Bone Anisotropy - Analytical and Finite Element Analysis

A human femur model, submitted to static loads, is analyzed through the utilization of three material constitutive relationships, namely: isotropic, transversally isotropic and orthotropic. The influence of bone anisotropy with respect to principal stress/strain distribution on human femur external surface was accessed through the use of analytical and finite element approaches. The models results show that the principal angles at a medial path bone surface have a good correlation with human femur bone lamellae angles.

Keywords:
Bone anisotropy; principal stresses; principal strains; principal angles

Authorship

Lucas Lisboa Vignoli

Department of Mechanical Engineering, Pontifical Catholic University of Rio de Janeiro, Rua Marques de São Vicente 225 - Gavea, Rio de Janeiro, RJ - CEP 22453-900 - Brazil. lucaslvig@gmail.comPontifical Catholic University of Rio de JaneiroBrazilRio de Janeiro, RJ, BrazilDepartment of Mechanical Engineering, Pontifical Catholic University of Rio de Janeiro, Rua Marques de São Vicente 225 - Gavea, Rio de Janeiro, RJ - CEP 22453-900 - Brazil. lucaslvig@gmail.com

Paulo Pedro Kenedi

PPEMM - Programa de Pós-Graduação em Engenharia Mecânica e Tecnologia de Materiais - CEFET/RJ - Av. Maracanã, 229 - Maracanã - RJ - CEP 20271-110 - Brazil. paulo.kenedi@cefet-rj.brCEFETBrazilRJ, BrazilPPEMM - Programa de Pós-Graduação em Engenharia Mecânica e Tecnologia de Materiais - CEFET/RJ - Av. Maracanã, 229 - Maracanã - RJ - CEP 20271-110 - Brazil. paulo.kenedi@cefet-rj.br

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (7)
Tables (2)
Formulas (36)

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Figure 1
The coordinates systems at a medial cross-section: (a) cartesian with forces and moments components and (b) cylindrical.

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Figure 2
Human femur static loading model.

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Figure 3
(a) Path created to extract the results and (b) Mesh used on the F.E. simulations.

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Figure 4
Orthotropic cortical tissue: Maximum principal stresses distribution at: (a) external surface and (b) path; Maximum principal strain distribution at: (c) external surface and (d) path.

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Figure 5
Comparative performance between analytical and numeric models for orthotropic cortical tissue at a medial external bone surface path: (a) principal stresses and (b) principal strains.

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Figure 6
Analytical model results, at a medial external bone surface path, for three different constitutive relations, for principal (a) stresses and (b) strains.

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Figure 7
Analytical model results, for three constitutive relations, at a medial external bone surface path, for (a) principal stress angles and (b) principal strain angles.

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Table 1
Loading forces and geometric constants. (* approximate measure).

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Table 2
Elastic mechanical properties. ¹(Kenedi and Riagusoff, 2015Kenedi, P.P., Riagusoff, I.I.T.(2015). Stress development at human femur by muscle forces, J Braz. Soc. Mech. Sci. Eng.,37(1), pp. 31-43.), ²(Yoon and Katz, 1976Yoon, H.S., Katz, J.L., (1976). Ultrasonic wave propagation in human cortical bone - II. Measurements of elastic properties and microhardness. ,J. Biomech., 9(7), pp. 459-464.) and³(Korsa and Mares, 2012Korsa, R., Mares, T. (2012). Numerical Identification of Orthotropic Coefficients of the Lamella of a Bone's Osteon, Bulletin of Applied Mechanics, 8(31), pp. 45-53.).

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Figure 1 The coordinates systems at a medial cross-section: (a) cartesian with forces and moments components and (b) cylindrical.

Figure 2 Human femur static loading model.

Figure 3 (a) Path created to extract the results and (b) Mesh used on the F.E. simulations.

Figure 4 Orthotropic cortical tissue: Maximum principal stresses distribution at: (a) external surface and (b) path; Maximum principal strain distribution at: (c) external surface and (d) path.

Figure 5 Comparative performance between analytical and numeric models for orthotropic cortical tissue at a medial external bone surface path: (a) principal stresses and (b) principal strains.

Figure 6 Analytical model results, at a medial external bone surface path, for three different constitutive relations, for principal (a) stresses and (b) strains.

Figure 7 Analytical model results, for three constitutive relations, at a medial external bone surface path, for (a) principal stress angles and (b) principal strain angles.

Table 1 Loading forces and geometric constants. (* approximate measure).

Joint reaction force	P₁(N)	(-1,062; -130; -2,800)
Abductors force	P₂ (N)	(430; 0; 1,160)
Iliopsoas force	P₃ (N)	(78; 560; 525)
Ilio-Tibial Tract force	P₄ (N)	(0; 0; -1,200)
P₁ point of application	A (mm)	(50.7; -2.7; 133)
P₂ point of application	B (mm)	(-13.5; -6.5; 115)
P₃ point of application	C (mm)	(18.8; -29.3; 58.7)
P₄ point of application	D (mm)	(-24.6; -4.2; 83)
cross section centroid	(mm)	(0.8; 2.4; -25)
R_e (external radius)	(mm)*	15.5
R_i (internal radius)	(mm)*	7.6

Table 2 Elastic mechanical properties. ¹(Kenedi and Riagusoff, 2015Kenedi, P.P., Riagusoff, I.I.T.(2015). Stress development at human femur by muscle forces, J Braz. Soc. Mech. Sci. Eng.,37(1), pp. 31-43.), ²(Yoon and Katz, 1976Yoon, H.S., Katz, J.L., (1976). Ultrasonic wave propagation in human cortical bone - II. Measurements of elastic properties and microhardness. ,J. Biomech., 9(7), pp. 459-464.) and³(Korsa and Mares, 2012Korsa, R., Mares, T. (2012). Numerical Identification of Orthotropic Coefficients of the Lamella of a Bone's Osteon, Bulletin of Applied Mechanics, 8(31), pp. 45-53.).

Material Properties	Isotropic¹	Transversally Isotropic²	Orthotropic³
E (GPa)	20	-	-
E₁ (GPa)	-	18.8	12
E₂ (GPa)	-	-	13.4
E₃ (GPa)	-	27.4	20
ν	0.3	-	-
ν₁₂	-	0.312	0.376
ν₁₃	-	0.193	0.222
ν₂₃	-	-	0.234
G₁₂ (GPa)	-	-	9.06
G₁₃ (GPa)	-	17.42	11.22
G₂₃ (GPa)	-	-	12.46

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