Fast and accurate elastic analysis of laminated composite plates via isogeometric collocation and an equilibrium-based stress recovery approach

TL;DR

The paper addresses efficiently predicting 3D stress states in laminated composite plates. It introduces a 3D isogeometric collocation framework with a homogenized single-element thickness and an equilibrium-based post-processing step to recover out-of-plane stresses, namely $ \sigma_{13}$, $ \sigma_{23}$, and $ \sigma_{33}$. The approach yields accurate displacements and in-plane stresses with substantially reduced DOFs compared with full layerwise models, demonstrated against the Pagano benchmark; the post-processing markedly improves $ \sigma_{13}$, $ \sigma_{23}$, and $ \sigma_{33}$, especially for large length-to-thickness ratios $S$ and many plies. The method enables fast, scalable analysis of laminated plates and suggests extensions to curved geometries and large deformations.

Abstract

A novel approach which combines isogeometric collocation and an equilibrium-based stress recovery technique is applied to analyze laminated composite plates. Isogeometric collocation is an appealing strong form alternative to standard Galerkin approaches, able to achieve high order convergence rates coupled with a significantly reduced computational cost. Laminated composite plates are herein conveniently modeled considering only one element through the thickness with homogenized material properties. This guarantees accurate results in terms of displacements and in-plane stress components. To recover an accurate out-of-plane stress state, equilibrium is imposed in strong form as a post-processing correction step, which requires the shape functions to be highly continuous. This continuity demand is fully granted by isogeometric analysis properties, and excellent results are obtained using a minimal number of collocation points per direction, particularly for increasing values of length-to-thickness plate ratio and number of layers.

Figures (9)

• Figure 1: Layerwise approach and homogenized single-element example of isogeometric shape functions for a degree of approximation equal to 4.
• Figure 2: Pagano's test case Pagano1970. Problem geometry and boundary conditions.
• Figure 3: Through-the-thickness stress solutions for the 3D Pagano problem Pagano1970 evaluated at $x_1=x_2=0.25L$. Case: plate with 3 layers and length-to-thickness ratio $S=20$ ($\boldsymbol{\leftrightline}$ Pagano's solution, homogenized single-element approach solution (without post-processing), $\boldsymbol{\times}$ post-processed solution).
• Figure 4: Through-the-thickness stress solutions for the 3D Pagano problem Pagano1970 evaluated at $x_1=x_2=0.25L$. Case: plate with 11 layers and length-to-thickness ratio $S=20$ ($\boldsymbol{\leftrightline}$ Pagano's solution, homogenized single-element approach solution (without post-processing), $\boldsymbol{\times}$ post-processed solution).
• Figure 5: Through-the-thickness $\bar{\sigma}_{13}$ profiles for several in plane sampling points. $L$ represents the total length of the plate, that for this case is $L=220\,\text{mm}$ (being $L=S\,t$ with $t=11\,\text{mm}$ and $S=20$), while the number of layers is 11 ($\boldsymbol{\leftrightline}$post-processed solution, $\boldsymbol{\times}$ analytical solution Pagano1970).

Theorems & Definitions (1)

• Remark 1