Linear Viscoelastic Theory Research Articles

Load transfer between permanent and dynamic networks due to stress gradients in nonlinear viscoelastic hydrogels

Load transfer between permanent and dynamic networks due to stress gradients in nonlinear viscoelastic hydrogels

Thermally activated viscoelasticity of cement paste: Minute-long creep tests and micromechanical link to molecular properties

The stiffness of cementitious materials decreases with increasing temperature. Herein, macroscopic samples of mature cement pastes are subjected at 20, 30, and 45 °C, respectively, to three-minutes-long creep compression experiments. The test evaluation is based on the linear theory of viscoelasticity and Boltzmann’s superposition principle. This yields macroscopic elastic and creep moduli as a function of temperature. A state-of-the-art multiscale model for creep homogenization of cement paste is extended to account for temperature-dependent elastic and creep moduli of the hydrate gel. This extension is based on results from published molecular simulations. Temperature-independent stiffness is assumed for cement clinker. Upscaling to the macroscale of cement paste yields elastic and creep moduli which agree well with the aforementioned experimental results. The Arrhenius-type activation energy of the creep modulus is found to be independent of scale, composition, and maturity, because of ineffective stress redistributions from creeping to non-creeping constituents. • Three-minutes creep tests of mature cement paste are performed at 20, 30, and 45 °C. • They give access to elastic and creep moduli rooted in linear viscoelasticity. • A multiscale model links these moduli to MD results at the hydrate level. • Across the scales, the activation energy of creep equals that of water.

Experimental study of Poisson's ratio in indirect tensile test mode for asphalt mixtures

• Poisson's ratio main curve of asphalt mixture was constructed. • The effect of different asphalt binder and mix types on Poisson's ratio was discussed. • The recommended values of Poisson's ratio for AC asphalt mixtures were proposed. To solve the problem of inaccurate Poisson's ratio in the design of asphalt pavement structures, based on the theory of complex modulus and linear viscoelasticity, the indirect tensile test was used to calculate and analyze Poisson’s ratio of asphalt mixtures. Furthermore, the test parameters and operational considerations of the indirect tensile test were clarified, and the conditions for judging whether the displacement derivative reaches stability under each test temperature were proposed. The indirect tensile test was carried out on five types of asphalt mixtures to analyze the effect of different asphalt and different mix grading types on Poisson's ratio, and the recommended values of Poisson's ratio for AC asphalt mixtures were proposed. The results of the study show that: the influence of temperature and loading frequency on Poisson’s ratio of AH-70 asphalt mixture is very obvious, showing that as the temperature increases and the frequency decreases, Poisson’s ratio gradually increases; its variation range is basically between 0.1 and 0.5, and with the increase of the mixture particle size, this variation range tends to decrease. Poisson’s ratio of SBS-modified asphalt mixtures is less affected by temperature changes and Poisson's ratio variation range is smaller, but at each frequency at each temperature, Poisson’s ratio changes with frequency are also more obvious. Under standard test conditions (20 °C, 10 Hz), Poisson’s ratio ranges from 0.20 to 0.35 when using AH-70 bitumen, with a recommended value of 0.28; when using SBS modified bitumen, Poisson’s ratio ranges from 0.15 to 0.30, with a recommended value of 0.23.

Inter-conversion of the generalized Kelvin and generalized Maxwell model parameters via a continuous spectrum method

• Constructed the continuous spectra ( H τ and L ρ ) via the direct and indirect methods. • Obtained the optimal ranges of retardation time and relaxation time. • Proposed a spectrum method to establish the inter-conversion between E k and D j Accurate determination of the Prony series parameters of the generalized Maxwell (GM) model and the generalized Kelvin (GK) model is crucial to the characterization of linear viscoelastic (LVE) properties of asphalt mixture and the prediction of in-service performance of asphalt pavement. However, existing methods may not be able to provide satisfactory accuracy due to their weakness of fully complying with the LVE theory or the involvement of unsolvable high-order nonlinear systems. To identify the Prony series parameters more efficiently and accurately, this paper developed a continuous spectrum method to establish the inter-conversion between the GM and GK model parameters. In the newly developed method, the complex modulus mater curves are first constructed through the conventional master curve models. The continuous relaxation and retardation spectra are then established using the proposed direct or indirect methods. The time and strength constants of the GM and GK models are subsequently calculated via an extended search method, based on which the Prony series parameters and the LVE variables of the asphalt mixture can be finally determined. Accuracy of the new method in determining the LVE variables was validated through a comparison of the calculated and measured LVE variables of asphalt mixture, while the equivalence between the GM and GK models established based on the new method was verified in different domains. Besides, the success implementation of the new method to the generalized Sigmodal model demonstrates the universality of the newly developed continuous spectrum method in determining the LVE variables of asphalt mixture.

Evaluating fatigue resistance based on viscoelastic properties of asphalt mixtures

ABSTRACT In this study, the fatigue resistance of different asphalt mixture types subjected to the same aging condition was evaluated using linear viscoelastic properties of asphalt mixtures. The relaxation spectra were determined from the dynamic modulus test based on the linear viscoelastic theory, and the relationship between the parameters of relaxation spectra and the parameters of the storage modulus master-curves were developed. The Texas overlay test was conducted to evaluate the fatigue resistance with the number of load cycle to failure used as the measure of fatigue resistance. From the corresponding results, it is shown that the inflection point frequency of the dynamic modulus master-curves could be used to assess the fatigue resistance compared to other parameters. The Glover-Rowe parameter also exhibited a fair relationship with the fatigue resistance of the asphalt mixtures. The effect of the shape parameter (γ) of the dynamic modulus master-curves was quantified but seemed not very sensitive to fatigue resistance for the asphalt mixtures evaluated. In the study, a new parameter (VECIndex) was developed to evaluate the fatigue resistance, which considered the inflection point frequency and shape of the dynamic modulus master-curves, both of which influence the linear viscoelastic properties and fatigue resistance of asphalt mixtures.

An improved method to establish continuous relaxation spectrum of asphalt materials

The relaxation spectrum plays a vital role in the study of linear viscoelastic (LVE) properties of asphalt materials; however, the current developed using methods are not accurate enough to characterize the LVE properties of asphalt materials. This paper aims to develop an improved method to determine the continuous relaxation spectrum of asphalt materials. A modified Christensen–Anderson–Marasteanu (CAM) model is first selected as the master curve model of the storage modulus, upon which the master curve model of loss modulus is then established using the numerical form of the exact Kramers-Kronig relation between the storage modulus and loss modulus. According to LVE conversions between the relaxation spectrum and storage modulus, the model of the continuous relaxation spectrum is subsequently derived and finally determined based on the same model parameters of the storage modulus and loss modulus models. This study demonstrates that the relaxation spectrum of both mixtures and binders constructed by the proposed method is compliant with LVE theory and physical causality. The generated relaxation spectrum is verified to be accurate enough to compare the storage modulus and loss modulus values recalculated by the relaxation spectrum and the corresponding values predicted by master curve models. The results also show that the approximate Kramers-Kronig relation between the storage modulus and loss modulus cannot be used to establish the relaxation spectrum of asphalt materials, especially for asphalt binders. In addition, the generated relaxation spectrum was utilized to construct the relaxation modulus of mixtures and to develop the stiffness modulus of binders, respectively. Since the relaxation modulus and creep stiffness are determined from the relaxation spectrum, both are consistent with LVE theory and physical causality.

Viscoelastic wave propagation and resonance in a metal–plastic bonded laminate

Stress wave propagation in a metal–plastic bonded laminate immersed in a fluid is investigated based on linear viscoelastic theory. The reflection spectrum is theoretically derived for normal wave incidence. For the incidence from the plastic layer, the amplitude of the reflection spectrum is shown to have local minima at the resonance frequencies of every single layer. Resonance in the metal layer leads to the destructive interference of the reflected wave components, while resonance in the plastic layer appears due to its viscoelastic damping. Experimental results obtained for bonded specimens are discussed from the viewpoint of the characterization for dissimilar joints.

Nonlinear rheological models and their application to describe the mechanical behavior of highly oriented polymer materials

The nonlinear viscoelasticity of uniaxial oriented polymer materials is considered. To explain the deformation mechanisms of oriented polymers and the possibility of predicting their mechanical behavior in various operating modes, new nonlinear rheological models have been proposed. The application of the simplest rheological model of a real viscoelastic solid to the description and explanation of the recovery process in polymer materials is studied. From the standpoint of rheology, a model of an ideal viscoelastic solid is introduced. Using the balance equation for the number of transitions through energy barriers, a method for calculating the new nonlinear rheological model has been proposed. To eliminate the shortcomings of the ideal viscoelastic solid model associated with the impossibility of predicting creep and stress relaxation modes for long times, a generalized rheological model of a real viscoelastic solid was obtained. In the corrected model, the simplest elements are connected in parallel, which means the presence of not one, but several energy barriers in the materials, the transmission through which have their own relaxation times. To describe the recovery processes in polymer materials, the model of an ideal viscoelastic solid is supplemented by a parallel connected elastic spring. An additional Hooke spring replaces the interfibrillar interaction between the individual elements of the structure and is responsible for possible obstacles during jump-like transitions through the energy barrier. Using the method of rheological models and describing interfibrillar bonds within the framework of the theory of elasticity, a constitutive equation was obtained for the case of recovery processes. Based on the constitutive equation of viscoelasticity for uniaxial oriented polymer materials, a new nonlinear highly elastic element is introduced, which replaces the Maxwell’s element in the theory of linear viscoelasticity. A new rheological model of parallel connection of elastic elements is shown. An explanation of the retardation of the recovery process in polymers is given. The simplest rheological model of a real viscoelastic solid is proposed, in which an elastic spring is responsible for interfibrillar bonds. A constitutive equation is obtained that describes the recovery process in polymers. This equation can be integrated by quadratures and gives a solution that is an analogue of the Newton-Leibniz formula for the proposed model. It is shown that the deformation recovery process in polymer does not depend on the level of initial deformation and the loading method. The obtained result is confirmed by experimental data for polyamide and polyethylene film yarns. When specifying a certain initial level of deformation, generalized recovery curves of these materials are obtained. The proposed simplest rheological model of a real viscoelastic solid makes it possible to predict the reducing properties of polymer materials. And also makes it possible to determine the height of the energy barrier and the magnitude of the elastic modulus in the model. Based on the new rheological models, it is planned in the future to consider the issues of modeling and forecasting different modes of deformation.

Simulation of multi – field coupling damping characteristics of magnetorheological damper in ring system by artificial intelligence method

A new kind of smart material called Magnetorheological (MR) has attracted a lot of attention among designers due to its mechanical properties that can alter in a controlled fashion by an external magnetic field. Regarding this issue, in the current work, for the first attempt, the dynamic stability of a sandwich shell made of MR core and graphene nanoplatelets reinforced composite (GPLRC) face sheets is presented. Using the role of the mixture and modified Halpin–Tsai model, the effective material properties of GPLRC face sheets are presented. Via Hamilton’s principle and linear viscoelastic theory, the governing equations of the current sandwich structure are developed. Also, compatibility equations for accurate modeling of the sandwich structure are presented. The discrete singular convolution integration method (DSC-IM) for prediction of nth-order derivatives of a sufficiently smooth function via weighted linear sum function of the amounts at each separated point in the throughout allowable input range is presented. For more verification, a deep neural network as an artificial intelligence method is presented. Finally, the results show that GPL’s weight fraction, viscoelastic foundation, the geometry of GPLs, and geometry of the structure have an important role in dynamic stability and modal loss factor parameter of sandwich shells.

Эффективные механические характеристики симметричного слоистого композита при различных условиях нагружения

Object and purpose of research. This paper discusses balanced and quasi-isotropic (in the reinforcement plane) symmetric layered composite structures made up by the layers of cloth-reinforced GRP. The purpose of this work was to demonstrate the necessity to justify the applicability of experimental results for effective mechanical parameters determined in the conditions of uniaxial tension/compression to the calculation of thin-walled layered composite structures that work in bending/twisting conditions. Materials and methods. The straining of layered composite structures is simulated as per the updated theory of first-order plates, the model of complex moduli and the principle of elastic-viscoelastic correspondence in linear viscoelasticity theory. Limit state predictions are based on Tsai-Wu tensor-polynomial strength criterion. Main results. This paper suggests the expressions that predict effective elastic constants, dissipation properties and strength limits for symmetric layered structures under investigation. The study shows that balanced symmetric structure made up by four layers of composite may be regarded, with the accuracy sufficient for engineering calculations, as an ortho-tropic material for all loading conditions. At the same time, symmetric quasi-isotropic (in the reinforcement plane) structure made up by thirty two composite layers must be regarded as orthotropic in case of tension/compression and monoclinic in case of bending/twisting. Conclusion. The study has shown the necessity to justify the application of experimental effective mechanical properties for uniaxial tension/compression to calculation of thin-walled layered composite structures exposed to bending/twisting.

Modeling a Weak Foundation in Interaction with Reinforced Sand Piles under a Low-Rise Building

One of the ways to increase the bearing capacity and stability of a water-saturated base by introducing a sand pile vertically reinforced along the contour with geosynthetic material (geogrid SSP 30 / 30-2.5) is experimentally substantiated. This constructive solution is used in low-rise construction. For the theoretical substantiation of the suggested method, it is proposed to model the interaction of a weak foundation and a reinforced sand pile on the basis of the linear theory of viscoelasticity. Calculation of vertical displacements of the pile and comparison with the results of in situ experiments is presented.

Rheological Relaxation of OSB Beams Reinforced with CFRP Composites.

An experimental and analytical approach to the relaxation problem of wood-based materials—OSB (Oriented Strand Boards—pressed wood-based composite panels) beams, including beams with CFRP (Carbon fiber reinforced polymer) tape composite reinforcement, is presented. It is a relevant engineering and scientific problem due to the fact that wood and wood-based materials, as well as composite reinforcements, are widely used in building constructions. Their rheological properties are very important and complicated to estimate. A 10 day long relaxation test of thick OSB beams without reinforcement and with CFRP tape was performed. A four-point bending test with five different bending levels was performed, during which the reduction of the loading force was measured. A five-parameter rheological model was used to describe the rheology of the beams. The equations of this model were calculated with the use of Laplace transform, whereas the values of the parameters were calculated based on the experimental relaxation curves. A high correlation between experimental and theoretical results was obtained. A beam reinforced with CFRP tape was treated as a system with a viscoelastic element (OSB) and an elastic element (CFRP), joined together without the possibility of slipping. The equations of the mathematical model were calculated based on the assumptions of the linear theory of viscoelasticity and the convolution integral. A good correlation between experimental and theoretical results was obtained. A significant redistribution of stresses was observed during the relaxation of the reinforced beam. The reinforced beams show a higher stiffness of approximately 63% and carry proportionally higher loads than unreinforced beams at the same deflection values.

Диссипативные свойства трехслойных композитных структур. 1. Постановка задачи

Object and purpose of research. The object under study is a sandwich plate with two rigid anisotropic layers and a filler of soft isotropic viscoelastic polymer. Each rigid layer is an anisotropic structure formed by a finite number of orthotropic viscoelastic composite plies of arbitrary orientation. The purpose is to develop a mathematical model of sandwich plate. Materials and methods. The mathematical model of sandwich plate decaying oscillations is based on Hamilton variational principle, Bolotin’s theory of multilayer structures, improved theory of the first order plates (Reissner-Mindlin theory), complex modulus model and principle of elastic-viscoelastic correspondence in the linear theory of viscoelasticity. In description of physical relations for rigid layers the effects of oscillation frequencies and ambient temperature are considered as negligible, while for the soft viscoelastic polymer layer the temperaturefrequency relation of elastic-dissipative characteristics are taken into account based on experimentally obtained generalized curves. Main results. Minimization of the Hamilton functional makes it possible to reduce the problem of decaying oscillations of anisotropic sandwich plate to the algebraic problem of complex eigenvalues. As a specific case of the general problem, the equations of decaying longitudinal and transversal oscillations are obtained for the globally orthotropic sandwich rod by neglecting deformations of middle surfaces of rigid layers in one of the sandwich plate rigid layer axes directions. Conclusions. The paper will be followed by description of a numerical method used to solve the problem of decaying oscillations of anisotropic sandwich plate, estimations of its convergence and reliability are given, as well as the results of numerical experiments are presented.

Indentation of a cylinder with different shape of the base into viscoelastic half-space

During developing medical equipment and processing results of experiments on the study of biomaterials, it is necessary to describe the process of interaction of the working tool with soft tissues. With this, it is important to take into account not only the mechanical properties of the tissue under study, but also the technical characteristics of the equipment, such as the shape of the tool surface and the speed of the interaction process. For this purpose, a contact model describing the indentation of an axisymmetric rigid cylinder with different foundation surface shape into viscoelastic half-space at a constant speed was constructed. The theory of linear viscoelasticity was used to describe the mechanical behavior of the half-space. A three-parameter model of a standard viscoelastic body was chosen as a viscoelastic model that defines the type of relaxation function of the half-space material. The distribution of contact pressure under the cylinder surface and the dependence of the applied load on the depth of indentation for various forms of the contact surface of the cylinder were obtained. The influence of the shape of the cylinder base surface, the indentation speed and the relaxation properties of the half space on the characteristics of the contact interaction was studied. For describing the mechanical behavior of biological tissues, a one-dimensional model of a standard viscoelastic body was also considered. For this simplified model, the distribution of the contact pressure under the surface of cylinder with different shape of the base and the dependence of the applied load on time were also obtained. The results obtained using the continuous model and the one-dimensional model were compared. It was shown when it is more appropriate to use the simplified models to describe the interaction of an instrument with biological tissue.

Constitutive modelling of rubbers: Mullins effect, residual strain, time-temperature dependence

Filled rubbers are an important part of engineering materials because they have unique properties such as large deformability, increased stiffness and energy dissipation ability. The complexity of the material model parameter identification and the consideration of essential behaviours such as temperature dependence or residual strain may be a great challenge during the constitutive modelling. In this paper, a hyper-visco-pseudo-elastic constitutive equation is introduced and applied to model the complex behaviour of filled rubbers by considering (i) the Mullins effect, (ii) the temperature effect, and (iii) the effect of residual strains. The time-independent material response is provided by the modified Dorfmann–Ogden pseudo-elastic material model, while the time and temperature dependence is taken into consideration by the Prony series-based linear viscoelastic theory and the time-temperature superposition principle. Furthermore, by assuming homogeneous deformations, numerical stress solutions are presented to perform the material model calibration simple and fast, even when considering uniaxial tension/compression and simple shear simultaneously. The material model can be adapted to handle any hyperelastic model, arbitrary damage and residual strain variable, user-defined time-temperature superposition function, and any number of Maxwell elements (spring-dashpot terms). Here, the model parameters are determined from uniaxial stress relaxation-recovery tests, as well as cyclic uniaxial tensile and cyclic simple shear tests of different strain rates. Finally, the theoretical results are compared with experimental data for an EPDM rubber filled with 50 phr of carbon black.

A New Representation for Viscoelastic Behavior of Materials in Two- and Three-Dimensional Problems

The main objective of this study is to develop a set of equations, in which any desirable rheological model can be employed to investigate the behavior of viscoelastic materials (VMs) in two- or three-dimensional settings within the scope of small deformations. Application of the differential-form constitutive equations in 3D problems has been mainly limited to the isotropic materials. On the other hand, the integral-form constitutive equations are mostly unfavorable in terms of computational costs. To the best knowledge of the authors, in general, there is no interrelation between the differential-form and the integral-form viscoelastic constitutive equations in the 3D theory of linear viscoelasticity. This study tends to link the differential and the integral forms of linear viscoelasticity in a general setting. As a consequence, for example, relaxation moduli of a VM can be formulated in terms of model parameters, i.e., spring constants and viscosity coefficients, of any desirable rheological model. Some numerical examples are also provided to support the efficiency of the proposed formulation in 2D and 3D loading scenarios.

GUIDELINES TO SIMULATE LINEAR VISCOELASTIC MATERIALS WITH AN ARBITRARY NUMBER OF CHARACTERISTIC TIMES IN THE CONTEXT OF ATOMIC FORCE MISCROSCOPY

We provide guidelines for modeling linear viscoelastic materials containing an arbitrary number of characteristic times, under atomic force microscopy (AFM) characterization. Instructions are provided to set up the governing equations that rule the deformation of the material by the AFM tip. Procedures are described in detail in the spirit of providing a simple handbook, which is accompanied by open-access code and workbook (Excel) sheets. These guidelines seek to complement the existing literature and reach out to a larger audience in the awareness of the interdisciplinary nature of science. Examples are given in the context of force-distance curves characterization within AFM, but they can be easily extrapolated to other types of contact characterization techniques at different length scales. Despite the simplified approach of this document, the algorithms described herein are built upon rigorous classical linear viscoelastic theory.

Magnetothermoelastic vibrations on a viscoelastic microbeam subjected to a laser heat source

The linear theory of viscoelasticity remains an important field of research like most solids and polymer materials when exposed to a vicious dynamic loading effect. This article introduces a new model for describing the behavior of thermoviscoelastic microbeams considering the effects of temperature change and the longitudinal magnetic field. The governing equations in this model are derived based on the Euler–Bernoulli beam theory, Kelvin–Voigt model of viscosity, the generalized thermoelasticity, and the classical Maxwell equations. The two ends of the microbeam are clamped and subjected to the influence of a laser pulse with a temporal intensity profile. The analytical solutions to the physical fields are evaluated using the Laplace transform and its inversion transforms are performed numerically. The thermo-viscoelastic responses of the microbeam are calculated numerically and investigated graphically. The effect of different parameters such as viscosity, laser intensity, and the magnitude of the magnetic field are studied in detail.

Use of Kramers–Kronig relations to construct the master curves of asphalt materials

Kramers–Kronig (K–K) relations have been widely used to construct the master curves of asphalt materials that satisfy the linear viscoelastic (LVE) theory. However, existing methods utilize approximate K–K relations to develop the master curves of viscoelastic variables, which lead to the inaccuracy of master curves. In addition, the resulting master curves cannot fully comply with physical causality. To overcome these deficiencies, this paper proposes two approaches to construct the master curves by using these exact K–K relations: (1) the exact K–K relations between the dynamic modulus and the phase angle, and (2) the exact K–Krelations between the storage modulus and loss modulus. By applying the trapezoid integral rule, the numerical form of the K–K relations between the dynamic modulus and phase angle and between the storage modulus and loss modulus is established, and the master curves of four viscoelastic variables are subsequently constructed. The results indicate that both of the developed methods are accurate in establishing master curves. The fitting errors of each viscoelastic variable are below 2.64%, and the $$R^{{2}}$$ value is larger than 0.96. More important, the developed master curves fully comply with the LVE theory and physical causality. Therefore, the two presented methods apply to developing the master curves of asphalt binders, asphalt mixtures, and even other viscoelastic materials.

Effect of flow-independent viscosity on the propagation of Rayleigh wave in porous media

Abstract The propagation characteristics of Rayleigh wave near the surface of fluid-saturated poroviscoelastic medium are investigated in this paper. The effective stress in porous media is defined by the linear viscoelasticity theory based on Biot model and Kelvin-Voigt model. The dispersion equations of Rayleigh wave are derived analytically and then solved numerically to obtain its propagation speed and attenuation coefficient in the poroviscoelastic solid half-space. The effects of relaxation time and viscous coupling on the dispersion and attenuation of the Rayleigh wave as well as the body waves are discussed. The results indicate that the flow-independent viscosity has important influence on the propagation characteristics of Rayleigh waves, compared with the flow-dependent effect of viscous coupling between the pore fluid and solid skeleton.