A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity Constraints
A generalized strain energy-based homogenization method for 2-D and 3-D cellular materials with and without periodicity constraints is proposed using Hill’s Lemma and the matrix method for spatial frames. In this new approach, the equilibrium equations are enforced at all boundary and interior nodes...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/10/1870 |
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| author | Ahmad I. Gad Xin-Lin Gao |
| author_facet | Ahmad I. Gad Xin-Lin Gao |
| author_sort | Ahmad I. Gad |
| collection | DOAJ |
| description | A generalized strain energy-based homogenization method for 2-D and 3-D cellular materials with and without periodicity constraints is proposed using Hill’s Lemma and the matrix method for spatial frames. In this new approach, the equilibrium equations are enforced at all boundary and interior nodes and each interior node is allowed to translate and rotate freely, which differ from existing methods where the equilibrium conditions are imposed only at the boundary nodes. The newly formulated homogenization method can be applied to cellular materials with or without symmetry. To illustrate the new method, four examples are studied: two for a 2-D cellular material and two for a 3-D pentamode metamaterial, with and without periodic constraints in each group. For the 2-D cellular material, an asymmetric microstructure with or without periodicity constraints is analyzed, and closed-form expressions of the effective stiffness components are obtained in both cases. For the 3-D pentamode metamaterial, a primitive diamond-shaped unit cell with or without periodicity constraints is considered. In each of these 3-D cases, two different representative cells in two orientations are examined. The homogenization analysis reveals that the pentamode metamaterial exhibits the cubic symmetry based on one representative cell, with the effective Poisson’s ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>v</mi><mo>¯</mo></mover></semantics></math></inline-formula> being nearly 0.5. Moreover, it is revealed that the pentamode metamaterial with the cubic symmetry can be tailored to be a rubber-like material (with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover><mo> </mo><mo>≅</mo><mn>0.5</mn></mrow></semantics></math></inline-formula>) or an auxetic material (with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>v</mi><mo>¯</mo></mover></semantics></math></inline-formula> < 0). |
| first_indexed | 2024-03-10T06:10:51Z |
| format | Article |
| id | doaj.art-d6fc31133edb44f69c9b6c35e5ec62ef |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T06:10:51Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-d6fc31133edb44f69c9b6c35e5ec62ef2023-11-22T20:10:21ZengMDPI AGSymmetry2073-89942021-10-011310187010.3390/sym13101870A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity ConstraintsAhmad I. Gad0Xin-Lin Gao1Department of Mechanical Engineering, Southern Methodist University, Dallas, TX 75275-0337, USADepartment of Mechanical Engineering, Southern Methodist University, Dallas, TX 75275-0337, USAA generalized strain energy-based homogenization method for 2-D and 3-D cellular materials with and without periodicity constraints is proposed using Hill’s Lemma and the matrix method for spatial frames. In this new approach, the equilibrium equations are enforced at all boundary and interior nodes and each interior node is allowed to translate and rotate freely, which differ from existing methods where the equilibrium conditions are imposed only at the boundary nodes. The newly formulated homogenization method can be applied to cellular materials with or without symmetry. To illustrate the new method, four examples are studied: two for a 2-D cellular material and two for a 3-D pentamode metamaterial, with and without periodic constraints in each group. For the 2-D cellular material, an asymmetric microstructure with or without periodicity constraints is analyzed, and closed-form expressions of the effective stiffness components are obtained in both cases. For the 3-D pentamode metamaterial, a primitive diamond-shaped unit cell with or without periodicity constraints is considered. In each of these 3-D cases, two different representative cells in two orientations are examined. The homogenization analysis reveals that the pentamode metamaterial exhibits the cubic symmetry based on one representative cell, with the effective Poisson’s ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>v</mi><mo>¯</mo></mover></semantics></math></inline-formula> being nearly 0.5. Moreover, it is revealed that the pentamode metamaterial with the cubic symmetry can be tailored to be a rubber-like material (with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover><mo> </mo><mo>≅</mo><mn>0.5</mn></mrow></semantics></math></inline-formula>) or an auxetic material (with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>v</mi><mo>¯</mo></mover></semantics></math></inline-formula> < 0).https://www.mdpi.com/2073-8994/13/10/1870cellular materialstrain energy-based homogenizationHill’s lemmapentamode metamaterialmatrix method for spatial frameseffective elastic properties |
| spellingShingle | Ahmad I. Gad Xin-Lin Gao A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity Constraints Symmetry cellular material strain energy-based homogenization Hill’s lemma pentamode metamaterial matrix method for spatial frames effective elastic properties |
| title | A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity Constraints |
| title_full | A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity Constraints |
| title_fullStr | A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity Constraints |
| title_full_unstemmed | A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity Constraints |
| title_short | A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity Constraints |
| title_sort | generalized strain energy based homogenization method for 2 d and 3 d cellular materials with and without periodicity constraints |
| topic | cellular material strain energy-based homogenization Hill’s lemma pentamode metamaterial matrix method for spatial frames effective elastic properties |
| url | https://www.mdpi.com/2073-8994/13/10/1870 |
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