Reissner Mindlin Research Articles

Numerical Simulations in Ultrasonic Guided Wave Analysis for the Design of SHM Systems—Benchmark Study Based on the Open Guided Waves Online Platform Dataset

Structural health monitoring (SHM) strategies based on ultrasonic guided waves are very promising regarding thin-walled lightweight structures. To study the performance of such systems, validated numerical analysis tools have to be used. For that procedure, a benchmark between two numerical methods will be presented. The first promising approach is the elastodynamic finite integration technique (EFIT). Miscellaneous research shows that its capability of capturing wave characteristics and interactions is advanced in various media and structures, including thin-walled composites. The second approach employs conventional shell-type finite elements following the Reissner–Mindlin theory for modelling layered composite structures. The advantage of using such finite element methods (FEM) is their high availability in general purpose simulation tools. As a reference model, the measurement data coming from the Open Guided Waves Project (OGW) was taken into account. The OGW dataset provides the experimental data of ultrasonic guided wave propagation in carbon fiber composite plates with an additional omega stringer. By using this contribution, this experiment was reproduced by simulation. The paper presents the results of a validation and motivates further research, such as in research related to the probability of detection analysis and numerical performance.

A variational multiscale stabilized finite element formulation for Reissner–Mindlin plates and Timoshenko beams

A variational multiscale stabilized finite element formulation for Reissner–Mindlin plates and Timoshenko beams

A Discontinuous Petrov–Galerkin Method for Reissner–Mindlin Plates

We present a discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model. Our method is based on a variational formulation that utilizes a Helmholtz decomposition of the shear force. It produces approximations of the primitive variables and the bending moments. For any canonical selection of boundary conditions the method converges quasi-optimally. In the case of hard-clamped convex plates, we prove that the lowest-order scheme is locking free. Several numerical experiments confirm our results.

Dynamic Analysis of Mindlin Plate Subjected to a Moving Mass-Spring-Damper System Using the Moving Element Method

The moving element method (MEM) is applied in this paper to study the dynamic behavior of the Mindlin plate subjected to a moving mass-spring-damper system. Reissner- Mindlin plate theory is utilized to study the plate structure. The plate is modeled by the Isoparametric quadrilateral nine-node element. The governing equation of motion is built into a coordinate system which moves with the moving mass-spring-damper system and is based on the principle of virtual work. The MEM is proposed to solve the governing equation of motion of the plate resting on a Pasternak foundation under the moving mass-spring-damper system including roughness of plate surface which is a dynamic source. Numerical results are investigated and verified by comparing with the published results. The effects of different parameters, such as spring stiffness, damping coefficient and velocity of the moving load system, plate thickness, foundation parameters and roughness of the plate surface on the dynamic response of the Mindlin plate are investigated.

Theoretical Analysis of Free Vibration and Transient Response of Rectangular Plate–Cavity System Under Impact Loading

AbstractThis paper presents a theoretical method to solve the free vibration and transient responses of a rectangular plate–cavity system. The spectral collocation method was used to solve the resonant frequencies and corresponding mode shapes of rectangular plates based on Kirchhoff thin plate and Mindlin–Reissner thick plate theories. A linear velocity potential function was employed to model the fluid pressure applied to the plate surface. Unlike in previous studies, it was not assumed that the wet-mode shapes were the same as the dry-mode ones. Rather, the wet modes were assumed to be the superposition of the dry modes; then, the resonant frequencies and corresponding mode shapes of the wet modes could be obtained by solving the equations of the coupled system by exploiting the orthogonality of dry modes. Using dry modes’ orthogonality and superposition of the wet modes, the transient responses of the rectangular plate–cavity system under impact loading can be solved. A method for estimating the resonant frequencies of the coupled system is proposed based on parametric studies to determine the influence of the fluid properties and plate materials on resonant frequencies. As a result, the resonant frequencies and transient responses obtained from the proposed theoretical methods are in excellent agreement with those obtained from finite element analysis.

High-fidelity analytical modeling of asymmetric CFRP composites using Reissner–Mindlin theory and hygroscopic degradation

Multi-stable carbon fiber-reinforced composites with asymmetric ply layouts have shown promising potential for creating multi-functional structural systems. To design and analyze these complex composites, high-fidelity and computationally efficient analytical models are crucial. To this end, Hyer and Dano formulated the first model four decades ago based on classical lamination theory (CLT). Surprisingly, despite many follow-up studies to refine this analytical approach, current models still follow the same fundamental ingredients used in Hyer and Dano’s original work. As a result, they all inherited the same limitations. For example, they ignore the interlaminar stresses and lack a rigorously calibrated hygroscopic term. To overcome these limitations, we propose a new analytical modeling approach by adopting the Reissner–Mindlin theory and hygroscopic degradation. This new model introduces in-plane rotations as two additional degrees of freedom and a combined expansion coefficient for the epoxy matrix materials. Compared to finite element simulations and experiment data, the new model can estimate the interlaminar stress reasonably well (especially regarding its concentration near the edges) and predict the laminates’ external shape more accurately than an equivalent Hyer’s model. The new model can also accurately predict the reduction in the laminate’s curvature and snap-through force in different design configurations, as well as their linear correlation to the moisture content levels. Overall, the results of this study present a fundamental advancement in the analytical modeling of asymmetric multi-stable composite laminates, offering a broad and versatile foundation for designing and analyzing the next generation of multi-functional composite structures.

Isogeometric analysis using G-spline surfaces with arbitrary unstructured quadrilateral layout

G-splines are a generalization of B-splines that deals with extraordinary points by imposing G1 constraints across their spoke edges, thus obtaining a continuous tangent plane throughout the surface. Using the isoparametric concept and the Bubnov–Galerkin method to solve partial differential equations with G-splines results in discretizations with global C1 continuity in physical space. Extraordinary points (EPs) are required to represent manifold surfaces with arbitrary topological genus. In this work, we allow both interior and boundary EPs and there are no limitations regarding how close EPs can be from each other. Reaching this level of flexibility is necessary so that splines with EPs can become mainstream in the design-through-analysis cycle of the complex thin-walled structures that appear in engineering applications. To the authors’ knowledge, the two EP constructions based on imposing G1 constraints proposed in this work are the first two EP constructions used in isogeometric analysis (IGA) that combine the following distinctive characteristics: (1) Only vertex-based control points are used and they behave as geometric shape handles, (2) any control point of the control net can potentially be an EP, (3) global C1 continuity in physical space is obtained without introducing singularities, (4) faces around EPs are not split into multiple elements, i.e., Bézier meshes with uniform element size are obtained, and (5) good surface quality is attained. The studies of convergence and surface quality performed in this paper suggest that G-splines are more suitable for IGA than EP constructions based on the D-patch framework. Finally, we have represented the stiffener, the inner part, and the outer part of a B-pillar with G-spline surfaces and solved eigenvalue problems using both Kirchhoff–Love and Reissner–Mindlin shell theories. The results are compared with bilinear quadrilateral meshes and excellent agreement is found between G-splines and conventional finite elements. In summary, G-splines are a viable alternative to design and analyze thin-walled structures using the same geometric representation so as to streamline the design-through-analysis cycle.

Multiscale modeling of laminated thin-shell structures with Direct FE[formula omitted

Shell finite element is computationally efficient and thereby favored in large-scale structural analysis. With the application of advanced materials, understanding the relation between micro-heterogeneities and structural behavior is important. To this end, we propose a multiscale method in the current contribution for modeling thin-walled fiber reinforced composite laminates with Mindlin–Reissner shell theory and Direct FE2. By exploiting the length scales in layered structure together with a shell formulation degenerated from continuum solid element, both in-plane and through-thickness periodicity can be assumed. A single-fiber unit cell attached to each Gauss point is employed as a representative volume element (RVE) rather than a through-thickness microstructure. In this way, the scale transitions are naturally dealt with using a first-order homogenization framework, in which the downscaling of transverse shear strain, commonly troublesome for the through-thickness RVE, is straightforward. The other merit inherited from Direct FE2 is that the two-scale problem is formulated in a single finite element analysis wherein the macro and micro DOFs are coupled based on the downscaling kinematics. It avoids the upscaling of the macroscopic stresses and tangents in the conventional nested FE2 scheme, which is challenging in the case of complex microstructural response. The developed method is then verified through several benchmarking examples, including a flat plate and two curve shells. Isotropic materials and composite laminates with homogeneous and heterogeneous RVEs have been examined. Two examples with plasticity and cohesive cracks are further presented to showcase its potential in modeling damage propagation in composite structures.

Mesh-free discretization of peridynamic shell structures and coupling model with isogeometric analysis

Peridynamic (PD) theory applies nonlocal framework to reform the equilibrium equation in fracture analysis. The mesh-free formulation of the PD model performs well in two- and three-dimensional formulations. A mesh-free discretization of the PD shell using Reissner–Mindlin theory is proposed to meet the demands of crack simulation in shell structures. A micro-beam bond with three translatory and three rotational degrees of freedom is applied to describe the in-shell planar, shearing, and bending deformations. A particle-based method for the coupling of isogeometric analysis (IGA) and PD shells is proposed to improve the computational efficiency. The smooth geometry of the shell structure is precisely described by nonuniform rational B-splines (NURBS). The zone around the crack is transformed from IGA mesh into PD particles by applying mesh-free PD discretization to account for the crack propagation. A particle equilibrium-based method is used for the coupling of IGA and PD zones. The proposed coupling model combines the advantages of both methods and successfully simulates crack growth in shell structures with better computational efficiency compared with the pure PD model. Numerical examples are conducted wherein the complex propagation and intersecting of cracks can be captured, thus proving the accuracy and effectiveness of the proposed method.

Coarse Mesh Superconvergence in Isogeometric Frequency Analysis of Mindlin–Reissner Plates with Reduced Integration and Quadratic Splines

A frequency accuracy study is presented for the isogeometric free vibration analysis of Mindlin–Reissner plates using reduced integration and quadratic splines, which reveals an interesting coarse mesh superconvergence. Firstly, the frequency error estimates for isogeometric discretization of Mindlin–Reissner plates with quadratic splines are rationally derived, where the degeneration to Timoshenko beams is discussed as well. Subsequently, in accordance with these frequency error measures, the shear locking issue corresponding to the full integration isogeometric formulation is elaborated with respect to the frequency accuracy deterioration. On the other hand, the locking-free characteristic for the isogeometric formulation with uniform reduced integration is illustrated by its superior frequency accuracy. Meanwhile, it is found that a frequency superconvergence of sixth order accuracy arises for coarse meshes when the reduced integration is employed for the isogeometric free vibration analysis of shear deformable beams and plates, in comparison with the ultimate fourth order accuracy as the meshes are progressively refined. Furthermore, the mesh size threshold for the coarse mesh superconvergence is provided as well. The proposed theoretical results are consistently proved by numerical experiments.

Locking alleviation technique for the peridynamic Reissner–Mindlin plate model: the developed reduced integration method

Locking alleviation technique for the peridynamic Reissner–Mindlin plate model: the developed reduced integration method

Analysis of shell structures by an improved 3-node triangular flat shell element with a bubble function and cell-based strain smoothing

An enhanced 3-node triangular flat shell element with 18 degrees of freedom according to the Reissner–Mindlin theory is presented in this paper. New cell-based smoothed (CS) bending strains were derived from the displacement approximations enriched by a cubic shape function for the rotations at the bubble node located at the centroid of the triangular element. The true drilling degrees of freedom of the proposed element, which is independent from user-defined coefficients, was achieved by cell-based smoothing the Allman’s membrane strains. The membrane and bending stiffness matrices were then efficiently determined by the line integration of the smoothing domains’ boundaries. The shear-locking removal MITC3+ technique was employed to interpolate the transverse shear strains. The outstanding performance of the developed 3-node triangular flat shell element, so-called the CS-MITC18+ shell element, was demonstrated through the static and free vibration analyses of several benchmark shell structures. Numerical results shown that the CS-MITC18+ flat shell element has the spatially isotropic and zero-energy mode properties, and provides competitive accuracy, convergence, and computational cost in comparison to other conventional and strain-smoothed 3-node triangular flat shell elements.

A discrete de Rham method for the Reissner–Mindlin plate bending problem on polygonal meshes

In this work we propose a discretisation method for the Reissner–Mindlin plate bending problem in primitive variables that supports general polygonal meshes and arbitrary order. The method is inspired by a two-dimensional discrete de Rham complex for which key commutation properties hold that enable the cancellation of the contribution to the error linked to the enforcement of the Kirchhoff constraint. Denoting by k≥0 the polynomial degree for the discrete spaces and by h the meshsize, we derive for the proposed method an error estimate in hk+1 for general k, as well as a locking-free error estimate for the lowest-order case k=0. The theoretical results are validated on a complete panel of numerical tests.

Tamamlayıcı Fonksiyonlar Yöntemi ile Fonksiyonel Derecelenmiş Sandviç Dairesel Plakların Eğilme Analizi

Bu çalışmada, Fonksiyonel Derecelenmiş (FD) malzemeli sandviç dairesel plakların eğilme davranışı teorik olarak araştırılmıştır. Malzeme özellikleri kalınlığı boyunca değişen sandviç dairesel plakların, öz tabakası veya çekirdeği izotropik homojen yüzey tabakları ise FD malzemeli olarak kabul edilmiştir. Ele alınan plakların statik davranışını idare eden denklemler Kirchhoff–Love ve Mindlin–Reissner plak teorilerine göre minimum toplam enerji prensibi yardımıyla kanonik halde elde edilmiştir. Elde edilen bu denklemlerin sayısal çözümleri için Tamamlayıcı Fonksiyonlar Yöntemi (TFY) uygulanmıştır. Bu araştırmada, malzeme değişim katsayılarının, yarıçap-kalınlık oranlarının, kayma deformasyon etkisinin ve farklı sınır koşullarının FD sandviç dairesel plakların eğilme davranışı üzerindeki etkileri parametrik olarak incelenmiştir. TFY’nin bu tür problemlere etkin bir şekilde uygulanabilirliği ve yöntemin doğruluğu, elde edilen sonuçların mevcut literatür ile karşılaştırılarak gösterilmiştir.

Physics-Based Self-Learning Recurrent Neural Network enhanced time integration scheme for computing viscoplastic structural finite element response

In the current study, we present an application to the class of deep learning known as the Physics Informed Neural Networks (PINNs), more specifically we develop a new implicit material integration scheme based on Recurrent Neural Network (RNN). In this study, we refer to it as the Recurrent Enhanced Implicit Integration Scheme (REIIS). We illustrate how to incorporate the constitutive relations into REIIS and explore its application to Lemaitre–Chaboche viscoplastic model. Whilst common applications in this class of deep learning utilize feed-forward neural networks (FFNN) for training, we propose a combination of the Long Term Short Memory (LSTM) network with FFNNs for an accurate representation of the state variables. Furthermore, with the present strategy, we combine the data-driven approach of replacing the material behavior with NN along with its ability to learn continuously through physical laws. To validate the approach, we test the results of the framework with the numerical solution obtained with Finite Element Method (FEM) for Reissner–Mindlin plate elements at different strain rates. We further observe that obeying physical constraints leads to improved robustness in the learning of REIIS and broadens the scope of integrating the NNs in FEM.

Wave attenuation in viscoelastic hierarchical plates

Phononic crystals (PCs) are periodic structures obtained by the spatial arrangement of materials with contrasting properties, which can be designed to efficiently manipulate mechanical waves. Plate structures can be modeled using the Mindlin–Reissner plate theory and have been extensively used to analyze the dispersion relations of PCs. Although the analysis of the propagating characteristics of PCs may be sufficient for simple elastic structures, analyzing the evanescent wave behavior becomes fundamental if the PC contains viscoelastic components. Another complication is that increasingly intricate material distributions in the unit cell of PCs with hierarchical configuration may render the calculation of the complex band structure (i.e., considering both propagating and evanescent waves) prohibitive due to excessive computational workload. In this work, we propose a new extended plane wave expansion formulation to compute the complex band structure of thick PC plates with arbitrary material distribution using the Mindlin–Reissner plate theory containing constituents with a viscoelastic behavior approximated by a Kelvin–Voigt model. We apply the method to investigate the evanescent behavior of periodic hierarchically structured plates for either (i) a hard purely elastic matrix with soft viscoelastic inclusions or (ii) a soft viscoelastic matrix with hard purely elastic inclusions. Our results show that for (i), an increase in the hierarchical order leads to a weight reduction with relatively preserved attenuation characteristics, including attenuation peaks due to locally resonant modes that present a decrease in attenuation upon increasing viscosity levels. For (ii), changing the hierarchical order implies in opening band gaps in distinct frequency ranges, with an overall attenuation improved by an increase in the viscosity levels.

Free vibration analysis of stiffened rectangular plate with cutouts using Nitsche based IGA method

The Nitsche based non-uniform rational B-splines (NURBS) isogeometric analysis (IGA) method is employed to study the free vibration of stiffened plate with holes. The plate with stiffeners is modeled by the Reissner–Mindlin theory and Timoshenko beam theory. Along the intersection lines between middle surface of plate and stiffeners, the plate is artificially divided into several non-overlapping NURBS patches, and stiffeners are located at the intersection lines. The holes are also constructed using several different non-overlapping NURBS patches along the common boundaries. Nitsche method is employed to glue different patches. Then, the free vibration analysis of stiffened plate with holes are conducted. One of the advantages of the presented method is that it is not necessary to identify the plate elements within which stiffener nodes located while the plate and stiffener need to be remeshed. The other advantage is that the presented method can be conveniently used to analyze the vibration of stiffened plate with multiple holes. Numerical examples are given to validate the correctness of presented method. The dimensionless frequencies and mode shapes of stiffened plate with holes or without holes are obtained by the present method and compared with the results that are available in the literature or those obtained using the ANSYS software. The results show that the effectiveness of the present method.

A stabilized quasi and bending consistent meshfree Galerkin formulation for Reissner–Mindlin plates

A stabilized quasi and bending consistent meshfree Galerkin formulation for Reissner–Mindlin plates

Numerical implementation for isogeometric analysis of thin-walled structures based on a Bézier extraction framework: nligaStruct

We present an open Octave/Matlab package nligaStruct, for linear and nonlinear isogeometric analysis of thin-walled structures. nligaStruct is an extension of NLIGA in aspect of isogeometric structural analysis. A framework for Bézier extraction of T-spline and NURBS is embedded in the simulation pipeline. Both Reissner–Mindlin and Kirchhoff–Love assumptions are considered in the case of static bending and free vibration analysis of thin-walled structures (plates and shells). The convergence behaviors of these two assumptions for both plate and shell are thoroughly studied and compared. The discrete formulations for large deformation of Kirchhoff–Love shells considering geometrical and material nonlinearity are detailedly derived and carefully implemented. Three types of hyperelastic materials, including St. Venant–Kirchhoff, incompressible neo-Hookean and compressible neo-Hookean, are employed in the large deformation of Kirchhoff–Love shells. A series of popular benchmark examples and their geometrical models are demonstrated in the package. The numerical results are validated by comparing with analytical solutions and that obtained from commercial software. The postprocessing is self-contained and implemented by tessellation of Bézier elements. Finally, complete source codes and the data obtained from numerical examples are fully provided and free to access.

Effective Inertia Coefficients Prediction and Cell Size Effects in Thickness Direction of Periodic Composite Plates

This paper develops a new homogenization method for free vibration problems of periodic composite plates. In this new method, three-dimensional (3D) periodic plates are equivalent to Reissner–Mindlin plates with both effective stiffnesses and effective inertia coefficients. The effective stiffnesses for the dynamic problems are the same as those for the static problems, and they can be achieved by the equivalence principle of macro- and microscopic internal virtual work. To fully take the inertia effects into account, the effective inertia coefficients including the effective translational, translational–rotational and rotational inertias are determined by the two-scale equivalence principle of kinetic energies under three rigid modes. In addition, cell size effects in the thickness direction of composite plates are investigated by using the proposed method and the asymptotic homogenization method (AHM). Numerical experiments validate the effectiveness of the proposed equivalent method for different scale factors, and show that the rotational inertia cannot be ignored for out-of-plane deformations, especially for higher-order modes. Besides, numerical comparisons show that the cell size effects are not negligible when using the AHM to analyze the out-of-plane deformations, and three or more repeated unit cells in the thickness direction are required to ensure accuracy.