Nonlinear Viscoelastic Material Research Articles

Sensitivity analysis of truck tire tread material properties for on-road applications

This research delves into the sensitivity analysis of a truck tire rubber compound concerning its impact on tire–road interaction characteristics. Initially, the study employs finite element analysis to model a 315/80R22.5 truck tire, which is subsequently validated through static and dynamic response assessments via various simulation tests. Following validation, the established tire model is utilized to conduct a sensitivity analysis of the tire rubber compound specifically applied on the tread. This analysis encompasses several material definitions, including Mooney–Rivlin, visco-Mooney–Rivlin, linear viscoelastic, and nonlinear viscoelastic materials. By exploring the effects of these material models, the research scrutinizes their influence on tire–road interaction characteristics across diverse operating conditions. The tire–road interaction characteristics include the rolling resistance coefficient, and the cornering force at operating conditions including the longitudinal speed, vertical load, and slip angle. This comprehensive investigation offers insights into the intricate relationship between tire composition and performance, thereby enhancing our understanding of tire behavior and informing potential advancements in tire technology.

A thermo-mechanically coupled constitutive model for semi-crystalline polymers at finite strains: Mechanical and thermal characterization of polyamide 6 blends

In the field of material modeling, thermoplastic polymers are often studied because of their complex material behavior and their prevalence in industry applications due to their low cost and wide range of applications. Nowadays, where reusability becomes more and more important, materials which can undergo reversible thermomechanical deformations are appealing for, e.g., the construction of car body components. To predict such complex forming processes with multiple influencing factors, such as temperature, strain rate or underlying material morphology, model formulations are needed that account for these influences simultaneously and are validated against experimental data. Unfortunately, up to now only a few contributions are available which consider all these phenomena. In addition, the range of process parameters considered is often narrow due to the experimental effort required for testing. This usually results in limited predictive capabilities of the model. To overcome these limitations, in this work, a thermo-mechanically coupled material model is developed that accounts for the underlying morphology in terms of the degree of crystallinity (DOC). The model formulation is derived in a thermodynamically consistent manner, incorporating coupled nonlinear visco-elastic and elasto-plastic material behavior at finite strains. To characterize and further validate the model, mechanical as well as thermal experiments are conducted for polyamide 6 (PA6). Here, a blending strategy of PA6 together with an amorphous co-polymer is introduced during specimen production to achieve a wider range of stable DOCs(approximately 15%). The model formulation is successfully applied to experimental results and its predictions are in good agreement with experimental observations.

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.

Viscoelasticity of hydrating shotcrete as key to realistic tunnel shell stress assessment with the New Austrian Tunneling Method

The New Austrian Tunneling Method (NATM) essentially rests on observational information concerning displacements measured in selected positions at the inner surface of shotcrete tunnel shells. The combination of these measurements with advanced material and structural mechanics, in the course of so-called hybrid methods, have successfully delivered, for more than 20 years, practically relevant estimations of internal and external forces and corresponding degrees of utilization. The reliability of the latter, however, may crucially depend on the used material model. Based on a recently proposed analytical structural mechanics model [Acta Mech 233, 2989–3019 (2022)], and focusing on the benchmark example of measurement cross section MC1452 of the Sieberg tunnel, driven in the 1990s in Miocene clay marl, the present paper compares the estimations of forces and degrees of utilization arising from differently refined constitutive concepts, namely (i) aging elasticity, (ii) aging linear viscoelasticity, and (iii) aging nonlinear viscoelasticity. It turns out that only the consideration of aging nonlinear viscoelastic material behavior provides access to realistic values for the degree of utilization, being lower than one. Simpler material models would indicate local material failure, which was not observed in situ.

Transient analysis of trusses considering nonlinear elastic and viscoelastic material models

Transient analysis of trusses considering nonlinear elastic and viscoelastic material models

A mechanics-informed deep learning framework for data-driven nonlinear viscoelasticity

A mechanics-informed, data-driven framework for learning the constitutive law of a nonlinear viscoelastic material from stress–strain data using deep artificial neural networks (ANNs) is proposed. In this framework, each ANN architecture is designed and its input and output are selected such that fundamental laws of mechanics and thermodynamics relevant to nonlinear viscoelasticity are strongly satisfied by the resulting data-driven constitutive model. The framework itself extends a previously developed mechanics-informed methodology for the data-driven constitutive modeling of nonlinear elasticity to account for the dependence of the stress on the entire strain history. It is grounded in an additive decomposition of the strain energy density and an internal variable approach to viscoelasticity. Collectively, these are characterized by three potentials whose ANN representations are taught the second law of thermodynamics, various stability conditions, consistency (preservation of rigid body modes), and the recovery of elastic limits. The proposed mechanics-informed deep learning framework for data-driven nonlinear viscoelasticity is illustrated and its performance is assessed for one academic problem where synthetic data is generated by a nonlinear convolution; a coupon-level application associated with a multi-scale homogenization of a viscoelastic woven fabric material; and two representative finite element analyses performed using the material law generated during the previous application, to highlight different features of the framework.

Rubber Comprehensive Constitutive Equation and the Prediction of Tire Temperature and Rolling Resistance

ABSTRACT Cooler running tires with a reduced rolling resistance is a consideration for both tire makers and automotive original equipment manufacturers. To investigate the mechanics of energy loss and temperature rise, a nonlinear viscoelastic model suitable for numerical analysis of rolling tires is developed and demonstrated. The rubber compounds are represented as nonlinear viscoelastic materials with temperature and frequency, strain, and strain history–dependent response. Several applications to different tire designs are provided to demonstrate the impact of this new material constitutive law in improving the quality of the numerical predictions over other methods. One application is a rolling tire model that was used to delineate the tire temperature distribution and the resultant rolling resistance for different three-dimensional tread patterns.

Modeling stress relaxation under conditions of cyclic deformation of nonlinear viscoelastic materials using the Heaviside function

The problem of calculating the stress relaxation of nonlinear viscoelastic materials under conditions of cyclic deformation is considered, which is given in the form of cyclic alternations of loading and unloading of equal amplitude in rectangular cycles, while the duration of half-cycles of loading and unloading coincide. The load program is implemented in the form of a sequence of elementary loads, which is specified using Heaviside's unit functions. A nonlinear creep model with time-independent nonlinearity of the Rabotnov model is used to describe the deformation process. The heredity kernels are given by Rabotnov's fractional-exponential function.

Existence and uniqueness for a viscoelastic Kelvin–Voigt model with nonconvex stored energy

We consider nonlinear viscoelastic materials of Kelvin–Voigt-type with stored energies satisfying an Andrews–Ball condition, allowing for nonconvexity in a compact set. Existence of weak solutions with deformation gradients in [Formula: see text] is established for energies of any superquadratic growth. In two space dimensions, weak solutions notably turn out to be unique in this class. Conservation of energy for weak solutions in two and three dimensions, as well as global regularity for smooth initial data in two dimensions are established under additional mild restrictions on the growth of the stored energy.

Experimental characterization and phenomenological modeling of nonlinear viscoelasticity, plasticity and damage of continuous carbon fiber-reinforced thermoplastics

The presented work addresses the experimental characterization and phenomenological modeling of the complex rate dependent mechanical response of continuous fiber-reinforced thermoplastics (cFRTP). The theoretical basis is given by the concept of an equilibrium curve of the material which provides a comprehensive framework for separating the mechanical response into a rate dependent overstress and a rate independent equilibrium stress. In long-term creep and relaxation tests, it is shown that relaxation tests provide several advantages over creep tests when determining the equilibrium response. A stepwise loading–unloading relaxation (SR) test is proposed which makes it possible to extract nonlinear viscoelasticity, damage and plasticity present in the material from a single test. The application to cFRTP is exemplified by using the proposed method for the experimental characterization of a continuous carbon fiber-reinforced polycarbonate (CF-PC). Based on the obtained experimental data, a phenomenological material model is developed which describes the mechanical response of a unidirectional ply. The model consists of an overstress branch addressing the nonlinear viscoelastic material response and an equilibrium branch that utilizes a coupled damage-plasticity approach. After calibrating the model using the experimental data, it is shown that the proposed model is able to accurately reproduce the mechanical response of cFRTP.

Thermomechanical response of nonlinear viscoelastic materials

In this article, the thermomechanical properties of the classical and nonclassical viscoelastic materials with particular emphasis on the entropy are studied in the framework of continuum thermodynamics. Specifically, considering the Clausius integral as “Part I” of the second law of thermodynamics in the classical formulation of continuum thermodynamics, the entropy function for various classes of nonclassical thermo-viscoelastic materials is shown to exist. This is achieved by employing the first law of thermodynamics and a scalar potential known as the Helmholtz free energy function. In addition, another function known as the internal rate of production of entropy defined for the classical and nonclassical thermo-viscoelastic materials is adopted to obtain a balance law for the entropy. Finally, upon linearizing the theory, a coupled system of differential equations is obtained for the nonclassical thermo-viscoelastic materials, which under some special circumstances, permits propagation of damped harmonic waves in the viscoelastic medium.

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.

Investigation of the relationship between tensile viscoelasticity and unloaded ultrasound shear wave measurements in ex vivo tendon

Mechanical properties of biological tissues are of key importance for proper function and in situ methods for mechanical characterization are sought after in the context of both medical diagnosis as well as understanding of pathophysiological processes. Shear wave elastography (SWE) and accompanying physical modelling methods provide valid estimates of stiffness in quasi-linear viscoelastic, isotropic tissue but suffer from limitations in assessing non-linear viscoelastic or anisotropic material, such as tendon. Indeed, mathematical modelling predicts the longitudinal shear wave velocity to be unaffected by the tensile but rather the shear viscoelasticity. Here, we employ a heuristic experimental testing approach to the problem to assess the most important potential confounders, namely tendon mass density and diameter, and to investigate associations between tendon tensile viscoelasticity with shear wave descriptors. Small oscillatory testing of animal flexor tendons at two baseline stress levels over a large frequency range comprehensively characterized tensile viscoelastic behavior. A broad set of shear wave descriptors was retrieved on the unloaded tendon based on high frame-rate plane wave ultrasound after applying an acoustic deformation impulse. Tensile modulus and strain energy dissipation increased logarithmically and linearly, respectively, with the frequency of the applied strain. Shear wave descriptors were mostly unaffected by tendon diameter but were highly sensitive to tendon mass density. Shear wave group and phase velocity showed no association with tensile elasticity or strain rate-stiffening but did show an association with tensile strain energy dissipation.The longitudinal shear wave velocity may not characterize tensile elasticity but rather tensile viscous properties of transversely isotropic collagenous tissues.

Multi-scale constitutive model of human trabecular bone

The present study aims to formulate a new multiscale constitutive model of human trabecular bone. The trabecular bone was modelled as a nonlinear viscoelastic material. The viscoelastic effects of single trabeculae were considered by means of a hereditary integral in which stress depends on time and strain, while the elastic response was described by the hyperelastic Mooney–Rivlin model. The cuboid bone sample was extracted from the femoral head during the hip replacement surgery. The material constants in the constitutive equation were identified based on the stress relaxation test performed on the cuboid sample and the microindentation tests performed on trabeculae using the curve-fitting procedure. The microindentation tests were performed using a spherical tip instead of Vickers or Berkovich tip to minimize plastic effects during trabecular deformation. In order to validate formulated constitutive model, results from a FE simulation of stress relaxation test and uniaxial compression test were compared to the results of the corresponding experiments conducted on a macroscopic bone sample. Good agreement was observed between numerical and experimental results. The viscoelastic behaviour predicted by the proposed constitutive equation corresponds well to the response of human trabecular bone under various types of load conditions. This demonstrates the high ability of our constitutive model to simulate the behaviour of trabecular bone on a micro- and macroscopic scale. Thus, we conclude that the model, which was formulated for a single trabecula, can be successfully applied to simulate mechanical behaviour of the tissue in a macroscale.

Mechanical Behavior of Human Cancellous Bone in Alveolar Bone under Uniaxial Compression and Creep Tests.

In the process of orthodontic treatment, the remodeling of cancellous bone in alveolar bone (in this paper, cancellous bone in alveolar bone is abbreviated as CBAB) is key to promoting tooth movement. Studying the mechanical behavior of CBAB is helpful to predict the displacement of teeth and achieve the best effect of orthodontic treatment. Three CBAB samples were cut from alveolar bone around the root apex of human teeth. A uniaxial compression test was used to study the transient elastic properties of CBAB. A creep test was used to study the time-dependent viscoelastic properties of CBAB. Both tests were carried out at the loading rates of 0.02 mm/min, 0.1 mm/min and 0.5 mm/min. The results revealed that CBAB is a nonlinear viscoelastic and hyperelastic material. The stress–strain curve obtained from the uniaxial compression test could be divided into three stages: the collapse stage of the front section, the exponential stage of the middle section and the almost linear stage of the rear end. According to the strain–time curve obtained from the compression creep test, a trend of increasing strain over time was relatively obvious within the first 30 s. After 200 s, the curve gradually tended to plateau. Four hyperelastic models and three viscoelastic models were used to fit the test data. Finally, the fifth-order polynomial hyperelastic model (coefficient of determination “R2 > 0.999”) was used to describe the hyperelastic properties of CBAB, and the seven-parameter model of the generalized Kelvin modified model (“R2 > 0.98”) was used to describe the viscoelastic properties of CBAB.

Viscoelastoplastic classification of cementitious suspensions: transient and non-linear flow analysis in rotational and oscillatory shear flows

Rheological modeling of special concretes such as ultra-high performance concretes (UHPCs), self-compacting concretes (SCCs), or environmentally friendly concretes with low clinker contents but containing many fine particles increases model complexity because high amounts of colloidal and fine particles and complex chemistry result in strongly non-Newtonian rheological behavior. Straight-forward viscoplastic rheological models such as the well-known Bingham- or Herschel-Bulkley approach do not fit the flow behavior on a large scale at and lead to boundary value problems in numerical flow simulations. Increased viscoelastic properties due to chemical admixtures are not described by pure viscoplasticity analysis methods. To fill this gap, our research presents viscoplastic and viscoelastic linear and non-linear rheological characterization methods for common and strongly non-Newtonian cement pastes and combines the methods for a comprehensive viscoelastoplastic modeling of cementitious building materials. We illustrate and discuss the benefits and boundaries of each measurement technique in dependence of cementitious paste complexity. Three solid volume fractions from phi = 0.45 to 0.55 and three different flowability ranges, adjusted through varying superplasticizer amounts, were investigated in respect of steady-state and transient yield stress, viscosity, and non-linear viscoelastic material properties. Our results reveal that with increasing solid volume fraction and polymer amount, viscoelastoplastic modeling describes rheological flow on a large flow scale more appropriately than common viscoplastic methods. The results show a new approach for a full quantitative and qualitative rheological classification of cementitious building materials and thus improve the understanding of densely packed colloidal cementitious suspensions. The findings serve as prospective guideline to model complex cementitious building materials both at low and high shear rates, and to find an appropriate rheological description at the boundaries of flow.Graphical abstract

Resonance-like enhancement of forced nonlinear diffusion as a nonequilibrium phase transition

We describe the phenomenon of a resonance-like, giant enhancement of diffusion in a basic model of nonlinear diffusion featured by a nonlinear in velocity friction and the corresponding multiplicative thermal noise. The model is consistent with thermal equilibrium in the absence of driving. Different from previous studies of this phenomenon, where the crucial nonlinearity originates from a periodic external potential while friction is linear, we focus on the case of a constant force driving, whereas the crucial nonlinearity stems from the friction. The basic model of such friction considered interpolates between linear viscous Stokes friction at small velocities and dry Coulomb-like friction at large velocities corresponding to a stress plateau in some nonlinear viscoelastic materials. Recently, a nonequilibrium phase transition to super-diffusion and super-transport was discovered within this basic model. We show that adding a tiny viscous friction part to major nonlinear friction regularizes in part this behavior. Diffusion becomes asymptotically normal. However, the phase transition translates into a giant enhancement of normal diffusion and mobility of particles at the transition point over the intuitively expected large force limit, where the linearization of friction occurs. Such a giant enhancement of diffusion is closely related to the largely enhanced kinetic temperature of the particles at and beyond the critical point. We provide analytical results obtained within an effective mass approximation which nicely agree with stochastic numerics.

Materials for Implantable Surface Electrode Arrays: Current Status and Future Directions.

Surface electrode arrays are mainly fabricated from rigid or elastic materials, and precisely manipulated ductile metal films, which offer limited stretchability. However, the living tissues to which they are applied are nonlinear viscoelastic materials, which can undergo significant mechanical deformation in dynamic biological environments. Further, the same arrays and compositions are often repurposed for vastly different tissues rather than optimizing the materials and mechanical properties of the implant for the target application. By first characterizing the desired biological environment, and then designing a technology for a particular organ, surface electrode arrays may be more conformable, and offer better interfaces to tissues while causing less damage. Here, the various materials used in each component of a surface electrode array are first reviewed, and then electrically active implants in three specific biological systems, the nervous system, the muscular system, and skin, are described. Finally, the fabrication of next-generation surface arrays that overcome current limitations is discussed.

Some open challenges in polymer physics*

AbstractThree subjects in polymer and soft matter physics are outlined in the present article. The first relates to concepts of an ideal glass transition. We describe work on ultra‐stable glass, either a 20‐million‐year‐old amber or a vapor deposited amorphous fluoropolymer, which examines the temperature dependence of the dynamics in a window between a low fictive temperature and the glass transition temperature. From this “finesse” of the problem of the geological time scales in sub‐glass temperature systems strong evidence that the divergence of the relaxation times at finite temperature from WLF‐types of extrapolation is not correct. The second topic is polymer nonlinear viscoelasticity as it relates to dynamic heterogeneity. We show results from mechanical hole burning experiments that suggest that dynamic heterogeneity arises from the nature of specific relaxation mechanisms rather than heterogeneous changes in, for example, the fictive temperature. Included is a discussion of large amplitude oscillatory shear testing and how it relates to characterization of nonlinear viscoelastic materials. The last topic addressed is the rheology of circular macromolecules. Here, we take a historical perspective and describe important new results from several laboratories that seem inconsistent. New works from our own studies on circular DNA are also discussed.

Free Energies for Nonlinear Materials with Memory

An exploration of representations of free energies and associated rates of dissipation for a broad class of nonlinear viscoelastic materials is presented in this work. Also included are expressions for the stress functions and work functions derivable from such free energies. For simplicity, only the scalar case is considered. Certain standard formulae are generalized to include higher power terms.It is shown that the correct initial procedure in this context is to specify the rate of dissipation as a positive semi-definite functional and then to determine the free energy from this, rather than the other way around, which would be the traditional approach.Particularly detailed versions of these formulae are given for the model with two memory contributions in the free energy, the first being the well-known quadratic functional leading to constitutive relations with linear history terms, while the second is a quartic functional yielding a cubic term for the stress function memory dependence. Also, the discrete spectrum model, for which each memory kernel is a sum of exponentials, is generalized from the quadratic functional representation for the free energy to that with the quartic functional included.Finally, a model is considered, allowing functional power series with an infinite number of terms for the free energy, rate of dissipation and stress function.