Viscoelastic Relaxation Function Research Articles

Direct integration of measured viscoelastic relaxation data in time-domain finite element simulations

The current approach to modeling viscoelastic materials in most commercial finite element packages is based on the General Maxwell Model, which views these materials as combinations of spring and dashpot elements. However, the data can be incorporated more directly into a transient finite element study by direct interpolation of the relaxation function. This work explores a linear interpolation scheme to the inclusion of viscoelastic relaxation functions on an example problem. The results show several benefits over the General Maxwell Model for transient studies. Included in the analysis are displacement solutions utilizing both approaches, relaxation function error calculations for both approaches, and parametric runtime studies comparing speed of calculation. The variation in computational flop counts is considered and an argument is made for the preference of the proposed approach.

Measuring the viscoelastic relaxation function of cells with a time-dependent interpretation of the Hertz-Sneddon indentation model

The Hertz-Sneddon elastic indentation model is widely adopted in the biomechanical investigation of living cells and other soft materials using atomic force microscopy despite the explicit viscoelastic nature of these materials. In this work, we demonstrate that an exact analytical viscoelastic force model for power-law materials, can be interpreted as a time-dependent Hertz-Sneddon-like model. Characterizing fibroblasts (L929) and osteoblasts (OFCOLII) demonstrates the model's accuracy. Our results show that the difference between Young's modulus EY obtained by fitting force curves with the Hertz-Sneddon model and the effective Young's modulus derived from the viscoelastic force model is less than 3%, even when cells are probed at large forces where nonlinear deformation effects become significant. We also propose a measurement protocol that involves probing samples at different indentation speeds and forces, enabling the construction of the average viscoelastic relaxation function of samples by conveniently fitting the force curves with the Hertz-Sneddon model.

On Determination of the Linear Viscoelastic Compliance and Relaxation Functions for Polymers in One Tensile Test

On Determination of the Linear Viscoelastic Compliance and Relaxation Functions for Polymers in One Tensile Test

A contact model for the normal force between viscoelastic particles in discrete element simulations

DEM modeling of granular materials composed of viscoelastic particles can provide valuable insights into the mechanical behavior of a wide range of engineering materials. In this paper, a new model for calculating the normal contact force between visoelastic spheres is presented based on contact mechanics that takes the mechanical behavior of the DEM particles into account. The model relies on an application of the viscoelastic correspondence principle to elastic Hertz contact. A viscoelastic relaxation function for the contact is defined and a generalized Maxwell material is used for describing this function. An analytical expression for the increment in contact force given an increment in overlap is derived leading to a computationally efficient model. The proposed model provides the analytical small deformation solution upon loading but provides an approximate solution at unloading. Comparisons are made with FEM simulations of contact between spheres of different sizes of equal and dissimilar materials. An excellent agreement is found between the model and the FEM simulations for almost all cases except at cyclic loading where the characteristic times of the viscoelastic behavior and the loading are similar.

Dynamics of Rouse chains undergoing head-to-head association and dissociation: Difference between dielectric and viscoelastic relaxation

Polymer chains having type-A dipoles parallel along the chain backbone exhibit slow dielectric relaxation reflecting their global motion. For monofunctionally head-associative Rouse chains having type-A dipoles, the Rouse equation of motion was combined with the association/dissociation kinetics to calculate the dielectric relaxation function, Φj(t) with j = 1 and 2 for unimer and dimer. Φ1(t) reflects the orientation of the end-to-end vector of the unimer, whereas Φ2(t) detects the orientation of two end-to-center vectors of the dimer (having symmetrical dipole inversion), both being in the direction of the applied electric field. The calculation was made by mapping the conformation of dissociating dimer onto the created unimer and vice versa with the aid of the Rouse eigenmode expansion of the bond vector of segments (Gaussian subchains). It turned out that Φj(t) is not influenced by the association/dissociation reaction. This result makes a striking contrast to the behavior of the viscoelastic relaxation function gj(t): gj(t) is strongly affected by the motional coupling between the unimer and dimer due to the reaction [H. Watanabe et al., Macromolecules 48, 3014–3030 (2015)]. This difference emerged because the dielectric Φj(t) corresponds to the vectorial first-moment average of the segmental bond vector at time t, u(n,t) with n being the segment index, whereas the viscoelastic gj(t) corresponds to the tensorial second-moment average. Because of this difference in the averaging moment, Φj(t) is subjected to cancelation in the conformational mapping but gj(t) is not, so that the reaction effect emerges only for gj(t). The experimental data of head-carboxylated high-cis polyisoprene chains (having the type-A dipoles), confirming this difference, are also presented.

A method for the determination of the viscoelastic relaxation function of reactive materials

Modern insensitive high explosives HE are heterogeneous materials consisting of a mixture of explosive grains within a polymeric binder matrix [1]. The mechanical material behaviour is strongly determined by this heterogeneous structure. The stiffness and strength comes mainly from the explosive grains whereas the softer polymeric binder is responsible for the viscoelastic behaviour. Depending on the mechanical loading regime different mechanisms are present and must be taken into account for the material modelling.

Fractional dynamics in silk: From molecular picosecond subdiffusion to macroscopic long-time relaxation.

Structural relaxations in humid silk fibers exposed to tensile stress have been reported to take place on a very wide range of time scales from a few milliseconds to several hours. The time-dependence of the measured tensile force following a quasi-instantaneously applied external strain on the fibers can be understood in terms of a fractional viscoelastic relaxation function introducing memory effects by which the mechanical state of a fiber depends on its tensile history. An analog fractional relaxation also gives rise to the subdiffusion observed on picosecond time scales, which governs the mobility of the amorphous polymer chains and adsorbed water on the molecular level. The reduction of the subdiffusive memory effect in stretched fibers compared to native fibers is consistent with the higher order of the polymers in the stretched state.

Natural fiber composite: Challenges simulating inelastic response in strain-controlled tensile tests

Problems occurring, when nonlinear time-dependent material model with parameters identified in creep tests is applied to simulate high-strain response in strain-controlled tests, are described and analyzed. Reasons for discrepancies with experimental loading curves are revealed. Presented numerical/experimental examples deal with three bio-based composites showing highly nonlinear behavior due to damage, nonlinear viscoelasticity and viscoplasticity. Schapery's approach for viscoelasticity and Zapas' model for viscoplasticity are used. The model is generalized to include microdamage effect. It is shown that the main problem in simulations at high stresses is the reliability of data from creep test for model identification in this region because creep rupture limits the available data region and extrapolation to higher stresses is rather uncertain. Alternative solution is to employ relaxation tests at high strains to obtain the missing information. However, it would work only in absence of viscoplastic strains: viscoelastic relaxation functions cannot be determined by maintaining constant total strain if viscoplastic-strain is developing. Based on sensitivity analysis of composite response to variations of the elastic modulus, damage, viscoelastic and viscoplastic parameters, suggestions are made for improving (further “tuning”) the model in high stress region by using tensile stress–strain curves in quasi-static loading.

Dielectric and Viscoelastic Behavior of Star-Branched Polyisoprene: Two Coarse-Grained Length Scales in Dynamic Tube Dilation

cis-Polyisoprene (PI) chain has the type A dipole parallel along the backbone so that its large-scale (global) motion results in not only viscoelastic but also dielectric relaxation. Utilizing this feature of PI, this paper examined dielectric and viscoelastic behavior of star PI probe chains (arm molecular weight 10–3Ma = 9.5–23.5, volume fraction υ1 = 0.1) blended in a matrix of long linear PI (M = 1.12 × 106). The constraint release (CR)/dynamic tube dilation (DTD) mechanism was quenched for those dilute probes entangled with the much longer matrix, as evidenced from coincidence of the frequency dependence of the dielectric and viscoelastic losses of the probe in the blend. Comparison of the probe data in the blend and in monodisperse bulk revealed that the star probe relaxation is retarded and broadened on blending and the retardation/broadening is enhanced exponentially with Ma. This result in turn demonstrates significant CR/DTD contribution to the dynamics of star PI in bulk. The magnitude of retardation was quantitatively analyzed within the context of the tube model, with the aid of the dielectrically evaluated survival fraction of the dilated tube, φ′(t), and the literature data of CR time, τCR. In the conventional molecular picture of partial-DTD, the tube is assumed to dilate laterally, but not coherently along the chain backbone. The corresponding lateral partial-DTD relationship between φ′(t) and the normalized viscoelastic relaxation function μ(t), μ(t) = φ′(t)/β(t) with β(t) being the number of entanglement segments per laterally dilated segment (that was evaluated from the φ′(t) and τCR data), held for the μ(t) and φ′(t) data of star PI in bulk. Nevertheless, the observed retardation of the star probe relaxation on blending was less significant compared to the retardation expected for the arm motion (retraction) along the laterally dilated tube in bulk PI. This result suggests that the relaxation time of the probe in bulk is governed by the longitudinal partial-DTD that occurs coherently along the chain backbone. In fact, the magnitude of retardation evaluated from the φ′(t) and τCR data on the basis of this longitudinal partial-DTD picture was close to the observation. These results strongly suggest that the star PI chains in monodisperse bulk have two different coarse-grained length scales: the diameter of laterally dilated tube that determines the modulus level and the diameter of longitudinally dilated tube that reflects the path length for the arm retraction and determines the relaxation time. Thus, the star PI chains in bulk appear to move along the longitudinally dilated tube that wriggles in the laterally dilated tube. This molecular scenario is consistent with the previous finding for bulk linear PI [Matsumiya et al. Macromolecules 2013, 46, 6067].

Viscoelastic representation of surface waves in patchy saturated poroelastic media

Wave-induced flow is observed as the dominated factor for P wave propagation at seismic frequencies. This mechanism has a mesoscopic scale nature. The inhomogeneous unsaturated patches are regarded larger than the pore size, but smaller than the wavelength. Surface wave, e.g., Rayleigh wave, which propagates along the free surface, generated by the interfering of body waves is also affected by the mesoscopic loss mechanisms. Recent studies have reported that the effect of the wave-induced flow in wave propagation shows a relaxation behavior. Viscoelastic equivalent relaxation function associated with the wave mode can describe the kinetic nature of the attenuation. In this paper, the equivalent viscoelastic relaxation functions are extended to take into account the free surface for the Rayleigh surface wave propagation in patchy saturated poroelastic media. Numerical results for the frequency-dependent velocity and attenuation and the time-dependent dynamical responses for the equivalent Rayleigh surface wave propagation along an interface between vacuum and patchy saturated porous media are reported in the low-frequency range (0.1–1,000 Hz). The results show that the dispersion and attenuation and kinetic characteristics of the mesoscopic loss effect for the surface wave can be effectively represented in the equivalent viscoelastic media. The simulation of surface wave propagation within mesoscopic patches requires solving Biot’s differential equations in very small grid spaces, involving the conversion of the fast P wave energy diffusion into the Biot slow wave. This procedure requires a very large amount of computer consumption. An efficient equivalent approach for this patchy saturated poroelastic media shows a more convenient way to solve the single phase viscoelastic differential equations.

Fundamental Investigation of Polymer Dynamics with Dielectric and Viscoelastic Methods

This paper briefly summarizes two representative examples of investigation of fundamental polymer dynamics with dielectric and viscoelastic methods: 1) Entangled dynamics of flexible polymers: For type-A polymers having the electrical dipoles parallel along the chain backbone, the entanglement dynamics is differently reflected in the dielectric and viscoelastic relaxation functions. A simple relationship between these functions was deduced for the case of full dynamic tube dilation (full-DTD). This full-DTD relationship was found to hold experimentally for monodisperse linear chains, but not for branched (star and Cayley-tree) chains and linear/linear blends. A refined relationship, referred to as partial-DTD, was developed by taking into account the time necessary for constraint release equilibration. This partial-DTD relationship was found to valid for the linear, branched, and blend systems. 2) Effect of carbon dioxide (CO2) on polymer dynamics: Pressurized CO2 dissolves into polymers to exhibit a plasticizing effect. Dielectric tests revealed that the local dynamics of polymer chains was accelerated by pressurized CO2 without changing relaxation mode distribution. This unusual plasticizing effect of pressurized CO2 for the local dynamics was attributed to extraordinary fast diffusion of CO2 compared to the segmental motion of the polymer chain. The concentration fluctuation of the polymers, being responsible for the relaxation mode broadening usually observed for plasticized polymer systems, is smeared by the fast CO2 molecules in the time scale of segmental motion. In contrast, pressurized CO2 behaves just as an ordinary solvent for the slow, global dynamics of polymer chains.

Creep Behavior of Passive Bovine Extraocular Muscle

This paper characterized bovine extraocular muscles (EOMs) using creep, which represents long-term stretching induced by a constant force. After preliminary optimization of testing conditions, 20 fresh EOM samples were subjected to four different loading rates of 1.67, 3.33, 8.33, and 16.67%/s, after which creep was observed for 1,500 s. A published quasilinear viscoelastic (QLV) relaxation function was transformed to a creep function that was compared with data. Repeatable creep was observed for each loading rate and was similar among all six anatomical EOMs. The mean creep coefficient after 1,500 seconds for a wide range of initial loading rates was at 1.37 ± 0.03 (standard deviation, SD). The creep function derived from the relaxation-based QLV model agreed with observed creep to within 2.7% following 16.67%/s ramp loading. Measured creep agrees closely with a derived QLV model of EOM relaxation, validating a previous QLV model for characterization of EOM biomechanics.

Viscoelastic and Dielectric Relaxation of a Cayley-Tree-Type Polyisoprene: Test of Molecular Picture of Dynamic Tube Dilation

For an entangled Cayley-tree-type cis-polyisoprene (CT-PI) composed of two classes of arms, trifunctionally branched inner arms each carrying two outer arms (Mi = 24.7 × 103 and Mo = 12.3 × 103 for each inner and outer arm, respectively), the molecular picture of dynamic tube dilation (DTD) was examined through viscoelastic and dielectric data. The global motion of the arms was dielectrically active because they had the so-called type A dipole parallel along their backbone. The dielectric relaxation function Φ(t) thus measured for CT-PI enabled us to experimentally evaluate the survival fraction of the dilated tube φ′(t), a key quantity in the DTD picture. In the conventional picture of full-DTD, relaxed portions of the arms are regarded as a simple solvent, thereby allowing the number of mutually equilibrated entanglement segments per dilated segment, β(t), to fully increase to that in a corresponding solution, βf-DTD(t) = {φ′(t)}−d with d = 1.3 for PI. The corresponding full-DTD relationship between φ′(t) and the normalized viscoelastic relaxation function μ(t), μf-DTD(t) = {φ′(t)}1+d, did not hold for the φ′(t) and μ(t) data of CT-PI. In contrast, in the molecular picture of partial-DTD, β(t) is determined through competition of the constraint release (CR) motion of the focused chain and the motion of surrounding chains, the former determining the maximum possible βCR(t) value in a given time scale, t. In this picture, the length and time scales are consistently coarse-grained on the basis of the CR mechanism. The partial-DTD relationship, μp-DTD(t) = φ′(t)/β*(t) with β*(t) = min[βf-DTD(t), βCR(t)], was found to hold for the φ′(t) and μ(t) data satisfactorily. In addition, analysis of the φ′(t) data unequivocally indicated that the inner arm hardly relaxed until the outer arm relaxation completed; i.e., the hierarchical relaxation occurred through partial-DTD. These results suggested the importance of the consistent coarse graining in molecular description of the dynamics of branched chains in general.

Quantitative comparison between dynamic structure factors obtained experimentally and those calculated with Doi-Onuki theory

This paper reports results of quantitative comparison between dynamic structure factors obtained experimentally and those calculated by using the Doi and Onuki (DO) theory for semidilute polymer solutions. The authors obtained the dynamic structure factors with dynamic light scattering (DLS) experiment while the dynamic structure factors were calculated by using DO theory with osmotic compressibility, viscoelastic relaxation function, and friction coefficient which are obtained independently of DLS experiment. Calculated dynamic structure factors agree with experimental ones well and can express the q-dependent fast modes and the q-insensitive slow mode which experimental ones show. The authors estimated the characteristic parameters, interdiffusion coefficient and cooperative diffusion coefficient, from experimental and calculated results by using the procedure proposed by Einaga and Fujita [Polymer 40, 565 (1999)]. The estimated parameters for the DLS experiment agree with those for the calculation. These agreements in dynamic structure factors and the parameters indicate that DO theory can describe well the relaxation processes of semidilute polymer solutions.

A Study on the Relationship of Frequency and Deformation for Viscoelastic Material

Viscoelastic materials are used in automobiles and other products. However, because of theoretical complexity, it has not been easy to put all the energy exhausted into the automotive viscoelastic materials. Since time is the most important factor in the study of viscoelastic material, creep and stress relaxation functions are very important. In this study, a bushing was selected for special viscoelastic material. A bushing is a device used in automotive suspension systems to cushion the force transmitted from the wheel to the frame of the vehicle. A bushing is essentially a hollow cylinder which is bonded to a solid metal shaft at its inner surface and a metal sleeve at its outer surface. The shaft is connected to the suspension and the sleeve is connected to the frame. The cylinder provides the cushion when it deforms due to relative motion between the shaft and sleeve. The relation between the force applied to the shaft or sleeve and their deformation is nonlinear and exhibits features of viscoelasticity. A force-displacement relation for a bushing is important for multi-body dynamics numerical simulations. Hence, an explicit force-displacement relation has been introduced. The relation is expressed in terms of a force relaxation function. With Pipkin-Rogers model, the direct relation of force and displacement that has been derived from experiment and numerical simulation, the sinusoidal displacement was chosen and the relation of frequency and deformation for the viscoelastic material was studied.

Test of Tube Dilation Picture at Short Times through Dynamic Scattering and Viscoelastic Experiments

The dynamic tube dilation (DTD) concept is tested for the viscoelastic relaxation function μ(t) and the coherent intermediate scattering function Ic(q,t) (q = the wave vector) of highly entangled linear chains. The analysis made for Ic and μ involves only the Gaussian nature of the chain and no other details of the chain dynamics. Simple relationships are derived between Ic(q,t) and μ(t) for the two cases of the fixed and dilated tubes. The range of time considered is longer than the Rouse time τR for contour length fluctuation but smaller than the terminal viscoelastic relaxation time τG, and the range of μ(t) is 1 > μ(t) > 0.5. The relationships are significantly different for the two cases. This difference should enable a test of the DTD picture at short t (but still longer than τR) through comparison of the Ic(q,t) and μ(t) data.

Relationship between Viscoelastic Relaxation Function and Segmental Displacement Function for Entangled Linear Chain in Early Stage of the Tube Dilation Process

For highly entangled linear chains, the mean-square displacement of the primitive chain segments [d(n,tJ] 2 (n = segment index) not including a contribution from a nondiffusive part of the contour length fluctuation (CLF) was analyzed on the basis of the tube concept. The analysis was made by just considering the Gaussian nature of the chains (without detailed assumptions about the chain motion). A simple analytic relationship between this [d(n,t)] 2 and the normalized viscoelastic relaxation function μ(t) was derived in a range of u(t) > 0.5 and at t longer than the Rouse time τ Rouse for CLF. This relationship was significantly affected by the dynamic tube dilation (DTD) mechanism. For example, the end segment exhibited d 2 = (R 2 )(1 - μ) in the absence of DTD and d 2 ≅ (R 2 )(1 - μ 1/2 )in the presence of DTD, with (R 2 ) being the mean-square end-to-end distance of the chain. This difference should enable a test of the DTD picture at short t (but still longer than τ Rouse ) through comparison of the viscoelastic and segmental displacement data.

Further Test of the Tube Dilation Process in Star-Branched cis-Polyisoprene: Role of Branching-Point Fluctuation

In current tube models for entangled polymer chains, the tube representing the topological constraint is considered to move with time. This tube motion results in constraint release (CR) as well as dynamic tube dilation (DTD), the latter occurring as a result of CR-equilibration of the chain. For the DTD process of type-A star chains, a particular relationship between the normalized dielectric and viscoelastic relaxation functions, Φ(t) and μ(t), can be derived under an assumption that a fluctuation amplitude of the star-branching point (BP) is smaller than the diameter of the dilated tube. This DTD relationship, μ(t) ≅ [Φ(t)]2, does not hold for the terminal relaxation of the stars (Watanabe et al. J. Polym. Sci. B Polym. Phys. 2000, 38, 1024). This result, suggesting failure of the DTD molecular picture, might have been also attributed to breakdown of the assumption utilized in the above derivation. Under this background, this study examined dielectric and viscoelastic properties for a series of type-A ...

Biphasic Poroviscoelastic Behavior of Hydrated Biological Soft Tissue

Hydrated biological soft tissue consists of a porous extracellular matrix (ECM) and an interstitial fluid. The poroelastic theory (Biot, 1962), which was originally developed for soil mechanics, has been widely used for mathematical modeling of such hydrated biological tissue. This theory assumes that the ECM is incompressible and purely elastic, and that the interstitial fluid is incompressible and inviscid. The overall viscoelasticity of the tissue is expressed as a result of the frictional interaction between the elastic porous matrix and the interstitial fluid. The poroelastic theory, also known as the biphasic theory (Mow et al., 1980) in the biomechanics field, has served well over the past 20 years as an excellent modeling tool for the interstitial fluid flow-dependent viscoelastic response of hydrated soft tissue. It has been demonstrated that hydrated soft tissue also possesses a significant intrinsic viscoelasticity, independent of the interstitial fluid flow. The biphasic poroviscoelastic (BPVE) theory, which was first introduced by Mak (1986a and 1986b), incorporates a viscoelastic relaxation function into the effective solid stress of the poroelastic theory thus accounting for both intrinsic fluid flow-independent and fluid flow-dependent viscoelasticity. The objective of the present study is to investigate the biphasic poroviscoelastic characteristics of hydrated soft tissue, with an emphasis on the relative contribution of fluid flow-dependent and fluid flow-independent viscoelasticity to the overall viscoelastic behavior of soft tissues.

Direct measurement of adhesion between viscoelastic polymers: A contact mechanical approach

We have measured the surface energies of several viscoelastic polymers, using contact mechanics methods. The materials studied were diblock copolymers of poly(ethylene)-poly(ethylene-propylene) (PE-PEP) having different PE volume fractions and molecular weights. The materials exhibit differing viscoelastic relaxation functions in the bulk. Surface energies were determined from contact mechanics experiments in the loading mode, analyzed by incorporating the bulk viscoelastic effects into the Johnson–Kendall–Roberts theory of adhesive elastic contacts. The samples were made of spherical caps of the materials formed by first melting, and then cooling to room temperature. The measured values of the surface energies are close to the reported value for that of the surface-active PEP block, based on prior contact angle measurements, indicating that reliable surface energy values can be extracted from contact mechanics experiments, suitably analyzed to account for viscoelastic behavior. We believe that this method may be generalizable to surface and interfacial energy studies of a wide range of viscoelastic polymers.