Thin Plate Limit Research Articles

Innovative approach to modeling propagative, non-propagative and complex modes of ultrasonic guided waves

In the present paper are developed a new hybrid-spectral approach that deals with the dispersion of propagative, non-propagative and complex ultrasonic guided waves in different types of waveguides. First, a mathematical formulation of the eigenvalue and eigenvector problem has been presented. Then, the hybrid approach based on eigenvector processing is presented. A comparison is then made with the frequency and wavenumber approaches, to highlight the strengths of our method. The dispersion curves of propagative, non-propagative and complex modes are plotted in 2D and 3D. From these representations, several observations on mode behavior have been highlighted, notably the phenomenon of conversion of non-propagative modes into propagative modes, and also conversions specific to imaginary modes. The braiding phenomenon between symmetrical and antisymmetrical modes was also analyzed. Single and multi-layer planar and cylindrical geometries were treated with different degrees of anisotropy. The accuracy of the results was checked by comparison with analytical dispersion curves for planar structures and based on the thin plate limit for cylindrical structures. The error evaluated is of the order of 10 − 5 . A computation time analysis has been carried out, showing that the approach developed is the fastest in comparison with those available in the literature. On the basis of the results obtained, the hybrid approach applied to eigenvalue problems resulting from a spectral formulation presents a very good alternative for obtaining dispersion curves. It is stable, convergent, easy to implement and requires very little computing time and it can generate all types of modes (propagative, non-propagative and complex), in contrast to existing approaches in the reference.

Alpha (α) assumed rotations and shear strains for spatially isotropic polygonal Reissner-Mindlin plate elements (αARS-Poly)

This paper proposes simple and efficient alpha assumed rotations and shear strains for polygonal plate elements, named αARS-Poly. In the αARS approach, an alternative assumption of the tangent rotations along element boundaries is applied by using the approximation of the rotations in Timoshenko's beam theory. Then, the quadratic term of this assumed field is linearly scaled up by adding an artificial positive scaling factor α>0. Through examination of the relative errors in the energy (s-) norm regarding α in numerical experiences, the value α=0.5 can be chosen as a general-fixed value that possibly achieves the optimal relative errors of the s-norm. The value α=0.5 seems to be not only mesh-independent but also problem-independent. The αARS-Poly element using α=0.5 passes all critical tests (spatially isotropic, zero-energy mode, and bending path tests) for a finite plate element which ensures the element orientation-independent property, solution stability, and free shear-locking in the thin plate limit. The implementation of the αARS-Poly element is straightforward through a unified form of the stiffness matrix for all arbitrary convex-shaped polygonal meshes. Numerical results show that the proposed element achieves high reliable and optimal results with uniform and excellent convergent rates in the static and free vibration analyses.

A Hybridized Mixed Approach for Efficient Stress Prediction in a Layerwise Plate Model

Building upon recent works devoted to the development of a stress-based layerwise model for multilayered plates, we explore an alternative finite-element discretization to the conventional displacement-based finite-element method. We rely on a mixed finite-element approach where both stresses and displacements are interpolated. Since conforming stress-based finite-elements ensuring traction continuity are difficult to construct, we consider a hybridization strategy in which traction continuity is relaxed by the introduction of an additional displacement-like Lagrange multiplier defined on the element facets. Such a strategy offers the advantage of uncoupling many degrees of freedom so that static condensation can be performed at the element level, yielding a much smaller final system to solve. Illustrative applications demonstrate that the proposed mixed approach is free from any shear-locking in the thin plate limit and is more accurate than a displacement approach for the same number of degrees of freedom. As a result, this method can be used to capture efficiently strong intra- and inter-laminar stress variations near free-edges or cracks.

First-order VEM for Reissner\u2013Mindlin plates

In this paper, a first-order virtual element method for Reissner–Mindlin plates is presented. A standard displacement-based variational formulation is employed, assuming transverse displacement and rotations as independent variables. In the framework of the first-order virtual element, a piecewise linear approximation is assumed for both displacement and rotations on the boundary of the element. The consistent term of the stiffness matrix is determined assuming uncoupled polynomial approximations for the generalized strains, with different polynomial degrees for bending and shear parts. In order to mitigate shear locking in the thin-plate limit while keeping the element formulation as simple as possible, a selective scheme for the stabilization term of the stiffness matrix is introduced, to indirectly enrich the approximation of the transverse displacement with respect to that of the rotations. Element performance is tested on various numerical examples involving both thin and thick plates and different polygonal meshes.

A new Rayleigh-like wave in guided propagation of antiplane waves in couple stress materials.

Motivated by the unexpected appearance of shear horizontal Rayleigh surface waves, we investigate the mechanics of antiplane wave reflection and propagation in couple stress (CS) elastic materials. Surface waves arise by mode conversion at a free surface, whereby bulk travelling waves trigger inhomogeneous modes. Indeed, Rayleigh waves are perturbations of the travelling mode and stem from its reflection at grazing incidence. As is well known, they correspond to the real zeros of the Rayleigh function. Interestingly, we show that the same generating mechanism sustains a new inhomogeneous wave, corresponding to a purely imaginary zero of the Rayleigh function. This wave emerges from 'reflection' of a bulk standing mode: This produces a new type of Rayleigh-like wave that travels away from, as opposed to along, the free surface, with a speed lower than that of bulk shear waves. Besides, a third complex zero of the Rayleigh function may exist, which represents waves attenuating/exploding both along and away from the surface. Since none of these zeros correspond to leaky waves, a new classification of the Rayleigh zeros is proposed. Furthermore, we extend to CS elasticity Mindlin's boundary conditions, by which partial waves are identified, whose interference lends Rayleigh-Lamb guided waves. Finally, asymptotic analysis in the thin-plate limit provides equivalent one-dimensional models.

A polygonal finite element method for plate analysis

A Reissner-Mindlin plate formulation on arbitrary polygonal meshes is proposed for plate analysis. We consider four barycentric shape function types named Wachspress, mean-value, Laplace and piecewise-linear and show its properties in numerical computation for Reissner-Mindlin plate problems. We then generalize an assumed strain field along sides of polygons under the enforcement of the Timoshenko’s beam assumption. The present approach is numerically verified by the bending patch test. It offers a general yet simple form for Reissner-Mindlin plate elements that is not only implementable for arbitrary polygonal meshes but also avoids transverse shear locking phenomenon at thin plate limit. The performance of the proposed elements is found through numerical examples. Our key contribution to this work is that the present formulation is established in a compact (or unified) form which is valid for triangular, quadrilateral and arbitrary polygonal meshes.

Modal analysis of laminates by a mixed assumed-strain finite element model

Fibre-reinforced plates and shells are finding an increasing interest in engineering applications; in most cases dynamic phenomena need to be taken into account. Consequently, effective and robust computational tools are sought in order to provide reliable results for the analysis of such structural models. In this paper the mixed assumed-strain laminated plate element, previously used for static analyses, has been extended to the dynamic realm. This model is derived within the framework of the so-called First-order Shear Deformation Theory (FSDT). What is peculiar in this assumed-strain finite element is that in-plane strain components are modeled directly; the corresponding stress components are deduced via constitutive law. By enforcing the equilibrium equations for each lamina, and taking continuity requirements into account, the out-of-plane shear stresses are computed and, finally, constitutive law provides the corresponding strains. The resulting global strain field depends only on a fixed number of parameters, regardless of the total number of layers. Since the proposed element is not locking-prone, even in the thin plate limit, and provides an accurate description of inter-laminar stresses, an extension to the dynamic range seems to be particularly attractive. The same kinematic assumptions will lead to the formulation of a consistent mass matrix. The element, developed in this way, has been extensively tested for several symmetric lamination sequences; comparison with available analytical solutions and with numerical results obtained by refined 3-D models are also presented.

Plastic Collapse Analysis of Mindlin–Reissner Plates Using a Stabilized Mesh-Free Method

This paper presents a numerical kinematic formulation for computation of collapse load of Mindlin–Reissner plates, that uses a stabilized mesh-free method in combination with second-order cone programming (SOCP). The kinematic formulation is discretized using a moving least squares approximation combined with a stabilized conforming nodal integration scheme, ensuring that shear locking problem at thin plate limit can be removed. The stabilized mesh-free based kinematic formulation is formulated as a conic problem so that it can be solved by an efficient primal-dual interior-point algorithm. To speed up computational progress, an adaptive refinement scheme using dissipation-based error indicator is also performed. The performance of the proposed numerical procedure is illustrated by examining several plates with arbitrary geometries and various boundary conditions.

Legendre spectral finite elements for Reissner–Mindlin composite plates

Legendre spectral finite elements (LSFEs) are examined in their application to Reissner–Mindlin composite plates for static and dynamic deformation on unstructured grids. LSFEs are high-order Lagrangian-interpolant finite elements whose nodes are located at the Gauss–Lobatto–Legendre quadrature points. Nodal quadrature is employed for mass-matrix calculations, which yields diagonal mass matrices. Full quadrature or mixed-reduced quadrature is used for stiffness-matrix calculations. Solution accuracy is examined in terms of model size, computation time, and memory storage for LSFEs and for quadratic serendipity elements calculated in a commercial finite-element code. Linear systems for both model types were solved with the same sparse-system direct solver. At their best, LSFEs provide many orders of magnitude more accuracy than the quadratic elements for a fixed measure (e.g., computation time). At their worst, LSFEs provide the same accuracy as the quadratic elements for a given measure. The LSFEs were insensitive to shear locking and were shown to be more robust in the thin-plate limit than their low-order counterparts.

Lower bound static approach for the yield design of thick plates

SUMMARY The present work addresses the lower bound limit analysis (or yield design) of thick plates under shear-bending interaction. Equilibrium finite elements are used to discretize the bending moment and the shear force fields. Different strength criteria, formulated in the five-dimensional space of bending moment and shear force, are considered, one of them taking into account the interaction between bending and shear resistances. The criteria are chosen to be sufficiently simple so that the resulting optimization problem can be formulated as a second-order cone programming problem (SOCP), which is solved by the dedicated solver MOSEK. The efficiency of the proposed finite element is illustrated by means of numerical examples on different plate geometries, for which the thin plate solutions as well as the pure shear solutions are accurately obtained as two different limit cases of the plate slenderness ratio. In particular, the proposed element exhibits a good behavior in the thin plate limit. Copyright © 2014 John Wiley & Sons, Ltd.

Electromagnetic semitransparentδ-function plate: Casimir interaction energy between parallel infinitesimally thin plates

We derive boundary conditions for electromagnetic fields on a $\delta$-function plate. The optical properties of such a plate are shown to necessarily be anisotropic in that they only depend on the transverse properties of the plate. We unambiguously obtain the boundary conditions for a perfectly conducting $\delta$-function plate in the limit of infinite dielectric response. We show that a material does not "optically vanish" in the thin-plate limit. The thin-plate limit of a plasma slab of thickness $d$ with plasma frequency $\omega_p^2=\zeta_p/d$ reduces to a $\delta$-function plate for frequencies ($\omega=i\zeta$) satisfying $\zeta d \ll \sqrt{\zeta_p d} \ll 1$. We show that the Casimir interaction energy between two parallel perfectly conducting $\delta$-function plates is the same as that for parallel perfectly conducting slabs. Similarly, we show that the interaction energy between an atom and a perfect electrically conducting $\delta$-function plate is the usual Casimir-Polder energy, which is verified by considering the thin-plate limit of dielectric slabs. The "thick" and "thin" boundary conditions considered by Bordag are found to be identical in the sense that they lead to the same electromagnetic fields.

A locking-free meshfree method for the simulation of shear-deformable plates based on a mixed variational formulation

The problem of shear-locking in the thin-plate limit is a well known issue that must be overcome when discretising the Reissner–Mindlin plate equations. In this paper we present a shear-locking-free method utilising meshfree maximum-entropy basis functions and rotated Raviart–Thomas-Nédélec elements within a mixed variational formulation. The formulation draws upon well known techniques in the finite element literature. Due to the inherent properties of the maximum-entropy basis functions our method allows for the direct imposition of Dirichlet (essential) boundary conditions, in contrast to methods based on moving least squares basis functions. We present benchmark problems that demonstrate the accuracy and performance of the proposed method.

Linear buckling analysis of cracked plates by SFEM and XFEM

In this paper, the linear buckling problem for isotropic plates is studied using a quadrilateral element with smoothed curvatures and the extended finite element method. First, the curvature at each point is obtained by a nonlocal approximation via a smoothing function. This element is later coupled with partition of unity enrichment to simplify the simulation of cracks. The proposed formulation suppresses locking and yields elements which behave very well, even in the thin plate limit. The buckling coefficient and mode shapes of square and rectangular plates are computed as functions of crack length, crack location, and plate thickness. The effects of different boundary conditions are also studied.

Buckling transition and boundary layer in non-Euclidean plates

Non-Euclidean plates are thin elastic bodies having no stress-free configuration, hence exhibiting residual stresses in the absence of external constraints. These bodies are endowed with a three-dimensional reference metric, which may not necessarily be immersible in physical space. Here, based on a recently developed theory for such bodies, we characterize the transition from flat to buckled equilibrium configurations at a critical value of the plate thickness. Depending on the reference metric, the buckling transition may be either continuous or discontinuous. In the infinitely thin plate limit, under the assumption that a limiting configuration exists, we show that the limit is a configuration that minimizes the bending content, among all configurations with zero stretching content (isometric immersions of the midsurface). For small but finite plate thickness, we show the formation of a boundary layer, whose size scales with the square root of the plate thickness and whose shape is determined by a balance between stretching and bending energies.

Strain smoothing in FEM and XFEM

We present in this paper recent achievements realised on the application of strain smoothing in finite elements and propose suitable extensions to problems with discontinuities and singularities. The numerical results indicate that for 2D and 3D continuum, locking can be avoided. New plate and shell formulations that avoid both shear and membrane locking are also briefly reviewed. The principle is then extended to partition of unity enrichment to simplify numerical integration of discontinuous approximations in the extended finite element method. Examples are presented to test the new elements for problems involving cracks in linear elastic continua and cracked plates. In the latter case, the proposed formulation suppresses locking and yields elements which behave very well, even in the thin plate limit. Two important features of the set of elements presented are their insensitivity to mesh distortion and a lower computational cost than standard finite elements for the same accuracy. These elements are easily implemented in existing codes since they only require the modification of the discretized gradient operator, B.

Effect of transverse shear strains on plates resting on elastic foundation using modified Vlasov model

A four-noded quadrilateral (PBQ4) and an eight-noded quadrilateral (PBQ8) plate bending element based on Mindlin plate theory are adopted for the analysis of thin and thick plates resting on elastic foundation using modified Vlasov model. The terms of vertical deflection stiffness matrix and shear deformation stiffness matrix of the subsoil are evaluated using finite element method, and presented in explicit forms. Selective reduced integration technique is used in addition to full integration technique for both the types of the elements to avoid shear-locking problem that occurs under the thin plate limit. It has been demonstrated that the performance of the eight-noded quadrilateral element is excellent for thin and thick plates on elastic foundation when selective-reduced integration technique is used. General conclusion can be drawn from the results that the effect of the shear strain on the behavior of the plate resting on subsoil is always quite small for free plates compare to the supported ones.

A locking-free meshless local Petrov–Galerkin formulation for thick and thin plates

In this paper, a locking-free meshless local Petrov–Galerkin formulation is presented for shear flexible thick plates, which remains theoretically valid in the thin-plate limit. The kinematics of a three-dimensional solid is used, instead of the conventional plate assumption. The local symmetric weak form is derived for cylindrical shaped local sub-domains. The numerical characteristics of the local symmetric weak form, in the thin plate limit, are discussed. Based on this discussion, the shear locking is theoretically eliminated by changing the two dependent variables in the governing equations. The moving least square interpolation is utilized in the in-plane numerical discretization for all the three displacement components. In the thickness direction, on the other hand, a linear interpolation is used for in-plane displacements, while a hierarchical quadratic interpolation is utilized for the transverse displacement, in order to eliminate the thickness locking. Numerical examples in both the thin plate limit and the thick plate limit are presented, and the results are compared with available analytical solutions.

A four-node hybrid assumed-strain finite element for laminated composite plates

Fibre-reinforced plates and shells are find- ing an increasing interest in engineering applications. Consequently, efficient and robust computational tools are required for the analysis of such structural models. As a matter of fact, a large amount of laminate finite el- ements have been developed and incorporated in most commercial codes for structural analysis. In this paper a new laminate hybrid assumed-strain plate element is derived within the framework of the First- order Shear Deformation Theory (i.e. assuming that par- ticles of the plate originally lying along a straight line which is normal to the undeformed middle surface re- main aligned along a straight lineduring the deformation process)and assumingperfect bondingbetween laminae. Thein-planecomponentsofthe(infinitesimal)strain ten- sor are interpolated and by making use of the constitu- tive law, the correspondingin-plane stress distribution is deduced for each layer. Out-of-plane shear stresses are then computed by integrating the equilibrium equations in each lamina, account taken of their continuityrequire- ments. Out-of-planeshear strains are finally obtained via the inverse constitutivelaw. The resulting global strain field depends on a fixed num- ber of parameters, regardless of the total number of lay- ers; 12 degrees of freedom are for instance assumed for the developed rectangular element. The proposed model does not suffer locking phenomena even in the thin plate limit and provides an accurate de- scription of inter-laminar stresses. Results are compared with both analytical and other finite element solutions.

AN UNSYMMETRIC STRESS FORMULATION FOR REISSNER-MINDLIN PLATES: A SIMPLE AND LOCKING-FREE RECTANGULAR ELEMENT

In the present paper a simple mixed-hybrid element for the linear analysis of Reissner-Mindlin plates is discussed. The element is derived from a modified Reissner functional and standard bilinear (isoparametric) interpolation for displacement and rotations is assumed whereas local stresses (rather than stress resultants and moments) are explicitly modelled. It is assumed that in plane shear stresses are not a priori symmetric. This choice allows to decouple the equilibrium equations, and involves introducing an in-plane infinitesimal rotation field, corresponding to drilling degrees of freedom. Out-of-plane shear stresses are then obtained such that equilibrium equations are exactly satisfied. The proposed element does not exhibit locking effects at all: i.e. the shear deformation energy is zero in the thin plate limit. Details of the formulation are provided, and the performances of the element are assessed with reference to well-established benchmark problems.

Generalized mixed variational methods for Reissner plate and its applications

Based on the mechanism of shear locking phenomenon and potential functional of Reissner plate bending problem, the generalized mixed variational principle for Reissner plate analysis is presented by parameterized Lagrange multiplier method. The proposed variational functional contains splitting factors which are able to adjust the shear potential energy and shear complementary energy components in it. The generalized mixed finite element formulation of bilinear quardrilateral element for Reissner plate bending analysis is established in terms of the new variational principle. The stiffness of the finite element model can be changed by the alteration of the splitting factors. Thus both the free of shear locking and higher accuracy are obtained by the choice of appropriate splitting factors. The most important is that this paper gives one self-adaptative way to choose the splitting factors for thin and moderately thick plates. This results in the comparative order of magnitude between the bending stiffness and shear stiffness for the arbitrary thickness. In the application of two-by-two exact Gaussian integration scheme to the proposed mixed element model, numerical examples show that free of locking is obtained even in the thin plate limit and high accuracy is given for moderately thick plate.