Linear Viscoelastic Theory Research Articles

A Comparative View of Becker, Lomnitz, and Lambert Linear Viscoelastic Models

We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum in the Becker and Lomnitz models, as well as novel integral representations for the retardation and relaxation spectra in the Lambert model. In order to derive these analytical expressions, we have used the analytical properties of the exponential integral and the Lambert W function, as well as the Titchmarsh’s inversion formula of the Stieltjes transform. In addition, we prove some interesting inequalities by comparing the different models considered, as well as the non-negativity of the retardation and relaxation spectral functions. This means that the complete monotonicity of the rate of creep and the relaxation functions is satisfied, as required by the classical theory of linear viscoelasticity.

Nonuniversal Dynamics of Hyperbranched Metallo-Supramolecular Polymer Networks by the Spontaneous Formation of Nanoparticles.

Various transient and permanent bonds are commonly combined in increasingly complex hierarchical structures to achieve biomimetic functions, along with high mechanical properties. However, there is a traditional trade-off between mechanical strength and biological functions like self-healing. To fill this gap, we develop a metallo-supramolecular polymer hydrogel based on the hyperbranched poly(ethylene imine) (PEI) backbone and phenanthroline ligands, which have unexpectedly high plateau modulus at low concentrations. Rheological measurements demonstrate nonuniversal metal-ion-specific dynamics, with significantly larger plateau moduli, longer relaxation times, and stronger temperature dependencies, compared to equivalent networks based on model-type telechelic precursors, which cannot be explained by the theory of linear viscoelasticity. TEM images reveal the in situ mineralization of metal ions, which nucleate by the ligand complexation and grow thanks to the spontaneous reducing effect of the PEI backbone. Evidently, the complex lifetime works against Ostwald ripening, resulting in the formation of thermodynamically stable smaller particles. This trend is followed by time-dependent network buildup measurements and is confirmed by a kinetic model for particle formation and aggregation. The spontaneous formation of particles with complex lifetime-dependent sizes can explain the nonuniversal dynamics through the interaction of polymer segments and particles at the nanoscale. This work describes how the polymer backbone can affect the strength and stability of supramolecular bonds, promising for combining high mechanical properties and self-healing comparable to natural tissues.

LONGITUDINAL-RADIAL VIBRATIONS OF A VISCOELASTIC CYLINDRICAL THREE-LAYER STRUCTURE

The paper considers a cylindrical three-layer structure of arbitrary thickness made of viscoelastic material. It consists of two external bearing layers and a middle layer, the materials of which are generally different. The problem of nonstationary longitudinal-radial vibrations of such a structure is formulated. Based on the exact solutions in transformations of the three-dimensional problem of the linear theory of viscoelasticity for a circular cylindrical three-layer body, a mathematical model of its nonstationary longitudinal-radial vibrations is developed. Equations are derived that allow, based on the results of solving the vibration equations, to determine the stress-strain state of a cylindrical structure and its layers in arbitrary sections. The results obtained allow for special cases of transition into cylindrical viscoelastic and elastic two-layer structures, as well as into homogeneous single-layer cylindrical structures and round rods.

An efficient and robust method for evaluating the low-temperature performance of asphalt binder based on DSR testing

The BBR test has received growing concerns regarding its applicability to field studies and special materials due to the large amount of long-term aged asphalt binder used in preparing asphalt binder beams and the time-consuming beam preparation and testing process. In recent years, the application of a dynamic shear rheometer (DSR) to investigate the low-temperature performance of asphalt binder has attracted widespread attention because it uses significantly less asphalt binder for testing and allows for easy specimen preparation. The objective of this study was to develop an efficient and robust method to evaluate the low-temperature performance of asphalt binder based on DSR testing. Frequency sweep tests were performed on twelve asphalt binders at temperatures ranging from 0 to 25 °C using the 8-mm parallel plate geometry. Two conversions were applied to frequency sweep test data in order to predict the flexural creep stiffness and m-value of asphalt binder, namely the conversion from frequency domain to time domain and the conversion from shear loading to bending loading. The 1S2P1D model was employed to construct the master curves of storage modulus and loss modulus. Afterwards, the flexural creep stiffness was predicted using the shear creep compliance master curve model, which was derived from the 1S2P1D model using the inverse Laplace transform, and the Poisson’s ratio, which was assumed to be constant. It was found that the predicted values of S(60) and m(60) were comparable to the counterparts measured by the BBR, which well demonstrated the feasibility of the proposed DSR-BBR conversion method. In comparison to existing analysis methods, the proposed conversion method strictly complied with the linear viscoelastic theory, and it enabled efficient and robust predictions because the prediction model of flexural creep stiffness had the same parameters as those of the 1S2P1D model, thus avoiding multiple data fittings.

An efficient viscoelastic multiscale model for investigation on the cure‐induced microscopic residual stresses of composite laminates

AbstractThe cure‐induced residual stresses have a significant influence on the deformation and mechanical properties of composites. Moreover, the cure‐induced residual stresses exhibit multi‐level and multiscale distributions due to the complexity of micro‐structure of composites and anisotropic characteristics. In this paper, a computationally efficient multiscale procedure is proposed to predict the microscopic residual stresses induced by cure temperature, cure shrinkage and macroscopic residual stresses, which is based to the linear viscoelastic theory combined with the micro‐mechanics method. The instantaneous microscopic residual stresses are expressed as a linear combination of the instantaneous temperature increment, degree of cure increment and macroscopic strain increment at all times of the cure process until to the current time. The proposed method is verified in comparison with the published results in literature and the microscale FEA at different length scales, respectively. The results show that the proposed method is about a thousand times faster than directly using FEA in the microscale RVE model once the influence coefficient matrices are determined. At length, the microscale stresses of all nodes of the entire macroscale model are examined by using the proposed method. It is found that the thermal‐chemical‐mechanical loads lead to serious microscopic stresses for the matrix and interphase constituents, especially for those due to the macroscopic residual stresses.Highlights Microscale residual stress is written as a linear combination of macroscale loads. Applying macroscale loads on microscale FEA model to obtain influence matrices. The presented model avoids repeatedly FEA microscopic to improve the efficiency. The online time required for the proposed model is much less than microscale FEA. The microscale stress increases the risk of matrix cracking and interface bonding.

The linear viscoelastic fracture theory applies to soft solids better when they are…viscoelastic

Over the last half-century, linear viscoelastic models for crack growth in soft solids have flourished but their predictions have rarely been compared to experiments. In fact, most available models are either very approximate or cast in forms which are not quite suitable for the analysis of actual data. Here, we propose a linear viscoelastic approach which consistently exploits the dynamic mechanical analysis data. We apply this method to four sets of results documenting fracture or adhesion rupture in soft solids with various degrees of viscoelasticity. For elastomers, the results reproduce the well-known inconsistency of the process zone size. In more viscoelastic systems, however, the present approach is able to match the measured velocity dependence of the rupture energy with physically acceptable process zone sizes. Moreover, our predictions agree with the damage zone sizes measured by mechanoluminescence. Building on these results, we discuss various issues arising when evaluating the linear viscoelastic contribution to the rupture/adhesion energy in soft solids: data quality, physical interpretation of the parameters, validity of simpler approximations and limitations of the present approach.

Enhancing nanoscale viscoelasticity characterization in bimodal atomic force microscopy.

Polymeric, soft, and biological materials exhibit viscoelasticity, which is a time dependent mechanical response to deformation. Material viscoelasticity emerges from the movement of a material's constituent molecules at the nano- and microscale in response to applied deformation. Therefore, viscoelastic properties depend on the speed at which a material is deformed. Recent technological advances, especially in atomic force microscopy (AFM), have provided tools to measure and map material viscoelasticity with nanoscale resolution. However, to obtain additional information about the viscoelastic behavior of a material from such measurements, theoretical grounding during data analysis is required. For example, commercially available bimodal AFM imaging maps two different viscoelastic properties of a sample, the storage modulus, E', and loss tangent, tan δ, with each property being measured by a different resonance frequency of the AFM cantilever. While such techniques provide high resolution maps of E' and tan δ, the different measurement frequencies make it difficult to calculate key viscoelastic properties of the sample such as: the model of viscoelasticity that describes the sample, the loss modulus, E'', at either frequency, elasticity E, viscosity η, and characteristic response times τ. To overcome this difficulty, we present a new data analysis procedure derived from linear viscoelasticity theory. This procedure is applied and validated by performing amplitude modulation-frequency modulation (AM-FM) AFM, a commercially available bimodal imaging technique, on a styrene-butadiene rubber (SBR) with known mechanical behavior. The new analysis procedure correctly identified the type of viscoelasticity exhibited by the SBR and accurately calculated SBR E, η, and τ, providing a useful means of enhancing the amount of information gained about a sample's nanoscale viscoelastic properties from bimodal AFM measurements. Additionally, being derived from fundamental models of linear viscoelasticity, the procedure can be employed for any technique where different viscoelastic properties are measured at different and discrete frequencies with applied deformations in the linear viscoelastic regime of a sample.

Viscoelastic response of gray matter and white matter brain tissues under creep and relaxation

Accurate measurement of the mechanical properties of brain tissue is of paramount importance for understanding its mechanics-biology relationship. Most published studies on brain viscoelasticity have been conducted using a single relaxation test, without validating the validity of linear viscoelasticity, which is insufficient to establish an accurate constitutive equation for brain tissue. We obtained the creep and relaxation profiles of fresh adult porcine white matter (N = 120) and gray matter (N = 56) under finite step-and-hold uniaxial compression, using a mechanical testing machine, with 16.67 mm/s loading rate and 80 s hold time. These curves were employed to determine viscoelastic properties and demonstrated an excellent fit with a concise power-law function. The average initial modulus for gray matter (GM) was 6.619 kPa, higher than that for white matter under transverse loading (WM-2D) at 5.579 kPa (p < 0.01), yet lower than that for white matter under axial loading (WM-1D) at 6.759 kPa (p = 0.0121). Notably, WM-2D exhibited the highest degree of fluidity (β = 0.216). Our findings reveal that gray matter behaves as a linear viscoelastic material with power-law creep compliance and relaxation modulus. Conversely, the creep and relaxation behavior of white matter deviates from the verification relationship derived from linear viscoelastic theory, indicating its nonlinearity. This fact underscores the inaccuracy of assuming a linear constitutive relationship to characterize the viscoelastic properties of white matter. By combining the power-law function with the experimentally obtained creep compliance and relaxation modulus, we offer a unique approach to determining the viscoelastic characteristics of brain tissue.

Trans-scale dynamic shear-lag model for the impact performance of fiber-reinforced composites

This study proposes a new trans-scale dynamic shear-lag model that integrates the strain gradient linear viscoelasticity theory to investigate the effects of size and viscous softening on the dynamic mechanical properties and energy absorption characteristics of fiber-reinforced composites with micro- and nano-structures. The proposed model clarifies how the strain gradient effect can decrease energy dissipation in the matrix. Our investigation suggests that both classical and higher-order viscosities of the submicron-scale matrix can synergistically optimize energy dissipation. Furthermore, parametric studies demonstrate that the attenuation efficiency of absorbed energy changes non-monotonically with microstructure and constituent properties, such as matrix thickness, strain gradient elasticity parameter, classical viscosity, and high-order viscosity. The trans-scale dynamic shear-lag model advances our understanding of the energy dissipation in the submicron-scale viscoelastic matrix of fiber-reinforced composites and provides valuable guidance for developing composites with micro- and nano-structures that display exceptional dynamic properties.

The effect of instrument inertia on the initiation of oscillatory flow in stress controlled rheometry

In a recent paper [Hassager, J. Rheol. 64, 545–550 (2020)], Hassager performed an analysis of the start up of stress-controlled oscillatory flow based on the general theory of linear viscoelasticity. The analysis provided a theoretical basis for exploring the establishment of a steady strain offset that is inherent to stress controlled oscillatory rheometric protocols. However, the analysis neglected the impact of instrument inertia on the establishment of the steady periodic response. The inclusion of the inertia term in the framework is important since it (i) gives rise to inertio-elastic ringing and (ii) introduces an additional phase shift in the periodic part of the response. Herein, we modify the expressions to include an appropriate inertial contribution and demonstrate that the presence of the additional terms can have a substantial impact on the time scale required to attain the steady state periodic response. The analysis is then applied to an aqueous solution of wormlike micelles.

Triaxial Deformation of the Goldwyer Gas Shale at In Situ Stress Conditions—Part II: Viscoelastic Creep/Relaxation and Frictional Failure

To understand the geomechanical implications of long-term creep (time-dependent deformation) response of gas shale, short-duration creep was recorded from laboratory triaxial tests on ten Goldwyer gas shale samples in the onshore Canning Basin at in situ stress conditions under constant differential axial stress. A simple power-law function captures primary creep behaviour involving elastic compliance constant B and time-dependent factor n. Experimental creep data revealed larger axial creep strain in clay and organic-rich rocks, than those dominated by carbonates. Anisotropic nature of creep was observed depending upon the direction of constant axial stress application (perpendicular or parallel to the bedding plane). Upon the application of linear viscoelastic theory on laboratory creep fitting coefficients, differential horizontal stress accumulation over a geological time scale was estimated from the viscoelastic stress relaxation concept. Further, this model was used to derive lithology-dependent least principal stress (Shmin) magnitude at depth for two vertical wells intersecting the Goldwyer gas shale formations. This newly proposed Shmin model was found to have a profound influence on designing hydraulic fracture simulation. Further, pore size distribution and specific surface area value SN2 were derived from low-pressure gas adsorption experiments. These physical properties along with weak mineral components were linked with creep constitutive parameters to understand the physical mechanisms of creep. A strong correlation was noted between SN2 and creep parameters n and B. Finally, an attempt was made to investigate how gas shale composition and failure frictional properties can influence shear fracturing.

Creep Behavior of CLT Beams with Finite Thickness Layers of Flexible Adhesives.

Creep behavior of Cross-Laminated-Timber (CLT) beams with a finite-thickness layer of flexible adhesives is investigated. Creep tests were carried out for all component materials as well as for the composite structure itself. Three-point bending creep tests were performed for spruce planks and for CLT beams, and uniaxial compression creep tests were performed for two flexible polyurethane adhesives: Sika® PS and Sika® PMM. All materials are characterized with the use of the three-element Generalized Maxwell Model. The results of creep tests for component materials were used in elaboration of the Finite Element (FE) model. The problem of linear theory of viscoelasticity was solved numerically with the use of the Abaqus software. Obtained results of Finite Element Analysis (FEA) are compared with experimental results.

Study on Nonlinear Viscoelastic Constitutive Equations for Rock

Numerical simulation is one of the effective tools for the long-term stability assessment of underground structures. The reliability improvement of the numerical simulation requires precise constitutive equations to reproduce time-dependent behavior of rock such as loading-rate dependence, creep, and relaxation. These time-dependent behaviors of rock have nonlinearity between stress and strain rate and are influenced by surrounding environment. A number of constitutive equations have been proposed in previous studies; however, the equations based on the theory of linear viscoelasticity have difficulty reproducing accurately the time-dependent behavior of rock, and most of the equations based on the theory of viscoplasticity have been applied to the limited loading condition such as creep. In this article, nonlinear viscoelastic constitutive equations for rock proposed by the authors were introduced, and their exact solutions and application to experimental results were summarized. These equations can reproduce nonlinear time-dependent behavior and not only brittle but also ductile failures of rock and can be applied to the behavior under various loading conditions and experiment environment. In the equations, compliance or inelastic strain is assumed to be increased with time, but the functions and their combinations are of great variety. Hence, the differences of the solutions and applicability of the equations were detailed, and in addition future subjects were presented.

A practical biaxial time-dependent material model of PVDF-coated polyester fabrics for structural applications

The utilization of air-supported fabric structures as formwork to construct concrete shells has attracted increasing attention in recent decades. An example of such an airform is the air-supported structure at Dahaize, which consists of a cylindrical part and a hemispherical part, and is 66 m in diameter and 65 m in height. The precise forming analysis of large-span air-supported structures requires the use of a reasonable material model that considers the biaxial stress condition and the creep behavior of the fabric. In this paper, a practical approach to modeling the biaxial time-dependent material behavior of composite fabrics is proposed based on the uniaxial linear viscoelastic theory and the plane stress orthotropic assumption. The multi-Kelvin physical model is developed to the two-dimensional form for the simulation of biaxial creep compliances. Biaxial tensile and biaxial creep tests are conducted on PVDF-coated polyester fabrics to calibrate the material parameters. The experimental stress conditions were determined according to the pre-estimated actual membrane stress distribution. The established material model is employed for the forming analysis of the Dahaize airform under conditions of varying creep times. The simulation results agree well with the on-site measured data, and this validates the accuracy and efficiency of the proposed material model.

MODELING AND RESEARCH OF PROCESSES FOR CREEP COMPLIANCE AND RELAXATION BASED ON FADING MEMORY CONCEPT

Using of rheonomic materials in engineering structures is one of the most important issues in determining strength and durability. The main task of the mechanics of a deformable solid is to simulate the processes of deformation of viscoelastic materials. Currently, there are quite well-developed theories and methods of viscoelasticity that allow to determine and describe the viscoelastic properties of materials. They differ between linear and nonlinear viscoelasticity. In the linear and nonlinear theory of viscoelasticity, such a task is reduced to finding creep and relaxation kernel. Creep and relaxation kernel are interconnected by a known integral relationship that establishes a relationship between stress, strain and time. The work is devoted to modeling of stress relaxation and sttrain of hereditary materials. Stress relaxation is described by a nonlinear integral equation with an Abel’s kernel. New efficient method has been proposed for determining of parameters (α, δ). Bisection method is used for obtaining of parameter α. Algorithm have been developed for calculating of parameters of α and δ.

Experimental Study on Mechanism, Aging and Fatigue Performance of Warm Mixing Speed Melting SBS Modified Asphalt Binders

In this study, two kinds of quick melting modifier SBS-T and SBS-W, as well as the traditional SBS modifier, were used in the optimization design of asphalt binders. The changes in material structure and fatigue properties of three polymer-modified asphalt after adding 3% Sasobit to warm mix agent were investigated. The feasibility of SBS-T and SBS-W in asphalt binder was discussed from the modification mechanism and fatigue properties. In order to reveal the modification mechanism, the interaction mechanism between the fast-melting SBS modifier and the base asphalt was characterized by Fourier transform infrared spectroscopy (FTIR) and X-ray diffraction (XRD). The temperature sensitivity and viscoelastic properties of SBS-T and SBS-W modified binders were determined by frequency scanning (FS). The fatigue properties of SBS-T and SBS-W modified binders were tested by linear amplitude scanning (LAS). The results of FTIR showed that there was no chemical reaction between the SBS-T and SBS-W and the base asphalt. XRD results showed that SBS-W-modified asphalt has stronger fluidity. The results of FS and LAS showed that the asphalt binder with Sasobit has good stiffness and elastic recovery ability, and the same SBS-T and SBS-W have better temperature sensitivity and deformation resistance. In addition, the fatigue life of asphalt binder under the linear viscoelastic continuous damage theory is increased 3.9 times by SBS-W.

Elastoviscoplasticity, hyperaging, and time-age-time-temperature superposition in aqueous dispersions of bentonite clay.

Clay slurries are both ubiquitous and essential in the oil exploration industry, and are most commonly employed as drilling fluids. Due to its natural abundance, bentonite clay is often the de facto choice for these materials. Understanding and predicting the mechanical response of these fluids is critical for safe and efficient drilling operations. However, rheological modeling of bentonite clay suspensions is complicated by the fact that thermally-driven microscopic arrangements of particle aggregates lead to a continual evolution of the viscoelastic properties and the yield stress of the suspension with time. Ergodic relations fundamental to linear viscoelastic theory, such as the Boltzmann superposition principle, do not hold in this scenario of 'rheological aging'. We present an approach for modeling the linear viscoelastic response of aging bentonite suspensions across a range of temperatures that is based on the transformation from laboratory time to an effective 'material time' domain in which time-translation invariance holds, and the typical relations of non-aging linear viscoelastic theory apply. In particular, we model the constitutive relationship between stress and strain-rate in the bentonite suspensions as fractional Maxwell gels with constant relaxation dynamics in the material time domain, in parallel with a non-aging Newtonian viscous contribution to the total stress. This approach is supported by experimental measurements of the stress relaxation and rapid time-resolved measurements of the linear viscoelastic properties performed using optimized exponential chirps. This data is then reduced to master curves in the material domain using time-age-time superposition to obtain best fits of the model parameters over a range of operating temperatures.

Torsional vibrations of a rotating viskoelastic rod

A homogeneous and isotropic round viscoelastic rod rotating around its axis of symmetry with a constant angular velocity is considered in a cylindrical coordinate system. It is believed that the behavior of the rod is described within the framework of the linear theory of viscoelasticity, where the relationship between stresses and deformations is given in the form of relations Boltzmann-Volterra. In this case, the condition of its reversibility is imposed on the kernel of the integral operator. The equations of motion of the rod concerning non-zero stress components are written, taking into account the centrifugal force caused by the rotation. It is assumed that torsional vibrations are caused by given stress on its surface. A general equation of torsional vibrations of such a rod is obtained, which is an integral-differential equation of infinitely high order for the main part of the torsional displacement. Limiting the general equations to the zero and first approximations, the equations of the second and fourth orders are obtained, which, in the case of the absence of rotation, exactly coincide with the known equations of other authors. The resulting refined equation of the fourth order in its structure considers the angular velocity of rotation, the deformation of the transverse shear, and the inertia of rotation. Based on the derived oscillation equations, a particular problem is solved to study the influence of rotation and viscoelastic properties of the material on the stress-strain state of the rod, according to the results of which graphs of the dependencies of elastic and viscoelastic changes on time at points of three different sections of the rod are constructed. A comparative analysis of the results obtained with the results of other authors is performed.

A new strict decay rate for systems of longitudinal $ m $-nonlinear viscoelastic wave equations

&lt;abstract&gt;&lt;p&gt;Recent years have been marked by a significant increase in interest in solving nonlinear equations that arise in various fields of natural science. This trend is associated with the creation of a new method of mathematical physics. The present study is devoted to the analysis of the propagation of $ m $-nonlinear viscoelastic waves equations in an unbounded domain. The physical properties are determined by the equations of the linear theory of viscoelasticity. This article shows the main effect and interaction between the different weak and strong damping terms on the behavior of solutions. We found, under a novel condition on the kernel functions, an energy decay rate by using an appropriate energy estimates.&lt;/p&gt;&lt;/abstract&gt;

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

Relaxation experiments are performed on single edge notch tension (SENT) and T shape specimens consisting of different variations of polyampholyte (PA) hydrogels. These specimens share a common feature of containing regions of high and low stress/strain. This feature enables us to probe the load transfer mechanisms between permanent and dynamic networks of different strengths. This load transfer mechanism is connected to viscoelastic behavior. PA gels are nonlinear viscoelastic, with time-dependent behavior controlled by the breaking and reforming of ionic bonds in the dynamic network. In contrast to the prediction of linear viscoelastic theory, the displacement and strain fields during stress relaxation are not fixed but change with time in a complex way depending on the strength of the dynamic bonds and the observation time window. These experimental results are explained by a nonlinear viscoelastic model. Additional physical insights are gained by appealing to the nonlinear rheology of dynamic bonds or stickers. Our result shows that load transfer between networks can be controlled by modifying the strength of dynamic bonds.