Numerical solution of non-classical beam and plate theories using di erential quadrature method

Abstract

For effcient design of the nano/micro scale structural systems, detailed analysis and through understanding of size-dependent mechanical behaviour at nano/micro scale is very critical. Various approaches have been used to investigate the mechanical behaviour of small scale structures, for instance, experimental approach, atomistic and molecular dynamics simulations, multi-scale modelling, etc. However, the application of these methods for practical problems have their own limitations, some are very cumbersome and expensive, others need high computational resources and remaining are mathematically involved. The non-classical continuum theories with micro-structural behaviour have proven to be very efficient alternative, which assures reasonable accuracy with less complexity and computational efforts as compared to other approaches. The non-classical theories are governed by higher order differential equations and introduce additional degrees of freedom (related to curvature and triple derivative of displacements) and material parameters to account for scale effects. A considerable amount of analytical work on beams and plates is conducted based on these theories, however, numerical treatment is limited to only few speci fic applications. The primary objective of this research is to develop a comprehensive set of novel and effcient differential quadrature-based elements for non-classical Euler-Bernoulli beam and Kirchhoff plate theories. Both strong and weak form differential quadrature elements are developed, which are fundamentally different in their formulation. The strong form elements are formulated using the governing equation and stress resultant equations, and the weak form elements are based on the variational principles. Lagrange interpolations are used to formulate the strong form beam elements, while the weak form beam elements are constructed for both Lagrange and Hermite interpolations. The plate elements (strong and weak) are developed using two different combinations of interpolation functions in the orthogonal directions, in the first choice, Lagrange interpolations are assumed in both orthogonal directions and in the second case Lagrange interpolation are assumed in one direction and Hermite in the another. The capability of these elements is demonstrated through non-classical Mindlin's simpli ed fi rst and second strain Euler-Bernoulli beam / Kirchhoff plate theories, which are governed by sixth and eighth order differential equations, respectively. The accuracy and applicability of the beam elements is veri fied for bending, free-vibration, stability, dynamic/transient and wave propagation analysis, and the plate elements for bending, free-vibration and stability analysis. The strong form differential quadrature element developed for first strain gradient Euler- Bernoulli theory demonstrated excellent agreement with the exact solutions with less number of nodes for static, free vibration and buckling analysis of prismatic and non-prismatic beams for different combinations of boundary conditions, loading and length scale parameters. Similar performance was demonstrated by the weak form quadrature element which was formulated using Hermite interpolation functions. The Lagrange interpolation based weak form quadrature element exhibited inferior performance as compared to the above two elements, and needed more number of nodes to obtain the accurate results. Good performance was shown by both strong and weak form differential quadrature elements for dynamic and wave propagation analysis. With fewer number of elements and nodes the velocity response and the group speeds were predicted accurately using these elements. Based on the finding it was concluded that the beam elements produced accurate results with reasonable number of nodes and can be effciently applied for different analysis of non-classical Euler-Bernoulli prismatic and non-prismatic beams for any choice of loading, boundary conditions and length scale parameters. The performance of strong and weak form beam elements developed for second strain gradient Euler-Bernoulli beam theory was also validated for static, free vibration, stability, dynamic and wave propagation analysis. Similar performance was demonstrated by the weak and strong form beam elements developed for second strain gradient Euler-Bernoulli beam theory. The strong form elements developed for fi rst strain gradient Kirchhoff plate theory demonstrated excellent performance for static bending, free vibration and stability analysis. Deflections, frequencies and buckling loads obtained using the single element with fewer number of nodes compare well with the exact solutions for different loading, boundary conditions and length scale values. The results obtained using the weak form quadrature elements also compared well with available literature results, however, for the plates which include one or more clamped edges need more number of nodes to obtain converged solutions as compared to the strong form elements. This aspect of weak form quadrature elements needs further investigation. Similar set of strong and weak form DQ elements developed for second strain gradient Kirchhoff plate theory also exhibited similar performance for static bending, free vibration and stability analysis.