Linear Theory Of Elastic Shells Research Articles

Numerical investigation on epi-off crosslinking effects on porcine corneas

Experimental inflation tests, conducted on 90 pig corneas before and after corneal collagen crosslinking (CXL) treatment, are simulated with the finite element method. The experimental sample consists of five groups of corneas treated with different UV-A irradiation times (2.5, 5, 10, 15, and 20 min) at constant irradiance 9 mW/cm2. The linear elastic shell theory is used to estimate the equivalent material stiffness of the corneas, revealing that it increases with the exposure time in CXL corneas. In the view of numerical simulations, a simple mechanical model assuming piecewise constant elastic modulus across the corneal thickness is introduced, to estimate the effective increment of the material stiffness in the anterior stroma and the effective depth of the stiffness increment. The two effective quantities are used in the finite element models to simulate the post-CXL tests. Numerical models are able to describe the mechanical effects of CXL in the cornea. The increment of equivalent material stiffness has to be ascribed to a localized increment of the material stiffness in the anterior layers of the cornea, while the posterior layers preserve the original material stiffness. According to the simplified model, the increment of the material stiffness of the anterior cornea increases with the irradiation dose, while the effective reinforcement depth decreases with the irradiation dose. This trend, predicted by a simple mechanical model by imposing equilibrium and compatibility, has been verified by the numerical calculations that captured the global mechanical response of the corneas in untreated and post-CXL conditions.

The resultant linear six‐field theory of elastic shells: What it brings to the classical linear shell models?

Basic relations of the resultant linear six‐field theory of shells are established by consistent linearization of the resultant 2D non‐linear theory of shells. As compared with the classical linear shell models of Kirchhoff‐Love and Timoshenko‐Reissner type, the six‐field linear shell model contains the drilling rotation as an independent kinematic variable as well as two surface drilling couples with two work‐conjugate surface drilling bending measures are present in description of the shell stress‐strain state. Among new results obtained here within the six‐field linear theory of elastic shells there are: 1) formulation of the extended static‐geometric analogy; 2) derivation of complex BVP for complex independent variables; 3) description of deformation of the shell boundary element; 4) the Cesáro type formulas and expressions for the vectors of stress functions along the shell boundary contour; 5) discussion on explicit appearance of gradients of 2D stress and strain measures in the resultant stress working.

Local stresses in the Janssen granular column

We study experimentally the distribution of local stresses in a granular material confined inside a vertical cylinder. We use an image correlation technique to measure the displacement field of the container induced by the forces exerted by the grains on the inner wall. We describe an optimization procedure based on the linear theory of elastic shells to deduce the distribution of these forces from the measured displacement field. They correspond to the stress field of the granular material close to the container's inner wall. We first confirm the validity of Janssen's description for various experiments, including the influence of the bead diameter and the effect of an additional mass on top of the granular column. We then apply this method to determine the stress field during the gravity-driven discharge of a silo through an aperture.

Viral Capsid Equilibrium Dynamics Reveals Nonuniform Elastic Properties

The long wavelength, low-frequency modes of motion are the relevant motions for understanding the continuum mechanical properties of biomolecules. By examining these low-frequency modes, in the context of a spherical harmonic basis set, we identify four elastic moduli that are required to describe the two-dimensional elastic behavior of capsids. This is in contrast to previous modeling and theoretical studies on elastic shells, which use only the two-dimensional Young's modulus (Y) and the bending modulus (κ) to describe the system. Presumably, the heterogeneity of the structure and the anisotropy of the biomolecular interactions lead to a deviation from the homogeneous, isotropic, linear elastic shell theory. We assign functional relevance of the various moduli governing different deformation modes, including a mode primarily sensed in atomic force microscopy nanoindentation experiments. We have performed our analysis on the T = 3 cowpea chlorotic mottle virus and our estimate for the nanoindentation modulus is in accord with experimental measurements.

Hamiltonian formalization of constitutive relations of the linear theory of elastic shells of revolution

Hamiltonian formalization of constitutive relations of the linear theory of elastic shells of revolution

Polarized Hessian covariant: Contribution to pattern formation in the Föppl–von Kármán shell equations

We analyze the structure of the Föppl–von Kármán shell equations of linear elastic shell theory using surface geometry and classical invariant theory. This equation describes the buckling of a thin shell subjected to a compressive load. In particular, we analyze the role of polarized Hessian covariant, also known as second transvectant, in linear elastic shell theory and its connection to minimal surfaces. We show how the terms of the Föppl–von Kármán equations related to in-plane stretching can be linearized using the hodograph transform and relate this result to the integrability of the classical membrane equations. Finally, we study the effect of the nonlinear second transvectant term in the Föppl–von Kármán equations on the buckling configurations of cylinders.

The intrinsic stiffness of single-wall carbon nanotubes

Single-wall carbon nanotubes have been frequently modeled as linear elastic thin shells. We have compared the atomistic-based shell theory for single-wall carbon nanotubes which was established directly from the interatomic potential to the classical linear elastic shell theory. It is shown that the constitutive relation is linear (within 2% error) only for strain up to 1%. The constitutive relation is approximately isotropic prior to deformation, but the degree of anisotropy increases rapidly as the deformation increases. The coupling between the stress and curvature, and between the bending moment and strain, which is neglected in the classical shell theory, is important for the constitutive behavior of single-wall carbon nanotubes.

An effective model for Lipschitz wrinkled arches

Within the framework of the Koiter's linear elastic shell theory, we study the limit model of a Lipschitz curved arch whose mid-surface is periodically waved. The magnitude and the period of the wavings are of the same order. To achieve the asymptotic analysis, we consider a mixed formulation, for which we perform a two-scale homogenization technique. We prove the convergence of the displacements, the rotation of the normal, and the membrane strain. From the limit formulation, we derive an effective model for curved critically wrinkled arches. It introduces two membrane strain functions—instead of one in the classical case—and exhibits a corrector membrane term to the coupling between the rotation of the normal and the membrane strain.

Local loads on a toroidal shell

In this study the effect of local loads applied on a sectorial toroidal shell (pipe bend) is considered. A linear elastic shell theory solution for local loads is first outlined. The solution corresponds to the case of a shell simply supported at the two ends. Detailed displacement and stress results are then given for a specific shell with loadings centred at three positions; the crown circles, the extrados, and the intrados. These results are compared with results for a corresponding cylindrical shell. The paper concludes with a table summarizing results for characteristic displacements and stresses in a number of shells, covering a wide range of geometric parameters.

Comparison of elastic solutions for three-point bending of curved pipes

In this paper the linear elastic shell theory problem of the three-point bending of a curved pipe is considered. Such a loading arises in the industrial pipe ram bending process. Summaries are given for solutions to this problem based on the Mushtari-Vlasov-Donnel (MVD) and Sanders linear shell theories. Numerical results for displacements and stresses are obtained using the two shell theories, and these are compared with results from the finite element method (FEM). The present study gives practical information about the behavior of curved pipes subjected to ram bending. As well it provides information about the solution characteristics of thin-shell theories in toroidal coordinates.

Axisymmetric Free Vibration of Thick Orthotropic Hemispherical Shells Under Various Edge Conditions

Axisymmetric free vibration of moderately thick polar orthotropic hemispherical shells are studied under the various boundary conditions with sliding, guided pin, clamped, and hinged edges. Based on the improved linear elastic shell theory with the transverse shear strain and rotatory inertia taken into account, the dynamic equlibrium equations are formulated and transformed into the displacement form in terms of mid-surface meridian and radial displacements and parallel circle cross-section rotation. These partial differential equations are solved by the Galerkin method using proper Legendre polynomials as admissible displacement functions with the aid of the orthogonality and a number of special integral relations. Natural frequencies and modes found from the eigenproblems are shown with reasonable results.

Axisymmetric Free Vibration of Orthotropic Complete Spherical Shells

This paper presents studies of the axisymmetric free vibration of moderately thick orthotropic complete spherical shells. The mathematical model is formulated in terms of the mid-surface meridian and radial displacements, and cross-section rotation of the meridians by the improved linear elastic shell theory with the effects of transverse shear strain and rotary inertia included. By the use of the Legendre polynomials, the exact solu tion from a set of three governing partial differential equations for the transversely iso tropic case and the series solution from the Ritz method for the polar orthotropic case are obtained. Natural frequencies and modes are found from the eigenproblems by using orthogonality and some special integrals with reasonable results.

Forces developed beneath hydrogel contact lenses due to squeeze pressure

The hydrodynamic squeeze pressure in the fluid film beneath hydrogel contact lenses fitted onto an axisymmetric model eye were measured. These pressures were due to the contact lens relaxing after deformation by an applied force of similar magnitude to the human eyelid force. The distribution of pressure for contact lenses typically fitted to human eyes was negative with respect to atmospheric pressure at the corneal apex and became less negative at the corneo-scleral limbus. The force that the contact lens applied to the cornea was determined by integrating the pressure distribution from the corneal apex to the limbus. This force varied from 6.0*10-4 N to -7.8*10-1 N depending on the thickness, elastic modulus and bearing relationship of the contact lens. An expression was derived to determine the pressure developed beneath the annulus of the hydrogel contact lens overlapping the cornea, in terms of the measured force over the cornea beneath the contact lens and the chord diameters of the contact lens and cornea. It was found that the deformation of hydrogel contact lenses on the model eye did not follow a linear elastic shell theory.

液体の入った円筒シェルの動的解析

It is the purpose of this report to formulate and solve theoretically the forced motion problems of cantilevered circular cylindrical shells partially filled with liquid subjected horizontal earthquake excitations. For containers, thin cylindrical shells are considered and the internal liquid is assumed to be ideal liquid. The fundamental equations are obtained based on the Donnell approximation, the linear elastic shell theory with small deflection and the linear potential flow theory. The unit mode displacement is obtained by approximating displacement for the axial direction as series of displacement functions for cantilevered beam and by applying the Rayleigh-Ritz method. Assuming that the pressure of internal liquid is devided into the convective pressure and impulsive pressure, the impulsive pressure is represented as the sum of pressures induced by rigid motion and elastic deformation of the shell. Since the frequencies of the first few dominant modes of liquid sloshing are usually much smaller than the frequency of liquid-shell system, the effect of elastic deformation of the shell to the convective pressure are neglected. Numerical computations proved that the response pressure considering elastic deformation of shell is larger than the pressure considering only rigid motion of shell.

Wave Propagation in the Linear Theory of Elastic Shells

The problem of wave propagation in elastic shells within the framework of a linear theory of a Cosserat surface is treated using the method of singular wave curves. The equations for determining the speeds of propagation and their associated wave mode shapes are obtained in a form involving the speeds of propagation in Cosserat plates and the curvature of the shell. A number of special cases in which the speeds and mode shapes simplify are considered. In particular, these special cases are shown to include as examples, certain systems of waves in elastic shells whose middle surfaces are the surface of revolution, the circular cylinder, the sphere, and the right helicoid.

Structural analysis and design of a nuclear power plant building for aircraft crash effects

The object of this investigation is to assess the effect of a large commercial airplane crashing perpendicularly on to the surface of a spherical reactor building dome. This investigation is related to a project currently in execution. Practical solutions of the postulated case, which vary in the degree of engineering effort used, are shown. Based on safety consideration the various solutions are discussed from the viewpoint of penetration, cracking and collapse modes of failure, where, primarily, the carrying capacity of the structure under an equivalent statical load is considered. The performed investigations include: 1. (a) Calculation of the failure load following the yield line theory; 2. (b) Calculation of the sectional forces using the linear-elastic shell theory and subsequent design by the ultimate strength method; 3. (c) Calculation of the failure load, establishing of the failure mechanism and distribution of sectional forces using the plastic shell theory; 4. (d) Calculation using a three-dimensional FEM program with plastic capabilities; this includes the collapse load, the failure mechanism and the distribution of sectional forces. A discussion of the resultant forces and the configutation of the critical section is given for the various methods used. The evaluation of the carrying capacity of the structure with respect to load is based on energy considerations. It is attempted to compare such results, to evaluate possible simplifications in the used solutions, and to give some recommendations for the practical design and for the development of structural details.

On Rotationally Symmetric Stress and Strain in Anisotropic Shells of Revolution

The present paper complements recent work on rotationally symmetric stress and strain in the linear theory of elastic shells of revolution. Departure here is from the two uncoupled systems of equilibrium and compatibility differential equations for what is called the problems of bending and of twisting, respectively. The object here is to reduce the complete set of equations once more to a system of two simultaneous second-order differential equations for the problem of bending alone of the polarorthotropic shell. (Author)

On the Equations of the Linear Theory of Elastic Conical Shells

In a recent article [19], it was shown that the equations of the linear theory of elastic shells of revolution under arbitrary loads, after a harmonic analysis in the polar angle of the base plane of the shell, can be reduced to two simultaneous fourth order ordinary differential equations for a stress function and a displacement function. The exact reduction procedure of [19] was applied to a circular cylindrical shell. Also mentioned was the possibility of a simplified procedure based on the fact that there is an inherent error in shell theory of the order h/R, where h is the shell thickness and R is a representative magnitude of the two principal radii of curvature. In the present paper, the exact and the simplified procedure of [19] will be applied to a conical shell frustrum. The simplified procedure leads to two differential equations as well as to auxiliary equations for the calculation of stress resultants and couples which differ from those of the exact procedure only in terms which are O(h/R). The differential equations obtained here are very similar in form to those of shallow shell theory [6,8] as well as to those of the Donnell type theory [15]. They do in fact reduce to the shallow shell equations when the slope angle /!? between a meridian and the base plane is very small. On the other hand, for /3 = z/2, they reduce to the “correct” equations for circular cylindrical shells of [16,19] while the results of [15] reduce only to the Donnell equation. Aside from their usefulness in obtaining the solution of specific problems, our results also delimit the range of applicability of the Donnell type equations for conical shells previously obtained in [5,15].

The linear elastic cosserat surface and shell theory

The linearized theory of an elastic Cosserat surface is discussed emphasizing its relevance to the classical problem of the linear theory of elastic shells (and plates) regarded as three dimensional bodies.

The Non‐Stationary Thermal Problem in the Linear Theory of Elastic Shells

AbstractThe statics of thermoelastic shells is considered with emphasis on a linear theory. The three‐dimensional heat conduction equation, with the dilatational effect included, is studied, and from it is deduced a pair of simultaneous equations, valid in the middle surface space. This pair of equations is shown to reduce to forms relevant to a Kirchhoff‐Love theory, a Green‐Zerna theory and a linearized Marguerre shallow‐shell theory.