Invariance and Ibragimov approach with Lie algebra of a nonlinear coupled elastic wave system
The propagation of elastic waves in hyperelastic materials is described by a nonlinear system of partial differential equations (PDEs) governing the material’s motion. Hyperelastic materials are characterized by a strain–energy density function that correlates material deformation with stored elasti...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Elsevier
2024-03-01
|
| Series: | Partial Differential Equations in Applied Mathematics |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666818124000263 |
| _version_ | 1827316892933554176 |
|---|---|
| author | Muhammad Usman Akhtar Hussain F.D. Zaman |
| author_facet | Muhammad Usman Akhtar Hussain F.D. Zaman |
| author_sort | Muhammad Usman |
| collection | DOAJ |
| description | The propagation of elastic waves in hyperelastic materials is described by a nonlinear system of partial differential equations (PDEs) governing the material’s motion. Hyperelastic materials are characterized by a strain–energy density function that correlates material deformation with stored elastic energy. For simplicity, we focus on the one-dimensional case where the displacement field, denoted as ϕ(x,t), signifies material deformation along the x-direction at position x and time t. The governing equation for elastic wave propagation in hyperelastic materials is derived from the strain–energy density function and associated stress–strain relations. In this study, Lie symmetry analysis is used to examine a nonlinear system of such equations relevant to the propagation of elastic waves in hyperelastic materials. The resulting equations are solved by applying Lie’s invariance criterion, yielding a ten-dimensional Lie algebra. An optimal system is derived from this algebra, allowing for the identification of invariant solutions under certain conditions. Additionally, the multiplier approach and Ibragimov’s new conservation laws are utilized to obtain the conservation laws for this coupled system of elastic waves. The outcome presented here is innovative and suggests utilizing the Lie symmetry method for investigating hyperelastic materials. |
| first_indexed | 2024-03-08T02:00:29Z |
| format | Article |
| id | doaj.art-0e165c32d5ec4d1fb1d04e55e4944fca |
| institution | Directory Open Access Journal |
| issn | 2666-8181 |
| language | English |
| last_indexed | 2024-04-24T23:23:27Z |
| publishDate | 2024-03-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Partial Differential Equations in Applied Mathematics |
| spelling | doaj.art-0e165c32d5ec4d1fb1d04e55e4944fca2024-03-16T05:09:35ZengElsevierPartial Differential Equations in Applied Mathematics2666-81812024-03-019100640Invariance and Ibragimov approach with Lie algebra of a nonlinear coupled elastic wave systemMuhammad Usman0Akhtar Hussain1F.D. Zaman2College of Electrical and Mechanical Engineering (CEME), National University of Science and Technology (NUST), H-12 Islamabad 44000, PakistanAbdus Salam School of Mathematical Sciences, Government College University, 68-B New Muslim Town, 54600 Lahore, Pakistan; Corresponding author.Abdus Salam School of Mathematical Sciences, Government College University, 68-B New Muslim Town, 54600 Lahore, PakistanThe propagation of elastic waves in hyperelastic materials is described by a nonlinear system of partial differential equations (PDEs) governing the material’s motion. Hyperelastic materials are characterized by a strain–energy density function that correlates material deformation with stored elastic energy. For simplicity, we focus on the one-dimensional case where the displacement field, denoted as ϕ(x,t), signifies material deformation along the x-direction at position x and time t. The governing equation for elastic wave propagation in hyperelastic materials is derived from the strain–energy density function and associated stress–strain relations. In this study, Lie symmetry analysis is used to examine a nonlinear system of such equations relevant to the propagation of elastic waves in hyperelastic materials. The resulting equations are solved by applying Lie’s invariance criterion, yielding a ten-dimensional Lie algebra. An optimal system is derived from this algebra, allowing for the identification of invariant solutions under certain conditions. Additionally, the multiplier approach and Ibragimov’s new conservation laws are utilized to obtain the conservation laws for this coupled system of elastic waves. The outcome presented here is innovative and suggests utilizing the Lie symmetry method for investigating hyperelastic materials.http://www.sciencedirect.com/science/article/pii/S2666818124000263Invariance criterionOptimal systemLie algebraIbragimov’s theoremHyperelastic material |
| spellingShingle | Muhammad Usman Akhtar Hussain F.D. Zaman Invariance and Ibragimov approach with Lie algebra of a nonlinear coupled elastic wave system Partial Differential Equations in Applied Mathematics Invariance criterion Optimal system Lie algebra Ibragimov’s theorem Hyperelastic material |
| title | Invariance and Ibragimov approach with Lie algebra of a nonlinear coupled elastic wave system |
| title_full | Invariance and Ibragimov approach with Lie algebra of a nonlinear coupled elastic wave system |
| title_fullStr | Invariance and Ibragimov approach with Lie algebra of a nonlinear coupled elastic wave system |
| title_full_unstemmed | Invariance and Ibragimov approach with Lie algebra of a nonlinear coupled elastic wave system |
| title_short | Invariance and Ibragimov approach with Lie algebra of a nonlinear coupled elastic wave system |
| title_sort | invariance and ibragimov approach with lie algebra of a nonlinear coupled elastic wave system |
| topic | Invariance criterion Optimal system Lie algebra Ibragimov’s theorem Hyperelastic material |
| url | http://www.sciencedirect.com/science/article/pii/S2666818124000263 |
| work_keys_str_mv | AT muhammadusman invarianceandibragimovapproachwithliealgebraofanonlinearcoupledelasticwavesystem AT akhtarhussain invarianceandibragimovapproachwithliealgebraofanonlinearcoupledelasticwavesystem AT fdzaman invarianceandibragimovapproachwithliealgebraofanonlinearcoupledelasticwavesystem |