Kirchhoff-Love Theory Research Articles

Construction of fundamental solution of static equations of medium-thickness isotropic plates

A fundamental solution of the elasticity theory equations for isotropic plates was obtained. To construct the two-dimensional elasticity theory equations, the approximation method of displacements, stresses and strains using Fourier series by Legendre polynomials on the transverse coordinate was used. This approach has allowed to take into account the transverse shear and normal stresses. Since the classical Kirchhoff-Love theory does not consider these stresses, the research based on the refined theories of the stress-strain state of isotropic plates under concentrated force actions is an urgent scientific and technical problem. The fundamental solution of these equations was found using the two-dimensional Fourier integral transform and the generalization method, built with a special G-function. The method allows to reduce the system of resolvent differential static equations of flat plates and shells to a system of algebraic equations. Then the inverse Fourier transform restores fundamental solution. Numerical studies that demonstrate behavior patterns of the stress-strain state components depending on the elastic constants of isotropic material were performed. The approach demonstrates the development of the refined theory of plates and shells based on the three-dimensional elasticity theory.

Asymptotic modeling of Signorini problem with Coulomb friction for a linearly elastostatic shallow shell

In 2002–2003, Paumier studied the Signorini problem with friction in the linear Kirchhoff–Love theory of plates using the convergence method. In 2008, Léger and Miara generalized this study to the case of linearized shallow shell but without friction. The purpose of this paper is to extend those results to the case of linearized shallow shell with a Coulomb friction law. Copyright © 2015 John Wiley & Sons, Ltd.

Nonlinear dynamic response analysis of localized damaged laminated composite structures in the context of component mode synthesis

The aim of this study is the prediction of the dynamic response of damaged laminated composite structures in the context of component mode synthesis. Hence, a method of damage localization of complex structures is proposed. The dynamic behavior of transversely isotropic layers is expressed through elasticity coupled with damage based on an existing macro model for cracked structures. The damage is located only in some regions of the whole structure, which is decomposed on substructures. The incremental linear dynamic governing equations are obtained by using the classical linear Kirchhoff-Love theory of plates. Then, considering the damage-induced nonlinearity, the obtained nonlinear dynamic equations are solved in time domain. However, a detailed finite element modelling of such structure on the scale of localized damage would generate very high computational costs. To reduce this cost, Component Mode Synthesis method (CMS) is used for modelling a nonlinear fine-scale substructure damaged, connected to linear dynamic models of the remaining substructures, which can be condensed and not updated at each iteration. Numerical results show that the mechanical properties of the structure highly change when damage is taken into account. Under an impact load, damage increases and reaches its highest value with the maximum of the applied load and then remains unchanged. Besides, the eigenfrequencies of the damaged structure decrease comparing with those of an undamaged one. This methodology can be used for monitoring strategies and lifetime estimations of hybrid complex structures due to the damage state is known in space and time.

Weak Dirichlet boundary conditions for trimmed thin isogeometric shells

Computer-aided design-based NURBS surfaces form the basis of isogeometric shell analysis which exploits the smoothness and higher continuity properties of NURBS to derive a suitable analysis model in an isoparametric sense. Equipped with higher order approximation capabilities the used NURBS functions focus increasingly on rotation-free shell elements which are considered to be difficult in the traditional finite element framework. The rotation-free formulation of shell elements is elegant and efficient but demands special care to enforce reliably essential translational and rotational boundary conditions which is even more challenging in the case of trimmed boundaries as common in CAD models. We propose a Nitsche-based extension of the Kirchhoff–Love theory to enforce weakly essential boundary conditions of the shell. We apply our method to trimmed and untrimmed NURBS structures and illustrate a good performance of the method with benchmark test models and a shell model from engineering practice. With an extension of the formulation to a weak enforcement of coupling constraints we are able to handle CAD-derived trimmed multi-patch NURBS models for thin shell structures.

Elastic Properties of the Annular Ligament of the Human Stapes--AFM Measurement.

Elastic properties of the human stapes annular ligament were determined in the physiological range of the ligament deflection using atomic force microscopy and temporal bone specimens. The annular ligament stiffness was determined based on the experimental load-deflection curves. The elastic modulus (Young's modulus) for a simplified geometry was calculated using the Kirchhoff-Love theory for thin plates. The results obtained in this study showed that the annular ligament is a linear elastic material up to deflections of about 100 nm, with a stiffness of about 120 N/m and a calculated elastic modulus of about 1.1 MPa. These parameters can be used in numerical and physical models of the middle and/or inner ear.

Comparison of methods to study the slab bending on multilayered foundations

An analytical method to solve the bending problem for an elastic slab resting on an elastic multilayered half-space is considered. This method is based on integral transforms and is compared with a finite element method. The slab is modeled according to the Kirchhoff—Love theory of thin plates. A number of numerical results are discussed for the case when the elasticity theory equations are used to model the slab behavior and are compared with the solution obtained on the basis of the thin plate theory. A computer program is developed to analyze the stress-strairi state of structures considered in this paper.

A rectangular shell element formulation with a new multi-resolution analysis

A multi-resolution rectangular shell element with membrane-bending based on the Kirchhoff-Love theory is proposed. The multi-resolution analysis (MRA) framework is formulated out of a mutually nesting displacement subspace sequence, whose basis functions are constructed of scaling and shifting on the element domain of basic node shape functions. The basic node shape functions are constructed from shifting to other three quadrants around a specific node of a basic element in one quadrant and joining the corresponding node shape functions of four elements at the specific node. The MRA endows the proposed element with the resolution level (RL) to adjust the element node number, thus modulating structural analysis accuracy accordingly. The node shape functions of Kronecker delta property make the treatment of element boundary condition quite convenient and enable the stiffness matrix and the loading column vectors of the proposed element to be automatically acquired through quadraturing around nodes in RL adjusting. As a result, the traditional 4-node rectangular shell element is a mono-resolution one and also a special case of the proposed element. The accuracy of a structural analysis is actually determined by the RL, not by the mesh. The simplicity and clarity of node shape function formulation with the Kronecker delta property, and the rational MRA enable the proposed element method to be implemented more rationally, easily and efficiently than the conventional mono-resolution rectangular shell element method or other corresponding MRA methods.

An extended isogeometric thin shell analysis based on Kirchhoff–Love theory

An extended isogeometric element formulation (XIGA) for analysis of through-the-thickness cracks in thin shell structures is developed. The discretization is based on Non-Uniform Rational B-Splines (NURBS). The proposed XIGA formulation can reproduce the singular field near the crack tip and the discontinuities across the crack. It is based on the Kirchhoff–Love theory where C1-continuity of the displacement field is required. This condition is satisfied by the NURBS basis functions. Hence, the formulation eliminates the need of rotational degrees of freedom or the discretization of the director field facilitating the enrichment strategy. The performance and validity of the formulation is tested by several benchmark examples.

Photoacoustic Signal Formation in Heterogeneous Multilayer Systems with Piezoelectric Detection

A new efficient model describing photoacoustic (PA) signal formation with piezoelectric detection is reported. Multilayer sandwich-like systems: heterogeneous studied structure—buffer layer—piezoelectric transducers are considered. In these systems, the buffer layer is used for spatial redistribution of thermoelastic force moments generated in the investigated structure. Thus, mechanical properties of this layer play a crucial role to ensure perfect control of the detected voltage formed on a piezoelectric transducer by contribution of different regions of the studied structure. In particular, formation of the voltage signal strongly depends on the point at which the thermoelastic source is applied. Therefore, use of relatively simple linear Green’s functions introduced in frames of the Kirchhoff–Love theory is chosen as an efficient approach for the PA signal description. Moreover, excellent agreement between the theoretical model and measured results obtained on a heterogeneous “porous silicon-bulk Si substrate” structure is stated. Furthermore, resolving of the inverse problem with fitting of the experimental curves by the developed model allows reliable evaluation of the thermal conductivity of the nanostructured porous silicon layer.

Singular Shear-Force States in Elementary Plate Theory

We show that the most classical Kirchhoff-Love theory of thin plates is compatible with the occurrence of a specific singular shear-force state in the interior of the body. It is well-known from Kirchhoff that, on the edge boundary of the plate, the specific shear-forces and the curve-gradient of the specific twisting-moments, measured per unit length, are statically inter-related. We observe and prove that a similar static equivalence holds for the edge boundary of any sub-body, and this allows many interpretations of the contact interactions that may take place between the parts of the plate. In particular, a specific shear-force acting on a smooth part of the edge boundary of a sub-body may depend upon its curvature, tending to a concentrated force at a sharp corner. The possibility of developing concentrated contact interactions is a general characteristic of non-simple continua, of which the theory of thin plates is but one representative example.

Coupling effects in elastic analysis of FGM composite plates by mesh-free methods

In this paper we shall investigate the static response of thin and/or thick elastic functionally graded (FG) plates. The spatial variation of material coefficients in the FG composite structures is determined by distribution of volume fractions of particular constituents. The attention is devoted to derivation of the special formulation of governing equations in FGM plates, which includes the Kirchhoff–Love theory (KLT) as well as the 1st and 3rd order shear deformation plate theory (SDPT). The power-law gradations across the plate thickness and along in-plane coordinates yield two different problems. To facilitate the numerical solution of rather complex governing equations, we propose the strong formulation combined with Moving Least Square (MLS) approximation for field variables. Several numerical examples are presented to investigate the accuracy, convergence of accuracy and computational efficiency of studied mesh-free formulation for boundary value problems with existing benchmark solutions. The coupling effects are studied via the response of FGM square plate to static loading within the KLT as well as the 1st and 3rd order SDPT.

Explicit finite deformation analysis of isogeometric membranes

NURBS-based isogeometric analysis was first extended to thin shell/membrane structures which allows for finite membrane stretching as well as large deflection and bending strain. The assumed non-linear kinematics employs the Kirchhoff–Love shell theory to describe the mechanical behaviour of thin to ultra-thin structures. The displacement fields are interpolated from the displacements of control points only, and no rotational degrees of freedom are used at control points. Due to the high order Ck (k≥1) continuity of NURBS shape functions the Kirchhoff–Love theory can be seamlessly implemented. An explicit time integration scheme is used to compute the transient response of membrane structures to time-domain excitations, and a dynamic relaxation method is employed to obtain steady-state solutions. The versatility and good performance of the present formulation are demonstrated with the aid of a number of test cases, including a square membrane strip under static pressure, the inflation of a spherical shell under internal pressure, the inflation of a square airbag and the inflation of a rubber balloon. The mechanical contribution of the bending stiffness is also evaluated.

Подкрепления в вариационно-разностном методе расчета оболочек сложной формы

We have already considered an introduction of reinforcements in the variational-difference method (VDM) of shells analysis with complex shape. At the moment only ribbed shells of revolution and shallow shells can be calculated with the help of developed analytical and finite-difference methods. Ribbed shells of arbitrary shape can be calculated only using the finite element method (FEM). However there are problems, when using FEM, which are absent in finiteand variational-difference methods: rigid body motion; conforming trial functions; parameterization of a surface; independent stress strain state. In this regard stiffeners are entered in VDM. VDM is based on the Lagrange principle — the principle of minimum total potential energy. Stress-strain state of ribs is described by the Kirchhoff-Clebsch theory of curvilinear bars: tension, bending and torsion of ribs are taken into account. Stressstrain state of shells is described by the Kirchhoff-Love theory of thin elastic shells. A position of points of the middle surface is defined by curvilinear orthogonal coordinates α, β. Curved ribs are situated along coordinate lines. Strain energy of ribs is added into the strain energy to account for ribs. A matrix form of strain energy of ribs is formed similar to a matrix form of the strain energy of the shell. A matrix of geometrical characteristics of a rib is formed from components of matrices of geometric characteristics of a shell. A matrix of mechanical characteristics of a rib contains rib’s eccentricity and geometrical characteristics of a rib’s section. Derivatives of displacements in the strain vector are replaced with finite-difference relations after the middle surface of a shell gets covered with a grid (grid lines coincide with the coordinate lines of principal curvatures). By this case the total potential energy functional becomes a function of strain nodal displacements. Partial derivatives of unknown nodal displacements are equated to zero in order to minimize the total potential energy. As an example a parabolic-sinusoidal shell with a stiffened hole is analyzed. It is shown that ribs have generally beneficial effect to the zone of the opening: cause a reduction in a modulus of a stress, but an eccentricity affects differently, so material properties and design solutions should be taken into account in an analysis.

Buckling phenomena in double curved cold-bent glass

Double curved anticlastic glazed surfaces are widely used for free-form façades and roofs of modern buildings. An effective technique consists in cold bending rectangular glass plies by twisting them with forces applied at the corners. The linear Kirchhoff–Love theory predicts that the deformed shape is a hyperbolic paraboloid, which preserves the straightness of the edges. However, experiments have provided evidence that a particular form of instability occurs above a certain limit of the distortion: one of the principal curvatures becomes dominant with respect to the other, the plate bulges into an asymmetric configuration and the edges are not any more straight. Here, a simple model is presented that, using a modified version of Mansfield׳s inextensional theory for thin plates, is able to interpret this phenomenon. Results are in good agreement with numerical experiments using large deflection theory. Moreover, the possibility of increasing the limit of the stable configuration by stiffening the edges is investigated.

Modeling orchestral crotales as thin plates

Orchestral crotales are designed in such a way that the overtones become less harmonic as the fundamental pitch increases. Deutsch, et al. used Kirchhoff-Love theory to show that by eliminating the central mass and choosing the correct ratio of inner to outer radius the overtones can be harmonically related (J. Acoust. Soc. Am. 116, 2427 (2004)). However, when a crotale was constructed using this design, the overtones were not harmonically related. We show that this lack of agreement between theory and experiment is due to the fact that shear motion of the inner boundary, which is neglected in classical thin-plate theory and thought to be unimportant, can significantly affect the resonance frequencies of plates even when they are extremely thin.

Subdivision shell elements with anisotropic growth

SUMMARYA thin shell finite element approach based on Loop's subdivision surfaces is proposed, capable of dealing with large deformations and anisotropic growth. To this end, the Kirchhoff–Love theory of thin shells is derived and extended to allow for arbitrary in‐plane growth. The simplicity and computational efficiency of the subdivision thin shell elements is outstanding, which is demonstrated on a few standard loading benchmarks. With this powerful tool at hand, we demonstrate the broad range of possible applications by numerical solution of several growth scenarios, ranging from the uniform growth of a sphere, to boundary instabilities induced by large anisotropic growth. Finally, it is shown that the problem of a slowly and uniformly growing sheet confined in a fixed hollow sphere is equivalent to the inverse process where a sheet of fixed size is slowly crumpled in a shrinking hollow sphere in the frictionless, quasistatic, elastic limit. Copyright © 2013 John Wiley & Sons, Ltd.

An asymptotic method for deriving the equations of the Reissner-Mindlin plate theory

The equations of the Kirchhoff-Love and Reissner-Mindlin plate theories are derived with the use of the asymptotic homogenization method.

A High Order Theory for Linear Thermoelastic Shells: Comparison with Classical Theories

A high order theory for linear thermoelasticity and heat conductivity of shells has been developed. The proposed theory is based on expansion of the 3-D equations of theory of thermoelasticity and heat conductivity into Fourier series in terms of Legendre polynomials. The first physical quantities that describe thermodynamic state have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby all equations of elasticity and heat conductivity including generalized Hooke's and Fourier's laws have been transformed to the corresponding equations for coefficients of the polynomial expansion. Then in the same way as in the 3D theories system of differential equations in terms of displacements and boundary conditions for Fourier coefficients has been obtained. First approximation theory is considered in more detail. The obtained equations for the first approximation theory are compared with the corresponding equations for Timoshenko's and Kirchhoff-Love's theories. Special case of plates and cylindrical shell is also considered, and corresponding equations in displacements are presented.

Blended isogeometric shells

We propose a new isogeometric shell formulation that blends Kirchhoff–Love theory with Reissner–Mindlin theory. This enables us to reduce the size of equation systems by eliminating rotational degrees of freedom while simultaneously providing a general and effective treatment of kinematic constraints engendered by shell intersections, folds, boundary conditions, the merging of NURBS patches, etc. We illustrate the blended theory’s performance on a series of test problems.

Nonlinear manifold learning for meshfree finite deformation thin‐shell analysis

SUMMARYCalculations on general point‐set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges. The absence of a mesh makes it nontrivial to decide if two neighboring points in the three‐dimensional embedding are nearby or rather far apart on the manifold. Furthermore, the topology of surfaces is generally not that of an open two‐dimensional set, ruling out global parametrizations. We propose a general and simple numerical method analogous to the mathematical theory of manifolds, in which the point‐set surface is described by a set of overlapping charts forming a complete atlas. We proceed in four steps: (1) partitioning of the node set into subregions of trivial topology; (2) automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods; (3) parametrization of the surface using smooth meshfree (here maximum‐entropy) approximants; and (4) gluing together the patch representations by means of a partition of unity. Each patch may be viewed as a meshfree macro‐element. We exemplify the generality, flexibility, and accuracy of the proposed approach by numerically approximating the geometrically nonlinear Kirchhoff–Love theory of thin‐shells. We analyze standard benchmark tests as well as point‐set surfaces of complex geometry and topology. Copyright © 2012 John Wiley & Sons, Ltd.