Elastic Shallow Shells Research Articles

From elastic shallow shells to beams with elastic hinges by Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-convergence

In this paper, we study the Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-limit of a properly rescaled family of energies, defined on a narrow strip, as the width of the strip tends to zero. The limit energy is one-dimensional and is able to capture (and penalize) concentrations of the midline curvature. At the best of our knowledge, it is the first paper in the Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-convergence field for dimension reduction that predicts elastic hinges. In particular, starting from a purely elastic shell model with “smooth” solutions, we obtain a beam model where the derivatives of the displacement and/or of the rotation fields may have jump discontinuities. Mechanically speaking, elastic hinges can occur in the beam.

On the problem of solvability of nonlinear boundary value problems for shallow isotropic shells of Timoshenko type in isometric coordinates

The solvability of a boundary value problem for a system, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with loose edges referred to isometric coordinates in the Timoshenko shear model and consists of five non-linear second-order partial differential equations under given non-linear boundary conditions, is studied. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in Sobolev space, the solvability of this equation is established with the help of the contraction mapping principle.

EQUILIBRIUM OF A NORMALLY LOADED ELASTIC SHALLOW CYLINDRICAL SHELL WITH DISLOCATIONS AND DISCLINATIONS

We consider the problem of the equilibrium of a normally loaded elastic shallow rectangular circular cylindrical shell, which contains sources of internal stress in the form of fields of continuously distributed edge dislocations and wedge disclinations or other sources, for example, thermal ones. Based on the modified system of Karman equations for the equilibrium of an elastic plate with dislocations and disclinations, a system of nonlinear equilibrium equations for the shell was constructed. The resulting system of equations differs from the Karman system of equilibrium equations for an elastic shallow cylindrical shell under the action of a normal load by the presence on the right side of the continuity equation of a nonzero function, called the incompatibility function, which is expressed through the densities of edge dislocations and wedge disclinations. The edges of the shell are freely clamped or hinged. In the absence of internal stress sources, this system transforms into a system of equilibrium equations for a shallow shell, and in the case of an infinitely large radius of curvature (and the presence of internal stress sources) into a modified system of Karman equilibrium equations for a flexible elastic plate with dislocations and disclinations. To solve the system of nonlinear shell equilibrium equations, the Newton – Kantorovich method in combination with the difference method is proposed. For the case of small values of the normal load and small values of the incompatibility function, linearization is carried out with respect to the deflection and stress function. As a result, a linear boundary value problem was obtained in the form of a system of differential equations for the deflection and the stress function, which is solved by the difference method. The solution of the linearized system is used as an initial approximation in the implementation of the Newton – Kantorovich method. Examples of numerical calculations of deflection and stress function for given values of normal load and incompatibility function are given, and corresponding graphs are constructed.

Nonconforming Finite Element Methods for Two-Dimensional Linearly Elastic Shallow Shell Model

Nonconforming Finite Element Methods for Two-Dimensional Linearly Elastic Shallow Shell Model

On the Optimality Conditions in the Weight Minimization Problem for a Shell of Revolution at a Given Vibration Frequency

We consider shallow elastic shells with a given circular boundary and seek an axisymmetric shell shape minimizing the weight at a given fundamental frequency of shell vibrations. Using the resulting formula for the linear part of the increment of the frequency functional, the multiplicity of the minimum natural frequency of vibrations of the shell is estimated. The Fréchet differentiability of the frequency functional is also established, and optimal conditions for minimizing the weight of the shell at a given fundamental vibration frequency are obtained.

Intrinsic Marguerre–von Kármán equations

We first show how the classical Marguerre–von Kármán equations modeling the deformation of a nonlinearly elastic shallow shell can be recast as equations whose sole unknowns are the bending moments and stress resultants inside the middle surface of the shell. Thus, these equations allow to compute the stresses inside the shell without having to compute first the displacement field. We then show that the boundary value problem formed by these new equations is well posed by establishing an existence theorem.

On the Existence of Solutions of Nonlinear Boundary Value Problems for a System of Differential Equilibrium Equations for Timoshenko-Type Shells in Isometric Coordinates

We prove the existence of solutions of a boundary value problem for a system of five nonlinear second-order partial differential equations with given nonlinear boundary conditions, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with free edges in the Timoshenko shear model referred to isometric coordinates. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in the Sobolev space, the solvability of which is established using the contraction mapping principle.

Erratum to: Natural Vibrations of Micropolar Elastic Flexible Plates and Shallow Shells

Erratum to: Natural Vibrations of Micropolar Elastic Flexible Plates and Shallow Shells

Natural Vibrations of Micropolar Elastic Flexible Plates and Shallow Shells

A mathematical model representing the dynamics of geometrically nonlinear (flexible) micropolar elastic thin plates in Cartesian and curvilinear coordinates is constructed (the approach is generalized to the case of micropolar flexible shallow shells as well). The model is developed under the assumption that the elastic deflection of a plate is comparable with the plate thickness but is small compared to the characteristic plate size in plan. Based on the given model of micropolar elastic flexible plates, the problem on free vibrations is solved for rectangular and circular plates and shallow shells. Effective manifestations of characteristic features of a micropolar material are considered in comparison with the corresponding classical material.

Shape derivatives of energy and regularity of minimizers for shallow elastic shells with cohesive cracks

The paper addresses the analysis of a model for a thin shallow linear elastic shell with a smooth vertical through crack that is based on the Kirchhoff–Love shell theory and accounts for cohesive forces acting between the crack faces. We follow the basic idea behind the Barenblatt theory assuming that the density of the total energy spent by the cohesive forces is the sum of longitudinal and transverse contributions, each of which in general is nonconvex. In order to eliminate nonphysical interpenetration of the crack faces, an inequality constraint that involves both the normal component of the longitudinal displacements and the normal derivative of the transverse deflection of the crack faces is imposed. We first prove the existence of minimizers for the total energy and study in detail the Euler–Lagrange system for them. Then we derive the left and right Eulerian shape derivatives of the minimal value of the total energy by developing a fully variational technique. Finally we apply the developed technique coupled with a difference quotient argument to obtain higher differentiability results in Besov and Sobolev spaces for the minimizers.

On the improved interior regularity of the solution of a fourth order elliptic problem modelling the displacement of a linearly elastic shallow shell lying subject to an obstacle

In this paper we show that the solution of an obstacle problem for linearly elastic shallow shells enjoys higher differentiability properties in the interior of the domain where it is defined.

Strength of thin-walled elastic building structures

The existence theorem is proved within the framework of the shear model by S.P. Timoshenko. The stress-strain state of elastic inhomogeneous isotropic shallow thin-walled shell constructions is studied. The stress-strain state of shell constructions is described by a system of the five equilibrium equations and by the five static boundary conditions with respect to generalized displacements. The aim of the work is to find generalized displacements from a system of equilibrium equations that satisfy given static boundary conditions. The research is based on integral representations for generalized displacements containing arbitrary holomorphic functions. Holomorphic functions are found so that the generalized displacements should satisfy five static boundary conditions. The integral representations constructed this way allow to obtain a nonlinear operator equation. The solvability of the nonlinear equation is established with the use of contraction mappings principle.

Growth and Non-Metricity in Föppl-von Kármán Shells

The non-homogeneous Foppl-von Karman equations for growing thin elastic shallow shells are revisited by deriving the inhomogeneity source terms directly from the non-metricity tensor associated with growth. This is in contrast with the existing literature where the source terms are obtained using the extensional and curvature growth strains after exploiting the additive decomposition of the total strain into its elastic and growth counterpart. Our framework not only establishes the additive decomposition but provides an unambiguous illustration of the geometric nature of growth in terms of a genuine material inhomogeneity measure given by the non-metricity tensor.

Nonlinear Limit Cycle Oscillation and Flutter Analysis of Clamped Curved Plates

This Paper studies the instability (flutter) and poststability limit cycle oscillations of elastic shallow shells in a supersonic gas flow. In a previous paper by the authors, a nonlinear dynamic aeroelastic analysis of streamwise square curved plates was made to compare with the experimental results of Anderson. In the present Paper, computational results are presented for a much larger range of several parameters, which are important for flutter and limit cycle oscillations including both spanwise and streamwise curved plates with various aspect ratios, various boundary conditions, and the effects of static pressure and in-plane mechanical loadings. No previous study has included such a wide range of parameters and parameter values. Von Kármán shallow shell theory, a modal expansion along with the Galerkin method, and the piston theory aerodynamic model for supersonic/hypersonic flow are employed to obtain these results.

A low-order mixed variational principle for the generalized Marguerre–von Kármán equations

We propose a mixed variational principle for deducing the generalized Marguerre–von Karman equations, governing the relatively large deflections of thin elastic shallow shells. These equations account for both non-flat stress-free configurations of the shell and inelastic strains. We implement this formulation by using $$C^0$$ interior penalty methods within the UFL language provided by the FEniCS project. We present two numerical examples, with the aim to discuss the role of the shallowness and the inelastic strain, comparing the results with the fully non-linear shell model a la Naghdi and the classical displacement formulation.

Buckling prognosis for thin elastic shallow shells

This article addresses several problems that are related to the elastic stability of thin shells and that are due to inconsistencies between the experimental data and predictions based on the equations of the shallow-shell theory. The above contradictions are solved within the three-dimensional nonlinear theory of elasticity. In particular, the dynamic approach enables a prediction of the moment of bifurcation for very thin shells under momentless stress through the analysis of asymptotically built two-dimensional equations. Appearance and development of the dents and patterns in initially ideal shells that precede the buckling are also explained in this article. The dents represent solitonic waves that are detectible by acoustic devices. Therefore, it is possible to reduce the risk of failure of thin shells by using acoustic devices to monitor their conditions. The article covers two types of experiments with thin shells when the loads are close to buckling. These experiments enable to assess the safety buckling factor under technical operation of shells.

Correlation of Experimental and Computational Results for Flutter of Streamwise Curved Plate

This work studies the instability (flutter) and poststability oscillations of an elastic shallow shell in a supersonic gas flow. The only available experimental results for flutter of a shallow shell with clamped edges are provided by Anderson. A comparison between theory and experiment is a novel and important contribution of this work. The nonlinear vibrations of a rectangular isotropic thin curved shell in supersonic airflow are studied to find the flutter speed and the limit cycle oscillations beyond the flutter boundary. Using the Galerkin method and the piston theory aerodynamic model for supersonic flow, the flutter and postflutter characteristics of the panel have been analyzed. Panel vibration frequencies and flutter dynamic pressure have been determined and compared with experimental results.

Application of Riemann–Hilbert Problem Solutions to a Study of Nonlinear Boundary Value Problems for Timoshenko Type Inhomogeneous Shells with Free Edges

The paper deals with the study of solvability of geometrically nonlinear boundary value problem for elastic shallow isotropic inhomogeneous shells with free edges within S. P. Timoshenko shear model. The problem is reduced to one nonlinear equation whose solvability is proved with the use of contracting mappings principle.

Justification and solvability of dynamical contact problems for generalized Marguerre–von Kármán shallow shells

AbstractIn this work, we consider a three–dimensional dynamical model of unilateral contact problems with Signorini conditions and Coulomb friction laws for nonlinearly elastic shallow shells with a specific class of boundary conditions of generalized Marguerre–von Kármán type. Using technics from formal asymptotic analysis, we show that the scaled three–dimensional solution still leads to a two–dimensional dynamical model with frictionless contact problems. Then, we solve the last problems, using penalization method.

A method of integral equations in nonlinear boundary-value problems for flat shells of the Timoshenko type with free edges

We prove the existence theorem for solutions of geometrically nonlinear boundary-value problems for elastic shallow isotropic homogeneous shells with free edges under shear model of S. P. Timoshenko. Research method consists in the reduction of the original system of equilibrium equations to a single nonlinear equation for the components of transverse shear deformations. The basis of this method are integral representations for the generalized displacements, containing an arbitrary holomorphic functions, which are determined by the boundary conditions involving the theory of one-dimensional singular integral equations.