Prediction of Cortical Bone Elastic Constants by a Two-Level Micromechanical Model Using a Generalized Self-Consistent Method

2005 ◽  
Vol 128 (3) ◽  
pp. 309-316 ◽  
Author(s):  
X. Neil Dong ◽  
X. Edward Guo
Keyword(s):  
Experimental Data ◽  
Cortical Bone ◽  
Elastic Constants ◽  
Micromechanical Model ◽  
Transversely Isotropic ◽  
Two Phase ◽  
Self Consistent ◽  
Isotropic Elasticity ◽  
Consistent Method ◽  
Haversian Canals

A two-level micromechanical model of cortical bone based on a generalized self-consistent method was developed to take into consideration the transversely isotropic elasticity of many microstructural features in cortical bone, including Haversian canals, resorption cavities, and osteonal and interstitial lamellae. In the first level, a single osteon was modeled as a two-phase composite such that Haversian canals were represented by elongated pores while the surrounding osteonal lamellae were considered as matrix. In the second level, osteons and resorption cavities were modeled as multiple inclusions while interstitial lamellae were regarded as matrix. The predictions of cortical bone elasticity from this two-level micromechanical model were mostly in agreement with experimental data for the dependence of transversely isotropic elasticity of human femoral cortical bone on porosity. However, variation in cortical bone elastic constants was greater in experimental data than in model predictions. This could be attributed to variations in the elastic properties of microstructural features in cortical bone. The present micromechanical model of cortical bone will be useful in understanding the contribution of cortical bone porosity to femoral neck fractures.

Green’s Functions for Two-Phase Transversely Isotropic Materials

1979 ◽  
Vol 46 (3) ◽  
pp. 551-556 ◽  
Author(s):  
Y.-C. Pan ◽  
T.-W. Chou
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Elastic Constants ◽  
Transversely Isotropic ◽  
Two Phase ◽  
Closed Form Solutions ◽  
Isotropic Materials ◽  
Function Solution ◽  
Present Solution ◽  
Plane Of Isotropy

Closed-form solutions are obtained for the Green’s function problems of point forces applied in the interior of a two-phase material consisting of two semi-infinite transversely isotropic elastic media bonded along a plane interface. The interface is parallel to the plane of isotropy of both media. The solutions are applicable to all combinations of elastic constants. The present solution reduces to Sueklo’s expression when the point force is normal to the plane of isotropy and (C11C33)1/2 ≠ C13 + 2C44 for both phases. When the elastic constants of one of the phases are set to zero, the solution can be reduced to the Green’s function for semi-infinite media obtained by Michell, Lekhnitzki, Hu, Shield, and Pan and Chou. The Green’s function solution of Pan and Chou for an infinite transversely isotropic solid can be reproduced from the present expression by setting the elastic constants of both phases to be equal. Finally, the Green’s function for isotropic materials can also be obtained from the present solution by suitable substitution of elastic constants.

An Analytical Modeling for Effective Thermal Conductivity of Multi-Phase Transversely Isotropic Fiberous Composites Using Generalized Self-Consistent Method

2012 ◽  
Vol 249-250 ◽  
pp. 904-909 ◽  
Author(s):  
Syed Aadil Hassan ◽  
Hassaan Ahmed ◽  
Asif Israr
Keyword(s):  
Thermal Conductivity ◽  
Effective Thermal Conductivity ◽  
Composite System ◽  
Volume Fraction ◽  
Transversely Isotropic ◽  
Balance Principle ◽  
Isotropic Composite ◽  
Self Consistent ◽  
Consistent Method ◽  
Multi Phase

In this paper a theoretical relationship for the effective thermal conductivity of a multiphase transversely isotropic composite system is obtained. The Generalized Self-Consistent Method and simple energy balance principle is employed to derive a more appropriate model. In the derivation, it is assumed that the orientation of fiber within the transversely isotropic composite system is unidirectional and surrounded by two different phases of porous and matrix phase. A combined effect of these three different phases on the effective thermal conductivity of the composite system in transverse direction is studied. The effect of the interfacial contact conductance between the fibers and porous medium is also considered. Results of effective thermal conductivity are plotted against volume fraction and conductance which shows extremely good agreement.

Micromechanical Model Prediction Overall Young Modulus Nanocomposites Polymer-Clay-Silica by Self Consistent Approach

2014 ◽  
Vol 980 ◽  
pp. 152-156
Author(s):  
Amar Mesbah ◽  
Krimo Azouaoui ◽  
Sid Ali Kaoua ◽  
Salah Boutaleb
Keyword(s):  
Mechanical Properties ◽  
Polymer Matrix ◽  
Elastic Constants ◽  
Model Prediction ◽  
Micromechanical Model ◽  
Young Modulus ◽  
Polyamide 6 ◽  
Micromechanical Approach ◽  
Self Consistent ◽  
Consistent Approach

In order to address the problem of stiffness and mechanical properties, a micromechanical approach for the prediction of the overall modulus of nanocomposites (Nylon-6/nanoclay/silica) using a self-consistent scheme based on the double-inclusion model and taking into account the different morphologies exfoliated or intercalated of the nanoparticles. Self-consistent approach that is used in our calculations was explained after reviewing the inclusion of Eshelby, in particular the double inclusion and while considering also the effect of constrained region, modeled as an interphase around reinforcements. Namely, polyamide 6 reinforced with clay platelets and silica particles. Several parameters on the Young's modulus of the composite were studied to see the effect of having mixed two or three reinforcements in polymer matrix. Finally, we demonstrated the process undertaken for the calculation of elastic constants of the material studied.

Mechanical Characterization of Graphite Platelet Nanocomposites Through Molecular Mechanics and Continuum Modeling

2006 ◽  
Author(s):  
J. Cho ◽  
J. J. Luo ◽  
I. M. Daniel
Keyword(s):  
Experimental Data ◽  
Molecular Mechanics ◽  
Elastic Constants ◽  
Strong Dependence ◽  
Micromechanical Model ◽  
Graphite Nanoplatelets ◽  
Inclusion Phase ◽  
Aspect Ratios ◽  
Molecular Force ◽  
Theoretical Predictions

Mechanical properties of nanocomposites consisting of epoxy substrate reinforced by randomly oriented graphite platelets are studied with Mori-Tanaka method in collaboration with molecular mechanics. Elastic constants of graphite nanoplatelets which are the inclusion phase of the micromechanical model are calculated based on their molecular force field. The calculated elastic constants are well compared with the both experimental data and other theoretical predictions in literatures. The results from Mori-Tanaka's method based on the graphite modulus calculation from molecular mechanics are found that nanocomposite moduli have strong dependence on the aspect ratios of reinforcing particles, but no direct size dependence. The predicted nanocomposite moduli compare favorably with modulus measurement of several graphite particles of various aspect ratios and sizes. The experimental data also shows that particle sizes have very weak effect on nanocomposite moduli.

Errors Induced by Off-Axis Measurement of the Elastic Properties of Bone

1988 ◽  
Vol 110 (3) ◽  
pp. 213-215 ◽  
Author(s):  
C. H. Turner ◽  
S. C. Cowin
Keyword(s):  
Cortical Bone ◽  
Bone Tissue ◽  
Cancellous Bone ◽  
Elastic Constants ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Misalignment Angle ◽  
Elastic Symmetry ◽  
Shear Moduli ◽  
Young’S Moduli

Misalignment between the axes of measurement and the material symmetry axes of bone causes error in anisotropic elastic property measurements. Measurements of Poisson’s ratio were strongly affected by misalignment errors. The mean errors in the measured Young’s moduli were 9.5 and 1.3 percent for cancellous and cortical bone, respectively, at a misalignment angle of 10 degrees. Mean errors of 1.1 and 5.0 percent in the measured shear moduli for cancellous and cortical bone, respectively, were found at a misalignment angle of 10 degrees. Although, cancellous bone tissue was assumed to have orthotropic elastic symmetry, the possibility of the greater symmetry of transverse isotropy was investigated. When the nine orthotropic elastic constants were forced to approximate the five transverse isotropic elastic constants, errors of over 60 percent were introduced. Therefore, it was concluded that cancellous bone is truly orthotropic and not transversely isotropic. A similar but less strong result for cortical bone tissue was obtained.

Elastic Properties of Human Osteon and Osteonal Lamella Computed by a Bidirectional Micromechanical Model and Validated by Nanoindentation

2015 ◽  
Vol 137 (8) ◽  
Author(s):  
Radim Korsa ◽  
Jaroslav Lukes ◽  
Josef Sepitka ◽  
Tomas Mares
Keyword(s):  
Cortical Bone ◽  
Elastic Properties ◽  
Elastic Constants ◽  
Micromechanical Model ◽  
Degenerative Diseases ◽  
Coordinate Systems ◽  
Inverse Homogenization ◽  
Curvilinear Coordinate ◽  
Anisotropic Elastic ◽  
Good Agreement

Knowledge of the anisotropic elastic properties of osteon and osteonal lamellae provides a better understanding of various pathophysiological conditions, such as aging, osteoporosis, osteoarthritis, and other degenerative diseases. For this reason, it is important to investigate and understand the elasticity of cortical bone. We created a bidirectional micromechanical model based on inverse homogenization for predicting the elastic properties of osteon and osteonal lamellae of cortical bone. The shape, the dimensions, and the curvature of osteon and osteonal lamellae are described by appropriately chosen curvilinear coordinate systems, so that the model operates close to the real morphology of these bone components. The model was used to calculate nine orthotropic elastic constants of osteonal lamellae. The input values have the elastic properties of a single osteon. We also expressed the dependence of the elastic properties of the lamellae on the angle of orientation. To validate the model, we performed nanoindentation tests on several osteonal lamellae. We compared the experimental results with the calculated results, and there was good agreement between them. The inverted model was used to calculate the elastic properties of a single osteon, where the input values are the elastic constants of osteonal lamellae. These calculations reveal that the model can be used in both directions of homogenization, i.e., direct homogenization and also inverse homogenization. The model described here can provide either the unknown elastic properties of a single lamella from the known elastic properties at the level of a single osteon, or the unknown elastic properties of a single osteon from the known elastic properties at the level of a single lamella.

Effective Thermal Conductivity of Composites with Different Particle Geometries and Interfacial Thermal Resistance

2010 ◽  
Vol 152-153 ◽  
pp. 269-273
Author(s):  
Mei Zhang ◽  
Peng Cheng Zhai
Keyword(s):  
Thermal Conductivity ◽  
Thermal Resistance ◽  
Effective Thermal Conductivity ◽  
Imperfect Interface ◽  
Interfacial Thermal Resistance ◽  
Surface Integral ◽  
Two Phase ◽  
Self Consistent ◽  
Consistent Method ◽  
Barrier Resistance

A new micromechanical method, the weighted residual self-consistent method (WRSCM) is developed to study the effective thermal conductivity of two-phase composites with different particle geometries in the presence of a thermal barrier resistance at the interface between constituents. The imperfect interface involves the continuity of the normal flux but allow for a finite temperature differences across the interface. Within the framework of self-consistent scheme, the effective thermal conductivity of two-phase composite is obtained using numerical iterative method on the basis of a surface integral of temperature over the imperfect interfaces. Numerical results show that for the given composite system, due to the existence of an interfacial thermal resistance, the particle geometries have significant impact on the effective thermal conductivity of composites.

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