Linear Viscoelastic Deformation Research Articles

A new effective phenomenological constitutive model for semi‐crystalline and amorphous polymers

AbstractA new phenomenological constitutive model is proposed for the prediction of the tensile and compressive behavior of semi‐crystalline and amorphous polymers. The new model modifies the phenomenological model previously proposed by Zhou and Mallick (ZM model) to correctly predict the complex behavior of thermoplastic materials including the linear viscoelastic deformation, the non‐linear viscoelastic deformation, the yielding, the post‐yield strain softening and the post‐yield strain hardening. A validation activity based on literature data has been carried out for polyether‐ether‐ketone (PEEK) and polycarbonate (PC) materials. The new model proved effective in fitting with high accuracy all the phases of the flow stress behavior for the considered materials, across a wide range of strain rates and temperature conditions. Finally, the comparison with Zhu et al., Duan et al., Nasraoui et al., Mulliken‐Boyce and Zhou‐Mallick models showed the better fitting performance of the proposed phenomenological model.Highlights Formulation of a new phenomenological model. Validation on mechanical test of semi‐crystalline and amorphous thermoplastics. Comparison with different phenomenological models present in the literature. Superior prediction results achieved compared to previously developed models.

Performance Assessment of High Reclaimed Asphalt Pavement Asphalt Mixtures with Recycling Agents

The increased use of reclaimed asphalt pavement (RAP) in asphalt mixtures has promoted the use of recycling agents (RAs) to mitigate the effect of oxidized materials on the performance of asphalt mixtures. This laboratory-based study assesses the linear viscoelastic (LVE), cracking resistance, and permanent deformation properties of recycled asphalt mixtures containing RAs. The dosage levels used for this study were determined based on the manufacturer’s recommendation to restore the blended binder system’s low-temperature performance grade (PGL) to −22°C. The study found that blending a performance grade (PG) 64-22 virgin binder with RAs in high RAP mixtures can yield a comparable LVE response to high RAP mixtures designed using a softer binder (PG 58-28). Moreover, the study revealed that the cracking performance of high RAP mixtures could be improved without jeopardizing the rutting potential of the material by using RAs at dosage levels selected to yield a blended binder PGL of −22°C.

Evaluation of Linear Deformation and Unloading Stiffness Characteristics of Asphalt Mixtures Incorporating Various Aggregate Gradations

Optimum stiffness and linear deformation in the unloading phase are fundamental properties of asphalt mixtures required for the durability of flexible pavements. In this research, blends of six different aggregate gradations were used for two base course (BC) and four wearing course (WC) asphalt mixtures. Stability and indirect tensile strength of resulting asphalt mixtures were evaluated to relate to viscoelastic unloading deformation and resilient moduli (instantaneous (MRI) and total (MRT)) at 25 °C using a 40/50 binder for 0.1 and 0.3 s load durations. Results indicated that an increase in coarse aggregate proportion from 48 to 70% for BC has shown a 12% and 14% increase in MRT for 0.1 and 0.3 s load durations, respectively, and an increase in coarse aggregate proportion from 41 to 57.5% for WC has caused a 26% and 20% increase in MRI for 0.1 and 0.3 s load durations, respectively. The same coarse aggregate proportions showed an increase in linear viscoelastic deformation at 0.1 s load duration from 54.6 to 68.2% for WC and from 53.0 to 62.7% for BC, whereas for 0.3 s load duration linear viscoelastic deformation increased from 58.1 to 69.1% for WC and 64.3 to 69.2% for BC. The findings of this study will assist in the selection of aggregate gradations to be used in wearing and base course asphalt mixtures for pavement design, construction and maintenance.

On the Influence of Viscoelastic Modeling in Fluid Flow Simulations of Gum Acrylonitrile Butadiene Rubber.

Computational fluid dynamics (CFD) simulation is an important tool as it enables engineers to study different design options without a time-consuming experimental workload. However, the prediction accuracy of any CFD simulation depends upon the set boundary conditions and upon the applied rheological constitutive equation. In the present study the viscoelastic nature of an unfilled gum acrylonitrile butadiene rubber (NBR) is considered by applying the integral and time-dependent Kaye–Bernstein–Kearsley–Zapas (K-BKZ) rheological model. First, exhaustive testing is carried out in the linear viscoelastic (LVE) and non-LVE deformation range including small amplitude oscillatory shear (SAOS) as well as high pressure capillary rheometer (HPCR) tests. Next, three abrupt capillary dies and one tapered orifice die are modeled in Ansys POLYFLOW. The pressure prediction accuracy of the K-BKZ/Wagner model was found to be excellent and insensitive to the applied normal force in SAOS testing as well as to the relation of first and second normal stress differences, provided that damping parameters are fitted to steady-state rheological data. Moreover, the crucial importance of viscoelastic modeling is proven for rubber materials, as two generalized Newtonian fluid (GNF) flow models severely underestimate measured pressure data, especially in contraction flow-dominated geometries.

A time-domain recursive method of SH-wave propagation through the filled fracture with linear viscoelastic deformation behavior

ABSTRACT The study of SH-wave across fractures can help uncover and improve the understanding of seismic wave propagation across geological media with fractures. Considering the viscoelastic behavior of a filled medium, a time-domain recursive method of SH-wave propagation through filled viscoelastic fractures was established. A displacement and stress discontinuous model and a differential constitutive relationship of the viscoelastic model were adopted to describe viscoelastic deformation behavior and mass of filled medium in fractures. Three viscoelastic models were utilized to analyze the effects of different viscoelastic models on wave propagation. The effects of frequency, incident angle, specific stiffness, and specific viscosity on the transmission and reflection coefficients and energy dissipation rate were discussed. Also, the influence of fracture slip behavior and the effects of filling mass on wave propagation were also addressed and the results indicated that the method established here was effective for investigating the propagation of SH-wave across viscoelastic filled fractures. Viscosity coupling might have caused a phase delay when the wave stress reached the shear strength and the filling mass had a subtle influence on SH-wave propagation through filled fractures with a thinner interlayer.

Study on Correction Method for Die Position Deviation Caused by Adhesive Tape Puncture

Accurate peeling off and transferring of microchips play a critical role in LED die sorting technology. Generally, sorting efficiency of more than 36 KUPH, and an alignment deviation less than 1 mil ( $25.4~\mu \text{m}$ ) is required. High precision position control and cooperation of multifactors should be implemented precisely. Among all the factors, the film deformation is an important one. As the sorting proceeds, dies are peeled off from substrates one by one, and the adhesive substrate is penetrated through a hole each time. As the number of stripped chips increases, viscoelastic deformation of tape surface takes place and tiny position shift of die adhered on tape arises. This shift may cause a negative impact on the peeling process. In this paper, linear viscoelastic deformation impact is tested with well-designed experiments. To evaluate the effect of tape deformation accurately for sorting the performance, stochastic Petri net model of the transferring system with dual-independent arms is formulated to analyze the influence of factors. On this basis, strategy on coordinating motion control and scheduling is proposed, rational velocity section is planned, and the active position compensation of supply platform is set, so that the die can be peeled off correctly and efficiently. Finally, to improve sorting performance, methods on active compensation of aligning platform are proposed and proven effective by an experimental validation.

Propagation of the Stress Wave Through the Filled Joint with Linear Viscoelastic Deformation Behavior Using Time-Domain Recursive Method

The dynamic behavior of filled joints is mostly controlled by the filled medium. In addition to nonlinear elastic behavior, viscoelastic behavior of filled joints is also of great significance. Here, a theoretical study of stress wave propagation through a filled rock joint with linear viscoelastic deformation behavior has been carried out using a modified time-domain recursive method (TDRM). A displacement discontinuity model was extended to form a displacement and stress discontinuity model, and the differential constitutive relationship of viscoelastic model was adopted to introduce the mass and viscoelastic behavior of filled medium. A standard linear solid model, which can be degenerated into the Kelvin and Maxwell models, was adopted in deriving this method. Transmission and reflection coefficients were adopted to verify this method. Besides, the effects of some parameters on wave propagation across a filled rock joint with linear viscoelastic deformation behavior were discussed. Then, a comparison of the time-history curves calculated by the present method with those by frequency-domain method (FDM) was performed. The results indicated that change tendencies of the transmission and reflection coefficients for these viscoelastic models versus incident angle were the same as each other but not frequency. The mass and viscosity coupling of filled medium did not fundamentally change wave propagation. The modified TDRM was found to be more efficient than the FDM.

Rheological Parameters as Affected by Water Tension in Subtropical Soils

Rheological parameters have been used to study the interaction between particles and the structural strength of soils subjected to mechanical stresses, in which soil composition and water content most strongly affect soil resistance to deformation. Our objective was to evaluate the effect of water tension on rheological parameters of soils with different mineralogical, physical, and chemical composition. Surface and subsurface horizons of four Oxisols, two Ultisols, one Alfisol, and one Vertisol were physically and chemically characterized; their rheological parameters were obtained from amplitude sweep tests under oscillatory shear on disturbed soil samples that were saturated and subjected to water tension of 1, 3, 6, and 10 kPa. In these samples, the rheological parameters linear viscoelastic deformation limit (γL), maximum shear stress (τmax), and integral z were determined. By simple regression analysis of the rheological parameters as a function of soil water tension, we observed increased mechanical strength with increasing water tension up to at least 6 kPa, primarily due to increased capillary forces in the soil. However, increased elasticity assessed by γL was not as expressive as the increase in structural rigidity assessed by τmax and integral z. Elastic deformation of the soil (γL) increases with the increase in the number of bonds among particles, which depend on the clay, total carbon, expansive clay mineral, and cation contents; however, maximum shear resistance (τmax) and structural stiffness (integral z) mainly increase with clay, kaolinite, and oxide content by increasing the strength of interparticle bonds. A decrease in mechanical strength occurs for water tension of 10 kPa (the lowest water content evaluated) in sandy horizons or in horizons with a high proportion of resistant microaggregates (pseudosand), when associated with low bulk density, due to fewer points of contact between soil particles and therefore lower capillary force.

Numerical Investigation of the Influence of Viscoelastic Deformation on the Pressure-Leakage Behavior of Plastic Pipes

AbstractIt has been well established that leakage from pipes is more sensitive to changes in pressure than the square root relationship predicted by the orifice equation. The main reason for this is that leak areas are not static, but vary with changes in system pressure. While previous studies have shown that the leak area is a linear function of pressure for elastic materials, the effect of the viscoelastic behavior of plastic pipes on the pressure-leakage response is not yet well understood. In this study, finite element analysis was used to investigate the effect of viscoelastic behavior on round holes and longitudinal cracks in both high density polyethylene (HDPE) and polyvinylchloride (PVC) pipes. The standard differential equation for linear viscoelastic deformation was then calibrated to the finite element results, resulting in equations that accurately describe the viscoelastic variation in leak areas as functions of time. The results also showed that the time-dependent behaviors of both pipe ma...

Multiscale testing-analysis of asphaltic materials considering viscoelastic and viscoplastic deformation

Fine aggregate matrix (FAM) is a phase consisting of asphalt binder, air voids, fine aggregates and fillers. It acts as a primary phase in evaluating the damage and deformation of entire asphalt concrete mixtures. The simplicity, repeatability and efficiency of the FAM testing make it a very attractive specification-type approach for evaluating the performance characteristics of the entire asphalt concrete mixtures. This study explores a linkage in the deformation characteristics between the two length scales: asphalt concrete mixture scale and its corresponding FAM scale. To that end, a simple creep-recovery test was conducted for both mixtures (i.e. asphalt concrete mixture and its corresponding FAM phase) at various stress levels. Test results were compared and analysed using Schapery’s single-integral viscoelastic theory and Perzyna-type viscoplasticity with a generalised Drucker–Prager yield surface. In particular, stress-dependent nonlinear viscoelastic and viscoplastic behaviours were characterised in addition to linear viscoelastic deformation characteristics, because the nonlinear viscoelastic and viscoplastic behaviours are considered significant in asphalt pavements that are subjected to heavy vehicle loads and elevated service temperatures. With a limited scope and test-analysis results at this stage, it was found that there is a strong link between the FAM and asphalt concrete in (linear and nonlinear) viscoelastic and viscoplastic deformation characteristics. This implies that the viscoelastic stiffness characteristics and viscoplastic hardening of typical asphalt concrete mixtures could be estimated or predicted from the simple FAM-based testing-analysis method, which can significantly reduce the experimental–analytical efforts required for asphalt concrete mixtures.

Interconversion Relationships for Completely Monotone Functions

In linear viscoelasticity, the analytically valid Volterra convolution interconversion relationships between the relaxation modulus $G$ and the corresponding creep compliance (retardation) modulus $J$ play a fundamental role. They allow $J$ to be determined from both theoretical and experimental estimates of $G$ and conversely. In order to guarantee conservation of energy for the related models of linear viscoelastic flow and deformation processes, some assumption such as the complete monotonicity of $G$ and $dJ/dt$ needs to be invoked. Interesting theoretical questions thereby arise about the analytic and structural properties of $G$ and $J$. Gross [\em Actualités Sci. Ind.} 1190, Hermann, Paris, 1953] appears to have been the first to derive analytical expressions for $G$ in terms of $J$ and conversely. However, the regularity invoked only guarantees their validity for a subset of all possible completely monotone functions. The purpose of this paper is an investigation of the extent to which these results extend to all completely monotone functions. This allows issues associated with the effect of perturbations in $G$ on $J$, and conversely, to be placed on a rigorous footing. In particular, it is shown, among other things, that $G$ (resp., $dJ/dt$) having absolutely continuous generating measure does not necessarily guarantee the same for $dJ/dt$ (resp., $G$). Our results have been derived by using the equivalent resolvent kernel equation that comes from a double differentiation of the interconversion equation. Consequently, they will hold more generally for resolvent kernel equations.

Mechanical Fluidity of Fully Suspended Biological Cells

Mechanical characteristics of single biological cells are used to identify and possibly leverage interesting differences among cells or cell populations. Fluidity—hysteresivity normalized to the extremes of an elastic solid or a viscous liquid—can be extracted from, and compared among, multiple rheological measurements of cells: creep compliance versus time, complex modulus versus frequency, and phase lag versus frequency. With multiple strategies available for acquisition of this nondimensional property, fluidity may serve as a useful and robust parameter for distinguishing cell populations, and for understanding the physical origins of deformability in soft matter. Here, for three disparate eukaryotic cell types deformed in the suspended state via optical stretching, we examine the dependence of fluidity on chemical and environmental influences at a timescale of ∼1 s. We find that fluidity estimates are consistent in the time and frequency domains under a structural damping (power-law or fractional-derivative) model, but not under an equivalent-complexity, lumped-component (spring-dashpot) model; the latter predicts spurious time constants. Although fluidity is suppressed by chemical cross-linking, we find that ATP depletion in the cell does not measurably alter the parameter, and we thus conclude that active ATP-driven events are not a crucial enabler of fluidity during linear viscoelastic deformation of a suspended cell. Finally, by using the capacity of optical stretching to produce near-instantaneous increases in cell temperature, we establish that fluidity increases with temperature—now measured in a fully suspended, sortable cell without the complicating factor of cell-substratum adhesion.

The use of stress sweep tests for asphalt mixtures nonlinear viscoelastic and fatigue damage responses identification

Asphaltic materials are known to present a behavior that can be approximated by the theory of viscoelasticity. For these materials it is essential to characterize fatigue damage. An important aspect therein is the separation between nonlinear viscoelastic and fatigue damage responses. This is a complex issue, since both nonlinearity and damage have a similar effect on the overall material mechanical behavior, i.e. decrease in the stiffness and increase in the phase angle. This paper presents an experimental and a mathematical procedure to separate the nonlinear viscoelastic from the fatigue damage response for asphaltic materials. Stress sweep tests were used to characterize a hot mixture asphalt at nine conditions (three temperatures and three frequencies). Once all strain values were obtained in a stress controlled sweep test, a statistical analysis was used to find the maximum stress that can be applied to the material without invoking the damage response. The results showed that the transition stress value is directly associated with material properties, the stiffness being an important factor in this result. Consequently, stress, temperature and frequency determine together the mechanical response of the material (linear or nonlinear viscoelastic, fatigue damage and/or plastic deformation). Results from this study can be associated with other fatigue damage approaches in order to better select the stress or strain amplitude that should be used in fatigue tests, and to eliminate the amount of energy that is dissipated in the nonlinear viscoelastic region.

The Great Myths of Rheology Part III: Elasticity of the Network of Entanglements

The dynamic data of polymer melts are analyzed in a novel way, presenting new correlations between the viscosity, G′ and G′′ (the elastic and loss moduli), and strain rate and the implications of the new formulas on our understanding of melt entanglement network elasticity are discussed. In the two previous articles of this series, Part I and Part II, we showed that the existing models valid in the linear viscoelastic deformation range were not adequate to extrapolate to the nonlinear regime, suggesting that the stability of the network of entanglements was at the center of the discrepancies. In this article, we introduce new tools for the analysis of the dynamic data and suggest new ideas for the understanding of melt deformation based on this different focus. In particular, we express classical concepts, such as shear-thinning, melt diffusion or melt elasticity and viscosity, in a different context, that of the existence of a dual-phase interaction, essential to our treatment of the statistics of interaction of the bonds responsible for the system coherence and cohesion. It is within this framework that viscoelasticity parameters emerge and the new view of the deformation of a polymer melt results in a different definition of the entanglement network.

Assessment of ZSV in Asphalt Concrete Using Shear Frequency Sweep Testing

This study elaborates on methods for assessing the material property called zero shear-rate viscosity (ZSV) in asphalt concrete based on shear frequency sweep testing. The purpose of ZSV assessment is for linear viscoelastic permanent deformation modeling. The basic approach was to fit parameters of the Laplace-transformed Burger’s model to the measured shear dynamic modulus data. This method produced multiple solutions, although an approximate measure of the minimum ZSV could be estimated. Therefore, extrapolation was necessary to assess the frequency-independent ZSV based on the measured frequency sweep data. The method was complemented with the simplified Cross extrapolation model that was successfully fitted to the measured data. The extended method also produced multiple solutions, although the minimum ZSV could be assessed unambiguously. The results were in accordance with expectations based on measured dynamic shear modulus and phase angle.

Articular human joint modelling

SUMMARYThe work reported in this paper encapsulates the theories and algorithms developed to drive the core analysis modules of the software which has been developed to model a musculoskeletal structure of anatomic joints. Due to local bone surface and contact geometry based joint kinematics, newly developed algorithms make the proposed modeller different from currently available modellers. There are many modellers that are capable of modelling gross human body motion. Nevertheless, none of the available modellers offer complete elements of joint modelling. It appears that joint modelling is an extension of their core analysis capability, which, in every case, appears to be musculoskeletal motion dynamics. It is felt that an analysis framework that is focused on human joints would have significant benefit and potential to be used in many orthopaedic applications. The local mobility of joints has a significant influence in human motion analysis, in understanding of joint loading, tissue behaviour and contact forces. However, in order to develop a bone surface based joint modeller, there are a number of major problems, from tissue idealizations to surface geometry discretization and non-linear motion analysis. This paper presents the following: (a) The physical deformation of biological tissues as linear or non-linear viscoelastic deformation, based on spring-dashpot elements. (b) The linear dynamic multibody modelling, where the linear formulation is established for small motions and is particularly useful for calculating the equilibrium position of the joint. This model can also be used for finding small motion behaviour or loading under static conditions. It also has the potential of quantifying the joint laxity. (c) The non-linear dynamic multibody modelling, where a non-matrix and algorithmic formulation is presented. The approach allows handling complex material and geometrical nonlinearity easily. (d) Shortest path algorithms for calculating soft tissue line of action geometries. The developed algorithms are based on calculating minimum ‘surface mass’ and ‘surface covariance’. An improved version of the ‘surface covariance’ algorithm is described as ‘residual covariance’. The resulting path is used to establish the direction of forces and moments acting on joints. This information is needed for linear or non-linear treatment of the joint motion. (e) The final contribution of the paper is the treatment of the collision. In the virtual world, the difficulty in analysing bodies in motion arises due to body interpenetrations. The collision algorithm proposed in the paper involves finding the shortest projected ray from one body to the other. The projection of the body is determined by the resultant forces acting on it due to soft tissue connections under tension. This enables the calculation of collision condition of non-convex objects accurately. After the initial collision detection, the analysis involves attaching special springs (stiffness only normal to the surfaces) at the ‘potentially colliding points’ and motion of bodies is recalculated. The collision algorithm incorporates the rotation as well as translation. The algorithm continues until the joint equilibrium is achieved. Finally, the results obtained based on the software are compared with experimental results obtained using cadaveric joints.

Nonlinear viscoelastic deformation of polypropylene threads under finite strains

An approach to the concretization of governing equations of state describing long-term deformation processes in multifilament polypropylene structures is proposed. Models of linear and nonlinear viscoelastic deformation of materials under finite strains are considered. The range of the applicability of the theory which uses the hypothesis of small strains has been determined, and the effectiveness of the proposed defining relations in the description of creep processes has been analyzed.

Short-term creep and strength of fibrous polypropylene structures

The short-term creep and strength of fibrous polypropylene structures are investigated. On the basis of these characteristics, we develop the models of linear and nonlinear viscoelastic deformation of materials, specify the fields of their applicability, and study criteria used for the evaluation of the static strength and durability of these composites.

A variable order constitutive relation for viscoelasticity

AbstractA constitutive relation for linear viscoelasticity of composite materials is formulated using the novel concept of Variable Order (VO) differintegrals. In the model proposed in this work, the order of the derivative is allowed to be a function of the independent variable (time), rather than a constant of arbitrary order. We generalize previous works that used fractional derivatives for the stress and strain relationship by allowing a continuous spectrum of non‐integer dynamics to describe the physical problem. Starting with the assumption that the order of the derivative is a measure of the rate of change of disorder within the material, we develop a statistical mechanical model that is in agreement with experimental results for strain rates varying more than eight orders of magnitude in value. We use experimental data for an epoxy resin and a carbon/epoxy composite undergoing constant compression rates in order to derive a VO constitutive equation that accurately models the linear viscoelastic deformation in time. The resulting dimensionless constitutive equation agrees well with all the normalized data while using a much smaller number of empirical coefficients when compared to available models in the literature.

A variable order constitutive relation for viscoelasticity

A constitutive relation for linear viscoelasticity of composite materials is formulated using the novel concept of Variable Order (VO) differintegrals. In the model proposed in this work, the order of the derivative is allowed to be a function of the independent variable (time), rather than a constant of arbitrary order. We generalize previous works that used fractional derivatives for the stress and strain relationship by allowing a continuous spectrum of non-integer dynamics to describe the physical problem. Starting with the assumption that the order of the derivative is a measure of the rate of change of disorder within the material, we develop a statistical mechanical model that is in agreement with experimental results for strain rates varying more than eight orders of magnitude in value. We use experimental data for an epoxy resin and a carbon/epoxy composite undergoing constant compression rates in order to derive a VO constitutive equation that accurately models the linear viscoelastic deformation in time. The resulting dimensionless constitutive equation agrees well with all the normalized data while using a much smaller number of empirical coefficients when compared to available models in the literature.