Linear Viscoelastic Theory Research Articles

Self-Healing Fish Gelatin/Sodium Montmorillonite Biohybrid Coacervates: Structural and Rheological Characterization

Complex coacervation driven by associative electrostatic interactions was studied in mixtures of exfoliated sodium-montmorillonite (Na(+)-MMT) nanoplatelets and fish gelatin, at a specific mixing ratio and room temperature. Structural and viscoelastic properties of the coacervate phase were investigated as a function of pH by means of different complementary techniques. Independent of the technique used, the results consistently showed that there is an optimum pH value at which the coacervate phase shows the tightest structure with highest elasticity. The solid-like coacervates showed an obvious shear-thinning behavior and network fracture but immediately recovered back into their original elastic character upon removal of the shear strain. The nonlinear mechanical response characterized by single step stress relaxation experiments revealed the same trend for the yield stress and isochronal shear modulus of the coacervates as a function of pH with a maximum at pH 3.0 and lower values at 2.5 and 3.5 pHs, followed by a very sharp drop at pH 4.0. Finally, small-angle X-ray scattering (SAXS) data confirmed that at pHs lower than 4.0 the coacervate phases were dense and structured with a characteristic length scale (ξ(SAXS)) of ~7-9 nm. Comparing the ξ(SAXS) with rheological characteristic length (ξ(rheol)) estimated from low-frequency linear viscoelastic data and network theory, it was concluded that both the strength of the electrostatic interactions and the conformation of the gelatin chains before and during of the coacervation process are responsible for the structure and rigidity of the coacervates.

On the viscoelastic characterization of the Jeffreys–Lomnitz law of creep

In 1958 Jeffreys proposed a power law of creep, generalizing the logarithmic law earlier introduced by Lomnitz, to broaden the geophysical applications to fluid-like materials including igneous rocks. This generalized law, however, can be applied also to solid-like viscoelastic materials. We revisit the Jeffreys-Lomnitz law of creep by allowing its power law exponent $\alpha$, usually limited to the range [0,1] to all negative values. This is consistent with the linear theory of viscoelasticity because the creep function still remains a Bernstein function, that is positive with a completely monotonic derivative, with a related spectrum of retardation times. The complete range $\alpha \le 1$ yields a continuous transition from a Hooke elastic solid with no creep ($\alpha \to -\infty$) to a Maxwell fluid with linear creep ($\alpha=1$) passing through the Lomnitz viscoelastic body with logarithmic creep ($\alpha=0$), which separates solid-like from fluid-like behaviors. Furthermore, we numerically compute the relaxation modulus and provide the analytical expression of the spectrum of retardation times corresponding to the Jeffreys-Lomnitz creep law extended to all $\alpha \le 1$.

Fractional viscoelastic models: master curve construction, interconversion, and numerical approximation

We use fractional viscoelastic models that result from the application of fractional calculus to the linear viscoelastic theory to characterize thermorheologically simple linear viscoelastic materials. Model parameters are obtained through an optimization procedure that simultaneously determines the time–temperature shift factors. We present analytical interconversion based on the fractional viscoelastic model between the main viscoelastic functions (relaxation modulus, creep compliance, storage modulus, and loss modulus) and the analytical forms of the relaxation and retardation spectra. We show that the fractional viscoelastic model can be approximated by a Prony series to any desired level of accuracy. This property allows the efficient determination of the fractional viscoelastic model response to any loading history using the well-known recursive relationships of Prony series models.

Applying of Integral Equation of Volterra for Determining the Section Forces in Composite Beam, Regarding Shrinkage of Concrete

During time, the equilibrium state of a steel-concrete mixed beam progresses under the concrete shrinkage. In this context, a contribution to the search of the time-dependant behaviour of mixed beams, including the shrinkage phenomenon of the concrete slab, is dealt with in this paper. The influence of the concrete shrinkage, taking into account the slab reinforcement contribution, on the strength of the mixed beams is taken into account using the linear concrete viscoelasticity theory. With the help of this theory we obtain an analytic model, based on two linear differential equations that result in the over-time-behaviour of the steel-concrete mixed beams. From this model we can determine, at any time t, the values of the normal force and bending moment due to the concrete shrinkage. So it allows us to give an account of the stress state evolution developed in the cross sections of the steel-concrete mixed beams. The results obtained from the application of the suggested model are completely comparable with the results derived by the method proposed by EUROCODE 4

Characterizing Permanent Deformation and Fracture of Asphalt Mixtures by Using Compressive Dynamic Modulus Tests

Permanent deformation and fracture may develop simultaneously when an asphalt mixture is subjected to a compressive load. The objective of this research is to separate viscoplasticity and viscofracture from viscoelasticity so that the permanent deformation and fracture of the asphalt mixtures can be individually and accurately characterized without the influence of viscoelasticity. The undamaged properties of 16 asphalt mixtures that have two binder types, two air void contents, and two aging conditions are first obtained by conducting nondestructive creep tests and nondestructive dynamic modulus tests. Testing results are analyzed by using the linear viscoelastic theory in which the creep compliance and the relaxation modulus are modeled by the Prony model. The dynamic modulus and phase angle of the undamaged asphalt mixtures remained constant with the load cycles. The undamaged asphalt mixtures are then used to perform the destructive dynamic modulus tests in which the dynamic modulus and phase angle of the damaged asphalt mixtures vary with load cycles. This indicates plastic evolution and crack propagation. The growth of cracks is signaled principally by the increase of the phase angle, which occurs only in the tertiary stage. The measured total strain is successfully decomposed into elastic strain, viscoelastic strain, plastic strain, viscoplastic strain, and viscofracture strain by employing the pseudostrain concept and the extended elastic-viscoelastic correspondence principle. The separated viscoplastic strain uses a predictive model to characterize the permanent deformation. The separated viscofracture strain uses a fracture strain model to characterize the fracture of the asphalt mixtures in which the flow number is determined and a crack speed index is proposed. Comparisons of the 16 samples show that aged asphalt mixtures with a low air void content have a better performance, resisting permanent deformation and fracture.

Viscoelastic Properties, Ionic Conductivity, and Materials Design Considerations for Poly(styrene-b-ethylene oxide-b-styrene)-Based Ion Gel Electrolytes

The viscoelastic properties and ionic conductivity of ion gels based on the self-assembly of a poly(styrene-b-ethylene oxide-b-styrene) (SOS) triblock copolymer (Mn,S = 3 kDa, Mn,O = 35 kDa) in the ionic liquid 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)amide ([EMI][TFSA]) were investigated over the composition range of 10–50 wt % SOS and the temperature range of 25–160 °C. The poly(styrene) (PS) end-blocks associate into micelles, whereas the poly(ethylene oxide) (PEO) midblocks are well-solvated by this ionic liquid. The ion gel with 10 wt % SOS melts at 54 °C, with the longest relaxation time exhibiting a similar temperature dependence to that of the viscosity of bulk PS. However, the actual values of the gel relaxation time are more than 4 orders of magnitude larger than the relaxation time of bulk PS. This is attributed to the thermodynamic penalty of pulling PS end-blocks through the PEO/[EMI][TFSA] matrix. Ion gels with 20–50 wt % SOS do not melt and show two plateaus in the storage modulus over the temperature and frequency ranges measured. The one at higher frequencies is that of an entangled network of PEO strands with PS cross-links; the modulus displays a quadratic dependence on polymer weight fraction and agrees with the prediction of linear viscoelastic theory assuming half of the PEO chains are elastically effective. The frequency that separates the two plateaus, ωc, reflects the time scale of PS end-block pull-out. The other plateau at lower frequencies is that of a congested micelle solution with PS cores and PEO coronas, which has a power law dependence on domain spacing similar to diblock melts. The ionic conductivity of the ion gels is compared to PEO homopolymer solutions at similar polymer concentrations; the conductivity is reduced by a factor of 2.1 or less, decreases with increasing PS volume fraction, and follows predictions based on a simple obstruction model. Our collective results allow the formulation of basic design considerations for optimizing the mechanical properties, thermal stability, and ionic conductivity of these gels.

Microindentation into an epoxy composition to assess the influence of aging on mechanical properties

AbstractThis work aims to apply the microindentation technique for assessing the influence of two aging factors—long‐term weathering and long‐term laboratory aging—on the time‐dependent mechanical properties of an epoxy composition. The linear theory of viscoelasticity was taken as a background for the problem, and the time‐independent Poisson ratio value was assumed to simplify the assessment. Two nano/microindenters were used (Hysitron Triboscan and Nano XP Indenter) at two different laboratories. Four special time‐dependent loading histories were applied: indentation under a step load, indentation under a constant load rate, indentation with a fixed depth of penetration, and indentation under a constant rate of penetration. The short‐term histories of the viscoelastic compliance of a common epoxy composition, affected by 5‐year weathering or laboratory aging, measured using a microindentation technique were compared to the data derived from standard macro measurements. The findings suggest that a qualitative assessment of the influence of the investigated aging effects on mechanical properties can be handled using short‐term microindentation data, but the data has to be freed of possible attendant factors, especially of the influence of polishing procedures accompanying the microindentation techniques, before comparing it with standard measurement data. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2012

Dynamic, Regional Mechanical Properties of the Porcine Brain: Indentation in the Coronal Plane

Stress relaxation tests using a custom designed microindentation device were performed on ten anatomic regions of fresh porcine brain (postmortem time <3 h). Using linear viscoelastic theory, a Prony series representation was used to describe the shear relaxation modulus for each anatomic region tested. Prony series parameters fit to load data from indentations performed to ∼10% strain differed significantly by anatomic region. The gray and white matter of the cerebellum along with corpus callosum and brainstem were the softest regions measured. The cortex and hippocampal CA1/CA3 were found to be the stiffest. To examine the large strain behavior of the tissue, multistep indentations were performed in the corona radiata to strains of 10%, 20%, and 30%. Reduced relaxation functions were not significantly different for each step, suggesting that quasi-linear viscoelastic theory may be appropriate for representing the nonlinear behavior of this anatomic region of porcine brain tissue. These data, for the first time, describe the dynamic and short time scale behavior of multiple anatomic regions of the porcine brain which will be useful for understanding porcine brain injury biomechanics at a finer spatial resolution than previously possible.

Anisotropic Viscoelastic Properties of Undamaged Asphalt Mixtures

A test protocol and a data analysis method are developed in this paper on the basis of linear viscoelastic theory to characterize the anisotropic viscoelastic properties of undamaged asphalt mixtures. The test protocol includes three nondestructive tests: (1) uniaxial com- pressive creep test, (2) indirect tensile creep test, and (3) the uniaxial tensile creep test. All three tests are conducted on asphalt mixture specimens at three temperatures (10, 20, and 30°C) to determine the tensile and compressive properties at each temperature and then to construct the master curve of each property. The determined properties include magnitude and phase angle of the compressive complex modulus in the vertical direction, magnitude and phase angle of the tensile complex modulus, and the magnitude and phase angle of the compressive complex modulus in the horizontal plane. The test results indicate that all tested asphalt mixtures have significantly different tensile properties from compressive properties. The peak value of the master curve of the tensile complex modulus phase angle is within a range from 65 to 85°, whereas the peak value of the compressive moduli phase angle in both directions ranges from 35 to 55°. In addition, the undamaged asphalt mixtures exhibit distinctively anisotropic properties in compression. The magnitude of the compressive modulus in the vertical direction is approximately 1.2 to ∼2 times of the magnitude of the compressive modulus in the horizontal plane. Dynamic modulus tests are performed to verify the results of the proposed test protocol. The test results from the proposed test protocol match well with those from the dynamic tests. DOI: 10.1061/(ASCE)TE.1943-5436.0000302. © 2012 American Society of Civil Engineers. CE Database subject headings: Asphalts; Mixtures; Anisotropy; Poisson ratio; Viscoelasticity; Material properties. Author keywords: Asphalt mixtures; Anisotropy; Viscoelasticity; Complex modulus; Poisson's ratio; Master curve.

Study of woven fabric shear behaviour

The paper presents the shear frame test with a new method of gripping the fabric sample to describe the shear behaviour of fabrics used in the garment industry under high shear strain conditions (maximum shear angles of 20°). This shear frame test was also used to compare between hysteresis curves given by the Kawabata evaluation system (KES) test, the bias extension test and the shear frame test for 10 woven fabrics. The linear viscoelastic theory was used to analyse the shear relaxation behaviour of the woven fabrics. In addition, in this study, the kernel function of the generalised Maxwell model has been used to describe the viscoelastic properties of the fabrics.

A method for determination of time‐ and temperature‐dependences of stress threshold of linear–nonlinear viscoelastic transition: Energy‐Based Approach

AbstractA methodology for determination of time‐ and temperature‐dependences of stress threshold of linear–nonlinear viscoelastic transition is proposed and validated by example of uniaxial creep of epoxy resin. Energy approach is applied for characterization of the region of linear viscoelasticity (LVE) and the threshold of LVE is given in the stress–strain representation as the master curve independent of time and temperature. Time‐ and temperature‐dependences of the stress threshold are calculated by extending LVE theory and time–temperature superposition principles (TTSP) to the energy relations. Reasonable agreement between experimental data and calculations is obtained. It is shown that number of tests required for characterization of LVE region in a wide range of test time and temperatures can be considerably reduced by applying the proposed methodology. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2011

Analysis of creep behavior in thermoplastics based on visco-elastic theory

Plastics and fiber-reinforced plastics (FRP) are used in the aerospace industry because of their mechanical properties. However, despite their excellent high-temperature mechanical properties, plastics and FRP eventually deform visco-elastically at high temperatures. Most of the research has focused on the creep behavior of FRPs, but few studies have investigated the linear visco-elastic behavior. Linear visco-elastic behavior and non-linear visco-elastic behavior occur with physical aging in these plastics. In this study, the non-linear visco-elastic behavior of plastics and FRP was investigated based on the bending creep deformation of polycarbonate (PC) and polyoxymethylene (POM). Moreover, the effects of the fiber volume fraction on the creep characteristics were investigated using glass fiber-reinforced polycarbonate (GFRPC). The creep deformation was calculated using the linear visco-elastic theory based on these effects, and comparison between experimental and estimated data showed that the creep analysis sufficiently predicted the creep behavior.

Investigation of creep effects in strengthened RC structures through double lap shear testing

The creep effects in RC strengthened structures are studied through measurements in double lap shear tests and numerical results. The experimental results were obtained from an original test procedure and measurement device. Short-term tests were performed at different temperatures and results were checked by means of long-term tests using the WLF theory and the time–temperature principle. In the second part of the paper, a numerical model is proposed based on an incremental form of the linear viscoelasticity theory and the FE method. Numerical results were compared with test data and a parametric study was carried out. These results show that the proposed model allows a prediction of time-dependent response of the double lap shear test specimen and the stress redistribution occurring between the different materials, along the bonding joint and through the width of the specimen.

Uniform $l^1$ Behavior of a Time Discretization Method for a Volterra Integrodifferential Equation with Convex Kernel; Stability

We study stability of a numerical method in which the backward Euler method is combined with order one convolution quadrature for approximating the integral term of the linear Volterra integrodifferential equation $\mathbf{u}'(t)+\int_{0}^{t}\beta(t-s)\mathbf{Au}(s)\,ds=0$, $t\geq0$, $\mathbf{u}(0)=\mathbf{u}_{0}$, which arises in the theory of linear viscoelasticity. Here $\mathbf{A}$ is a positive self-adjoint densely defined linear operator in a real Hilbert space, and $\beta(t)$ is locally integrable, nonnegative, nonincreasing, convex, and $-\beta'(t)$ is convex. We establish stability of the method under these hypotheses on $\beta(t)$. Thus, the method is stable for a wider class of kernel functions $\beta(t)$ than was previously known. We also extend the class of operators $\mathbf{A}$ for which the method is stable.

Transient and steady-state nanoindentation creep of polymeric materials

The transient and steady-state nanoindentation creep of polymeric materials was investigated. The creep model is used to explain the experimental data of transient and steady-state creep dominated by viscoelastic deformation and power-law creep deformation, respectively. The Burgers viscoelastic model was used to interpret the transient creep in polymers under nano-indentation. Explicit expression for the displacement of transient creep was derived using the correspondence principle of linear viscoelasticity theory. The power law of strain rate-stress relation was used to explain the creep displacement during the steady state. Three polymers of poly(methyl methacrylate), hydroxyethyl methacrylate copolymer, and the fast-cure acrylic resin were used to measure the nanoindentation creep. The transient creep data are in good agreement with the predictions from the Burgers viscoelastic model. The creep displacement is mainly attributed to the viscous flow of the Kelvin element, and the computed values of viscosities ( η 1,c, η 2,c) increase with decreasing preloading rate. By comparing the steady-state creep data with the power law of strain rate-stress relation, the stress exponents for the above polymeric materials were quantitatively determined.

Boundary value problems of the theory of viscoelasticity for Kelvin‐Voight materials

AbstractIn this paper, the classical Kelvin‐Voight model of the linear theory of viscoelasticity is considered and the following results are obtained: the fundamental solution of the equation of steady vibrations is constructed, the basic properties of plane waves and elastopotentials are established, the uniqueness theorem of the internal and external boundary value problems (BVPs) are proved, the existence theorems for classical solutions of the BVPs by means of the potential method and the theory of two‐dimensional singular integral equations are proved. (© 2010 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Smoothing Asphalt Concrete Complex Modulus Test Data

A computational approach is offered for smoothing asphalt concrete complex modulus test results in a way that forces compliance with linear viscoelastic theory and thermorheological simplicity. Essentially, it is based on presupposing the shape of the relaxation spectrum with calculations carried out in the wicket domain. The suggested scheme calls for the determination of six free parameters—three of which are associated with the mechanical (viscoelastic) response, while the remaining three are related to the time-temperature shifting properties and include (as unknown) the reference temperature level. In the evaluation process, it is shown how the first three parameters can be obtained directly from the measurements and separately from the other three, while the determination of the latter set follows as a second independent step. This decoupled process simplifies the analysis and includes additional advantages. In this paper, the smoothing approach is described and discussed, and its appropriateness and relevancy for asphalt concrete materials is demonstrated using a sample data set.

On material velocities and non-locality in the thermo-mechanics of continua

Following is a discussion of several issues surrounding the continuum modeling of discrete particulate systems, particularly the uncertainty in the definition of continuum-level material particles and velocities. The work is motivated in part by various proposals for the introduction of supplemental velocity fields into the thermo-mechanics of single-component fluids. A review and modification are given of the relevant continuum field equations, based on the Eulerian rather than the conventional Lagrangian description, and a connection is made to the well-known statistical mechanics of Kirkwood and coworkers for point-particles. The Noetherian technique of Green and Naghdi is employed to derive the momentum balance directly from the particle-level energy balance, and it is shown that non-barycentric effects in particle motion engender stress asymmetry. It is conjectured that this and related effects are generally bound up with higher-gradient effects and the breakdown of the simple-material model for continuum thermo-mechanics. To illustrate this, a theory of non-local linear viscoelasticity is presented and compared with Brenner’s “bi-velocity hydrodynamics” and Müller’s “extended thermodynamics” for heat flux and stress. These models are shown to represent expansions in spatial wave vector, representing a Knudsen number, and temporal frequency, representing a Deborah number. Suggestions are made for further work to explore the associated effects of couple-stress and the consequences of departures of entropy flux from heat flux.

Rheological behavior and mechanical properties of high‐density polyethylene blends with different molecular weights

AbstractThe dynamic rheological and mechanical properties of the binary blends of two conventional high‐density polyethylenes [HDPEs; low molecular weight (LMW) and high molecular weight (HMW)] with distinct different weight‐average molecular weights were studied. The rheological results show that the rheological behavior of the blends departed from classical linear viscoelastic theory because of the polydispersity of the HDPEs that we used. Plots of the logarithm of the zero shear viscosity fitted by the Cross model versus the blend composition, Cole–Cole plots, Han curves, and master curves of the storage and loss moduli indicated the LMW/HMW blends of different compositions were miscible in the melt state. The tensile yield strength of the blends generally followed the linear additivity rule, whereas the elongation at break and impact strength were lower than those predicted by linear additivity; this suggested the incompatibility of the blends in solid state. © 2010 Wiley Periodicals, Inc. J Appl Polym Sci, 2010

Acoustic response characteristics of unsaturated porous media

By employing the plane wave analysis method, the dispersion equations associated with compressional and shear waves using Santos’s three-phase poroelastic theory were driven. Considering the reservoir pressure, the high frequency corrections and the coupling drag of two fluids in pores, the influences of frequency and gas saturation on the phase velocities and the inverse quality factors of four body waves predicted by Santos’s theory were discussed in detail. The theoretical velocities of the fast compressional and shear waves were compared with the results of the low and high frequency experiments from open publications, respectively. The results showed that they are in good agreement in the low frequency case rather than in the high frequency case. In the latter case, several popular poroelastic models were considered and compared with the experimental data. In the models, the results of White’s theory fit the experimental data, but the parameter b in White’s model has a significant impact on the results. Under the framework of the linear viscoelasticity theory, the attenuation mechanism of Santos’s model was extended, and the comparisons between the experimental and theoretical results were also made with respect to attenuation. For the case of water saturation less than 90%, the extended model makes good predictions of the inverse quality factor of shear wave. There is a significant difference between the experimental and theoretical results for the compressional wave, but the difference can be explained by the experimental data available.