Mathematical justification of the obstacle problem for a piezoelectric shallow shell

Article type: Research Article

Authors: Yan, Guan^{; *} | Miara, Bernadette

Affiliations: Mathematics & Science College, Shanghai Normal University, Shanghai 200234, China. E-mails: gypyly@aliyun.com, bernadette.miara@gmail.com

Correspondence: [*] Corresponding author. E-mail: gypyly@aliyun.com.

Abstract: In this paper we properly justify the modeling of a thin piezoelectric shallow shell in unilateral contact with a rigid plane. Starting from the three-dimensional nonlinear Signorini problem, we establish the convergence of the displacement field and of the electric potential as the thickness of the shell goes to zero. More precisely we obtain that the transverse mechanical displacement field coupled with the in-plane components solve an obstacle problem described new piezoelectric characteristics. We also investigate the very popular case of cubic crystals and show that, for two-dimensional shallow shells, the coupling piezoelectric effect disappears.

Keywords: Signorini problem, obstacle problem, asymptotic analysis, shallow shell

DOI: 10.3233/ASY-171411

Journal: Asymptotic Analysis, vol. 102, no. 1-2, pp. 71-97, 2017