Variational Statement Research Articles

Mixed schemes of finite element method for non-standard boundary value problems of the nonlinear theory of thin elastic shells

Variational statements of equilibrium problems for a thin elastic shell within a geometrically nonlinear theory of mean bending for different types of principal boundary conditions, including non-classical, are given. Two classes of shell models are considered: (1) a geometrically nonlinear equilibrium model for an anisotropic shell of a material obeying generalized Hooke’s law; (2) a geometrically and physically nonlinear equilibria model for a shallow shell. Sufficient conditions for their generalized solvability in the corresponding Sobolev spaces are derived. In the case of the non-shallow shell the implicit function theorem is used. In the case of the shallow shell the variation problem is investigated using the generalized Weierstrass principle. Mixed finite element methods based on the use of the second derivatives of the deflection as auxiliary variables are constructed for approximate solutions of these problems. Sufficient conditions for solvability of the corresponding discrete problems are obtained. The convergence of approximate solutions is investigated. Accuracy estimates in the case of sufficiently smooth solutions of the original problems are given. Additional conditions are proposed for the problem on the shallow shell to ensure of implementation of the inequalities of the type of strong monotonicity and Lipschitz-continuity of the differential operator.

Optimal Path Planning for an Object in a Random Search Region

This paper considers the planning problem of an object’s optimal path that passes through a random search region. The problem specifics consist in that the search region is a priori unknown but the algorithmic and technical characteristics of search means are known. Two variational statements of the integral risk minimization problem with and without path length constraints are suggested. Program modules for their numerical solution are also developed.

STRESS-STRAIN STATE OF HYDROVOLUMETRIC DRIVES ELEMENTS IN CONTACT

The work describes methods of analysis of stress-strain state of hydrovolumetric drives elements that are in contact interaction. A variational problem statement is proposed. Modifications of boundary element method are developed. The analysis focused on several structural factors that have the most profound impact on the deformed state of the most loaded elements of the drive. Those include the number of cylinders in the rotor and the geometry of tracks on the internal surface of the stator ring along wich the spherical pistons are running. Hydrostatic load from the oil under the piston in any particular cylinder as well as the adjacent cylinders lead to non-uniform radial defomations. The contact size and contact pressure distribution depend on the initial gap between stator surface and the spherical piston. The influence of the running track profile on the contact ineraction has been determined by numerical analysis. The acquired database serves for the parameter justification and design recommendation for the radial hydrovolumetric drives.

Variational Asymptotic Based Shear Correction Factor for Isotropic Circular Tubes

An analytical closed-form expression is derived for the shear correction factor of beams with annular cross sections. The beam theory is formulated using the variational asymptotic method, which uses inherently occurring small parameters to convert a variational statement from three-dimensional elasticity to an engineering beam theory without any ad hoc assumptions on the beam kinematics. Comparisons are made with existing solutions in the literature; then, the critical speeds of a rotor are shown to depend not only on the inclusion of transverse shear but also on the choice of the shear correction factor, establishing its importance. The motivation of this work is twofold: first, rotors that can be modeled as beams with annular cross sections are often encountered, and this expression for the shear correction factor can be readily incorporated into any rotor dynamics solver; and second, closed-form expressions, when available, always provide more insight into structural mechanics as opposed to numerical solutions.

Bending and free vibration of multilayered functionally graded doubly curved shells by an improved layerwise theory

An improved layerwise theory is established for the bending and vibration responses of multi-layered functionally graded doubly curved shells with material properties varying gradually and continuously across the whole shell thickness. The theory accounts for shear deformation and normal strain effects by assuming for each layer, a displacement field with zigzag first-order in-plane displacements and through-the-thickness parabolic transverse displacement. Moreover, it is assumed a stress field satisfying the loading conditions on the external shell faces and the continuity conditions of the internal stresses at the layers interfaces, so, there is no need for introducing in the present formulation any shear correction factor. For this purpose, a mixed variational statement is used. The continuity conditions for the displacements at the layers interfaces are used to decrease the degrees of freedom of the theory. Bending and free vibration problems are solved for FGM single- and three-layered sandwich open cylindrical and spherical shells with completely simply supported or clamped edges. Comparisons for some present results with numerical results obtained by other authors due to advanced 2D and 3D elasticity solutions are made showing the high efficiency of the present theory in predicting the bending and vibration parameters. Graphics are presented to demonstrate the importance of the normal strain effect for the static bending and free vibration of the considered shells.

A variational approach to vibrations of laminated composite plates with a line hinge

A Ritz variational approach for the analysis of free vibration of plates with internal hinges distributed along a straight line is presented. The plate is modeled with the first order shear deformation theory and symmetric stacking sequences. The line with hinges is modeled by using subdomain decomposition of the plate coupled with penalty techniques, which causes an increment of the variational statement to assure the continuity conditions along the contact boundary of the plate's subdomains. In order to obtain an indication of the accuracy of the developed mathematical model and the proposed technique, some cases available in the literature have been considered. Original results are presented for solution of representative symmetric multilayered plates showing the effects of the line hinge parameters on the natural frequencies and mode shapes.

О РЕГУЛЯРНОСТИ РЕШЕНИЯ В ЗАДАЧЕ О РАВНОВЕСИИ ПЛАСТИНЫ ТИМОШЕНКО, СОДЕРЖАЩЕЙ НАКЛОННУЮ ТРЕЩИНУ

Исследуется задача о равновесии упругой трансверсально-изотропной пластины (модели Тимошенко), содержащей сквозную наклонную трещину. Считается что трещина не выходит на внешнюю границу. В исходном состоянии предполагается, что противоположные берега соприкасаются друг с другом. При этом трещина описывается с помощью поверхности, которая удовлетворят определенным предположениям. На кривой, задающей трещину в срединной плоскости, ставится краевое условие в виде неравенства, описывающее непроникание противоположных берегов трещины. На внешней границе заданы однородные условия Дирихле. Установлена локальная дополнительная гладкость решения по сравнению с заданной в вариационной формулировке при определенных условиях на поверхность, задающую трещину. Доказана бесконечная дифференцируемость функции решения при дополнительных предположениях на функцию, задающую внешние нагрузки, а также на значения функций перемещений вблизи кривой, описывающей трещину. The equilibrium problem for an transversely isotropic elastic plate (Timoshenko model) with an inclined crack is studied. It is supposed that the crack does not touch the external boundary. For initial state, we assume that opposite crack faces are in contact with each other on a frictionless crack surface. Herewith, the crack is described with the use of a surface satisfying certain assumptions. On the crack curve defining the crack in the middle plane, we impose a nonlinear boundary condition as an inequality describing the nonpenetration of the opposite crack faces. It is assumed that on the exterior boundary of the cracked elastic plate the homogeneous Dirichlet boundary conditions are prescribed. We establish additional smoothness of the solution in comparison with that given in the variational statement. We prove that the solution functions are infinitely smooth under additional assumptions on the function of external loads and the functions of displacements near the curve describing the inclined crack.

A micromechanics theory for homogenization and dehomogenization of aperiodic heterogeneous materials

Based on the recently discovered mechanics of structure genome, a micromechanics theory is developed for computing the effective properties and local fields of aperiodic heterogeneous materials. This theory starts with expressing the displacements of the heterogeneous material in terms of those of the corresponding homogeneous material and fluctuating functions. Integral constraints are introduced so that the kinematics including both displacements and strains of the homogeneous material can be defined as the average of those of the heterogeneous material. The principle of minimum information loss is used along with the variational asymptotic method to formulate the governing variational statement for the micromechanics theory. As this theory does not require conditions applied on the boundaries, it can handle microstructures of arbitrary shapes. This theory can also straightforwardly model periodic materials by enforcing the equality of the fluctuating functions on periodic edges. This theory provides a rational approach for avoiding the difficulty of creating periodic meshes and automatically captures finite dimension effects. Furthermore, this theory can model heterogeneous materials with partial periodicity. We have used several examples to verify and demonstrate the capability of this theory.

Optimal Control under a Decrease in the Thermal-Field Intensity Based on Selection of the Heterogeneous-Construction Structure in the Variational Formulation

The application of variational methods to the problems of optimal control for a decrease in the intensity of temperature oscillations of the environment on the basis of the choice of the physical and geometrical structure of heterogeneous constructions is proposed. The necessary optimality conditions are developed for the optimal-design problems under study in the variational statement. On the basis of the constructive analysis of the necessary optimality conditions, the qualitative regularities of the optimal heterogeneous structures are established, which makes it possible to estimate the efficiency of the existing heat-shielding constructions and to find in which direction it is necessary to improve their heat-shielding ability to achieve the ultimate possibilities for a decrease in the temperature-action intensity.

A multi-scale homogenization model for fine-grained porous viscoplastic polycrystals: I – Finite-strain theory

We make use of the recently developed iterated second-order homogenization method to obtain finite-strain constitutive models for the macroscopic response of porous polycrystals consisting of large pores randomly distributed in a fine-grained polycrystalline matrix. The porous polycrystal is modeled as a three-scale composite, where the grains are described by single-crystal viscoplasticity and the pores are assumed to be large compared to the grain size. The method makes use of a linear comparison composite (LCC) with the same substructure as the actual nonlinear composite, but whose local properties are chosen optimally via a suitably designed variational statement. In turn, the effective properties of the resulting three-scale LCC are determined by means of a sequential homogenization procedure, utilizing the self-consistent estimates for the effective behavior of the polycrystalline matrix, and the Willis estimates for the effective behavior of the porous composite. The iterated homogenization procedure allows for a more accurate characterization of the properties of the matrix by means of a finer “discretization” of the properties of the LCC to obtain improved estimates, especially at low porosities, high nonlinearties and high triaxialities. In addition, consistent homogenization estimates for the average strain rate and spin fields in the pores and grains are used to develop evolution laws for the substructural variables, including the porosity, pore shape and orientation, as well as the “crystallographic” and “morphological” textures of the underlying matrix. In Part II of this work has appeared in Song and Ponte Castañeda (2018b), the model will be used to generate estimates for both the instantaneous effective response and the evolution of the microstructure for porous FCC and HCP polycrystals under various loading conditions.

On the finite element formulation for second-order strain gradient nonlocal beam theories

ABSTRACTFinite element (FE) formulation of nanobeams using second-order positive/negative strain gradient theories is presented. The variational statement for beam, written using Hamilton's principle involving virtual work of stress/moment resultants, with C1 continuity requirement leads to unsymmetric nonlocal terms whereas the symmetrization of the nonlocal terms leads to C2 continuity requirement. The C2 continuous FE with proper boundary conditions is accurate for the modeling of nanobeams using second-order strain gradient theories. The negative strain gradient model predicts results matching with the experimental investigations on microbeams. The strain gradient theory based plate FE is also formulated by extending C2 continuous beam FE formulation.

Vibrations of Elastic Shells of Revolution Partially Filled with Ideal Liquid

We propose an algorithm for the numerical analyses of vibrations of elastic shells of revolution partially filled with ideal incompressible liquids. In the solution of this problem, the wave motions of liquid on its free surface are taken into account. The solution of the problem of hydroelasticity is based on the application of the method of decomposition of the domain of integration of equations of the theory of shells with the use of the variational statement of the problem and on the approximate construction of the operator inverse to the operator of the hydrodynamic part of the problem. We construct a generalized functional of displacements of the shell for which the role of conditions of matching for the solutions in different subdomains is played by the natural boundary conditions. The obtained numerical results are compared with the existing exact solutions of the analyzed problem for a shell in the form of a straight circular cylinder.

Modeling of Plane Arrays Using a Variational Approach

The variational statement of synthesis problem is generalized in order to account the additional requirements to the synthesized radiation pattern (RP) and field distribution in the specified points of near zone. For this aim, the minimizing functional is supplemented by term providing the possibility to minimize the values of field in these points; creating the deep zeros in the RP for the certain angular coordinates is realized too. The approach foresees reduction of an explicit formula for field values in a near zone. The results of computational modeling testify the possibility to create zeros in the given RP and to minimize the values of field in a near zone of plane arrays in a great extent.

Optimal stiffness distribution in preliminary design of tubed-system tall buildings

This paper presents an optimal pattern for distributing stiffness along a framed tube structure through an analytic equation, which may be used during the preliminary design stage. Most studies in this field are computationally intensive and time consuming, while a hand-calculation method, as presented here, is a more suitable tool for sensitivity analyses and parametric studies. Approach in development of the analytic model is to minimize the mean compliance (external work) for a given volume of material. A variational statement of the problem is made, and a specified deformation-profile is obtained as the necessary condition for a minimum; enforcing this condition, stiffness is then computed. Due to some near-zero values for stiffness, the problem is modified by considering a lower bound constraint. To deal with this constraint, the design domain is assumed to be divided into two zones of constant stiffness and constant curvature; and the problem is restated in terms of these concepts. It will be shown that this methodology allows for easy computation of stiffness through an analytic and dimensionless equation, valid in any system of units. To show practicality of the proposed method, a tubed-system structure with uniform stiffness distribution is redesigned using the proposed model. Comparative analyses of the results reveal that in addition to simplicity of the proposed method, it provides a rather high degree of accuracy for real-world problems.

Component-wise analysis of laminated structures by hierarchical refined models with mapping features and enhanced accuracy at layer to fiber-matrix scales

ABSTRACTModern methodologies for the analysis of composite structures are demanded to satisfy the accuracy requirements at different scales, from micro to macro and possibly in a global/local sense. In this domain, the present paper proposes a hierarchical, component-wise approach for the linear static analysis of layered structures. By employing the Carrera Unified Formulation and a variational statement, finite element arrays of refined beam models are expressed in terms of fundamental nuclei. Legendre-based polynomials are utilized to implement, in a hierarchical form, higher-order beam kinematics. Also, curved cross-section geometries are formulated in a correct and consistent manner through mapping blending functions. Mapping and refined kinematics beam models are, thus, inherently combined to give the component-wise method, according to which each component of the structure (e.g., layer, fibers, and matrix) is modeled independently and with its geometric and mechanical characteristics. This approach is demonstrated to provide enhanced accuracy for the analysis of composite structures, from layer scale to fiber-matrix level, but still with low computational costs.

Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide

We consider the problem on normal waves in an inhomogeneous waveguide structure reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The inhomogeneity of the dielectric filling and the occurrence of the spectral parameter in the transmission conditions necessitate giving a special definition of what a solution of the problem is. To find the solution, we use the variational statement of the problem. The variational problem is reduced to the study of an operator function. We study the properties of the operator function needed for the analysis of its spectral properties. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of the operator function on the complex plane are proved.

Free-edge stress fields in cylindrically curved symmetric and unsymmetric cross-ply laminates under bending load

In this paper, an analysis method for the determination of displacement, strain and stress fields in cylindrically curved cross-ply laminates under bending load is presented. The considered cross-ply laminates may be either symmetrically or unsymmetrically laminated and are clamped at one end while the other end is loaded by an evenly distributed bending moment. The analysis method employs a layerwise plane strain approach in the inner regions of the laminate in which the stresses in each layer are represented by adequate formulations for Airy’s stress function. In the regions of the free laminate edges where significant three-dimensional and possibly singular interlaminar stress fields are to be expected, the plane-strain approach is upgraded by a layerwise displacement-based formulation wherein the physical laminate layers are discretized into a number of mathematical layers with respect to the thickness direction. The governing differential equations for the unknown additional displacement functions with respect to the width coordinate in the form of the Euler-Lagrange equations stemming from the underlying variational statement can be solved exactly and eventually lead to an eigenvalue problem that needs to be solved numerically. Usage of continuity conditions between the individual laminate layers and formulation of adequate boundary conditions at the free edges in an integral sense then lead to complete representations for displacements, strains and stresses at every location in the considered laminate. While the analysis approach relies on a discretization of the laminate into a number of mathematical layers with respect to the thickness direction and further requires a numerical solution of a quadratic eigenvalue problem, it provides closed-form analytical solutions concerning the width direction and can thus be classified as being a semi-analytical solution. The presented analysis method is compared to the results of comparative finite element simulations and is shown to be in good agreement, however with only a fraction of the computational effort that is required for according finite element simulations.

Free vibration characteristics of nanoscaled beams based on nonlocal integral elasticity theory

In the present article, the total potential energy principle and the nonlocal integral elasticity theory have been used to develop a novel finite element method for studying the free vibration behavior of nano-scaled beams. The formulations are based on Euler-Bernoulli beam theory and this method is able to properly analyze the free vibration of beams with various boundary conditions. By implementing the variational statements, the eigenvalue problem of the free vibration is obtained. The validation investigation is pursued by comparing the results of the current study with those available in the literature. The effects of nonlocal parameter, geometry parameters and boundary conditions on the free vibration of the Euler-Bernoulli beam are then studied.

Pretwisted beam subjected to thermal loads: A gradient thermoelastic analogue

ABSTRACTIt is well known from the classical torsion theory that the cross section of a prismatic beam subjected to end torsional moments will rotate and warp in the longitudinal direction. Rotation is depicted through the angle of twist per unit length and depends in general on the position along the length of the beam, while the warping function addresses the longitudinal distortion of the unrotated cross sections. In the present study, we consider a prismatic beam that possesses an initial twist which is constant along its length. A thermal field is present along the beam and its ends are loaded with axial forces and torsional moments. The governing equilibrium equations and the corresponding boundary conditions were obtained using an energy variational statement. A one-dimensional gradient thermoelastic analogue is developed. The advantageous aspect of the present study is that the additional (and peculiar) boundary conditions required by the gradient elasticity theory and the related microstructural lengths, analogous to micromechanical lengths, emerge in a natural way from the geometrical characteristics of the beam cross section and the material properties. We have examined various examples with different cross sections and loads to demonstrate the applicability of the model to the design of special yarns useful in smart textiles and thermally activated microdrilling actuators.

Numerical-Analytical Method of Studying Creep and Sustained Strength Characteristics of a Multilayer Shell

The problem of stress-strain state, creep, and sustained strength computations for multilayer shallow shells of medium thickness, free form in plan was discussed. The variational statement is done within the refined theory of shells. The solution method for the nonlinear initial boundary creep problem based on combining the R-functions, Ritz, and Runge-Kutta–Merson methods was elaborated. An example of creep and sustained strength calculations is given for a two-layer cylindrical shell simulating the thermal barrier coating.