Formulation For Large Deformation Analysis Research Articles

New insight into large deformation analysis of stretch-based and invariant-based rubber-like hyperelastic elastomers

A nonlinear finite element formulation for large deformation analysis of hyperelastic membranes in Total Lagrangian framework is developed. In many cases, it is convenient to utilize the invariant-based strain energy density function to analyze the structures when employing the finite element method. However, in this contribution, both invariant-based and stretch-based formulations are developed. Formulations are completely general and not restricted to axisymmetric loading and geometry of membrane. Furthermore, initially flat and non-flat membrane can be investigated by this method. Incompressible as well as compressible materials can be modeled. A method for imposing the plane stress condition to the formulation for compressible materials is presented. Moreover, isotropic and anisotropic materials can be considered. Some problems are analyzed and results compared to the available ones in the literature. The comparison between solution based on invariant-based method and stretch-based method with exact fourth-order elasticity tensor obtained in this work demonstrates that the stretch-based method derived in this paper works very well. The presented examples demonstrate that the developed formulations have no restriction or simplification with respect to the modeled object geometry and inherent anisotropy.

Recent Advances in the Absolute Nodal Coordinate Formulation: Literature Review From 2012 to 2020

Abstract Absolute nodal coordinate formulation (ANCF) is a nonincremental nonlinear finite element procedure that has been successfully applied to the large deformation analysis of multibody systems for more than two decades. Although a comprehensive review on ANCF was conducted by Gerstmayr et al. (2013, “Review on the Absolute Nodal Coordinate Formulation for Large Deformation Analysis of Multibody Systems,” J. Comput. Nonlinear Dyn., 8(3), p. 031016), significant theoretical developments have been made since then at a much faster pace to improve the element accuracy and computational efficiency. In order to overview recent advances in ANCF simulation capabilities that are not covered in the first review paper, this paper aims to conduct a comprehensive review of 259 papers concerning ANCF published from 2012 to 2020. It is shown that the ANCF element library has grown substantially for beam, plate/shell, solid elements, eliminating drawbacks of ANCF elements developed earlier. The application areas have extended, especially in the aerospace field, and the enhanced ANCF simulation capabilities have been demonstrated in solving challenging engineering problems. Research efforts have been made continually to integrate computer-aided design (CAD) and analysis with ANCF elements. Furthermore, computational improvements and multiphysics simulations have become major research topics for ANCF. It is also demonstrated that the accurate ANCF geometry description can be exploited to facilitate structural optimization of multibody systems.

A meshfree formulation for large deformation analysis of flexoelectric structures accounting for the surface effects

In this work, we present a compactly supported radial basis function (CSRBF) based meshfree method to analyse geometrically nonlinear flexoelectric nanostructures considering surface effects. Flexoelectricity is the polarization of dielectric materials due to the gradient of strain, which is different from piezoelectricity in which polarization is dependent linearly on strain. The surface effects gain prominence as the size of the structure tends to nanoscale and so their consideration is inevitable when flexoelectric nanostructures are analysed. First, the proposed meshfree formulation is validated and the influence of nonlinear strain terms on the energy conversion ability of flexoelectric beams made of a non-piezoelectric material like cubic Strontium Titanate is studied. Subsequently, the meshfree formulation for nonlinear flexoelectricity is extended to include nonlinear surface effects. It is determined that the surface effects can have notable influence on the output flexoelectric voltage of nano-sized cantilever structures in the nonlinear regime.

Finite deformation analysis of visco-hyperelastic materials via solid tetrahedral finite elements

An alternative solid finite element formulation for large deformation analysis of viscoelastic materials is proposed. This new approach is based on positions and makes possible a robust implementation of an isoparametric solid tetrahedral that presents no locking when dealing with complex stress, strain and strain rate for general structural analysis. A consistent way to write internal variables that accounts for finite viscoelastic strains is proposed. In this alternative methodology the neo-Hookean hyperelastic law is taken into account together with the Zener viscoelastic model. The evolution law is described in terms of a rate equation involving the viscous right Cauchy-Green stretch tensor. The study is dedicated to homogeneous materials under isothermal and quasi-static conditions. The nonlinear solution procedure is performed via the Newton-Raphson iterative technique and the backward-Euler method.Four illustrative examples involving large viscoelastic strains are analyzed: uniaxial tension, simple shear, buckling of a clamped column and partially loaded block. The present formulation can reproduce creep, stress relaxation and viscoelastic rate dependent stiffening at large strains, which are usually observed in polymeric materials. Even for very complex stress, strain and strain rates the mesh refinement of the proposed methodology leads to more accurate results, avoiding general locking problems. The effect of the viscosity parameter on the material response and the evolution of viscous stretches over time are also highlighted in the results.

Review on the Absolute Nodal Coordinate Formulation for Large Deformation Analysis of Multibody Systems

The aim of this study is to provide a comprehensive review of the finite element absolute nodal coordinate formulation, which can be used to obtain efficient solutions to large deformation problems of constrained multibody systems. In particular, important features of different types of beam and plate elements that have been proposed since 1996 are reviewed. These elements are categorized by parameterization of the elements (i.e., fully parameterized and gradient deficient elements), strain measures used, and remedies for locking effects. Material nonlinearities and the integration of the absolute nodal coordinate formulation to general multibody dynamics computer algorithms are addressed with particular emphasis on visco-elasticity, elasto-plasticity, and joint constraint formulations. Furthermore, it is shown that the absolute nodal coordinate formulation has been applied to a wide variety of challenging nonlinear dynamics problems that include belt drives, rotor blades, elastic cables, leaf springs, and tires. Unresolved issues and future perspectives of the study of the absolute nodal coordinate formulation are also addressed in this investigation.

A Novel Meshfree Method for Large Deformation Analysis of Elastic and Viscoelastic Bodies without Using Background Cells

A Galerkin-type new meshfree formulation for large deformation analysis of elastic and viscoelastic bodies with a quasi-implicit time advancing scheme was proposed. The proposed method uses floating stress-points for the domain integration instead of background cell integration or nodal integration. The nodes and stress-points are initially arranged in the domain in a similar way of finite elements with coarseness and fineness. The number of stress-points is set greater than that of the nodes so that zero-energy modes don't arise unlike nodal integration methods without stabilization. Shape functions and their derivatives at each stress-points are defined by moving least squares (MLS) approximation with variable support radii and are updated in every time steps. A scaling-typed integration correction was introduced for the satisfaction of integration constraints. A patch test and beam bending analyses were performed to verify and evaluate the accuracy of the proposed method. Additionally, a press forming-like analysis that is difficult to be analysed with finite element methods without adaptive meshing technique was performed and successfully converged.

An arbitrary Lagrangian-Eulerian finite element method for incompressible hyperelasticity

In this paper, we present an arbitrary Lagrangian-Eulerian (ALE) formulation for large deformation analysis of incompressible hyperelastic materials, which are often employed for modeling rubber-like materials. Conventional ALE formulations, which have been successfully applied to not only fluids but also solid materials with rate constitutive equations, are usually described in velocity-based variables. Thus, in the application of such formulations to hyperelasticity, the velocity needs to be integrated in the time variable to obtain the deformation gradient necessarily appearing in hyperelastic material modeling. However, such integration is expensive and difficult to evaluate precisely. From such aspects, we propose a new ALE finite element formulation which directly incorporates the total deformation to avoid integration of velocity-based variables and can deal with arbitrary mesh motion by considering the variation of geometric mappings. Some basic numerical results are also given to illustrate the validity of our formulation.

Large Deformation Analysis of Elastic Plane Block using the Green-Rate Constitutive Equation

In large deformation analysis, the constitutive equation utilized will play an important role in finite-element solutions. Most of the finite-element codes employ hypoelasticity based on the Jaumann stress rate for largely deformed elastoplastic behavior of metal alloys. The present paper shows the finite-element formulation for large deformation analysis in the elastic range using the different alternate objective stress rate of Green, which has not been studied in detail in previous works from the viewpoint of numerical analysis. Three kinds of shear deformations of a two-dimensional block in plane strain conditions are taken as examples to demonstrate the effects of the constitutive equations employed, and also discussed are the effects of different kinds of the isoparametric shape functions and the Gaussian integration techniques projected on the finite-element solutions.

A mixed variational formulation for large deformation analysis of plates

A mixed variational formulation for large deformation analysis of plates is introduced. In this formulation the equilibrium and compatibility equations are satisfied identically by means of stress functions and displacement components, respectively, and the constitutive equations are satisfied in a least square sense. An example is solved and the results are compared with those available in the literature. Further, the functional is particularized for buckling analysis of plates and a simple example is solved to illustrate the theory.

Application of a Mixed Variational Principle to Buckling Analysis of Circular Cylinders

AbstractA mixed variational formulation for large deformation analysis of circular cylinders is introduced. In this formulation the equilibrium and compatibility equations are satisfied identically by means of stress functions and by the use of displacement components, respectively, and the constitutive equations are satisfied in a least squares sense. The variational functional is linearized in the neighborhood of an initial equilibrium for the stability analysis of cylinders. Two classical examples are solved to illustrate the theory.

A consistent Eulerian formulation for large deformation analysis with reference to metal-extrusion process

A critical discussion of the different formulation methods for the finite element analysis of non-linear problems is given. The discussion is concerned mainly with the differences between updated Lagrangian and Eulerian formulations, and with the specific nature and basic characteristics of each. A consistent Eulerian formulation is developed by means of the virtual work principle in an explicit form. Differences between the presented formulation and similar formulations as well as the attempted Eulerian formulations are discussed according to the classification of formulation methods. To demonstrate the applicability and the effectiveness of the Eulerian finite element analysis using a fixed mesh in space, a metal-extrusion problem has been solved. In this approach, the mesh is maintained fixed in space and the increment of stress tensors for a forward incremental step are added to a set of interpolated stress tensors. Then, these stresses are interpolated back to obtain the state of stress of the body-points momentarily occupying the fixed integration points of the mesh.

A large deformation formulation for shell analysis by the finite element method

A total Lagrangian formulation for large deformation analysis of shells by the finite element method is presented. The development of the model is based upon the three dimensional field equations. To permit solution of shell problems without numerical difficulties, a special discretization in the thickness direction is employed. The displacement field of the shell element is represented by the displacement on the shell midsurface together with the relative displacement on the shell top surface, without resorting to the more complicating finite rotation parameters. Consistent linearization of the discretized balance equations is used to establish a Newton-Raphson solution scheme. The versatility and accuracy of the present shell element are demonstrated by solving several numerical examples.

Some variational formulations for large deformation analysis of plates

A mixed variational formulation for large deformation analysis of plates is introduced. In this formulation the equilibrium and compatibility equations are satisfied identically by means of stress functions and displacement components, respectively, and the constitutive equations are satisfied in a least square sense. An example is solved and the results are compared with those available in the literature.