Startseite Nonlinear Forced Vibration Analysis of FG Cylindrical Nanopanels Based on Mindlin’s Strain Gradient Theory and 3D Elasticity
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Nonlinear Forced Vibration Analysis of FG Cylindrical Nanopanels Based on Mindlin’s Strain Gradient Theory and 3D Elasticity

  • Y. Gholami , R. Ansari EMAIL logo , R. Gholami ORCID logo EMAIL logo und H. Rouhi
Veröffentlicht/Copyright: 3. März 2020
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International Journal of Nonlinear Sciences and Numerical Simulation
Aus der Zeitschrift International Journal of Nonlinear Sciences and Numerical Simulation Band 21 Heft 6

Abstract

A numerical approach is used herein to study the primary resonant dynamics of functionally graded (FG) cylindrical nanoscale panels taking the strain gradient effects into consideration. The basic relations of the paper are written based upon Mindlin’s strain gradient theory (SGT) and three-dimensional (3D) elasticity. Since the formulation is developed using Mindlin’s SGT, it is possible to reduce it to simpler size-dependent theories including modified forms of couple stress and strain gradient theories (MCST & MSGT). The governing equations is derived and directly discretized via the variational differential quadrature technique. Then, a numerical solution technique is employed to study the nonlinear resonance response of nanopanels with various edge conditions under a harmonic load. The impacts of length scale parameter, material and geometrical parameters on the frequency–response curves of nanopanels are investigated. In addition, comparisons are provided between the predictions of MSGT, MCST and the classical elasticity theory.

Keywords: FG cylindrical nanopanel; nonlinear forced vibration; strain gradient theory; three-dimensional elasticity; numerical solution
PACS: 65-XX; 65Pxx; 65Nxx; 74-XX; 74Hxx

Appendix: Discretization on Space Domain

By considering fx as a variable function on the domain x1,…,xN, the approximated value of its rth-order derivative at a specified point xi can be computed as follows:

(63)drfxdxr=∑j=1NDijrfxj

where the weighting coefficients of rth-order derivative Dijr can be obtained by a recursive process as the following relation:

Ez=Ec−EmVfz+Em,

Now, for a 3D variable function fx,θ,z on intervals x1,…,xN, θ1,…,θM and z1,…,zP, one can approximate its derivatives by means of GDQ method and Kronecker tensor product signifying by ⊗. For example, one can write

(65)∂4fx,θ,z∂x∂θ2∂z=Dz1⊗Dθ2⊗Dx1f‾l.

where f‾l is defined as the following column vector

(66)f¯l=[f(x1,θ1,zl),…,f(xN,θ1,zl),f(x1,θ2,zl),…,f(xN,θ2,zl),…,f(x1,θM,zl),…,f(xN,θM,zl)]T,l=[1, … ,P].

In which the number of grid points through the x-, θ –and z –axes are, respectively, denoted by N, M and P.

Furthermore, the grid point through the coordinate axes are considered as follows:

(67)xi=12(1−cosi−1N−1π),i=1,2,3, … ,Nθj      =12(1−cosj−1M−1π), j=1,2,3, … ,Mzk      =12(1−cosk−1P−1π), k=1,2,3, … ,P

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Received: 2018-10-30
Accepted: 2020-02-02
Published Online: 2020-03-03
Published in Print: 2020-10-25

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. Nonlinear Forced Vibration Analysis of FG Cylindrical Nanopanels Based on Mindlin’s Strain Gradient Theory and 3D Elasticity
  4. Numerical Solutions Based on a Collocation Method Combined with Euler Polynomials for Linear Fractional Differential Equations with Delay
  5. A Remanufacturing Duopoly Game Based on a Piecewise Nonlinear Map: Analysis and Investigations
  6. Development of a Momentum Transfer Coefficient in Particle Horizontal Channel Flow
  7. Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations
  8. On the Dynamics and Control of Fractional Chaotic Maps with Sine Terms
  9. A stable class of modified Newton-like methods for multiple roots and their dynamics
  10. Experimental and numerical studies on failure and energy absorption of composite thin-walled square tubes under quasi-static compression loading
  11. Existence of periodic solutions with minimal period for fourth-order discrete systems via variational methods
  12. Derivative free iterative methods with memory having higher R-order of convergence
https://doi.org/10.1515/ijnsns-2018-0333
Schlagwörter für diesen Artikel
FG cylindrical nanopanel; nonlinear forced vibration; strain gradient theory; three-dimensional elasticity; numerical solution

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. Nonlinear Forced Vibration Analysis of FG Cylindrical Nanopanels Based on Mindlin’s Strain Gradient Theory and 3D Elasticity
  4. Numerical Solutions Based on a Collocation Method Combined with Euler Polynomials for Linear Fractional Differential Equations with Delay
  5. A Remanufacturing Duopoly Game Based on a Piecewise Nonlinear Map: Analysis and Investigations
  6. Development of a Momentum Transfer Coefficient in Particle Horizontal Channel Flow
  7. Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations
  8. On the Dynamics and Control of Fractional Chaotic Maps with Sine Terms
  9. A stable class of modified Newton-like methods for multiple roots and their dynamics
  10. Experimental and numerical studies on failure and energy absorption of composite thin-walled square tubes under quasi-static compression loading
  11. Existence of periodic solutions with minimal period for fourth-order discrete systems via variational methods
  12. Derivative free iterative methods with memory having higher R-order of convergence
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