Isotropic Elastic Shell Research Articles

A molecular based anisotropic shell model for single-walled carbon nanotubes

Single-walled carbon nanotubes (SWCNTs) are frequently modeled as isotropic elastic shells. However, there are obvious evidences showing that SWCNTs exhibit remarkable chirality induced anisotropy that should not be neglected in some cases. In this paper, we derive the closed-form expressions for the anisotropic elastic properties of SWCNTs using a molecular mechanics model. Based on these anisotropic elastic properties, we develop a molecular based anisotropic shell model (MBASM) for predicting the mechanical behavior of SWCNTs. The explicit expressions for the coupling of axial, circumferential, and torsional strains, the radial breathing mode frequency, and the longitudinal and torsional wave speeds are obtained. We show that the MBASM is capable of predicting the effects of size and chirality on these quantities. The efficiency and accuracy of the MBASM are validated by comparisons of the present results with the existing results.

Continuum modeling of boron nitride nanotubes

Boron nitride nanotubes display unique properties and have many potential applications.A finite-deformation shell theory is developed for boron nitride nanotubes directly fromthe interatomic potential to account for the effect of bending and curvature. Itsconstitutive relation accounts for the nonlinear, multi-body atomistic interactions,and therefore can model the important effect of tube chirality and radius. Thetheory is then used to determine whether a single-wall boron nitride nanotubecan be modeled as a linear elastic isotropic shell. Instabilities of boron nitridenanotubes under different loadings (e.g., tension, compression, and torsion) are alsostudied. It is shown that the tension instability of boron nitride nanotubes ismaterial instability, while the compression and torsion instabilities are structuralinstabilities.

Phenomenological invariant-based finite-element model for geometrically nonlinear analysis of thin shells

A Kirchhoff–Love type curved triangular finite element is proposed for geometrically nonlinear analysis of elastic isotropic shells undergoing small strains but large displacements. The finite-element formulation is based on the expression of the strain energy in terms of invariants of the strain and curvature-change tensors of the shell middle surface. The element sides are chosen as three independent directions for determining the strains and curvature changes. The emphasis is put on improvement of the bending behavior of the element so that the element is able to undergo finite curvature changes. Recursive relations are obtained for exactly calculating the coefficients of the first- and second-order variations of the strain energy of the finite element which are necessary to formulate the equilibrium and stability conditions of the discrete model of a shell. A shell finite element with 15 degrees of freedom is developed and tested. Numerical examples are presented to demonstrate the accuracy and mesh convergence of the finite-element solutions.

The G. D. Q. method for the harmonic dynamic analysis of rotational shell structural elements

This paper deals with the modal analysis of rotational shell structures by means of the numerical solution technique known as the Generalized Differential Quadrature (G. D. Q.) method. The treatment is conducted within the Reissner first order shear deformation theory (F. S. D. T.) for linearly elastic isotropic shells. Starting from a non-linear formulation, the compatibility equations via Principle of Virtual Works are obtained, for the general shell structure, given the internal equilibrium equations in terms of stress resultants and couples. These equations are subsequently linearized and specialized for the rotational geometry, expanding all problem variables in a partial Fourier series, with respect to the longitudinal coordinate. The procedure leads to the fundamental system of dynamic equilibrium equations in terms of the reference surface kinematic harmonic components. Finally, a one-dimensional problem, by means of a set of five ordinary differential equations, in which the only spatial coordinate appearing is the one along meridians, is obtained. This can be conveniently solved using an appropriate G. D. Q. method in meridional direction, yielding accurate results with an extremely low computational cost and not using the so-called "delta-point" technique.

Infinitesimal Isometric Deformations of a Pseudospherical Shell

We consider a thin linear elastic isotropic shell of constant thickness. We assume that it undergoes deformations of the Kirchhoff–Love type and derive some explicit formulas for certain infinitesimal isometric deformations of a pseudospherical shell by considering a linearized version of this problem.

Elastic response of a layered cylinder subjected to diametral loading

The elastic stress field produced in an arbitrarily layered cylinder subjected to diametrically opposed loads is examined and subsequently employed to investigate the transverse strengths of tungsten-cored boron and carbon-cored silicon carbide fibers measured from diametral compression tests. The displacement formulation of linear elasticity and the local/global stiffness matrix method are used to obtain solutions for stress and displacement components in a composite cylinder consisting of annular linear elastic isotropic shells under the assumption of plane strain. It is found that the presence of the carbon and tungsten cores in the ceramic fibers gives rise to a tensile stress concentration at the core/outer shell interface under diametral compressive loading. This is shown to contribute to a significant loss of apparent transverse strength. The solution also provides insight into first-stage consolidation of alloy-coated ceramic fibers. Elevated hoop stresses are shown to occur at fiber-fiber contacts that result in a reduction in the yield coefficient from that of homogeneous alloy fibers. Thus, the initiation of the yield mechanism of consolidation can be accomplished at lower applied loads when a ceramic fiber is present.

On the numerical solution of spherically symmetric problems using the theory of a cosserat surface

A numerical solution of spherically symmetric problems is formulated by dividing the spherical region into N spherical shells and modelling each of these shells as a Cosserat surface. By coupling the motion of each of these shells with that of its neighbours we obtain a system of ordinary differential equations for functions of time only. Both the displacements and the contact stresses at the common boundaries are obtained. Specific constitutive equations are developed for linear motion of an isotropic elastic shell and the resulting mass and stiffness matrices are shown to be different from those obtained using the displacement approach and Galerkin methods. Comparison between Cosserat, Galerkin and exact solutions are shown for three examples which include both static and dynamic problems. In each of these examples, the Cosserat solution is superior to the Galerkin solution.

Numerical analysis of stressed state of thin shells with curvilinear holes

Many problems of stress concentration have alreadybeen solved for elastic isotropic shells with holes of various shapes. Theoretical results, pertaining primarily to shallow spherical, cylindrical, and conical shells with circular holes, were obtained by methods involving separation of variables in the classical formulation [6]. For shells with noncircular curvilinear (elliptical, square, triangular, and others with round corners) holes was used the method of perturbation of the contour shape according to Guz' [3]. The most nearly complete survey of theoretical and experimental studies made so far concerning the problem of stress concentration in isotroFicand orthotropic elastic and inelastic shells can be found in [4], the method~ used and solutions to specific problems in linear or nonlinear formulation being expounded in the all-encompassing monograph [5].

Free and forced finite amplitude oscillation of a spherical shell of radially isotropic elastic material

A spherical elastic shell with radial transverse isotropy is considered. The periods of finite amplitude radial oscillation of the shell have been obtained in two cases, namely (i) when both the surfaces of the shell are free from traction, and (ii) when the shell boundaries are uniformly loaded in such a way that the pressure difference between inner and outer surfaces is constant with respect to time. It is observed that for free oscillation to take place it is necessary to impose a new restriction on the strain energy function in addition to those already obtained for finite amplitude oscillation of an isotropic elastic shell. In the forced oscillation case however the required conditions are of the same form as in the corresponding case for isotropic media.

On the response functions of isotropic elastic shells

On the response functions of isotropic elastic shells

Circumferentially sinusoidal stress and strain in shells of revolution

The system of equations of the linear theory of isotropic elastic shells of revolution for stress and strain sinusoidal in the polar angle of the base plane with period 2π is reduced to two simultaneous second order ordinary differential equations which are remarkably similar to the corresponding equations for symmetric bending. The reduction makes extensive use of the static geometric analogy to simplify the analysis and allows for the possibility of a non-periodic displacement field. The problem of asymmetric twisting and bending of ring shell sectors associated with the non-periodic displacement state is formulated for the first time within the framework of shell theory.

Strain energy expression for large deformations of isotropic elastic shells subjected to arbitrary temperature distribution

A general strain energy expression for large deflections of thin isotropic elastic shells is derived, including the effects of heating. With this expression, the thermal buckling and post-buckling behavior of heated shells may be treated. The results are specialized for circular cylindrical shells, for spherical shells and for right-circular conical shells. Important applications of thetheory presented occur with high-speed aircraft and rocket structures and with nuclear reactor structures.

Optimum thickness transitions for cylindrical pressure vessels with hemispherical heads.

An analysis is presented for two new thickness transition configurations for cylindrical pressure vessels with hemispherical heads. These configurations, which extend on both sides of the juncture between the shells, are the versine variation and the bilinear variation. The bilinear transition is shown to result in lighter designs while holding overstress to a negligible amount. Nondimensional curves are presented which show the over-all vessel weight for both types of transition. Results are obtained by solving numerically a system of secondorder differential equations applicable to thin elastic isotropic shells of variable thickness, using an established decoupling criterion for spherical and cylindrical shells under edge loads.

Berechnung von elastisch orthogonal bewehrten maxwell-Schalen mit kleinem bewehrungsanteil (z.B. Stahlbeton-Schalen) — teil I

The constitutive equations for shallow shells of viscoelastic (Maxwell) material with elastic reinforcement are derived. With respect to reinforced concrete shells shrinkage is considered. (Constitutive equations here mean relations between stress resultants, as membrane forces, bending and twisting moments, and strains or distortions of the shell middle surface, and further, formulae for the determination of the stress resultants related to the elastic and to the viscoelastic part of the section, respectively.) Concerning strain geometry, the usual assumptions of the theory of elastic shells are made. The reinforcement shall be orthogonal, but may be unsymmetrical to the middle surface and locally variable. Influences of transverse contraction are not considered. The derivations are based on the tri-dimensional constitutive equations for viscoelastic media of the type of reinforced concrete as established by D. Hilliges, who proved, that such a medium behaves in a different manner under shearing stresses than under normal stresses. This characteristic behaviour is also expressed in the constitutive shell equations given in this paper. The elastic and inelastic strains of a compound shell uder bending moments and normal forces are much smaller than those of a corresponding shell without reinforcement, and there is a decrease of the stress resultants of the viscoelastic part of the section (relaxation) and a corresponding increase of the stress resultants of the elastic part, whereas under shearing and twisting loads the (elastic and inelastic) rigidity of a compound shell element is only slightly higher than that of an element without reinforcement, and there is no remarkable shift of the stresses from the viscoelastic to the elastic part of the section. Using the derived constitutive equations there are given the displacement equations of the orthotropic Maxwell shell. Exact solutions can only be found for some simple cases (i.e. for a simply supported rectangular plate). Therefore these equations are transformed in a way that perturbation calculation can be applied. Thus the solution is reduced to the solution of the corresponding elastic problem and to the finding of particular integrals. This method of analysis of orthotropic Maxwell shells is applied to the following examples: circular cylindrical shell, rectangular plate, and circular disk with locally variable reinforcement. It is found that partly the stress resultants are considerably different (more than 20%) from those of an elastic isotropic shell and after creeping even different from those of the corresponding orthotropic elastic shell. It may even occur, that the stress resultants in the viscoelastic component of the material act against the reinforcement stress resultants. The present work represents an extension and generalization of derivations by R. Trostel for the case of compound plates.

The Strain-Energy Expression for Thin Elastic Shells

Abstract A strain-energy expression is derived for thin isotropic elastic shells in terms of the displacements of the middle surface of the shell. This expression is confined to small-deflection theory, and the condition of plane stress previously used in the theory of thin shells is retained. A simplified expression is also obtained by the introduction of the Kirchhoff-Love hypothesis, and the relative merits of these two expressions are discussed. The strain-energy expression is applied to the thin circular cylinder, and the result is compared with various strain-energy expressions developed by previous authors.

A Strain-Energy Expression for Thin Elastic Shells

Abstract A derivation is given for the strain energy of an isotropic elastic shell whose radii of curvature are sufficiently large that strains may be assumed to vary linearly throughout the thickness. The work of Love (1) has been the only previous general investigation which expresses the strain energy in terms of the displacements of the middle surface. The effects of the tangential displacements upon the energy due to bending are found to differ appreciably from Love’s results in the first-order terms. As in the classical large-deflection theory of flat plates, quadratic terms in the derivatives of the normal deflection are retained in the strain tensor, but quadratic terms which involve the tangential displacements are neglected. Special forms of the general energy expression derived in this paper are given for shells in the shapes of flat plates, circular cylinders, elliptical cylinders, ellipsoids of revolution, and spheres. These applications, as well as certain intuitive observations, provide checks on the theory.