M3(1) Tissue Mechanics 1

describing tissues

biomechanics lab often describe tissues using anatomical and structural terminology

• e.g. proximal. mid-substance, distal

mid substance

the central portion of the tissue b/w its attachments

• often where uniform material properties are measured or where injuries commonly occur

hip capsule

strong sleeve of dense irregular connective tissue that encloses the hip joint

• provides stability and protection for the hip joint

donor tissues

if a hip capsule needs a graft, the iliotibial band (ITB) or Achillies tendon are commonly selected as donor grafts

iliotibial band

broad, flat, tough band made of dense regular connective tissue

achilles tendon

thick and strong, one of the strongest tendons in the body made of dense regular connective tissue

material properties

inherent characteristics of the tissue/material itself; allows us to describe the tissue w/o altering it

• aka physical properties

mechanical properties

how a tissue/material behaves under a load

• aka mechanical response

tissue loading seven factors

1. magnitude

2. location

3. direction

4. duration

5. frequency

6. variability

7. rate of force application

magnitude

refers to how big the force/load is

location

where on the tissue or structure the load is applied

direction

the vector of the force relative to the tissue

duration

how long the force is applied for

frequency

how often the loading occurs

variability

refers to whether the load is always the same or changes

rate of force application

how quickly the force is applied

types of loading

loading can be separated into two categories

• axial/linear

• off-axis

axial

forces that act along the tissue’s long axis

1. compression

2. tension

3. shear

compression

force pushes the ends of the tissue towards each other

tension

fore pulls the ends of the tissue away from each other

shear

forces act parallel to the tissue surface (sliding forces)

off axis

forces that are not aligned with the tissues long axis

1. bending

2. torsion

bending

a combination of tension on one side and compression on the other

• at the very center of the material no stress occurs (ever)

torsion

produced by opposite rotational forces applied at each end

resistance to bending

resistance to bending is determined by area moment of inertia (I)

resistance to bending equation

the larger the radius / thickness, the more resistant the material is to bending

for a solid cylinder

• I (area moment of inertia) = πr⁴ / 4

for a hollow cylinder

• I (area moment of inertia) = π(R⁴ - r⁴) / 4

resistance to torsion

resistance to torsion is the polar moment of inertia (J)

resistance to torsion equation

the larger the radius / thickness, the more resistant the material is to torsion

for a solid cylinder

• J = π(r⁴) / 2

for a hollow cylinder

• J = [π(R⁴ - r⁴] / 2

load deformation graph

using a load-deformation graph we can measure a tissues deformation for a given load, also used to calculate stiffness (k)

• x-axis = deformation

• y-axis = load

stiffness

how much a whole structure resists deformation under a given force, represented by “k” (N/mm)

• instantaneous stiffness: k = load / deformation

• average stiffness (slope): k = change in load / change in deformation

better to use slope whenever possible for a more accurate measure of k

stress

measure of how much internal force a material experiences when an external load is applied, represented by “σ” (Pa, N/m²)

• σ = force / cross-sectional area

• σ = F / A

stress example

objects with larger CSA experience less stress with the same amount of force applied

CSA variability

if a tissue varies in CSA along its length, the stress it experiences may also vary as well

strain

strain is the deformation a material undergoes in response to stress, represented by “ε” (unitless, x100%)

• ε = ΔL / L₀

• ε = (L_final - L_initial) / L₀

strain example

longer objects stretch more in absolute units for the same strain, but if the absolute stretch is the same as a shorter object, the longer object’s strain is smaller

• long object experiences much less strain because the elongation is small relative to its original length

stress strain curve

stress–strain curve shows how a material deforms (strain) in response to applied force (stress)

Youngs modulus

a measure of a material’s stiffness — how much it resists deformation under stress, represented by “E”

• it is the slope of the linear (elastic) region of a stress-strain curve

E = change in stress (σ) / change in strain (ε)

Youngs modulus example

Intrinsic property → depends only on the material, not its shape or size.

• High E → stiff material (resists stretching, e.g., bone, tendon).

• Low E → compliant material (easily deforms, e.g., skin, ligaments)

deflection

how much bending there is in an object

Poissons effect

describes how a material changes shape in directions perpendicular to the direction of loading

• ε_a axial strain = change in original width

• ε_t transverse strain = change in original length

Poissons ratio

a material property that quantifies the Poisson effect

• different materials have a different ratio

stress strain curve components

1. elastic region

2. linear limit

3. elastic limit

4. yield point

5. plastic region

6. ultimate stress

7. rupture

elastic region

the region of a stress-strain curve where the tissue deforms under load but can return to its original shape once the load is removed

• determined by Hooke’s law

Hooke’s law

stress = stiffness (Elastic Modulus) x strain

• stress is proportional to strain and stiffness

  • if the tissue is very stiff (big E), a small strain creates big stress

  • if the tissue is very compliant (small E), a small strain causes less stress

linear limit

relationship b/w stress and strain becomes no longer proportional

• applying stress here induces higher magnitude of strain

elastic limit

the limit at which a material can still return to its original shape once the load is removed

yield point

the point at which the material just begins to deform permanently

plastic region

the part of the stress-strain curve where the tissue undergoes permanent deformation

ultimate stress

the maximum stress a tissue can withstand before it starts to fail

• the highest peak on a stress-strain curve

rupture

the point at where the tissue breaks apart

• line (stress) goes to 0 because the tissue can’t carry any more load

tissue mechanical properties

way in which biological tissues behave when forces are applied to them

• linear mechanical properties

• non-linear mechanical properties

linear mechanical properties

how a tissue behaves when a force is applied to it, specifically in the range where the relationship b/w stress and strain is linear (e.g. elastic region)

• stress proportional to strain

non linear mechanical properties

how a tissue behaves when a force is applied to it, specifically in the range where the relationship b/w stress and strain is non-linear (e.g. plastic region)

• viscoelastic and hyper elastic responses

ductility and brittleness

refers to how much a tissue can strain (deform) before it breaks when a stress is applied to it

• being ductile is not to be confused with strength (achieving a higher stress/force)

area under the stress-strain curve

the energy absorbed by the tissue as it deforms

• the area under the stress-strain curve and load-deformation curve

resiliency

ability of a material or tissue to absorb energy when deformed elastically and then return it when unloaded

• area under the elastic region ONLY on the stress-strain curve

toughness

ability of a material or tissue to absorb energy up to fracture

• entire area under the stress-strain curve

modulus of resilience

the amount of energy per unit volume a material or tissue can absorb elastically without permanent deformation

• U_r = modulus of resistance

• σ_y = stress at yield point

• E = Young’s modulus (stiffness)

• U_r = σ_y² / 2E

viscoelastic properties

most biological tissues are viscoelastic meaning they show both elastic (solid-like) and viscous (fluid-like) behaviour

key viscoelastic properties

1. strain rate dependency

2. hysteresis

3. creep

4. stress relaxation

strain rate dependency

tissue response depends on how fast you apply the load

• fast loading → tissue becomes stiffer (steeper slope)

• slow loading → more deformation (shallower slope)

hysteresis

energy lost during loading and unloading cycle

• on a stress-strain curve the loading and unloading curves form a loop (area inside = energy lost as heat)

low hysteresis

normal for quick returning tissues → less energy lost

high hysteresis

normal for damped response (slow return) tissues → more energy lost

creep

gradual deformation under constant stress (force) over time

• time dependent

• bi phasic

e.g. intervertebral discs decompress when doing a deadhang

time dependent

for creep, the amount of deformation depends on how long stress is applied

• the longer the tissue is stretched, the more it is deformed

bi phasic creep

when creep happens, it occurs in two phases

1. rapid phase — initial loading results in quick mechanical response (stretching)

2. slow phase — constant load results in continued deformation (CREEP)

stress relaxation

gradual decrease in stress under constant strain

• bi phasic

bi phasic stress relaxation

1. rapid phase — initial deformation resulting in immediate mechanical response

2. slow phase — constant strain results in decrease in stress

material fatigue and failure

occurs when multiple loading cycles are applied

two potential results:

1. adaptation/equilibrium

2. failure

adaptation

early in repeated loading, the material might adapt

• for tissues this may mean strengthening, thickening, or reorganizing fibers to better handle the load

• initial load is experienced but deformation plateaus over time

failure

if loading continues beyond the material’s tolerance

• micro-damage accumulates and eventually the material fails (fractures, tears, or otherwise loses its ability to support the load)