Homogenized trigonal models for biomechanical applications description copia

“Homogenized trigonal models for biomechanical applications”

Relatore:Ch.mo Prof. Ing.

MASSIMILIANO FRALDI

Correlatore:GIANPAOLO PERRELLA

Candidato:CIERVO MARCO

Matr. 691/939

Università degli Studi di Napoli Federico II

FACOLTA’ DI INGEGNERIA

CORSO DI LAUREA ININGEGNERIA BIOMEDICA

(CLASSE DELLE LAUREE IN INGEGNERIA DELL’INFORMAZIONE n.9)

Description

Ligaments: Anterior Cruciate Ligament (ACL)

Healing ACL ruptures: 6-cord wire-rope scaffold

Analytical model: George A. Costello’s theory of the wire rope

Homogenized Trigonal model: example of development

Hierarchical structures: importance in biomechanics

Anterior Cruciate LigamentLigaments hierarchical structure

Viscoelastic behaviour

The AP displacement of the tibia was determined by the displacement of the origin of the tibial coor-dinate system on the APaxis of the tibia. Tibial rotation was determined by the projection of the medial–lateral (ML) axis of the tibia onto the

A. Guadagno,Relatore Ch.mo Prof. Ing. A. Pepino e Correlatore Ing. A. RanavoloNapoli, 2004/2005.

VALUTAZIONE DELLE FORZE COMPRESSIVE E DI TAGLIO ALL’ARTICOLAZIONE DEL GINOCCHIO CON SISTEMA DI ANALISI COMPUTERIZZATA MULTIFATTORIALE DEL MOVIMENTO,

J Biomech. 2010 July 20; 43(10): 2039–2042. doi:10.1016/j.jbiomech.2010.03.015.A Knee-Specific Finite Element Analysis of the Human Anterior

Cruciate Ligament Impingement against the FemoralIntercondylar Notch

Hyung-Soon Park1,†, Chulhyun Ahn,†, David T. Fung, Yupeng Ren, and Li-QunZhang

ACL Kinematics

Flexion: 46.3 deg.Abduction: 0 deg.

External rotation: 0 deg.

Flexion: 44.8 deg.Abduction: 10.0 deg.

External rotation: 29.1 deg.

ACL Kinematics

The AP displacement of the tibia was determined by the displacement of the origin of the tibial coor-dinate system on the APaxis of the tibia. Tibial rotation was determined by the projection of the medial–lateral (ML) axis of the tibia onto the

Journal of Biomechanics 38 (2005) 293–298Interactions between kinematics and loading during

walking for thenormal and ACL deficient knee

Thomas P. Andriacchia, Chris O. Dyrby

Altman’s Scaffold

Biomaterials 23 (2002) 4131–4141Silk matrix for tissue engineered anterior cruciate

ligamentsGregory H. Altmana, Rebecca L. Horana, Helen H.

Lua, Jodie Moreaua, Ivan Martinb,John C. Richmondc, David L. Kaplana

Analytical Model

h1 h3h2core wires

axial tension in the wiretwisting moment in the wireshear force in the wire, along the local cooordinates system directionsbending moment in the wire in x and y directions, along the local coordina-tes system curvature of the wire in x and y directions, along the local coordinates system twist per unit length of the wire

THN, N’G, G’

κ, κ’

т

Kinematics of a wire

Mechanical Engineering Series, Springer - Theory of Wke Rope, 2nd ed. - George A. Costello

3

//t

k kF AEk kM ERεε εβ

βε ββ

εβ

=

Simple straigth strand

Δαξw = ξc - Tg α

core radiusRcRwαR = Rc+2Rwr = Rc+RwEvξc = ε

ξwβr = Tg α - Δα + νr

Rc ξc + Rw ξwTg α

β т = r βr = R β

Δт’ = 1 - 2 Sin2αr

Δα + Rc ξc + Rw ξwr2ν Sin α Cos α

Δκ’ = - 2 Sin α Cos αr

Δα + Rc ξc + Rw ξwr2ν Cos2 α curve variation

Gw’ = E Rw4 Δκ’ �4

Hw = E Rw4 Δт’ �4 (1 + v)Nw’ = Hw Cos2 α

r- Gw’ Sin α Cos α

rTw = � E Rw2 ξw

Fw = mw (Tw Sin α + Nw’ Cos α) Mw = mw (Hw Sin α + Gw’ Cos α + r Tw Cos α + r Hw Sin α )

Fc = � E Rc2 ξc Mc = E Rc4 т’ �4 (1 + v)

Geometrical characteristics

Deformationscore axial strain

angle of twist per unit lenght

Material propertiesYoung ModulusPoisson’s ratio

Core Loads

Wire Loads

wire radiuswire helix anglestrand total radiusstrand helical radius

wire axial strain

helical rototional strain

total rototional strain

Costitutive assumptions

Hipothesis of small displacements: Δα<<1, ξ<<1

30 Fibers

1 Bundle

Rf = 10.4067 10-5 m Equivalent fiber radius

R1 = [19 ± 2.8]10-6 mSilk fibroin average radius

On the right pilot-scale manu-facturing equipment for the fabrication of silk wire-rope matrices

Two particluars (B) and (C) of (i) and (iii), showing the extraction of the fibroins

On the left a close-up view of (i): the twisting machines showing the motor controlled spring-loaded clamps.

Biomaterials 24 (2003) 401–416Silk-based biomaterials

Gregory H. Altman,Frank Diaz,Caroline Jakuba,Tara Calabro,Rebecca L. Horan,

Jingsong Chen,Helen Lu,John Richmond, David L. Kaplan

6 Bundles

Rch3 = R2 Rwh3 = R3αh3


E R34 Sin αh3 �2(2 + ν)Cos2 αh3Ach1wh2wh3 =

Gych1wh2wh3’ = Ach1wh2wh3 Δκch1wh2wh3’

Loads

ξch1wh2wh3βrch1wh2wh3 = Tg αh3- Δαch1wh2wh3 + ν

rh3 R3 ξch1wh2wh3 + R2 ξch1wh2ch3

Tg αh3

Δтch1wh2wh3’ = 1 - 2 Sin2αh3rh3

Δαch1wh2wh3 + R3 ξch1wh2wh3 + R2 ξch1wh2ch3rh32

ν Sin αh3 Cos αh3

Δκch1wh2wh3’ = - 2 Sin αh3 Cos αh3rh3

Δαch1wh2wh3 +rh32ν Cos2 αh3R3 ξch1wh2wh3 + R2 ξch1wh2ch3

ξch1wh2ch3 = ξch1wh2βrch1wh2h3 = Δтch1wh2’ Rh3

ξch1wh2wh3 Deformations

Rch3 = R2 Rwh3 = R3αh3

ξwh1wh2wh3βrwh1wh2h3 = Tg αh3 - Δαwh1wh2wh3 + νrh3

R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3Tg αh3

Δтwh1wh2wh3’ = 1 - 2 Sin2αh3rh3

Δαwh1wh2wh3 + R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3rh32


Δκwh1wh2wh3’ = - 2 Sin αh3 Cos αh3rh3

Δαwh1wh2wh3 +rh32ν Cos2 αh3R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3

ξwh1wh2ch3 = ξwh1wh2

E R34 Sin αh3 �2(2 + ν)Cos2 αh3Awh1wh2wh3 =

Gywh1wh2wh3’ = Awh1wh2wh3 Δκwh1wh2wh3’

βrwh1wh2h3 = Δтwh1wh2’ Rh3

ξwh1wh2wh3


Loads

Deformations

6 Bundles

3 Strands

Rch2 = 0 Rwh2rh2 =Rwh2 αh2

13 Sin2 αh2

1 +


mh3 E R34 Sin αh3 �2(2 + ν)Cos2 αh3Ach1wh2 = + E R24 �

4

Tch1wh2=Fch1wh2h3Hch1wh2=Mch1wh2h3

Fch1ch2=0Mch1ch2=0

Loads

ξch1wh2βrch1wh2 = Tg αh2 - Δαch1wh2 + νrh2

2 R3 ξch1wh2wh3 + R2 ξch1wh2ch3Tg αh2

Δтch1wh2’ = 1 - 2 Sin2αh2rh2

Δαch1wh2+ 2 R3 ξch1wh2wh3 + R2 ξch1wh2ch3rh22


Δκch1wh2’ = - 2 Sin αh2 Cos αh2rh2

Δαch1wh2 +rh22ν Cos2 αh22 R3 ξch1wh2wh3 + R2 ξch1wh2ch3

ξch1ch2 = ξch1βrch1h2 = Δтch1’ Rh2

Deformationsξch1wh2

ξwh1wh2

3 Strands

Rch2 = 0 Rwh2rh2 =Rwh2 αh2

13 Sin2 αh2

1 +


mh3 E R34 Sin αh3 �2(2 + ν)Cos2 αh3Awh1wh2 = + E R24 �

4

Twh1wh2=Fwh1wh2h3Hwh1wh2=Mwh1wh2h3

Fwh1ch2=0Mwh1ch2=0

Loads

ξwh1wh2βrwh1wh2 = Tg αh2 - Δαwh1wh2 + νrh2

2 R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3Tg αh2

Δтwh1wh2’ = 1 - 2 Sin2αh2rh2

Δαwh1wh2+ 2 R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3rh22


Δκwh1wh2’ = - 2 Sin αh2 Cos αh2rh2

Δαwh1wh2 +rh22ν Cos2 αh22 R3 ξwh1wh2wh3 + R2 ξwh1wh2ch3

ξwh1ch2 = ξwh1βrwh1h2 = Δтwh1’ Rh2

Deformationsξwh1wh2

6 Cords

ξwh1βh1

ξwh1βrh1 = Tg αh1 - Δαh1 + νrh1

4 R3 ξwh1wh2wh3 + 2 R2 ξwh1wh2ch3 + 4 R3 ξch1wh2wh3 + 2 R2 ξch1wh2ch3Tg αh1

т = rh1 βrh1 = Rh1 βh1

Δтh1’ = 1 - 2 Sin2αh1rh1

Δαh1 + 4 R3 ξwh1wh2wh3 + 2 R2 ξwh1wh2ch3 + 4 R3 ξch1wh2wh3 + 2 R2 ξch1wh2ch3rh12


Δκh1’ = - 2 Sin α Cos αr

Δα +r2ν Cos2 α4 R3 ξwh1wh2wh3 + 2 R2 ξwh1wh2ch3 + 4 R3 ξch1wh2wh3 + 2 R2 ξch1wh2ch3

Rch1Rwh1αh1


Fwh1 = mh1 (Twh1 Sin αh1 + Nwh1’ Cos αh1)Mwh1 = mh1 (Hwh1 Sin αh1 + Gwh1’ Cos αh1 + rh1 Twh1 Cos αh1 + rh1 Hwh1 Sin αh1)

Awh1 = mh2 Awh1wh2Gywh1 = Awh1 Δκh1’

Hwh1 = Mh1Twh1 = Fh1 Loads

Deformationsξch1 = ε

Analytical model Results and validation

0.05 0.10 0.15 0.20 0.25

200

300

400

500

600

Calculated

G. Vunjak-Novakovic et al.

Displacement (%)

Load

(N)

Root Mean Square Error Percentage

2

0.023

ni i

i i

x xx

RMSEPn

− = =

∑

Annu. Rev. Biomed. Eng. 2004. 6:131–56 TISSUE ENGINEERING OF LIGAMENTS

G. Vunjak-Novakovic1, Gregory Altman2,3, Rebecca Horan2 and David L. Kaplan2

1 Massachusetts Institute of Technology, Harvard-MIT Division of Health Sciences and Technology, Cambridge, Massachusetts 02139; email: [email protected]

2 Department of Biomedical Engineering, Tufts University, Medford,Massa-chusetts 02155; email: [email protected], [email protected], [email protected]

3 Tissue Regeneration, Inc., Medford, Massachusetts 02155

ΔL

Homogenized modelCostello’s constants Trigonal constants [Pa]

S11=5.02232*1010

S12=3.34821*1010

S22=5.02232x1010

S33=1.06738x109

S34=1.68011*109

S44=2.86704*107

S55=2.86704*107

S66=8.37054*109

kεε=0.52972

kββ=0.0160126

kεβ=0.0566074

kβε=0.122818⇔

11 12 13 14

12 22 23 24

33 13 23 33 34 33

3 14 24 34 44 3

3 55 56 3

56 66

0 00 00 0

.0 0 2

0 0 0 0 20 0 0 0 2

rr rr

r r

r r

S S S SS S S SS S S SS S S S

S SS S

θθ θθ

θ θ

θ θ

σ εσ εσ εσ εσ εσ ε

= ⋅

3

//t

k kF AEk kM ERεε εβ

βε ββ

εβ

=

Displacements along the z-axis [m] Rotations around the z-axis [deg]

0.10 0.15 0.20 0.25

200

300

400

500

600 Analytical model

Homogenized model

Displacement (%)

Load

(N)

Homogenized model Results and validation


2

0.086

ni i

i i

x xx

RMSEPn

− = =

∑


2

0.028

ni i

i i

x xx

RMSEPn

− = =

∑

Displacement (%)R

otat

ion

(deg

)0.10 0.15 0.20 0.25

100

80

60

40

Analytical model

Homogenized model

Conclusions andfuture developments

NON LINEAR BEHAVIOUR DEVELOPMENT

3D FEM JOINT IMPLEMENTATION WITH ON-SITE TESTS

IMPROVEMENT OF CURRENT SCAFFOLDS

DESIGN OF NEW STRUCTURES

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