Large amplitude oscillatory shear rheology of three different shear-thickening particle dispersions

We present a large amplitude oscillatory shear rheology (LAOS) investigation of three different shear-thickening particle dispersions - fumed silica in polyethylene oxide (FLOC), fumed silica in polypropylene glycol (HydroC), and cornstarch in water (JAM). These systems shear-thicken by three different mechanisms - shear-induced formation of particle clusters flocculated by polymer bridging, hydrocluster formation, and jamming. The viscoelastic non-linearities of the three fluids were studied as a function of strain and strain-rate space through the use of Lissajous-Bowditch curves and local nonlinear viscoelastic moduli of an oscillatory shear cycle. The nonlinear behaviors of the three fluids were compared and contrasted to understand the nonlinear shear-thickening mechanism of each. Both HydroC and JAM dispersions were found to exhibit strong strain stiffening of the elastic moduli and strain thickening of the loss moduli behavior associated with possible hydrocluster formation and particle jamming. However, the FLOC dispersion, in contrast, showed strong strain softening and strain thinning behavior at large strain amplitudes associated with yielding of the microstructure. The expected thickening of the loss modulus of FLOC in LAOS with increasing strain was not observed even though viscosity of FLOC was found to shear-thicken in steady-shear measurements. This disagreement is likely due to very large strain amplitudes required for shear-thickening to occur by shear-induced polymer bridging mechanism. The hypothesis was confirmed through stress growth experiments. Conversely, the HydroC and JAM dispersions required relatively small applied strains for shear-thickening to occur by hydrocluster and jamming mechanism. The comparison of local intra-cycle nonlinearity through Lissajous-Bowditch plots and nonlinear viscoelastic parameters indicated that the elastic nonlinearities of all three systems are primarily driven by a strong dependence on the magnitude of the applied strain-rates within an oscillatory cycle rather than the amplitude of the applied strain. A close inspection of the LAOS data reveals strong differences in the viscoelastic nonlinearities of these three different shear-thickening dispersions which can be used to create a nonlinear rheological fingerprint for each and offers valuable new insights into the nonlinear dynamics associated with each of the shear-thickening mechanisms.     

TTS in LAOS: validation of time-temperature superposition under large amplitude oscillatory shear

The validation of time-temperature superposition of non-linear parameters obtained from large amplitude oscillatory shear is investigated for a model viscoelastic fluid. Oscillatory time sweeps were performed on a 11?wt.% solution of high molecular weight polyisobutylene in pristane as a function of temperature and frequency and for a broad range of strain amplitudes varying from the linear to the highly non-linear regime. Lissajous curves show that this reference material displays strong non-linear behaviour when the strain amplitude is exceeding a critical value. Elastic and viscous Chebyshev coefficients and alternative non-linear parameters were obtained based on the framework of Ewoldt et al. (J Rheol 52(6):1427?C1458, 2008) as a function of temperature, frequency and strain amplitude. For each strain amplitude, temperature shift factors a _T (T) were calculated for the first order elastic and viscous Chebyshev coefficients simultaneously, so that master curves at a certain reference temperature T _ref were obtained. It is shown that the expected independency of these shift factors on strain amplitude holds even in the non-linear regime. The shift factors a _T (T) can be used to also superpose the higher order elastic and viscous Chebyshev coefficients and the alternative moduli and viscosities onto master curves. It was shown that the Rutgers-Delaware rule also holds for a viscoelastic solution at large strain amplitudes.     

On the blast resistance of laminated glass

Blast resistant glazing systems typically use laminated glass to reduce the risk of flying glass debris in the event of an explosion. Laminated glass has one or more bonded polymer interlayers to retain glass fragments upon fracture. With good design, the flexibility of the interlayer and the adhesion between layers enable laminated glass to continue to resist blast after the glass layers fracture. This gives protection from significantly higher blast loads when compared to a monolithic pane. Full-scale open-air blast tests were performed on laminated glass containing a polyvinyl butyral (PVB) interlayer. Test windows of size 1.5 m × 1.2 m were secured to robust frames using structural silicone sealant. Blast loads were produced using charge masses of 15 kg and 30 kg (TNT equivalent) at distances of 10–16 m. Deflection and shape measurements of deforming laminated glass were obtained using high-speed digital image correlation. Measurements of loading at the joint, between the laminated glass and the frame, were obtained using strain gauges. The main failure mechanisms observed were the cohesive failure of the bonded silicone joint and delamination between the glass and interlayer at the pane edge. A new finite element model of laminated glass is developed and calibrated using laboratory based tests. Predictions from this model are compared against the experimental results.     

Scattering of acoustic waves by elastic and viscoelastic obstacles immersed in a fluid

A scattering or T-matrix approach is presented for studying the scattering of acoustic waves by elastic and viscoelastic obstacles immersed in a fluid. A Kelvin-Voigt model is used to obtain the complex elastic moduli of the viscoelastic solid. The T-matris formulation is somewhat complicated because the wave equations and fields are quite different in the solid and fluid regions and are coupled by continuity conditions at the interface. We have obtained fairly extensive numerical results for prolate and oblate spheroids for a variety of scattering geometries. The backscattering, bistatic, absorption and extinction cross-section are presented as a function of the frequency of the incident wave.     

A linear viscoelastic model for solid polyethylene

Stress relaxation tests have been carried out on a blue, pipe grade PE 80 medium density polyethylene (BP Chemicals), to provide thermo-viscoelastic rheology for use in calculating thermal stresses in pipe production. Stresses up to 4 MPa were used, with strains up to about 2%, in tests at temperatures from 23° to 90°C. Within this range a linear viscoelastic model was applicable, provided the initial ramp strain rate was less than 7×10^–5 s^–1. The stress relaxation data was fitted directly by a model incorporating an elastic response to volumetric strains, and a generalised linear solid model, consisting of two Maxwell elements and a purely elastic element in parallel, for deviatoric strains. Arrhenius type temperature dependence of relaxation times and shear moduli is found, and within experimental accuracy the temperature dependence of all these model parameters is the same. As a consequence, and provided that the duration of the strain ramp is sufficiently short relative to relaxation times, the model leads to time-temperature superposition of the relaxation moduli, using the same shift factor on both the response magnitude and time axes.     

Rheological probing of structure and pigment-resin interactions in magnetic paints

Dynamic rheological measurements are used to probe the structure of a model magnetic paint which contains iron oxide magnetic pigment particles, a PVC wetting resin and an organic solvent. As the wetting resin concentration, , increases, the viscoelastic moduli, and , go through a minimum at which corresponds to the saturated adsorption of the resin. At the paint is highly elastic, thixotropic and brittle ( sharply drops, G′′ goes through a strong maximum at high strain), which indicates the existence of a strong flocculated network. At the moduli increase due to accumulation of free resin. However, at high the paint becomes less elastic, less brittle and less thixotropic, which indicates that the structure formation in this case is dominated by small aggregates rather than a network. Received: 29 July 1996 Accepted: 29 September 1997     

Influence of sliding interfaces on the response of a layered viscoelastic medium under a moving load

This article presents a method to compute the response of a viscoelastic layered half-space to a moving load when interlayer slip is considered. The Navier equations of equilibrium are solved for each layer in the frequency domain. The solution in the spatial coordinate system is subsequently obtained by means of Fast Fourier Transform and quadrature rules applied to integrable singularities. Following the global solution technique, the developed method compiles all the interface and the boundary conditions within a global matrix and it solves a unique linear system per couple of wave numbers. This method proves to be effective and is validated in an elastic case by comparison with the ALIZE-LCPC software that implements the Burmister axisymmetric solution. The influence of the interface sliding condition on the response of a layered viscoelastic medium is studied through an application to pavement structures. In this application, the effect of the load speed on vertical and horizontal profiles of the longitudinal strain and the normal stress is analyzed. It is shown, inter alia, that the maximum extension in the medium is not systematically observed at the location of an interface and that, as expected, low speeds and interlayer slip are more damaging to the structure when either a strain or a stress criterion is considered.     

Non-linear viscoelastic behavior of fumed silica suspensions

Suspensions of fumed silica exhibit a wide range of rheological properties depending on the nature and magnitude of the interparticle forces. In a non-polar fluid, the particles interact through hydrogen bonding and can form a three-dimensional network. The microstructure formation is responsible for the non-linear viscoelastic behavior of fumed silica suspensions, even at very small strain. These non-linear rheological properties have been studied in small amplitude oscillatory experiments as a function of particle size, surface treatment of particles, suspending medium polarity and solids concentration. The non-linear viscoelastic behavior is characterized by a non-sinusoidal waveform of the signal response. For suspensions in a non-polar fluid, both the elastic and the loss moduli are shown to be sensitive to the strain amplitude: the elastic modulus is decreasing with increasing strain whereas the loss moduli is initially increasing with strain. We have chosen to examine the dissipated energy which is clearly related to the breakup of the suspension structure. A comparison of model predictions and the experimental data shows the limitations of these models, recently proposed in the literature to describe the behavior of colloidal suspensions. Received: 9 March 1998 Accepted: 17 November 1998     

Stresses in rotating heterogeneous viscoelastic composite cylinders with variable thickness

An analytical solution is presented for the rotation problem of a two-layer composite elastic cylinder under a plane strain assumption. The external cylinder has variable-thickness formulation, and is made of a heterogeneous orthotropic material. It contains a fiber-reinforced viscoelastic homogeneous isotropic solid core of uniform thickness. The thickness and elastic properties of the external cylinder are taken as power functions of the radial direction. By the boundary and continuity conditions, the radial displacement and stresses for the rotating composite cylinder are determined. The effective moduli and Illyushin’s approximation methods are used to obtain the viscoelastic solution to the problem. The effects of heterogeneity, thickness variation, constitutive, time parameters on the radial displacement, and stresses are investigated.     

Generalized Maxwell model for the viscoelastic behavior of a soda-lime-silica glass under low frequency shear loading

The linear viscoelastic behavior of a soda-lime-silica glass under low frequency shear loading is investigated in the glass transition range. Using the time-temperature superposition technique, the master curves of the shear dynamic relaxation moduli are obtained at a reference temperature of 566°C. A method to determine the viscoelastic constants from dynamic relaxation moduli is proposed. However, some viscoelastic constants cannot be directly measured from the experimental curves and others cannot be precisely obtained due to non-linearity effects at very low frequencies. The generalized Maxwell model is investigated from the experimental dynamic moduli without fixing the viscoelastic constants. A set of parameters is shown to be in good agreement with the experimental dynamic relaxation moduli, but does not give the correct values of the viscoelastic constants of the investigated glass. The soda-lime-silica glass exhibits a non-linear viscoelastic behavior at very low stress level which is usually observed for organic glasses. This non-linear behavior is questioned.     

A nonhomogeneous nonlocal elasticity model

Nonlocal elasticity with nonhomogeneous elastic moduli and internal length is addressed within a thermodynamic framework suitable to cope with continuum nonlocality. The Clausius–Duhem inequality, enriched by the energy residual, is used to derive the state equations and all other thermodynamic restrictions upon the constitutive equations. A phenomenological nonhomogeneous nonlocal (strain difference-dependent) elasticity model is proposed, in which the stress is the sum of two contributions, local and nonlocal, respectively governed by the standard elastic moduli tensor and the (symmetric positive-definite) nonlocal stiffness tensor. The inhomogeneities of the elastic moduli and of the internal length are conjectured to be each the cause of additional attenuation effects upon the long distance particle interactions. The increased attenuation effects are accounted for by means of the standard attenuation function, but with the standard spatial distance replaced by a suitably larger equivalent distance, and with the spatially variable internal length replaced by the largest value within the domain. Formulae for the computation of the equivalent distance are heuristically suggested and illustrated with numerical examples. The solution uniqueness of the continuum boundary-value problem is proven and the related total potential energy principle given and employed for possible nonlocal-FEM discretizations. A bar in tension is considered for a few numerical applications, showing perfect numerical stability, provided the free energy potential is positive definite.     

Aging, rejuvenation, and thixotropy in yielding magnetorheological fluids

The yielding behavior of dilute magnetorheological (MR) fluids has been investigated using creep–recovery tests. At very low stress levels, MR fluids behave in the linear viscoelastic regime as demonstrated by the fact that the instantaneous strain equals the instantaneous (elastic) recovery. In this region, gap-spanning field-induced structures support the stress levels applied. Upon increasing the stress value, the MR fluid evolves towards a nonlinear viscoelastic response. Here, the retarded elastic and viscous strain decrease, and the plastic contribution to the instantaneous strain grows probably due to the appearance of unattached field-induced structures. A larger stress value results in a viscoplastic solid behavior with negligible retarded and viscous strain and a fully plastic instantaneous strain. Finally, a plastic fluid behavior is found when the stress value is larger than the so-called yield stress. MR fluids exhibit an intermediate behavior between non-thixotropic (simple) and highly thixotropic model yield stress fluids.     

In search of physical meaning: defining transient parameters for nonlinear viscoelasticity

A complete set of model-independent viscoelastic functions for understanding responses to transient nonlinear rheological tests is presented, using large-amplitude oscillatory shear strain as a model nonlinear protocol. The derivation makes no assumptions about symmetries, and is therefore applicable to the responses to any input, allowing researchers to unambiguously define time-dependent moduli, viscosities, compliances, fluidities, and normal stress coefficients. A legend for interpreting the dynamic trajectories in modulus space is provided, along with explicit definitions of the rates at which the moduli change. These provide a quantitative mechanism to identify when, and by how much, a material response stiffens, softens, thickens, or thins while being deformed. In addition to providing analytical expressions for the moduli, the derivation requires the definition of a conceptually new term. This means there exist three, not two, time-dependent nonlinear viscoelastic functions by which any response can be fully described. The third function accounts for nonlinear properties such as yield stresses and the shifting of the strain equilibrium. This complete analysis scheme is unique in making a distinction between the strains in the lab and material frames. The quantitative sequence of physical process analysis, which is fully developed in this work, allows for comprehensive physical interpretations of responses to transient deformations of any kind to be made, including the steady alternance responses to large-amplitude oscillatory shear (LAOS), time-dependent oscillatory shear startup responses, and thixotropic and anti-thixotropic responses.     

Exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder

We find closed-form solutions for axisymmetric plane strain deformations of a functionally graded circular cylinder comprised of an isotropic and incompressible second-order elastic material with moduli varying only in the radial direction. Cylinder's inner and outer surfaces are loaded by hydrostatic pressures. These solutions are specialized to cases where only one of the two surfaces is loaded. It is found that for a linear through-the-thickness variation of the elastic moduli, the hoop stress for the first-order solution (or in a cylinder comprised of a linear elastic material) is a constant but that for the second-order solution varies through the thickness. The radial displacement, the radial stress and the hoop stress do not depend upon the second-order elastic constant but the hydrostatic pressure and hence the axial stress depends upon it. When the two elastic moduli vary as the radius raised to the power two or four, the radial and the hoop stresses in an infinite space with a pressurized cylindrical cavity equal the pressure in the cavity. For an affine variation of the elastic moduli, the hoop stress in an internally loaded cylinder made of a linear elastic isotropic and incompressible material at the point is the same as that in a homogeneous cylinder. Here R_in and R_ou equal, respectively, the inner and the outer radius of the undeformed cylinder and R the radial coordinate of a point in the unstressed reference configuration.     

An Eulerian formulation for large deformations of anisotropic elastic and viscoelastic solids and viscous fluids

An Eulerian formulation has been developed for the constitutive response of a group of materials that includes anisotropic elastic and viscoelastic solids and viscous fluids. The material is considered to be a composite of an elastic solid and a viscous fluid. Evolution equations are proposed for a triad of vectors m _i that represent the stretches and orientations of material line elements in the solid component. Evolution equations for an orthonormal triad of vectors s _i are also proposed to characterize anisotropy of the fluid component. In particular, for an elastic solid it is shown that the material response is totally characterized by the functional form of the strain energy and by the current values of m _i , which are measurable in the current state of the material. Moreover, it is shown that the proposed Eulerian formulation removes unphysical arbitrariness of the choice of the reference configuration in the standard formulation of constitutive equations for anisotropic elastic solids.     

Rate form of the Eshelby and Hill tensors

Expressions are derived for the rates of change of the S and P tensors for transformed homogeneous inclusions in an anisotropic comparison medium undergoing prescribed changes of its elastic moduli. General results are obtained for ellipsoids and then reduced to yield explicit expressions in terms of the Stroh eigenvalues for cylindrical and disk-shaped inclusions in anisotropic solids and for spherical inclusions in isotropic solids. Applications are illustrated by solving the rate problem for an inhomogeneity in a large volume of a comparison medium, which is shown to be readily adaptable to standard averaging techniques for predictions of rates of change of overall moduli of composite materials experiencing evolution of phase moduli.     

Homogénéisation d'un milieu élastique fortement hétérogèneHomogenization of a strongly heteregeneous elastic medium

We consider the homogenization of an elastostatic problem in a strongly heterogeneous periodic medium made of two connected components having comparable tensors of elastic moduli, separated by a third medium (soft layer), the thickness of which is of the same order ε than the basic periodicity cell, and such that its elastic moduli tensor becomes infinitely small following a rate ε^r, r>0. If r?2, we identify the homogenized problem. Otherwise, we have to assume moreover that there are no volume forces in the third medium. To cite this article: M. Mabrouk, A. Boughammoura, C. R. Mecanique 330 (2002) 543–548.     

Bending of a fiber-reinforced viscoelastic composite plate resting on elastic foundations

Composite structures on an elastic foundation are being widely used in engineering applications. Bending response of inhomogeneous viscoelastic plate as a composite structure on a two-parameter (Pasternak’s type) elastic foundation is investigated. The formulations are based on sinusoidal shear deformation plate theory. Trigonometric terms are used in the present theory for the displacements in addition to the initial terms of a power series through the thickness. The transverse shear correction factors are not needed because a correct representation of the transverse shear strain is given. The interaction between the plate and the foundation is included in the formulation with a two-parameter Pasternak’s model. The effective moduli and Illyushin’s approximation methods are used to derive the viscoelastic solution. The effects played by foundation stiffness, plate aspect ratio, and other parameters are presented.     

Integral formulation and self-consistent modelling of elastoviscoplastic behavior of heterogeneous materials

Summary Based on projection operators, an integral formulation is proposed for elastoviscoplastic heterogeneous materials. The problem requires a complete mechanical formulation, including the static equilibrium property concerning the known field σ, in addition to the classical field equations concerning the unknown fields ɛ˙ and σ˙. The formulation leads to an integral equation, in which elasticity and viscoplasticity effects interact through an homogeneous elastoviscoplastic medium with elastic moduli C and viscoplastic moduli B. To approximate the integral equation, the self-consistent scheme is followed. In order to obtain consistent approximation conditions, we introduce fluctuations of elastic and viscoplastic strain rate fields with respect to known kinematically compatible fields. It results in a strain rate concentration relation connecting the strain rate at each point to the macroscopic loading conditions and the local stress field. The results are presented and compared with other models and with experimental data in the case of a two-phase material. Received 26 August 1997; accepted for publication 2 July 1998     

Static and free vibration analysis of laminated glass beam on viscoelastic supports

Static behavior and free vibration analysis of laminated glass beam on viscoelastic supports are performed. For the static case, an analytical way is developed for analyzing and optimization of laminated glass beam with general restraints at the boundaries. In the case of free linear vibrations, the modal properties of the glass are determined using a finite element method which is a powerful tool in the design of support damping treatment of a sandwich glass for passive vibration control.