Nonlinear Shell Equations Research Articles

A Weak Form Implementation of Nonlinear Axisymmetric Shell Equations With Examples

Abstract We present a weak form implementation of the nonlinear axisymmetric shell equations. This implementation is suitable to study the nonlinear deformations of axisymmetric shells, with the capability of considering a general mid-surface shape, non-homogeneous (axisymmetric) mechanical properties and thickness variations. Moreover, given that the weak balance equations are arrived to naturally, any external load that can be expressed in terms of an energy potential can, therefore, be easily included and modeled. We validate our approach with existing results from the literature, in a variety of settings, including buckling of imperfect spherical shells, indentation of spherical and ellipsoidal shells, and geometry-induced rigidity (GIR) of pressurized ellipsoidal shells. Whereas the fundamental basis of our approach is classic and well established, from a methodological view point, we hope that this brief note will be of both technical and pedagogical value to the growing and dynamic community that is revisiting these canonical but still challenging class of problems in shell mechanics.

On the shell theory on the nanoscale with surface stresses

Below we discuss the derivation of the governing nonlinear shell equations taking into account the surface stresses. The surface effects are significant for the modeling of some structures as nanofilms, nanoporous materials and other nano-size structures. In particular, the surface stresses are responsible for the size effect, i.e. dependence of the material properties on the specimen size. The theory of elasticity with surface stresses is applied to the modeling of shells with nano-scaled thickness. It will be shown that the resultant stress and couple stress tensors can be represented as a sum of two terms. The first term in the sum depends on the stress distribution in the bulk material while the second one relates to the surface stresses. Hence, the resultant stress and couple stress tensors are linear functions with respect to the surface stresses. As an example the effective stiffness properties of a linear elastic Cosserat shells taking into account the surface stresses are presented.

Hadamard well-posedness for a class of nonlinear shallow shell problems

This paper is concerned with the nonlinear shallow shell model introduced in 1966 by W.T. Koiter in [On the nonlinear theory of thin elastic shells. III, Nederl. Akad. Wetensch. Proc. Ser. B 69 (1966) 33–54, Section 11] and later studied in [M. Bernadou, J.T. Oden, An existence theorem for a class of nonlinear shallow shell problems, J. Math. Pures Appl. (9) 60(3) (1981) 285–308]. We consider a version of this model which is based upon the intrinsic shell modeling techniques introduced by Michel Delfour and Jean-Paul Zolésio. We show existence and uniqueness of both regular and weak solutions to the dynamical model and that the solutions are continuous with respect to the initial data. While existence and uniqueness of regular solutions to nonlinear dynamic shell equations has been known, full Hadamard well-posedness of weak solutions, as shown in this paper, is a new result which solves an old open problem in the field.

Justification of two-dimensional nonlinear dynamic shell equations of Koiter's type

In this paper we give the justification of a two-dimensional nonlinear dynamic shells model of Koiter's type.

An Introduction to Differential Geometry with Applications to Elasticity

Preface Chapter 1. Three-dimensional differential geometry: 1.1. Curvilinear coordinates, 1.2. Metric tensor, 1.3. Volume, areas, and lengths in curvilinear coordinates, 1.4. Covariant derivatives of a vector field, 1.5. Necessary conditions satisfied by the metric tensor the Riemann curvature tensor, 1.6. Existence of an immersion defined on an open set in R3 with a prescribed metric tensor, 1.7. Uniqueness up to isometries of immersions with the same metric tensor, 1.8. Continuity of an immersion as a function of its metric tensor Chapter 2. Differential geometry of surfaces: 2.1. Curvilinear coordinates on a surface, 2.2. First fundamental form, 2.3. Areas and lengths on a surface, 2.4. Second fundamental form curvature on a surface, 2.5. Principal curvatures Gaussian curvature, 2.6. Covariant derivatives of a vector field defined on a surface the Gauss and Weingarten formulas, 2.7. Necessary conditions satisfied by the first and second fundamental forms: the Gauss and Codazzi-Mainardi equations Gauss' theorema egregium, 2.8. Existence of a surface with prescribed first and second fundamental forms, 2.9. Uniqueness up to proper isometries of surfaces with the same fundamental forms, 2.10.Continuity of a surface as a function of its fundamental forms Chapter 3. Applications to three-dimensional elasticity in curvilinear coordinates: 3.1. The equations of nonlinear elasticity in Cartesian coordinates, 3.2. Principle of virtual work in curvilinear coordinates, 3.3. Equations of equilibrium in curvilinear coordinates covariant derivatives of a tensor field, 3.4. Constitutive equation in curvilinear coordinates, 3.5. The equations of nonlinear elasticity in curvilinear coordinates, 3.6. The equations of linearized elasticity in curvilinear coordinates, 3.7. A fundamental lemma of J.L. Lions, 3.8. Korn's inequalities in curvilinear coordinates, 3.9. Existence and uniqueness theorems in linearizedelasticity in curvilinear coordinates Chapter 4. Applications to shell theory: 4.1. The nonlinear Koiter shell equations, 4.2. The linear Koiter shell equations, 4.3. Korn's inequality on a surface, 4.4. Existence and uniqueness theorems for the linear Koiter shell equations covariant derivatives of a tensor field defined on a surface, 4.5. A brief review of linear shell theories References Index.

Vibration of a hemispherical shell gyro excited by an electrostatic field

In this paper the stability and precession phenomenon of the modal pattern of a rotating hemispherical shell gyro excited parametrically by electrostatic forces is investigated. Nonlinear modal equations of the rotating shell under small strain and moderate rotation are derived by following Niordson's thin shell theory. The method of multiple time scales is used to determine quantitatively the vibration displacement of the shell from which the precession rate of the modal vibration pattern can be extracted. Floquet theory is employed to study the stability of the analytical solution. Numerical simulation verifies the analytical results. Recent work by Matthews and Rybak (1), and Chou and Chang (2,3) showed that the hemispherical shell gyroscope becomes attractive and competitive in comparison with other types of gyros. The work- ing principle is that the modal vibration pattern of the shell moves when the shell is rotated about its axis and this movement provides a measure of the applied rate of turn. An important design feature of the hemispherical resonator gyroscope is that one is able to excite the shell without having a direct mechan- ical coupling between the shell and its supporting frame. In order to maintain the precessing vibration modal pattern, an electrostatic excitation is employed. Linear analysis (3) of a rotating hemispherical shell gyro excited parametrically by electrostatic forces showed that the vibration amplitude is decayed to zero in the stable region and becomes divergent in the unstable region under excitation. Therefore the geometrically nonlinear (4) effects of the shell under small strain and moderate rotation (5) are studied. Reissner (6) derived the geometrical nonlinear shell equations with Kirchhoff-Love assumption. Since that, many people continuously improved the nonlinear shell theories (7). In this paper, the nonlinear modal equations of the rotating shell under small strain and moderate rotation are derived by following Niordson's (8) thin shell theory. The method of multiple time scales (9) is used to determine quantitatively the vibration amplitude of the shell from which the precessional rate of the modal vibration pattern can be extracted. Floquet (10) theory is employed to study the stability of the analytical solution. Numerical simulation verifies the analytical results.

Theoretical and numerical studies of nonlinear shell equations

We study the solution field M of a parameter dependent nonlinear two-point boundary value problem presented by Troger and Steindl [H. Troger, A. Steindl, Nonlinear Stability and Bifurcation Theory, Springer, Wien, New York, 1991]. This problem models the buckling of a thin-walled spherical shell under a uniform external static pressure. The boundary value problem is formulated as an abstract operator equation T( x, λ)=0 in appropriate Banach spaces. By exploiting the equivariance of T, we obtain detailed informations about the structure of M . These theoretical results are used to compute efficiently interesting parts of M with numerical standard techniques. Bifurcation diagrams, a stability diagram and pictures of deformed shells are presented.

曲げと圧縮による円筒殻の座屈

This paper presents results of an experimental study on the buckling of circular cylindrical shells loaded simultaneously with a bending moment and a compressive force. A high precision loading apparatus was designed, by which both the torque and the axial compression can be applied equally on both ends of the cylinder, so that the state of the bending under axial compression is established in the cylinder. Test specimens were made from polyester films, which had length-to-diameter ratios of 2, 5 and 10 and a diameter-to-thickness ratio of 300. In the experiment, the axial force was applied to induce axial stress varying in the range from 0 to 50% of the classical buckling stress. It has been shown that the buckling occurs when the bending stress added to the compressive stress approaches to the classical buckling stress. The experimental buckling stresses were compared with theoretical ones obtained by solving Budiansky's non-linear shell equations numerically.

Stokes flow in an elastic tube—a large-displacement fluid-structure interaction problem

SUMMARY Viscous flow in elastic (collapsible) tubes is a large-displacement fluid-structure interaction problem frequently encountered in biomechanics. This paper presents a robust and rapidly converging procedure for the solution of the steady three-dimensional Stokes equations, coupled to the geometrically non-linear shell equations which describe the large deformations of the tube wall. The fluid and solid equations are coupled in a segregated method whose slow convergence is accelerated by an extrapolation procedure based on the scheme’s asymptotic convergence behaviour. A displacement control technique is developed to handle the system’s snap-through behaviour. Finally, results for the tube’s post-buckling deformation and for the flow in the strongly collapsed tube are shown. © 1998 John Wiley & Sons, Ltd.

Nondimensional Parameters for Geometric Nonlinear Effects in Pressurized Cylinders With Axial Cracks

The effects of geometric nonlinearity on the response of axially cracked cylindrical shells under internal pressure are investigated in a general way. Using the Donnell-Mushtari-Vlasov nonlinear shell equations, the nonlinear response is shown to depend on two nondimensional parameters: the geometrical parameter λ, which is a function of the cylinder geometries and crack length, and the loading parameter η, which depends on the applied pressure, material properties, and cylinder geometries. To assess the applicability of such parameters, nonlinear analyses of different cylindrical configurations were performed using the STAGS finite element code. The results show that the two parameters are able to characterize the nonlinear response of such cylinders. Effects of nonlinearity are then presented in the form of an iso-nonlinear plot showing the percentage difference between the linear and nonlinear stress intensification factors. Using the iso-nonlinear plot, the importance of geometric nonlinearity can thus be assessed once the cylinder geometries, loading parameters, and material properties are known.

On a stress resultant geometrically exact shell model. Part VII: Shell intersections with [formula omitted]-DOF finite element formulations

If the mid-surface of a shell is smooth, the classical theory describes the orientation of the director field attached to the mid-surface by two independent (rotational) degrees of freedom. At a shell intersection, where this smoothness assumption no longer holds, it is shown that the director field in the full nonlinear continuum shell equations must be necessarily described by three degrees of freedom. This added degree of freedom, however, is totally unrelated to the so-called drill rotation, widely used as a means of tackling the shell intersection problem. A computational procedure involving a trivial modification of the global singularity-free update procedure described in Part III of this work is described, which completely resolves the shell intersection problem without introducing ‘drill springs’ or related ad-hoc devices. The proposed approach leaves unchanged standard finite element formulations in terms of 5 DOF/node, affects only the global update formulae and exhibits excellent performance, as illustrated by representative numerical simulations.

部分荷重を受ける球形シェルの非保存的安定性解析

The purpose of the-present study is to show theoretically that dynamic instability takes place in a clamped spherical shell under a partially distributed normal follower force. Large deflection and stability of shallow and deep spherical shells are studied. The axisymmetric equilibrium states are solved through a Newton-Raphson technique on discretized nonlinear shell equations by means of finite difference method. The stability of the respective equilibrium states on the axisymmetric fundamental equilibrium path is examined by the dynamic method. In this study, it is shown that transition from stability to instability takes place not only in the form of static instability, but also in the form of dynamic instability.

The elastic analysis of a dent on pressurised pipe

The elastic analysis of a dent on a pressurised cylinder is investigated. The predetermined shapes of the dent are grouped into three categories according to the physical geometry of the dent width, namely the local dent, the short dent, and the long dent. The induced bending stresses associated with the dent are evaluated using Sanders's non-linear shell theory. The geometry of the dent is treated as an initial geometric imperfection and incorporated in the expressions of strain. The non-linear cylindrical shell equations thus obtained are linearised and then solved by a Fourier series expansion technique. Comparison of theoretical results with the finite element approach is made for a pipe of assumed dimensions and pressure. Good agreement of the results is observed. The analytical approach of this paper is suitable for pipe or cylinder with a dent depth of up to five times the wall thickness.

Large Deflection Stability of Spherical Shells With Ring Loads

Large deflections of shallow and deep spherical shells under ring loads are studied. The axisymmetric problem is solved through a Newton-Raphson technique on discretized nonlinear shell equations. Comparison of computed load-deflection curves to experimental data from both thick and thin shells generally shows good agreement in peak loads and the type of instability. For a point load, the load increases monotonically with deflection; as the ring radius increases, transition-type (snap-through) and then local buckling occurs. In addition, the pre- and post-buckled mechanical behaviors of the shell are examined.

Derivation of geometrically nonlinear finite shell elements via tensor notation

AbstractTo develop geometrically nonlinear, doubly curved finite shell elements the basic equations of nonlinear shell theories have to be transferred into the finite element model. As these equations in general are written in tensor notation, their implementation into the finite element matrix formulation requires considerable effort. The present paper will demonstrate how to derive the nonlinear element matrices directly from the incrementally formulated nonlinear shell equations using a tensor‐oriented procedure. This enables the numerical realization of all structural responses, e.g. the calculation of pre‐ and post‐buckling branches in snap‐through analysis and especially in bifurcation analysis, including the detection of critical points and the consideration of geometric imperfections. To avoid loss of accuracy care is taken for a realistic computation of the geometric properties as well as of the external loads. Finally, the developed family of shell elements will be presented and its efficiency will be demonstrated by some applications to linear and geometrically nonlinear structural phenomena.

Large deformation analysis of ellipsoidal head pressure vessels

Among the most common pressure vessels, the elliptical head pressure vessel stands out to be the second best in order of preferable stress distribution at the head-cylinder junction—the best being hemispherical head. But, like others, the deformation at the head-cylinder junction of this pressure vessel is so large that the stress distribution obtained from linear shell theory is far off from reality. The nonlinear shell equations of ellipsoidal head pressure vessels are solved here using the multisegment method of integration, developed by Kalnins and Lestingi[11]. Reissner's nonlinear equations for the axisymmetric deformation of shells of revolution, after specialization for ellipsoidal shells and further necessary modification for avoiding geometrical singularity at the apex of the head are used in this analysis. It is shown that the linear theory consistently over estimates the stresses at the head-cylinder junction of ellipsoidal head pressure vessels.

Imperfection sensitivity of fiber-reinforced, composite, thin cylinders

The imperfection sensitivity of thin cylindrical shells, made out of fiber-reinforced composite material and subjected to uniform axial compression, and the effects upon it of certain parameters, are investigated. The methodology is based on linear constitutive relations, nonlinear kinematic shell equations (Donnell-type) and the usual lamination theory. The laminate consists of orthotropic laminae, stacked in a general manner (asymmetric laminate). The uniform axial compression is applied eccentrically, and the geometrically imperfect cylindrical shell can be supported in various ways at the boundaries. In this investigation a number of parametric studies are performed. The scope of these studies is to establish the effect of (a) in-plane and transverse boundary conditions and (b) load eccentricity, on the imperfection sensitivity of typical boron/epoxy laminated cylindrical shells with various stacking sequences of laminate. The sensitivity is established by calculating critical loads for various imperfection amplitudes and shapes.

The Effect of Length: Diameter Ratio on Collapse of Casing

This paper presents an analysis of the cross-sectional collapse of a cylinder of finite length loaded simultaneously by an axial tension (which may be zero) and external pressure. The calculation is based on Sanders’ nonlinear shell equations with plasticity introduced via the concept of effective stress from a uniaxial tension test. The finite cylinder is an appropriate model of oil well casing as it undergoes quality control testing in the steel mill where the edges of the cylinder are usually fixed in the case of nonzero axial load and free in the case of zero axial load. However, in field application, the length: diameter ratio of casing is such that the cylinder may be considered infinite. Guidelines contained herein permit prediction of the collapse resistance of field casing from the results of mill tests performed on short samples.

Nonlinear stress-strain and strength response of axisymmetric bimodulus composite material shells

Bimodulus materials are those which exhibit different material response when stressed in compression rather than tension. Strength predictions of laminated composites can be affected by bimodulus material response and nonlinear stress-strain behavior in individual laminae. Most applications of phenomenological failure models to laminated composites assume that lamina yield and ultimate lamina strength are the same. In addition, bimodulus material response is frequently neglected when applying failure models to predict laminate strength. This paper presents an investigation of the effects which two bimodulus material models have on strength predictions for graphite-epoxy composites. Nonlinear response of composite laminae are modeled in an approximate manner by employing piecewise linear and spline representations for transverse compression and shear stress-strain curves. The Tsai-Wu and Sandhu failure criteria are applied to. predict failure of thin, laminated shells. The strength predictions are based on numerical solutions to Sanders' axisymmetric nonlinear shell equations. Numerical predictions for the strength and stress-strain response of internally pressurized rings are compared to experimental data cited from the literature.

Influence of initial geometrical imperfections on vibrations of axially compressed stiffened cylindrical shells

The vibrations of stiffened cylindrical shells having axisymmetric or asymmetric initial geometrical imperfections and axial preload are analyzed. The analysis is based on a solution of the von Kárman-Donnell non-linear shell equations, an “exact” solution of the compatibility equation, and a first order approximation by the Galerkin method of the equilibrium equation. The stiffeners are closely spaced and “smeared” stiffener theory is employed. The results of an extensive parametric study carried out on shells similar to those used in vibration and buckling tests at the Technion show that stiffening of the shell will lower the imperfection-sensitivity of its free vibrations, but the decrease depends on the type of stiffening (stringers or rings), the mode shapes of the vibration and the imperfection, the stiffener strength and eccentricity. The imperfection-sensitivity decrease, caused by the stiffeners, is greater for vibration mode shapes with high imperfection-sensitivity than for other vibration mode shapes. The sensitivity differences between stringer and ring-stiffened shells depend especially on the vibration and the imperfection mode shapes, and on their coupling. Small imperfections change the natural frequencies of stiffened shells in the same directions as for isotropic shells, but to a smaller extent. The frequency dependence on the external load is also strongly affected by the imperfection mode shape. The results correlate well with earlier ones for isotropic shells.