Chapter 3 - Joint and Materials Mechanics

Artificial hips

Shoe impact tester

Joint Motion

• Anatomical Position• Planes

– Sagittal

– Frontal

– Transverse

Mobility and ROM

• ROM joint & person specific

• Injuries: excessive ROM

• Factors affecting ROM:– Shape and geometry of

articulating surfaces

– Joint capsule and ligaments

– Surrounding muscles

– Apposition of body parts

• Joint Stability

Lever Systems

• Rigid rod fixed at point to which two forces are applied

• 1st class

• 2nd class

• 3rd class

• Functions applied force effective speed

R F

RF

FR

Instantaneous Joint Center

• Caused by asymmetries in the joint motion

• Basic movements– rotation

– sliding

– rolling

Moment of Force & Joint Motion

• Moment = F ·d• Moment = muscular

activity, essential for controlling joint motion

• Theory: actions at joints can be represented by the resultant joint force and the resultant joint moment

Resultant Joint Force vs Bone-on-Bone Forces

• RJF: Net force across the joint produced by bone, ligaments, muscle etc.

• Bone-on-Bone: more complex calculation

Material Mechanics

• Rigid body mechanics: body segments are considered rigid structures (non-deformable)

– fixed center of mass

– homogeneous material

• Used to analyze movements

• Easier to model and provide a reasonable approximation

Deformable solids

• Segment or tissue analyzed undergoes deformation

• More complicated analysis and difficult to model

Material Properties

• Basic properties– Size

– Shape

– Area

– Volume

– Mass

• Derived

– Density

– Centroid

Stress

• Stress (): internal resistance to an external load– Axial (compressive or

tensile) =F/A

– Shear = F/A (parallel or tangential forces)

• Units Pascal (Pa) = 1Nm2

Axial

Shear

Strain

• Change in shape or deformation ()

• Absolute strain• Relative strain

L/Lo

Stress &Strain

• Stress-strain ratio: stiffness or compliance of the material– E = /

• Linear material– Hooke’ law: = E·

• Biological material non-linear due to its tissue fluid component (viscoelastic properties)

A

B

Uniaxial Loading• Simplest form: forces applied along

single line typically the primary axis

– compressive

– tensile

– shear

• Stress-strain curve

– Linear region (B)

– elastic limit (C)

– yield point (D)

– ultimate stress (E)

– Rupture (F)

– Energy stored (area)

B

C

D

E

F

Poisson’s effect

• When a body experiences an uniaxial load its axial & transverse dimensions will change,

• v = -(t/ a)

Force

Multiaxial Loading

• Deformation in all three directions

• Net effect of the strains

• Shear stresses

Bending

• Long bones: beams• Compressive stress:

inner portion• Tensile stress: outer

portion• Max stresses near the

edges, less near the neutral axis

C

T

axis

x=(Mb·y)/I

yaxis

Bending Moments

• Shear stresses max at neutral axis and zero at the surface

= (Q·V)/(I ·b)• Q= area moment• V= vertical shear force

h

b

Q

y

Bending

• Three point bending– failure at middle

– ski boot fracture

• Four point bending– failure at the weakest

point between two inside forces

Bending

• Cantilever bending• Compressive force

acting off-center from long axis

Torsion

• Twisting action applied to a structure

• Resistance about long axis determined by polar moment of inertia

• J=[·(r4o-r4

i)]/2

• Shear stress along the shaft =(T·r)/J

• Twist angle: =(T·l)/(G·J)

Torsion

• Larger radius of the shaft, greater resistance

• Stiffer the material harder to deform

• In addition to shear stress, normal stress (tensile & compressive) are produced in a helical path (spiral fractures)

r

Viscoelasticity

• Provided by the fluid component in biological tissue

• Resistance to flow• Affects stress-strain• Increase in strain rate

produces-increases stiffness of the material

Viscoelasticity

• Pure elastic material– strain energy returned

– no energy loss

• Viscoelastic tissues– lose energy due to heat

– energy is not returned immediately

– Resilient

– Dampened

• Hysteresis: area representing energy lost

Load

unlo

ad

Elastic

Non

Viscoelasticity

• Creep response• Stress-relaxation

response• Effects of strain-rate

on stress relaxation

Time

creep

Material Fatigue & Failure

• Fatigue: repeated loads above a certain threshold

• Continued loading: failure

• First cycle effect: shift in mechanical response

1 2 3 n

Initial cycle effect

Material failure

• Distribution of stresses– Discontinuity (stress risers)

• fractures sites

• screws

• osteotendinous junctions

• Ductile vs Brittle materials

• Failure theories– maximal normal stress

– maximal shear stress

– maximal energy distortion

Biomechanical Modeling & Simulation

• Model: representation of one or more of an object’s or system’s characteristics using mathematical equations

• Goal– improve understanding of a

system

• Simulation:process of using validated model

Biomechanical Modeling & Simulation

• Physical model: simulates actual conditions, crash test dummies

• Mathematical or computer model: conditions are represented using mathematical equations

Why use a model?

• Easy to duplicate • Easy to make change

in the system• Time • Economic factor

How to select a model?

• What questions is being posed?

– Type: molecular, tissue, organ etc.

– Deformable or rigid– finite or

continuoum– static, quasi static

or dynamic

– Linear or nonlinear– 2D or 3D– Determined or

stochastic– kinematics or

kinetics– inverse or direct

Model & Simulations

• Models are simplifications of actual situations

• Model and simulation are as good as the data use as input

• Stability of the model (range of values)

Finite-element modeling

• Structures are represented as simple blocks assembled to form complex geometrical structures

• Connected at poinst (nodes) forming a mathetical representation of the structure

• Forces are applied at the structure and stress and strain are predicted

• Complex and requires a great deal of computing power

Finite modeling

Rheological Models

• Study of deformation and flow of matter

• Use to model biological tissue

• Interrelate stress, strain, and strain rate

• Three types

– Linear spring

– dashpot

– frictional

• Linear spring

– elastic properties of tissue

Rheological Models

• Dashpot– loading response that is

strain rate dependent

– fluid viscosity (newtonian fluid)

= ·

• Frictional element• Combinations of

models

Strain rate