Linear Viscoelastic Theory Research Articles

On the solutions in the linear theory of micropolar viscoelasticity

In the present paper the linear theory of micropolar viscoelasticity is considered. The explicit expression of fundamental solution of the system of equations of steady vibrations is constructed by means of elementary functions and its basic properties are established. The Green's formulas in the considered theory are obtained. The formulas of integral representations of Somigliana-type of regular vector and regular (classical) solution are presented. The representation formulas of Galerkin-type solution of the system of nonhomogeneous equations and of the general solution of the system of homogeneous equations by means of eight metaharmonic functions are presented. The completeness of these solutions is proved.

Dynamic properties of an axially moving sandwich beam with magnetorheological fluid core

Dynamic properties and vibration suppression capabilities of an axially moving sandwich beam with a magnetorheological fluid core were investigated in this study. The stress–strain relationship for the magnetorheological fluid was described by a complex shear modulus using linear viscoelasticity theory. First, the dynamic model of an axially moving magnetorheological fluid beam was derived based on Hamilton’s principle. Then, the natural frequency of the sandwich beam for the first mode was determined. Later, the effects of the speed of the axial movement, axial force, applied magnetic field, skin–core thickness ratio, and their combination on the dynamic properties of the sandwich beam with a magnetorheological fluid core were investigated. It was found that these parameters have significant effects on the dynamic properties of the sandwich beam. Moreover, the results indicate that the active control ability of magnetic field has been influenced by the axial force, moving speed, and increasing skin–core thickness ratio.

Transverse vibration of nematic elastomer Timoshenko beams.

Being a rubber-like liquid crystalline elastomer, a nematic elastomer (NE) is anisotropic viscoelastic, and displays dynamic soft elasticity. In this paper, the transverse vibration of a NE Timoshenko beam is studied based on the linear viscoelasticity theory of nematic elastomers. The governing equation of motion for the transverse vibration of a NE Timoshenko beam is derived. A complex modal analysis method is used to obtain the natural frequencies and decrement coefficients of NE beams. The influences of the nematic director rotation, the rubber relaxation time, and the director rotation time on the vibration characteristic of NE Timoshenko beams are discussed in detail. The sensitivity of the dynamic performance of NE beams to director initial angle and relaxation times provides a possibility of intelligent controlling of their dynamic performance.

Additive Manufacturing of a Cylindrical Arch of Viscoelastic Aging Material Under Gravity Action at various Modes of the Process

The engineering problem of successive construction of a heavy circular vault (a cylindrical arch) on a smooth rigid horizontal base is studied. The constructed vault manifests properties of creep and aging. The construction goes by layer-by-layer buildup of a prefabricated residual stress-free vaulted preform originally installed on the base. A buildup process in which a large number of thin layers of additional material are successively attached to the interior cylindrical surface of the arch is considered. The process is modeled as a continuous accretion process, that is, an infinitely thin material layer is added to the solid on every infinitesimal time interval. The added material is assumed not to be prestressed. The sliding mounting of the vault to the base is assumed. The case of plane strain of the vault is examined. The initial-boundary value problems describing deformation process of the structure in question before, during and after its accretion within the framework of the linear theory of viscoelasticity of accreted solids are stated and solved analytically in series and quadratures. Numerous numeral calculations for different time modes of the viscoelastic vault constructing process are performed. The influence of the temporal characteristics of the process on obtained stress state of the constructed vault is investigated. Some practically relevant phenomena are discovered. Requirement of taking into account the gravity forces in the course of the whole constructing process but not only in the terminal structure configuration is shown convincingly.

Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges

This study investigates the size-dependent quasistatic response of a nonlinear viscoelastic microelectromechanical system (MEMS) under an electric actuation. To have this problem in view, the deformable electrode of the MEMS is modelled using cantilever and doubly-clamped viscoelastic microbeams. The modified couple stress theory in conjunction with Bernoulli–Euler beam theory are used for mathematical modeling of the size-dependent instability of microsystems in the framework of linear viscoelastic theory. Simultaneous effect of electrostatic actuation including fringing field, residual stress, mid-plane stretching and Casimir and van der Waals intermolecular forces are considered in the theoretical model. A single element of the standard linear solid element is used to simulate the viscoelastic behavior. Based on the extended Hamilton’s variational principle, the nonlinear governing integro-differential equation and boundary conditions are derived. Thereafter, a new generalized differential-integral quadrature solution for the nonlinear quasistatic response of electrically actuated viscoelastic micro/nanobeams under two different boundary conditions; doubly-clamped microbridge and clamped-free microcantilever. The developed model is verified and a good agreement is obtained. Finally, a comprehensive study is conducted to investigate the effects of various parameters such as material relaxation time, durable modulus, material length scale parameter, Casimir force, van der Waals force, initial gap and beam length on the pull-in response of viscoelastic microbridges and microcantilevers in the framework of viscoelasticity.

Fractional model of concrete hereditary viscoelastic behaviour

The evaluation of creep effects in concrete structures is addressed in the literature using different predictive models, supplied by specific codes, and applying the concepts of linear viscoelastic theory with ageing. The expressions used in the literature are mainly based on exponential laws, which are introduced in the integral expression of the Boltzmann principle; this approach leads to the need of finding approximated numerical solutions of the viscoelastic response. In this study, the hereditary fractional viscoelastic model is applied to concrete elements, underlining the convenience of using creep or relaxation functions expressed by power laws. The full reciprocal character of creep and relaxation, described through the power laws, is shown in Laplace domain, and the basic theorems of linear viscoelastic theory are re-written. In order to simplify the solution of fractional expressions, the Grunwald–Letnikov approach is followed and a discretization of fractional operators is applied through a simple matrix procedure. Two cases are examined: The first is the application of the fractional model to the creep laws supplied by international codes, finding the power law which approximates in the best way the creep curve given by the B3 predictive model. The second one is the application to the experimental data of an actual concrete mixture, for which the corresponding power law is found and introduced in the fractional model. A best-fitting procedure is applied to obtain the two parameters of power law in order to describe the hereditary character of concrete creep; afterwards, ageing is interpreted through the variation of coefficients with time. A third application on a concrete beam shows the suitability of the proposed model to the estimation of the delayed deformations of actual concrete structures.

External boundary value problems in the quasi static theory of viscoelasticity for Kelvin‐Voigt materials with double porosity

AbstractIn the present paper the linear quasi static theory of viscoelasticity for Kelvin‐Voigt materials with double porosity is considered. The basic external boundary value problems (BVPs) of steady vibrations in this theory are formulated. The uniqueness and existence theorems for regular (classical) solutions of the BVPs are proved by using of the potential method (boundary integral equations method) and the theory of singular integral equations. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Partial slip contact of a rigid pin and a linear viscoelastic plate

This paper analyzes partial slip contact problems in the theory of linear viscoelasticity under a wide variety of loading conditions, including cyclic (fretting) loads, using a semi-analytical method. Such problems arise in applications like metal-polymer contacts in orthopedic implants. By using viscoelastic analogues of Green’s functions, the governing equations for viscoelastic partial-slip contact are formulated as a pair of coupled Singular Integral Equations (SIEs) for a conforming (pin-plate) geometry. The formulation is entirely in the time-domain, avoiding Laplace transforms. Both Coulomb and hysteretic effects are considered, and arbitrary load histories, including bidirectional pin loads and remote plate stresses, are allowed. Moreover, the contact patch is allowed to advance and recede with no restrictions. Viscoelasticity necessitates the application of the stick-zone boundary condition in convolved form, and also introduces additional convolved gap terms in the governing equations, which are not present in the elastic case. Transient as well as steady-state contact tractions are studied under monotonic ramp-hold, unload-reload, cyclic bidirectional (fretting) and remote plate loading for a three-element solid. The contact size, stick-zone size, indenter approach, Coulomb energy dissipation and surface hoop stresses are tracked during fretting.Viscoelastic fretting contacts differ from their elastic counterparts in notable ways. While they shakedown just like their elastic counterparts, the number of cycles to attain shakedown states is strongly dependent on the ratio of the load cycle time to the relaxation time. Steady-state cyclic bulk hysteretic energy dissipation typically dominates the cyclic Coulomb dissipation, with a more pronounced difference at slower load cycling. However, despite this, it is essential to include Coulomb friction to obtain accurate contact stresses. Moreover, while viscoelastic steady-state tractions agree very well with the elastic tractions using the steady-state shear modulus in load-hold analyses, viscoelastic fretting tractions in shakedown differ considerably from their elastic counterparts. Additionally, an approximate elastic analysis misidentifies the edge of contact by as many as 7 degrees in fretting, showing the importance of viscoelastic contact analysis. The SIE method is not restricted to simple viscoelastic networks and is tested on a 12-element solid with very long time scales. In such cases, the material is effectively always in a transient state, with no steady edge-of-contact. This has implications for fretting crack nucleation.

Magnetic field and frequency dependent LVE limit characterization of magnetorheological elastomer

Magnetorheological elastomer (MRE) based semi-active isolators have the potential to replace conventional passive isolators to achieve wide frequency range isolation. The effectiveness of MRE isolator depends on the control strategies developed based on viscoelastic constitutive relations. The theory of linear viscoelasticity is the basis for viscoelastic constitutive relations which can predict the material behavior within a certain strain limit referred as linear viscoelastic (LVE) limit. Beyond the LVE limit, the performance of MRE semi-active isolator exacerbates as the control strategies turns out to be ineffective. In the present study, variation in LVE limit of MRE with the magnetic field and frequency is investigated through forced vibration tests. To exclude the effect of terminal non-linearity on the measurement, the blocked transfer stiffness method described in the ISO 10846-2 is adopted. The results revealed that the LVE limit of MRE is strongly dependant on the magnetic field and exhibited a weak dependency on the operating frequency. Under magnetized state, the transition from linear to non-linear behavior of MRE is at lower strain levels indicating the increased friction energy dissipation at particle–matrix interface.

Time-Dependent Response of Mass Continuous Solid Phase Media by Integral Form of Constitutive Equations - Mathematical Modeling

Mathematical modeling of boundary value problems in linear theory of viscoelasticity. Definitions and basic principles in the mathematical modeling theory. Constitutive functional and its transformation into a form of Stieltjes integral. Application of theory of algebraic sets and corresponding subsets. Riesz theorem of representation and its application for derivation of constitutive equations. Integral operator forms of stress-strain relationships for a solid-phase continuous media.

Viscoelastic Effects in Solid Phase Continuous Media - Stress-Strain Relations

Mathematical modeling of boundary value problems in linear theory of viscoelasticity. Definitions and basic principles in the mathematical modeling theory. Constitutive functional and its transformation into a form of Stieltjes integral. Application of theory of algebraic sets and corresponding subsets. Riesz theory of representation and its application for derivation of constitutive equations. Integral and differential operator forms of stress-strain relationships for a solid-phase continuous media.

Analysis of viscoelastic Bernoulli–Euler nanobeams incorporating nonlocal and microstructure effects

The present study introduces an analytical–computational model to simulate the effects of different simultaneous aspects on the behavior of nanobeams. The first one deals with the space nonlocality interaction and taking into account the microstructure effects, which has been formulated by using the nonlocal couple-stress elasticity. The second factor deals with the memory-dependent effect and has been investigated in the framework of linear viscoelasticity theory. It is the first time to apply the coupled effects of the microstructure and long-range interactions between the particles, to reflect the size-dependency of viscoelastic structures. Bernoulli–Euler nanobeam is taken as a vehicle to present the details of the proposed model. Eringen nonlocal elasticity and the modified couple-stress theory are used to formulate the two phenomena of long-range cohesive interaction and the microstructure local rotation effects, respectively. Boltzmann superposition viscoelastic model, endowed by Wiechert series, is used to simulate the linear behavior of isotropic, homogeneous and non-aging viscoelastic materials. The extended Hamilton’s principle is applied to formulate the analytical model of mechanical behavior of the nonlocal couple-stress nanobeam. The model has been verified and some results are compared with those published in the literature and a good agreement has been obtained. It is shown that the material-length scale parameter, nonlocal parameter, viscoelastic relaxation time and length-to-thickness ratio have a significant effect on the bending response of viscoelastic nanobeams with various boundary conditions.

Mechanics of a Dual Cross-Link Gel with Dynamic Bonds: Steady State Kinetics and Large Deformation Effects

We develop a three-dimensional continuum theory for a dual cross-link gel with permanent and transient bonds. This theory connects the breaking and re-forming kinetics of the transient bonds to the large deformation behavior of the gel. Although based on similar principles as our previous theory for the same gel, the new theory addresses a limitation of the previous theory and agrees equally well with experimental data. Specifically, in the new theory we assume that breaking and re-forming kinetics of transient bonds reaches a steady state. This assumption leads to fewer material parameters and enables the new theory to capture the healing process of the dual cross-link gel. In addition, the steady state assumption also allows the new theory to reduce to classical linear viscoelastic theory in the limit of infinitesimal strains, where the kinetics of bond breaking and re-formation can be linked directly to the relaxation function. We also extend the theory to account for strain stiffening under very large...

Application of viscoelastic continuum damage approach to predict fatigue performance of Binzhou perpetual pavements

For this study, the Binzhou perpetual pavement test sections constructed in Shandong Province, China, were simulated for long-term fatigue performance using the layered viscoelastic pavement analysis for critical distresses (LVECD) finite element software package. In this framework, asphalt concrete was treated in the context of linear viscoelastic continuum damage theory. A recently developed unified fatigue failure criterion that defined the boundaries of the applicable region of the theory was also incorporated. The mechanistic modeling of the fatigue mechanisms was able to accommodate the complex temperature variations and loading conditions of the field pavements in a rigorous manner. All of the material models were conveniently characterized by dynamic modulus tests and direct tension cyclic fatigue tests in the laboratory using cylindrical specimens. By comparing the obtained damage characteristic curves and failure criteria, it is found that mixtures with small aggregate particle sizes, a dense gradation, and modified asphalt binder tended to exhibit the best fatigue resistance at the material level. The 15-year finite element structural simulation results for all the test sections indicate that fatigue performance has a strong dependence on the thickness of the asphalt pavements. Based on the predicted location and severity of the fatigue damage, it is recommended that Sections 1 and 3 of the Binzhou test sections be employed for perpetual pavement design.

Mechanical, dielectric and structural characterization of cross-linked PEG-diacrylate/ethylammonium nitrate ionogels

We report the preparation and characterization of cross-linked ionogels (x-IGs) composed of polyethylene glycol diacrylate (PEGDA) and the protic ionic liquid, ethylammonium nitrate (EAN). The cross-linking process has a huge effect on the mechanical properties of the solutions (forming stiff solids from Newtonian solutions) and a minimum penalty on the ionic conductivity. The interdependence of the mechanical and dielectric properties with the network structure of the x-IGs was studied using three experimental probes: torsion and compression mechanical testing, dielectric spectroscopy and small angle neutron scattering. The microstructure, the mechanical strength and the conductivity of the x-IGs depend strongly on the polymer concentration and weakly on the temperature. High modulus and relatively low conductivities are associated to small cross-link junction lengths, ξ, observed in concentrated samples, whereas large ξ values, observed in dilute samples, result in high conductivities and relatively low modulus. The topological restriction to ionic transport (i.e., to conductivity) is quantified by the obstruction factor, which increases monotonically with ξ, while the shear modulus exhibits a power law behavior, G∼ξ−3, in accordance to linear viscoelastic theory.

Coupled Orientation and Stretching of Chains in Mesoscale Models of Polydisperse Linear Polymers in Startup of Steady Shear Flow Simulations

Polydisperse linear polymers are studied in startup of steady shear flow simulations using dissipative particle dynamics. The results show that with an increase in polydispersity the stress overshoot declines while the steady‐state stress increases. Various physical characteristics of the systems are studied including frequency of nonbonded interactions, gyration radius data, flow alignment angles, and average bond lengths. The patterns in the data suggest higher forces are necessary to orient and stretch long chain fractions in the flow direction. Relaxation modulus data prove the broad range of relaxation mechanisms in polydisperse systems. Linear viscoelasticity theory is used to quantify the relaxation spectrum. The results indicate an increase in the longest relaxation time in systems with higher polydispersity. The steady‐state shear viscosity results show higher viscosities with an increase in polydispersity at all shear‐rates. The good agreement of the characteristic behaviors of modeled polydisperse polymers with experiments is encouraging for future work. image

樹脂を含む四層積層体の熱負荷過程における反り変形予測

The warp deformation behavior for various viscoelastic four-layer laminated structure consisting of epoxy resin, printed board and steel which modeled electronic devices caused by a series of thermal load from heating to cooling was examined by experiment and viscoelastic stress analysis based on linear viscoelastic theory. As a result, it was clarified that the warp deformation behavior of laminated structure containing resin was influenced by thickness and viscoelastic properties of epoxy resin and printed board such as modulus of longitudinal elasticity and thermal expansion coefficient depending on temperature and time, and that the warp deformation behavior of viscoelastic four-layer laminated structure was determined by the balance between flexural rigidity of laminated structure and bending moment caused with the thermal expansion or shrinkage of constituent materials. The warp deformation behavior of laminated structure containing resin could be predicted in general by analysis in consideration of the characteristic of viscoelasticity.

Time-Temperature Superposition in Linear and Non-linear Domain

Long-term behavior of polymers and their composites is related mainly to their time-dependent mechanical properties, i.e. material functions or master curves, usually obtained by time-temperature superposition (tTS) principle. The “weakest” point utilizing tTS lies in “manual or hand” shifting procedure, which can lead to over- or under-estimated life span of polymeric materials. This problem was solved by using closed form mathematical solution called Closed-Form-Shifting (CFS) algorithm which completely removes issues of “manual or hand” shifting procedure. However, it is important to acknowledge that tTS is well established only within the realm of linear viscoelastic theory. Results have shown that utilizing tTS in non-linear domain, where time-dependent properties become dependent on magnitude of external load, my lead to construction of misleading master curves, i.e. overestimated long-term behavior of polymers that could consequently cause premature failure of polymeric products.

Use of linear viscoelastic theory to predict resilient behavior of unbound granular materials

This paper presents a methodology to estimate the stress-strain relationship of an unbound aggregate base using linear viscoelastic theory. Current Mechanistic-Empirical (ME) pavement design procedure adopts the resilient modulus concept to explain the behavior of granular materials for flexible pavement design. The resilient modulus is a stress dependent material property of granular materials that is different from strength. Although California Bearing Ratio (CBR) test results (i.e., stress and strain) can be used to estimate the strength of a granular material, it is not possible to estimate the resilient modulus directly. Therefore, it is necessary to estimate stress along with strain changes. The convolution integral enables the stress to be estimated from the given strain changes only if the relaxation modulus is measured. Aggregate specimens prepared from two different sources in Georgia were subjected to the relaxation modulus test. From the test data, the time-dependent stress due to a known strain rate was computed as a convolution integral of the strain. The computed stress-strain relationship was compared with that from the resilient modulus (MR) test. The results indicate that the stress-strain relationships from the MR test and the convolution integral are similar with nearly the same slopes when horizontal stress is assumed to be approximately 45% of vertical stress. This observation supports the use of the proposed methodology by state highway agencies to validate the MR test results for quality control and quality assurance of aggregate base material selection for pavement design and construction.

The spectral theory of anisotropic linear viscoelasticity of polymers

The spectral theory of anisotropic linear viscoelasticity of polymer liquids is developed on the basis of the spectral decomposition of the relaxation tensor, which characterizes their anisotropic rheological properties. A definition of fading memory for anisotropic liquids is given. The oscillating shear and tension flows of liquids with uniaxial anisotropy are considered. The dependence of their dynamic characteristics on the direction is studied.