Yavuz Başar

Refined shear-deformation models for composite laminates with finite rotations

Abstract For arbitrary multilayered shell structures made particularly of composite material layers a refined finite-rotation theory with seven independent displacement variables is developed, approximating the displacement field by a cubic series expansion of thickness coordinates. This model allows a quadratic shear deformation distribution across the thickness. Procedures are given permitting a unique determination of the first order displacement term in the case of finite rotations. Kinematic relations are formulated in two alternative forms suitable for both classical and isoparametric finite element formulations. The constitutive relations presented model orthotropic material properties varying arbitrarily across the thickness. This third order single-layer theory is then transformed, by introducing further constraints, into three simplified models : a third order theory with five independent displacement variables, a Mindlin-Reissner type theory and a Kirchhoff-Love type theory. These four models differ, however, from each other essentially in the constraints imposed on the first and third order displacement variables : a significant advantage for a unified finite element development. Finally, the Mindlin-Reissner type theory is generalized to a layer-wise model being the most predictive one in dealing with local interlaminar effects. The theoretical models are transformed into adequate finite shell elements and then compared by means of appropriate examples concerning their prediction capability. Also examples are given demonstrating their applicability to finite-rotation phenomena.

A consistent theory of geometrically non-linear shells with an independent rotation vector

Abstract For shells undergoing finite rotations, a general theory is formulated in terms of consistent displacement and force variables. An independent rotation vector is used for the description of the deformation state. The strain-displacement equations are obtained considering shear deformations. These relations are then transformed by a variational procedure into consistent equilibrium equations and boundary conditions, the validity of which is also confirmed by an independent two-dimensional derivation. The paper closes with the physical interpretation of the force variables and the formulation of the constitutive equations.

Finite-rotation elements for the non-linear analysis of thin shell structures

Abstract For the numerical analysis of shells undergoing finite rotations doubly curved finite shell elements are developed via the displacement formulation. The derivation starts from a consistent finite-rotation shell theory which is transformed by a variational procedure into an incremental formulation. Thus, the non-linearity can be treated by an incremental-iterative technique. The non-linear element matrices are obtained by a tensor-oriented procedure permitting a direct transformation of the initial equations into efficient numerical models. Unlike in the usual procedure, the KIRCHHOFF-LOVE assumption is treated as a subsidiary condition at the element level. This computer-oriented approach permits the elimination of the dependent rotational degrees of freedom without loss of accuracy. Finally, some examples are presented to demonstrate the ability of the resulting finite elements to deal with finite-rotation problems.

Shear deformation models for large-strain shell analysis

Abstract The objective is the theoretical and numerical simulation of large-strain phenomena of rubber-like shells by means of shear deformation models. The development starts with a general applicable shell model constructed on the basis of a quadratic displacement approximation which involves two thickness stretching parameters. This model is then coupled with incompressible material models of Mooney-Rivlin and neo-Hookean types. Material incompressibility is described by two-dimensional constraints considered at the element level as subsidiary conditions. A special care is given to the stress prediction in the presence of large-strains. After transformation of the theoretical model into an incremental formulation a four-node isoparametric finite element is derived. Examples are finally given to demonstrate the ability of this model to deal with very strong deformations and to predict the related stresses.

Finite element formulation of the Ogden material model with application to rubber‐like shells

The present contribution proposes a variational procedure for the numerical implementation of the Ogden material model. For this purpose the strain energy density originally formulated in terms of the principal stretches is transformed as variational quantities into the invariants of the right Cauchy–Green tensor. This formulation holds for arbitrary three-dimensional deformations and requires neither solving eigenvalue problems nor co-ordinate system transformations. Particular attention is given to the consideration of special cases with coinciding eigenvalues. For the analysis of rubber-like shells this material model is then coupled with a six parametric shells kinematics able to deal with large strains and finite rotations. The incompressibility condition is considered in the strain energy, but it is additionally used as 2-D constraint for the elimination of the stretching parameter at the element level. A four node isoparametric finite element is developed by interpolating the transverse shear strains according to assumed strain concept. Finally, examples are given permitting to discuss the capability of the finite element model developed concerning various aspects.

Finite-element analysis of hyperelastic thin shells with large strains

The objective of this contribution is the development of theoretical and numerical models applicable to large strain analysis of hyperelastic shells confining particular attention to incompressible materials. The theoretical model is developed on the basis of a quadratic displacement approximation in thickness coordinate by neglecting transverse shear strains. In the case of incompressible materials this leads to a three-parametric theory governed solely by mid-surface displacements. The material incompressiblity is expressed by two equivalent equation sets considered at the element level as subsidiary conditions. For the simulation of nonlinear material behaviour the Mooney-Rivlin model is adopted including neo-Hookean materials as a special case. After transformation of nonlinear relations into incremental formulation doubly curved triangular and quadrilateral elements are developed via the displacement method. Finally, examples are given to demonstrate the ability of these models in dealing with large strain as well as finite rotation shell problems.

Finite-rotation shell elements via mixed formulation

Starting from a tensorial five-parametric finite-rotation shell theory a family of mixed finite elements is developed on the basis of a Reissner-Mindlin type functional. The family developed contains 4-node and 9-node quadrilateral shell elements. In each of them the displacement approximation is combined with various force variable interpolations in order to improve flexibility for numerical applications. The so-called difference vector occurring in the shell theory is expressed in terms of new rotational degrees of freedom which permit a unique determination of this variable in every deformed position. The corresponding constraints are then satisfied at the element level numerically. Due to the underlying theory the numerical models developed are able to predict the physical 2D force variables accurately. Their capability to deal with strongly nonlinear situations is demonstrated by several examples where numerical results due to Kirchhoff-Love type elements are also included for a systematical comparison.

Free-vibration analysis of thin/thick laminated structures by layer-wise shell models

Abstract The objective of this contribution is the free-vibration analysis of thick/thin shell structures by means of finite element (FE) models permitting the consideration of arbitrary complex through thickness distribution of transversal shear deformations as well as thickness stretching in any desired accuracy order and presenting therefore an alternative to 3D FE formulation. The development basis is a refined shell kinematics, where the displacement field is approximated by a quadratic polynomial. Numerically attractive aspect of this approach is the multiplicative decomposition of the first-order term and inclusion of a quadratic stretching parameter. This formulation is then extended to a layer-wise model to develop in a further step, an isoparametric four-node element by means of assumed strain concept. A large number of examples are investigated, showing the performance of the element proposed particularly in dealing with thick structures. The influence of transverse strains on the response is also discussed by adequate examples.

On an isoparametric finite-element for composite laminates with finite rotations

For composite laminates consisting of an arbitrary number of orthotropic laminae first a finite-rotation theory is presented as basis of isoparametric finite-element formulations. The derivation is achieved by a Reissner-Mindlin type kinematic assumption which allows a constant shear deformation distribution across the thickness. The constitutive equations are presented in a general form such that orthotropic material behaviour with material axes varying arbitrarily across the thickness may easily be considered in numerical implementation, also when using curvilinear coordinates. Special attention is taken to predict the force distribution in the deformed shell structure. This theory is then transformed into a four-node isoparametric assumed-strain finite element. Unlike in the degeneration approach, interpolation polynomials are introduced directly for rotation variables determining the deformed position of the unit normal vector. The capability of the finite element developed to deal with strongly nonlinear situations is demonstrated by many examples. Also numerical results are presented permitting a systematical comparison of classical and isoparametric approaches concerning the numerical efficiency.

Theory and Finite-Element Formulation for Shell Structures Undergoing Finite Rotations

Abstract For the analysis of shell structures with finite displacements and rotations a consistent shear-deformation theory is derived and discussed under different aspects. Unlike in the earlier formulations, the constraint for the so-called difference vector is replaced by new conditions with which this variable can be determined clearly in all nonlinear range. The paper continues with the finite-element implementation of the theory presented, using a mixed-formulation. The finite-element family developed on the basis of HELLINGER-REISSNER functional consists of quadrilateral shell elements with 4 and 9 nodes. Their efficiency is due to the exact enforcement of the above mentioned constraints at the element level. The ability of the finite elements to predict the displacements and the real force distribution is finally demonstrated by some strongly nonlinear examples.