Totally Ergodic Generalised Matrix Equilibrium States have the Bernoulli Property

We show that every totally ergodic generalised matrix equilibrium state is $$\psi $$ ψ -mixing with respect to the natural partition into cylinders and hence is measurably isomorphic to a Bernoulli shift in its natural extension. This implies that the natural extensions of ergodic generalised matrix equilibrium states are measurably isomorphic to Bernoulli processes extended by finite rotations. This resolves a question of Gatzouras and Peres in the special case of self-affine repelling sets with generic translations.

BEAM ELEMENT UNDER FINITE ROTATIONS

The present work focuses on the 2-D formulation of a nonlinear beam model for slender structures that can exhibit large rotations of the cross sections while remaining in the small-strain regime. Bernoulli-Euler hypothesis that plane sections remain plane and perpendicular to the deformed beam centerline is combined with a linear elastic stress-strain law.The formulation is based on the integrated form of equilibrium equations and leads to a set of three first-order differential equations for the displacements and rotation, which are numerically integrated using a special version of the shooting method. The element has been implemented into an open-source finite element code to ease computations involving more complex structures. Numerical examples show a favorable comparison with standard beam elements formulated in the finite-strain framework and with analytical solutions.

Analyzing FG shells with large deformations and finite rotations

Purpose The purpose of this study is dedicated to use an efficient mixed strain finite element approach to develop a three-node triangular shell element. Moreover, large deformation analysis of the functionally graded material shells is the main contribution of this research. These target structures include thin or moderately thick panels. Design/methodology/approach Due to reach these goals, Green–Lagrange strain formulation with respect to small strains and large deformations with finite rotations is used. First, an efficient three-node triangular degenerated shell element is formulated using tensorial components of two-dimensional shell theory. Then, the variation of Young’s modulus through the thickness of shell is formulated by using power function. Note that the change of Poisson’s ratio is ignored. Finally, the governing linearized incremental relation was iteratively solved using a high potential nonlinear solution method entitled generalized displacement control. Findings Some well-known problems are solved to validate the proposed formulations. The suggested triangular shell element can obtain the exact responses of functionally graded (FG) shell structures, without any shear locking, instabilities and ill-conditioning, even by using fewer numbers of the elements. The obtained outcomes are compared with the other reference solutions. All findings demonstrate the accuracy and capability of authors’ element for analyzing FG shell structures. Research limitations/implications A mixed strain finite element approach is used for nonlinear analysis of FG shells. These structures are curved thin and moderately thick shells. Small strains and large deformations with finite rotations are assumed. Practical implications FG shells are mostly made curved thin or moderately thick, and these structures have a lot of applications in the civil and mechanical engineering. Social implications The social implication of this study is concerned with how technology impacts the world. In short, the presented scheme can improve structural analysis ways. Originality/value Developing an efficient three-node triangular element, for geometrically nonlinear analysis of FG doubly-curved thin and moderately thick shells, is the main contribution of the current research. Finite rotations are considered by using the Taylor’s expansion of the rotation matrix. Mixed interpolation of strain fields is used to alleviate the locking phenomena. Using fewer numbers of shell elements with fewer numbers of degrees of freedom can reduce the computational costs and errors significantly.

Noncommutativity of Finite Rotations and Definitions of Curvature and Torsion

The geometry of a space curve, including its curvature and torsion, can be uniquely defined in terms of only one parameter which can be the arc length parameter. Using the differential geometry equations, the Frenet frame of the space curve is completely defined using the curve equation and the arc length parameter only. Therefore, when Euler angles are used to describe the curve geometry, these angles are no longer independent and can be expressed in terms of one parameter as field variables. The relationships between Euler angles used in the definition of the curve geometry are developed in a closed-differential form expressed in terms of the curve curvature and torsion. While the curvature and torsion of a space curve are unique, the Euler-angle representation of the space curve is not unique because of the noncommutative nature of the finite rotations. Depending on the sequence of Euler angles used, different expressions for the curvature and torsion can be obtained in terms of Euler angles, despite the fact that only one Euler angle can be treated as an independent variable, and such an independent angle can be used as the curve parameter instead of its arc length, as discussed in this paper. The curve differential equations developed in this paper demonstrate that the curvature and torsion expressed in terms of Euler angles do not depend on the sequence of rotations only in the case of infinitesimal rotations. This important conclusion is consistent with the definition of Euler angles as generalized coordinates in rigid body dynamics. This paper generalizes this definition by demonstrating that finite rotations cannot be directly associated with physical geometric properties or deformation modes except in the cases when infinitesimal-rotation assumptions are used.

Influence of Geometric Nonlinearities on the Asynchronous Modes of an Articulated Prestressed Slender Structure

Synchronous modal oscillations, characterized by unisonous motions for all physical coordinates, are well known. In turn, asynchronous oscillations lack a general definition to address all the associated features and implications. It might be thought, at first, that asynchronicity could be related to nonsimilar modes, which might be associated with phase differences between displacement and velocity fields. Due to such differences, the modes, although still periodic, might not be characterized by stationary waves so that physical coordinates might not attain their extreme values at the same instants of time, as in the case of synchronous modes. Yet, it seems that asynchronicity is more related to frequency rather than phase differences. A more promising line of thought associates asynchronous oscillations to different frequency contents over distinct parts of a system. That is the case when, in a vibration mode, part of the structure remains at rest, that is, with zero frequency, whereas other parts vibrate with non-null modal frequency. In such a scenario, localized oscillations would explain modal asynchronicity. When the system parameters are properly tuned, localization may appear even in very simple models, like Ziegler's columns, shear buildings, and slender structures. Now, the latter ones are recast, but finite rotations are assumed, in order to verify how nonlinearity affects existing linear asynchronous modes. For this purpose, the authors follow Shaw–Pierre's invariant manifold formulation. It is believed that full understanding of asynchronicity may apply to design of vibration controllers, microsensors, and energy-harvesting systems.

Numerical implementation of the Murnaghan material model in ABAQUS/Standard

The paper presents a numerical implementation of the Murnaghan material model (M) [1] in the finite element method software ABAQUS / Standard v. 6.14 [2]. The UHYPER user subroutine is employed, which is suitable for the class of isotropic hyperelastic models [3]. As a special case of the M model, the Saint Venant-Kirchhoff (SVK) model is considered [4]. Formal verification on the basis of elementary tests is performed. Among others, a special attention is paid to a simple shear deformation. In all tested types of deformation, analytical values confirms results based on the finite element procedure within assumed numerical precision and accuracy. It should be noted that the stored-energy function of the M and SVK models do not meet any requirements of the mathematical theory of non-linear elasticity [4, 5]. Therefore, these models are suitable for relatively small deformations, while there are no restrictions on finite rotations. As an example of applications, a tube under axial compression is considered in two cases. Various starting parameters for the Riks procedure [6, 7] are adopted to obtain different solutions of corresponding boundary value problem. Material parameters of steel are considered according to Lurie [8].