Boštjan Brank

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.

On large deformations of thin elasto-plastic shells: Implementation of a finite rotation model for quadrilateral shell element

A large-deformation model for thin shells composed of elasto-plastic material is presented in this work, Formulation of the shell model, equivalent to the two-dimensional Cosserat continuum, is developed from the three-dimensional continuum by employing standard assumptions on the distribution of the displacement held in the shell body, A model for thin shells is obtained by an approximation of terms describing the shell geometry. Finite rotations of the director field are described by a rotation vector formulation. An elasto-plastic constitutive model is developed based on the von Mises yield criterion and isotropic hardening. In this work, attention is restricted to problems where strains remain small allowing for all aspects of material identification and associated computational treatment, developed for small-strain elastoplastic models, to be transferred easily to the present elasto-plastic thin-shell model. A finite element formulation is based on the four-noded isoparametric element. A particular attention is devoted to the consistent linearization of the shell kinematics and elasto-plastic material model, in order to achieve quadratic rate of asymptotic convergence typical for the Newton-Raphson-based solution procedures. To illustrate the main objective of the present approach-namely the simulation of failures of thin elastoplastic shells typically associated with buckling-type instabilities and/or bending-dominated shell problems resulting in formation of plastic hinges-several numerical examples are presented, Numerical results are compared with the available experimental results and representative numerical simulations.

Nonlinear shell problem formulation accounting for through-the-thickness stretching and its finite element implementation

We first review some recent and current research works attributing to a very significant progress on shell problem theoretical foundation and numerical implementation attained over a period of the last several years. We then discuss theoretical formulations of shell model accounting for the trough-the-thickness stretching, which allows for large deformations and direct use of 3d constitutive equations. Three different possibilities for implementing this model within the framework of the finite element method are examined, one leading to 7 nodal parameters and the remaining two to 6 nodal parameters. Comparisons are performed of the 7- parameter shell model with no simplification of kinematic terms and 7-parameter shell model which exploits usual simplifications of Green-Lagrange strain measures. Comparisons are also presented of two different ways of implementing the incompatible mode method for reducing the number of shell model parameters to 6. One implementation uses additive decomposition of the strains and the other additive decomposition of the deformation gradient. Several numerical examples are given to illustrate performance of the shell elements developed herein.

On the relation between different parametrizations of finite rotations for shells

In this work we present interrelations between different finite rotation parametrizations for geometrically exact classical shell models (i.e. models without drilling rotation). In these kind of models the finite rotations are unrestricted in size but constrained in the 3‐d space. In the finite element approximation we use interpolation that restricts the treatment of rotations to the finite element nodes. Mutual relationships between different parametrizations are very clearly established and presented by informative commutative diagrams. The pluses and minuses of different parametrizations are discussed and the finite rotation terms arising in the linearization are given in their explicit forms.

On non-linear dynamics of shells : implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-directior is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton-Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm.

Dynamics and time-stepping schemes for elastic shells undergoing finite rotations

In this work we study the time-stepping schemes for shell models, which describe the shell-director vector motion by the finite rotations. Different possibilities for choosing director rotations are examined and their relationships are cast in terms of the commutative diagram. The Newmark time-stepping schemes, making use of different rotation parameters, are then developed. The mid-point scheme modified to either conserve or dissipate the total energy is further examined. Several numerical simulations are presented to illustrate the performance of each developed scheme.

Model adaptivity for finite element analysis of thin or thick plates based on equilibrated boundary stress resultants

Purpose – The purpose of this paper is to address error-controlled adaptive finite element (FE) method for thin and thick plates. A procedure is presented for determining the most suitable plate model (among available hierarchical plate models) for each particular FE of the selected mesh, that is provided as the final output of the mesh adaptivity procedure. Design/methodology/approach – The model adaptivity procedure can be seen as an appropriate extension to model adaptivity for linear elastic plates of so-called equilibrated boundary traction approach error estimates, previously proposed for 2D/3D linear elasticity. Model error indicator is based on a posteriori element-wise computation of improved (continuous) equilibrated boundary stress resultants, and on a set of hierarchical plate models. The paper illustrates the details of proposed model adaptivity procedure for choosing between two most frequently used plate models: the one of Kirchhoff and the other of Reissner-Mindlin. The implementation details are provided for a particular case of the discrete Kirchhoff quadrilateral four-node plate FE and the corresponding Reissner-Mindlin quadrilateral with the same number of nodes. The key feature for those elements that they both provide the same quality of the discretization space (and thus the same discretization error) is the one which justifies uncoupling of the proposed model adaptivity from the mesh adaptivity. Findings – Several numerical examples are presented in order to illustrate a very satisfying performance of the proposed methodology in guiding the final choice of the optimal model and mesh in analysis of complex plate structures. Originality/value – The paper confirms that one can make an automatic selection of the most appropriate plate model for thin and thick plates on the basis of proposed model adaptivity procedure.

Constrained finite rotations in dynamics of shells and Newmark implicit time-stepping schemes

Purpose – Aims to address the issues pertaining to dynamics of constrained finite rotations as a follow-up from previous considerations in statics. Design/methodology/approach – A conceptual approach is taken. Findings – In this work the corresponding version of the Newmark time-stepping schemes for the dynamics of smooth shells employing constrained finite rotations is developed. Different possibilities to choose the constrained rotation parameters are discussed, with the special attention given to the preferred choice of the incremental rotation vector. Originality/value – The pertinent details of consistent linearization, rotation updates and illustrative numerical simulations are supplied.

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.

A Path-Following Method Based on Plastic Dissipation Control

A path-following method that is based on controlling incremental plastic dissipation is presented. It can be applied for analysis of an elasto-plastic solid or structure. It can be also applied for complete failure computation of a solid or structure that is performed by using a material failure model. In this work, we applied it for computations with the embedded-discontinuity finite elements that use rigid-plastic cohesive laws with softening to model material failure process. The most important part of the path-following method is the constraint function. Several constraint functions are derived and proposed for geometrically nonlinear small strain elasto-plasticity with linear isotropic hardening. The constraint functions are also derived for the embedded-discontinuity finite elements. In particular, they are derived for 2-d solid (and frame) embedded-discontinuity finite elements that describe cohesive stresses (or forces and moments) in the discontinuity curve (or point) by rigid-plasticity with softening. Numerical examples are presented in order to illustrate performance of the discussed path-following method.