The Extreme Values of Young’s Modulus and the Negative Poisson’s Ratios of Rhombic Crystals
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,...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-07-01
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| Series: | Crystals |
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| Online Access: | https://www.mdpi.com/2073-4352/11/8/863 |
| _version_ | 1797524276147912704 |
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| author | Valentin A. Gorodtsov Dmitry S. Lisovenko |
| author_facet | Valentin A. Gorodtsov Dmitry S. Lisovenko |
| author_sort | Valentin A. Gorodtsov |
| collection | DOAJ |
| description | 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 ((CH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>NCH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>COO·(CH)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>(COOH)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>, I, SC(NH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>, (CH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>NCH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>COO·H<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>BO<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>, Cu-14 wt%Al, 3.0wt%Ni, NH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>4</mn></msub></semantics></math></inline-formula>B<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>5</mn></msub></semantics></math></inline-formula>O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>8</mn></msub><mo>·</mo></mrow></semantics></math></inline-formula>4H<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>O, NH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>4</mn></msub></semantics></math></inline-formula>HC<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>4</mn></msub><mo>·</mo></mrow></semantics></math></inline-formula>1/2H<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>O, C<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>6</mn></msub></semantics></math></inline-formula>N<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>H<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>6</mn></msub></semantics></math></inline-formula> and CaSO<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>4</mn></msub></semantics></math></inline-formula>) 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 BeAl<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>4</mn></msub></semantics></math></inline-formula> crystal. More rigid materials were revealed among tetragonal (PdPb<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>; 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. |
| first_indexed | 2024-03-10T08:55:02Z |
| format | Article |
| id | doaj.art-ecccea08dc104b5dbef2b7cc19cbc21e |
| institution | Directory Open Access Journal |
| issn | 2073-4352 |
| language | English |
| last_indexed | 2024-03-10T08:55:02Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
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| series | Crystals |
| spelling | doaj.art-ecccea08dc104b5dbef2b7cc19cbc21e2023-11-22T07:16:02ZengMDPI AGCrystals2073-43522021-07-0111886310.3390/cryst11080863The Extreme Values of Young’s Modulus and the Negative Poisson’s Ratios of Rhombic CrystalsValentin A. Gorodtsov0Dmitry S. Lisovenko1Ishlinsky Institute for Problems in Mechanics RAS, Prosp. Vernadskogo 101-1, 119526 Moscow, RussiaIshlinsky Institute for Problems in Mechanics RAS, Prosp. Vernadskogo 101-1, 119526 Moscow, RussiaThe 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 ((CH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>NCH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>COO·(CH)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>(COOH)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>, I, SC(NH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>, (CH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>)<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>NCH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>COO·H<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>BO<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>, Cu-14 wt%Al, 3.0wt%Ni, NH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>4</mn></msub></semantics></math></inline-formula>B<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>5</mn></msub></semantics></math></inline-formula>O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>8</mn></msub><mo>·</mo></mrow></semantics></math></inline-formula>4H<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>O, NH<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>4</mn></msub></semantics></math></inline-formula>HC<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>4</mn></msub><mo>·</mo></mrow></semantics></math></inline-formula>1/2H<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>O, C<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>6</mn></msub></semantics></math></inline-formula>N<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>3</mn></msub></semantics></math></inline-formula>H<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>6</mn></msub></semantics></math></inline-formula> and CaSO<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>4</mn></msub></semantics></math></inline-formula>) 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 BeAl<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>4</mn></msub></semantics></math></inline-formula> crystal. More rigid materials were revealed among tetragonal (PdPb<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula>; 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.https://www.mdpi.com/2073-4352/11/8/863rhombic crystalsYoung’s moduluselasticitycrystalsauxetics |
| spellingShingle | Valentin A. Gorodtsov Dmitry S. Lisovenko The Extreme Values of Young’s Modulus and the Negative Poisson’s Ratios of Rhombic Crystals Crystals rhombic crystals Young’s modulus elasticity crystals auxetics |
| title | The Extreme Values of Young’s Modulus and the Negative Poisson’s Ratios of Rhombic Crystals |
| title_full | The Extreme Values of Young’s Modulus and the Negative Poisson’s Ratios of Rhombic Crystals |
| title_fullStr | The Extreme Values of Young’s Modulus and the Negative Poisson’s Ratios of Rhombic Crystals |
| title_full_unstemmed | The Extreme Values of Young’s Modulus and the Negative Poisson’s Ratios of Rhombic Crystals |
| title_short | The Extreme Values of Young’s Modulus and the Negative Poisson’s Ratios of Rhombic Crystals |
| title_sort | extreme values of young s modulus and the negative poisson s ratios of rhombic crystals |
| topic | rhombic crystals Young’s modulus elasticity crystals auxetics |
| url | https://www.mdpi.com/2073-4352/11/8/863 |
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