On Boussinesq’s Problem for a Power-Law Graded Elastic Half-Space on Elliptical and General Contact Domains
The indentation of a power-law graded elastic half-space by a rigid counter body is considered in the framework of linear elasticity. Poisson’s ratio is assumed to be constant over the half-space. For indenters with an ellipsoidal power-law shape, an exact contact solution is derived, based on the g...
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| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Materials |
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| Online Access: | https://www.mdpi.com/1996-1944/16/12/4364 |
| _version_ | 1797593754687766528 |
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| author | Emanuel Willert |
| author_facet | Emanuel Willert |
| author_sort | Emanuel Willert |
| collection | DOAJ |
| description | The indentation of a power-law graded elastic half-space by a rigid counter body is considered in the framework of linear elasticity. Poisson’s ratio is assumed to be constant over the half-space. For indenters with an ellipsoidal power-law shape, an exact contact solution is derived, based on the generalizations of Galin’s theorem and Barber’s extremal principle for the inhomogeneous half-space. As a special case, the elliptical Hertzian contact is revisited. Generally, elastic grading with a positive grading exponent reduces the contact eccentricity. Fabrikant’s approximation for the pressure distribution under a flat punch of arbitrary planform is generalized for power-law graded elastic media and compared with rigorous numerical calculations based on the boundary element method (BEM). Very good agreement between the analytical asymptotic solution and the numerical simulation is obtained for the contact stiffness and the contact pressure distribution. A recently published approximate analytic solution for the indentation of a homogeneous half-space by a counter body, whose shape slightly deviates from axial symmetry but is otherwise arbitrary, is generalized for the power-law graded half-space. The approximate procedure for the elliptical Hertzian contact exhibits the same asymptotic behavior as the exact solution. The approximate analytic solution for the indentation by a pyramid with square planform is in very good agreement with a BEM-based numerical solution of the same problem. |
| first_indexed | 2024-03-11T02:13:00Z |
| format | Article |
| id | doaj.art-f4f4e4c1cbb646f882f6579001f4369b |
| institution | Directory Open Access Journal |
| issn | 1996-1944 |
| language | English |
| last_indexed | 2024-03-11T02:13:00Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Materials |
| spelling | doaj.art-f4f4e4c1cbb646f882f6579001f4369b2023-11-18T11:25:13ZengMDPI AGMaterials1996-19442023-06-011612436410.3390/ma16124364On Boussinesq’s Problem for a Power-Law Graded Elastic Half-Space on Elliptical and General Contact DomainsEmanuel Willert0Institute of Mechanics, Technische Universität Berlin, Sekr. C8-4, Straße des 17. Juni 135, 10623 Berlin, GermanyThe indentation of a power-law graded elastic half-space by a rigid counter body is considered in the framework of linear elasticity. Poisson’s ratio is assumed to be constant over the half-space. For indenters with an ellipsoidal power-law shape, an exact contact solution is derived, based on the generalizations of Galin’s theorem and Barber’s extremal principle for the inhomogeneous half-space. As a special case, the elliptical Hertzian contact is revisited. Generally, elastic grading with a positive grading exponent reduces the contact eccentricity. Fabrikant’s approximation for the pressure distribution under a flat punch of arbitrary planform is generalized for power-law graded elastic media and compared with rigorous numerical calculations based on the boundary element method (BEM). Very good agreement between the analytical asymptotic solution and the numerical simulation is obtained for the contact stiffness and the contact pressure distribution. A recently published approximate analytic solution for the indentation of a homogeneous half-space by a counter body, whose shape slightly deviates from axial symmetry but is otherwise arbitrary, is generalized for the power-law graded half-space. The approximate procedure for the elliptical Hertzian contact exhibits the same asymptotic behavior as the exact solution. The approximate analytic solution for the indentation by a pyramid with square planform is in very good agreement with a BEM-based numerical solution of the same problem.https://www.mdpi.com/1996-1944/16/12/4364functionally graded materialsnormal contactelliptical contactsalmost axisymmetric contactsFabrikant’s approximationpower-law indenters |
| spellingShingle | Emanuel Willert On Boussinesq’s Problem for a Power-Law Graded Elastic Half-Space on Elliptical and General Contact Domains Materials functionally graded materials normal contact elliptical contacts almost axisymmetric contacts Fabrikant’s approximation power-law indenters |
| title | On Boussinesq’s Problem for a Power-Law Graded Elastic Half-Space on Elliptical and General Contact Domains |
| title_full | On Boussinesq’s Problem for a Power-Law Graded Elastic Half-Space on Elliptical and General Contact Domains |
| title_fullStr | On Boussinesq’s Problem for a Power-Law Graded Elastic Half-Space on Elliptical and General Contact Domains |
| title_full_unstemmed | On Boussinesq’s Problem for a Power-Law Graded Elastic Half-Space on Elliptical and General Contact Domains |
| title_short | On Boussinesq’s Problem for a Power-Law Graded Elastic Half-Space on Elliptical and General Contact Domains |
| title_sort | on boussinesq s problem for a power law graded elastic half space on elliptical and general contact domains |
| topic | functionally graded materials normal contact elliptical contacts almost axisymmetric contacts Fabrikant’s approximation power-law indenters |
| url | https://www.mdpi.com/1996-1944/16/12/4364 |
| work_keys_str_mv | AT emanuelwillert onboussinesqsproblemforapowerlawgradedelastichalfspaceonellipticalandgeneralcontactdomains |