Nonlinear Viscoelastic Material Model Research Articles

Performance Evaluation of Nonlinear Viscoelastic Materials using Finite Element Method

This research paper applies the finite element method as a methodology to evaluate the structural performance of nonlinear viscoelastic solids. A finite element algorithm was built and developed to simulate the mathematical nonlinear viscoelastic material behavior based on incremental constitutive equations. The derived Equation of the incremental constitutive included the complete strain and stress histories. The Schapery’s nonlinear viscoelastic material model was integrated within the displacement-based finite element environment to perform the analysis. A modified Newton-Raphson technique was used to solve the nonlinear part in the resultant equations. In this work, the deviatoric and volumetric strain–stress relations were decoupled, and the hereditary strains were updated at the end of each time increment. It is worth mentioning that the developed algorithm can be effectively employed for all the permissible values of Poisson’s ratio by using a selective integration procedure. The algorithm was tested for a number of applications, and the results were compared with some previously published experimental results. A small percentage error of (1%) was observed comparing the published experimental results. The developed algorithm can be considered a promising numerical tool that overcomes convergence issues, enhancing equilibrium with high-accuracy results.

Transient analysis of trusses considering nonlinear elastic and viscoelastic material models

Transient analysis of trusses considering nonlinear elastic and viscoelastic material models

A non-linear viscoelastic material model with progressive damage based on microstructural evolution and phase transition in polycrystalline ice for design against ice impact

This study presents a nonlinear viscoelastic material model incorporating a progressive damage framework with an iterative algorithm for glacial/freshwater polycrystalline ice subject to compressive impact load induced by ice-structure interaction. The damage model accounts for microcracking, dynamic recrystallisation, pressure melting, and high-shear elastic failure with a pressure- and rate-dependent convex failure locus. The constitutive laws are written in Fortran and implemented as a vectorised user material (VUMAT) in Abaqus with three different numerical methods, Lagrangian FEM, coupled FEM-SPH, and ALE-FEM. The constitutive model together with the implemented numerical methods are validated against two different types of laboratory-scale physical tests, indentation of cone-shaped ice and triaxial creep. The proposed model implemented with the coupled FEM-SPH method enables simulation of the cyclic transition from solid-like intact ice to the fluid-like pulverised/granular substance, progressively extruded with viscous rheology.

Influence of Waveforms on Fatigue Crack Growth Characteristics of Tire Tread Rubber using Finite Element Analysis

ABSTRACT Rubber products are typically subjected to cyclic fatigue loading in service. During prolonged exposure to cyclic loading, damages initiate at intrinsic defect sites at microscopic levels and subsequently propagate, leading to catastrophic failure. Therefore, the material that offers better resistance to fatigue crack growth (FCG) is suitable for a durable product. FCG characteristics of rubber compounds depend on many factors, such as constituent material (rubber, filler, etc.), environment, and operational conditions (loading amplitude, loading pattern, etc.). To simulate the realistic service condition of a product, the choice of loading pattern is a key factor and has emerged as a very important research topic in recent times. The present work focuses on the effect of loading pattern on FCG characteristics of tire tread rubber compounds. In the present study, FCG characteristics of a 100% natural rubber (NR) compound were measured on a tear and fatigue analyzer (TFA, Coesfeld, Germany) using double edge notched pure shear specimen. Fatigue loading was applied using sinusoidal and pulse waveforms over a wide range of tearing energy levels. Pulse mode recorded a very high crack growth rate (∼2 times) compared to sine mode at equivalent peak energy levels. In order to understand the mechanics of the higher crack growth rate in pulse mode, finite element analysis (FEA) of a pure shear specimen was performed wherein FCG experimental conditions were used as boundary conditions. FE analyses were carried out using both linear and nonlinear viscoelastic material models. Nonlinear viscoelastic FEA results revealed that viscous energy dissipation at the crack tip is much lower in the case of pulse mode, which is in support of higher the FCG rate in pulse mode as observed in the experiments.

Application of computationally efficient TR-BDF2 scheme for the finite element implementation of explicit non-linear viscoelastic models for filled elastomers

This paper presents the implementation of a nonlinear viscoelastic multi-configurational rate type material model within the general framework of the commercial finite element package Abaqus (Implicit). A user-defined subroutine, UMAT for continuum elements, is employed to implement a model similar to the one developed and extensively validated experimentally by Devendiran et al. One of the major challenges of using an implicit finite element (FE) solver is the description of material Jacobian, which may be difficult to obtain in the form of an analytical equation for multi-configurational models. Hence, numerical approximations are used to implement the consistent material Jacobian used in the global Newton iterations. We have used a Jacobian that is formulated by perturbing the deformation gradient, by extending an idea developed by Miehe and widely implemented in hyperelasticity. In order to determine the multiple intermediate configurations that keep evolving with time, this study also examines the computational efficiency of a fully second order integrator in comparison to traditional first order BDF integrator. A highly efficient TR-BDF2 integrator, originally developed for simulating transients of silicon VLSI devices, is utilised to determine the “current relaxed configuration”, which to the author’s knowledge is the first implementation in literature for rate type viscoelastic constitutive equations. Coupling the numerically approximated Jacobian and TR-BDF2 integrator, the UMAT is generic and can easily be modified for various combinations of stored energy potential functions and rate of dissipation equations. The UMAT is validated against material point solution of the constitutive model in MATLAB; and is found to be exact within a tight tolerance. The implemented UMAT is used to determine the rolling resistance of a Grosch wheel which demonstrates the practical application of such a material model focussed towards the workflows for tyre analysis.

A comparative finite element analysis of maxillary expansion with and without midpalatal suture viscoelasticity using a representative skeletal geometry

The goal of this investigation was to adapt and incorporate a nonlinear viscoelastic material model representative of the midpalatal suture’s viscoelastic nature into finite element analysis simulations of maxillary expansion treatment. Step-wise displacements were applied to a partial skull geometry to simulate treatment using an expansion screw appliance. Four simulation cases were considered for the midpalatal and intermaxillary sutures: 1. Neglecting suture tissue; 2. Linear elastic properties; 3. Viscoelastic properties; 4. A fused intermaxillary and viscoelastic midpalatal suture. Results from simulations indicated that removal of suture tissue and inclusion of viscoelastic properties resulted in the same maxillary displacement following 29 activations of 0.125 mm applied directly to the maxilla; however, assuming a fused intermaxillary suture significantly changed maxillary displacement patterns. Initial stress results within the suture complex were significantly influenced by the inclusion of suture viscoelasticity as compared to linear elastic properties. The presented study demonstrates successful incorporation of suture viscoelasticity into finite element analysis simulations of maxillary expansion treatment, and elucidates the appropriateness of various suture material property assumptions depending desired research outcomes.

On the Development of Creep Laws for Rubber in the Parallel Rheological Framework

ABSTRACT It is widely known that filler-reinforced rubber material in tires shows a very complicated material behavior when subjected to cyclic loadings. One of the most interesting effects for rolling tires is the nonlinear rate-dependent behavior, which is implicitly linked to the amplitude dependency of dynamic stiffness (Payne effect) at a given frequency and temperature. This effect, however, cannot be described by a conventional linear viscoelastic constitutive law, e.g., the Prony series model. Several nonlinear viscoelastic material models have been proposed in the last decades. Among others, Lapczyk et al. (Lapczyk, I., Hurtado, J. A., and Govindarajan, S. M., “A Parallel Rheological Framework for Modeling Elastomers and Polymers,” 182nd Technical Meeting of the Rubber Division of the American Chemical Society, Cincinnati, Ohio, October 2012) recently proposed a quite general framework for the class of nonlinear viscoelasticity, called parallel rheological framework (PRF), which is followed by Abaqus. The model has an open option for different types of viscoelastic creep laws. In spite of the very attractive nonlinear rate-dependency, the identification of material parameters becomes a very challenging task, especially when a wide frequency and amplitude range is of interest. This contribution points out that the creep law is numerically sound if it can be degenerated to the linear viscoelastic model at a very small strain amplitude, which also significantly simplifies model calibration. More precisely, the ratio between viscoelastic stress and strain rate has to converge to a certain value, i.e., the viscosity in a linear viscoelastic case. The creep laws implemented in Abaqus are discussed in detail here, with a focus on their fitting capability. The conclusion of the investigation consequently gives us a guideline to develop a new creep law in PRF. Here, one creep law from Abaqus that meets the requirements of our guideline has been selected. A fairly good fit of the model is shown by the comparison of the simulated complex modulus in a wide frequency and amplitude range with experimental results.

Characterization and modelling the mechanical behaviour of poly(l-lactic acid) for the manufacture of bioresorbable vascular scaffolds by stretch blow moulding

Bioresorbable Vascular Scaffolds (BVS) manufactured from poly (l-lactic acid) (PLLA) offer an alternative to metal scaffolds for the treatment of coronary heart disease. One of the key steps in the manufacture of these scaffolds is the stretch blow moulding process where the PLLA is biaxially stretched above glass transition temperature (Tg), inducing biaxial orientation and thus increasing ductility, strength and stiffness. To optimise the manufacture and performance of these scaffolds it is important to understand the influence of temperature and strain rate on the constitutive behaviour of PLLA in the blow moulding process. Experiments have been performed on samples of PLLA on a custom built biaxial stretch testing machine to replicate conditions typically experienced during blow moulding i.e. in a temperature range from 70 °C to 100 °C and at strain rates of 1 s−1, 4 s−1 and 16 s−1 respectively. The data is subsequently used to calibrate a nonlinear viscoelastic material model to represent the deformation behaviour of PLLA in the blow moulding process. The results highlight the significance of temperature and strain rate on the yielding and strain hardening behaviour of PLLA and the ability of the selected model to capture it.

The effect of material density on load rate sensitivity in nonlinear viscoelastic material models

This work presents the results of a numerical sensitivity analysis of material density on crack propagation in nonlinear viscoelastic softening material models. Two basic material models are analyzed: Maxwell and Kelvin material models, under the loading with changing rate (slow or fast impact loading). The material is discretized as a lattice model, and the analysis is performed on simple examples consisting of several lattice bars. The mathematical description is based on the theory of dynamical systems, i.e., it is a system of nonlinear differential equations for Kelvin, and a system of nonlinear differential-algebraic equations for Maxwell material model. Sensitivity of displacements on mass and load rate accompanies the analysis.

Viscoelasticity of spinal cord and meningeal tissues

Compared to the outer dura mater, the mechanical behavior of spinal pia and arachnoid meningeal layers has received very little attention in the literature. This is despite experimental evidence of their importance with respect to the overall spinal cord stiffness and recovery following compression. Accordingly, inclusion of the mechanical contribution of the pia and arachnoid maters would improve the predictive accuracy of finite element models of the spine, especially in the distribution of stresses and strain through the cord’s cross-section. However, to-date, only linearly elastic moduli for what has been previously identified as spinal pia mater is available in the literature. This study is the first to quantitatively compare the viscoelastic behavior of isolated spinal pia-arachnoid-complex, neural tissue of the spinal cord parenchyma, and intact construct of the two. The results show that while it only makes up 5.5% of the overall cross-sectional area, the thin membranes of the innermost meninges significantly affect both the elastic and viscous response of the intact construct. Without the contribution of the pia and arachnoid maters, the spinal cord has very little inherent stiffness and experiences significant relaxation when strained. The ability of the fitted non-linear viscoelastic material models of each condition to predict independent data within experimental variability supports their implementation into future finite element computational studies of the spine. Statement of SignificanceThe neural tissue of the spinal cord is surrounded by three fibrous layers called meninges which are important in the behavior of the overall spinal-cord-meningeal construct. While the mechanical properties of the outermost layer have been reported, the pia mater and arachnoid mater have received considerably less attention. This study is the first to directly compare the behavior of the isolated neural tissue of the cord, the isolated pia-arachnoid complex, and the construct of these individual components. The results show that, despite being very thin, the inner meninges significantly affect the elastic and time-dependent response of the spinal cord, which may have important implications for studies of spinal cord injury.

Advanced Rebar Formulation within a Thermo‐Mechanically Coupled Tire Simulation

AbstractAs shown in various experiments, coupling between the thermal and the mechanical field of a rolling tire in contact with a rigid surface can be observed. An increase of its temperature over time when rolling with a constant velocity can be seen. This phenomenon is caused by the viscoelastic material properties, where the dissipated energy will lead to a heating of the rubber. Also the constitutive behaviour will change, when the temperature increases. A nonlinear viscoelastic material model is chosen to model an amplitude dependency of the time dependent behaviour of the rubber, proposed by Bergström and Boyce [1].Another important part of a tire are the reinforcements, which are embedded inside the rubber. These consist mostly of steel or polyester. These materials are not known for a dissipative behaviour and will not contribute to the heating of the tire. However, they also change their properties with increasing temperature and have a significant influence on the overall tire properties. Even without a contribution to the heating, the reinforcements will change its conductivity and have to be taken into account.For an accurate and efficient way to model the thermo‐mechanical coupling, an in‐house framework [2] will be presented, by use of the Arbitrary Langrangian Eulerian formulation and a commercial code. It is assumed, that the temperature in circumferential direction will stay constant and instant coupling effects are negligibly small. Therefore, the problem is split into a thermal and a mechanical part, and the coupling quantities (dissipated energy and temperature) are updated in a sequential manner.A common practise to model the reinforcements in tires is to use smeared layers, which leads to some difficulties. First of all, the modelling is only valid for a really small spacing between the bars. Another problem is, that the smeared layers have to fit with the nodes of the rubber matrix. Because of that, the mesh has to be refined, which leads to an increasing computation time and can cause numerical difficulties, due to too small elements.In this contribution, a different way of modelling the rebars within a thermo‐mechanical coupled algorithm is presented without the assumption of a smeared layer and treat them as single rebars. In numerical examples, the advantages of the proposed method are presented and the difference to the classical approach is shown. Also, the influence of the reinforcements on the stiffness, heat generation and rolling resistance of the tire, will be discussed. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Proper Orthogonal Decomposition–Radial Basis Function Surrogate Model-Based Inverse Analysis for Identifying Nonlinear Burgers Model Parameters From Nanoindentation Data

This study explores the application of a proper orthogonal decomposition (POD) and radial basis function (RBF)-based surrogate model to identify the parameters of a nonlinear viscoelastic material model using nanoindentation data. The inverse problem is solved by reducing the difference between finite element simulation-trained surrogate model approximation and experimental data through genetic algorithm (GA)-based optimization. The surrogate model, created using POD–RBF, is trained using finite element (FE) data obtained by varying model parameters within a parametric space. Sensitivity of the model parameters toward the load–displacement output is utilized to reduce the number of training points required for surrogate model training. The effect of friction on simulated load–displacement data is also analyzed. For the obtained model parameter set, the simulated output matches well with experimental data for various experimental conditions.

A thermo-viscoelastic approach for the characterization and modeling of the bending behavior of thermoplastic composites – Part II

A proper description of the bending behavior is crucial to obtain accurate forming simulations, especially for continuous fiber-reinforced thermoplastic composites. These materials exhibit a highly temperature and bending-curvature dependent bending stiffness. These dependencies make the property challenging to characterize with conventional characterization methods, and therefore require novel techniques. The first part of the study has shown how Dynamic Mechanical Analysis and a rheometer-based method can be used to examine viscoelastic bending behavior. This subsequent part focuses on combining their advantages in a universal characterization method, which provides an accurate description of the bending behavior over a broad temperature range, including the phase transition of recrystallization. Dynamic isothermal experiments as well as dynamic experiments over defined temperature ranges were conducted. The aforementioned experiments were reconstructed in simulations, employing the non-linear viscoelastic material model from the first part of the study, to evaluate the characterization method and to further validate the model.

A nonlinear viscoelastic iceberg material model and its numerical validation

This article proposes a nonlinear viscoelastic iceberg material model. A nonlinear Burgers’ model in which Kelvin and Maxwell units are strain rate- and stress-dependent is adopted for the iceberg material. The strain rate effect is considered in this model based on the experimental results. The stress of the iceberg model grows linearly (in log form) with increasing strain rate before reaching the transition strain rate, after which the stress remains rather constant. A damage function that reflects the microstructure changes and severe fractures in ice is adopted as the failure criterion. The iceberg model is implemented using implicit integration Crank–Nicolson method and is incorporated in the commercial software LS-DYNA by a user-defined material. Laboratory-scale experiments, creep experiments and constant strain rate experiments, and reality-scale experiment, iceberg–rigid steel plate collisions, are simulated to validate the proposed iceberg material model. Simulated time–strain curves are compared with the results of creep experiments. In the constant strain rate experiments, the strain–stress curves for brittle and ductile failure and ultimate triaxial strength of the ice model are analysed. Area–pressure curves and contact force–displacement relations are investigated for different impact speeds in iceberg–steel plate collisions. The contact force is also studied in view of the kinetic energy of icebergs. The numerical results show that the proposed iceberg material model yields reasonably good results.

Virtual characterization and macroscopic material modeling of a carbon fiber-reinforced PA6 UD composite

Despite the increasing use of continuously fiber-reinforced composites with thermoplastic matrices in load-bearing structural applications, published research on their deformation behavior in general and their long-term behavior in particular is very limited. As a result, adequate material models to describe their time- and direction-dependent behavior on a macroscopic scale are lagging. The present study uses micromechanical representative volume element models and numerical homogenization techniques to analyze the material behavior of a carbon fiber-reinforced PA6 material as a function of load direction and time. The matrix is modeled as viscoelastic–viscoplastic. Based on the results of the representative volume element model, a homogenized macroscopic modeling approach is proposed. It is shown that the time dependency of the unidirectional composite is highly anisotropic. Plastic deformation occurs in the matrix, but may be neglected in the homogenized composite at laminate-relevant strain levels. The proposed homogenized orthotropic, nonlinear viscoelastic material model adequately captures the behavior of the heterogeneous unidirectional composite.

Creep characterization of gels and nonlinear viscoelastic material model

In this paper, we examine gel creep behavior and develop a material model for useful and simple numerical simulation of this behavior. This study has three stages and aims: (1) gel creep behavior is examined; (2) the material model is determined and the material constants are identified; and (3) the versatility of the material model and the constants are evaluated. The creep behavior is found to be independent of the initial stress level in the present experiment. Thus, the viscoelastic model proposed by Simo is selected, and its material constants are identified using the results of creep tests. Moreover, from the results of numerical calculations and experiments, it is found that the chosen material model has good reproducibility, predictive performance and high versatility.

Analysis of nonlinear shear stress–strain response of unidirectional GF/EP composite

Reasons for nonlinear stress–strain curves in shear of unidirectional glass fibre composite are analysed. Laminate with stacking sequence [45/−45]s is used in tensile quasi-static as well as in tensile creep and strain recovery tests to study the development of viscoelastic and viscoplastic shear strains in local coordinates of the ply. It is shown that Zapa’s integral representation of viscoplasticity is applicable for this material and methodology for parameter determination is given. Schapery’s nonlinear viscoelastic material model was used for shear response characterisation and the nonlinearity parameters’ dependence on the shear stress was determined and described by fitting functions. Microdamage development is quantified by measuring axial modulus and Poisson’s ratio of the laminate. The obtained nonlinear viscoelastic, viscoplastic model with included effect of microdamage was successfully used to predict the nonlinear shear stress–strain curve in strain controlled tensile loading.

A 3D Meso-Scale Analysis of Angle-Ply Laminates

A 3D two-cell representation of laminates with periodic fiber arrays is established and the corresponding periodic boundary conditions are presented. The matrix is described by a nonlinear viscoelastic material model and the fiber by an elastic one. Matrix cracking is modeled by a “smeared crack” method. The modeling of material nonlinearity, crack initiation, and growth, with appropriate periodic boundary conditions, are incorporated into a finite element model (FEM). Numerical results for five different angle-ply laminates under tensile loading are presented. The results are found to be in fair agreement with the available test data.

Experiment and numerical simulation on delamination during the laminated steel sheet forming processes

Laminated steel sheets, which have advantages in reducing vibration and noise, are widely used in home appliances, automotive components and building structures. With polymer layer laminated in steel sheets, the unique failure mode—delamination may occur in the forming process if the process parameters are defined improperly. In the presented work, firstly, the delamination is investigated by U channel forming in bending mode. Then, a step-bottom square cup drawing is also conducted to study the delamination in the deep drawing of the laminated sheet metal in more complex stress state. Nonlinear visco-elastic material model is applied to describe the mechanical behavior of polymer layer. Cohesive element and continuum shell elements are utilized to discretize the polymer layer and the outer steel sheets, respectively. The results of U channel forming indicate that increasing forming speed somewhat decreases the tendency of delamination, and increasing blank holding force (BHF) significantly diminishes the occurrence of delamination. Meanwhile, the results of the step-bottom square cup drawing reveal that the wrinkling of facial sheets often induces delamination. Increasing the BHF and the frictional coefficient prevents the occurrence of wrinkling and the delamination induced by wrinkling. However, when wrinkling is suppressed, further rising in BHF and frictional coefficient will increase the risk of direct delamination.

Experimental Parameter Estimation Method for Nonlinear Viscoelastic Composite Material Models: An Application on Arterial Tissue

This study is aimed at setting a method of experimental parameter estimation for large-deforming nonlinear viscoelastic continuous fibre-reinforced composite material model. Specifically, arterial tissue was investigated during experimental research and parameter estimation studies, due to medical, scientific and socio-economic importance of soft tissue research. Using analytical formulations for specimens under combined inflation/extension/torsion on thick-walled cylindrical tubes, in vitro experiments were carried out with fresh sheep arterial segments, and parameter estimation procedures were carried out on experimental data. Model restrictions were pointed out using outcomes from parameter estimation. Needs for further studies that can be developed are discussed.