Dynamic Finite Element Formulation Research Articles

Three-dimensional Non-linear Finite Element Analysis of Impact Response and Damage in Laminated Composite Shells

The impact response and the impact-induced damage in curved composite laminates subjected to transverse impact by a metallic impactor are studied using three-dimensional non-linear transient dynamic finite element formulation. A layered version of isoparametric eight-noded hexahedral element with incompatible modes is developed which incorporates geometrical non-linearity based on total Langragian approach. The non-linearity of both strain displacement relation and contact loading are simultaneously solved using Newton-Raphson incrementaliterative method. Impact-induced damages (matrix cracking and delamination) are predicted using appropriate three-dimensional stress-based failure criteria. Some example problems of graphite/epoxy laminated cylindrical shells with variation of important parameters such as impactor velocity, shell curvature, laminate dimension and fibre orientation of plies are solved and the influence of geometrical non-linear effect on both impact response and resulting damages is demonstrated.

Nitsche method for contact with Coulomb friction: Existence results for the static and dynamic finite element formulations

We study the Nitsche-based finite element method for contact with Coulomb friction considering both static and dynamic situations. We provide existence and/or uniqueness results for the discretized problems under appropriate assumptions on physical and numerical parameters. In the dynamic case, existence and uniqueness of the space semi-discrete problem holds for every value of the friction coefficient and the Nitsche parameter. In the static case, if the Nitsche parameter is large enough, existence is ensured for any friction coefficient, and uniqueness can be obtained provided that the friction coefficient is below a bound that depends on the mesh size. These results are complemented by a numerical study.

Investigation on the Thermoforming of Pmsq-Hdpe for the Manufacture of a NACA Profile of Small Dimensions.

Unmanned aerial vehicles (UAVs) or drones are attracting increasing interest in the aviation industry, both for military and civilian applications. The materials used so far in the manufacture of UAVs are wood, plastic, aluminum and carbon fiber. In this regard, a new family of high-density polyethylene (HDPE) nanocomposites reinforced with polymethylsilsesquioxane nanoparticles (PMSQ), with mechanical performances significantly superior to those of pure HPDE, has been prepared by a fusion-combination process. Their viscoelastic properties were determined by oscillatory shear tests and their viscoelastic behavior characterized by the Lodge integral model. Then, the Lagrangian formulation and the membrane theory assumption were used in the explicit implementation of the dynamic finite element formulation. For the forming phase, we considered the thermodynamic approach to express the external work in terms of closed volume. In terms of von Mises stress distribution and thickness in the blade, the results indicate that HDPE-PMSQ behaves like virgin HDPE. Furthermore, its materials, for all intents and purposes, require the same amount of energy to form as HDPE.

On the geometrically nonlinear analysis of sandwich shells with viscoelastic core: A layerwise dynamic finite element formulation

The objective of this work is to present a finite element formulation for dynamic analysis of sandwich shells with viscoelastic core under large deformation. The present study is based on an incremental updated Lagrangian approach together with the Newmark integration scheme. The viscoelastic constitutive model which is used to define the behavior of the core, comes from the Riesz theorem and the corresponding creep functions are estimated using Dirichlet-Prony series. Also, the viscoelastic deferred strain is derived in an appropriate incremental form using the state variables. The employed layerwise shell element which is based on zig-zag theory has eight nodes on its mid layer. What’s more, in addition to three translational and two rotational degrees of freedom per node, the upper and lower layers are allowed to rotate independently relative to their neighbor layers. Thus, the damping effect of the viscoelastic core could be well described within the shear deformable displacement field. The presented nonlinear formulation is then implemented to a nonlinear finite element program to be appraised by solving different kinds of problems. The obtained numerical results correlate well with those available in the literature.

Data-Driven Adaptive Iterative Learning Method for Active Vibration Control Based on Imprecise Probability

A data-driven adaptive iterative learning (IL) method is proposed for the active control of structural vibration. Considering the repeatability of structural dynamic responses in the vibration process, the time-varying proportional-type iterative learning (P-type IL) method was applied for the design of feedback controllers. The model-free adaptive (MFA) control, a data-driven method, was used to self-tune the time-varying learning gains of the P-type IL method for improving the control precision of the system and the learning speed of the controllers. By using multi-source information, the state of the controlled system was detected and identified. The square root values of feedback gains can be considered as characteristic parameters and the theory of imprecise probability was investigated as a tool for designing the stopping criteria. The motion equation was driven from dynamic finite element (FE) formulation of piezoelectric material, and then was linearized and transformed properly to design the MFA controller. The proposed method was numerically and experimentally tested for a piezoelectric cantilever plate. The results demonstrate that the proposed method performs excellent in vibration suppression and the controllers had fast learning speeds.

New Frequency‐Dependent Trigonometric Interpolation Functions for the Dynamic Finite Element Analysis of Thin Rectangular Plates

The Dynamic Finite Element (DFE) formulation is a superconvergent, semianalytical method used to perform vibration analysis of structural components during the early stages of design. It was presented as an alternative to analytical and numerical methods that exhibit various drawbacks, which limit their applicability during the preliminary design stages. The DFE method, originally developed by the second author, has been exploited heavily to study the modal behaviour of beams in the past. Results from these studies have shown that the DFE method is capable of arriving at highly accurate results with a coarse mesh, thus, making it an ideal choice for preliminary stage modal analysis and design of structural components. However, the DFE method has not yet been extended to study the vibration behaviour of plates. Thus, the aim of this study is to develop a set of frequency‐dependent, trigonometric shape functions for a 4‐noded, 4‐DOF per node element as a basis for developing a DFE method for thin rectangular plates. To this end, the authors exploit a distinct quasi‐exact solution to the plate governing equation and this solution is then used to derive the new, trigonometric basis and shape functions, based on which the DFE method would be developed.

Dynamic finite element formulations for moderately thick plate vibrations based on the modified Mindlin theory

In this paper, the modified Mindlin theory is used for the construction of the dynamic stiffness matrix, the flexibility matrix, and the transfer matrix of a thick plate simply supported at two opposite edges. The modified Mindlin theory operates with bending deflection as the basic variable for the determination of the total (bending+shear) deflection and the angles of rotation. It is shown that the appropriate application of the constructed matrices to various boundary conditions leads to a determined formulation of the eigenvalue problem. As a result, the problem can be treated using ordinary algorithms for linear eigenvalue problems. Therefore the application of the relatively complex Wittrick-Williams algorithm, developed for the transcendental eigenvalue problems with the unusual forms of non-zero determinant is avoided. Using this technique, dynamic finite elements can be obtained in a simpler form than that based on the application of the conventional Mindlin theory. All phenomena related to the dynamic finite element application are investigated in the case of an axial bar vibration in a transparent analytical way. Furthermore, the application of the developed thick plate elements is illustrated through several numerical examples. Examples include using single elements as well as an assembly of elements. In the examples of single elements transcendental eigenfunctions are derived. Also a dynamic beam finite element is considered as a special case of the plate finite strip which exhibits no variation in the transverse direction.

A Framework for Extension of Dynamic Finite Element Formulation to Flexural Vibration Analysis of Thin Plates

Dynamic Finite Element formulation is a powerful technique that combines the accuracy of the exact analysis with wide applicability of the finite element method. The infinite dimensionality of the exact solution space of plate equation has been a major challenge for development of such elements for the dynamic analysis of flexible two-dimensional structures. In this research, a framework for such extension based on subset solutions is proposed. An example element is then developed and implemented in MATLAB® software for numerical testing, verification, and validation purposes. Although the presented formulation is not exact, the element exhibits good convergence characteristics and can be further enriched using the proposed framework.

A Dynamic Finite Element for Coupled Extensional-Torsional Vibration of Uniform Composite Thin-Walled Beams

A Dynamic Finite Element (DFE) formulation for the free vibration analysis of extension-torsion coupled uniform composite thin-walled beams is presented. Employing the exact solutions of the differential equations governing the uncoupled vibrations of a uniform beam element, the analytical expressions for extensional and torsional dynamic trigonometric shape functions are derived. By exploiting the principle of virtual work and the frequency-dependent shape functions, the element dynamic stiffness matrix is developed. The application of the theory is demonstrated by a Circumferentially Uniform Stiffness (CUS) composite circular tube for which the influence of ply fibre-angle on the natural frequencies is studied. A variety of CUS configurations are studied and the correctness of the theory and the superiority of the proposed DFE over the conventional FEM methods are confirmed by numerical checks and the published results.

Static and Dynamic Micropolar Linear Elastic Beam Finite Element Formulation, Implementation, and Analysis

AbstractStarting with static and dynamic micropolar linear plane stress elasticity, and applying Timoshenko beam kinematics with axial stretch, a mixed micropolar small-strain beam finite element (FE) formulation results. The mixed formulation is shown to be convergent upon mesh refinement under static and dynamic loading. The acceleration form of the Newmark family of time integration methods is applied to integrate the coupled hyperbolic linear governing equations. Instantaneous axial and transverse step forces are applied and released to analyze the free longitudinal and transverse vibrations with the mixed formulation FE implementation. The transverse displacement and rotational degrees of freedom (DOF) are coupled, but the axial displacement is decoupled from the rotational DOF because the first area moment of inertia Q is zero. Applied sinusoidal axial and transverse forces lead to axial and transverse displacement and rotational wave patterns that are a combination of low and high frequency waves. ...

Dynamic finite element formulation for Cosserat elastic plates

SummaryA displacement and rotation‐based dynamic finite element formulation for Cosserat plates is discussed in detail in this paper. Special attention is given to the validation of the element through adequate benchmarks and patch tests. The choice of the interpolation functions and the order of integration of the stiffness and the mass matrices are extensively argued. The possibility of local system deficiencies is explored, and a shear locking investigation specifically elaborated for Cosserat plates is carried out. It is shown how the present formulation has interesting computational properties as compared to a classical continuum‐based formulation and how it can provide suitable results despite the use of reduced integration. An example of application of the finite element is given, in which the natural frequencies of a masonry panel modelled by means of discrete elements are computed and compared with the finite elements solution. Copyright © 2014 John Wiley & Sons, Ltd.

Dynamic analysis of strain localization in water‐saturated clay using a cyclic elasto‐viscoplastic model

SUMMARYA computational framework is presented for dynamic strain localization and deformation analyses of water‐saturated clay by using a cyclic elasto‐viscoplastic constitutive model. In the model, the nonlinear kinematic hardening rule and softening due to the structural degradation of soil particles are considered. In order to appropriately simulate the large deformation phenomenon in strain localization analysis, the dynamic finite element formulation for a two‐phase mixture is derived in the updated Lagrangian framework. The shear band development is shown through the distributions of viscoplastic shear strain, the axial strain, the mean effective stress, and the pore water pressure in a normally consolidated clay specimen. From the local stress–strain relations, more brittleness is found inside the shear bands than outside of them. The effects of partially drained conditions and mesh‐size dependency on the shear banding are also investigated. The effect of a partially drained boundary is found to be insignificant on the dynamic shear band propagation because of the rapid rate of applied loading and low permeability of the clay. Using the finer mesh results in slightly narrower shear bands; nonetheless, the results manifest convergency through the mesh refinement in terms of the overall shape of shear banding and stress–strain relations. Copyright © 2013 John Wiley & Sons, Ltd.

Dynamic Models of Piled Foundations

The vibration behavior of piled foundations is an important consideration in fields such as earthquake engineering, construction, machine-foundation design, offshore structures, nuclear energy, and road and rail development. This paper presents a review of the past 40 years' literature on modeling the frequency-dependent behavior of pile foundations. Beginning with the earliest model of a single pile, adapted from those for embedded footings, it charts the development of the four pile-modeling techniques: the “dynamic Winkler-foundation” approach that uses springs to represent the effect of the soil; elastic-continuum-type formulations involving the analytical solutions for displacements due to a subsurface disk, cylinder, or other element; boundary element methods; and dynamic finite-element formulations with special nonreflecting boundaries. The modeling of pile groups involves accounting for pile-soil-pile interactions, and four such methods exist: interaction factors; complete pile models; the equivalent pier method; and periodic structure theory. Approaches for validating pile models are also explored.

Electromechanical instability on dielectric polymer surface: Modeling and experiment

We present a dynamic finite element formulation for dielectric elastomers that significantly alleviates the problem of volumetric locking that occurs due to the incompressible nature of the elastomers. We accomplish this by modifying the Q1P0 formulation of Simo et al. [1], and adapting it to the electromechanical coupling that occurs in dielectric elastomers. We demonstrate that volumetric locking has a significant impact on the critical electric fields that are necessary to induce electromechanical instabilities such as creasing and cratering in dielectric elastomers, and that the locking effects are most severe in problems related to recent experiments that involve significant constraints upon the deformation of the elastomers. We then compare the results using the new Q1P0 formulation to that obtained using standard 8-node linear and 27-node quadratic hexahedral elements to demonstrate the capability of the proposed approach. Finally, direct comparison to the recent experimental work on the creasing instability on dielectric polymer surface by Wang et al. [2] is presented. The present formulation demonstrates good agreement to experiment for not only the critical electric field for the onset of the creasing instability, but also the experimentally observed average spacing between the creases.

Hybrid Finite Element Formulation for Electrostrictive Materials: Dynamic Analysis of Beam

A nonlinear electromechanical coupled dynamic finite element formulation for electrostrictive materials is proposed. This formulation includes the quadratic dependence of strain with polarization, valid at a constant temperature and excludes hysteresis. The present formulation uses linear finite element analysis along with the numerical solution of the nonlinear constitutive equation using Newton-Raphson technique only within each electrostrictive elements. Therefore this formulation is capable of handling nonlinearity only on the electrostrictive domain instead of making the whole finite element analysis as nonlinear. The Newton-Raphson technique is specially modified in order to guarantee the convergence of the solution. A simple technique for obtaining the initial guess of the solution for Newton-Raphson technique is also proposed which gives faster convergence of the solution. The assumption made in most of the finite element formulations, that polarization is approximately equal to electric displacement has been relaxed. The proposed finite element formulation is computationally very efficient than any other available nonlinear dynamic finite element formulation for electrostrictive materials and it proves its very high computational efficiency especially in case of electrostrictive patches embedded in large structures.

Analysis of impact response and damage in laminated composite cylindrical shells undergoing large deformations

The impact behaviour and the impact-induced damage in laminated composite cylindrical shell subjected to transverse impact by a foreign object are studied using three-dimensional non-linear transient dynamic finite element formulation. A layered version of 20 noded hexahedral element incorporating geometrical non-linearity is developed based on total Langragian approach. Non-linear system of equations resulting from non-linear strain displacement relation and non-linear contact loading are solved using Newton-Raphson incremental-iterative method. Some example problems of graphite/epoxy cylindrical shell panels are considered with variation of impactor and laminate parameters and influence of geometrical non-linear effect on the impact response and the resulting damage is investigated.

Numerical modeling and simulation of pile in liquefiable soil

Presented in this paper is numerical methodology to model and simulate behavior of piles in liquefiable soils. Modeling relies on use of validated elasto-plastic material model for soil skeleton, verified fully coupled porous media (soil skeleton) – pore fluid (water) dynamic finite element formulation, and detailed load staging of FEM models. A bounding surface elastic–plastic sand model that accounts for fabric change is used to model soil skeleton, while a fully coupled, dynamic, inelastic formulation ( u – p – U ) is used to model soil and water displacement and pore water pressures. Much attention is paid to accurate staged loading of the models, which start from a zero state of stress and strain for a soil without a pile, followed by application of self-weight, then by excavation and pile installation with application of pile self-weighting. Finally, seismic loading is applied followed by time to dissipate excess pore pressures that have developed. A total of six cases were modeled and simulated varying slope inclination, presence of pile–column and boundary condition for pile–column system. Presented are interesting and useful results that are used to deepen our understanding of behavior of soil–pile–column systems during liquefaction (lateral deformations, pile pinning effect, ground settlement). Moreover, detailed description of modeling is used to emphasize the availability and use of high fidelity modeling tools for simulating effects of liquefied soil on soil–structure systems.

Off‐fault plasticity and earthquake rupture dynamics: 2. Effects of fluid saturation

We present an analysis of inelastic off‐fault response in fluid‐saturated material during earthquake shear rupture. The analysis is conducted for 2‐D plane strain deformation using an explicit dynamic finite element formulation. Along the fault, linear slip‐weakening behavior is specified, and the off‐fault material is described using an elastic‐plastic description of the Drucker‐Prager form, which characterizes the brittle behavior of rocks under compressive stress when the primary mode of inelastic deformation is frictional sliding of fissure surfaces, microcracking and granular flow. In this part (part 1), pore pressure changes were neglected in materials bordering the fault. In part 2, we more fully address the effects of fluid saturation. During the rapid stressing by a propagating rupture, the associated undrained response of the surrounding fluid‐saturated material may be either strengthened or weakened against inelastic deformation. We consider poroelastoplastic materials with and without plastic dilation. During nondilatant undrained response near a propagating rupture, large increases in pore pressure on the compressional side of the fault decrease the effective normal stress and weaken the material, and decreases in pore pressure on the extensional side strengthen the material. Positive plastic dilatancy reduces pore pressure, universally strengthening the material. Dilatantly strengthened undrained deformation has a diffusive instability on a long enough timescale when the underlying drained deformation is unstable. Neglecting this instability on the short timescale of plastic straining, we show that undrained deformation is notably more resistant to shear localization than predicted by neglect of pore pressure changes.

The Use Of Trigonometric Interpolation Functions For Vibration AnalysisOf Beam Structures - Bridging Gap Between FEM And Exact Formulations

The small linear harmonic oscillations of linearly elastic coupled beam structures are addressed. The Finite Element Method (FEM), conventional Hermite beam formulation, and the resulting element matrices are then briefly discussed. The so-called exact Dynamic Stiffness Matrix (DSM) formulation for the longitudinal vibration of axially loaded beams is presented. Finally, the Dynamic Finite Element (DFE) approach is introduced and its application to the axial vibration of beams is displayed. The comparison is made between the standard static and (frequency dependent) Dynamic beam shape functions. The DFE formulation, combines the generality of the FEM and the high precision provided by DSM methods. The weighting functions and shape functions are evaluated referring to the appropriate exact DSM formulation. The DFE approach can be advantageously extended to more complex cases which distinguishes this method from the DSM method.

Preliminary assessment of sandwich plates subject to blast loads

The question motivating the present study is whether metal sandwich plates with sufficiently strong cores are able to sustain substantially larger blast loads than monolithic solid plates of the same material and total mass. Circular plates clamped at their edges are considered under blast loads large enough to produce substantial deflections. The material is elastic–perfectly plastic. Material strain-rate dependence and fracture are neglected. A dynamic finite element formulation for elastic–plastic solids is employed to analyze the plate response. Uniformly distributed blast impulses are considered. As a basis for comparison, complete results are obtained for solid plates for both zero-period and finite-period impulses. Similar computations are carried out for a set of sandwich plates having tetragonal truss cores. The potential for superior strength and energy absorbing capacity of the sandwich plates is demonstrated compared with solid plates having the same mass. The importance of both the strength and energy absorbing capacity of the core are highlighted for superior blast resistance. Proposals for further research are made.