Two-field formulations for isogeometric Reissner–Mindlin plates and shells with global and local condensation

In this paper, mixed formulations are presented in the framework of isogeometric Reissner–Mindlin plates and shells with the aim of alleviating membrane and shear locking. The formulations are based on the Hellinger-Reissner functional and use the stress resultants as additional unknowns, which have to be interpolated in appropriate approximation spaces. The additional unknowns can be eliminated by static condensation. In the framework of isogeometric analysis static condensation is performed globally on the patch level, which leads to a high computational cost. Thus, two additional local approaches to the existing continuous method are presented, an approach with discontinuous stress resultant fields at the element boundaries and a reconstructed approach which is blending the local control variables by using weights in order to compute the global ones. Both approaches allow for a static condensation on the element level instead of the patch level. Various numerical examples are investigated in order to verify the accuracy and effectiveness of the different approaches and a comparison to existing elements that include mechanisms against locking is carried out.

Assumed strain finite element for natural frequencies of bending plates

Purpose This paper aims to describe the formulation of a new finite element by assuming the strain field rather than the displacement field and by using the Reissner–Mindlin plate theory for the free vibration analysis of bending plates. This quadrilateral element consists of four-nodes and twelve degrees of freedom. The suggested element is based on assumed functions of the strain field that satisfy the compatibility equation. Design/methodology/approach After the proposition of the new element, several numerical tests for plates with regular and distorted meshes are presented to assess the performance of the new element. In addition, a parametric study is carried out to analyze the effects of biaxial loads on the natural frequencies of square plates with various boundary conditions. Detailed discussions are proposed after each benchmark problem. Findings The formulated element has verified the shear locking test and passes the patch test. The obtained results from the developed element show an excellent accuracy and fast convergence, and the natural frequencies are in excellent agreement when compared with analytical and other available numerical solutions. Originality/value The present element is simple in its formulation and has been proven to be applicable to thin or thick plate situations with sufficient accuracy. This element with full integration is free from shear locking, however, the numerical results provided by the standard four-node plate element R4 element show locking phenomena in thin plates. In addition to these features, the imposition of the compatibility conditions and the rigid body modes allow obtaining a finite element with higher-order terms for displacements field, which can increase the performance of the finite elements.

Development of a shear forming envelope for carbon fibre non-crimp fabrics

Our research aims to develop a shear forming envelope for the preforming of textiles, a critical step in the manufacture of fibre-reinforced composite materials. This paper demonstrates the progress towards this aim by conducting picture frame tests to empirically determine the locking angle of non-crimp fabrics with different fibre orientations. While conventional shear tests typically utilise woven textile samples with orthogonal fibre directions of 0°/90°, the investigation of non-crimp fabrics, especially with non-standard fibre orientations, is less common. As a result, there is little knowledge about the shear deformation behaviour of these fabric types, despite their relevance to the aerospace industry. In this study, the shear locking angles of various carbon fibre non-crimp fabrics are investigated, gradually reducing the relative fibre angles of the textile materials from ±45° to ±22.5°. Previously, it was observed that unidirectional 0° reinforcement layers induce draping defects when forming multiaxial non-crimp fabric stacks into curved aerospace stiffeners. Their substitution by reinforcements with smaller cross-ply angles such as ±30° resulted in better formability and reduced defects. It is however unclear, how the shear locking angle decreases with more acute cross-ply angles. Here, we report for the first time a correlation between the fibre orientation of the non-crimp fabric and its shear locking angle. The resulting shear forming envelope provides composite design and manufacturing guidance for an enhanced utilisation of the advantageous but anisotropic properties of carbon fibre textiles.

Analysis and Modeling of Wrinkling in Composite Forming

Different approaches for the simulation of wrinkling during forming of textile reinforcements are presented. It is shown that 3D finite element modeling requires the consideration of an additional bending stiffness of the fibers. In shell-type modeling, the bending stiffness is important because it conditions the size of the wrinkles. Different methods to take into account the bending stiffness independently of the tensile stiffness are presented. The onset and development of wrinkles during forming is a global problem that concerns all deformation modes. It is shown in examples that the shear locking angle is not sufficient to conclude about the development of wrinkles. This article highlights the two points common to the different cases of wrinkling of continuous fiber textile reinforcements: the quasi-inextensibility of the fibers and the possible slippage between the fibers. It presents and compares different approaches to consider these two aspects. The simulation of the simultaneous forming of multilayered textile reinforcements makes it possible to see the influence of the orientation of different plies which is an important factor with regard to wrinkling.

A Gradient Stable Node-Based Smoothed Discrete Shear Gap Method for Analysis of Reissner–Mindlin Plates

In this paper, a gradient stable node-based smoothed discrete shear gap method (GS-DSG) using 3-node triangular elements is presented for Reissner–Mindlin plates in elastic-static, free vibration, and buckling analyses fields. By applying the smoothed Galerkin weak form, the discretized system equations are obtained. In order to carry out the smoothing operation and numerical integration, the smoothing domain associated with each node is defined. The modified smoothed strain with gradient information is derived from the Hu–Washizu three-field variational principle, resulting in the stabilization terms in the system equations. The stabilized discrete shear gap method is also applied to avoid transverse shear-locking problem. Several numerical examples are provided to illustrate the accuracy and effectiveness. The results demonstrate that the presented method is free of shear locking and can overcome the temporal instability issues, simultaneously obtaining excellent solutions.

Assumed stress quasi-conforming formulation for static and free vibration analysis of symmetric laminated plates

Purpose Numerous finite elements are proposed based on analytical solutions. However, it is difficult to find the solutions for complicated governing equations. This paper aims to present a novel formulation in the framework of assumed stress quasi-conforming method for the static and free vibration analysis of anisotropic and symmetric laminated plates. Design/methodology/approach Firstly, an initial stress approximation ruled by 17 parameters, which satisfies the equilibrium equations is derived to improve the performance of the constructed element. Then the stress matrix is treated as the weighted function to weaken the strain-displacement equations. Finally, the Timoshenko’s laminated composite beam functions are adopted as boundary string-net functions for strain integration. Findings Several numerical examples are presented to show the performance of the new element, and the results obtained are compared with other available ones. Numerical results have proved that the new element is free from shear locking and possesses high accuracy for the analysis of anisotropic and symmetric laminated plates. Originality/value This paper proposes a new QC element for the static and free vibration analysis of anisotropic and symmetric laminated plates. In contrast with the complicated analytical solutions of the equilibrium equations, an initial stress approximation ruled by 17 parameters is adopted here. The Timoshenkos laminated composite beam functions are introduced as boundary string-net functions for strain integration. Numerical results demonstrate the new element is free from shear locking and possesses high accuracy for the analysis of anisotropic and symmetric laminated plates.