Free vibration analysis of orthotropic doubly-curved shallow shells based on the gradient elasticity

Abstract The comprehensive free vibration analysis of doubly-curved shallow shells which are made of an orthotropic material is presented. The size effect is taken into consideration using the gradient elasticity theory. Novozhilov’s linear shallow shell theory is used and it is assumed that the shell is simply supported. The governing equations of the doubly-curved shallow shells with consideration of the length scales are developed. Analytical solutions to the equations are proposed to obtain the frequencies of the shells. Numerical results are presented for various shell curvatures including spherical, cylindrical and hyperbolic paraboloidal shells and compared with those available in the literature.

[1]  S. Park,et al.  Analytical Solution for a Pressurized Thick-Walled Spherical Shell Based on a Simplified Strain Gradient Elasticity Theory , 2009 .

[2]  Shenjie Zhou,et al.  Static and dynamic analysis of micro beams based on strain gradient elasticity theory , 2009 .

[3]  Mohamad S. Qatu,et al.  Vibration of doubly curved shallow shells with arbitrary boundaries , 2012 .

[4]  E. Aifantis,et al.  Gradient elasticity and flexural wave dispersion in carbon nanotubes , 2009 .

[5]  Alavandi Bhimaraddi,et al.  Free vibration analysis of doubly curved shallow shells on rectangular planform using three-dimensional elasticity theory , 1991 .

[6]  Francesco Tornabene,et al.  Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations , 2011 .

[7]  A. Giannakopoulos,et al.  Variational analysis of gradient elastic flexural plates under static loading , 2010 .

[8]  S. K. Park,et al.  Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem , 2007 .

[9]  D. Beskos,et al.  Wave propagation in and free vibrations of gradient elastic circular cylindrical shells , 2012 .

[10]  Marco Amabili,et al.  A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells , 2010 .

[11]  R. D. Mindlin.  Second gradient of strain and surface-tension in linear elasticity , 1965 .

[12]  A. Eringen.  On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves , 1983 .

[13]  D. Beskos,et al.  Stability analysis of gradient elastic circular cylindrical thin shells , 2009 .

[14]  E. Aifantis,et al.  On Some Aspects in the Special Theory of Gradient Elasticity , 1997 .

[15]  Elias C. Aifantis,et al.  Exploring the applicability of gradient elasticity to certain micro/nano reliability problems , 2008 .

[16]  M. Baghani,et al.  Strain gradient elasticity solution for functionally graded micro-cylinders , 2013, 1305.3307.

[17]  D. E. Beskos,et al.  Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates , 2008 .

[18]  Kyoko Nagai,et al.  Modal interaction in chaotic vibrations of a shallow double-curved shell-panel , 2008 .

[19]  K. M. Liew,et al.  Vibration of doubly-curved shallow shells , 1996 .

[20]  R. D. Mindlin,et al.  On first strain-gradient theories in linear elasticity , 1968 .

[21]  Jae-Hung Han,et al.  Vibration characteristics and snapping behavior of hygro-thermo-elastic composite doubly curved shells , 2009 .

[22]  Harm Askes,et al.  The Representative Volume Size in Static and Dynamic Micro-Macro Transitions , 2005 .

[23]  K. A. Lazopoulos,et al.  Nonlinear strain gradient elastic thin shallow shells , 2011 .

[24]  K. Chandrashekhara,et al.  Free vibrations of anisotropic laminated doubly curved shells , 1989 .

[25]  Marco Amabili,et al.  Nonlinear Vibrations and Stability of Shells and Plates , 2008 .

[26]  H. Askes,et al.  Elasticity Theories with Higher-order Gradients of Inertia and Stiffness for the Modelling of Wave Dispersion in Laminates , 2007 .

[27]  E. Aifantis,et al.  Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results , 2011 .

[28]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[29]  Mohamad S. Qatu,et al.  Natural frequencies for cantilevered doubly-curved laminated composite shallow shells , 1991 .

[30]  E. Aifantis,et al.  On some applications of gradient elasticity to composite materials , 2001 .

[31]  K. Liew,et al.  Vibration of Thick Doubly-Curved Stress Free Shallow Shells of Curvilinear Planform , 1997 .