Exact Shell Theory Research Articles

3D hygro-elastic shell model for the analysis of composite and sandwich structures

The present work proposes the study of the hygrometric loading effects in the static analysis of multilayered composite and sandwich plates and shells. The employed model is based on a general exact 3D shell theory valid for one-layered and multilayered sandwich and composite plates, cylinders and cylindrical/spherical shell panels. The employed 3D equilibrium equations are developed in an orthogonal mixed curvilinear coordinate system for spherical shells in order to obtain the other simpler geometries as particular cases. A layer-wise approach is employed and equilibrium equations are solved in the thickness direction via the exponential matrix method. The partial derivatives in the in-plane directions are exactly calculated by assuming simply-supported edges and harmonic forms for the variables. The presented 3D shell model is extended to hygro-elastic cases by considering moisture content amplitudes imposed at the external surfaces of the structures in steady-state conditions. The moisture content profile through the thickness of the structures can be included in the 3D shell model using three different methodologies: it can be calculated by solving the steady-state 3D or 1D version of the Fick diffusion law, or it can be “a priori” assumed as linear through the thickness. Therefore, the developed system includes three non-homogeneous second order differential equilibrium equations that are solved after the introduction of opportune mathematical layers to consider the curvature effects through the thickness. The reduction to a first order differential equation system is obtained by redoubling the number of variables. The exponential matrix method is employed to accurately define both general and particular solutions. The implemented 3D hygro-elastic shell model is firstly validated and then it is employed with confidence for new benchmarks where the effects of thickness ratio, geometry, lamination scheme, material, thickness layer and imposed moisture content values at the external surfaces are investigated for composite and sandwich plates and shells. Showed results demonstrate the importance of an accurate developing of the elastic part of the 3D shell model combined with an appropriate evaluation of the moisture content profile through the thickness. Assumed linear moisture content profiles are correct only for thin one-layered isotropic structures, the use of the 1D Fick diffusion law allows only the correct definition of the material layer effect, the use of its 3D version allows the correct definition of both material and thickness layer effects.

On nonlocal geometrically exact shell theory and modeling fracture in shell structures

In this work, we have developed a nonlocal geometrically-exact shell theory and its computational formulation, which can model fracture and crack growth in shell structures under finite deformations. We have derived the local form of the nonlocal balance laws for a nonlocal continuum, which are instrumental in developing the nonlocal stress-resultant based geometrically-exact shell theory. This approach is particularly efficient for modeling material damage and structural failure process with strong discontinuities. A meshfree Galerkin weak formulation is developed for the nonlocal geometrically-exact shell theory by using nonlocal differential operators. Several numerical examples, including finite deformation and fractures, are conducted to verify the effectiveness of the present method. The numerical results demonstrate that the proposed nonlocal shell theory is an accurate and robust method to model the failure of shell structures.

Comparing quantum, molecular and continuum models for graphene at large deformations

In this paper, the validity and accuracy of three interatomic potentials and the continuum shell model of Ghaffari and Sauer [1] are investigated. The mechanical behavior of single-layered graphene sheets (SLGSs) under uniaxial stretching, biaxial stretching and pure bending is studied for this comparison. The validity of the molecular and continuum models is assessed by direct comparison with density functional theory (DFT) data available in the literature. The molecular simulations are carried out employing the MM3, Tersoff and REBO + LJ potentials. The continuum formulation uses an anisotropic hyperelastic material model in the framework of the geometrically exact Kirchhoff-Love shell theory and isogeometric finite elements. Results from the continuum model are in good agreement with those from DFT. The results from the MM3 potential agree well up to the point of material instability, whereas those from the REBO + LJ and Tersoff potentials agree only for small deformations. Only the Tersoff potential is found to yield auxetic response in SLGSs under uniaxial stretch. Additionally, the transverse vibration frequencies of a pre-stretched graphene sheet and a carbon nanocone are obtained using the continuum model and molecular simulations with the MM3 potential. The variations of the frequencies from these approaches agree within an error of ≈5%.

A computational framework for incompressible electromechanics based on convex multi-variable strain energies for geometrically exact shell theory

In this paper, a new computational framework for the analysis of incompressible Electro Active Polymer (EAP) shells subjected to large strains and large electric fields is presented. Two novelties are incorporated in this work. First, the variational and constitutive frameworks developed by the authors in recent publications (Gil and Ortigosa, 2016; Ortigosa and Gil, 2016; Ortigosa et al., 2016) in the context of three-dimensional electromechanics are particularised/degenerated to the case of geometrically exact shell theory. This formulation is computationally very convenient as EAPs are typically used as thin shell-like components in a vast range of applications. The proposed formulation follows a rotationless description of the kinematics of the shell, enhanced with extra degrees of freedom corresponding to the thickness stretch and the hydrostatic pressure, critical for the consideration of incompressibility. Different approaches are investigated for the interpolation of these extra fields and that of the electric potential across the thickness of the shell. Crucially, this allows for the simulation of multilayer and composite materials, which can display a discontinuous strain distribution across their thickness. As a second novelty, a continuum degenerate approach allows for the consideration of complex three-dimensional electromechanical constitutive models, as opposed to those defined in terms of the main strain measures of the shell. More specifically, convex multi-variable (three-dimensional) constitutive models, complying with the ellipticity condition and hence, satisfying material stability for the entire range of deformations and electric fields, are used for the first time in the context of shell theory.

Shear Effects on Energy Dissipation From an Elastic Beam on a Rigid Foundation

This work describes the energy dissipation arising from microslip for an elastic shell incorporating shear and longitudinal deformation resting on a rough-rigid foundation. This phenomenon is investigated using finite element (FE) analysis and nonlinear geometrically exact shell theory. Both approaches illustrate the effect of shear within the shell and observe a reduction in the energy dissipated from microslip as compared to a similar system neglecting shear deformation. In particular, it is found that the shear deformation allows for load to be transmitted beyond the region of slip so that the entire interface contributes to the load carrying capability of the shell. The energy dissipation resulting from the shell model is shown to agree well with that arising from the FE model, and this representation can be used as a basis for reduced order models that capture the microslip phenomenon.

Geometrically exact displacement-based shell theory

The deformed geometry often is the most important information for applications of highly flexible plates/shells, and a geometrically exact shell theory should be displacement-based in order to directly and exactly describe any greatly deformed geometry. The main challenges of modeling a shell undergoing large deformation are how to describe its deformed reference plane and its differential element's large rotations and how to derive objective strains in terms of global displacements and rotations that contain both elastic straining and rigid-body movement. This paper presents a truly geometrically exact displacement-based shell theory without singularity problems. The theory fully accounts for geometric nonlinearities, all possible initial curvatures, and extensionality by using Jaumann strains and stresses, exact coordinate transformation, and orthogonal virtual rotations. Moreover, transverse shear deformations are accounted for by using a high-order shear deformation theory. The derived fully nonlinear strain–displacement relations enable geometrically exact forward analysis (obtaining the deformed geometry under a set of known loads) and inverse analysis (obtaining the required loads for a desired deformed geometry). Several numerical examples are used to demonstrate the accuracy and capabilities of the geometrically exact shell theory. Moreover, different theoretical and numerical problems of other geometrically nonlinear shell theories are shown to be mainly caused by the use of Mindlin plate theory to account for transverse shears, Green–Lagrange strains to account for geometric nonlinearities, and/or Euler and Rodrigues parameters to model large rotations.

A high order theory for functionally graded axisymmetric cylindrical shells

In this paper a high order theory for functionally graded (FG) axisymmetric cylindrical shells based on the expansion of the axisymmetric equations of elasticity for functionally graded materials (FGMs) into Fourier series in terms of Legendre's polynomials is presented. Starting from the axisymmetric equations of elasticity, the stress and strain tensors, the displacement, traction and body force vectors are expanded into Fourier series in terms of Legendre's polynomials in the thickness coordinate. In the same way the material parameters that describe the functionally graded material properties are also expanded into Fourier series. All equations of the linear elasticity including Hooke's law are transformed into the corresponding equations for the Fourier series expansion coefficients. Then a system of differential equations in terms of the displacements and the boundary conditions for the Fourier series expansion coefficients is obtained. In particular the first and second order approximations of the exact shell theory are considered in more details. The obtained boundary-value problems are solved by the finite element method (FEM) with COMSOL Multiphysics and MATLAB software. Numerical results are presented and discussed.

Applications of point interpolation method for spatial general shells structures

The meshfree point interpolation method (PIM) based on polynomial function is presented for the static analysis of spatial general shell structures. The formulation of the discrete system equations is derived from stress resultant geometrically exact shell theory of shear flexible shells based on the Cosserat surface. The PIM technique constructs its interpolation functions through a set of arbitrarily distributed points in the problem domain and its shape function has the delta function property. Hence, the implementation of essential boundary conditions can be imposed with ease as in conventional finite element method. An improved matrix triangularization algorithm (MTA) is proposed to obtain proper node enclosure and basis selection in order to overcome the singularity problem in the point interpolation method using a polynomial basis. It is found that the shape functions are generally not compatible to ensure the compatibility of the solution and lead to non-conforming PIM (NPIM) when the energy principles are used to formulate PIM. To obtain compatible shape functions, the enforcement of compatibility is needed along the common edges of the background cells in the problem domain and a background cell-based nodal selection is utilized, which lead to conforming PIM (CPIM). Both NPIM and CPIM are evaluated in the paper. Several benchmark problems for shells are analyzed to demonstrate the validity and efficiency of the NPIM and CPIM. The phenomena of shear locking and membrane locking are illustrated by showing the membrane and shear energies as fractions of the total energy. The convergences and performances are investigated for both conforming and non-conforming PIMs. It is also shown that the NPIM and CPIM with the improved MTA are very easy to implement and very efficient in constructing shape functions in comparison with the Element-Free Galerkin method.

Element-free Galerkin method for static and dynamic analysis of spatial shell structures

The implementation of the element free Galerkin method (EFG) for static and free vibration analysis of general shell structures is presented in this paper. The formulation of the discrete system equations is derived from the governing equations of stress resultant geometrically exact shell theory based on the Cosserat surface. A discrete singularity-free mapping between the five- and the six-degrees-of-freedom formulation is constructed by exploiting the geometry connection between the orthogonal group and the unit sphere. Moving least-squares approximation is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometry. Discrete system equations are obtained by incorporating these interpolations into the Galerkin weak form. The formulation is verified through numerical examples of static stress analysis and frequency analysis of spatial shell structures. The phenomena of shear locking and membrane locking are illustrated by showing the membrane and shear energy as fractions of the total energy. Essential boundary conditions are efficiently imposed through a penalty technique for both static analysis and frequency analysis. The formulation is tested on several benchmark problems and the results compare favorably with closed-form solutions and that of finite element analyses.

Numerical integration of the stiff dynamics of geometrically exact shells: an energy‐dissipative momentum‐conserving scheme

AbstractThis paper presents a new family of time‐stepping algorithms for the integration of the dynamics of non‐linear shells. We consider the geometrically exact shell theory involving an inextensible director field (the so‐called five‐parameter shell model). The main characteristic of this model is the presence of the group of finite rotations in the configuration manifold describing the deformation of the solid. In this context, we develop time‐stepping algorithms whose discrete solutions exhibit the same conservation laws of linear and angular momenta as the underlying physical system, and allow the introduction of a controllable non‐negative energy dissipation to handle the high numerical stiffness characteristic of these problems. A series of algorithmic parameters for the different components of the deformation of the shell (i.e. membrane, bending and transverse shear) fully control this numerical dissipation, recovering existing energy‐momentum schemes as a particular choice of these algorithmic parameters. We present rigorous proofs of the numerical properties of the resulting algorithms in the full non‐linear range. Furthermore, it is argued that the numerical dissipation is introduced in the high‐frequency range by considering the proposed algorithm in the context of a linear problem. The finite element implementation of the resulting methods is described in detail as well as considered in the final arguments proving the aforementioned conservation/dissipation properties. We present several representative numerical simulations illustrating the performance of the newly proposed methods. The robustness gained over existing methods in these stiff problems is confirmed in particular. Copyright © 2002 John Wiley & Sons, Ltd.

Exact shell theory analysis of free vibrations of submerged thin spherical shells

By the introduction of two displacement functions, the non-axisymmetric free vibrations of a complete thin isotropic spherical shell submerged in a compressible fluid medium are successfully investigated. It is found that there exist two classes of free vibrations: the first class is not affected by the ambient fluid while the second is. It is further proved that the frequency equations can be expressed in terms of polynomial. For the second class, it is also demonstrated that only complex frequencies exist except for the case of n = 1, for which the trivial solution Ω = 0 emerges. For n = 0, 1 and 2, the frequency equations of the second class are investigated numerically and the effects of various relative parameters are discussed. The small damping coefficient method is also discussed finally in the paper.

On hybrid stress, hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom

The paper is concerned with the finite element formulation of a recently proposed geometrically exact shell theory with natural inclusion of drilling degrees of freedom. Stress hybrid finite elements are contrasted by strain hybrid elements as well as enhanced strain elements. Numerical investigations and comparison is carried out for a four-node element as well as a nine-node one. As far as the four-node element is concerned it is shown that the stress hybrid element and the enhanced strain one are equivalent. The hybrid strain formulation corresponds to the hybrid stress formulation only in shear dominated problems, that is the case of the plate. © 1998 John Wiley & Sons, Ltd.

On hybrid stress, hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom

On hybrid stress, hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom

Stress Resultant Geometrically Exact Shell Theory for Finite Rotations and Its Finite Element Implementation

This article reviews the significant progress on shell problem theoretical foundation and numerical implementation attained over a period of the last several years. First, a careful consideration of the three-dimensional finite rotations is given including the choice of optimal parameters, their admissible variations and the much revealing relationship between different parameters. A non-conventional derivation of the stress resultant shell theory is presented, which makes use of the virtual work principle and local Cartesian frames. The presented derivation introduces no simplifying hypotheses regarding the shell balance equations, hence the resulting shell theory is referred to as being geometrically exact. The strain measures energy-conjugate to the chosen stress resultants are identified and the nature of the stress resultants with respect to the three-dimensional stress tensor is explained along with the resulting constitutive restrictions. Comments are made regarding a rather useful extension of the shell theory which accounts for the rotational degree of freedom about the director, the so-called drilling rotation. A linear shell theory is obtained as a very useful by-product of the present work, by linearizing the present nonlinear shell theory about the reference configuration. It is shown that this non-conventional approach not only clarifies an often confusing derivation of the linear shell theory, but also leads to a novel linear shell theory capable of delivering significantly improved results and essentially exact solutions to the standard linear benchmark problems. Another important aspect of the nonlinear shell problem solution, the finite element approximation of the shell theory, is also discussed. The model problem of assumed shear strain interpolation is used to illustrate that numerical implementation which preserves the salient features of the theoretical formulation often brings an improved final result. For the selected rotation parameterization and the finite element interpolation, the issues of the consistent linearization of the nonlinear shell problem are addressed. In a number of numerical simulations, the latter is proved to play a crucial role not only in ensuring the robust performance of the Newton solution procedure, but also in linear and nonlinear buckling problems of shells. Several directions for future research are pointed out and some contemporary works of special interest are listed. There are 127 references at the end of the article.

Finite element analysis of smooth, folded and multi-shell structures

The paper is concerned with the nonlinear theory and finite element analysis of shell structures with an arbitrary geometry, loading and boundary conditions. A complete set of shell field equations and side conditions (boundary and jump conditions) is derived from the basic laws of continuum mechanics. The developed shell theory includes the so-called drilling couples as well as the drilling rotation. It is shown that this property is crucial in the analysis of irregular shell structures, such as those containing folds, branches, column supports, stiffeners, etc. The relevant variational principles with relaxed regularity requirements are also presented. These principles provide the mathematical basis for the formulation of various classes of shell finite elements. The developed finite elements include a displacement/rotation based Lagrange family, a stress resultant based mixed and a semi-mixed family as well as so-called assumed strain elements. All elements have six degrees of freedom at each node, three translational and three rotational ones, including the drilling rotation formulated on the foundation of an exact (in defined sense) shell theory. As such, they are equally applicable to smooth as well as to irregular shell structures. The general applicability of the developed elements is illustrated through an extensive numerical analysis of the representative test examples. In order to obtain a still deeper insight into the problem a Lagrange family of standard degenerated shell elements with five degrees of freedom per node and an element with six degrees of freedom per node based on the von Karman plate theory are considered as well. The presented numerical results include complex plate and doubly-curved shell structures. Linear and non-linear solutions with a pre- and post-buckling analysis are discussed.

Large Axisymmetric Deformation of a Cylindrical Shell Conveying a Viscous Flow

Large axisymmetric deformations of collapsible tubes conveying a viscous flow are examined. Geometrically nonlinear Lagrangian shell theory is used to describe the deformation of the tube. The fluid flow is modelled using lubrication theory. The coupled fluid-solid problem is solved numerically using an FEM technique. In order to explain the mechanisms involved in the tube deformation, the effects of bending stiffness, wall shear stress and axial pre-stretch are examined in detail. The dependence of the tube resistance on the volume flux is investigated for two different experimental set-ups (constant pressure at either upstream or downstream end of the collapsible tube). Finally, the exact nonlinear shell theory used in this paper is compared to Sanders' moderate rotation theory and an improvement to his theory is suggested.