Rigorous derivation of active plate models for thin sheets of nematic elastomers. (October 2020)

• Record Type: Journal Article
• Authors: Agostiniani, Virginia; DeSimone, Antonio
• Is Part Of: Mathematics and mechanics of solids. Volume 25:Number 10(2020)
• Page Start: 1804
• Page End: 1830
• Publication Date: 2020-10
• Subjects: Nematic elastomers -- dimension reduction -- plate theory; Materials -- Mechanical properties -- Periodicals; Solids -- Periodicals; Materials science -- Mathematics -- Periodicals; 620.11205
• DOI: 10.1177/1081286517699991
• Languages: English
• ISSNs: 1081-2865

Abstract

In the context of finite elasticity, we propose plate models describing the spontaneous bending of nematic elastomer thin films due to variations along the thickness of the nematic order parameters. Reduced energy functionals are deduced from a three-dimensional description of the system using rigorous dimension reduction techniques, based on the theory of Γ-convergence. The two-dimensional models are non-linear plate theories, in which deviations from a characteristic target curvature tensor cost elastic energy. Moreover, the stored energy functional cannot be minimised to zero, thus revealing the presence of residual stresses, as observed in numerical simulations. Three nematic textures are considered: splay-bend and twisted orientations of the nematic director, and a uniform director perpendicular to the mid-plane of the film, with variable degree of nematic order along the thickness. These three textures realise three very different structural models: one with only one stable spontaneously bent configuration, a bistable model with two oppositely curved configurations of minimal energy, and a shell with zero stiffness to twisting.