Signorini problem as a variational limit of obstacle problems in nonlinear elasticity

Francesco Maddalena; Danilo Percivale; Franco Tomarelli; Francesco Maddalena; Danilo Percivale; Franco Tomarelli

doi:10.3934/mine.2024012

Mathematics in Engineering

2024, Volume 6, Issue 2: 261-304. doi: 10.3934/mine.2024012

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Research article

Signorini problem as a variational limit of obstacle problems in nonlinear elasticity

1.
Politecnico di Bari, Dipartimento di Meccanica, Matematica, Management, via Re David 200, Bari 70125, Italy
2.
Università degli Studi di Genova, Dipartimento di Ingegneria Meccanica, Energetica, Gestionale e dei Trasporti (DIME), Piazzale Kennedy, Fiera del Mare, Padiglione D, Genova 16129, Italy
3.
Politecnico di Milano, Dipartimento di Matematica, Piazza Leonardo da Vinci 32, Milano 20133, Italy

Received: 02 February 2024 Revised: 06 March 2024 Accepted: 06 March 2024 Published: 14 March 2024

An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.
- calculus of variations,
- Signorini problem,
- linear elasticity,
- nonlinear elasticity,
- finite elasticity,
- Gamma-convergence,
- asymptotic analysis,
- unilateral constraint
Citation: Francesco Maddalena, Danilo Percivale, Franco Tomarelli. Signorini problem as a variational limit of obstacle problems in nonlinear elasticity[J]. Mathematics in Engineering, 2024, 6(2): 261-304. doi: 10.3934/mine.2024012

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Abstract

An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.

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