Strain gradient differential quadrature finite element for moderately thick micro‐plates

Abstract

Summary In this study, we integrate the advantages of differential quadrature method (DQM) and finite element method (FEM) to construct a C1‐type four‐node quadrilateral element with 48 degrees of freedom (DOF) for strain gradient Mindlin micro‐plates. This element is free of shape functions and shear locking. The C1‐continuity requirements of deflection and rotation functions are accomplished by a fourth‐order differential quadrature (DQ)‐based geometric mapping scheme, which facilitates the conversion of the displacement parameters at Gauss‐Lobatto quadrature (GLQ) points into those at element nodes. The appropriate application of DQ rule to non‐rectangular domains is proceeded by the natural‐to‐Cartesian geometric mapping technique. Using GLQ and DQ rules, we discretize the total potential energy functional of a generic micro‐plate element into a function of nodal displacement parameters. Then, we adopt the principle of minimum potential energy to determine element stiffness matrix, mass matrix, and load vector. The efficacy of the present element is validated through several examples associated with the static bending and free vibration problems of rectangular, annular sectorial, and elliptical micro‐plates. Finally, the developed element is applied to study the behavior of freely vibrating moderately thick micro‐plates with irregular shapes. It is shown that our element has better convergence and adaptability than that of Bogner‐Fox‐Schmit (BFS) one, and strain gradient effects can cause a significant increase in vibration frequencies and a certain change in vibration mode shapes.

Variational formulation and differential quadrature finite element for freely vibrating strain gradient Kirchhoff plates

“…From Figure 6, one can see that the relative errors of our element are significantly lower than those of the classical element except for the case of related to the first pair of meshes (i.e.,8 × 8and 15 × 15). The above observation reveals that the present element has better convergence than the classical Kirchhoff type, which is consistent with the conclusions drawn in [86, 88].…”

“…Ishaquddin and Gopalakrishnan [81, 82] presented a weak form QEM for the simplified strain gradient Euler‐Bernoulli beam and Kirchhoff plate models. More recently, Zhang and his co‐workers [83–89] proposed gradient elasticity‐related DQFEs for MSGT/MCST‐based Euler‐Bernoulli, Timoshenko and Reddy beam models, and Mindlin and Kirchhoff plate models and showed the efficacy of their developed methods through comparison with the standard FEM and DQM. However, C 2 ‐continuous DQFE for gradient elastic Kirchhoff plate models [40, 41, 71] has not been reported yet.…”

Multi‐patch variational differential quadrature method for shear‐deformable strain gradient plates

“…It is well known that in its original form (both strong and weak form implementations), the GDQ method has its limitations in the presence of problems with complex solution domains. In the case of irregular convex domains, a mapping technique was proposed to enhance the applications of the approach 14–18 . The first attempts for extending the GDQ method to irregular domains can be found in Reference 19.…”

“…In the case of irregular convex domains, a mapping technique was proposed to enhance the applications of the approach. [14][15][16][17][18] The first attempts for extending the GDQ method to irregular domains can be found in Reference 19. Besides, Tornabene and his colleagues integrated the accuracy of the GDQ method with the element-wise idea of the FE method to propose the generalized differential quadrature finite element method (GDQFEM) to improve the applicability of the GDQ technique in more complex problems.…”

Theoretical modeling on the combination resonance of size-dependent microbeams

“…The linear elastic behavior of microbeams is investigated at first (Kong et al, 2008; Lei et al, 2013, 2017, 2019a; Zhang et al, 2020a), combined with length scale parameter involved theories such as the modified couple stress theory (Park and Gao, 2006), strain gradient theory (Yang et al, 2002; Zhang et al, 2020b, 2020c, 2019), and nonlocal theory (Lei et al, 2019b). Li et al (Li et al, 2019c, 2018) studied the vibration of titanium, nickel, and copper microbeams; the resonant frequencies of the first to third mode were found to be size-dependent.…”