Nonlinear Viscoelastic Solids 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.

Inertial evolution of non-linear viscoelastic solids in the face of (self-)collision

Inertial evolution of non-linear viscoelastic solids in the face of (self-)collision

Branching during axial compression histories of solid cylinders of time-dependent materials

This work considers the uniaxial compression of a solid circular cylinder of time-dependent material. Initially, the cylinder undergoes a homogeneous compression history. The purpose is to determine a time when this deformation history can form a new inhomogenous branch. Material time dependence is described by a nonlinear single-integral constitutive equation that relates the stress in the material to its deformation history. A criterion is developed for determining branching times using a Pipkin–Rogers constitutive equation for nonlinear incompressible isotropic viscoelastic solids. For the purpose of numerical examples, material parameters are chosen so that the material acts as a neo-Hookean material in its short-time response and a softer one in its long-time equilibrium response. Examples show that characteristic relaxation times and deformation histories influence the time to branch and the corresponding compressive stretch and compressive force.

Cavity expansion in nonlinear viscoelastic solids: A nonlinear dynamic study

Dynamic cavity expansion has been studied for hyperelastic materials since the work of Knowles and Jakub in 1965. It has recently drawn new attention for potential applications in measuring the mechanical properties of soft polymer or biological tissues. These soft materials, however, are intrinsically rate-dependent, and the study of dynamic cavitation in nonlinear viscoelastic solid is relatively scarce. In this article, we utilize the established two-potential nonlinear viscoelastic framework to study the cavity behavior in infinite solid. Differently, we further introduced the surface energy into account and pay extra attention to the oscillation of the cavity surface based on nonlinear dynamics. The cavity behavior and stress change under a monotonically increasing pressure, creep experiment, and relaxation experiment are investigated. Considering surface effects, we notice the cavity surface has an oscillation process of expansion and shrink before infinitely large expansion before rupture. The oscillation amplitude gradually diminishes with time due to the energy consumption of viscosity. Regarding the viscoelastic material as a system with diminishing shear modulus, we analyze the stability of the cavity surface and find it can be either a stable center point or an unstable saddle point. The results of this study are helpful for understanding the cavitation phenomenon in viscoelastic solids caused by dynamic loading.

Acoustoelastic analysis of soft viscoelastic solids with application to pre-stressed phononic crystals

The effective dynamic properties of specific periodic structures involving rubber-like materials can be adjusted by pre-strain, thus facilitating the design of custom acoustic filters. While nonlinear viscoelastic behaviour is one of the main features of soft solids, it has been rarely incorporated in the study of such phononic media. Here, we study the dynamic response of nonlinear viscoelastic solids within a ‘small-on-large’ acoustoelasticity framework, that is we consider the propagation of small amplitude waves superimposed on a large static deformation. Incompressible soft solids whose behaviour is described by the Fung–Simo quasi-linear viscoelasticity theory (QLV) are considered. We derive the incremental equations using stress-like memory variables governed by linear evolution equations. Thus, we show that wave dispersion follows a strain-dependent generalised Maxwell rheology. Illustrations cover the propagation of plane waves under homogeneous tensile strain in a QLV Mooney–Rivlin solid. The acoustoelasticity theory is then applied to phononic crystals involving a lattice of hollow cylinders, by making use of a dedicated perturbation approach. In particular, results highlight the influence of viscoelastic dissipation on the location of the first band gap. We show that dissipation shifts the band gap frequencies, simultaneously increasing the band gap width. These results are relevant to practical applications of soft viscoelastic solids subject to static pre-stress.

Dimensional changes during shear without normal tractions (the Poynting effect) in nonlinear viscoelastic fiber-reinforced solids

When a rectangular block of a nonlinear material is subjected to a simple shearing deformation, specific normal tractions are required to ensure that the distances between the faces of the block, i.e. its dimensions, do not change. This work investigates the time-dependent dimensional changes during shear in the absence of these normal tractions (the Poynting effect) that occur in a block composed of an incompressible nonlinearly viscoelastic fiber-reinforced solid. The material is modeled using the Pipkin–Rogers nonlinear single integral constitutive equation for viscoelasticity. This constitutive equation is used because (1) it exhibits the essential features of nonlinear viscoelasticity; (2) it is straightforward to include the material symmetry restrictions due to the reinforcing fibers. A system of nonlinear Volterra integral equations is formulated for the dimensional changes in the block. Numerical solutions are presented for the case when the standard reinforcing model for nonlinearly elastic fiber-reinforced materials is incorporated in the Pipkin–Rogers constitutive framework. The results illustrate how the time-dependent dimensional changes depend on the fiber orientation and the viscoelastic properties of the fibers relative to those of the matrix.

До питання врахування впливу третього інваріанту девіатора напружень на процес довготривалого деформування нелінійно-в’язкопружних матеріалів

The problem on the influence of stressed state on the process of long-term deformation of nonlinear viscoelastic materials under the simple and quasi-simple modes of loading by introduction of the function with the parameter of Lode angle into the defining equations is considered. The mentioned function is determined by analysis of base experimental data obtained from the base experiments on axial tension and pure torsion. Physical and mechanical properties of nonlinear viscoelastic solids are defined by the correspondence between the invariants of deformation tensors and tensions according to the modified nonlinear Rabotnov’s model for viscoelasticity. The heredity kernels are given by the fractional-exponential function. The constructed defining equations are verified experimentally for the problems of determination of nonlinear creep deformations under combined loading applied to the thin-walled tubular elements made of polyethylene of high density and low pressure polyethylene. As a result of juxtaposition of experimental data and calculations it is a stated that allowing for the type of stressed state improves their agreement qualitatively and quantitatively.

Analysis of steady-state frictional rolling contact problems in Schapery–nonlinear viscoelasticity

In the context of an updated Lagrangian formulation, a computational model is developed for analyzing the steady-state frictional rolling contact problems in nonlinear viscoelastic solids. Schapery's nonlinear viscoelastic model is adopted to simulate the viscoelastic behavior. In addition to the material nonlinearity, the model accounts for geometrical nonlinearities, large displacements, and rotations with small strains. To satisfy the steady-state rolling contact condition, a spatially dependent incremental form of the viscoelastic constitutive equations is derived. Consequently, the dependence on the past history of the strain rate in the stress–strain law is expressed in terms of the spatial variation of the strain. The contact conditions are exactly satisfied by employing the Lagrange multiplier approach to enforce the contact constraints. The classical Coulomb's friction law is used to simulate friction. The developed model is verified and compared and good agreement is obtained. The applicability of the developed model is demonstrated by analyzing the steady-state rolling contact response of viscoelastically walled-wheel over rigid foundation. Moreover, the obtained results show remarkable effects of the rotational velocity and the viscoelastic material parameters on the mechanical response of steady-state frictional rolling contact.

Modelling stress-affected chemical reactions in non-linear viscoelastic solids with application to lithiation reaction in spherical Si particles

This paper aims at modelling stress-affected chemical reactions in spherical particles by adopting the chemo-mechanical framework based on the chemical affinity tensor and combining it with the finite-strain non-linear viscoelastic constitutive model. The model is applied to the chemical reaction between lithium (Li) ions and silicon (Si), which has been considered as promising successor to graphite for use as active material in lithium-ion battery (LIB) anodes. However, during charging of LIBs, Si enters into the chemical reaction with Li ions, causing large volumetric expansion of Si particles, which leads to the emergence of mechanical stresses, which, in turn, can affect the kinetics of the chemical reaction even up to the reaction arrest. In this paper, the propagation of the reaction front separating the chemically transformed and the untransformed phases is modelled, and the coupled stress-diffusion-reaction problem is solved using the finite element approach. The model predicts the retardation and the locking of the chemical reaction in Si depending on the values of the chemical energy parameter, which corresponds to experimental observations.

Modeling longitudinal wave propagation in nonlinear viscoelastic solids with softening

A model for longitudinal wave propagation in rocks and concrete is presented. Such materials are known to soften under a dynamic loading, i.e. the speed of sound diminishes with forcing amplitudes. Also known as slow dynamics, the softening of the material is not instantaneous. Based on continuum mechanics with internal variables of state, a new formulation is proposed, which accounts for nonlinear Zener viscoelasticity and softening. A finite-volume method using Roe linearization is developed for the system of partial differential equations so-obtained. The method is used to carry out resonance simulations, and its performance is assessed in the linear viscoelastic case. Qualitative agreement with experimental results of nonlinear ultrasound spectroscopy (NRUS) and dynamic acousto-elastic testing (DAET) is obtained.

Rate type constitutive equations for fiber reinforced nonlinearly vicoelastic solids using spectral invariants

In this paper we are interested in developing constitutive equations for fiber-reinforced nonlinearly viscoelastic solids. It has been shown that constitutive equations for such bodies can be expressed in terms of a complete minimal set of 18 classical invariants associated with deformation and fiber orientation. In this paper, we give an alternative formulation using a set of spectral invariants. It is shown via the use of spectral invariants that only 11 of the 18 classical invariants are independent. We analyze the spectral invariants for two illustrative deformation gradients: (i) simple tension, and (ii) simple shear.

Deformations of infinite slabs of non-linear viscoelastic solids containing an elliptic hole

In this paper we study the state of stress and strain in infinite elastic slabs of nonlinear viscoelastic solids containing elliptic holes subject to an uni-axial as well as a bi-axial state of stress. The geometry affords one to get some inkling concerning the states of stress and strain in bodies containing a crack by obtaining the limit of the solutions as the aspect ratio (in this case the ratio of the minor axis to the major axis) of the ellipse tends to zero. We consider two classes of non-linear viscoelastic bodies, the classical incompressible Kelvin–Voigt solid (Thomson in R Soc Lond 14:289–297, 1865; Voigt in Ann Phys 283(12):671–693, 1892) and a generalization of a compressible model due to Gent (Rubber Chem Technol 69(1):59–61, 1996). We also study for the sake of comparison the case of a nonlinear neo-Hookean elastic solid with an elliptic hole.

Dynamic cavitation and relaxation in incompressible nonlinear viscoelastic solids

Closed form analytical solutions for the dynamic expansion of a spherical void in a finite, spherically symmetric, nonlinearly viscoelastic medium are presented. We account for finite strains and application of load, both internally, at the cavity wall, and at the external surface. Hence, the present solution may apply to a wide range of known physical phenomena with examples from biomechanics, as in the appearance of cavitation bubbles due to traumatic brain injury or expansion of soft tissue due to an internal growth, to fracture initiation in soft materials and softening of seismic bearings in large scale structures. Specifically it is suggested that the present closed form analytical relations can be facilitated to measure the local viscoelastic properties of the material through controlled relaxation experiments.

The Mechanism of Moisture Transport in a Gel Cylinder

In drying of gels, the analysis of the process must encompass both heat and mass transfer aspects as well as deformation, usually very strong, of the material in the process. Deformation is almost always assumed to be linearly dependent on moisture content without considering an internal stress or pressure build-up. Since there is a great commercial interest in production of non-deformed dry gels by convective ambient pressure drying, it was interesting to investigate the behavior of such materials in drying. Convective drying would eliminate the expensive supercritical. In this work several types of gels were synthesized and dried convectively in a lab scale. Since drying induces the internal stress formation responsible for shrinkage and cracking, great care was taken to measure deformation and internal pressure development in gel during drying. Cylindrical gel samples 35 mm in diameter were dried convectively in air at 30, 40, and 50 °C. The hydrogels were made of PVP, agar-agar, and silica. Different types of behavior of the gels during drying was observed: almost uniform deformation in the case of PVP, case-hardening for agar-agar gels, and cracking for silica hydrogels. This behavior directly results from different mechanical properties of the investigated gels. It was also observed that the internal pressure changes from initially overpressure to underpressure in the course of drying which is also a result of internal stress build up. The experimental data may be used for validation of mechanical models of drying of non-linear visco-elastic solids.

A numerical solution for contact problem with finite deformation in nonlinear Schapery viscoelastic solids

The present paper presents an incremental finite element model for the solution of the two dimensional quasi-static frictionless nonlinear viscoelastic contact problems with large deformations. The nonlinear viscoelasticity is simulated using the Schapery’s single-integral viscoelastic model with stress-dependent properties. The updated Lagrangian formulation is used to model the material and geometrical nonlinearity of the problem. To eliminate the need to store the entire strain histories, the constitutive equations are transformed into an incremental recursive form. The contact constraints are enforced by using the Lagrange multiplier method where the normal contact forces are treated as independent variables. The resulting nonlinear equilibrium equations are solved by the Newton–Raphson method in an incremental-iterative procedure. The capability of the developed numerical incremental-iterative solution procedure is demonstrated by analysing two different contact problems with different nature.

Nonlinear Viscoelastic Fracture Mechanics Using Boundary Elements

The boundary element methodology is applied to the fracture mechanics of non-linear viscoelastic solids. The adopted non-linear model is based on the ‘free volume’ concept, which is introduced into the relaxation moduli entering the linear viscoelastic relations through a time shift depending on the volumetric strain. Nonlinearity generates an irreducible domain integral into the original boundary integral equation governing the behaviour of linear viscoelastic solids. This necessitates the evaluation of domain strains, which relies on a non-standard differentiation of an integral with a strong kernel singularity. A time domain formulation based on constant shape functions over boundary elements and domain cells is computer-implemented through a numerical integration algorithm. The effectiveness of the developed numerical tool is demonstrated through the analysis of a plate with a central crack. The predicted stress field around the crack tip is compared with respective results obtained by the finite element method.

On a Special Class of Nonlinear Viscoelastic Solids

We investigate some possible nonlinear generalizations of the Kelvin—Voigt viscoelastic models. We use the usual idealization of creep and recovery experiments to discuss the mechanical significance of some constitutive requirements and we show that in the case of a shear-rate dependent viscosity localization of the solution is possible under a simple constitutive characterization.

Response of Anisotropic Nonlinearly Viscoelastic Solids

Despite the technological relevance of anisotropic nonlinear viscoelastic solids, little effort has been expended in the development of specific constitutive theories. In this study we develop a constitutive model for describing the nonlinear response of anisotropic viscoelastic solids that might be well suited to describe the response of biological and geological solids. The model is an integral model that takes into account the history of deformation of the body. Using the model a few boundary value problems are studied, namely the time dependent extension and shearing of such bodies.

On thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic solids with application to biomechanics

In this paper, we are interested in developing thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic bodies, in particular for transversely isotropic nonlinearly viscoelastic solids, in isothermal processes. It follows from results in the theory of algebraic invariants that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants associated with deformation and fiber orientation: 10 of them are isotropic invariants and 8 of them are associated with the deformation and the orientation of the fiber. Among the 8 anisotropic invariants just 6 are related to the viscoelastic response. The terms in the Cauchy stress tensor associated to these 6 invariants are analyzed with respect to thermodynamical consistency, and we obtain restrictions for the corresponding constitutive coefficients. This framework is applied to viscoelastic potentials within the context of biomaterials.

Elastomers - An Engineering Material

Abstract Elastomers are engineering materials. Defined as nonlinear viscoelastic solids; applications are normally deformations under twenty five (25) percent. Conventional engineering values of shear modulus G and Young's modulus E can be determined in this relatively linear section of the stress-strain relationships. The viscoelastic description indicates a time/temperature relation response. Treloar notes that G is both the shear modulus value and the elastic constant. The high bulk modulus allows us to assume E=3G thus we have several approaches to develop G and utilize the equation G=NκT=ρRT/Mc, (hereby ρ is density, R gas constant, Mc average chain molecular weight, T temperature Kelvin). This allows us to monitor aging by changes in Mc that occur due to network aging. The aging process will be monitored by strain energy density changes, compressive stress-relaxation (CSR) and dynamic mechanical rheologic testing (DMRT)