Adnan Ibrahimbegovic

On finite element implementation of geometrically nonlinear Reissner's beam theory: three-dimensional curved beam elements

Abstract Finite element implementation of the three-dimensional finite-strain beam theory of Reissner is considered in this work. In contrast with some earlier works on the subject, discussed here are the beam elements whose reference axes are arbitrary space-curved lines. We have shown that an improved representation of curved reference geometry significantly increases the accuracy of the results. However, it also makes the choice of non-locking finite element interpolations somewhat more delicate. A hierarchical displacement interpolation proposed here is proved to be capable of eliminating both shear and membrane locking phenomena. A very satisfying non-locking performance is demonstrated for a set of problems in nonlinear elasto-statics.

On the choice of finite rotation parameters

In this work we discuss some aspects of the three-dimensional finite rotations pertinent to the formulation and computational treatment of the geometrically exact structural theories. Among various possibilities to parameterize the finite rotations, special attention is dedicated to a choice featuring an incremental rotation vector. Some computational aspects pertinent to the implementation of the Newton iterative scheme and the Newmark time-stepping algorithm applied to solving these problems are examined. Representative numerical simulations are presented in order to illustrate the performance of the proposed formulation.

Finite rotations in dynamics of beams and implicit time-stepping schemes

We examine theoretical and computational aspects of three-dimensional finite rotations pertinent to the dynamics of beams. The model problem chosen for consideration is the Reissner beam theory capable of modelling finite strains and finite rotations in geometrically exact manner. Special emphasis is placed on clarifying the geometry aspects, finite rotation updates and the associated linearization procedure pertaining to different choices of rotation parameters. The latter is shown to play an important role in constructing the optimal implementation of a time-stepping scheme.

Stress resultant geometrically nonlinear shell theory with drilling rotations—Part I. A consistent formulation

Abstract In this work we present a consistent theoretical framework for a novel stress resultant geometrically nonlinear shell theory. The main feature of the present shell theory development, which stands in contrast with the classical developments in the shell theory, is the presence of a rotation component around the shell normal (so called drilling rotation) in the description of shell finite rotations. The relationship of the proposed theory with a finite deformation theory of a three-dimensional continuum with independent rotation field is clearly identified. The latter is shown to be an important link which facilitates the proper choice of the shell constitutive model, and a proper construction of the regularized form of the theory capable of supporting the drilling rotations. The corresponding linearized form of the present shell theory is discussed in the closure.

Strong coupling methods in multi-phase and multi-scale modeling of inelastic behavior of heterogeneous structures

In this work we address several issues pertaining to efficiency of the computational approach geared towards modeling of inelastic behavior of a heterogeneous structure, which is represented by a multi-scale model. We elaborate in particular upon the case where the scales remain coupled throughout the computations, implying a constant communication between the finite element models employed at each scale, and only briefly comment upon our treatment of inelastic analysis of a more classical case where the scales can be separated. We also discuss different manners of representing a complex multi-phase microstructure within the framework of the finite element model constructed at that scale, selecting a model problem of two-phase material where each phase has potentially different inelastic behavior. Several numerical examples are given to further illustrate the presented theoretical considerations.

On the role of frame-invariance in structural mechanics models at finite rotations

In this work we re-examine so called geometrically exact models for structures, such as beams, shells or solids with independent rotation field, with respect to invariance under superposed rigid body motion. A special attention is given to clarifying the issues pertaining to the finite element implementation which guarantees that the invariance of the continuum problem is preserved by its discrete approximation. Several numerical simulations dealing with finite rotations of structural models are presented in order to further illustrate and confirm the given theoretical considerations.

Quadrilateral finite elements for analysis of thick and thin plates

Abstract We discuss two quadrilateral plate elements applicable in the analysis of both thick and thin plates. The elements are based on Reissner-Mindlin plate theory and an enhanced displacement interpolation, which enables the consistent loading vector to be constructed. The constraint on the constant shear strain is enforced explicitly thus eliminating the shear locking phenomena in the analysis of thin plates. As a by-product of this work, we take a new look at a well-known discrete Kirchhoff plate element.

Use of incompatible displacement modes for the calculation of element stiffnesses or stresses

Abstract The addition of incompatible displacement modes to lower-order displacement-based elements is re-evaluated. Recent research has indicated that a simple numerical correction can be applied to the shape functions in order that the constant-strain patch test is passed. In this paper a new method of stress recovery is presented in which incompatible modes are introduced. A least-square approximation is used to calculate element stresses which are in microscopic equilibrium and tend to be in global equilibrium with the applied nodal loads on the finite element assemblage. Also, a consistent and robust method for the evaluation of thermal stresses is presented. The numerical methods presented are general and can be applied to all displacement-based finite elements. The basic formulation and examples which are presented in this paper are in three-dimensional elasticity using the eight-node isoparametric elements. Accuracy of the element is illustrated and it is demonstrated that both displacements and stresses are almost identical to those produced by Pians hybrid stress elements.

Stress resultant geometrically nonlinear shell theory with drilling rotations—Part II. Computational aspects

Abstract In this work we discuss some details of the numerical implementation of the geometrically nonlinear shell theory presented in Part I. Two possibilities to represent finite rotations, with an orthogonal matrix and with a rotation vector, are examined in detail, along with their mutual relationship in both spatial and material description. The issues pertinent to the consistent linearization procedure corresponding to these rotation parameterizations are also carefully considered. The geometrically nonlinear method of incompatible modes is extended to the nonlinear shell theory under consideration, and used to provide a 4-node shell element with enhanced performance. A rather extensive set of numerical examples in nonlinear elastostatics is solved in order to corroborate the non-locking performance of the incompatible-mode-based shell elements. The examples include not only analyses of simple shell structures undergoing very large displacements and rotations, but also cases of strong practical interest, such as non-smooth shell structures and shell structures with stiffeners.

Microscale and mesoscale discrete models for dynamic fracture of structures built of brittle material

In this work we present the discrete models for dynamic fracture of structures built of brittle materials. The models construction is based on Voronoi cell representation of the heterogeneous structure, with the beam lattice network used to model the cohesive and compressive forces between the neighboring cells. Each lattice component is a geometrically exact shear deformable beam which can describe large rigid body motion and the most salient fracture mechanisms. The latter can be represented through the corresponding form of the beam constitutive equations, which are derived either at microscale with random distribution of material properties or at a mesoscale with average deterministic values. The proposed models are also placed within the framework of dynamics, where special attention is paid to constructing the lattice network mass matrix as well as the corresponding time-stepping schemes. Numerical simulations of compression and bending tests is given to illustrate the models performance.