Extreme values of Young’s modulus of tetragonal crystals

2021 ◽  
Vol 154 ◽  
pp. 103724
Author(s):  
Valentin A. Gorodtsov ◽  
Valentin G. Tkachenko ◽  
Dmitry S. Lisovenko
Keyword(s):  
Young’S Modulus ◽  
Extreme Values ◽  
Young's Modulus ◽  
Tetragonal Crystals

scholarly journals The Extreme Values of Young’s Modulus and the Negative Poisson’s Ratios of Rhombic Crystals

Crystals ◽  
2021 ◽  
Vol 11 (8) ◽  
pp. 863
Author(s):  
Valentin A. Gorodtsov ◽  
Dmitry S. Lisovenko
Keyword(s):  
Young’S Modulus ◽  
Poisson’S Ratio ◽  
Extreme Values ◽  
Young's Modulus ◽  
Sufficient Conditions ◽  
Poisson's Ratio ◽  
Cubic Crystals ◽  
Special Cases ◽  
Function Of Two Variables ◽  
Necessary And Sufficient

The extreme values of Young’s modulus for rhombic (orthorhombic) crystals using the necessary and sufficient conditions for the extremum of the function of two variables are analyzed herein. Seven stationary expressions of Young’s modulus are obtained. For three stationary values of Young’s modulus, simple analytical dependences included in the sufficient conditions for the extremum of the function of two variables are revealed. The numerical values of the stationary and extreme values of Young’s modulus for all rhombic crystals with experimental data on elastic constants from the well-known Landolt-Börnstein reference book are calculated. For three stationary values of Young’s modulus of rhombic crystals, a classification scheme based on two dimensionless parameters is presented. Rhombic crystals ((CH3)3NCH2COO·(CH)2(COOH)2, I, SC(NH2)2, (CH3)3NCH2COO·H3BO3, Cu-14 wt%Al, 3.0wt%Ni, NH4B5O8·4H2O, NH4HC2O4·1/2H2O, C6N2O3H6 and CaSO4) having a large difference between maximum and minimum Young’s modulus values were revealed. The highest Young’s modulus among the rhombic crystals was found to be 478 GPa for a BeAl2O4 crystal. More rigid materials were revealed among tetragonal (PdPb2; maximum Young’s modulus, 684 GPa), hexagonal (graphite; maximum Young’s modulus, 1020 GPa) and cubic (diamond; maximum Young’s modulus, 1207 GPa) crystals. The analytical stationary values of Young’s modulus for tetragonal, hexagonal and cubic crystals are presented as special cases of stationary values for rhombic crystals. It was found that rhombic, tetragonal and cubic crystals that have large differences between their maximum and minimum values of Young’s modulus often have negative minimum values of Poisson’s ratio (auxetics). We use the abbreviated term auxetics instead of partial auxetics, since only the latter were found. No similar relationship between a negative Poisson’s ratio and a large difference between the maximum and minimum values of Young’s modulus was found for hexagonal crystals.

On the Extreme Values of Young’s Modulus, the Shear Modulus, and Poisson’s Ratio for Cubic Materials

1998 ◽  
Vol 65 (3) ◽  
pp. 786-787 ◽  
Author(s):  
M. Hayes ◽  
A. Shuvalov
Keyword(s):  
Shear Modulus ◽  
Young’S Modulus ◽  
Poisson’S Ratio ◽  
Extreme Values ◽  
Young's Modulus ◽  
Positive Definite ◽  
Poisson's Ratio ◽  
Stored Energy ◽  
Maximum Value ◽  
Cubic Materials

For homogeneous cubic elastic materials with positive definite stored energy it is shown that the maximum and minimum values of Young’s modulus E are related to the maximum and minimum values of the shear modulus G through the simple connection 1/Gmin−1/Gmax=3(1/Emin−1/Emax). It is deduced that the ratio of compliances −s12/s44 is the maximum value of Poisson’s ratio v in the cubic materials with a positive parameter χ=2s11−2s12−s44, and the minimum value of ν in the cubic materials with negative χ.

Young's modulus surface and Poisson's ratio curve for tetragonal crystals

2008 ◽  
Vol 17 (5) ◽  
pp. 1565-1573 ◽  
Author(s):  
Zhang Jian-Min ◽  
Zhang Yan ◽  
Xu Ke-Wei ◽  
Ji Vincent
Keyword(s):  
Young’S Modulus ◽  
Poisson’S Ratio ◽  
Young's Modulus ◽  
Poisson's Ratio ◽  
Ratio Curve ◽  
Tetragonal Crystals

Mechanical properties of FeAlSi powders prepared by mechanical alloying from different initial feedstock materials

2019 ◽  
Vol 107 (2) ◽  
pp. 207 ◽  
Author(s):  
Jaroslav Čech ◽  
Petr Haušild ◽  
Miroslav Karlík ◽  
Veronika Kadlecová ◽  
Jiří Čapek ◽  
...  
Keyword(s):  
Mechanical Properties ◽  
Mechanical Alloying ◽  
Young’S Modulus ◽  
Milling Time ◽  
Young's Modulus ◽  
Element Model ◽  
X Ray Diffraction ◽  
X Ray ◽  
Powder Particles ◽  
Complete Homogenization

FeAl20Si20 (wt.%) powders prepared by mechanical alloying from different initial feedstock materials (Fe, Al, Si, FeAl27) were investigated in this study. Scanning electron microscopy, X-ray diffraction and nanoindentation techniques were used to analyze microstructure, phase composition and mechanical properties (hardness and Young’s modulus). Finite element model was developed to account for the decrease in measured values of mechanical properties of powder particles with increasing penetration depth caused by surrounding soft resin used for embedding powder particles. Progressive homogenization of the powders’ microstructure and an increase of hardness and Young’s modulus with milling time were observed and the time for complete homogenization was estimated.

Degradation of Rocks, Through Cracking Caused by Differential Thermal Expansion, in Relation to Nuclear Waste Repositories

1981 ◽  
Vol 6 ◽  
Author(s):  
J.R. Mclaren ◽  
R.W. Davidge ◽  
I. Titchell ◽  
K. Sincock ◽  
A. Bromley
Keyword(s):  
Thermal Expansion ◽  
Grain Boundary ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Crack Density ◽  
Granitic Rocks ◽  
Multiphase Materials ◽  
Thermal Expansion Behaviour ◽  
Waste Repositories ◽  
Expansion Behaviour

ABSTRACTHeating to temperatures up to 500°C, gives a reduction in Young's modulus and increase in permeability of granitic rocks and it is likely that a major reason is grain boundary cracking. The cracking of grain boundary facets in polycrystalline multiphase materials showing anisotropic thermal expansion behaviour is controlled by several microstructural factors in addition to the intrinsic thermal and elastic properties. Of specific interest are the relative orientations of the two grains meeting at the facet, and the size of the facet; these factors thus introduce two statistical aspects to the problem and these are introduced to give quantitative data on crack density versus temperature. The theory is compared with experimental measurements of Young's modulus and permeability for various rocks as a function of temperature. There is good qualitative agreement, and the additional (mainly microstructural) data required for a quantitative comparison are defined.

Is it necessary to calculate Young’s modulus in AFM nanoindentation experiments regarding biological samples?

2020 ◽  
Vol 12 ◽  
Author(s):  
S.V. Kontomaris ◽  
A. Malamou ◽  
A. Stylianou
Keyword(s):  
Young’S Modulus ◽  
Biological Samples ◽  
Young's Modulus ◽  
Reference Sample ◽  
Nonlinear Behavior ◽  
Accurate Method ◽  
Accurate Solution ◽  
Hertz Model ◽  
Work Done

Background: The determination of the mechanical properties of biological samples using Atomic Force Microscopy (AFM) at the nanoscale is usually performed using basic models arising from the contact mechanics theory. In particular, the Hertz model is the most frequently used theoretical tool for data processing. However, the Hertz model requires several assumptions such as homogeneous and isotropic samples and indenters with perfectly spherical or conical shapes. As it is widely known, none of these requirements are 100 % fulfilled for the case of indentation experiments at the nanoscale. As a result, significant errors arise in the Young’s modulus calculation. At the same time, an analytical model that could account complexities of soft biomaterials, such as nonlinear behavior, anisotropy, and heterogeneity, may be far-reaching. In addition, this hypothetical model would be ‘too difficult’ to be applied in real clinical activities since it would require very heavy workload and highly specialized personnel. Objective: In this paper a simple solution is provided to the aforementioned dead-end. A new approach is introduced in order to provide a simple and accurate method for the mechanical characterization at the nanoscale. Method: The ratio of the work done by the indenter on the sample of interest to the work done by the indenter on a reference sample is introduced as a new physical quantity that does not require homogeneous, isotropic samples or perfect indenters. Results: The proposed approach, not only provides an accurate solution from a physical perspective but also a simpler solution which does not require activities such as the determination of the cantilever’s spring constant and the dimensions of the AFM tip. Conclusion: The proposed, by this opinion paper, solution aims to provide a significant opportunity to overcome the existing limitations provided by Hertzian mechanics and apply AFM techniques in real clinical activities.

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