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 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.

Effective Elastic Constants of Fiber-Eutectics and Transformation Particles Composite Ceramic

2010 ◽  
Vol 177 ◽  
pp. 182-185 ◽  
Author(s):  
Bao Feng Li ◽  
Jian Zheng ◽  
Xin Hua Ni ◽  
Ying Chen Ma ◽  
Jing Zhang
Keyword(s):  
Elastic Modulus ◽  
Elastic Constant ◽  
Effective Stress ◽  
Elastic Constants ◽  
Transverse Isotropy ◽  
Analytical Formula ◽  
Composite Ceramic ◽  
Phase Model ◽  
Composite Ceramics ◽  
Effective Elastic Constants

The composite ceramics is composed of fiber-eutectics, transformation particles and matrix particles. First, the recessive expression between the effective stress in fiber-eutectic and the flexibility increment tensor is obtained according to the four-phase model. Second, the analytical formula which contains elastic constant of the fiber-eutectic is obtained applying Taylor’s formula. The eutectic is transverse isotropy, so there are five elastic constants. Third, the effective elastic constants of composite ceramics are predicted. The result shows that the elastic modulus of composite ceramic is reduced with the increase of fibers fraction and fibers diameter.

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).

scholarly journals Extremely small twist elastic constants in lyotropic nematic liquid crystals

2020 ◽  
Vol 117 (44) ◽  
pp. 27238-27244 ◽  
Author(s):  
Clarissa F. Dietrich ◽  
Peter J. Collings ◽  
Thomas Sottmann ◽  
Per Rudquist ◽  
Frank Giesselmann
Keyword(s):  
Liquid Crystals ◽  
Elastic Constant ◽  
Elastic Constants ◽  
Nematic Phase ◽  
Elastic Moduli ◽  
Building Blocks ◽  
Lyotropic Liquid Crystals ◽  
Order Of Magnitude ◽  
Typical Range ◽  
Special Case

Recent measurements of the elastic constants in lyotropic chromonic liquid crystals (LCLCs) have revealed an anomalously small twist elastic constant compared to the splay and bend constants. Interestingly, measurements of the elastic constants in the micellar lyotropic liquid crystals (LLCs) that are formed by surfactants, by far the most ubiquitous and studied class of LLCs, are extremely rare and report only the ratios of elastic constants and do not include the twist elastic constant. By means of light scattering, this study presents absolute values of the elastic constants and their corresponding viscosities for the nematic phase of a standard LLC composed of disk-shaped micelles. Very different elastic moduli are found. While the splay elastic constant is in the typical range of 1.5 pN as is true in general for thermotropic nematics, the twist elastic constant is found to be one order of magnitude smaller (0.30 pN) and almost two orders of magnitude smaller than the bend elastic constant (21 pN). These results demonstrate that a small twist elastic constant is not restricted to the special case of LCLCs, but is true for LLCs in general. The reason for this extremely small twist elastic constant very likely originates with the flexibility of the assemblies that are the building blocks of both micellar and chromonic lyotropic liquid crystals.

Diffuse scattering and phason modes in the i-AlPdMn quasicrystalline phase

2005 ◽  
Vol 220 (12) ◽  
Author(s):  
Marc de Boissieu ◽  
Sonia Francoual
Keyword(s):  
Elastic Constant ◽  
Elastic Constants ◽  
Intensity Distribution ◽  
Diffuse Scattering ◽  
Hydrodynamic Theory ◽  
Quasicrystalline Phase ◽  
X Rays ◽  
X Ray ◽  
Long Wavelength ◽  
Diffusive Modes

AbstractWe review results obtained in the study of the diffuse scattering in the i-AlPdMn quasicrystal. Most of the diffuse scattering is the result of long wavelength phason modes. The shape and intensity distribution of the diffuse scattering is well reproduced using the generalised elasticity theory and two phason elastic constants. The temperature dependence of the diffuse scattering indicates a softening of the phason elastic constant as the temperature is lowered. Using coherent X-rays and photo-correlation X-ray spectroscopy, it is shown that phason modes are collective diffusive modes, in agreement with the hydrodynamic theory of long wavelength fluctuations in quasicrystals.

Elastic constants. II

1977 ◽  
Vol 357 (1690) ◽  
pp. 297-307
Keyword(s):  
Energy Density ◽  
Elastic Constant ◽  
Initial Stress ◽  
Elastic Constants ◽  
Interatomic Potential ◽  
Uniform Strain ◽  
Partial Derivatives ◽  
Constant Tensor ◽  
Derivatives Of

In the previous paper of this series we derived expressions for the initial stress and the elastic constant tensor for a crystal in terms of the partial derivatives of the energy density with uniform strain or sublattice displacement. In this paper we shall develop these equations further by considering the most general form of interatomic potential energies.

Determination of X-ray Elastic Constants in a Ti-14Al-21Nb Nb Alloy and a Ti-14Al-21Nb/SiC Metal Matrix Composite

1990 ◽  
Vol 34 ◽  
pp. 689-698 ◽  
Author(s):  
J. Jo ◽  
R. W. Hendricks ◽  
W. D. Brewer ◽  
Karen M. Brown
Keyword(s):  
Residual Stress ◽  
Plastic Deformation ◽  
Elastic Constant ◽  
Theoretical Calculation ◽  
Metal Matrix Composite ◽  
Elastic Constants ◽  
Metal Matrix ◽  
Intrinsic Value ◽  
X Ray

Residual stress values in a material are governed by the measurements of the atomic spacings in a specific crystallographic plane and the elastic constant for that plane. It has been reported that the value of the elastic constant depends on microstructure, preferred orientation, plastic deformation and morphology [1], Thus, the theoretical calculation of the elastic constant may deviate from the intrinsic value for a real alloy.

Elastic anomalies of Hg2X2 compounds

1992 ◽  
Vol 70 (9) ◽  
pp. 745-751
Author(s):  
K. S. Viswanathan ◽  
J. C. Jeeja Ramani
Keyword(s):  
Elastic Constants ◽  
Fourth Order ◽  
Order Parameter ◽  
Zero Point ◽  
Stability Conditions ◽  
Point Energy ◽  
The Third ◽  
Limit Formula ◽  
The Stability ◽  
Elastic Anomalies

The anomalies of the second-, third-, and fourth-order elastic constants are considered for the phase transition of Hg2X2 type of compounds. Expressions are obtained for the equilibrium values of the order parameters in the ferroelastic phase from the stability conditions. The fluctuation in the order parameter is evaluated from the Landau–Khalatnikov equation. An expression is derived for the shift in the zero-point energy in the low-temperature ferroelastic phase and the specific heat anomaly. It is shown that these are proportional to (T − T)2 and (T − Tc), respectively. All the anomalies of the second-order elastic (SOE) constants are obtained from a single general formula, and relations among them are established. The temperature variation of the SOE constants in the limit [Formula: see text] is discussed. Similarly, expressions are derived for the anomalies of the third- and fourth-order elastic constants. In the limit [Formula: see text] it is shown that these constants diverge as [Formula: see text] and [Formula: see text], respectively.

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