On a remarkable geometric-mechanical synergism based on a novel linear eigenvalue problem

The vertices of two specific eigenvectors, obtained from a novel linear eigenvalue problem, describe two curves on the surface of an N-dimensional unit hypersphere. N denotes the number of degrees of freedom in the framework of structural analysis by the Finite Element Method. The radii of curvature of these two curves are 0 and 1. They correlate with pure stretching and pure bending, respectively, of structures. The two coefficient matrices of the eigenvalue problem are the tangent stiffness matrix at the load level considered and the one at the onset of loading. The goals of this paper are to report on the numerical verification of the aforesaid geometric-mechanical synergism and to summarize current attempts of its extension to combinations of stretching and bending of structures.

An analogy between analytical, approximate and numerical methods in nonlinear buckling of functionally graded columns

Purpose The purpose of this study is to investigate the post-buckling analysis of functionally graded columns by using three analytical, approximate and numerical methods. A pre-defined function as an initial assumption for the post-buckling path is introduced to solve the differential equation. The finite difference method is used to approximate the lateral deflection of the column based on the differential equation. Moreover, the finite element method is used to derive the tangent stiffness matrix of the column. Design/methodology/approach The non-linear buckling analysis of functionally graded materials is carried out by using three analytical, finite difference and finite element methods. The elastic deformation and Euler-Bernoulli beam theory are considered to establish the constitutive and kinematics relations, respectively. The governing differential equation of the post-buckling problem is derived through the energy method and the calculus variation. Findings An incremental iterative solution and the perturbation of the displacement vector at the critical buckling point are performed to determine the post-buckling path. The convergence of the finite element results and the effects of geometric and material characteristics on the post-buckling path are investigated. Originality/value The key point of the research is to compare three methods and to detect error sources by considering the derivation process of relations. This comparison shows that a non-incremental solution in the analytical and finite difference methods and an initial assumption in the analytical method lead to an error in results. However, the post-buckling path in the finite element method is traced by the updated tangent stiffness matrix in each load step without any initial limitation.

Quasistatic Fracture using Nonliner-Nonlocal Elastostatics with Explicit Tangent Stiffness Matrix

We apply a nonlinear-nonlocal field theory for numerical calculation of quasistatic fracture. The model is given by a regularized nonlinear pairwise (RNP) potential in a peridynamic formulation. The potential function is given by an explicit formula with and explicit first and second derivatives. This fact allows us to write the entries of the tangent stiffness matrix explicitly thereby saving computational costs during the assembly of the tangent stiffness matrix. We validate our approach against classical continuum mechanics for the linear elastic material behavior. In addition, we compare our approach to a state-based peridynamic model that uses standard numerical derivations to assemble the tangent stiffness matrix. The numerical experiments show that for elastic material behavior our approach agrees with both classical continuum mechanics and the state-based model.The fracture model is applied to produce a fracture simulation for a ASTM E8 like tension test. We conclude with an example of crack growth in a pre-cracked square plate. For the pre-cracked plate, we investigated {\it soft loading} (load in force) and {\it hard loading} (load in displacement). Our approach is novel in that only bond softening is used as opposed to bond breaking. For the fracture simulation we have shown that our approach works with and without initial damage for two common test problems.

Analysis of Elastic–Plastic Problems Using the Improved Interpolating Complex Variable Element Free Galerkin Method

A numerical model for the two-dimensional nonlinear elastic–plastic problem is proposed based on the improved interpolating complex variable element free Galerkin (IICVEFG) method and the incremental tangent stiffness matrix method. The viability of the proposed model is verified through three elastic–plastic examples. The numerical analyses show that the IICVEFG method has good convergence. The solutions using the IICVEFG method are consistent with the solutions obtained from the finite element method using the ABAQUS program. Moreover, the IICVEFG method shows greater computing precision and efficiency than the non-interpolating meshless methods.

Consistent Implicit Time Integration for Viscoplastic Modelingof Subsidence above Hydrocarbon Reservoirs

The viscoplastic model proposed by Vermeer and Neher in 1999 is still currently used in the oil and gas industry for subsidence modeling, to predict the deformation of the ground surface induced by hydrocarbon withdrawal from underground reservoirs. Even though several different implementations of this model have been proposed in the literature, also very recently, a consistent fully implicit implementation is still missing, probably due to the technical difficulties involved in the rigorous derivation of the analytical tangent matrix. To fill this gap and to provide an effective tool to the engineering community, a fully implicit backward Euler integration is proposed and validated in this work. The consistent expression of the tangent stiffness matrix is also derived analytically, and its validation strategy is described in detail. The model was implemented in a commercial finite element code through a user-defined material subroutine. The advantages of the proposed implicit formulation in terms of stability with respect to an explicit formulation were assessed and validated. The examples include studies at material point level and at field scale for a case study of subsidence in a synthetic reservoir.

Structural Mechanics of Negative Stiffness Honeycomb Metamaterials

The development of multi-stable structural forms has attracted considerable attention in the design of architected multi-materials, metamaterials, and morphing structures, as a result of some unusual properties such as negative stiffness and, possibly, negative Poisson's ratio. Multi-stability is achieved through a morphological change of shape upon loading, and in doing so multi-stable structures undergo transitions from one equilibrium state to another. This paper investigates the structural performance of the negative stiffness honeycomb (NSH) metamaterials made of double curved beams which are emerging in various applications such as sensors, actuators, and lightweight impact protective structures with structural tunability and recoverability. An analytical treatment is pursued using the Euler–Lagrange theorem and the stability of the honeycomb has been studied. Based on a static analysis of the nonlinear elastic system, the developed tangent stiffness matrix and ensuing deformation curve were assessed through multiple phases of deformation. The closed-form solution was in good agreement with the numerical finite element (FE) model at different bistability ratios. It was shown that the bistability ratio had a pronounced effect on the overall response of the honeycomb and the desired negativity in the stiffness matrix could be achieved with high bistability ratios.

Numerical Investigation on the Nonlinear Dynamics of Anisotropic Breathing Cracked Rotor Systems

The nonlinear breathing crack behaviors and anisotropy of the bearing are important sources of severe vibration of rotor systems. However, the rotor system considering both of these factors has not gained sufficient attention in the existing studies. In this paper, the nonlinear dynamics of such anisotropic breathing cracked rotor system is investigated based on three-dimensional finite element model (FEM). Firstly, the equations of motion of the rotor system are established in the rotating frame to facilitate the modeling of the breathing crack. The fixed-interface component mode synthesis (CMS) is used to reduce the system’s degrees of freedom (DOFs). Then, in the process of solving the equations by harmonic balance method (HBM) and Newton-Raphson method, an original method for fast calculating tangent stiffness matrix is proposed. Finally, the effects of the crack depth, the anisotropy of bearing and relative angle between bearings on the nonlinear dynamics of the system are studied. The results show that the breathing behavior will complicate the vibration and introduce additional transverse stiffness. The increase of crack depth will deteriorate the vibration. The anisotropy and relative angle of bearing will lead to the splitting and merging of the resonant peaks, respectively.

3-D Simulation of Iceberg Towing Operations: Cable Modeling and Frictional Contact Formulation Using Finite Element Analysis

Ice management in regions of offshore development with icebergs present includes re-direction of icebergs by means of towing. The prevention of tow-rope slippage and iceberg rolling due to hydrostatic instability are essential for an effective and safe operation. The ability to simulate any particular towing operation in the field, prior to attempting it, would provide some measure of assurance of its feasibility. In addition, such a model will allow optimization of towing configuration and application by showing optimal tow direction, maximum force and rate of force application; selection of single tow line or net; and optimum net configuration. The objective of the described work is to develop a simulation tool that can be used for such application. A previous project funded by Hibernia Management and Development Company Ltd. (HMDC) gathered 3-dimensional profile data on 29 icebergs off the East coast of Canada; and further data collection has been ongoing. The present project utilizes these profiles as valuable input in the development of the model for simulating single-line and net tows. This paper presents the first phase of development of a 3-D dynamic iceberg towing model that evolved from an earlier 2-D static version. The current iteration applies the ‘Total Lagrangian’ Finite-Element Method (FEM) to model the cable-and-rope structure between the towing vessel and iceberg, and a contact model that includes sticking and sliding friction between the rope/net and iceberg. The iceberg is modeled as a rigid surface mesh and is fully constrained against motion during the current phase of development, while the cables and ropes are modeled as elastic bar elements with translational inertia and velocity-squared fluid drag. The contact elements consist of penalty springs with proportional damping, and appropriate values of these are found to be critical for numerical stability of the solution. As well, due to the large difference in stiffness values between the heavy tow cable and buoyant ropes, special attention is given to obtaining the initial tangent stiffness matrix of the cable-and-rope structure. The FE dynamic equations of motion are solved implicitly in the time domain using a combination of full and modified Newton-Raphson iteration. Simulations of contact initiation between the rope and iceberg for single-loop and net configurations are presented, as well as slipping during particular single-loop tows. Current challenges and opportunities for further development are discussed, including improving computational speed, implementing iceberg motion, adding wind and wave forces, and validating rope-ice friction characteristics through small-scale iceberg towing response in a laboratory.