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TABLE 1. Mathematical relationships in homothetic and a quasi-homothetic transformation.

 
   

Mathematical Relationship

Homothetic transformation

Lineal entities

equation t1for i=1 to 3

Surface entities

equation t2 with i j

Quasi-homothetic transformation

Lineal entities

equation t3

equation t4

Surface entities

equation t5

 

 TABLE 2. Equations of forces in a scaled model B with reference to model A. AA is the area of the reference model, AB the area of the scaled model, tA is the thickness of the reference model and tB the thickness of the scaled model.

 
 

Stress state constant

Displacements constants

Plane Stress

equation t6

equation t7

Plane Strain

equation t8

equation t9

TABLE 3. Thickness and surface of the four scaled models of Connochaetes taurinus used in Case 1.

 

Scaled model

Thickness
(t)
in mm

Area
(A)
in mm2

1 (Reference)

1,00

20291,00

2

10,00

5072,75

3

6,00

20291,00

4

24,50

1371,67

 

TABLE 4. Thickness and a of C. taurinus, A. buselaphus, H. niger and K. vardoni used in Case 2.

 

Model

Thickness
(t)
in mm

Area
(A)
in mm2

C. taurinus (Reference)

20,88

20282,00

A. buselaphus

16,97

17897,00

H. niger

19,67

20742,00

K. vardoni

14,22

10617,00

 

nogueJordi Marcé-Nogué
Departament de Resistència de Materials i Estructures a l'Enginyeria
Universitat Politècnica de Catalunya
08222, Terrassa
Spain
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Jordi Marcé-Nogué (Igualada, 1979) has Bachelor degree of Industrial Engineering issued from Universitat Politècnica de Catalunya (Spain) in 2003 and PhD degree Resistència de Materials i Computational Biomechanics from Universitat Politècnica de Catalunya in 2009.

He started as a research fellow in the Departament de Estructures a l’Enginyeria and from 2006 he has being a assistant teacher in Escola Tècnica Superior d’Enginyeria i Aeronàutica de Terrassa (UPC) and a researcher at Laboratory for the Innovation of Structures and Materials (LITEM)

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demiguelDaniel DeMiguel
Institut Català de Paleontologia Miquel Crusafont
Edifici ICP
Campus de la UAB s/n, 08193
Cerdanyola del Vallès
Spain
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Daniel DeMiguel has a degree in Geology from the University of Zaragoza, and a PhD degree in Science (2009) from the University of Zaragoza focused on the biomechanics of the dentition and the study of dental wear of fossil ruminants from the Neogene of the Iberian Range basins. Shortly after defending his thesis, he joined the Department of "Neogene and Quaternary fauna" of the Institut Català de Paleontologia Miquel crusafont ICP as a postdoctoral researcher. Nowadays, he works in this center as a "Juan de la Cierva" researcher since 2012.

His main research interests are in the area of evolutionary paleoecology, covering a wide variety of problems related to the biological evolution of herbivorous mammals and the climatic evolution of the Iberian Neogene basins.

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fortunyJosep Fortuny
Institut Català de Paleontologia Miquel Crusafont
Edifici ICP
Campus de la UAB s/n, 08193
Cerdanyola del Vallès
Spain
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Josep Fortuny (La Roca del Vallès, 1980) has a Degree of Biology (Zoology) at the Universitat Autònoma de Barcelona, Barcelona, Catalonia (2006) and a PhD of Geology by the Universitat de Barcelona, Barcelona, Catalonia (2011) about the early tetrapods from the Permian and Triassic periods and its paleobiology using biomechanical approaches.

He started in paleontology as scholar in Institut de Paleontologia de Sabadell and after he got a predoctoral contract. Currently, he has a postdoctoral position in the Institut Català de Paleontologia Miquel Crusafont (Barcelona, Catalonia) and he is the coordinator of the Virtual Paleontology Research Group. His research focus on the paleobiology of early tetrapods using non invasive tecniques as CT scanners and biomechanical approaches as Finite Element Analysis (FEA).

Dr. Fortuny is member of the Society of Vertebrate Palaeontology (SVP) and the Association des Géologues du Permien et du Trias (AGPT).

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de estebanSoledad de Esteban-Trivigno
Transmitting Science, Gardenia 2
Piera, 08784
Spain;
Institut Català de Paleontologia Miquel Crusafont
Edifici ICP
Campus de la UAB s/n
08193, Cerdanyola del Vallès
Spain
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Soledad De Esteban-Trivigno (Balcarce, 1975) obtained her PhD degree in Biology at the University of Valencia in 2008. From 2008 to 2010 she was Postdoc at the Paleobiology department in The Institut Català de Paleontologia. Nowadays she is Academic Director at Transmitting Science and Associate Researcher at the Institut Català de Paleontologia. 

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gilLluís Gil
Departament de Resistència de Materials i Estructures a l'Enginyeria
Universitat Politècnica de Catalunya
08222, Terrassa
Spain
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Lluís Gil (Barcelona, 1966) achieved a civil engineering degree in 1992 and a PhD from Universitat Politecnica de Catalunya UPC in 1997.

He currently is associate professor at UPC in the field of aerospace structures. Now is the Director of research of the Laboratory for the Innovation of Structures and Materials (LITEM). Co-author of 27 international journal and 45 conference contributions and 1 patent. Interested in modal analysis, composites and new applications of recycled materials. 

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FIGURE 1. Homothetic and quasi-homothetic transformation.

 figure 1

FIGURE 2. Boundary conditions, forces applied in the reference model A and the scaled model B and location of points P and Q in the jaw.

 figure 2

FIGURE 3. Von Mises Stress distribution (in MPa) and Displacement distribution (in mm) in the four scaled models of Connochaetes taurinus analyzed for plane stress and plane strain.

 figure 3

FIGURE 4. Von Mises Stress distribution (in MPa) and Displacement distribution (in mm) in the four different models for plane stress and plane strain when a unitary force is applied in all the models.

 figure 4

FIGURE 5. Von Mises Stress distribution (in MPa) and Displacement distribution (in mm) in the four different models for plane stress and plane strain when the quasi-homotetic transformation was applied.

 figure 5

 

TABLE S1. Numerical results of Von Mises Stress (MPa) and displacements (mm) in the points P and Q for the four different scaled models of C. taurinus.

 

Scaled models

Plane Stress - Stress state constant

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

1 (Reference)

0,073340

0,020745

0,00511430

0,00433380

2

0,073890

0,020980

0,00258460

0,00219120

3

0,073340

0,020745

0,00511430

0,00433380

4

0,072596

0,020499

0,00132320

0,00112100

 

Plane Stress – Displacements constants

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

1 (Reference)

0,073340

0,020745

0,00511430

0,00433380

2

0,149780

0,042537

0,00516910

0,00438250

3

0,073340

0,020745

0,00511430

0,00433380

4

0,279220

0,078844

0,00510800

0,00431160

 

Plane Strain - Stress state constant

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

1 (Reference)

0,064205

0,017786

0,00431340

0,00365400

2

0,065554

0,018242

0,00217950

0,00184720

3

0,064205

0,017786

0,00431340

0,00365400

4

0,063551

0,017571

0,00111610

0,00111610

Case

Plane Strain – Displacements constants

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

1 (Reference)

0,064205

0,017786

0,00431340

0,00365400

2

0,131110

0,036483

0,00435900

0,00369440

3

0,064205

0,017786

0,00431340

0,00365400

4

0,244430

0,067580

0,00429270

0,00363570

 

TABLE S2. Numerical values of Von Mises Stress (MPa) and displacements (mm) in the points P and Q for C. taurinus, A. buselaphus, H. niger and K. vardoni when all the models have applied a unitary force (1N).

 

Model

Plane Stress - Stress state constant

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

C. taurinus (Reference)

0,003300

0,000923

0,000238

0,000201

A.buselaphus

0,000113

0,003397

0,000175

0,000213

H. niger

0,002467

0,001511

0,000167

0,000116

K. vardoni

0,000134

0,002422

0,000135

0,000129

 

Plane Stress – Displacements constants

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

C. taurinus (Reference)

0,003300

0,000923

0,000238

0,000201

A.buselaphus

0,000113

0,003397

0,000175

0,000213

H. niger

0,002467

0,001511

0,000167

0,000116

K. vardoni

0,000134

0,002422

0,000135

0,000129

 

Plane Strain - Stress state constant

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

C. taurinus (Reference)

0,064205

0,017786

0,000238

0,000201

A.buselaphus

0,001607

0,048940

0,002498

0,003036

H. niger

0,042460

0,025511

0,002771

0,001919

K. vardoni

0,001733

0,029973

0,001623

0,001550

 

Plane Strain – Displacements constants

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

C. taurinus (Reference)

0,064205

0,017786

0,000238

0,000201

A.buselaphus

0,001607

0,048940

0,002498

0,003036

H. niger

0,042460

0,025511

0,002771

0,001919

K. vardoni

0,001733

0,029973

0,001623

0,001550

 

TABLE S3. Numerical values of Von Mises Stress (MPa) and displacements (mm) in the points P and Q for C. taurinus, A. buselaphus, H. niger and K. vardoni when all the models have applied a force value according to the scaled relationships.

 

Model

Plane Stress - Stress state constant

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

C. taurinus (Reference)

0,003300

0,000923

0,00023754

0,00020140

A.buselaphus

0,000086

0,002594

0,00013342

0,00016224

H. niger

0,002350

0,001440

0,00015908

0,00011037

K. vardoni

0,000066

0,001193

0,00006657

0,00006376

 

Plane Stress – Displacements constants

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

C. taurinus (Reference)

0,003300

0,000923

0,00023754

0,00020140

A.buselaphus

0,000092

0,002761

0,00014202

0,00017270

H. niger

0,002324

0,001423

0,00015729

0,00010913

K. vardoni

0,000091

0,001649

0,00009201

0,00008812

 

Plane Strain - Stress state constant

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

C. taurinus (Reference)

0,064205

0,017786

0,00431340

0,00365400

A.buselaphus

0,001510

0,045975

0,00234620

0,00285220

H. niger

0,042940

0,025799

0,00280220

0,00194110

K. vardoni

0,001254

0,021686

0,00117410

0,00112170

 

Plane Strain – Displacements constants

Von Mises Stress [MPa] in P

Von Mises Stress [MPa] in Q

Displacement [mm] in P

Displacement [mm] in Q

C. taurinus (Reference)

0,064205

0,017786

0,00431340

0,00365400

A.buselaphus

0,001607

0,048940

0,00249750

0,00303610

H. niger

0,042460

0,025511

0,00277090

0,00191940

K. vardoni

0,001733

0,029973

0,00162280

0,00155040

 
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Quasi-homothetic transformation for comparing the mechanical performance of planar models in biological research

Jordi Marcé-Nogué, Daniel DeMiguel, Josep Fortuny, Soledad de Esteban-Trivigno, and Lluís Gil

Plain Language Abstract

Finite Element Analysis is a powerful tool creating and analysing the physical behaviour of computational models. To compare the behaviour of vertebrate structures that differ in size and shape, every analysis requires standardization of the initial conditions of the model. In this paper we present a new procedure adapting two-dimensional models of vertebrate structures to easily compare them only modifying the value of the biomechanical forces applied. This approach is shown to be extremely useful when exploring the effect of the shape in front of the strength and the stiffness of vertebrate bone structures that can be simplified to two-dimensional models.

Resumen en Castellano

Transformación casi-homotética para la comparación del comportamiento mecánico de modelos bidimensionales en la investigación biológica

En los últimos años, el potencial de los análisis por elementos finitos (FEA) como técnica analítica en el campo de la investigación biológica ha sido muy notorio. A pesar de su gran capacidad, no siempre se puede utilizar en la comparación del comportamiento mecánico de modelos que difieren tanto en el tamaño como en la forma. En este trabajo se presenta una metodología sencilla e innovadora para escalar modelos bidimensionales de elementos finitos de varias especies de bóvidos existentes que difieren en tamaño y forma. El método se basa en la modificación de los valores de las fuerzas aplicadas al modelo usando transformaciones casi-homotéticas y teniendo en cuenta las particularidades de la formulación de la elasticidad plana (tensión plana y deformación plana). Se demuestra que este enfoque es muy útil para explorar el efecto de la forma en frente a la resistencia y la rigidez en las estructuras óseas de los vertebrados. Así, el concepto definido como "casi- homotético" es una nueva propuesta para utilizar en modelos bidimensionales de elementos finitos de estructuras biológicas (especialmente en vertebrados) que se pueden modelar siguiendo la formulación de la elasticidad plana.

Palabras clave: FEA; transformación homotética; elasticidad plana; tamaño; morfología; mecánica de medios continuos

Traducción: los autores

Resum en Català

Transformació quasi-homotètica per a la comparació del comportament mecànic de models bidimensionales en la recerca biològica

En els darrers anys, el potencial dels Anàlisis per Elements Finits (FEA) com a tècnica analítica en el camp de la recerca biològica ha estat molt notori. Tot i la seva gran capacitat, no sempre es pot utilitzar en la comparació del comportament mecànic de models que difereixen tan en la mida com en la forma. En aquest treball es presenta una metodologia senzilla i innovadora per escalar models bidimensionals d'elements finits per a diverses espècies de bòvids existents que difereixen en mida i forma. El mètode es basa en la modificació dels valors de les forces aplicades al model utilitzant transformacions quasi-homotètiques i tenint en compte les particularitats de la formulació de l'elasticitat plana (tensió plana i deformació plana). Es demostra que aquest enfocament és molt útil per explorar l'efecte de la forma en front de la resistència i la rigidesa en les estructures òssies dels vertebrats. Així, el concepte definit com a "quasi-homotètic" és una nova proposta per a utilitzar en models bidimensionals d'elements finits d'estructures biològiques (especialment en vertebrats) que es poden modelar seguint la formulació de l'elasticitat plana.

Paraules clau: FEA; transformació homotètica; elasticitat plana; mida; morfologia; mecànica de medis continus

Traducció: els autors

Résumé en Français

Transformation quasi homothétique pour comparer les performances mécaniques des modèles planaires dans la recherche biologique

Le potentiel de l'analyse par éléments finis (FEA) comme une technique d'analyse pour la recherche biologique a été largement mis en évidence au cours des dernières années. En dépit de son grand pouvoir, seulement dans le meilleur des cas peut-on comparer le comportement des modèles qui diffèrent en taille et en forme. Ici, une nouvelle procédure facile à échelonner les modèles FE d'élasticité plane est présentée pour plusieurs espèces de bovidés existantes qui diffèrent significativement en taille et en morphologie. La méthode est basée sur la modification des valeurs des forces appliquées en tenant compte des particularités des modèles d'élasticité plane (déformation plane et les équations de tension plane) à l'aide de transformations quasi- homothétiques . Cette approche s'avère très utile lors de l'exploration de l'effet de la forme devant la force et la rigidité des structures osseuses des vertébrés. Ainsi, le concept quasi homothétique est une nouvelle et intéressante proposition pour être utilisé dans les modèles d'élasticité planes des structures biologiques, qui peuvent être modélisés comme des modèles d'éléments finis bidimensionnels, et plus particulièrement chez les vertébrés.

Mots-clés: analyse par éléments finis ; transformation homothétique ; élasticité planaire ; taille ; morphologie ; mécanique des milieux continus

Translator: Kenny J. Travouillon

Deutsche Zusammenfassung

In progress

Translator: Eva Gebauer

Arabic

365 arab

Translator: Ashraf M.T. Elewa