A component-free Lagrangian finite element formulation for large strain elastodynamics

Standard finite element formulation and implementation in solid dynamics at large strains usually relies upon and indicial-tensor Voigt notation to factorized the weighting functions and take advantage of the symmetric structure of the algebraic objects involved. In the present work, a novel component-free approach, where no reference to a basis, axes or components is made, implied or required, is adopted for the finite element formulation. Under this approach, the factorisation of the weighting function and also of the increment of the displacement field, can be performed by means of component-free operations avoiding both the use of any index notation and the subsequent reorganisation in matrix Voigt form. This new approach leads to a straightforward implementation of the formulation where only vectors and second order tensors in $${\mathbb {R}}^3$$ R 3 are required. The proposed formulation is as accurate as the standard Voigt based finite element method however is more efficient, concise, transparent and easy to implement.

Numerical Simulation of Flame Retardant Polymers Using a Combined Eulerian–Lagrangian Finite Element Formulation

Many polymer-made objects show a trend of melting and dripping in fire, a behavior that may be modified by adding flame retardants (FRs). These affect materials properties, e.g., heat absorption and viscosity. In this paper, the effect of a flame retardant on the fire behavior of polymers in the UL 94 scenario is studied. This goal is achieved essentially by applying a new computational strategy that combines the particle finite element method for the polymer with an Eulerian formulation for air. The sample selected is a polypropylene (PP) with magnesium hydroxide at 30 wt.%. For modelling, values of density, conductivity, specific heat, viscosity, and Arrhenius coefficients are obtained from different literature sources, and experimental characterization is performed. However, to alleviate the missing viscosity at a high temperature, three viscosity curves are introduced on the basis of the viscosity curve provided by NIST and the images of the test. In the experiment, we burn the specimen under the UL 94 condition, recording the process and measuring the temperature evolution by means of three thermocouples. The UL 94 test is solved, validating the methodology and quantifying the effect of FR on the dripping behavior. The numerical results prove that well-adjusted viscosity is crucial to achieving good agreement between the experimental and numerical results in terms of the shape of the polymer and the temperature evolution inside the polymer.

Benchmark for the Coupled Magneto-Mechanical Boundary Value Problem in Magneto-Active Elastomers

Within this contribution, a novel benchmark problem for the coupled magneto-mechanical boundary value problem in magneto-active elastomers is presented. Being derived from an experimental analysis of magnetically induced interactions in these materials, the problem under investigation allows us to validate different modeling strategies by means of a simple setup with only a few influencing factors. Here, results of a sharp-interface Lagrangian finite element framework and a diffuse-interface Eulerian approach based on the application of a spectral solver on a fixed grid are compared for the simplified two-dimensional as well as the general three-dimensional case. After influences of different boundary conditions and the sample size are analyzed, the results of both strategies are examined: for the material models under consideration, a good agreement of them is found, while all discrepancies can be ascribed to well-known effects described in the literature. Thus, the benchmark problem can be seen as a basis for future comparisons with both other modeling strategies and more elaborate material models.

Steady Motion of a Slack Belt Drive: Dynamics of a Beam in Frictional Contact With Rotating Pulleys

We seek the steady-state motion of a slack two-pulley belt drive with the belt modeled as an elastic, shear-deformable rod. Dynamic effects and gravity induce significant transverse deflections due to the low pre-tension. In analogy to the belt-creep theory, it is assumed that each contact region between the belt and one of the pulleys consists of a single sticking and a single sliding zone. Based on the governing equations of the rod theory, we for the first time derive the corresponding boundary value problem and integrate it numerically. Furthermore, a novel mixed Eulerian–Lagrangian finite element scheme is developed that iteratively seeks the steady-state solution. Finite element solutions are validated against semi-analytic results obtained by numerical integration of the boundary value problem. Parameter studies are conducted to examine solution dependence on the stiffness coefficients and the belt pre-tension.

Effect of Structure on Response of a Three-Dimensional-Printed Photopolymer-Particulate Composite Under Intermediate Strain Rate Loading

The thermo-mechanical response of an additively manufactured photopolymer-particulate composite under conditions of macroscopic uniaxial compression without lateral confinement at overall strain rates of 400–2000 s−1 is studied. The material has a direct-ink-written unidirectional structure. Computations are performed to quantify the effects of microstructure attributes including anisotropy, defects, and filament size on localized deformation, energy dissipations, and temperature rises. To this effect, an experimentally informed Lagrangian finite element framework is used, accounting for finite-strain elastic–plastic deformation, strain-rate effect, failure initiation and propagation, post-failure internal contact and friction, heat generation due to friction and inelastic bulk deformation, and heat conduction. The analysis focuses on the material behavior under overall compression. Despite relatively low contribution to overall heating, friction is localized at fracture sites and plays an essential role in the development of local temperature spikes unknown as hotspots. The microstructural attributes are found to significantly affect the development of the hotspots, with local heating most pronounced when loading is transverse to the filaments or when the material has higher porosities, stronger inter-filament junctions, or smaller filament sizes. Samples with smaller filament sizes undergo more damage, exhibit higher frictional dissipation, and develop larger hotspots that occur primarily at failure sites.

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.