Nonlinearly Elastic Shells Research Articles

Numerical treatment of a geometrically nonlinear planar Cosserat shell model

We present a new way to discretize a geometrically nonlinear elastic planar Cosserat shell. The kinematical model is similar to the general six-parameter resultant shell model with drilling rotations. The discretization uses geodesic finite elements (GFEs), which leads to an objective discrete model which naturally allows arbitrarily large rotations. GFEs of any approximation order can be constructed. The resulting algebraic problem is a minimization problem posed on a nonlinear finite-dimensional Riemannian manifold. We solve this problem using a Riemannian trust-region method, which is a generalization of Newton's method that converges globally without intermediate loading steps. We present the continuous model and the discretization, discuss the properties of the discrete model, and show several numerical examples, including wrinkling of thin elastic sheets in shear.

A rod model with thin-walled flexible cross-section: Extension to 3D motions and application to 3D foldings of tape springs

A planar rod model with flexible cross-section has been proposed and discussed recently in literature (Guinot et al., 2012; Picault et al., 2014). This 1D model is especially suitable for the modeling of tape springs, which develop localized folds through the flattening of the cross-section, and represents a good alternative to 2D shell models which are numerically hard to perform. However, due to its planar nature, it does not account for the coupling between bending and twisting that is noticed experimentally. In this context, we propose an extension of this rod model to 3D motions that includes torsional warping of the cross-section, based on postulated assumptions that are approximations. Starting from a complete non-linear elastic shell model, we introduce an original kinematics inspired from the elastica model and from Vlassov’s theory (Vlassov, 1962) in order to describe in-plane and out-of-plane changes of the cross-section shape with few parameters. In the specific case of shallow tape springs, approximate expressions of the strain and kinetic energies are derived by performing an analytical integration over the cross-section. The 1D tape spring rod model eventually involves only eight kinematic variables: seven to describe the translation and the rotation of the cross-section (parameterized by a unit quaternion) and one for the shape of the cross-section which is assumed to remain circular. Expressions of the potential and kinetic energies are then introduced in a suitable finite element software that can perform an automatic differentiation to solve the problem. Two static cases are treated. The first one, a tape spring clamped at one end and submitted at the other end to a follower twisting moment, shows that the model account for the bending-twisting couplings in large displacements and large rotations. The second one, a novel test named the transverse bending test, illustrates the ability of the model to account for complex scenarios of 3D foldings, involving bending and twisting as well as the creation of folds, their migration along the tape and their duplication. Quantitative comparisons to numerical results obtained with a shell finite element model are shown.

Radially Symmetric Steady States of Nonlinearly Elastic Plates and Shells

This paper treats the subject of its title, showing that despite its apparent simplicity, it is very rich: The existence, multiplicity, and qualitative behavior of solutions for problems accounting for the live loads due to hydrostatic pressure and centrifugal force depend critically on the material properties of the bodies, physically reasonable refined descriptions of which are given and examined here with great care, and on the nature of boundary conditions. The treatment here, giving new and sharp results, employs several different mathematical tools, ranging from phase-plane analysis to the mathematically more sophisticated direct methods of the calculus of variations, fixed-point theorems, and global continuation methods, each of which has different strengths and weaknesses for handling intrinsic difficulties in the mechanics. The practical importance of these results is that for each problem involving live loads they furnish thresholds in material response delimiting materials for which solutions are ill behaved. A mathematical or numerical study limited to a particular class of materials may dangerously indicate well-behaved solutions when there are other realistic materials for which solutions are ill behaved.

Some Differential Geometric Relations in the Elastic Shell

The theory of the elastic shells is one of the most important parts of the theory of solid mechanics. The elastic shell can be described with its middle surface; that is, the three-dimensional elastic shell with equal thickness comprises a series of overlying surfaces like middle surface. In this paper, the differential geometric relations between elastic shell and its middle surface are provided under the curvilinear coordinate systems, which are very important for forming two-dimensional linear and nonlinear elastic shell models. Concretely, the metric tensors, the determinant of metric matrix field, the Christoffel symbols, and Riemann tensors on the three-dimensional elasticity are expressed by those on the two-dimensional middle surface, which are featured by the asymptotic expressions with respect to the variable in the direction of thickness of the shell. Thus, the novelty of this work is that we can further split three-dimensional mechanics equations into two-dimensional variation problems. Finally, two kinds of special shells, hemispherical shell and semicylindrical shell, are provided as the examples.

Strain solitary waves in a thin-walled waveguide

A mathematical model is proposed to describe bulk longitudinal waves in a nonlinearly elastic thin-walled cylindrical shell. The equation of motion for the longitudinal displacement is derived. In case of the homogeneous shell, this equation is reduced to the doubly dispersive equation for the linear longitudinal strain component and provides a solitary wave solution. Results of the first experimental observation of the bulk strain soliton in a duct-like shell are presented, and both the wave amplitude and velocity are estimated.

A planar rod model with flexible cross-section for the folding and the dynamic deployment of tape springs: Improvements and comparisons with experiments

A planar rod model with flexible cross-section has been recently proposed in literature (Guinot et al., 2012). This model is especially suitable for the modeling of tape springs, which develop localized folds due to the flattening of the cross-section. Starting from a complete non-linear elastic shell model, original kinematics assumptions (inspired from the elastica model) have been made to describe the important in-plane changes of the cross-section shape. In the present work, the choice of the position of the rod reference line is discussed. This choice plays an important role in the overall behavior because of the large changes of the cross-section shape. We show that the model published in Guinot et al. (2012) can be improved by considering the centerline as the rod reference line. This enhanced model is then validated through quantitative comparisons with experimental results of dynamic deployments taken from literature.

A new approximate model of nonlinearly elastic flexural shell and its numerical computation

In this article, we construct a two‐dimensional model for the nonlinearly elastic flexural shell using differential geometry and tensor analysis under the assumption that flexural energy is dominant, that is, the metric of middle surface remains invariant. We conduct a numerical experiment for special shell—a portion of cylinder shell, which is applied to the forces along the opposite direction of the normal vector. The displacements distribution of all points in the middle surface when the shell deforms is obtained. Numerical experiment results are consistent with the theory, which proves the validity of the proposed model. We then compare the proposed model and Ciarlet's model with 3D model, which proves that the proposed model is more approximate to 3D model than Ciarlet's. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1727–1739, 2014

Isogeometric shell discretizations for flexible multibody dynamics

This work aims at including nonlinear elastic shell models in a multibody framework. We focus our attention to Kirchhoff–Love shells and explore the benefits of an isogeometric approach, the latest development in finite element methods, within a multibody system. Isogeometric analysis extends isoparameteric finite elements to more general functions such as B-splines and NURBS (Non-Uniform Rational B-Splines) and works on exact geometry representations even at the coarsest level of discretizations. Using NURBS as basis functions, high regularity requirements of the shell model, which are difficult to achieve with standard finite elements, are easily fulfilled. A particular advantage is the promise of simplifying the mesh generation step, and mesh refinement is easily performed by eliminating the need for communication with the geometry representation in a CAD (Computer-Aided Design) tool. Target applications are wind turbine blades and twist beam rear suspensions. First numerical examples demonstrate an impressive convergence behavior of the isogeometric approach even for a coarse mesh, while offering substantial savings with respect to the number of degrees of freedom.

Orientation-preserving condition and polyconvexity on a surface: Application to nonlinear shell theory

We propose in this paper a definition of a “polyconvex function on a surface”, inspired by the definitions set forth in other contexts by J. Ball (1977) [3] and by J. Ball, J.C. Currie, and P.J. Olver (1981) [5]. When the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function, we show that it is possible to assume in addition that this function is coercive for appropriate Sobolev norms and that it satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the “surface analogue” of the orientation-preserving condition of J. Ball. We then show that a functional with such a polyconvex integrand is weakly lower semi-continuous, a property which eventually allows to establish the existence of minimizers. We also indicate how this new approach compares with the classical nonlinear shell theories, such as those of W.T. Koiter and P.M. Naghdi.

Existence Result for a Dynamical Equations of Generalized Marguerre-von Kármán Shallow Shells

In a recent work in the static case, Gratie (Appl. Anal. 81:1107–1126, 2002) has generalized the classical Marguerre-von Karman equations studied by Ciarlet and Paumier in (Comput. Mech. 1:177–202, 1986), where only a portion of the lateral face of the shallow shell is subjected to boundary conditions of von Karman type, while the remaining portion is subjected to boundary conditions of free edge. Then Ciarlet and Gratie (Math. Mech. Solids 11:83–100, 2006) have established an existence theorem for these equations. In Chacha et al. (Rev. ARIMA 13:63–76, 2010), we extended formally these studies to the dynamical case. More precisely, we considered a three-dimensional dynamical model for a nonlinearly elastic shallow shell with a specific class of boundary conditions of generalized Marguerre-von Karman type. Using technics from formal asymptotic analysis, we showed that the scaled three-dimensional solution still leads to two-dimensional dynamical boundary value problem called the dynamical equations of generalized Marguerre-von Karman shallow shells. In this paper, we establish the existence of solutions to these equations using a compactness method of Lions (Quelques Methodes de Resolution des Problemes aux Limites non Lineaires, Dunod, Paris, 1969).

Математическое и компьютерное моделирование динамики нелинейных волн в физически нелинейных упругих цилиндрических оболочках, содержащих вязкую несжимаемую жидкость

Математическое и компьютерное моделирование динамики нелинейных волн в физически нелинейных упругих цилиндрических оболочках, содержащих вязкую несжимаемую жидкость

Asymptotic analysis of nonlinearly elastic shells “mixed approach”

The aim of this paper is to extend the results obtained by Miara [Arch. Ration. Mech. Anal. 142 (1998), 331–353] and Lods and Miara [Arch. Ration. Mech. Anal. 142 (1998), 335–374] on the asymptotic analysis of nonlinearly elastic shells to the nonhomogeneous anisotropic case. To this end, we use the Hellinger–Reissner variational principle.

A notion of polyconvex function on a surface suggested by nonlinear shell theory

Combining the definitions set forth by J. Ball in 1977 and by J. Ball, J.C. Currie, and P.J. Olver in 1981, we propose in this Note a definition of a “polyconvex function on a surface”. When the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function, we show that it is possible to assume in addition that this function is coercive for appropriate Sobolev norms and that it satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the “surface analogue” of the orientation-preserving condition of J. Ball. We then show that a functional with such a polyconvex integrand is weakly lower semi-continuous, which eventually allows to establish the existence of minimizers.

On exact expressions of the bending tensor in the nonlinear theory of thin shells

Some equivalent exact expressions of the bending tensor in the nonlinear theory of thin shells are reviewed. It is noted that the bending tensor, proposed by Shen et al. (2010) [X.Q. Shen, K.T. Li, Y. Ming “The modified model of Koiter’s type for the nonlinearly elastic shells”, Appl. Math. Model. 34 (2010) 3527–3535] as a third-degree polynomial of displacements, is an approximate expression, not the exact one. Then integrability of the fourth kinematic boundary condition, associated with two different but equivalent exact expressions of the bending tensor, is briefly discussed. Finally, a few modified definitions of the bending tensor proposed in the literature are recalled. Within the first-approximation theory they all lead to energetically equivalent models of elastic shells.

Limiting state of a multilayered nonlinearly elastic long cylindrical shell under the action of nonuniform external pressure

The buckling of a long multilayered nonlinearly elastic shell made of different materials and subject to the action of external pressure is investigated. The load is not hydrostatic and greatly varies in value and direction. Neglecting the effect of end fastening of the shell, the problem is reduced to an analysis of the loss of load-carrying ability of a ring of unit width separated from the shell. The solution is based on a variational method of mixed type formulated for heterogeneous nonlinearly elastic bodies, taking into account the geometrical nonlinearity, in a combination with the Rayleigh–Ritz method. The initial analysis is reduced to solving the Cauchy problem for a nonlinear ordinary differential equation resolved for the derivative. Numerically, using the Runge–Kutta method, the effect of the number of layers and of the parameter of nonuniformity of the external pressure on the critical buckling force is revealed. The urgency and importance of the problem are connected with the research of reserves in the saving of materials with a simultaneous possibility of increasing the load-carrying ability of a structure.

Equilibrium of an Elastic Spherical Shell Filled with a Heavy Fluid under Pressure

This paper considers the problem of equilibrium of a nonlinearly elastic spherical shell filled with a heavy fluid and resting on a smooth, absolutely rigid, flat surface. The weight of the shell is assumed to be negligible in comparison with the weight of the fluid filling it. The contact region with the supporting plane is one of the unknowns in the problem. Equilibrium equations for a membrane shell are obtained in an accurate nonlinear formulation. Stresses and strains of a shell made of a Mooney–Rivlin material are numerically investigated. The results are compared with calculation results for the case of inflation of a spherical shell ignoring the weight of the fluid filling. The effect of the fluid weight on shell strains and stresses is estimated.

The modified model of Koiter’s type for nonlinearly elastic shell

In this paper, we proposed a modified model of Koiter’s type for nonlinearly elastic shell. The change of metric tensor and the change of curvature tensor play an important role in constructing the linearly and nonlinearly elastic shell model of Koiter’s type. The approximate expressions of them once were proposed by Ciarlet. In this paper, the exact full expressions of the change of metric tensor and the change of curvature tensor are provided by tensor analysis. The former coincides with Ciarlet’s expression. And the latter is more exact than Ciarlet’s expression. Thus the modified model is better than Ciarlet’s model. At the same time, a numerical experiment of special hemispherical shell is provided to validate the modified model of Koiter’s type.

Discussion about parameterization in the asymptotic numerical method: Application to nonlinear elastic shells

The Asymptotic Numerical Method (ANM) is a family of algorithms for path following problems based on the computation of truncated vectorial series with respect to a path parameter “ a” [B. Cochelin, N. Damil, M. Potier-Ferry, Méthode Asymptotique Numérique, Hermès-Lavoisier, Paris, 2007]. In this paper, we discuss and compare three concepts of parameterizations of the ANM curves i.e. the definition of the path parameter “ a”. The first concept is based on the classical arc-length parameterization [E. Riks, Some computational aspects of the stability analysis of nonlinear structures, Computer Methods in Applied Mechanics and Engineering, 47 (1984) 219–259], the second is based on the so-called local parameterization [W. C. Rheinboldt, J. V. Burkadt, A Locally parameterized continuation, Acm Transaction on mathematical Software, 9 (1983) 215–235; R. Seydel, A Tracing Branches, World of Bifurcation, Online Collection and Tutorials of Nonlinear Phenomena ( http://www.bifurcation.de), 1999; J. J. Gervais, H. Sadiky, A new steplength control for continuation with the asymptotic numerical method, IAM, J. Numer. Anal. 22, No. 2, (2000) 207–229] and the third is based on a minimization condition of a rest [S. Lopez, An effective parametrization for asymptotic extrapolation, Computer Methods in Applied Mechanics and Engineering, 189 (2000) 297–311]. We demonstrate that the third concept is equivalent to a maximization condition of the ANM step lengths. To illustrate the performance of these proposed parameterizations, we give some numerical comparisons on nonlinear elastic shell problems.

The Interaction of the 3D Navier–Stokes Equations with a Moving Nonlinear Koiter Elastic Shell

We study a moving boundary value problem consisting of a viscous incompressible fluid moving and interacting with a nonlinear elastic solid shell. The fluid motion is governed by the Navier–Stokes equations, while the shell is modeled by the nonlinear Koiter shell model, consisting of bending and membrane tractions as well as inertia. The fluid is coupled to the solid shell through continuity of displacements and tractions (stresses) along the moving material interface. We prove existence and uniqueness of solutions in Sobolev spaces.

Geometrically Exact Nonlinear Extended-Reissner/Mindlin Shells: Fundamentals, Finite Element Formulation, Elasticity

Extended Reissner/Mindlin shell kinematics are used to develop an exact shell that makes no further approximations on the deformation or strain. The formulation is fully tensorial. A three-dimensional isoparametric element is used as the reference configuration. The FE approximation introduces nodes on the mid-surface of the isoparametric element, with the thickness having a separately defined interpolation. The variational formulation is based on the nominal stress tensor. Isotropic linear and nonlinear elastic shells are constructed that use p-type interpolation along the director. The formulation is exact, resulting in nonlinear elements showing better characteristics in comparison to other such elements.