BIOL 47506750: Biology of the ECM: Mechanics I: Stress, Strain and the 1D Mechanics of Biological Ti

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BIOL 4750/6750 Biology of the ECMMechanics
I Stress, Strain and the 1-D Mechanics of
Biological Tissues

• January 23, 2009
• Prof. David T. Corr

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Preliminary 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

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Mechanics 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
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Mechanics 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

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Internal Forces
Deformable solid (arbitrary shape)
subjected to external loads (F1, F2, F3,
F4) pass section through center view internal
resultant forces and moment
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Internal Forces

• Distribution of actual internal forces

internal force distribution
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Internal Forces

• Resultant
• Internal Force (FR)
• Moment (Mro)
• Represent the resultant effect of
• actual force distribution

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Internal Forces

• Differential area element ?A
• ?F internal force vector acting on ?A
• Resolve force into
• components
• ?Fn normal force
• ?Ft tangential force

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Normal 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
-
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Shear Stress

• SHEAR STRESS (? )
• Intensity of force, or force per unit area,
  acting tangent to area ?A
• Mathematically,

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Cartesian 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

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Cartesian 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
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Average 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

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Average 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

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Average Normal Stress
External force

• Average normal stress eqn. valid in uniform
• deformation region
• Localized distortions near ends
• Points of external load application

External force
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Effect of Localized Load
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Average 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
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Examples Single Shear
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Strain in Deformable Bodies

• Deformations due to loading
• Normalized to specimen geometry

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Strain
Deformable solid (arbitrary shape) subjected
to loading
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Strain
Normal Strain (e) elongation (or
contraction) of a line segment per unit length
Normal direction (AB) change in length Ds -
Ds
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Strain
Deformable solid (arbitrary shape) subjected
to loading
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Strain
Shear Strain (g) change in angle between
two, originally perpendicular, line segments
Change in angle CAB undeformed p/2
deformed q
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General State of Strain
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General 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)

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Part II 1-D Mechanical Behavior of Materials
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Mechanics 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

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Applied 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.

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Mechanical Testing Modes
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Mechanics 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 (?)
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Mechanics Ductile Materials
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Stress-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

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Common 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

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Putting 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

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Non-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
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Non-linear Elasticity Collagenous Tissues
tendon ligament skin passive
muscle connective tissue
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Non-linear Elasticity Collagenous Tissues
Tendon structure Non-homogeneous Orthotropic(?)
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Why 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.
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Why is this important?
Develop new material to replace tendon (or
ligament) identical Youngs modulus
identical failure stress
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Why is this important?
?1
?1
?1
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Time 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

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Creep
Stress Relaxation
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Creep
Additional strain over time w/o changing the
applied load
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Stress Relaxation
Pressure or Tension
Decrease in load over time w/o changing the
applied strain
Volume or Length
Time (min)
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Biphasic 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
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Biphasic Materials - Example

• Articular Cartilage
• Comprised of
• cells (chondrocytes)
• glycosaminoglycans (GAGs)
• collagen matrix
• GAGs bind with water to maintain tissue hydration

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Biphasic MaterialsResponse to Step Load
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Biphasic 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

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Strain Rate Dependence
Some materials exhibit different responses to
loading depending on the rate at which they are
loaded.

• stiffness
• yield
• strength

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Example Collagenous Materials

• Ligaments
• crimped strands/fibers
• cross-linked fibers
• low strain rate, fibers stretch, cross links
  break first
• high strain rate, fibers fail.

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Example Strain rate in Bone

• Bones
• Ligaments

As strain rate (e)? stiffness (E) ?
yield stress (sy) ? ultimate stress (sult)
? brittleness ? Energy to failure
constant
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Example Strain Rate in Bone
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Activation 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)
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Activation Contractile Effects
Youngs Modulus, E (Pa)
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Thank You