Bone Group Presentation 22306 Bone Biomechanics Relating Mechanics

Bone Group Presentation 2/23/06 Bone Biomechanics: Relating Mechanics Concepts to Bone Presented by: Jeffrey M. Leismer, MEng Edward K. Walsh, Ph. D

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?

Agenda Ed & 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

Overview BONE MECHANICS Failure (result of loading) Mechanics Mechanical Testing (effects of forces on a body) (response to loading) Statics Dynamics (equilibrium of forces and moments) (bodies in motion) To Prof. Walsh Kinematics Kinetics (displacements, velocities, and accelerations) (forces responsible for motion)

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 System To Prof. Walsh

Vocabulary • Load (N; lbf) • Deformation (mm; in) • Stress (N/m 2=Pa; psi) • Strain (mm/mm; in/in) • Moment/Torque (N*m=J; in-lb) • Moment of Inertia (mm 4; in 4)

Overview Failure (result of loading) Mechanics Mechanical Testing (effects of forces on a body) (response to loading) Statics (equilibrium of of forces and (equilibrium moments) Dynamics (bodies in motion) Kinematics Kinetics (displacements, velocities, and accelerations) (forces responsible for motion)

Statics Overview

Statics • Equilibrium of forces ∑F=0 • Equilibrium of moments ∑M=0 • *Bicep curl example To Jeff

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

useful angles y Static Analysis What Use Now the let’s happens figure plug to intofind some F if we the realistic increase forcesvalues theset moments moment and solve arm, infor each the d? zero direction forces Sum the forces and moments and them equal to + ∑Fx=0 =Rx+F*cos(b) Rx=-F*cos(b) + ∑Fy=0 =F*sin(b)-W+Ry Ry=W-F*sin(b) + ∑Mb=0 =W*L*sin(q)-d*F*sin(f) F=W*L*sin(q)/[d*sin(f)] q f b=90°-q+f q If W=20 lbf, L=14 in, d=0. 5 in, q=70°, and f=50°: b=70° F=687 lbf Rx=-235 lbf Ry=626 lbf If d=1 in F=343 lbf Rx=-117 lbf Ry=303 lbf To Prof. Walsh Resolve muscle force vector into x and y components x

Overview Failure (result of loading) Mechanics Mechanical Testing (effects of forces on a body) (response to loading) Statics Dynamics (equilibrium of forces and moments) (bodies in motion) Kinematics Kinetics (displacements, velocities, and accelerations) (forces responsible for motion)

Mechanics of Materials Overview

Mechanics of Materials stress cube (can be used to show the state of stress at a point) state of stress syy rigid body sxx Types of stresses and their equations Normal Stresses: Bending sb=M*c/I Axial s=F/A Shear Stresses: Torsion t=T*r/J Transverse Shear To Jeff tyz szz txy txz txy szz tyz sxx t=V/A syy

Mechanics of Materials -To find the stresses at point ‘e’: -Make a cut at ‘e’ -Remove all components to the right of the cut -Replace the removed section with reaction forces and a moment at point ‘e’

Mechanics of Materials Solve for the reaction forces and moment Wy=W*cos(q) ; Wx=W*sin(q) + ∑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 q=70°: M=89 in-lb y x Simplify analysis by rotating the coordinate system and force vectors

Mechanics of Materials Bending stress at ‘e’ due to moment ‘M’ sb=M*c/I c=ro I=p*(ro 4 -ri 4)/4 For M=89 in-lb, ro=0. 75 in, ri=0. 25 in I=0. 245 in 4 failure strength (bending) sb=272 lb/in 2 = 272 psi << sf=30, 250 psi Normal stress at ‘e’ due to Rx s. N=Rx/A s. N=4. 4 psi A=p*(ro 2 -ri 2)=1. 57 in 2 y Rx=Wx=W*cos(q)=6. 8 lbf Cross-section of bone at ‘e’ Shear stress at point ‘e’ due to Ry s. N=Ry/A s. N=12 psi Ry=Wy=W*sin(q)=18. 8 lbf x 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. To Prof. Walsh

Stress-Strain Relationship: Constitutive Law • Anisotropy • Orthotropy Hooke’s Law {s}=[C]{ε } where [C] is the stiffness matrix {ε }=[S]{s}, where [S] is the compliance matrix Inverse relationship [S]=[C]-1 (21 elastic constants) (9 elastic constants) • Transverse Isotropy • Isotropy Material properties Elastic modulus = E Poisson’s ratio = u Shear modulus = G To Jeff (5 elastic constants) (2 elastic constants)

Overview Failure (result of loading) Mechanical Testing Mechanics (response to loading) (effects of forces on a body) Statics Dynamics (equilibrium of forces and moments) (bodies in motion) Kinematics Kinetics (displacements, velocities, and accelerations) (forces responsible for 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. Ultimate Load − − − X S U Displacement Fracture Ultimate Displacement

Overview Failure (result of loading) Mechanical Testing Mechanics (response to loading) (effects of forces on a body) Statics Dynamics (equilibrium of forces and moments) (bodies in motion) Kinematics Kinetics (displacements, velocities, and accelerations) (forces responsible for 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

Manatee Bone Fracture Study • Aims − Characterize manatee rib bone − Determine anisotropic fracture properties − Predict the anisotropic stress intensity factors (KII, KIIIIII) using finite element methods and fracture analysis software

Manatee Bone Fracture Study proximal 1 3 2 Tests Rib bone distal Specimens Measured Properties Elastic Moduli and Poisson’s Ratios E 1, E 2, E 3, u 23, u 12 Tension Shear Moduli G 23, G 12 Torsion crack tip Compact Tension Stress Intensity Factors, Fracture Toughness 1) KI, KIII, KIC 2) KI, KIII, KIC 3) KI, KIII, KIC

Manatee Bone Fracture Study • Visual Image Correlation (VIC) 2 cameras take simultaneous pictures of the specimen as it is loaded Correlation software maps the specimen surfaces from the images to digitized 3 D space Images of the loaded specimen are used to digitally measure deformations relative the reference photo of the undeformed specimen

Manatee Bone Fracture Study Hooke’s Law Orthotropic Compliance Matrix Resulting Strains Due to Applied Stresses -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 3 D 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

PROFESSOR WALSH ANDTHE I WILL NOW TAKE WE HOPE YOU ENJOYED PRESENTATION THE REMAINING TIME TO ANSWER YOUR QUESTIONS Resources • Contact Info: − Computational solid mechanics lab (103 MAE-C) − Email Jeff: jeffleismer@gmail. com − Email Ed: ekw@mae. ufl. edu • Books: − Bone Mechanics Handbook (Cowin, 2001) − Mechanical Testing of Bone (An & Draughn, 2000) WE WOULD APPRECIATE YOUR FEEDBACK, SO PLEASE EMAIL US