Part I – Theory of Mixtures

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

http://link.springer.com/content/pdf/10.1007/978-3-642-69569-8_2.pdf

Theory of Elastic Stability

On certain inconsistencies in berger equations for large deflections of plastic plates

Free and forced response of beams and plates undergoing moderately large amplitude steady state oscillations is determined using the averaging method of Ritz. The supports of these structural elements are assumed immovable during bending and the resulting membrane stress is the source of non-linearity in these problems. The application of Ritz's method to solve the governing non-linear partial differential equations yields non-linear algebraic equations. These algebraic equations are solved by forming the Bigradient Matrix for the polynomials. The response relations are represented graphically. Comparison of these results with the existing results for free oscillations shows remarkable agreement.

Non-linear vibrations of beams and plates

Nonlinear Transverse Vibrations of a Spinning Disk

Es werden mittels einer Verallgemeinerung der von-Karmanschen Gleichungen auf den dynamischen und orthotropen Fall die nichtlinearen Schwingungen elastisch orthotroper Kreisplatten untersucht. Die Bewegungsgleichung wird mittels des Galerkin-Verfahrens von den Ortsvariablen befreit. Als Beispiel wird Einspannung des Randes unter Erhaltung der radialen Verschieblichkeit angenommen, und es wird unter diesen Randbedingungen der Einfluss der Anisotropie und der relativen Amplitude auf die Eigenschwingungen sowie den Spannungszustand studiert. Als Sonderfall wird der nichtlineare Gleichgewichtszustand diskutiert.

Nonlinear vibrations of elastic circular plates exhibiting rectilinear orthotropy

Propagation of longitudinal waves in isotropic homogeneous elastic plates is studied in the context of the linear theory of nonlocal continuum mechanics. To determine the nonlocal moduli, the dispersion equation obtained for the plane longitudinal waves in an infinite medium is matched with the parallel equation derived in the theory of atomic lattice dynamics. Using the integroalgebraic representation of the stress tensor and the Fourier transform, the system of two coupled differential field equations is solved in the standard manner giving the frequency equations for the symmetric and antisymmetric wave modes. It is found that the short wave speed in the Poisson medium differs by about 13 percent from the speed established in the classical theory. A numerical example is given.

On the Nonlocal Theory of Wave Propagation in Elastic Plates

Es werden unter Benutzung der Naherung von Berger vereinfachte Differentialgleichungen fur grosse Deformationen elastischer Schalen hergeleitet und auf kreiszylindrische Schalen spezialisiert Der rechtwinklige Streifen mit beliebigem Seitenverhaltnis und der unendlich lange Streifen werden, beide mit Randauflage, im Detail behandelt Die numerischen Ergebnisse fur den quadratischen Ausschnitt werden mit anderen be kannten Losungen verglichen

Certain approximate analyses of large deflections of cylindrical shells

TWO-PHASE POROELASTIC MATERIAL IS TAKEN AS A MODEL OF THE LIVING BONE IN THE SENSE THAT THE OSSEOUS TISSUE IS TREATED AS A LINEAR, ISOTROPIC, PERFECTLY ELASTIC SOLID, AND THE FLUID SUBSTANCES FILLING THE PORES AS A PERFECT FLUID. USING BIOT'S EQUATIONS, DERIVED IN HIS CONSOLIDATION THEORY, FOUR COUPLED GOVERNING DIFFERENTIAL EQUATIONS FOR THE PROPAGATION OF HARMONIC LONGITUDINAL WAVES IN CIRCULARLY CYLINDRICAL BARS OF POROELASTIC MATERIAL ARE DERIVED. A LONGER MANIPULATION REDUCES THE TASK OF SOLUTION TO A SINGLED ORDINARY DIFFERENTIAL EQUATION WITH VARIABLE COEFFICIENTS AND A REGULAR SINGULAR POINT. THE EQUATION IS SOLVED BY FROBENIUS' METHOD. THREE BOUNDARY CONDITIONS ON THE CURVED SURFACE OF THE BAR, EXPRESSING THE ABSENCE OF EXTERNAL LOADING AND THE PERMEABILITY OF THE SURFACE, SUPPLY A SYSTEM OF THREE LINEAR EQUATIONS IN THREE UNKNOWN COEFFICIENTS. A NONTRIVIAL SOLUTION OF THE SYSTEM GIVES TWO PHASE VELOCITIES OF PROPAGATION OF LONGITUDINAL WAVES IN AGREEMENT WITH THE FINDING OF BIOT FOR AN INFINITE MEDIUM. A SIMPLIFICATION OF THE PURELY ELASTIC CASE YIELDS THE ELEMENTARY CLASSICAL RESULT FOR THE LONGITUDINAL WAVES. /AUTHOR/

J. L. Nowinski

Papers

Nonlinear Transverse Vibrations of a Spinning Disk

Nonlinear vibrations of elastic circular plates exhibiting rectilinear orthotropy

On the Nonlocal Theory of Wave Propagation in Elastic Plates

Certain approximate analyses of large deflections of cylindrical shells

Propagation of Longitudinal Waves in Circularly Cylindrical Bone Elements