Introduction. 1. Nonlinear theories of elasticity of plates and shells 2. Nonlinear theories of doubly curved shells for conventional and advanced materials 3. Introduction to nonlinear dynamics 4. Vibrations of rectangular plates 5. Vibrations of empty and fluid-filled circular cylindrical 6. Reduced order models: proper orthogonal decomposition and nonlinear normal modes 7. Comparison of different shell theories for nonlinear vibrations and stability of circular cylindrical shells 8. Effect of boundary conditions on a large-amplitude vibrations of circular cylindrical shells 9. Vibrations of circular cylindrical panels with different boundary conditions 10. Nonlinear vibrations and stability of doubly-curved shallow-shells: isotropic and laminated materials 11. Meshless discretization of plates and shells of complex shapes by using the R-functions 12. Vibrations of circular plates and rotating disks 13. Nonlinear stability of circular cylindrical shells under static and dynamic axial loads 14. Nonlinear stability and vibrations of circular shells conveying flow 15. Nonlinear supersonic flutter of circular cylindrical shells with imperfections.

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Nonlinear Vibrations and Stability of Shells and Plates

Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction

This book presents the most recent advances on the mechanics of soft and composite shells and their nonlinear vibrations and stability, including advanced problems of modeling human vessels (aorta) with fluid-structure interaction. It guides the reader into nonlinear modelling of shell structures in applications where advanced composite and complex biological materials must be described with great accuracy. To achieve this goal, the book presents nonlinear shell theories, nonlinear vibrations, buckling, composite and functionally graded materials, hyperelasticity, viscoelasticity, nonlinear damping, rubber and soft biological materials. Advanced nonlinear shell theories, not available in any other book, are fully derived in a simple notation and are ready to be implemented in numerical codes. The work features a blend of the most advanced theory and experimental results, and is a valuable resource for researchers, professionals and graduate students, especially those interested in mechanics, aeronautics, civil structures, materials, bioengineering and solid matter at different scales.

Nonlinear Mechanics of Shells and Plates in Composite, Soft and Biological Materials

The magnetostrictive strain-applied field curve typical of giant magnetostrictive materials is nonlinear and it is affected notably by prestress and temperature. A constitutive model to describe these properties is suggested in this paper. The model considered here is built on the Gibbs free energy function G(σij,Mk,T). Thermodynamic relations are used to obtain the constitutive expressions. Results based on the model show that the nonlinear property of magnetostrictive strain, the magnetization, and Young’s modulus under different prestress and temperature are in excellent agreement with experimental data. Moreover, the proposed model is convenient to be used in engineering applications since the parameters referred to the model can all be easily determined by experiments.

A nonlinear constitutive model of magneto-thermo-mechanical coupling for giant magnetostrictive materials

This paper is concerned with a geometrically non-linear solid shell element to analyse piezoelectric structures. The finite element formulation is based on a variational principle of the Hu-Washizu type and includes six independent fields: displacements, electric potential, strains, electric field, mechanical stresses and dielectric displacements. The element has eight nodes with four nodal degrees of freedoms, three displacements and the electric potential. A bilinear distribution through the thickness of the independent electric field is assumed to fulfill the electric charge conservation law in bending dominated situations exactly. The presented finite shell element is able to model arbitrary curved shell structures and incorporates a 3D-material law. A geometrically non-linear theory allows large deformations and includes stability problems. Linear and non-linear numerical examples demonstrate the ability of the proposed model to analyse piezoelectric devices.

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H.S. Tzou

Papers

Active control of nonlinear piezoelectric circular shallow spherical shells