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

We discuss a theoretical formulation of shell model accounting for through-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. The 7-parameter shell model with no simplification of kinematic terms is compared to the 7-parameter shell model which exploits usual simplifications of the Green–Lagrange strains. Two different ways of implementing the incompatible mode method for reducing the number of parameters to 6 are presented. One implementation uses an additive decomposition of the strains and the other an additive decomposition of the deformation gradient. Several numerical examples are given to illustrate performance of the shell elements developed herein

On boundary layer in the Mindlin plate model: Levy plates

This work is related to the bending problem of thick rectangular Levy plates. Series solution for the Mindlin (thick) plate model is obtained and represented as a sum of the Kirchhoff (thin) plate model solution, the ``shear terms'' and the ``boundary layer terms''. Hard- and soft-simple supported, hard- and soft-clamped and free boundary conditions are considered. In order to detect plate regions where Kirchhoff model is good enough, and plate regions where Mindlin model should be used, a model error indicator is introduced. Several examples are presented, illustrating the difference between the Mindlin and the Kirchhoff results, the strengths of boundary layers for different boundary conditions, accuracy of several possible model error indicators and dependence of results on plate thickness. (C) 2007 Elsevier Ltd. All rights reserved

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

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

Stress resultant plasticity for shells revisited

In this work, we revisit the stress resultant elastoplastic geometrically exact shell finite element formulation that is based on the Ilyushin–Shapiro two-surface yield function with isotropic and kinematic hardening. The main focus is on implicit projection algorithms for computation of updated values of internal variables for stress resultant shell elastoplasticity. Four different algorithms are derived and compared. Three of them yield practically identical final results, yet they differ considerably in computational efficiency and implementation complexity, since they solve different sets of equations and they use different procedures that choose active yield surfaces. One algorithm does not provide acceptable accuracy. It turns out that the most simple and straightforward algorithm performs surprisingly well and efficiently. Several numerical examples are presented to illustrate the Ilyushin–Shapiro stress resultant shell formulation and the numerical performance of the presented integration algorithms

On stress resultant plasticity and viscoplasticity for metal plates

In this work we derive elastoplastic and elastoviscoplastic finite element formulations for stress resultant bending analysis of thin metal plates. The principle of maximum plastic dissipation is used to obtain the ingredients of the small strain stress resultant plate elastoplasticy with state variables describing general isotropic and linear kinematic hardening. The ingredients of the plate stress resultant elastoviscoplasticy are further obtained by using the penalty-like form of the principle of maximum plastic dissipation. Such an approach enables single framework for numerical implementation of both considered inelastic stress resultant plate material models. The implementation is based on the spectral decomposition algorithm. For spatial discretization we use simple and robust quadrilateral finite element. A set of numerical examples is presented to illustrate the approach and to discuss the accuracy of the stress resultant inelastic plate formulations. (c) 2007 Elsevier B.V. All rights reserved

Constrained finite rotations in dynamic 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. \ud \ud Design/methodology/approach – A conceptual approach is taken. \ud \ud 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. \ud \ud Originality/value – The pertinent details of consistent linearization, rotation updates and illustrative numerical simulations are supplied.\u

On Prediction of 3d Stress State in Elastic Shell by Higher-order Shell Formulations

In this work we study the accuracy of modem higher-order shell finite element formulations in computation of 3d stress state in elastic shells. In that sense we compare three higher-order shell models: (i) with seven dislacement-like kinematic parameters, and (ii, iii) with six displacement-like kinematic parameters plus one strain-like kinematic parameter introduced by two different versions of enhanced assumed strain (EAS) concept. The finite element approximations of all shell models are based on 4-node quadrilateral elements. Geometrically nonlinear and consistently linearized forms of considered formulations are given. Several numerical examples are presented, where computed stresses are compared with analytical solutions. It was found that through-the-thickness variation of some (non-dominant) stress tensor components, including through-the-thickness normal stress, may be computed very inaccurately. The reliable representation for those stresses can be interpreted only if the ``layer-wise'' averaging or the through-the-thickness averaging is performed

Quadrilateral Finite Element with Embedded Strong Discontinuity for Failure Analysis of Solids

We present a quadrilateral finite element with discontinuous displacement fields that can be used to model material failure in 2d brittle and ductile solids. The element provides mesh-objective results. The element's kinematics can represent linear displacement jumps along the discontinuity line in both normal and tangential directions to the line. The cohesive law in the discontinuity line is based on rigid-plasticity model with softening. The material of the bulk of the element is described by hardening plasticity model. Static condensation of the jump-in-displacements kinematic parameters is made, which provides standard form of the element stiffness matrix. However, in order to make the discontinuity growth algorithm more robust, the continuity of the failure line between the elements is enforced. Several numerical tests show that the element can describe constant and linear separation modes without spurious transfer of the stresses. Other numerical examples represent failure of pure concrete, composite and metal 2d solids

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. \ud \ud 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. \ud \ud 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. \ud \ud 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.\u