Specimen Specific Multiscale Model for the Anisotropic Elastic Properties of Human Cortical Bone Tissue

2007 ◽  
Author(s):  
Justin M. Deuerling ◽  
Weimin Yue ◽  
Alejandro A. Espinoza ◽  
Ryan K. Roeder
Keyword(s):  
Cortical Bone ◽  
Elastic Properties ◽  
Structural Information ◽  
Transversely Isotropic ◽  
Longitudinal Axis ◽  
Orientation Distribution ◽  
Tissue Properties ◽  
Apatite Crystals ◽  
Micromechanical Models ◽  
Anisotropic Elastic

The elastic constants of cortical bone are orthotropic or transversely isotropic depending on the anatomic origin of the tissue. Micromechanical models have been developed to predict anisotropic elastic properties from structural information. Many have utilized microstructural features such as osteons, cement lines and Haversian canals to model the tissue properties [1]. Others have utilized nanoscale features to model the mineralized collagen fibril [2]. Quantitative texture analysis using x-ray diffraction techniques has shown that elongated apatite crystals exhibit a preferred orientation in the longitudinal axis of the bone [3]. The orientation distribution of apatite crystals provides fundamental information influencing the anisotropy of the extracellular matrix (ECM) but has not been utilized in existing micromechanical models.

Modeling anisotropic static elastic properties of soft mudrocks with different clay fractions

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. MR27-MR37 ◽  
Author(s):  
Biao Li ◽  
Ron C. K. Wong
Keyword(s):  
Elastic Properties ◽  
Void Ratio ◽  
Volume Fraction ◽  
Clay Fraction ◽  
Transversely Isotropic ◽  
Orientation Distribution ◽  
Simplified Approach ◽  
Effective Medium Approach ◽  
Anisotropic Elastic ◽  
Colorado Shale

We have quantified the effects of clay fraction and fabric on the static elastic properties of soft mudrocks with emphasis on microlevel mechanisms. Soft mudrocks are treated as a mixture of nonclay minerals and clay-water composites. We have devised a simplified approach to estimate the fabric orientation distribution of soft mudrocks based on measured parameters such as clay fraction and porosity. A single parameter (fabric angle) that characterizes the fabric orientation distribution of soft mudrocks is related to the void ratio of clay-water composites. The static transversely isotropic (TI) elastic properties of soft mudrocks are modeled using an anisotropic differential effective medium approach. The effect of variation in fabric orientation distribution on the TI elastic parameters of clay-water composites is studied by applying the Voigt approximation. With an increase of clay fraction, soft mudrocks have decreasing trends in the deformation moduli because some nonclay minerals are replaced by clay-water composites. However, the deformation moduli of clay-water composites could increase when there is more anisotropy in the fabric due to an increase in the clay fraction. Thus, the correlations between anisotropic elastic moduli and volume fraction of clay-water composites will display some fluctuations. Such nonlinear relationships are validated against published experimental data on Colorado shale samples from the Western Canadian Sedimentary Basin.

RELATIONS OF ANISOTROPIC ELASTIC MODULI TO DENSITY AND CT NUMBER IN BOVINE CORTICAL BONE

2008 ◽  
Vol 20 (03) ◽  
pp. 139-143 ◽  
Author(s):  
Jui-Ting Hsu ◽  
Ming-Tzu Tsai ◽  
Heng-Li Huang
Keyword(s):  
Elastic Modulus ◽  
Bone Density ◽  
Cortical Bone ◽  
Elastic Moduli ◽  
Transversely Isotropic ◽  
Longitudinal Direction ◽  
Coefficient Of Determination ◽  
Ct Number ◽  
Noninvasive Technique ◽  
Anisotropic Elastic

It would be useful to be able to determine the mechanical properties of bone using a noninvasive technique such as computed tomography (CT). However, in contrast to cancellous bone tissue, quantifying the elastic modulus of cortical bone from bone density and CT number has not been investigated extensively. This study measured the elastic moduli of cortical bone from eight bovine femora in the longitudinal, circumferential, and radial directions using mechanical compressive testing. Before testing, the CT number and wet apparent bone density were obtained. The experimentally determined coefficient of determination between CT number and bone density was around 0.6. Bone density was a good predictor of the elastic modulus of cortical bone in the longitudinal direction (r2 > 0.79), but it could not be used to predict the elastic moduli in the circumferential (r2 < 0.4) and radial (r2 < 0.22) directions. The coefficient of determination between CT number and the elastic modulus in the longitudinal direction was higher than 0.41. However, the correlations between CT number and elastic moduli were weak in the circumferential (r2 < 0.21) and radial (r2 < 0.19) directions. Moreover, the elastic modulus was much higher in the longitudinal direction than the circumferential and radial directions, and hence cortical bone can be considered a transversely isotropic property.

Anisotropic elastic properties of nanocrystalline nickel thin films

2005 ◽  
Vol 20 (5) ◽  
pp. 1186-1193 ◽  
Author(s):  
D.C. Hurley ◽  
R.H. Geiss ◽  
M. Kopycinska-Müller ◽  
J. Müller ◽  
D.T. Read ◽  
...  
Keyword(s):  
Elastic Properties ◽  
Transversely Isotropic ◽  
Acoustic Microscopy ◽  
Indentation Modulus ◽  
Polycrystalline Nickel ◽  
Nickel Thin Films ◽  
Microtensile Testing ◽  
Measurement Results ◽  
Oriented Polycrystalline ◽  
Anisotropic Elastic

The elastic properties of a nickel film approximately 800 nm thick were measured with nanoindentation, microtensile testing, atomic force acoustic microscopy (AFAM), and surface acoustic wave (SAW) spectroscopy. Values for the indentation modulus (220–223 GPa) and Young’s modulus (177–204 GPa) were lower than predicted for randomly oriented polycrystalline nickel. The observed behavior was attributed to grain-boundary effects in the nanocrystalline film. In addition, the different measurement results were not self-consistent when interpreted assuming elastic isotropy. Agreement was improved by adopting a transversely isotropic model corresponding to the film’s 〈111〉 preferred orientation and reducing the elastic moduli by 10–15%. The SAW spectroscopy results indicated that the film density was 1–2% lower than expected for bulk nickel, consistent with models for nanocrystalline materials. Similar reductions in modulus and density were observed for two additional films approximately 200 and 50 nm thick using AFAM and SAW spectroscopy. These results illustrate how complementary methods can provide a more complete picture of film properties.

Free‐surface boundary conditions for elastic staggered‐grid modeling schemes

Geophysics ◽  
2002 ◽  
Vol 67 (5) ◽  
pp. 1616-1623 ◽  
Author(s):  
Rune Mittet
Keyword(s):  
Free Surface ◽  
Rayleigh Wave ◽  
Elastic Properties ◽  
Transversely Isotropic ◽  
Staggered Grid ◽  
Surface Boundary ◽  
Surface Boundary Conditions ◽  
Propagating Waves ◽  
Simple Implementation ◽  
Anisotropic Elastic

I present an accurate, stable, and simple implementation of the elastic free surface for staggered‐grid modeling schemes. The method is based on the presumption that the elastic Hooke's tensor on the free surface can be taken as similar to a transversely isotropic medium. The anisotropic elastic properties are then analyzed. The final scheme is isotropic, but with modified elastic properties on the free surface. The proposed scheme shows good agreement with an analytical solution. It is shown that the Rayleigh wave can not be properly resolved at two gridpoints per shortest wavelength if the shortest wavelength is calculated as the Rayleigh‐wave velocity over the maximum frequency. The Rayleigh wave is exponentially damped away from the free surface and requires denser sampling than purely propagating waves.

Dependence of Elastic Properties of Human Femoral Cortical Bone on Porosity

2015 ◽  
Author(s):  
Maryam Khosroshahi ◽  
Fred Barez ◽  
Amer El-Hage ◽  
James Kao
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Finite Element Model ◽  
Cortical Bone ◽  
Elastic Properties ◽  
Transversely Isotropic ◽  
Element Model ◽  
Element Analysis ◽  
Age Related ◽  
The Finite Element Model

Hip fracture is one of the most serious and common health problems among elderly which may lead to permanent disability or death. Hip fracture commonly occurs in the femoral bone, the major bone in the hip joint. Microscopic age-related changes in the structure of cortical bone is one of the factors that is considered to be partially responsible for the increase of fracture risk in elderly. It is of great interest to develop a predictable model of such fractures for the aging population in preparation of a suitable therapy. These micro structural changes influence mechanical properties and, therefore, behavior of bone and are critical to understand risk and mechanics of fracture of bone. Correlation between cortical bone strength and porosity, as a microscopic structural factor, has been examined frequently as a function of age and/or porosity. These studies have investigated the effect of porosity experimentally and have not studied the effect of porosity independently from other structural factors such as bone mineral density. In this study effect of porosity on elastic properties of human femoral cortical bone was studied independently using finite element analysis assuming transversely isotropic behavior in terms of elastic properties with the axis of elastic properties along the longitudinal axis of femur shaft. In this study, published standard mechanical tests for transversely isotropic materials were simulated using finite element computer simulation on models with different porosities. The developed finite element model utilized material properties based on the best fit regression in previously published articles. Pores’ size, shape and distribution were also modeled based on previous experimental studies. The finite element model, in general, predicted behavior of five independent elastic mechanical properties, namely, longitudinal Young’s modulus, transverse poisson’s ratio, transverse shear modulus, transverse Young’s modulus and longitudinal poisson’s ratio, as a function of porosity. Furthermore, effect of porosity on the elastic properties across various age groups was investigated using published data on age-related changes in bone porosity. Mathematical models based on Finite Element Analysis results have been developed using linear least square regression. These models show negative linear relationship between studied elastic properties of human femoral cortical bone and porosity. The Finite Element Analysis results compared well with the previously published experimental data. Furthermore, the results obtained show the elastic properties as functions of age for females and males. The predicted values for elastic properties are lower for men compared to women of age 20 to 40 years old. However, after the age of 44, elastic properties of femoral cortical bone for men are higher than women. The Finite Element Model developed in this study will help to create a clinical bone model for the prediction of fracture risk or the selection of suitable therapy in orthopedic surgery.

Determination of Elastic Constants for Human Femurs

1979 ◽  
Vol 101 (3) ◽  
pp. 193-197 ◽  
Author(s):  
V. G. Lappi ◽  
M. S. King ◽  
I. Le May
Keyword(s):  
Elastic Properties ◽  
Elastic Constants ◽  
Isotropic Material ◽  
Rotational Symmetry ◽  
Transversely Isotropic ◽  
Longitudinal Axis ◽  
Transversely Isotropic Material ◽  
Axis Of Symmetry ◽  
Plane Of Isotropy

The elastic properties of the bone constituting human femurs have been determined from measurements of the velocities of ultrasonic compressional and shear waves through wet, embalmed bone samples. The bone has been shown to be a transversely isotropic material with the axis of symmetry parallel to the longitudinal axis of the bone. The values of the elastic constants were determined to be: c11=6860±330MPaE3=5500MPac12=2700±570MPaE1=4990MPac13=3760±1570MPaν31=0.39c33=8480±760MPaν12=0.20c44=2240±180MPaG31=2240MPa where the 3-axis is that of rotational symmetry and the 1- and 2-axes are in the plane of isotropy.

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.

On the effective anisotropic elastic properties of porous hydroxyapatite, porous collagen, and cortical bone: A homogenization scheme with percolation threshold concept

2018 ◽  
Vol 24 (4) ◽  
pp. 1091-1102 ◽  
Author(s):  
Mai-Ba Vu ◽  
Tuan Nguyen-Sy
Keyword(s):  
Percolation Threshold ◽  
Cortical Bone ◽  
Elastic Properties ◽  
Organic Content ◽  
Homogenization Theory ◽  
High Porosity ◽  
Porous Hydroxyapatite ◽  
Threshold Concept ◽  
Homogenization Scheme ◽  
Anisotropic Elastic

The objective of this study is to model the effective anisotropic elastic properties of porous hydroxyapatite, wet collagen, and cortical bone by an advanced homogenization scheme with a percolation threshold concept. The theoretical basis of the anisotropic homogenization theory is first presented. A homogenization scheme with a percolation threshold concept is then introduced and validated against experimental data for porous hydroxyapatite as well as bone after decollagenization. It is also validated on a porous collagen that is a result of the demineralization of bone. Even though aligned collagen fibers are considered, similar values of the elastic stiffnesses [Formula: see text] and [Formula: see text] were found for demineralized bone due to its very high porosity. Finally the proposed method is used to model cortical bone as a mixture of hydroxyapatite mineral and soft organic content that is in turn a mixture of collagen fiber and pores filled by water. Good agreement between modeled and measured data is observed. The model presented herein is simpler than existing multi-scale homogenization schemes in the literature, but its results fit very well with the experimented trends.

scholarly journals Anisotropic elastic properties of human femoral cortical bone and relationships with composition and microstructure in elderly

2019 ◽  
Vol 90 ◽  
pp. 254-266 ◽  
Author(s):  
Xiran Cai ◽  
Hélène Follet ◽  
Laura Peralta ◽  
Marc Gardegaront ◽  
Delphine Farlay ◽  
...  
Keyword(s):  
Cortical Bone ◽  
Elastic Properties ◽  
Anisotropic Elastic
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