Bone Biomechanics: Relating Mechanics Concepts to Bone Presented by: Jeffrey M. Leismer, MEng Edward...

Bone Biomechanics:

Relating Mechanics Concepts

to BonePresented by:

Jeffrey M. Leismer, MEng

Edward K. Walsh, PhD

Bone Group Presentation 2/23/06

Introduction

• Definitions−Statics, dynamics, mechanics of materials,

failure

• Concepts to be learned−What are stresses and strains?−How can knowledge of loads and deformations

be used to obtain stresses and strains?−How does bone fail?

• Audience background/interests?

AgendaEd & Jeff_________________________________________(25

minutes)• Bone mechanics overview• Mechanical influences on the skeleton• Statics• Mechanics of materials• Stress-strain relationship• Mechanical testing• Failure modes in bone

Jeff_______________________________________________(5 minutes)• Application: Manatee bone fracture study

BONE MECHANICSOverview

Mechanics(effects of forces on a body)

Mechanical Testing(response to loading)

Statics(equilibrium of forces and

moments)

Dynamics(bodies in motion)

Kinematics(displacements, velocities,

and accelerations)

Kinetics(forces responsible for

motion)

Failure(result of loading)

BONE MECHANICS

To Prof. Walsh

Bone Mechanics Overview

To Jeff

Mechanical Influences on the Skeleton• Internal/external loading factors

−Loading site, direction, magnitude, speed, repetition, duration

• Physiological loading concepts−Muscle forces−Tendon and ligament attachment

points−Moment arms (*bicep curl example)

Mechanical Lever SystemMechanical Lever System

To Prof. Walsh

Vocabulary

• Load (N; lbf)

• Deformation (mm; in)

• Stress (N/m2=Pa; psi)

• Strain (mm/mm; in/in)

• Moment/Torque (N*m=J; in-lb)

• Moment of Inertia (mm4; in4)

Overview




moments)



and accelerations)


motion)



moments)

Statics Overview

Statics

• Equilibrium of forces∑F=0

• Equilibrium of moments∑M=0

• *Bicep curl example

To Jeff

Static AnalysisStatic Analysis

-To solve for the muscle force, remove rigid body ‘bc’ and replace the section with reaction forces at ‘b’

If W=20 lbf, L=14 in, d=0.5 in, =70°, and =50°:

=70°F=687 lbfRx=-235 lbfRy=626 lbf

If d=1 inF=343 lbfRx=-117 lbfRy=303 lbf

=Rx+F*cos() Rx=-F*cos()

=F*sin()-W+Ry Ry=W-F*sin()

=W*L*sin()-d*F*sin() F=W*L*sin()/[d*sin()]

+ ∑Fx=0

+ ∑Fy=0

+ ∑Mb=0

=90°-+

y

x

Static Analysis

Resolve muscle force vector into x and y components

useful angles

Sum the forces and moments and set them equal to zeroUse the figure to find the forces and moments in each directionNow let’s plug in some realistic values and solve for the forcesWhat happens to F if we increase the moment arm, d?

To Prof. Walsh

Overview




moments)



and accelerations)


motion)



Mechanics of Materials Overview

xx

yy

zzxyxy

yzyz

xzxzxz

Mechanics of Materials

Types of stresses and their equationszz

xx

yy

yzyz

xzxz

xyxy

rigid body

stress cube

Normal Stresses:Bendingb=M*c/IAxial =F/AShear Stresses:

Torsion =T*r/JTransverse Shear =V/A

state of stress

(can be used to show the state of stress at a point)

To Jeff

Mechanics of MaterialsMechanics of Materials

-To find the stresses at point ‘e’:

-Make a cut at ‘e’

-Replace the removed section with reaction forces and a moment at point ‘e’

-Remove all components to the right of the cut


x

y

Wy=W*cos() ; Wx=W*sin()

+ ∑Fx=0=Rx

+ ∑Fy=0=-Wy+Ry Ry=Wy

+ ∑M=0=Wy*(L-d)-M M=Wy*(L-d)

If W=20 lbf, L=14 in, d=1 in, and =70°:M=89 in-lb

Simplify analysis by rotating the coordinate system and force vectors

Solve for the reaction forces and moment


Cross-section of bone at ‘e’

x

y

Bending stress at ‘e’ due to moment ‘M’b=M*c/I

Normal stress at ‘e’ due to RxN=Rx/A N=4.4 psi A=*(ro2-ri2)=1.57 in2

Rx=Wx=W*cos()=6.8 lbf

Shear stress at point ‘e’ due to RyN=Ry/A N=12 psiRy=Wy=W*sin()=18.8 lbf

c=roI=*(ro4-ri4)/4

For M=89 in-lb, ro=0.75 in, ri=0.25 inI=0.245 in4

b=272 lb/in2 = 272 psifailure strength (bending)<< f=30,250 psi

To Prof. Walsh

The stresses found above were calculated for a point at the top of the cross-section. The stresses will be lower at any other point about the cross-section.

Stress-Strain Relationship: Constitutive LawHooke’s Law

{}=[C]{ε} where [C] is the stiffness matrix

{ε}=[S]{}, where [S] is the compliance matrix

Inverse relationship[S]=[C]-1

Material propertiesElastic modulus = EPoisson’s ratio = Shear modulus = G

S

s11

s12

s13

s14

s15

s16

s12

s22

s23

s24

s25

s26

s13

s23

s33

s34

s35

s36

s14

s24

s34

s44

s45

s46

s15

s25

s35

s45

s55

s56

s16

s26

s36

s46

s56

s66

s11

•Anisotropy

(21 elastic constants)

S

1

E1

12

E1

13

E1

0

0

0

21

E2

1

E2

23

E2

0

0

0

31

E3

32

E3

1

E3

0

0

0

0

0

0

1

G23

0

0

0

0

0

0

1

G31

0

0

0

0

0

0

1

G12

E

•Orthotropy


•Transverse Isotropy

S

1

E1

21

E1

23

E1

0

0

0

21

E1

1

E1

23

E1

0

0

0

31

E3

31

E3

1

E3

0

0

0

0

0

0

1

G23

0

0

0

0

0

0

1

G23

0

0

0

0

0

0

2 212

E1

E


•Isotropy

S

1

E

E

E

0

0

0

E

1

E

E

0

0

0

E

E

1

E

0

0

0

0

0

0

2 2E

0

0

0

0

0

0

2 2E

0

0

0

0

0

0

2 2E

E

(2 elastic constants)To Jeff

Overview




moments)



and accelerations)


motion)



Mechanical Testing of Bone

• Handling considerations−Hydration, temperature, strain rate

• Types of tests−Tension/compression, bending, torsion, shear,

indentation, fracture, fatigue, acoustic

• Equipment−Mechanical testing machine, deformation

measurement system, recording instrumentation (load-deflection)

• Other considerations−Specimen size & orientation, species, sampling

location

Mechanical Testing• Outcome measures (uniaxial test)

− Ultimate load: reflects integrity of bone structure− Stiffness: related to mineralization− Work to failure: energy required to break bone− Ultimate displacement: inversely related to

brittleness− Etc.

X

Loa

d

Displacement

FractureUltimate

Load

UltimateDisplacement

SU

Overview




moments)



and accelerations)


motion)



Failure of Bone

• Failure modes−Ductile Overload Fracture

• Failure results from loading bone in excess of its failure strength

−Brittle Fracture• Stress is intensified at sharp corners (micro-cracks or voids)

and results in fracture without exceeding the failure strength of bone

−Creep• Slow, permanent deformation resulting from application of a

sustained, sub-failure magnitude load (*Silly Putty™)

−Fatigue• Failure due to repetitive loading below the failure strength

of bone (a.k.a. stress fractures)

How can this information be put to use?

• EXAMPLE−Manatee Bone Fracture Study

• 25% of all manatees die as a result of collisions with watercraft

• Reducing boat speed in manatee zones can greatly reduce the energy of impact in the event of a collision

• Previous researchers correlated the energy associated with traveling at various speeds in a small boat to the energy required to fracture manatee bone

• One of the goals of my dissertation work is to build on this information to further reinforce speed restrictions in manatee zones so that this docile creature can remain in existence for future generations to admire

• EXAMPLE−Manatee Bone Fracture Study

• 25% of all manatees die as a result of collisions with watercraft

• Reducing boat speed in manatee zones can greatly reduce the energy of impact in the event of a collision

• Previous researchers correlated the energy associated with traveling at various speeds in a small boat to the energy required to fracture manatee bone

• One of the goals of my dissertation work is to build on this information to further reinforce speed restrictions in manatee zones so that this docile creature can remain in existence for future generations to admire

How can this information be put to use?

Manatee Bone Fracture Study• Aims

−Characterize manatee rib bone−Determine anisotropic fracture

properties−Predict the anisotropic stress intensity

factors (KI,KII,KIII) using finite element methods and fracture analysis software

• Aims−Characterize manatee rib bone−Determine anisotropic fracture

properties−Predict the anisotropic stress intensity

factors (KI,KII,KIII) using finite element methods and fracture analysis software

Manatee Bone Fracture Study

Tension

Torsion

Compact Tension


Elastic Moduli and Poisson’s RatiosE1, E2, E3, 23, 13, 12

Shear ModuliG23, G13, G12

Stress Intensity Factors,Fracture Toughness1) KI, KII, KIII, KIC

2) KI, KII, KIII, KIC

3) KI, KII, KIII, KIC

crack tip

proximal

3

2 distal

1

Specimens Measured PropertiesTests

Rib bone

Manatee Bone Fracture Study• Visual Image Correlation (VIC)


Correlation software maps the specimen surfaces from the images to digitized 3D space

2 cameras take simultaneous pictures of the specimen as it is loaded

Images of the loaded specimen are used to digitally measure deformations relative the reference photo of the undeformed specimen


S

1

E1

12E1

13E1

0

0

0

12E2

1

E2

23E2

0

0

0

13E3

23E3

1

E3

0

0

0

0

0

0

1

G23

0

0

0

0

0

0

1

G13

0

0

0

0

0

0

1

G12

Orthotropic Compliance Matrix

ε1

ε2

ε3

ε4

ε5

ε6

1

E11

12E2

213E3

3

12E1

11

E22

23E3

3

13E1

123E2

21

E33

1

G234

1

G135

1

G126

Resulting Strains Due to Applied Stresses

ε S

Hooke’s Law

-Six experiments are run, each with the application of only a single component of stress-From the measured strain, we can calculate all of the orthotropic elastic constants-The elastic constants are used as input to a finite element model for further analysis

Manatee Bone Fracture Study• Finite Element Analysis (FEA)

• Computational Fracture Analysis−Crack opening displacements (COD’s) from FEA

are used to determine the 3D anisotropic stress intensity factors in a specimen

−Numerical results are compared with those from experiment to determine the predictive capacity of the model for fracture analyses

Resources

• Contact Info:−Computational solid mechanics lab (103 MAE-C)−Email Jeff: [email protected]−Email Ed: [email protected]

• Books:−Bone Mechanics Handbook (Cowin, 2001)−Mechanical Testing of Bone (An & Draughn,

2000)

WE HOPE YOU ENJOYED THE PRESENTATIONPROFESSOR WALSH AND I WILL NOW TAKETHE REMAINING TIME TO ANSWER YOUR QUESTIONS

WE WOULD APPRECIATE YOUR FEEDBACK, SO PLEASE EMAIL US

Bone Biomechanics: Relating Mechanics Concepts to Bone Presented by: Jeffrey M. Leismer, MEng Edward...

Documents

Transcript of Bone Biomechanics: Relating Mechanics Concepts to Bone Presented by: Jeffrey M. Leismer, MEng Edward...