A nonlinear viscoelastic bushing element in multibody dynamics

This paper presents a formulation for incorporating nonlinear viscoelastic bushing elements into multibody systems. The formulation is based on the assumption that the relaxation function can be expressed as a sum of functions which are nonlinear in deformation and exponentially decreasing in time. These forces can represent elastomeric mounts or bushings in automotive suspension systems. The numerical implementation of the nonlinear viscoelastic bushing model into a general purpose rigid multibody dynamics code is described, and an extension of the formulation is also presented wherein component flexibility is included. Model validation was performed by comparing experimental data to simulation results obtained using the nonlinear viscoelastic model and a nonlinear elastic model. The experimental data were obtained at the Center's facilities by testing an automotive lower control arm/bushing system, subjected to a simulated road load event. The comparison demonstrates the better load prediction capability of the viscoelastic bushing model compared to the conventional model.     

A nonlinear fractional viscoelastic material model for polymers

In this contribution a test scheme based on tensile tests at different velocities, relaxation experiments and deformation controlled loading and unloading processes with intermediate relaxations has been used to experimentally characterize the nonlinear, inelastic material behavior. Based on the experimental observations a small strain nonlinear fractional viscoelastic material model is derived. In order to use the model within a finite element analysis, the constitutive equations have been generalized for the multiaxial case. The experimental test scheme and the fractional viscoelastic material model are subsequently applied to characterize and compute the mechanical behavior of the thermoplastic Polypropylene. After the identification of the material parameters several uniaxial and multiaxial simulations have been carried out and compared with experimental results.     

A phenomenological constitutive model for the nonlinear viscoelastic responses of biodegradable polymers

We formulate a constitutive framework for biodegradable polymers that accounts for nonlinear viscous behavior under regimes with large deformation. The generalized Maxwell model is used to represent the degraded viscoelastic response of a polymer. The large-deformation, time-dependent behavior of viscoelastic solids is described using an Ogden-type hyperviscoelastic model. A deformation-induced degradation mechanism is assumed in which a scalar field depicts the local state of the degradation, which is responsible for the changes in the material’s properties. The degradation process introduces another timescale (the intrinsic material clock) and an entropy production mechanism. Examples of the degradation of a polymer under various loading conditions, including creep, relaxation and cyclic loading, are presented. Results from parametric studies to determine the effects of various parameters on the process of degradation are reported. Finally, degradation of an annular cylinder subjected to pressure is also presented to mimic the effects of viscoelastic arterial walls (the outer cylinder) on the degradation response of a biodegradable stent (the inner cylinder). A general contact analysis is performed. As the stiffness of the biodegradable stent decreases, stress reduction in the stented viscoelastic arterial wall is observed. The integration of the proposed constitutive model with finite element software could help a designer to predict the time-dependent response of a biodegradable stent exhibiting finite deformation and under complex mechanical loading conditions.     

A viscoelastic constitutive model with nonlinear evolutionary internal variables

Summary Based on the internal variable theory, a viscoelastic constitutive model of a highly deformable continuous medium is proposed. A set of second rank tensorial internal state variables corresponding to Biot's strain is introduced, and a nonlinear evolution law for these internal variables is suggested. The proposed model may be considered as an extension of the network theory of rubber elasticity to take the viscous effects into account. In order to verify the validity of the present model, uniaxial tension tests for HDPE are carried out at different strain rates. The prediction of the present model shows a good agreement with the experimental data. Finally, a discussion of the present constitutive model is given. It is found that the present constitutive model is more flexible to describe the strain rate sensitivity of polymeric materials in a wide range of strain rates.     

A variational constitutive framework for the nonlinear viscoelastic response of a dielectric elastomer

We formulate a variational constitutive framework that accounts for nonlinear viscous behavior of electrically sensitive polymers, specifically Dielectric Elastomers (DEs), under large deformation. DEs are highly viscoelastic and their actuation response is greatly affected in dynamic applications. We used the generalized Maxwell model to represent the viscoelastic response of DE allowing the material to relax with multiple mechanisms. The constitutive updates at each load increment are obtained by minimizing an objective function formulated using the free energy and electrostatic energy of the elastomer, in addition to the viscous dissipation potential of the dashpots in each Maxwell branch. The model is then used to predict the electromechanical instability (EMI) of DE. The electro-elastic response of the DE is verified with available analytical solutions in the literature and then the material parameters are calibrated using experimental data. The model is integrated with finite element software to perform a variety of simulations on different types of electrically driven actuators under various electromechanical loadings. The electromechanical response of the DE and the critical conditions at which EMI occurs were found to be greatly affected by the viscoelasticity. Our model predicts that under a dead load EMI can be avoided if the DE operates at a high voltage rate. Subjected to constant, ramp and cyclic voltage, our model qualitatively predicts responses similar to the ones obtained from the analytical solutions and experimental data available in the literature.     

Nonlocal beam model for nonlinear analysis of carbon nanotubes on elastomeric substrates

Postbuckling, nonlinear bending and nonlinear vibration analyses are presented for single-wall carbon nanotubes (SWCNTs) resting on a two-parameter elastic foundation in thermal environments. The SWCNT is modeled as a nonlocal nanobeam which contains small scale effects. The elastomeric substrate with finite depth is modeled as a two-parameter elastic foundation. The thermal effects are included and the material properties of both SWCNTs and the substrate are assumed to be temperature-dependent. The governing equation that includes beam–foundation interaction is solved by a two-step perturbation technique. The numerical results reveal that the small scale parameter e₀a reduces the postbuckling equilibrium paths, the static large deflections and natural frequencies of SWCNTs resting on an elastic foundation. The results also reveal that the effect of the small scale parameter is significant for compressive buckling, but less pronounced for static bending and marginal for free vibration of SWCNTs resting on an elastic foundation.     

A new class of quasi-linear models for describing the nonlinear viscoelastic response of materials

In this short note, we develop a new class of “quasi-linear” viscoelastic models wherein the linearized strain is expressed in terms of a nonlinear measure of the stress. The class of models that is developed could be regarded as counterpart to the class of models referred to popularly as “quasi-linear” models, proposed by Fung to describe the response of viscoelastic bodies; however, now the strain is expressed as an integral of a nonlinear measure of the stress. The class of models that are developed can describe response that cannot be described by the class of models proposed by Fung, and moreover, these models are more reasonable from the point of view of causality.     

Modelling the nonlinear viscoelastic response of asphalt binders

In a recent article, Narayan et al. [Narayan, S.P.A., et al., 2012b. Nonlinear viscoelastic response of asphalt binders: experimental study of relaxation of torque and normal force in torsion. Mechanics Research Communications, 43, 66–74] recorded both torque and normal force in torsional relaxation experiments on asphalt binders. The data are three-dimensional and thus, require interpretation using three-dimensional constitutive relations. In this article, we develop such a three-dimensional nonlinear viscoelastic model for asphalt binder. The predictions of the model fit the experimental data reasonably well. While there are already some three-dimensional constitutive relations available in the literature for asphalt binders which can describe the normal forces that are developed during torsion, they do not capture the relaxation behaviour presented by Narayan et al. (2012b). This new model, however, captures most of the key features of the experimental data including the development of normal forces in torsion and the relaxation behaviour. In this respect, the new model performs considerably better than the existing models.     

A constitutive model for compressible elastomeric solids

A non-linear thermo-elastic constitutive model for the large deformations of isotropic materials is formulated. This model is specialized to account for the physics and thermodynamics of the elastic deformation of rubber-like materials, and based on these molecular considerations a constitutive model for compressible elastomeric solids is proposed. The new constitutive model generalizes the incompressible and isothermal model of Arruda and Boyce (1993) to include the compressibility and thermal expansion of these materials. The model is fit to existing experimental data on vulcanized natural rubbers to determine the material parameters for the rubbers examined. The fit between the simple model and the data is found to be very good for large stretches and moderate volume changes.List of symbols x\s=f(p) Deformation function - p Material point of a body in a reference configuration - x Place occupied by material point p in the current configuration - F(p)\eq(\t6/\t6p) f(p) Deformation gradient - J\s=det F\s>0 Determinant of F - F\s=RU\s=VR Polar decompositions of F - U, V Right and left stretch tensors; positive definite and symmetric - R Rotation tensor; proper orthogonal - U= _1–1 ³ ₁ ² r₁r₁ Spectral representation of U - V= ₁₌₁ ³ _t ² 1_t1₁ Spectral representation of V - _t > 0 Principal stretches - {r_i} Right principal basis - {l_i} Left principal basis - C\s=F ^T F, B\s=FF ^T Right and left Cauchy-Green strain tensors - \gq\s>0 Absolute temperature - \ge Internal energy density/unit reference volume - \gh Entropy density/unit reference volume - \gy\s=\ge\t-\gq\gh Helmholtz free energy/unit reference volume     

A uniaxial nonlinear viscoelastic constitutive model with damage for M30 gun propellant

The nonlinear viscoelastic mechanical response of a conventional tank gun propellant, M30, is modeled using a “modified superposition integral” that incorporates the effects of microstructural fracture damage. Specifically, a linear, time-dependent kernel is convolved with the first-time derivative of a power-law function of stress and a damage “softening” that accounts for damage evolution by a microcrack growth mechanism. The microcrack damage function is a master curve formed from shifted isothermal, compressive, uniaxial constant strain rate (0.01 s⁻¹ to 420 s⁻¹) data on solid, right-circular cylinders of M30 gun propellant. An attractive feature of the model is its ability to predict work-softening behavior under conditions of monotonically increasing deformation. Time-dependent predictions of stress versus time, failure stress versus failure time, and failure stress versus strain rate, quantitatively agree with experimental results from constant strain rate tests on the propellant. Theoretical predictions of time-dependent stresses for Heaviside and “ballistic-like” strain histories are also provided.     

A nonlinear viscoelastic constitutive model taking into account of physical aging

Mechanics of Time-Dependent Materials - Polymers exhibit viscoelastic behavior: their mechanical response depends on the loading time, or on the loading frequency. In addition, if a polymer...     

A formulation to model the nonlinear viscoelastic properties of the vascular tissue

Nearly all soft tissues, including the vascular tissue, present a certain degree of viscoelastic material response, which becomes apparent performing multiple relaxation tests over a wide range of strain levels and plotting the resulting stress relaxation curves, nonlinear viscoelasticity of the tissue. Changes in relaxation rate at each strain may occur at multiple strain levels. A constitutive formulation considering the particular features of the vascular tissue, such as anisotropy, together with these nonlinear viscoelastic phenomena is here presented and used to fit stress?Cstretch curves from experimental relaxation tests. This constitutive model was used to fit several data set of in vitro experimental stress relaxation tests performed on ovine and porcine aorta. The good fitting of the experimental data shows the capability of the model to reproduce the viscoelastic response of the vascular tissue.     

Oscillatory torsional flow of a viscoelastic solid

 We show that a viscoelastic solid, modelled by a three-dimensional analogue of the Kelvin-Meyer-Voigt equation (the neo-Hookean rubber-like body and the Oldroyd-B element in parallel), has an exact similarity solution in the torsional flow geometry, including inertia. Numerical results obtained by a finite difference method for the oscillatory torsional flow indicate that the flow loses its stability at high amplitudes of oscillations. This is partly explained by a linear stability analysis of a simpler flow generated by a constant angular displacement. Received 10 December 2001 / Accepted 12 March 2002     

Propagation of torsional waves in a viscoelastic layer over an inhomogeneous half space

This paper presents a theoretical study on propagation of torsional surface waves in a homogeneous viscoelastic isotropic layer with Voigt type viscosity over an inhomogeneous isotropic infinite half space. The non-homogeneity in half space is assumed to arise due to exponential variation in shear modulus and density. A closed-form solution has been obtained for the displacement in the layer as well as for a infinite half space. The dispersion and absorption relations for an torsional wave under the assumed geometry have been found. Numerical results are presented for propagation characteristics in terms of a number of non-dimensionalized parameters and have been produced graphically. This study investigates the effect of various parameters, namely non-homogeneity parameter, internal friction, the layer width and complex wave number on dissipation function and phase velocity of the torsional wave. Results in some special cases are also compared with existing solutions available from analytical methods, which show a close agreement.     

Single integral representations for nonlinear viscoelastic solids

The finite linear viscoelastic solid and single integral representation with nonlinear dependence on history are investigated in uni-axial stress. Both integral kernels in the stress formulation are determined by single-step constant strain tests, and both kernels in the strain formulation are determined by single-step constant stress tests. Single integral stress and strain formulations are not equivalent. The stress histories required to maintain constant strain-rate for both models are determined from the Volterra integral equations given by the strain formulations once their kernels are determined by constant stress tests. However, known constant strain-rate response does not determine the kernels. Examples are presented to show that variation of the kernel within a given qualitative shape can lead to different shapes of constant strain-rate response, so that both constant stress and constant strain-rate tests may be necessary to deduce the optimum single integral approximation, in preference to multi-step stress tests. It is shown that the apparently simpler finite linear viscoelastic model requires a far lengthier numerical algorithm to solve the Volterra equation, and leads to non-unique and physically unacceptable response, in comparison with the more flexible nonlinear history dependence which yields unique acceptable responses.     

A model for anisotropic viscoelastic damage in composites

A model for continuous damage combined with viscoelasticity is proposed. The starting point is the formulation connecting the elastic properties to the tensor of damage variables. A hardening law associated with the damage process is identified from available experimental information and the rate-type constitutive equations are derived. This elastic damage formulation is used to formulate an internal variable approximation to viscoelastic damage in the form of a non-linear Kelvin chain. Elastic and viscoelastic equations are implemented into a finite element procedure. The code is verified by comparison with closed-form solutions in simplified configurations, and validated by fitting results of experimental creep tests.     

A micromechanical model for a viscoelastic cohesive zone

A micromechanical model for a viscoelastic cohesive zone is formulated herein. Care has been taken in the construction of a physically-based continuum mechanics model of the damaged region ahead of the crack tip. The homogenization of the cohesive forces encountered in this region results in a damage dependent traction-displacement law which is both single integral and internal variable-type. An incrementalized form of this traction-displacement law has been integrated numerically and placed within an implicit finite element program designed to predict crack propagation in viscoelastic media. This research concludes with several example problems on the response of this model for various displacement boundary conditions.     

Computational model for predicting nonlinear viscoelastic damage evolution in materials subjected to dynamic loading

Many inelastic solids accumulate numerous cracks before failure due to impact loading, thus rendering any exact solution of the IBVP untenable. It is therefore useful to construct computational models that can accurately predict the evolution of damage during actual impact/dynamic events in order to develop design tools for assessing performance characteristics. This paper presents a computational model for predicting the evolution of cracking in structures subjected to dynamic loading. Fracture is modeled via a nonlinear viscoelastic cohesive zone model. Two example problems are shown: one for model validation through comparison with a one-dimensional analytical solution for dynamic viscoelastic debonding, and the other demonstrates the applicability of the approach to model dynamic fracture propagation in the double cantilever beam test with a viscoelastic cohesive zone.     

A nonlinear model of the response of primary probes for selective electromagnetic nondestructive testing

We propose a method for the construction of a nonlinear multiparameter model of the response of a system of electromagnetic nondestructive testing of materials and products. The method is based on the approximation of functions of many variables (solutions of direct problems for an individual system) by many-dimensional algebraic polynomials. The model is designed for the mathematical simulation in the investigation and development of the methods and means for the solution of inverse problems of multiparameter nondestructive testing. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 6, pp. 93–103, November–December, 1996.