Assessment of second Piola–Kirchhoff and Cauchy stress tensors in finite rotation sandwich and laminated shells under non-conservative pressure loads

Nonlinear analysis of laminated composite shells with piezoelectric patches under displacement‐dependent pressure loads

“…The base vectors of the deformed SaS of the nth layer are represented in the orthonormal basis 𝐞 𝑖 [32] )𝑢…”

“…n− and n+ are the unit normal vectors to the deformed outer surfaces; p− and p+ are the loads per unit area of the deformed surface acting in the normal direction to the deformed outer surfaces, which allows us to correctly describe the Cauchy stress tensor as discussed in [32]; Q− and Q+ are the charge densities per unit area on the outer surfaces; 𝑐 − 𝛼 and 𝑐 + 𝛼 are the components of the shifter tensor on the outer surfaces. In the mixed variational Equations ( 22)-( 24), matrix notations are introduced ] T ,…”

“…2𝑟 𝑢 ] T are the displacement vectors of interfaces and inner SaS of the nth layer at element nodes, and using Equations ( 13), (32), and (37), we obtain…”

“…Foundations of the nonlinear theory of shells subjected to displacementdependent loads can be found in [12][13][14][15]. Most studies use isoparametric shell and brick elements [16][17][18][19][20][21][22][23][24][25][26][27], whereas works [28][29][30][31][32] deal with GeX solid-shell elements on the basis of higher-order single-layer and layer-wise theories. In this paper, we extend the results [9,32] to laminated composite shells with piezoelectric patches subjected to displacement-dependent pressure loads.…”

“…Most studies use isoparametric shell and brick elements [16][17][18][19][20][21][22][23][24][25][26][27], whereas works [28][29][30][31][32] deal with GeX solid-shell elements on the basis of higher-order single-layer and layer-wise theories. In this paper, we extend the results [9,32] to laminated composite shells with piezoelectric patches subjected to displacement-dependent pressure loads. To address this issue, the GeX four-node solid-shell element based on the layer-wise SaS-based shell theory has been developed.…”