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 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

Multi-scale computational model for failure analysis of metal frames that includes softening and local buckling

In this work we present a new modelling paradigm for computing the complete failure of metal frames by combining the stress-resultant beam model and the shell model. The shell model is used to compute the material parameters that are needed by an inelastic stress-resultant beam model; therefore, we consider the shell model as the meso-scale model and the beam model as the macro-scale model. The shell model takes into account elastoplasticity with strain-hardening and strain-softening, as well as geometrical nonlinearity (including local buckling of a part of a beam). By using results of the shell model, the stress-resultant inelastic beam model is derived that takes into account elastoplasticity with hardening, as well as softening effects (of material and geometric type). The beam softening effects are numerically modelled in a localized failure point by using beam finite element with embedded discontinuity. The original feature of the proposed multi-scale (i.e. shell-beam) computational model is its ability to incorporate both material and geometrical instability contributions into the stress-resultant beam model softening response. Several representative numerical simulations are presented to illustrate a very satisfying performance of the proposed approach

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

An embedded crack model for failure analysis of concrete solids

We present a quadrilateral fi�nite element with an embedded crack that can be used to model tensile fracture in two-dimensional concrete solids and the crack growth. The element has kinematics that can represent linear jumps in both normal and tangential displacements along the crack\ud line. The cohesive law in the crack is based on rigid-plasticity with soft-ening. The required material data for the concrete failure analysis are the constants of isotropic elasticity and the mode I softening curve. The results of two well known tests are presented in order to illustrate very satisfying performance of the presented approach to simulate failure of concrete solids

Recent developments in nonlinear shell theory with finite rotations and finite deformations

In this paper we discuss a theoretical formulation of a fully nonlinear shell model, capable of representing finite rotations and finite strains. The latter imposes that one should account for through-the-thickness stretching, which allows for direct use of 3D constitutive equations from classical continuum model. Three different possibilities for implementing this kind of shell model within the framework of the finite element method are examined, the first 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. A couple of numerical examples are given to illustrate performance of the shell elements developed herein

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. 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

Overview of the numerical methods for the modelling of rock mechanics problems

Počeci numeričkih metoda sežu u rane 1960-e. Već tada je bilo jasno da numeričke metode mogu biti uspješno upotrijebljene za različita inženjerska i znanstvena područja, uključujući i primjenu u mehanici stijena. Ubrzan razvoj računala je omogućavao razvoj numeričkih metoda i rješavanje računalno zahtjevnijih sustava. Takav razvoj doveo je do velikog broja različitih metoda i pristupa koji se mogu svrstati u dvije skupine: metode kontinuuma i metode diskontinuuma. Određene zadaće zahtijevaju prednosti oba pristupa, što je dovelo do razvoja kombiniranih konačno-diskretnih metoda. Prvi cilj ovog rada je predstavljanje numeričkih metoda i pristupa koji se koriste za rješavanje zadaća u mehanici stijena, kao i kratko objašnjenje osnovnih teorijskih postavki svake od metoda. Drugi cilj je osvrt na primjenjivost pojedine metode u mehanici stijena.The numerical methods have their origin in the early 1960s and even at that time it was noted that numerical methods can be successfully applied in various engineering and scientific fields, including the rock mechanics. Moreover, the rapid development of computers was a necessary background for solving computationally more demanding problems and the development process of the methods in general. Thus, we have many different methods presently, which can be separated into two main branches: continuum and discontinuum-based numerical methods. Some problems require the strengths of both main approaches which brought the hybrid continuum/discontinuum methods. The first goal of this paper is to present the state of the art numerical methods and approaches for solving the rock mechanics problems, as well as to give the brief explanation about the theoretical background of each method. The second goal is to emphasise the area of applicability of the methods in rock mechanics