A stress recovery procedure for solving geometrically non-linear problems in the mechanics of a deformable solid by the finite element method

A stress recovery procedure is presented for non-linear and linearized problems, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the Lagrange variational equation written in the initial configuration using an asymmetric Piola–Kirchhoff stress tensor. Vectors of the forces reduced to the mesh points are constructed using the displacements at the mesh points found by solving this equation and for the known stiffness matrices of the elements. On the other hand, these forces at the mesh points are defined in terms of unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The values of the Piola–Kirchhoff stress tensor of the first kind at the mesh points are determined using the values found for the distributed forces on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations for the initial configuration. The linearized representation of this tensor enables all the derivatives of the increment in the strain vector with respect to the coordinates to be found without invoking the operation of differentiation. The particular features of the use of the stress recovery procedure are demonstrated for a plane problem in the non-linear theory of elasticity.     

Automatic energy–momentum conserving time integrators for hyperelastic waves

An energy–momentum conserving time integrator coupled with an automatic finite element algorithm is developed to study longitudinal wave propagation in hyperelastic layers. The Murnaghan strain energy function is used to model material nonlinearity and full geometric nonlinearity is considered. An automatic assembly algorithm using algorithmic differentiation is developed within a discrete Hamiltonian framework to directly formulate the finite element matrices without recourse to an explicit derivation of their algebraic form or the governing equations. The algorithm is illustrated with applications to longitudinal wave propagation in a thin hyperelastic layer modeled with a two-mode kinematic model. Solution obtained using a standard nonlinear finite element model with Newmark time stepping is provided for comparison.     

Modeling of nonlinear viscoelastic contact problems with large deformations

Contact problems are one of the most important engineering problems. These problems become much more tedious when one of the contacting bodies behaves nonlinear viscoelasticity and large deformations. This paper presents an incremental-iterative finite element model for the analysis of two dimensional quasistatic frictionless contact problems. Nonlinear viscoelastic behavior and large deformations are considered. The Schapery’s single-integral creep model with stress-dependent properties is used for nonlinear viscoelasticity. The constitutive equations are transformed into an incremental form resulting in a recursive relationship. Thereby, the need to store the entire strain histories is eliminated, except that from the previous time increment. The updated Lagrangian formulation is used to model the material and geometrical nonlinearities. Also, the Lagrange multiplier method is adopted to enforce the contact constraints. The converged solution is obtained using the Newton–Raphson iterative technique. The developed model has been verified with the previously published works and found a good agreement with them. To demonstrate the efficient capability of the developed computational model, three contact problems with different nature are analyzed.     

Multiscale simulations of concrete mechanical tests

In civil engineering, computational modeling is widely used in the design process at the structural level. In contrast to that, an automated support for the selection or design of construction materials is currently not available. Specification of material properties and model parameters has a strong influence on the results. Therefore, an uncoupled two-step approach is employed to provide relatively quick and reliable simulations of concrete (mortar) tests. First, the Mori–Tanaka method is utilized to include the majority of small aggregates and air voids. The strain incremental form of MT approach serves for the prediction of material properties subsequently used in the finite element simulations of mechanical tests.     

Finite element simulation of deformation behaviour of cellular rubber components

This paper presents a new prospect of investigating the mechanical behaviour of cellular rubber using porous hyperelastic material model. There are number of hyperelastic material models to describe the behaviour of homogeneous elastomer, but very few to characterise the complex properties of cellular rubber. The analysis of dependence of material behaviour on pore density using the new material model is supported with experiments to characterise the actual material behaviour. The new material model which is based on Danielsson et al [1] decouples the influence of porosity from the mechanical properties of the solid material by introducing volume fraction of the pores as an explicit scalar variable. The finite element simulations are then followed by experiments on complex model to validate the material model. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multilevel Preconditioners for Mixed Methods for Second Order Elliptic Problems

A new approach for constructing algebraic multilevel preconditioners for mixed finite element methods for second order elliptic problems with tensor coefficients on general geometry is proposed. The linear system arising from the mixed methods is first algebraically condensed to a symmetric, positive definite system for Lagrange multipliers, which corresponds to a linear system generated by standard nonconforming finite element methods. Algebraic multilevel preconditioners for this system are then constructed based on a triangulation of the domain into tetrahedral substructures. Explicit estimates of condition numbers and simple computational schemes are established for the constructed preconditioners. Finally, numerical results for the mixed finite element methods are presented to illustrate the present theory.     

Experimental Investigations on and Modelling of the Transient Phenomena of the Payne-Effect.

The wide majority of industrially–used rubber is filled with a considerable amount of active fillers like carbon black or silica. Due to this, the material is strengthend and mechanical key features like stiffness and strength are significantly increased. In contrast to unfilled rubber, filled elastomers show a pronounced amplitude dependence, which is widely known as Fletcher–Gent or Payne effect. Besides that, some recently published works show a significant history dependence of this effect with distinctive relaxation phenomena. In the present work, some experiments on typical tyre rubber compounds with focus on these amplitude dependent phenomena are presented. On this basis, an appropriate thermodynamic consistent phenomenological material model of finite viscoelasticity is introduced. In order to incorporate the history dependent phenomena of the amplitude dependence, this model is generalized with intrinsic time scales on the basis of inner structure variables, which are a measure of the materials microstructure. The performance of the model is critically demonstrated by a few simulation results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical evaluation of the temperature field of steady-state rolling tires

Rubber is the main component of pneumatic tires. The tire heating is caused by the hysteresis effects due to the deformation of the rubber during operation. Tire temperatures can depend on many factors, including tire geometry, inflation pressure, vehicle load and speed, road type and temperature and environmental conditions. The focus of this study is to develop a finite element approach to computationally evaluate the temperature field of a steady-state rolling tire. For simplicity, the tire is assumed to be composed of rubber and body-ply. The nonlinear mechanical behavior of the rubber is characterized by a Mooney–Rivlin model while the body-ply is assumed to be linear elastic material. The coupled effects of the inflation pressure and vehicle loading are investigated. The influences of body-ply stiffness are studied as well. The simulation results show that loading is the main factor to determine the temperature field. The stiffer body-ply causes less deformation of rubber and consequently decreases the temperature.     

On the Excitation of Rolling Tires from the Road Surface Texture

Rolling tires are excited from the contact with the rough road surface to vibrations, which cause rolling noise. A two scale approach is suggested, where at the macro–scale the vibration of the rolling tire structure is modeled by quite detailed finite element methods. The road surface is described using measured textures. A fine resolution finite element discretization of the tread rubber is performed in order to resolve the asperity contact. The material properties are described by a non–linear viscoelastic rubber model. The tread patch is enforced to approach the rough surface in a transient dynamics manner. From these investigations an enveloping surface profile is reconstructed to be used for the excitation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A macroscopic consitutive model on induced anisotropy for polymers with weighting functions

The alignment of polymer chains is a well known microstructural evolution effect due to straining of polymers. This has a drastic influence on the macroscopic properties of the initially isotropic material, such as a pronounced strength in the loading direction of stretched films. Experiments on strain induced anisotropy at room temperature are analyzed by optical measurements. For modeling the effect of strain induced anisotropy a macroscopic constitutive model is presented. As a key idea, weighting functions are introduced to represent a strain-softening/hardening-effect to account for induced anisotropy. These functions represent the ratio between the total strain rate and a structural tensor. In this way, material parameters are used as a sum of weighted direction related quantities. In the finite element examples we simulate the cold-forming of amorphous thermoplastic films below the glass transition temperature subjected to different re-loading directions. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimation of temperature in rubber-like materials using non-linear finite element analysis based on strain history

A finite element procedure for hyper-elastic materials such as rubber has been developed to estimate the temperature rise during cyclic loading. The irreversible mechanical work developed in rubber has been used to determine the heat generation rate for carrying out thermal analysis. The evaluation of the heat energy is dependent on the strains. The finite element analysis assumes Green–Lagrangian strain displacement relations, Mooney–Rivlin strain energy density function for constitutive relationship, incremental equilibrium equations, and Total Lagrangian approach and the stress and strain of the rubber-like materials are evaluated using a degenerated shell element with assumed strain field technique, considering both material and geometric non-linearities. A transient heat conduction analysis has been carried out to estimate the temperature rise for different time steps in rubber-like materials using Galerkin's formulations. A numerical example is presented and the computed temperature values for various load steps agree closely with the experimental results reported in the literature.     

Two formulations of elastoplastic problems and the theoretical determination of the location of neck formation in samples under tension

Two formulations of elastoplastic problems in the mechanics of deformable solids with finite displacements and deformations are investigated. The first of these is formulated starting from the classical geometrically non-linear equations of the theory of elasticity and plasticity, in which the components of the Cauchy–Green strain tensor, associated with the components of the conditional stress tensor by physically non-linear relations according to flow theory in the simplest version of their representation, are taken as a measure of the deformations. The second formulation is based on the introduction of the true tensile and shear strains which, according to Novoshilov, are associated with the components of the true stresses by physical relations of the above-mentioned form. It is shown that, in the second version of the formulation of the problem, the use of the corresponding equations, complied taking account of the elastoplastic properties of the material with correct modelling of the ends of cylindrical samples and the method of loading (stretching) them, enables the location of the formation of a neck to be determined theoretically and enables the initial stage of its formation to be described without making any assumptions regarding the existence of initial irregularities in the geometry of the samples.     

Material model based on NURBS response surfaces

The development of new and complex models is typically required to gain an understanding of advanced materials and structures. The accuracy of a structural response obtained from finite element analysis is highly dependent on the extent to which a material model represents actual material behavior. This study proposes the application of rational surfaces to provide a smooth response surface for membrane material behavior. Constitutive material tensor is calculated by using the derivatives of rational surfaces. The surfaces possess two axes of strain and an axis of stress. The methodology is validated based on classical hyperelastic models. A set of results from an aluminum testing program illustrates the response surface construction procedure from the test results. The results indicate that it is easy to implement an extension of finite element codes to include the response surface approach based on NURBS.     

An atomic-continuum multiscale modeling approach for interfacial thermal behavior between materials

The aim of this paper is to provide a systematic method to perform interfacial thermal behavior between materials. A multiscale modeling method is proposed to investigate the interfacial thermal properties about copper nano interface structure. The interface stress element (ISE) method is set as a coupling button to a span-scale model combined with molecular dynamics (MD) and finite element (FE) methods. The handshake regions can simulate the structure transfer properties between the transition with MD and ISE, ISE and FE. The multiscale model is used to calculate the interfacial thermal characters under different temperatures. Some examples about numerical experiments with copper materials demonstrate the performance of MD–ISE–FE multiscale model is more successful compared with the approach applying MD–FE model. The results indicate that the accuracy of the MD–ISE–FE model is higher than that of MD–FE mode. This investigation implies a potential possibility of multiscale analysis from atomic to continuum scales.     

Nonlinear finite element analysis and modeling of a precast hybrid beam–column connection subjected to cyclic loads

In this work, a detailed three-dimensional (3D) nonlinear finite element model is developed to study the response and predict the behavior of precast hybrid beam–column connection subjected to cyclic loads that was tested at the National Institute of Standards and Technology (NIST) laboratory. The precast joint is modeled using 3D solid elements and surface-to-surface contact elements between the beam/column faces and interface grout in the vicinity of the connection. The model takes into account the pre-tension effect in the post-tensioning strand and the nonlinear material behavior of concrete. The model response is compared with experimental test results and yielded good agreement at all stages of loading. Fracture of the mild-steel bars resulted in the failure of the connection. In order to predict this failure mode, stress and strain fields in the mild-steel bars at the beam–column interface were generated from the analyzed model. Such fields of stresses and strains are hard to measure in experimental testing. In addition, the magnitude of the force developed in the post-tensioning steel tendon was also monitored and it was observed that it did not yield during the entire loading history. Successful finite element modeling will provide a practical and economical tool to investigate the behavior of such connections.     

Anisotropic multiplicative elastoplasticity for the prediction of earings in sheet forming

In this work, the simulation of earings in cup drawing by means of a recently developed anisotropic combined hardening material model is discussed. The model represents a multiplicative formulation of anisotropic elastoplasticity in the finite strain regime with nonlinear kinematic and isotropic hardening. Plastic anisotropy is described by the use of second-order structure tensors as additional arguments in the representation of the yield function and the plastic flow rule. The evolution equations are integrated by a form of the exponential map that preserves the plastic volume and the symmetry of the internal variables. Finite element simulations of cylindrical cup drawing processes are performed by means of ABAQUS/Standard where the discussed material model has been implemented into a user-defined reduced integration solid-shell element. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High-order tetrahedral finite elements applied to large deformation analysis of functionally graded rubber-like materials

In this paper, high-order tetrahedral finite elements are employed to analyze structures and solids composed of functionally graded rubber-like materials under finite displacements, finite strains, statically applied forces and isothermal conditions. In order to do so, the following concepts are used: geometrically nonlinear analysis, Green–Lagrange strain tensor, second Piola–Kirchhoff stress tensor, hyperelastic constitutive relations, isoparametric solid tetrahedral finite elements of any order of approximation, and functionally graded materials. The equilibrium of the body is achieved via the Principle of the Stationary Total Potential Energy. The elements are fully integrated via Gaussian quadratures, and the resultant processing time is reduced by means of parallel techniques. To solve the nonlinear system of equations, the Newton–Raphson iterative procedure is employed.The proposed formulation is validated by benchmark problems such as: the Cook’s membrane and the thick cylinder. Other interesting simulation, the Cook’s block is proposed in order to evaluate high strain gradient situations. The results show that, in the context of the present study, locking-free behavior is obtained with simple mesh refinement.     

A new mixed finite element method for the Stokes problem

We propose a new mixed formulation of the Stokes problem where the extra stress tensor is considered. Based on such a formulation, a mixed finite element is constructed and analyzed. This new finite element has properties analogous to the finite volume methods, namely, the local conservation of the momentum and the mass. Optimal error estimates are derived. For the numerical implementation of this finite element, a hybrid form is presented. This work is a first step towards the treatment of viscoelastic fluid flows by mixed finite element methods.