Matrix-Tensor Mathematics in Orthotropic Elasticity

2009 ◽  
pp. 39-39-20
Author(s):  
B. A. Jayne ◽  
S. K. Suddarth
Keyword(s):  
Orthotropic Elasticity

Estimate of the Elastic Property of Microtubule Based on a Multiscale Model

2011 ◽  
Vol 211-212 ◽  
pp. 545-549 ◽  
Author(s):  
Yu Zhou Sun ◽  
Jin Yan Wang ◽  
Bin Gao ◽  
Li Wu Chang
Keyword(s):  
Energy Density ◽  
Elastic Property ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Multiscale Model ◽  
Future Application ◽  
Geometrical Parameters ◽  
Orthotropic Elasticity ◽  
Representative Cell ◽  
Tubular Shape

This paper presents a multiscale method to estimate the elastic property of the 13_3 microtubule. A microtubule is viewed as being formed by rolling up its planar encounter into the tubular shape, and the transformation is written into a set of equations with three geometrical parameters. The representative cell is chosen as an unit of two pairs of tubulin momoners. With the appropriate longitudinal and lateral interactive potentials, the strain energy density is calculated by deviding the cell’s energy by its spacial volume. The elastic property of microtubule is determined by minimizing the strain energy density, and the elastic constants are calculated in the thereotical scheme of orthotropic elasticity due to the geometrical meanings of the induced parameters. The advantage of the proposed method and its future application are discussed.

scholarly journals The Effect of the Anisotropy of Single Crystal Silicon on the Frequency Split of Vibrating Ring Gyroscopes

Micromachines ◽  
2019 ◽  
Vol 10 (2) ◽  
pp. 126 ◽  
Author(s):  
Zhengcheng Qin ◽  
Yang Gao ◽  
Jia Jia ◽  
Xukai Ding ◽  
Libin Huang ◽  
...  
Keyword(s):  
Single Crystal ◽  
Elastic Properties ◽  
Single Crystal Silicon ◽  
Coordinate Systems ◽  
Glass Technology ◽  
Crystal Silicon ◽  
Orthotropic Elasticity ◽  
Frequency Split ◽  
Direction Cosines ◽  
Simulation Results

This paper analyzes the effect of the anisotropy of single crystal silicon on the frequency split of the vibrating ring gyroscope, operated in the n = 2 wineglass mode. Firstly, the elastic properties including elastic matrices and orthotropic elasticity values of (100) and (111) silicon wafers were calculated using the direction cosines of transformed coordinate systems. The (111) wafer was found to be in-plane isotropic. Then, the frequency splits of the n = 2 mode ring gyroscopes of two wafers were simulated using the calculated elastic properties. The simulation results show that the frequency split of the (100) ring gyroscope is far larger than that of the (111) ring gyroscope. Finally, experimental verifications were carried out on the micro-gyroscopes fabricated using deep dry silicon on glass technology. The experimental results are sufficiently in agreement with those of the simulation. Although the single crystal silicon is anisotropic, all the results show that compared with the (100) ring gyroscope, the frequency split of the ring gyroscope fabricated using the (111) wafer is less affected by the crystal direction, which demonstrates that the (111) wafer is more suitable for use in silicon ring gyroscopes as it is possible to get a lower frequency split.

On the Solution of Plane, Orthotropic Elasticity Problems by an Integral Method

1972 ◽  
Vol 39 (3) ◽  
pp. 801-808 ◽  
Author(s):  
R. Benjumea ◽  
D. L. Sikarskie
Keyword(s):  
Stress Field ◽  
Integral Equations ◽  
Influence Function ◽  
Elastic Solid ◽  
Isotropic Case ◽  
Fredholm Equations ◽  
The Real ◽  
Orthotropic Elasticity ◽  
Elasticity Problems ◽  
Quarter Space

The present paper is concerned with the application of integral equation techniques to problems in plane orthotropic elasticity. Two approaches for solving such problems are outlined, both of which are characterized by embedding the real body in a “fictitious” body for which the appropriate influence functions are known. Fictitious tractions are then introduced such that the boundary conditions on the real body are satisfied. This results in a coupled set of integral equations in the fictitious traction components. Once these are found the unknowns, i.e., stresses, etc., are found in a straightforward manner. The difficulty is in introducing the fictitious traction field such that the resulting integral equations are useful computationally, i.e., are Fredholm equations of the second rather than the first kind. A sufficient condition for this is that the fictitious traction field is applied to the boundary of the real body. The two approaches just mentioned differ in the choice of influence function used, in one case the influence function being singular in the field and the other singular on the boundary. A solution method already exists in the isotropic case using the boundary influence function [3]. An alternate formulation is presented using an internal influence function which is shown to have computational advantages in the anisotropic (orthotropic) case. To illustrate the methods, the stress field is found in a “truncated” orthotropic quarter space, under the condition of a given traction on the truncated surface, traction-free elsewhere. This problem is of interest in certain Rock Mechanics calculations, e.g., to a first approximation the stress field is that due to a rigid wedge penetrating a brittle, orthotropic elastic solid (prior to chip formation).

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