Linear Viscoelastic Theory Research Articles

Theory of linear viscoelasticity of semiflexible rods in dilute solution

We present a theory of the linear viscoelasticity of dilute solutions of freely draining, inextensible, semiflexible rods. The theory is developed expanding the polymer contour about a rigid rod reference state, in a manner that respects the inextensibility of the chain, and is asymptotically exact in the rodlike limit where the polymer length L is much less than its persistence length Lp. In this limit, the relaxation modulus G(t) exhibits three time regimes: At very early times, less than a time τ∥∝L8/Lp5 required for the end-to-end length of a chain to relax significantly after a deformation, the average tension induced in each chain and G(t) both decay as t−3/4. Over a broad range of intermediate times, τ∥≪t≪τ⊥, where τ⊥∝L4/Lp is the longest relaxation time for the transverse bending modes, the end-to-end length decays as t−1/4, while the residual tension required to drive this relaxation and G(t) both decay as t−5/4. As later times, the stress is dominated by an entropic orientational stress, giving ...

Completely monotone fading memory relaxation modulii

In linear viscoelasticity, the fundamental model is the Boltzmann caual integral equation which defines how the stress σ(t) at time t depends on the earlier history of the shear rate via the relaxation modulus (kernel) G (t). Physical reality is achieved by requiring that the form of the relaxation modulus G (t) gives the Boltzmann equation fading memory, so that changes in the distant past have less effect now than the same changes in the more recent past. A popular choice, though others have previously been proposed and investigated, is the assumption that G (t) be a completely monotone function. This assumption has much deeper ramifications than have been identified, discussed or exploited in the rheological literature. The purpose of this paper is to review the key mathematical properties of completely monotone functions, and to illustrate how these properties impact on the theory and application of linear viscoelasticity and polymer dynamics. A more general representation of a completely monotone function, known in the mathematical literature, but not the rheological, is formulated and discussed. This representation is used to derive new rheological relationships. In particular, explicit inversion formulas are derived for the relationships that are obtained when the relaxation spectrum model and a mixing rule are linked through a common relaxation modulus.

On the Effect of Coarse Aggregate Fraction and Shape on the Rheological Properties of Self-Compacting Concrete

Abstract Rheological measurements have been undertaken to illustrate the applicability on fresh self-compacting concrete of a model for the relative viscosity and yield stress of suspensions. The model is based on linear viscoelasticity and classical theory for composite materials and it takes into account the amount, shape (aspect ratio of ellipsoids), and maximum packing density of the particles (here aggregates), as well as the rheological properties of the matrix (here mortar). Self-compacting concretes made from different types of coarse aggregates: spheres (glass beads), sea dredged, crushed, and a mix of sea dredged and crushed aggregates, have been tested in a coaxial cylinder concrete rheometer (BML). The investigations indicate that the aspect ratio, angularity, and surface texture of aggregates affect the viscosity and yield stress. The magnitude of these effects depends on the property in question, viscosity and yield stress.

Nonlinear viscoelastic response of vocal-fold tissues

Previous rheological measurements of the viscoelastic shear properties of vocal-fold tissues have focused on the linear regime, in the small strain region typically with γ0≤1.0%. This imposed limit was necessary in order for the theory of linear viscoelasticity to be valid, yielding dynamic shear data that can be applicable for the biomechanical modeling of small-amplitude vocal-fold oscillation. Nonetheless, as the physiological range of phonation does involve more than small-amplitude oscillation, the large strain viscoelastic behaviors of vocal-fold tissues are equally important and remain to be quantified. This paper reports preliminary measurements of some of these viscoelastic behaviors in large strain shear. Excised sheep vocal-fold mucosal tissues were subject to stress relaxation, constant stress, and constant strain rate tests in a controlled-strain torsional rheometer. Results showed that vocal-fold tissues demonstrate nonlinear viscoelastic response in shear, including stress relaxation that is dependent on strain and strain creep that is dependent on stress. These findings cannot be adequately described by Y. C. Fung’s quasilinear viscoelasticity formulation, which assumes strain dependence and time dependence to be separable. A more general constitutive model is being developed to better characterize the observed nonlinear response.

Maximum and minimum free energies for a linear viscoelastic material

Certain results about free energies of materials with memory are proved, using the abstract formulation of thermodynamics, both in the general case and as applied within the theory of linear viscoelasticity. In particular, an integral equation for the strain continuation associated with the maximum recoverable work from a given linear viscoelastic state is shown to have a unique solution and is solved directly, using the Wiener-Hopf technique. This leads to an expression for the minimum free energy, previously derived by means of a variational technique in the frequency domain. A new variational method is developed in both the time and frequency domains. In the former case, this approach yields integral equations for both the minimum and maximum free energies associated with a given viscoelastic state. In the latter case, explicit forms of a family of free energies, associated with a given state of a discrete spectrum viscoelastic material, are derived. This includes both maximum and minimum free energies.

Nonlinear ligament viscoelasticity.

Ligaments display time-dependent behavior, characteristic of a viscoelastic solid, and are nonlinear in their stress-strain response. Recent experiments (25) reveal that stress relaxation proceeds more rapidly than creep in medial collateral ligaments, a fact not explained by linear viscoelastic theory but shown by Lakes and Vanderby (17) to be consistent with nonlinear theory. This study tests the following hypothesis: nonlinear viscoelasticity of ligament requires a description more general than the separable quasilinear viscoelasticity (QLV) formulation commonly used. The experimental test for this hypothesis involves performing both creep and relaxation studies at various loads and deformations below the damage threshold. Freshly harvested, rat medial collateral ligaments (MCLs) were used as a model. Results consistently show a nonlinear behavior in which the rate of creep is dependent upon stress level and the rate of relaxation is dependent upon strain level. Furthermore, relaxation proceeds faster than creep; consistent with the experimental observations of Thornton et al. (25) The above results from rat MCLs are not consistent with a separable QLV theory. Inclusion of these nonlinearities would require a more general formulation.

Material Design in Physical Modelling Sound Synthesis

This paper deals with designing material parameters for physical models. Using the linear theory of viscoelasticity, we show that the sound signature of a material is a frequency-damping characteristic function. Some qualitative features of the sound signature of linear hypothetical materials are stated. Physical modelling is then organised in two independent steps: shape modelling within the spring-mass model paradigm and material modelling according to material sound signatures. Any conservative model can then be covered with a viscoelastic dress in order to represent an arbitrary shaped material.

Measurement of stress and strain during tensile testing of gellan gum gels: effect of deformation speed

Tensile tests were carried out on gellan gum gels of concentration 2% in order to observe the relation between deformation speed and deformation properties such as tensile modulus and rupture strain. The primary objective of this experiment is to show the validity of the measurement for the actual strain and strain rate during tensile tests. The tensile modulus decreased with decreasing deformation speed, showing the viscoelastic character of the gel, while the stress–strain curve was linear. The results were interpreted in terms of linear viscoelasticity theory. The instantaneous modulus was calculated and compared with the storage modulus found in the literature. A guide value of the relaxation time was also calculated and shown to be much larger than the observation time.

Prediction of the 3-D effective damping matrix and energy dissipation of viscoelastic fiber composites

A new method for calculating the viscoelastic damping matrix of fiber composites is presented in this paper. Firstly, on the lamina scale, the energy dissipation formula is obtained by using a damping matrix and the theory of linear anisotropic viscoelasticity. Then on the laminate scale, a 3-D effective damping (loss) matrix is obtained by using the energy method. In terms of the 3-D effective damping matrix, the laminate energy loss and specific damping capacity (SDC) can be calculated by using the effective stresses and strains. The key to the application of the method lies in the derivation of the effective damping matrix from the constituent lamina damping coefficients, stiffness coefficients and volume fraction and the laminate effective stiffness and compliance coefficients. Based on the effective damping matrix method, the energy loss of viscoelastic fiber composites under arbitrary loading conditions can be calculated. This method has the advantages of generality, simplicity and directness. Several numerical examples are given to illustrate the accuracy of the model.

Evolution of Motion of a Binary Planet

A model of a binary planet, consisting of a material point of small mass and a deformable viscoelastic sphere, is suggested. The center of mass of the binary planet moves in the gravitational field of a central body in the plane, which contains planets forming the binary planet. A deformable spherical planet rotates around the axis orthogonal to the plane of planetary motion. Planet deformations are described by the linear theory of viscoelasticity. It is shown that with an appropriate approximation of the gravitational potential, there is a class of quasicircular orbits, when the eccentricities of an orbit of the center of mass of a binary planet and an orbit, describing mutual planet motion, are equal to zero. The further evolution of motion is investigated in this class of orbits with the use of the canonical Poincare–Andoyer variables. Corresponding averaged equations are found, and phase pictures are constructed; the stability of stationary solutions is investigated on the basis of equations in variations. For the Solar system planets with their satellites, forming binary planets, the application of the presented model allows us to conclude that satellites sooner or later will fall on the corresponding planets.

Equibiaxial extensional flow of polymer melts via lubricated squeezing flow. I. Experimental analysis

The technique of lubricated squeezing flow is evaluated using experiments in both constant strain rate and constant stress flows of two low-density polyethylene melts. Experimental parameters that include the lubricant viscosity and sample aspect ratio are systematically varied to examine their effect on the viability of the technique in the linear viscoelastic regime by comparing measured quantities to those predicted by finite linear viscoelastic theory. An evaluation is also made by comparing viscosities measured using the lubricated squeezing flow technique and Meissner's rotating clamp (MAD) rheometer in the non-linear regime. In both cases, deviations between the expected and measured viscosities were observed, indicating that the technique is not applicable to large strains for constant strain rate and constant stress flows. It is suggested that this limit is the result of lubricant thinning.

Description of a set of solutions of a dissipation inequality for third-order relaxation systems

For the class of linear dynamic systems indicated in the title, a variety of the third degree is determined and it is proved that the set of solutions of a dissipation inequality is characterized by a limited convex component of this variety. Singular varieties are determined, and the relationship between them and the maximum and minimum elements of the set of solutions for the dissipation inequality is established. The results obtained can find application in the theory of linear viscoelasticity and the theory of generalized thermodynamic systems with memory.

Calculation of viscoelastic properties of edible films: application of three models

The viscoelastic properties of edible films can provide information at the structural level of the biopolymers used. The objective of this work was to test three simple models of linear viscoelastic theory (Maxwell, Generalized Maxwell with two units in parallel, and Burgers) using the results of stress relaxation tests in edible films of myofibrillar proteins of Nile Tilapia. The films were elaborated according to a casting technique and pre-conditioned at 58% relative humidity and 22ºC for 4 days. The testing sample (15mm x 118mm) was submitted to tests of stress relaxation in an equipment of physical measurements, TA.XT2i. The deformation, imposed to the sample, was 1%, guaranteeing the permanency in the domain of the linear viscoelasticity. The models were fitted to experimental data (stress x time) by nonlinear regression. The Generalized Maxwell model with two units in parallel and the Burgers model represented the relaxation curves of stress satisfactorily. The viscoelastic properties varied in a way that they were less dependent on the thickness of the films.

Theory of linear viscoelasticity of cholesteric liquid crystals

The theory of linear viscoelasticity of rod-like cholesteric liquid crystals subjected to small-amplitude oscillatory shear flow is formulated and applied to the cholesteric helix along the flow, velocity gradient, and vorticity directions. Expressions for the zero- and infinite-frequency viscosities are derived and their ordering is predicted. Based on the classical ordering of the Miesowicz shear viscosities and anisotropies of torque coefficients, it is found that the largest (smallest) zero-frequency viscosity obtains with the helix along the flow (gradient) direction. In addition, the difference between the zero- and infinite-frequency viscosities is found to be sensitive to the helix orientation, such that it is largest (smallest) when the helix is along the flow (gradient) direction. The complex viscosity corresponds to a viscoelastic material with a single relaxation time. The relaxation time depends on the Frank elastic constants involved in the deformation, such that when the helix is along the vorticity it is twist dependent, and splay–bend otherwise. The strength of the viscoelasticity is largest (smallest) when the helix is along the flow (gradient) direction. The hard-rod theory of Doi is used to confirm the predicted dependence of the strength of the viscoelastic response on the cholesteric helix orientation.

Dynamic mechanical testing of the creep and relaxation properties of polyvinylidene fluoride

The paper concerns material testing and characterizations of the time-dependent properties of polyvinylidene fluoride (PVDF), a thin film piezoelectric polymer. The study is based on the assumptions of linear viscoelastic theory of hereditary type and time-temperature equivalence principle. A developed experimental program involves quasi-static creep tests and dynamic mechanical tests. The results of this study validate the assumptions of linear viscoelasticity in application to PVDF thin films and provide consistent empirical database regarding the creep and relaxation properties of PVDF in the material directions parallel and perpendicular to the orientation of the aligned molecular chains of the polymer.

The Method of Asymptotic Expansion for Plate Problem in the Linear Theory of Viscoelasticity

AbstractThe method of asymptotic expansion originally developed for elasticity problems is generalized to linear stic problems. Three‐dimensional, unsteady problems corresponding to a class of viscoelastic plates are considy employing the Laplace transform, the original problem is converted to an equivalent elastic problem, where mptotic expansion method is used. An application of this method to the Maxwell model is also presented.

Crack resistance of a viscoelastic orthotropic body under plane strain

A problem of the linear theory of viscoelasticity for an orthotropic body with a crack under plane strain is solved. The expression for the viscoelastic opening of the crack is obtained based on the Volterra principle and the method of continued fractions. The safe load is evaluated. The effect of the degree of material anisotropy on the safe-load domain is shown by examples of specific materials.

Linear Viscoelasticity of Dilute Polymer Solutions in a Viscoelastic Solvent

A theory of linear viscoelasticity is developed for dilute polymer solutions in viscoelastic solvents. The non-Markovian dynamics of a macromolecule with hydrodynamic interaction is analyzed on the basis of a generalized Langevin equation with relaxed friction. The relaxation spectrum (relaxation times and elastic moduli) of the solutions is calculated for a simple viscoelastic model of solvent with a single relaxation time. The theory predicts that at high frequencies the dynamic viscosity of such solutions is less than the zero-shear-rate solvent viscosity.

Incremental analysis of time-dependent effects in composite structures

An efficient numerical method is presented, to account for time effects in composite structures. The method is based on the theory of linear viscoelasticity and on the finite element method. The global behaviour of structural elements made of several viscoelastic materials (composite beams, layered plates or shells) is expressed in terms of the generalized variables of the structure mechanics. The incremental nature of the formulation results from the choice of a Dirichlet’s series to express the constitutive law of the materials. A specific algorithm which allows for a process of construction by phases is implemented in a finite element program. The use of the method is illustrated through two computation examples: a prestressed concrete beam previously tested over a 5-year period of creeping, and an existing cooling tower which is a thin reinforced concrete shell. The method provides valuable information about time effects that may be used either while designing new structures, or for the diagnosis of existing structures.

Rheology of acrylonitrile–butadiene–styrene polymer melts and viscoelastic constitutive models

Experimental shear and tensile stress growth coefficients for startup of steady shear and uniaxial elongation for a commercial grade acrylonitrile–butadiene–styrene (ABS) polymer melt are presented. One differential [Phan-Thien and Tanner (PTT)] and two Kaye–Bernstein–Kearsley–Zapas (K-BKZ) type single integral [Wagner and Papanastasiou–Scriven–Macosko (PSM)] nonlinear viscoelastic constitutive models were fit to shear and extensional experimental data. A comparison of the fit quality was performed, and the PTT model was found to result in the best overall quality fit, although deviations from linear viscoelastic behavior at small extensional strains could not be adequately described by the linear viscoelasticity theory without a time correction. Experimental data for stress relaxation after cessation of both steady shear and uniaxial elongation were compared with three constitutive model predictions in order to evaluate their anticipated accuracy in the prediction of residual stresses. The PTT model provided better predictions of ABS relaxational behavior than K-BKZ models except for shear stress relaxation in the nonlinear regime. Qualitative differences between experimental data and model predictions were found to be present for all models tested at large shear and extensional strains.