Kirchhoff Thin Plate Research Articles

Small-scale indentation of an elastic coated half-space: The effect of compliant substrate

Under consideration in this paper is the axisymmetric problem of unilateral frictionless indentation of a thin stiff film bonded to a compliant substrate, which is assumed to be a homogeneous isotropic elastic half-space. The previously developed first-order asymptotic model for the small-scale indentation, when the contact radius is small compared to the film thickness, is compared with approximate solutions based on the Kirchhoff thin plate theory with satisfactory results.

Axisymmetric time-harmonic response of a surface-stiffened transversely isotropic half-space

This study deals with the elastodynamic response of a surface-stiffened transversely isotropic half-space subjected to a buried time-harmonic normal load. The half-space is reinforced by a Kirchhoff thin plate on its surface. By virtue of a displacement potential function and appropriate time-harmonic Green’s functions of transversely isotropic half-spaces, a robust solution corresponding to two plate-medium bonding assumptions, namely (a) frictionless interface, and (b) perfectly bonded interface is obtained for the first time. All elastic fields of the problem are expressed explicitly in the form of semi-infinite line integrals. Results of some limiting cases including isotropic materials, static loading, and surface loading are recovered from the obtained solutions and subsequently have been verified with those available in the literature. Effects of anisotropy, depth of loading, bonding assumption, and frequency of excitation on the results are precisely discussed. Based on the proposed numerical scheme, some plots of practical importance are depicted.

Stress intensity factors of a rhombic hole with symmetric cracks under uniform transverse thin plate bending

A thin plate bending problem (Kirchhoff thin plate) is solved analytically for an infinite plate with a rhombic hole with symmetric cracks at the both corners. The loading is a uniform transverse bending at infinity. The stress distribution and the stress intensity factors of the bending are obtained for a range of corner angles and crack lengths of the rhombic hole. Approximate expressions of the stress intensity factor are proposed for some corner angles and Poisson’s ratios using the stress distribution expressed by the roots of the characteristic equation of the sharp notched plate before the crack initiation. The precision of the expressions are investigated. Characteristic factors for the approximate expressions are stated. A rational mapping function of a sum of fractional expressions and complex stress functions are used for the stress analysis.

Stability of Axially Moving Piezolaminated Viscoelastic Plate Subjected to Follower Force

The stability of the moving viscoelastic plate with the piezoelectric layer subjected to uniformly distributed tangential follower force is investigated. The force excited by the piezoelectric layer due to external voltage is modeled as the follower tensile force. The differential equation of the axially moving viscoelastic rectangular plate with piezoelectric layer subjected to uniformly distributed tangential follower force is formulated on the basis of the Kirchhoff thin plate theory and the two-dimensional viscoelastic differential constitutive relation. The complex eigenvalue equations are established by the differential quadrature method. Via numerical calculation, the curves of real parts and imaginary parts of dimensionless complex frequencies versus uniformly distributed tangential follower force and dimensionless moving speed are obtained. The effects of nonconservative force, dimensionless axially moving speed, and dimensionless applied voltages on the stability of axially moving nonconservative viscoelastic plate with piezoelectric layer are analyzed.

Dynamic responses of a saturated poroelastic half-space generated by a moving truck on the uneven pavement

A truck–pavement–ground coupling model was established to study the dynamic responses of a saturated poroelastic half-space generated by a moving heavy truck on the uneven pavement. The ground was simulated as a fully saturated poroelastic half-space governed by Biot’s theory. The overlying pavement was simplified as a Kirchhoff thin plate. With the assumption of a sinusoidal pavement surface, the dynamic wheel–pavement force was obtained through a linear Hertizian contact model. The numerical results showed that this dynamic load could make considerable contributions to the stress and excess pore water pressure responses in the ground. Furthermore, the effective stress path of the soil unit beneath the pavement caused by the moving truck was firstly calculated and presented. It was found that the differences between the total stress path and the effective stress path became significant as the truck speed increased, thus the effective stress path was more suitable than total stress path to reflect the stress history of soil elements in the saturated ground during the passage of high-speed traffics.

Hydroelastic response of a floating runway to cnoidal waves

The hydroelastic response of mat-type Very Large Floating Structures (VLFSs) to severe sea conditions, such as tsunamis and hurricanes, must be assessed for safety and survivability. An efficient and robust nonlinear hydroelastic model is required to predict accurately the motion of and the dynamic loads on a VLFS due to such large waves. We develop a nonlinear theory to predict the hydroelastic response of a VLFS in the presence of cnoidal waves and compare the predictions with the linear theory that is also developed here. This hydroelastic problem is formulated by directly coupling the structure with the fluid, by use of the Level I Green-Naghdi theory for the fluid motion and the Kirchhoff thin plate theory for the runway. The coupled fluid structure system, together with the appropriate jump conditions are solved in two-dimensions by the finite-difference method. The numerical model is used to study the nonlinear response of a VLFS to storm waves which are modeled by use of the cnoidal-wave theory. Parametric studies show that the nonlinearity of the waves is very important in accurately predicting the dynamic bending moment and wave run-up on a VLFS in high seas.

Decomposition of in-plane shear in thin-walled members

In this paper in-plane (membrane) shear deformations in thin-walled members are discussed. Unlike many earlier works on the effect of shear deformations, the actual work is based on shell model rather than beam model. Thus, the presented work has no intention to develop a new shear-deformable beam theory, by adding some assumed shear deformations to classical beam degrees of freedom, but the actual research considers the thin-walled member as a set of connected flat plates, the displacement field of which is determined by nodal displacements and shape functions. The goal here is to make a full and meaningful modal decomposition within the given displacement field, i.e., to determine sub-fields within the general displacement field so that deformations within the sub-fields would correspond to some desired buckling or deformation modes. The presented work is therefore the direct extension of constraint finite strip method (cFSM), however, this time having a special focus on the shear modes. Though the presented shear modes are initiated by cFSM, and the presentation utilizes the terminology of cFSM, it is to highlight here that the proposed modal decomposition of in-plane shear deformations is not limited to buckling analysis or to the finite strip method. Indeed, the proposed shear space decomposition can be used in any case when the analysed member is thin-walled, built up from flat plates, is modelled by shell-type elements, and the member is prismatic. Since the flexural behaviour of the plates is assumed to follow Kirchhoff thin plate theory, no out-of-plane shear is considered.

Development of quadrilateral spline thin plate elements using the B-net method

The quadrilateral discrete Kirchhoff thin plate bending element DKQ is based on the isoparametric element Q8, however, the accuracy of the isoparametric quadrilateral elements will drop significantly due to mesh distortions. In a previouswork, we constructed an 8-node quadrilateral spline element L8 using the triangular area coordinates and the B-net method, which can be insensitive to mesh distortions and possess the second order completeness in the Cartesian coordinates. In this paper, a thin plate spline element is developed based on the spline element L8 and the refined technique. Numerical examples show that the present element indeed possesses higher accuracy than the DKQ element for distorted meshes.

Numerical manifold space of Hermitian form and application to Kirchhoff's thin plate problems

SUMMARYFor second‐order problems, where the behavior is described by second‐order partial differential equations, the numerical manifold method (NMM) has gained great success. Because of difficulties in the construction of the H 2‐regular Lagrangian partition of unity subordinate to the finite element cover; however, few applications of the NMM have been found to fourth‐order problems such as Kirchhoff's thin plate problems. Parallel to the finite element methods, this study constructs the numerical manifold space of the Hermitian form to solve fourth‐order problems. From the minimum potential principle, meanwhile, the mixed primal formulation and the penalized formulation fitted to the NMM for Kirchhoff's thin plate problems are derived. The typical examples indicate that by the proposed procedures, even those earliest developed elements in the finite element history, such as Zienkiewicz's plate element, regain their vigor. Copyright © 2013 John Wiley & Sons, Ltd.

A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.

A geophysical application of an elastostatic contact problem

This paper develops an analytical solution to the problem of the interaction between an elastic sealing geologic layer that is embedded within elastically dissimilar geologic media of semi-infinite extent. The axisymmetric mechanics of the elastic sealing layer during the application of pressurization of circular regions located in both halfspace regions is evaluated using an interaction analysis based on the classical theory of elasticity. In particular, the flexural behavior of the elastic sealing layer is modeled as an elastic plate that satisfies the Germain–Poisson–Kirchhoff thin plate theory. Integral expressions are developed for the deflections and stresses in the sealing layer, which are essential for establishing its sealing integrity. Specific numerical results are presented for the case where a single halfspace region is subjected to pressurization over a circular area of finite thickness.

Simplified description of out-of-plane waves in thin annular elastic plates

Dispersion relations are derived for the out-of-plane wave propagation in planar elastic plates with constant curvature using the classical Kirchhoff thin plate theory. The dispersion diagrams and the mode shapes are compared with their counterparts for a straight plate strip and the role of curvature is assessed for plates with unconstrained edges. Elementary Bernoulli–Euler theory for a beam of rectangular cross-section with the circular shape of its axis is also employed to analyze the wave guide properties of this structure in its out-of-plane deformation. The applicability range of the elementary beam theory is validated. The wave finite element method in the formulation of the three-dimensional elasticity theory is used to ensure that the comparison of dispersion diagrams is performed in the frequency range, where the classical thin plate theory is valid. Thus, the paper summarizes the effects brought to the propagation of out-of-plane waves in thin elastic plates by their constant curvature and the models of these plates.

Exact Relationships between Classical and Sinusoidal Theories for FGM Plates

Exact relationships between the bending solution of the classical (Kirchhoff) thin plate theory and the sinusoidal shear deformation thick plate theory for homogeneous and functionally graded material plates are derived by using the mathematical similarity of governing equations of the two theories, and the basis of load equivalence. It is assumed in the analysis that the mechanical properties of the functionally graded plates vary continuously through the thickness of the plate and obey a simple power law distribution of the volume fraction of the constituents. Effects of material gradient property and shear deformation on the bending of functionally graded plates are discussed in the framework of the sinusoidal shear deformation plate theory. The boundary conditions at two edges of the functionally graded plate are simply supported while the remaining two edges having the same boundary conditions are, namely, simply supported, clamped, or free. Several examples are presented to illustrate the use and accuracy of these relationships. The effects of many parameters on the deflections and stresses are investigated.

Green’s functions of a surface-stiffened transversely isotropic half-space

Green’s functions of a transversely isotropic half-space overlaid by a thin coating layer are analytically obtained. The surface coating is modeled by a Kirchhoff thin plate perfectly bonded to the half-space. With the aid of superposition technique and employing appropriate displacement potential functions, the Green’s functions are expressed in two parts; (i) a closed-form part corresponding to the transversely isotropic half-space with surface kinematic constraints, and (ii) a numerically evaluated part reflecting the interaction between the half-space and the plate in the form of semi-infinite integrals. Some limiting cases of the problem such as surface-stiffened isotropic half-space, Boussinesq and Cerruti loadings, and extremely flexible and rigid plates are also studied. For the classical Cerruti problem in transversely isotropic materials, the effects of incompressibility are highlighted. Numerical results are provided to show the effects of material anisotropy, relative stiffness factor, and load buried depth. The obtained Green’s functions play a key role in treating further mixed-boundary-value problems in surface stiffened transversely isotropic half-spaces.

Dynamic stability of functionally graded materials skew plates subjected to uniformly distributed tangential follower forces

The dynamic stability of functionally graded material (FGM) skew thin plate subjected to uniformly distributed tangential follower force is investigated. The material properties are assumed to vary continuously through their thickness according to a power-law distribution of the volume fractions of the plate constituents based on the Voigt model. In skew coordinate system, the differential equations of motion of the FGM skew plate subjected to uniformly distributed tangential follower force are derived by the Kirchhoff thin plate theory, and the different boundary conditions are obtained of the plate for arbitrary curve edges. By eliminating the in-plane displacement components on the neutral plane, the differential equations of motion can be expressed in terms of deflection only. Then the equations are discretized by the differential quadrature method, and the curves of real parts and imaginary parts of the first second-order dimensionless complex frequencies vs. uniformly distributed tangential follower force are obtained. The effects of the gradient index, skew angle and aspect ratio on the instability type and the corresponding critical load of the non-conservation FGM skew plate are analyzed.

Effect of temperature on pull-in voltage and natural frequency of an electrostatically actuated microplate

In this paper a model based on distributed parameters has been presented to study temperature effects on the mechanical behavior of an electrostatically actuated microplate. This work consists of two major parts. First part deals with the effect of temperature, stretching and residual stresses on the static instability of an electrostatically actuated microplate. To do this, the governing nonlinear integro differential equation has been derived using Kirchhoff thin plate theory and linearized using step-by-step linearization method (SSLM). The obtained linearized differential equation has been discretized applying finite difference method (FDM). The results obtained have been compared to other existing experimental results and good agreement is observed. In the second part, in order to study the natural or eigenfrequencies of the system, small vibrations of the electrostatically deflected microplate about the equilibrium position have been studied. Here, the governing linear eigenvalue partial differential equation has been solved using a Galerkin based reduced-order model, and the natural frequencies of the microplate have been determined. The results show that temperature changes and residual stress have considerable effects on the system characteristics such as pull-in voltage and natural frequencies.

Closed Form Expression for the Vibration Problem of a Transversely Isotropic Magneto-Electro-Elastic Plate

A closed form expression for the transverse vibration of a magnetoelectroelastic (MEE) thin plate is derived, and the exact solution for the free vibration of a two-layered BaTiO3–CoFe2O4 composite is obtained. Based on the Kirchhoff thin plate theory, the bending problem of a transversely isotropic MEE rectangular plate is investigated, and the governing equation in terms of the transverse displacement is then presented in a rather compact form. The material coefficients for such MEE plate are expressed uniquely by the volume fraction (vf) of the two-layered BaTiO3–CoFe2O4 composite, which indicates a transversely isotropic MEE medium. The natural frequencies of such MEE plate are evaluated analytically, and the effects of different volume fractions on the natural frequency are further discussed.

A COMPLETE STABILITY THEORY FOR THE KIRCHHOFF THIN PLATE UNDER ALL KINDS OF ACTIONS

A complete stabi1ity theory for a plate can be constructed by an incremental virtual work equation describing instability effects induced by a1l kinds of actions. Besides, this incremental virtual work equation should satisfy the rigid body rule, i.e., it should objectively obey the rigid body rule no matter what coordinates systems are adopted. In this paper, a complete nonlinear stability theory for the Kirchhoff thin plate is proposed by using the principle of virtual work and the update Lagrangian formulation. Then, a rigid body motion testing method is developed for examining the incremental virtual work equation. In developing such a theory, three key procedures are especially considered. First of all, the virtual strain energy contributed from all six nonlinear strain components are clearly identified and then, two actions on the effective transverse edge per unit length, namely the Kirchhoff's forces and moment per unit length in the current1y deformed configuration (2C state) are especially considered here in contrast to be ignored in previous literatures. Finally, nonlinear terms of the virtual work done by boundary moments per unit length in the 2C state are also derived. Advantages of this new theory not only come from passing the rigid body rule, which is seldom found in the nonlinear theory of the plate, but also owing to the completeness of the proposed theory.

Diffraction of flexural waves by cracks in orthotropic thin elastic plates. Part II. Far field analysis

The scattered field arising from diffraction of a plane flexural wave by a semi-infinite crack in an orthotropic Kirchhoff thin plate is analysed. The crack is aligned with a principal direction of the material, so that two of the plate's three planes of symmetry are preserved. An asymptotic approximation is derived via the method of steepest descents, and explicit expressions are given for the most significant contributions. The effects of anisotropy upon the scattered field are made clear, and numerical results are presented for several typical engineering materials.

Hp-Version a priori Error Analysis of Interior Penalty Discontinuous Galerkin Finite Element Approximations to the Biharmonic Equation

We consider the symmetric formulation of the interior penalty discontinuous Galerkin finite element method for the numerical solution of the biharmonic equation with Dirichlet boundary conditions in a bounded polyhedral domain in $$\mathbb{R}^d, d \geqslant 2$$ . For a shape-regular family of meshes consisting of parallelepipeds, we derive hp-version a priori bounds on the global error measured in the L2 norm and in broken Sobolev norms. Using these, we obtain hp-version bounds on the error in linear functionals of the solution. The bounds are optimal with respect to the mesh size h and suboptimal with respect to the degree of the piecewise polynomial approximation p. The theoretical results are confirmed by numerical experiments, and some practical applications in Poisson---Kirchhoff thin plate theory are presented.