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Geometrically nonlinear Hessian eigenmode decomposition for local stability analysis of thin-walled structures

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Abstract

A nonlinear geometric formulation based on position and unconstrained vector is originally proposed to evaluate the local stability of thin-walled members. The Hessian is decomposed into linear and geometric equivalent installments. As the Hessian matrix exhibits a definite positive quadratic form, stability condition for large displacements requires that the Hessian matrix positivity be verified at each load small steps. This condition must be established by evaluating the smallest eigenvalue signal at the critical point imminence. The positional finite element method is formulated from a total Lagrangian reference. The mechanical system equilibrium is guaranteed by the total potential energy stationary principle. The constitutive law of Saint–Venant–Kirchhoff is obtained from the linear relationship between the second Piola–Kirchhoff stress tensor and Green–Lagrange deformation tensor. Some examples demonstrate the applicability of the methods.

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Acknowledgements

The authors acknowledge Federal University for Latin American Integration (UNILA) for the work environment and shared resources.

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Authors and Affiliations

  1. Federal University for Latin-American Integration, Tancredo Neves Av., 6731, POBox: 2102, Foz do Iguaçu, Paraná, 85867-970, Brazil

    Aref K. L. Kzam

  2. São Carlos School of Engineering, University of São Paulo, Trabalhador São Carlense Av., 400, São Carlos, São Paulo, 13566-960, Brazil

    Humberto B. Coda

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  1. Aref K. L. Kzam
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Correspondence to Aref K. L. Kzam.

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Kzam, A.K.L., Coda, H.B. Geometrically nonlinear Hessian eigenmode decomposition for local stability analysis of thin-walled structures. Engineering with Computers 36, 1627–1641 (2020). https://doi.org/10.1007/s00366-019-00785-2

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