Modified Mindlin plate theory and shear locking-free finite element formulation

A Simple High-Order Shear Deformation Triangular Plate Element with Incompatible Polynomial Approximation

“…As described in Senjanović et al [4], the shear-locking problem could be well and naturally solved because the bending angles of rotation and shear angles are treated as independent variables in HSDTs. The regular full integration can be applied to make HSDTPEs valid for the thick-thin plates for the computation of Equation (42), for instance, seven quadrature points for each triangular element [57], four Gauss quadrature points for transverse direction z (where the analytical integration can also be applied for the direction z), and four Gauss quadrature points for the local direction s of each fictitious layer are used for the integration of the TrSDTPE using the third-order shear function g(z) [58][59][60].…”

“…The plate elements derived from the FSDT only require C 0 continuity in approximation fields, have the advantages of physical clarity and simplicity of application [3], and hence were widely accepted and used to model thick-thin plates by scientists and engineers. Unfortunately, the FSDT elements suffer from the shear-locking problem when the thickness to length ration of the plate becomes very small, due to inadequate dependence among transverse deflection and rotations using an ordinary low-order finite element [4]. Quite a large number of techniques have been developed to overcome this problem, such as the assumed shear strain approach, the discrete Kirchhoff/Mindlin representation, the mixed/hybrid formulation, and the reduced/selected integration [5][6][7][8][9][10][11][12][13][14][15].…”

“…In recent years, High-order Shear Deformation Theories (HSDTs) have gained extensive recognition and made quite great progress [4,. Based on polynomial or non-polynomial transverse shear functions, various HSDTs have been proposed to avoid the use of a shear correction factor, and to better predict the shear behavior of the plate, for instance, the third-order shear deformation theory [17,18], the fifth-order shear deformation theory [19], the exponential shear deformation theory [20], the hyperbolic shear deformation theory [21], and the combined or mixed HSDTs [22,23].…”

“…Please see Thai and Kim [16] and Caliri et al [24] for a comprehensive review of HSDTs. In HSDTs, the bending angles of rotation and shear angles can be treated as independent variables, and the shear-locking problem encountered in FSDT can be well-solved [4]. In [30][31][32][33][34][35][36][37][38][39][40][41][42][43], two well-know HSDTs named as equivalent single layer (ESL) and layer-wise (LW) models are developed to evaluate the effective mechanical behavior of composite structures correctly.…”

“…In [30][31][32][33][34][35][36][37][38][39][40][41][42][43], two well-know HSDTs named as equivalent single layer (ESL) and layer-wise (LW) models are developed to evaluate the effective mechanical behavior of composite structures correctly. The accuracy and reliability of HSDTs have been illustrated by numerous examples in the literature [4,17,[25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. However, the general requirement of C 1 continuity in approximation fields in HSDTs brings difficulties for the numerical implementation of the standard Finite Element Method (FEM), which is similar to that of the classic Kirchhoff-Love plate theory.…”