Reissner Mindlin Research Articles

Brittle fracture on plates governed by topological derivatives

PurposeIn the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.Design/methodology/approachThe Francfort–Marigo damage theory is adapted to the Kirchhoff and Reissner–Mindlin plate bending models. Then, the topological derivative method is used to minimize the associated Francfort–Marigo shape functional. In particular, the whole damaging process is governed by a threshold approach based on the topological derivative field, leading to a notable simple algorithm.FindingsNumerical simulations are driven in order to verify the applicability of the proposed method in the context of brittle fracture modeling on plates. The obtained results reveal the capability of the method to determine nucleation and propagation including bifurcation of multiple cracks with a minimal number of user-defined algorithmic parameters.Originality/valueThis is the first work concerning brittle fracture modeling of plate structures based on the topological derivative method.

Numerical modeling for viscoelastic sandwich smart structures bonded with piezoelectric materials

Regarding to viscoelastic laminated piezo smart plates and shells, electro-mechanically coupled finite element (FE) formations are derived based on zig-zag hypothesis for vibration and damping analyses. Three lamination configurations are considered in the model, where both elastic layers and viscoelastic layers are assumed to be Reissner–Mindlin plate. The quadratic element is adopted for FE modeling, each node has seven mechanical degrees of freedom (DOFs), which considers the complex shear modulus of viscoelastic materials. The analytical model is first validated by FE simulation of viscoelastic sandwich plates. Later, the effects of various parameters on the damping performance of piezo-viscoelastic laminated cylindrical structures are carried out, e.g. boundary conditions, reinforcement orientation angle, thickness of the viscoelastic core and structural curvature. The present FE modeling and analyses provide a powerful tool to accurately investigate the behaviors of viscoelastic laminated structures.

Static Analysis of Stiffened Shells Using an Edge-Based Smoothed MITC3 (ES-MITC3) Method

A novel approach for solving the stiffened shell structures by using an edge-based smoothed MITC3 finite element method (ES-MITC3) is presented in this paper. The ES-MITC3 method is an efficient finite element method by combining the edge-based smoothed finite element method (ES-FEM) with the original MITC3 triangular element to not only significantly improve the accuracy but also overcome the shear-locking phenomenon in the Reissner–Mindlin shell analysis. In this study, the ES-MITC3 method is applied for shell structures and then reinforced by stiffeners based on the Timoshenko beam theory to achieve more durability and strength structures. The transverse displacements of the shell structures and stiffeners at the contact positions are assumed compatible. Numerical results of the ES-MITC3 element are compared with those of available other numerical results to demonstrate a good convergence and accuracy of the present method.

Computational analysis of the nonlinear vibrational behavior of perforated plates with initial imperfection using NURBS-based isogeometric approach

Abstract In this article, an isogeometric analysis through NURBS basis functions is presented to study the nonlinear vibrational behavior of perforated plates with initial imperfection. In this regard, the governing equations of plate dynamics, as well as the displacement–strain relations, are derived using the Mindlin–Reissner plate theory by considering von Karman nonlinearity. The geometry of the structure is formed by selecting the order of NURBS basis functions and the number of control points according to the physics of the problem. Since similar basis functions are utilized to estimate the accurate geometry and displacement field of the domain, the order of the basic functions and the number of control points are optimized for the proper approximation of the unknown field variables. By utilizing the energy approach and Hamilton principle and discretizing the equations of motion, the vibrational response of the perforated imperfect plate is extracted through an eigenvalue problem. The results of linear vibrations, geometrically nonlinear vibrations, and nonlinear vibrations of imperfect plates are separately validated by considering the previously reported findings, which shows a satisfactory agreement. Thereafter, a coefficient of the first mode shape is considered as the initial imperfection and the vibrational analysis is reexamined. Furthermore, the nonlinear vibrations of the perforated plate with initial imperfection are analysed using an iterative approach. The effects of the perforated hole, initial imperfection, and geometric nonlinearity are also addressed and discussed.

Thermal buckling adaptive multi-patch isogeometric analysis of arbitrary complex-shaped plates based on locally refined NURBS and Nitsche’s method

Plate structures suffering thermal environments are often encountered in practice. In this paper, thermal buckling behavior of complex-shaped plates by an adaptive multi-patch isogeometric analysis (IGA) based on locally refined NURBS and Nitsche’s method is studied. Kinematic equations are derived using Reissner–Mindlin plate theory, while complex geometries of plates are represented with multiple patches, where the connection between two adjacent patches is constructed through Nitsche’s method. To conduct adaptive local refinement, structural mesh refinement strategy is combined with a posterior error estimator, which is defined according to the recovered stresses of the first thermal buckling mode. Numerical examples of plates with simple and complex geometries are considered. The accuracy of critical buckling temperature rise obtained from this study is verified against reference solutions.

Strain Gradient Theory Based Dynamic Mindlin-Reissner and Kirchhoff Micro-Plates with Microstructural and Micro-Inertial Effects

In this study, a dynamic Mindlin–Reissner-type plate is developed based on a simplified version of Mindlin’s form-II first-strain gradient elasticity theory. The governing equations of motion and the corresponding boundary conditions are derived using the general virtual work variational principle. The presented model contains, apart from the two classical Lame constants, one additional microstructure material parameter g for the static case and one micro-inertia parameter h for the dynamic case. The formal reduction of this model to a Kirchhoff-type plate model is also presented. Upon diminishing the microstructure parameters g and h, the classical Mindlin–Reissner and Kirchhoff plate theories are derived. Three points distinguish the present work from other similar published in the literature. First, the plane stress assumption, fundamental for the development of plate theories, is expressed by the vanishing of the z-component of the generalized true traction vector and not merely by the zz-component of the Cauchy stress tensor. Second, micro-inertia terms are included in the expression of the kinetic energy of the model. Finally, the detailed structure of classical and non-classical boundary conditions is presented for both Mindlin–Reissner and Kirchhoff micro-plates. An example of a simply supported rectangular plate is used to illustrate the proposed model and to compare it with results from the literature. The numerical results reveal the significance of the strain gradient effect on the bending and free vibration response of the micro-plate, when the plate thickness is at the micron-scale; in comparison to the classical theories for Mindlin–Reissner and Kirchhoff plates, the deflections, the rotations, and the shear-thickness frequencies are smaller, while the fundamental flexural frequency is higher. It is also observed that the micro-inertia effect should not be ignored in estimating the fundamental frequencies of micro-plates, primarily for thick plates, when plate thickness is at the micron scale (strain gradient effect).

Parametric study using a curved shell finite element to dynamic analysis of footbridge under rhythmic loading

When designing structural systems, the relation between form and structure is not always considered, because architects and engineers work independently. Structures with mutual collaboration such as surface-active (e.g., shells) when well-designed can optimize specific behaviors. This work proposes an analysis of some parameters that influence the dynamic behavior of footbridges under rhythmic loading. The variation of these parameters allows defining different mass distribution along the footbridge with a constant total weight. The relation between form and structure was also analyzed in this parametric study considering two architectural models for the footbridges. The parametric study proposed required a dynamic analysis of shells with variable thickness. Therefore, the formulation of a curved shell finite element is presented. This element is based on a degenerate three-dimensional solid element and is restricted to the behavior of shell under the Reissner–Mindlin approach. Two classic examples from the literature and the analytical solution of a long cylindrical shell under membrane and bending behavior were used in the validation of the curved shell finite element used in numerical analysis. From the proposed parametric study, results are presented for the parameters that define a better response in relation to the vibration fundamental frequencies of the footbridge and its maximum accelerations due to a rhythmic loading. It is concluded that the form of the curved shell supporting the flat slab significantly affects the footbridge dynamic behavior.

Buckling of Shell Panels Made of Fiberglass and Reinforced with an Orthogonal Grid of Stiffeners

The paper presents an approach to the stress-strain and buckling analysis in fiberglass cylindrical and conical panels reinforced from the concave side with an orthogonal grid of stiffeners. A mathematical model of the Timoshenko (Mindlin–Reissner) type is used. Transverse shears and geometric nonlinearity are taken into account. The stiffeners are introduced in two ways: using the method of refined discrete introduction and the method of structural anisotropy. We use a computational algorithm based on the Ritz method and the best parameter continuation method. We also provide buckling load values and make a comparison between two types of approaches to account for stiffeners, which shows good convergence.

Assumed strain finite element for natural frequencies of bending plates

PurposeThis 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/approachAfter 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.FindingsThe 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/valueThe 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.

A Smoothed GFEM Based on Taylor Expansion and Constrained MLS for Analysis of Reissner–Mindlin Plate

Based on the Taylor Expansion and constrained moving least square function, a smoothed GFEM (SGFEM) is proposed in this paper for static, free vibration and buckling analysis of Reissner–Mindlin plate. The displacement function based on SGFEM is composed of classical linear finite element shape function and nodal displacement function, which are obtained by introducing the gradient smoothed meshfree approximation in Taylor expansion of nodal displacement function. A constrained moving least square function is proposed for constituting meshfree nodal displacement function. The merits of the proposed SGFEM, including high accuracy, rapid error convergence, insensitive to mesh distortion, free of shear-locking problem, no extra DOFs and temporal stability, etc., are demonstrated by several typical examples and comparisons with other numerical methods.

Mindlin-Reissner Analytical Model with Curvature for Tunnel Ventilation Shafts Analysis

The formulation and analytic solution of a new mathematical model with constitutive curvature for analysis of tunnel ventilation shaft wall is proposed. Based on the Mindlin–Reissner theory for thick shells, this model also takes into account the shell constitutive curvature and considers an expression of the shear correction factor variable (αn) in terms of the thickness (h) and the radius of curvature (R). The main advantage of the proposed model is that it has the possibility to analyze thin, medium and thick tunnel ventilation shafts. As a result, two comparisons were made: the first one, between the new model and the Mindlin–Reissner model without constitutive curvature with the shear correction factor αn=5/6 as a constant, and the other, between the new model and the tridimensional numerical models (solids and shells) obtained by finite element method for different slenderness ratios (h/R). The limitation of the proposed model is that it is to be formulated for a general linear-elastic and axial-symmetrical state with continuous distribution of the mass.

Multi-patch local mesh refinement XIGA based on LR NURBS and Nitsche’s method for crack growth in complex cracked plates

The objective of this article is to simulate crack growth of complex Mindlin–Reissner plates by developing an adaptive multi-patch extended isogeometric analysis (XIGA). Nitsche’s method is used to treat continuity between multi-patches or the coupling of non-conforming meshes, exactly describing the geometry of complex plates. The computational meshes in XIGA are independent of the cracks by introducing some enrichment functions into the displacement approximation based on the partition of unity, thus it is convenient in modeling evolution of crack. The locally refined non-uniform rational B-splines (LR NURBS), which have the properties of local refinement and exact description of geometry, are used to construct geometric model and taken as the shape functions in XIGA. To implement the adaptive local refinement, a method based on recovery technique by Zienkiewicz and Zhu is proposed. Based on the Mindlin–Reissner plate theory, interaction integral method is employed to calculate stress intensity factors (SIFs) and the maximum circumferential stress criterion is used to determine the crack propagation direction. Several numerical examples are carried out to demonstrate the performance of the adaptive multi-patch XIGA for simulating crack growth in complex plates.

Shape‐free arbitrary polygonal hybrid stress/displacement‐function flat shell element for linear and geometrically nonlinear analyses

AbstractA high‐performance shape‐free arbitrary polygonal hybrid stress/displacement‐function flat shell finite element method is proposed for linear and geometrically nonlinear analyses of shells. First, an arbitrary polygonal Mindlin–Reissner plate element and an arbitrary polygonal membrane element with drilling degrees of freedom are constructed based on hybrid displacement‐function and hybrid stress‐function methods, respectively. Both elements have only two corner nodes along each edge. Second, by assembling the plate and the membrane elements, an arbitrary polygonal flat shell element is constructed. Third, based on the corotational method, a proper best‐fit corotated frame for geometrically nonlinear polygonal elements is designed. By updating the analytical trial functions of shell element in each increment step, the original linear flat shell element is generalized to a geometrically nonlinear model. Numerical examples show that the new element possesses excellent performance for both linear and geometrically nonlinear analyses, and possesses outstanding flexibility in dealing with complex loading distributions and mesh shapes.

An adaptive XIGA with locally refined NURBS for modeling cracked composite FG Mindlin–Reissner plates

This paper is devoted to numerical investigations on mechanical behavior of cracked composite functionally graded (FG) plates. We thus develop an efficient adaptive approach in terms of the extended isogeometric analysis (XIGA) enhanced by locally refined non-uniform rational B-spline (LR NURBS) for natural frequency and buckling analysis of cracked FG Mindlin–Reissner plates. In this setting, the crack geometries, which are described by the level sets, are independent of the computational mesh; and the LR NURBS basis functions, which have the abilities of local refinement and modeling the complex shape, are used as the shape functions of XIGA. According to the recovered stresses of the first buckling or free vibration mode, a posteriori error estimator for driving the adaptive process is defined. The accuracy can be effectively improved through adaptive local refinements, demonstrated through numerical experiments with complex geometries. Compared with uniform global refinement, XIGA based on adaptive local refinement has the characteristics of high precision at low cost and fast convergence rate. The effects of some factors such as crack length and location, plate thickness, gradient index, boundary condition, loading type, etc. on critical buckling loads and natural frequencies are investigated.

Mixed dimensional isogeometric FE-BE coupling analysis for solid–shell structures

In this paper, the collocation-based isogeometric boundary element method (IGABEM) is combined with the isogeometric Reissner–Mindlin shell elements to conduct the mixed dimensional solid–shell coupling analysis. On the basis of taking advantages of the geometric smoothness and exactness of isogeometric analysis (IGA), this method can reduce the computational scale of the solid part and eliminate the procedure of constructing analysis-suitable volumetric discretizations. In addition, this method only needs to provide the surface CAD (computer-aided design) meshes of the whole coupling model, so it has great potential in achieving close link with CAD systems. For the coupling implementation, the stiffness formula of BE subdomains is formed by condensing the unknown tractions, which simplifies the implementation of coupling constraints. That is to say, only the displacement compatibility equations on the interface are required, without explicitly establishing the traction equilibrium conditions. Two coupling approaches are developed and studied. One is the collocation-based direct strict kinematic coupling based on the deformation hypothesis of the shell across the section, and another is the weak coupling formulation in an integral sense based on work equality. The accuracy and convergence properties of the presented coupling method are investigated through numerical examples including a complex impeller blade.

Development of an analytical framework for viscoelastic corrugated-core sandwich plates and validation against FEM

In this study, an analytical procedure for the bending problem of a viscoelastic sandwich plate with a corrugated core is presented. Reissner–Mindlin plate theory and N-termed Prony series are employed to define the elastic and time-dependent contributions of the governing equations, respectively. Three different corrugation patterns, i.e., rectangular, trapezoidal, and triangular, are examined. Moreover, the structure is analyzed under both simply support and clamp boundary conditions. The calibrated material parameters of polymethyl methacrylate (PMMA) for the Generalized Maxwell rheological model are employed to show the viscoelastic response of the structure. A 3D finite element simulation of the problem is also conducted to confirm the accuracy of the analytical formulation. The two well-known creep and stress relaxation phenomena of the viscoelastic materials are examined for the mentioned corrugation cores and both boundary conditions analytically and numerically. The time-dependent dimensionless deflection and resultant von Mises stress distributions are provided. Besides, the variation of the results with various rise-times and applied load are studied in detail. The von Mises stress contours of the upper surface of the structure at the end of the creep test are also presented. The finite element method outcomes verify the analytical results with excellent compatibility. The proposed analytical procedure can be used as an efficient tool to study the effects of various parameters such as material, geometrical constants, and corrugation pattern on bending of viscoelastic sandwich plates with corrugated core problems for design and optimization, which involves a high number of simulations.

In-Plane Shear-Axial Strain Coupling Formulation for Shear-Deformable Composite Thin-Walled Beams

This paper presents an improved description of the in-plane strain coupling in Librescu-type shear-deformable composite thin-walled beams (CTWB). Based on existing descriptions for Euler-type CTWB, an analogous formulation for shear-deformable CTWB is here developed by building, via the Mindlin–Reissner theory and an orthotropic constitutive law of the shell wall, an alternate equation for the in-plane shear force which effectively couples the axial and shear in-plane strains. It is observed that this strain coupling formulation includes some of the transversal (out-of-plane) shear strain terms, thus also functioning as a path for transferring transversal shear energy to the in-plane strain field and therefore improving shear-deformability. The performance of the new CTWB model is compared against that of previously available CWTB (i.e. Euler-type with strain coupling and Timoshenko-type without strain coupling) for several aspect ratios, fibre-orientations and laminate types. Error measures are calculated by comparing several relevant stiffness coefficients and displacement shapes to reference results provided by corresponding 3D shell-based ANSYS finite-element models. Results indicate that for cases involving significant shear energy (i.e. short aspect ratios) and/or in-plane shear-axial strain coupling (i.e. off-axis or asymmetric/unbalanced laminates), the new CTWB model proposed in this work can attain an accuracy level comparable to that associated to more sophisticated models, two to three orders of magnitude larger, at a fraction of the computational cost.

Galerkin formulations of isogeometric shell analysis: Alleviating locking with Greville quadratures and higher-order elements

We propose new quadrature schemes that asymptotically require only four in-plane points for Reissner–Mindlin shell elements and nine in-plane points for Kirchhoff–Love shell elements in B-spline and NURBS-based isogeometric shell analysis, independent of the polynomial degree p of the elements. The quadrature points are Greville abscissae associated with pth-order B-spline basis functions whose continuities depend on the specific Galerkin formulations, and the quadrature weights are calculated by solving a linear moment fitting problem in each parametric direction. The proposed shell element formulations are shown through numerical studies to be rank sufficient and to be free of spurious modes. The studies reveal comparable accuracy, in terms of both displacement and stress, compared with fully integrated spline-based shell elements, while at the same time reducing storage and computational cost associated with forming element stiffness and mass matrices and force vectors. The high accuracy with low computational cost makes the proposed quadratures along with higher-order spline bases, in particular polynomial orders, p=5 and 6, good choices for alleviating membrane and shear locking in shells.

Bending waves in a semi-infinite piezoelectric plate with edge coated by a metal strip plate

This work considers the propagation of bending waves along the free edge of a semi-infinite piezoelectric plate perfectly bonded with a metal strip plate of the same thickness within the framework of the first-order Reissner–Mindlin refined plate theory. The three-dimensional coupled equations for displacement fields are solved with the help of the solution to electric potential for a plate of infinite extent. The exact dispersion relation of bending edge waves is obtained. The propagation of bending edge waves in such a composite plate demonstrates multi-mode due to the presence of the metal strip plate. This wave has properties analogous to the surface wave in an elastic half-space covered by a thin layer within the theory of plane strain. The number of propagation modes depends on the width–thickness ratio of the metal plate and increases as this width–thickness ratio increases. The phase velocity of the bending edge wave in a piezoelectric composite plate is much lower than that of pure piezoelectric plate. Effect of the dimension of the metal strip plate on dispersion curves of bending edge waves is limited.

Efficient Theoretical and Numerical Methods for Solving Free Vibrations and Transient Responses of a Circular Plate Coupled With Fluid Subjected to Impact Loadings

Abstract This paper presents an easy-to-use theoretical method and an efficient numerical method for solving free vibrations and transient responses of a circular plate coupled with fluid subjected to impact loadings and provides insights into various coupling cases with these developed methods. The Kirchhoff plate theory, Mindlin–Reissner plate theory, and the linear velocity potential function are used. The wet mode of the coupled system is described as the superposition of dry modes of the plate, which has been considered in few studies. The natural frequencies and corresponding mode shapes are solved using the orthogonality of dry modes. The transient responses of the plate are then solved using the superposition of the wet modes and the orthogonality of dry modes. To validate the theoretical results, an efficient and flexible finite element method is proposed and verified by comparing with commercial software. The four-node mixed interpolation of the tensorial component quadrilateral plate finite element (MITC4) and the eight-node acoustic pressure element are used to model the plate and the fluid, respectively. The theoretical and numerical methods provide reliable and accurate results. Parametric studies are performed to investigate the influence of geometric sizes, plate material properties, and fluid properties on the natural frequencies of the coupled system. A coupling parameter of fluid–structure interaction is proposed. The nondimensional added virtual mass incremental (NAVMI) factor decreases as the coupling parameter increases. Besides, the influence of fluid on wet modes of the plate decreases with the order.