Meshless Modelling of Laminate Mindlin Plates under Dynamic Loads

Milan Zmindak,Daniel Riecky

doi:10.26552/com.c.2012.3.24-31

Abstract

Collocation method and Galerkin method have been dominant in the existing meshless methods. A meshless local Petrov-Galerkin (MLPG) method is applied to solve laminate plate problems described by the Reissner-Mindlin theory for transient dynamic loads. The Reissner-Mindlin theory reduces the original three-dimensional (3-D) thick plate problem to a two-dimensional (2-D) problem. The bending moment and the shear force expressions are obtained by integration through the laminated plate for the considered constitutive equations in each lamina. The weak-form on small subdomains with a Heaviside step function as the test functions is applied to derive local integral equations. After performing the spatial MLS approximation, a system of ordinary differential equations of the second order for certain nodal unknowns is obtained. The derived ordinary differential equations are solved by the Houbolt finite-difference scheme as a time-stepping method.

Highlights

Composite materials are common engineering materials used in a wide range of applications
It is well known that the classical thin plate theory of Kirchhoff gives rise to certain non-physical simplifications mainly related to the omission of the shear deformation and the rotary inertia, which become more significant for increasing thickness of the plate
The effects of shear deformation and rotary inertia are taken into account in the Reissner-Mindlin plate bending theory [6]

Summary

Introduction

Composite materials are common engineering materials used in a wide range of applications. Continuity can be tuned to a desired value Their results showed an excellent convergence, their formulation is not applicable to shear deformable plate/shell problems. Where the material stiffness coefficients c(ikjm) l are assumed to be homogeneous for the k-th lamina It can be seen from equation (2) that the strains are continuous throughout the plate thickness. The constitutive equation for orthotropic materials and plane stress problem are given for example in [23, 26]. Substituting constitutive equations and (2) into moment and force resultants (5) allows the expression of the bending moments Mαβ and shear forces Qα for α, β ϭ 1, 2, in terms of rotations and lateral displacements of the orthotropic plate. Throughout the analysis, Greek indices vary from 1 to 2, and the dots over a quantity indicate differentiations with respect to time τ

Local Petrov-Galerkin Weak-Form

Numerical Implementation

Numerical Examples

Conclusion