Shell Director Research Articles

A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition

A curved non-isoparametric Reissner–Mindlin shell element is developed for analyzing thin-walled structures. The standard kinematic description of the element requires the calculation of the director vector. To address this demand accurately, similar to isogeometric analysis (IGA), the geometry is defined by utilization of the non-uniform rational B-splines (NURBS) imported directly from computer-aided design (CAD) files. Then, shape functions of the Legendre spectral element method (SEM) are used to interpolate the displacements. Consequently, the shell director vector and Jacobian of the transformation are calculated properly according to the presented formulation. On the other hand, in Legendre SEM combined with Gauss–Lobatto–Legendre quadrature, the integration points and the element nodes coincide. Thus, the easily computable local coordinate systems at the integration points can be used directly as nodal basis systems. A separate calculation of nodal basis systems at control points, which is the source of either complexity or error in IGA shells, is not required. Given the condition number of the stiffness matrix in the developed method, super high-order elements can also be used. Very high order p-refined elements are used in addition to h-refinement of the mesh to show the capability of higher order elements to analyze problems without mesh refinement. The validity and convergence rate of the method are investigated and verified through various cases of h- and p-refinement in challenging obstacle course problems.

A co-rotational triangular finite element for large deformation analysis of smooth, folded and multi-shells

A six-node co-rotational curved triangular shell finite element with a novel rotation treatment for folded and multi-shell structures is presented. Different from other co-rotational triangular element formulations, rotations are not represented by axial (pseudo) vectors, but by components of polar (proper) vectors, of which additivity and commutativity lead to symmetry of the tangent stiffness matrices in both local and global coordinate systems. In the co-rotational local coordinate system, the two smallest components of the shell director are defined as the nodal rotational variables. Similarly, the two smallest components of each director in the global coordinate system are adopted as the global rotational variables for nodes located either on smooth shells or away from non-smooth shell intersections. At intersections of folded and multi-shells, global rotational variables are defined as three selected components of an orthogonal triad initially oriented along the global coordinate system axes. As such, the vectorial rotational variables enable simple additive update of all nodal variables in an incremental-iterative procedure, resulting in significant enhancement in computational efficiency for large deformation analysis. To alleviate membrane and shear locking phenomena, an assumed strain method is employed in obtaining the element tangent stiffness matrices and the internal force vector. The effectiveness of the presented co-rotational triangular shell element formulation is verified by analyzing several benchmark problems of smooth, folded and multi-shell structures undergoing large displacements and large rotations.

Realignment of Liquid Crystal Shells Driven by Temperature-Dependent Surfactant Solubility.

We investigate dynamic director field variations in shells of the nematic liquid crystal (LC) compound, 4-cyano-4′-pentylbiphenyl, suspended in and containing immiscible aqueous phases. The outer and inner shell interfaces are stabilized by the cationic surfactant, cetyl trimethyl ammonium bromide (CTAB), and by the water soluble polymer, poly(vinyl alcohol) (PVA), respectively. PVA and surfactant solutions normally promote tangential and orthogonal alignments, respectively, of the LC director. The rather high Krafft temperature of CTAB, TK ≈ 25 °C, means that its solubility in water is below the critical micelle concentration at room temperature in most labs. Here, we study the effect of cooling/heating past TK on the LC shell director configuration. Within a certain concentration range, CTAB in the outer aqueous phase (and PVA in the inner) switches the LC director field from hybrid to uniformly orthogonal upon cooling below TK. We argue that the effect is related to the migration of the surfactant through the fluid LC membrane into the initially surfactant-free aqueous PVA solution, triggered by the drastically reduced water solubility of CTAB at T < TK. The results suggest that LC shells can detect solutes in the continuous phase, provided there is sufficient probability that the solute migrates through the LC into the inner aqueous phase.

Implicit dynamic analysis using an isogeometric Reissner–Mindlin shell formulation

SummaryIn isogeometric analysis, identical basis functions are used for geometrical representation and analysis. In this work, non‐uniform rational basis splines basis functions are applied in an isoparametric approach. An isogeometric Reissner–Mindlin shell formulation for implicit dynamic calculations using the Galerkin method is presented. A consistent as well as a lumped matrix formulation is implemented. The suitability of the developed shell formulation for natural frequency analysis is demonstrated by a numerical example. In a second set of examples, transient problems of plane and curved geometries undergoing large deformations in combination with nonlinear material behavior are investigated. Via a zero‐thickness stress algorithm for arbitrary material models, a J2‐plasticity constitutive law is implemented. In the numerical examples, the effectiveness, robustness, and superior accuracy of a continuous interpolation method of the shell director vector is compared with experimental results and alternative numerical approaches. Copyright © 2016 John Wiley &amp; Sons, Ltd.

An isogeometric Reissner-Mindlin shell element for dynamic analysis considering geometric and material nonlinearities

The present approach deals with the dynamical analysis of thin structures using an isogeometric Reissner-Mindlin shell formulation. Here, a consistent and a lumped mass matrix are employed for the implicit time integration method. The formulation allows for large displacements and finite rotations. The Rodrigues formula, which incorporates the axial vector is used for the rotational description. It necessitates an interpolation of the director vector in the current configuration. Two concept for the interpolation of the director vector are presented. They are denoted as continuous interpolation method and discrete interpolation method. The shell formulation is based on the assumption of zero stress in thickness direction. In the present formulation an interface to 3D nonlinear material laws is used. It leads to an iterative procedure at each integration point. Here, a J2 plasticity material law is implemented. The suitability of the developed shell formulation for natural frequency analysis is demonstrated in numerical examples. Transient problems undergoing large deformations in combination with nonlinear material behavior are analyzed. The effectiveness, robustness and superior accuracy of the two interpolation methods of the shell director vector are investigated and are compared to numerical reference solutions.

A three-dimensional progressive failure model for laminated composite plates subjected to transverse loading

A novel, three-dimensional computational model is presented for the simulation of progressive failure in laminated composites subjected to out-of-plane loading conditions. Solid-like shell elements are used to model the thin plies of the laminate. In order to model mesh independent matrix cracking, a discontinuous solid-like shell element (DSLS) is utilized. A partition of unity approach is exploited to incorporate the discontinuity in the shell mid-surface, shell director and internal stretching field. A shell interface model is presented for the modeling of delamination damage. The model allows for the computationally efficient simulation of delamination and the evaluation of a consistently linearized tangent stiffness matrix for large deformation problems, which is essential for convergence. To model the coupled response of matrix cracking and delamination under large deformations, a computational framework is developed. The combined modeling of matrix cracking and delamination is achieved without incorporation of additional degrees of freedom. Numerical results are presented to show the performance and several distinct features of the model to simulate progressive failure in laminated composite shell structures.

Analysis of fracture in thin shells by overlapping paired elements

A finite element methodology for evolution of cracks in thin shells using mid-surface displacement and director field discontinuities is presented. We enrich the mid-surface displacement and director fields of a discrete Kirchhoff–Love quadrilateral element using a piecewise decomposition of element kinematics, which leads to a basis that is a variant of the one used in the extended finite element method. This allows considerable simplifications in the inclusion of the shell director field. A cohesive law is employed to represent the progressive release of the fracture energy. In contrast with previous works, we retain the original quadrature points after the formation of a crack, which, in combination with an elasto-plastic multiplicative decomposition of the deformation gradient, avoids the previously required internal variable mapping during crack evolution. Results are presented for large strain elastic and elasto-plastic crack propagation.

On the physical meaning of the dynamical equations governing geometrically-exact multilayer shells

In this paper, the physical meaning of the equations governing geometrically-exact multilayer shells is provided through a totally new derivation by means of dynamic equilibrium consideration, starting from a 3-D continuum setting in curvilinear coordinates. The theory is valid for large deformation and large overall motion, as characteristic of geometrically exact formulations. The dynamic equilibrium derivation offers a clear physical insight into the meaning of the terms in the equations of motion; such insight was not afforded by previous derivations based on the principle of virtual power. In particular, we obtain directly the balance of angular momentum in true resultant couples, without the need to use the constitutive restriction as in our previous papers. Moreover, we also obtain here new expressions for the inertia operators that will greatly simplify the computational formulation, when compared to our previous work. In addition, an analytical justification of the inertial operators is also provided. The present formulation is independent of any kinematic assumption along the shell director, and is in particular valid for the case of shells with variable-length director (e.g., for modeling through-the-thickness deformation) and higher-order kinematic assumptions.

Nonlinear shell models with seven kinematic parameters

We discuss design of nonlinear finite rotation shell model with seven kinematic displacement-like parameters, which are: three displacements of the middle surface, two rotations of the shell director, and two through-the-thickness stretching parameters. From the theoretical side we examine several possibilities for constructing the enriched kinematic field, which leads to different higher-order 7-parameter shell formulations. From the finite element implementation side a shell director interpolation is identified which eliminates the “curvature thickness locking”. Numerical examples are presented in order to compare different formulations and to illustrate the performance of the developed finite elements.

A formulation of the non linear discrete Kirchhoff quadrilateral shell element with finite rotations and enhanced strains

This paper presents a new formulation of the non-linear discrete Kirchhoff quadrilateral shell element applicable for the analysis of geometrically nonlinear structures undergoing finite rotations. The shell director is directly interpolated and the exact linearization of the discreet form of the equilibrium equations is derived in closed form. The consistent tangent stiffness matrix is symmetric and is given explicitly in this paper. Two or three rotational variables are used at each node. To improve the in-plane deformation enhanced incompatible modes are introduced. The formulation is then illustrated by a comprehensive set of numerical experiments selected from the literature.

One point quadrature shell element with through-thickness stretch

A general purpose one point quadrature shell element accounting for through-thickness deformation is developed. In the shell, a complete 3-D constitutive law is introduced, leading to a 7-parameter theory which explicitly accounts for thickness change and also for a linear variation of thickness stretch. An interpolation scheme for the shell director is developed to avoid thickness locking. The developed shell element covers flexible warping behavior by using a local nodal coordinate system at each node, which is updated with second order accuracy. A physical stabilization scheme for zero energy modes is employed based on the decomposition of the strain field into constant and linear terms with respect to the natural coordinates. The rigid body projection is applied to treat rigid body rotations effectively. Linear and nonlinear patch tests including elasto-plasticity and contact are performed and the results are compared with analytical or previously reported results. The results are also compared with those of 5-parameter shell elements, in order to show that there is no significant deterioration in accuracy, especially for thin shell applications.

Incompressibility at large strains and finite-element implementation

This contribution is concerned with the consideration of material incompressibility at large strains and proposes various methods for the enforcement of the corresponding constraint into finite-rotation shell models. The incompressibility condition can be expressed in terms of displacement as well as strain variables and is considered by means of three different procedures in the numerical implementation. As kinematic hypothesis a quadratic assumption with respect to the thickness coordinate is used in which the corresponding directors are decomposed into two stretch parameters and a common inextensible unit vector. Various constitutive laws holding for incompressible isotropic hyperelasticity are considered and directly coupled with shell equations through a numerical thickness integration. A 4-node isoparametric shell element is developed parameterizing the inextensible shell director in terms of rotation variables in the framework of an up-dated rotation formulation. Finally, several examples are analysed to identify the most effective procedure for modelling isochoric deformations in thin-walled structures.

A triangular finite shell element based on a fully nonlinear shell formulation

This work presents a fully nonlinear six- parameter (3 displacements and 3 rotations) shell model for finite deformations together with a triangular shell finite element for the solution of the resulting static boundary value problem. Our approach defines energeti- cally conjugated generalized cross-sectional stresses and strains, incorporating first-order shear deformations for an inextensible shell director (no thickness change). Finite rotations are treated by the Euler-Rodrigues formula in a very convenient way, and alternative parameterizations are also discussed herein. Condensation of the three-dimen- sional finite strain constitutive equations is performed by applying a mathematically consistent plane stress condi- tion, which does not destroy the symmetry of the linear- ized weak form. The results are general and can be easily extended to inelastic shells once a stress integration scheme within a time step is at hand. A special displace- ment-based triangular shell element with 6 nodes is fur- thermore introduced. The element has a nonconforming linear rotation field and a compatible quadratic interpo- lation scheme for the displacements. Locking is not ob- served as the performance of the element is assessed by several numerical examples, which also illustrate the robustness of our formulation. We believe that the com- bination of reliable triangular shell elements with powerful mesh generators is an excellent tool for nonlinear finite element analysis.

On composite shell models with a piecewise linear warping function

A multilayered shell model accounting for a piecewise linear (i.e. zig-zag) distribution of displacements through the laminate thickness is discussed. The model has seven unknown kinematic variables: three displacements of the middle surface, two rotations of the shell director and two displacements associated with the wrinkling of the laminate cross-sections. The initial transverse shear stress field is introduced, and the constitutive relations are then relaxed in the framework of the variational principle. Finite element solutions obtained with this kind of model are compared with the analytical solutions for the case of cylindrical shell bending.

Kinematics of finite rotation shells with in-plane twist parameter

The paper concerns shell equations derived from a 3D mixed formulation with explicit rotations. The constitutive equation is postulated in terms of the relaxed Biot stress and right stretch strain. The shell kinematics is based on the Reissner hypothesis and on a constant/linear approximation of the rotation tensor with respect to the thickness coordinate. The rotation around the shell director is parametrized by the drill and in-plane twist parameters, with the latter one associated with the linear approximation. The in-plane twist parameter is new, and hence consequences of its presence for the shell rotation constrains and strain measures are examined, and compared to the formulation with the drill rotation only. In the presented formulation, the drill rotation is retained explicitly, and can be recovered from the drill rotation constraint, while the in-plane twist parameter is an internal parameter of the theory. In this way the enhanced shell kinematics does not increase the number of degrees of freedom of a discrete model.

A new shell element accounting for through-thickness deformation

A new 7-parameter shell model is presented for thick shell applications. The model accounts explicitly for the thickness change in the shell, as well as the normal stress and strain through the shell thickness. Large deformations are accounted for by using the second Piola–Kirchhoff stress and the Green–Lagrange strain tensors. An assumed transverse shear strain interpolation is used to avoid shear locking. Two new interpolation schemes for the shell director are developed to avoid thickness locking. These interpolations are implemented and their consistent linearization is derived. Guidelines are developed for neglecting some of the quadratic terms in the consistent stiffness matrix to minimize computational time. The thick shell element performance is tested to show that the higher order terms result in improved accuracy. It also demonstrates that for thin shells, there is no significant deterioration in accuracy, compared with traditional 5-parameter shell elements.

Composite laminates: nonlinear interlaminar stress analysis by multi-layer shell elements

An essential problem in dealing with composite laminates is an accurate prediction of interlaminar stresses playing an important role in the design, particularly in the failure analysis of these structures. The objective of the present paper is to develop a multi-layer shell-element family capable to deal with this aspect in the presence of strong nonlinearities. First a multi-director shell theory is presented on the basis of a quadratic approximation of the displacement field. A particular attention is given to the parametrization of the inextensible shell director in order to make the formulation accessible to finite rotations. This is accomplished by using Euler-angles as well as an updated rotation formulation. The single layer theory is then coupled with a multi-layer concept, which has been shown to be very predictive in dealing with complex through-thickness stress distributions. The constitutive relations are formulated so as to consider the particularities of composite materials. After standard linearization, shell equations are transformed into multi-layer finite shell elements using assumed strain concept and enhanced strain formulation as stabilization algorithms. The performance of the finite elements concerning various aspects is discussed on adequate examples.

Generalized Energy–Momentum Method for non-linear adaptive shell dynamics

In the present study, a generalization of the Energy–Momentum Method, denoted by Generalized Energy–Momentum Method, applied to the non-linear dynamics of shells will be developed within the framework of the Generalized- α Method. This algorithmic environment contains the unconditionally stable Energy–Momentum Method and its numerically damped version as well as the classical Newmark and α-methods as special cases. In order to control the size of the time steps of the integration scheme with respect to accuracy and efficiency, an adaptive time stepping procedure based on local a posteriori error estimation will be improved for non-linear dynamical systems and applied to the proposed class of algorithms. The spatial discretization is realized by an eight noded finite shell element of Reissner/Mindlin type including an extensible shell director field permitting the application of three-dimensional material laws. The original formulation of this finite element will be developed for non-linear dynamic analysis and adapted for the employment within the introduced energy conserving/decaying time integration scheme.

On implementation of a nonlinear four node shell finite element for thin multilayered elastic shells

A simple non-linear stress resultant four node shell finite element is presented. The underlying shell theory is developed from the three dimensional continuum theory via standard assumptions on the displacement field. A model for thin shells is obtained by approximating terms describing the shell geometry. In this work the rotation of the shell director is parameterized by the two Euler angles, although other approaches can be easily accomodated. A procedure is provided to extend the presented approach by including the through-thickness variable material properties. These may include a general non-linear elastic material with varied degree of orthotropy, which is typical for fibre reinforced composites. Thus a simple and efficient model suitable for analysis of multilayered composite shells is attained. Shell kinematics is consistently linearized, leading to the Newton-Raphson numerical procedure, which preserves quadratic rate of asymptotic convergence. A range of linear and non-linear tests is provided and compared with available solutions to illustrate the approach.