scholarly journals The Anisotropic Elastic Constants of Cancellous Bone

2000 ◽  
Author(s):  
Stephen C. Cowin
Keyword(s):  
Data Analysis ◽  
Elastic Constant ◽  
Data Base ◽  
Cancellous Bone ◽  
Elastic Constants ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Variable Composition ◽  
Elastic Symmetry ◽  
Anisotropic Elastic

Abstract A method of data analysis for a set of elastic constant measurements is applied to an excellent data base for cancellous bone. For these materials the identification of the type of elastic symmetry is complicated by the variable composition of the material. The data analysis method permits the identification of the type of elastic symmetry to be accomplished independent of the examination of the variable composition. This method of analysis may be applied to any set of elastic constant measurements, but is illustrated here by application to an extraordinary data base of cancellous bone elastic constants. The solid volume fraction or bulk density is the compositional variable for the elastic constants of these natural materials. The final results are the solid volume fraction dependent orthotropic Hooke’s law for cancellous bone.

The Dependence of the Elastic Constants of Cancellous Bone Upon Volume Fraction

1999 ◽  
Author(s):  
Guoyu Yang ◽  
Stephen C. Cowin ◽  
Jesper Kabel ◽  
Anders Odgaard ◽  
Bert van Rietbergen ◽  
...  
Keyword(s):  
Cancellous Bone ◽  
Elastic Constants ◽  
Volume Fraction ◽  
Explicit Representation ◽  
Clinical Test ◽  
In Vivo Evaluation ◽  
Elastic Symmetry ◽  
Human Cancellous Bone ◽  
Anisotropic Elastic

Abstract A simple and explicit representation for the anisotropic elastic constants of human cancellous bone was determined using a new method of analysis on a data base consisting of 141 human cancellous bone specimens. The method of analysis shows that human cancellous bone has orthotropic elastic symmetry at the 95% confidence level and it also provides expressions for all the orthotropic elastic constants as functions of the volume fraction of solid only; no stereological measures of trabecular architecture are involved. This result should be employed to improve the in vivo evaluation of cancellous bone stiffness and strength from a standard clinical test.

scholarly journals A CAS Approach to Handle the Anisotropic Hooke’s Law for Cancellous Bone and Wood

2014 ◽  
Vol 2014 ◽  
pp. 1-28
Author(s):  
Sandeep Kumar
Keyword(s):  
Data Analysis ◽  
Elastic Constant ◽  
Cancellous Bone ◽  
Elastic Constants ◽  
Fourth Order ◽  
Elasticity Tensor ◽  
Eigenvalues And Eigenvectors ◽  
Hardwood Species ◽  
Maple Code ◽  
Softwood Species

The present research entirely relies on the Computer Algebric Systems (CAS) to develop techniques for the data analysis of the sets of elastic constant data measurements. In particular, this study deals with the development of some appropriate programming codes that favor the data analysis of known values of elastic constants for cancellous bone, hardwoods, and softwood species. More precisely, a “Mathematica” code, which has an ability to unfold a fourth-order elasticity tensor is discussed. Also, an effort towards the fabrication of an appropriate “MAPLE” code has been exposed, that can calculate not only the eigenvalues and eigenvectors for cancellous bone, hardwoods, and softwood species, but also computes the nominal average of eigenvectors, average eigenvectors, average eigenvalues, and the average elasticity matrices for these materials. Further, using such a MAPLE code, the histograms corresponding to average elasticity matrices of 15 hardwood species have been plotted and the graphs for I, II, III, IV, V, and VI eigenvalues of each hardwood species against their apparent densities are also drawn.

scholarly journals Determination of anisotropic elastic parameters from morphological parameters of cancellous bone for osteoporotic lumbar spine

2021 ◽  
Author(s):  
Christoph Oefner ◽  
Elena Riemer ◽  
Kerstin Funke ◽  
Michael Werner ◽  
Christoph-Eckhard Heyde ◽  
...  
Keyword(s):  
Finite Element ◽  
Lumbar Spine ◽  
Cancellous Bone ◽  
Elastic Constants ◽  
Bone Volume ◽  
Human Bone ◽  
Homogenization Theory ◽  
Elastic Parameters ◽  
Anisotropic Elastic

AbstractIn biomechanics, large finite element models with macroscopic representation of several bones or joints are necessary to analyze implant failure mechanisms. In order to handle large simulation models of human bone, it is crucial to homogenize the trabecular structure regarding the mechanical behavior without losing information about the realistic material properties. Accordingly, morphology and fabric measurements of 60 vertebral cancellous bone samples from three osteoporotic lumbar spines were performed on the basis of X-ray microtomography (μCT) images to determine anisotropic elastic parameters as a function of bone density in the area of pedicle screw anchorage. The fabric tensor was mapped in cubic bone volumes by a 3D mean-intercept-length method. Fabric measurements resulted in a high degree of anisotropy (DA = 0.554). For the Young’s and shear moduli as a function of bone volume fraction (BV/TV, bone volume/total volume), an individually fit function was determined and high correlations were found (97.3 ≤ R2 ≤ 99.1,p < 0.005). The results suggest that the mathematical formulation for the relationship between anisotropic elastic constants and BV/TV is applicable to current μCT data of cancellous bone in the osteoporotic lumbar spine. In combination with the obtained results and findings, the developed routine allows determination of elastic constants of osteoporotic lumbar spine. Based on this, the elastic constants determined using homogenization theory can enable efficient investigation of human bone using finite element analysis (FEA).

Yield Behavior of Bovine Cancellous Bone

1989 ◽  
Vol 111 (3) ◽  
pp. 256-260 ◽  
Author(s):  
C. H. Turner
Keyword(s):  
Bone Density ◽  
Yield Point ◽  
Cancellous Bone ◽  
Volume Fraction ◽  
Yield Criterion ◽  
Solid Volume Fraction ◽  
Significant Negative Correlation ◽  
Yield Strain ◽  
Yield Behavior ◽  
Structural Density

The compressive yield strain was measured for 61 specimens of bovine cancellous bone from three distal femora. There was no significant relationship (p = 0.08, R2 = 0.051) between yield strain and the degree of trabecular orientation. There was a significant positive correlation (p<0.00001, R2 = 0.319) between yield strain and structural (apparent) density and significant negative correlation (p<0.0025, R2 = 0.145) between yield strain and bone density. Yield strain correlated best with bone solid volume fraction Vv (εy = 0.592 + 1.446vv, R2 = 0.337). The quantity, yield strain, is highly dependent on specific definitions of the yield point and the point of zero strain. For this study the yield point was defined by a 0.0003 offset criterion, and the point of zero strain was defined as the point where the tangent at 15 percent of yield crosses zero. The results using these definitions were compared with results using yield strain values determined by other definitions of the yield point and zero strain. The correlations between yield strain and trabecular orientation, structural density and bone density changed very little for differing definitions of yield. The results suggest that yield strain in cancellous bone is isotropic or independent of textural anisotropy, so the yield behavior may be characterized by a maximum strain yield criterion. The results also suggest that the primary mode of yield in cancellous bone is buckling of the trabeculae.

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.

INVERSION OF ANISOTROPIC ELASTIC CONSTANTS AND MUD SPEED USING BOREHOLE SONIC MODES

2020 ◽  
Author(s):  
Ting Lei ◽  
◽  
Romain Prioul ◽  
Adam Donald ◽  
Edgar Ignacio Velez Arteaga ◽  
...  
Keyword(s):  
Elastic Constants ◽  
Anisotropic Elastic

Ciliary Flow of Casson Nanofluid with the Influence of MHD having Carbon Nanotubes

2020 ◽  
Vol 16 ◽  
Author(s):  
Adel Alblawi ◽  
Saba Keyani ◽  
S. Nadeem ◽  
Alibek Issakhov ◽  
Ibrahim M. Alarifi
Keyword(s):  
Carbon Nanotubes ◽  
Velocity Profile ◽  
Pressure Gradient ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Pressure Rise ◽  
Shear Stresses ◽  
Left Wall ◽  
Coupled Equations ◽  
The Right

Objective: In this paper, we consider a model that describes the ciliary beating in the form of metachronal waves along with the effects of Magnetohydrodynamic fluid over a curved channel with slip effects. This work aims at evaluating the effect of Magnetohydrodynamic (MHD) on the steady two dimensional (2-D) mixed convection flow induced in carbon nanotubes. The work is done for both the single wall nanotube and multiple wall nanotube. The right wall and the left wall possess a metachronal wave that is travelling along the outer boundary of the channel. Methods: The wavelength is considered as very large for cilia induced MHD flow. The governing linear coupled equations are simplified by considering the approximations of long wavelength and small Reynolds number. Exact solutions are obtained for temperature and velocity profile. The analytical expressions for the pressure gradient and wall shear stresses are obtained. Term for pressure rise is obtained by applying Numerical integration method. Results: Numerical results of velocity profile are mentioned in a table form, for various values of solid volume fraction, curvature, Hartmann number [M] and Casson fluid parameter [ζ]. Final section of this paper is devoted to discussing the graphical results of temperature, pressure gradient, pressure rise, shear stresses and stream functions. Conclusion: Velocity profile near the right wall of the channel decreases when we add nanoparticles into our base fluid, whereas an opposite behaviour is depicted near the left wall due to ciliated tips whereas the temperature is an increasing function of B and ߛ and decreasing function of ߶.

scholarly journals Quantification of shear viscosity and wall slip velocity of highly concentrated suspensions with non-Newtonian matrices in pressure driven flows

2021 ◽  
Author(s):  
Patrick Wilms ◽  
Jan Wieringa ◽  
Theo Blijdenstein ◽  
Kees van Malssen ◽  
Reinhard Kohlus
Keyword(s):  
Shear Stress ◽  
Liquid Phase ◽  
Shear Rate ◽  
Shear Viscosity ◽  
Slip Velocity ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Wall Slip ◽  
Flow Index ◽  
Concentrated Suspensions

AbstractThe rheological characterization of concentrated suspensions is complicated by the heterogeneous nature of their flow. In this contribution, the shear viscosity and wall slip velocity are quantified for highly concentrated suspensions (solid volume fractions of 0.55–0.60, D4,3 ~ 5 µm). The shear viscosity was determined using a high-pressure capillary rheometer equipped with a 3D-printed die that has a grooved surface of the internal flow channel. The wall slip velocity was then calculated from the difference between the apparent shear rates through a rough and smooth die, at identical wall shear stress. The influence of liquid phase rheology on the wall slip velocity was investigated by using different thickeners, resulting in different degrees of shear rate dependency, i.e. the flow indices varied between 0.20 and 1.00. The wall slip velocity scaled with the flow index of the liquid phase at a solid volume fraction of 0.60 and showed increasingly large deviations with decreasing solid volume fraction. It is hypothesized that these deviations are related to shear-induced migration of solids and macromolecules due to the large shear stress and shear rate gradients.

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