A mixed operator approach to peridynamics

Federico Cluni; Vittorio Gusella; Dimitri Mugnai; Edoardo Proietti Lippi; Patrizia Pucci; Federico Cluni; Vittorio Gusella; Dimitri Mugnai; Edoardo Proietti Lippi; Patrizia Pucci

doi:10.3934/mine.2023082

Mathematics in Engineering

2023, Volume 5, Issue 5: 1-22. doi: 10.3934/mine.2023082

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A mixed operator approach to peridynamics

1.
Dipartimento di Ingegneria Civile ed Ambientale, Università degli Studi di Perugia, Via G. Duranti 93, 06125 Perugia, Italy
2.
Dipartimento di Scienze Ecologiche e Biologiche, Università degli Studi della Tuscia, Largo dell'Università, 01100 Viterbo, Italy
3.
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy

Received: 16 March 2023 Revised: 03 April 2023 Accepted: 03 April 2023 Published: 11 April 2023

In the present paper we propose a model describing the nonlocal behavior of an elastic body using a peridynamical approach. Indeed, peridynamics is a suitable framework for problems where discontinuities appear naturally, such as fractures, dislocations, or, in general, multiscale materials. In particular, the regional fractional Laplacian is used as the nonlocal operator. Moreover, a combination of the fractional and classical Laplacian operators is used to obtain a better description of the phenomenological response in elasticity. We consider models with linear and nonlinear perturbations. In the linear case, we prove the existence and uniqueness of the solution, while in the nonlinear case the existence of at least two nontrivial solutions of opposite sign is proved. The linear and nonlinear problems are also solved by a numerical approach which estimates the regional fractional Laplacian by means of its singular integral representation. In both cases, a numerical estimation of the solutions is obtained, using in the nonlinear case an approach involving a random variation of an initial guess of the solution. Moreover, in the linear case a parametric analysis is made in order to study the effects of the parameters involved in the model, such as the order of the fractional Laplacian and the mixture law between local and nonlocal behavior.
- peridynamics,
- fractional Laplacian,
- mixed local and nonlocal operators,
- variational methods,
- numerical methods
Citation: Federico Cluni, Vittorio Gusella, Dimitri Mugnai, Edoardo Proietti Lippi, Patrizia Pucci. A mixed operator approach to peridynamics[J]. Mathematics in Engineering, 2023, 5(5): 1-22. doi: 10.3934/mine.2023082

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Abstract

In the present paper we propose a model describing the nonlocal behavior of an elastic body using a peridynamical approach. Indeed, peridynamics is a suitable framework for problems where discontinuities appear naturally, such as fractures, dislocations, or, in general, multiscale materials. In particular, the regional fractional Laplacian is used as the nonlocal operator. Moreover, a combination of the fractional and classical Laplacian operators is used to obtain a better description of the phenomenological response in elasticity. We consider models with linear and nonlinear perturbations. In the linear case, we prove the existence and uniqueness of the solution, while in the nonlinear case the existence of at least two nontrivial solutions of opposite sign is proved. The linear and nonlinear problems are also solved by a numerical approach which estimates the regional fractional Laplacian by means of its singular integral representation. In both cases, a numerical estimation of the solutions is obtained, using in the nonlinear case an approach involving a random variation of an initial guess of the solution. Moreover, in the linear case a parametric analysis is made in order to study the effects of the parameters involved in the model, such as the order of the fractional Laplacian and the mixture law between local and nonlocal behavior.

References

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