues
tendon ligament skin passive
muscle connective tissue
37Non-linear Elasticity Collagenous Tissues
Tendon structure Non-homogeneous Orthotropic(?)
38Why is this important?
Determines how the tissue behaves in
physiologically relevant range of
motion Matching material properties w/o concern
for relevant operational range can lead to
enormous errors Example Tissue engineering
tendon and Youngs modulus.
39Why is this important?
Develop new material to replace tendon (or
ligament) identical Youngs modulus
identical failure stress
40Why is this important?
?1
?1
?1
41Time Dependence
- Viscoelastic materials
- Exhibit creep
- increasing strain under the same load.
- response characterized by Creep Compliance, J(t).
- Exhibit stress relaxation
- decreasing stress under same deformation.
- response characterized by Relaxation Modulus,
G(t). - Collagen based tissue (including bone) can be
modeled as VE material - Collagen and polymers that exhibit VE behavior
- molecules stretch over time
- cross-links stretch over time
42Creep
Stress Relaxation
43Creep
Additional strain over time w/o changing the
applied load
44Stress Relaxation
Pressure or Tension
Decrease in load over time w/o changing the
applied strain
Volume or Length
Time (min)
45Biphasic Materials
- Biphasic materials have a permeable solid
component and a liquid component. - eg. cartilage
- meniscus
- intervertebral disk
- Mechanical properties of biphasic materials are
dictated by both constituents. - Response is highly time dependent.
- When Loaded
- initial response is dictated by permeability
(porosity of solid, viscosity of fluid) - steady-state response is dictated by material
properties of solid.
Confined compression
46Biphasic Materials - Example
- Articular Cartilage
- Comprised of
- cells (chondrocytes)
- glycosaminoglycans (GAGs)
- collagen matrix
- GAGs bind with water to maintain tissue hydration
47Biphasic MaterialsResponse to Step Load
48Biphasic Materials - Biphasic Theory
- When compressed, fluid escapes out of pores
- Friction at high speeds, little at low speeds
- Less force to deform initially at low speeds
compared to high speeds - Once all fluid has escaped, solid material
properties dictate response
49Strain Rate Dependence
Some materials exhibit different responses to
loading depending on the rate at which they are
loaded.
50Example Collagenous Materials
- Ligaments
- crimped strands/fibers
- cross-linked fibers
- low strain rate, fibers stretch, cross links
break first - high strain rate, fibers fail.
51Example Strain rate in Bone
As strain rate (e)? stiffness (E) ?
yield stress (sy) ? ultimate stress (sult)
? brittleness ? Energy to failure
constant
52Example Strain Rate in Bone
53Activation Contractile Effects
Muscle has the unique ability to produce force as
well as resist applied forces. Tissue response
and mechanical properties change significantly
when activated
Force (N)
Time (sec)
54Activation Contractile Effects
Youngs Modulus, E (Pa)
55Thank You