Effective Time Theory Research Articles

Continuum Mechanics Models for Describing the Creep and Physical Aging Behaviors of Laminated Composites

Two types of carbon fiber-reinforced epoxy composite laminates are chosen for long-term tensile creep tests under different temperatures and load levels. Their time-dependent and non-monotonic deformations indicate clearly both temperature effect and physical aging effect. To characterize these viscoelastic behavior, two phenomenological constitutive models and one physical model are developed. The linear viscoelastic model based on the Boltzmann superposition principle is able to describe reasonably the deformations at relatively lower stress levels and temperatures. The nonlinear viscoelastic model of Schapery’s single-integral form, together with a usage of effective time theory, could describe nicely all the effects of temperature, stress, and physical aging. The physical model based on Ngai’s coupling mechanism concept is further combined with the framework of Schapery’s nonlinear viscoelastic theory, which may provide certain physical understanding about the effect of aging behavior on long-term creep deformation of the laminated composites. Numerical modelling by finite element method are implemented, and comparisons between the experimental and simulation results are demonstrated.

Characterization of nonlinear viscoelastic behaviors of CF/EP laminates with consideration to physical aging effect under thermo-mechanical loading

The characterization of the time-dependent deformation of fiber-reinforced polymer laminates is critical when evaluating the reliability of composite structures over long-term service lifetime. Several isothermal tensile creep tests under different temperatures and mechanical load levels for two types of carbon fiber-reinforced epoxy composite laminates are conducted in this work. The results indicate that the composites deformed in a time-dependent way and their axial strains are non-monotonic, which implies that both the creep temperature effect and physical aging effect come into play and the creep deformation was affected by the physical aging within the range of loads and temperatures involved in the tests. To formulate this kind of time-dependent behavior, a phenomenological nonlinear viscoelastic constitutive model of single-integral form was proposed and the effective time theory was introduced to describe the effects of temperature, stress, and physical aging. The measured strain was departed into two parts, one is the creep strain and the other is caused by the physical aging effect. The model was implemented within a finite element software by employing a recursive-iterative algorithm and was verified by comparing the simulation results with the experimental ones.

The influence of physical ageing on the in-plane shear creep compliance of 5HS C/PPS

Thermoplastic polymer-matrix composites, such as carbon woven fabric reinforced poly(phenylene sulphide) (C/PPS), are increasingly used in the aircraft industry. Primary structural applications, however, are limited due to uncertainty concerning the long-term behaviour. Recent work indicated a progressive creep response over time, which would render these materials unusable for such applications. However, the effect of physical ageing was neglected, which is well known to alleviate the creep behaviour and hence physical ageing is rigorously included in this study on the long-term creep response of C/PPS. Short-term tensile creep tests in the bias direction were performed at temperatures of 50, 60, 65, 70, 75 and 80^{circ}mbox{C} to obtain a master curve using the time–temperature superposition principle. Ordinary horizontal shifting failed to produce a smooth curve and therefore three alternative approaches were used and compared. The physical ageing rate was, however, characterised with horizontal shifting only at 50^{circ}mbox{C} and was implemented by means of the effective time theory (Struik, 1977) to correct the momentary master curves for the influence of physical ageing. The resulting predictions are more realistic and demonstrate that the structural changes in a material reduce the creep rate over time. Hence, the long-term creep compliance tends to increase asymptotically towards a finite value, in contrast to the unbounded momentary response.

Simulation study of N-hit SET variation in differential cascade voltage switch logical circuits

The advancement in the process leads to more concern about the Single Event (SE) sensitivity of the Differential Cascade Voltage Switch Logic (DCVSL) circuits. The simulation results indicate that the Single Event Transient (SET) generated at the DCVSL gate is much larger than that at the ordinary CMOS gate, and their SET variation is different. Based on charge collection, in this paper, the effective collection time theory is proposed to set forth the SET pulse generated at the DCVSL gate. Through 3D TCAD mixed-mode simulation in 65 nm twin-well bulk CMOS process, the effects on SET variation of device parameters such as well contact size and environment parameters such as voltage are investigated.

Effect of Tensile Strain on Thermal Properties of Graphene

ABSTRACTWe have employed a semicontinuum model to investigate the effect of tensile strain on thermal properties of graphene. Analytical expressions derived by Nihira and Iwata for phonon dispersion relations and vibrational density of states are employed, based on the semicontinuum model proposed by Komatsu and Nagamiya. The thermal conductivity is computed within the framework of Callaway’s effective relaxation time theory. It is found that thermal properties of graphene are quite sensitive to tensile strain. In the presence of tensile strain, the specific heat increases but the thermal conductivity decreases.

Thermal conductivity of graphene and graphite

The thermal conductivities of graphene and graphite are computed within the framework of Callaway's effective relaxation time theory. Analytical expressions derived by Nihira and Iwata for phonon dispersion relations and vibrational density of states are employed, based on the semicontinuum model proposed by Komatsu and Nagamiya. The conductivity of graphene is predicted to be higher than the in-plane conductivity of graphite for all temperatures. Incorporation of the ${}^{13}$C isotope can be expected to produce significant reduction in the conductivity of graphene in the temperature range 50--300 K. In the presence of tensile strain on graphene, the specific heat increases, but the conductivity can decrease or increase depending on the level of the purity and temperature of the sample.

Phonon conductivity in graphene

The phonon conductivity in suspended graphene and in graphene nanoribbons has been studied within the framework of Callaway’s effective relaxation time theory. The conductivity expression has been computed by employing analytical expressions for phonon dispersion relations and vibrational density of states based on the semicontinuum model by Nihira and Iwata. It is found that the Normal-drift contribution to the conductivity, arising from the consideration of the momentum conserving nature of three-phonon Normal processes, is very important for explaining the magnitude as well as the temperature dependence of the experimentally measured results for suspended graphene.

Hyper-Aging Dynamics of Nanoclay Suspension

Aqueous suspension of nanoclay Laponite undergoes structural evolution as a function of time, which enhances its elasticity and relaxation time. In this work, we employ an effective time approach to investigate long-term relaxation dynamics by carrying out creep experiments. Typically, we observe that the monotonic evolution of elastic modulus shifts to lower aging times, while maxima in viscous moduli get progressively broader for experiments carried out on a later date after preparation (idle time) of the nanoclay suspension. Application of effective time theory produces a superposition of all the creep curves irrespective of their initial state. The resulting dependence of the relaxation time on aging time shows very strong hyper-aging dynamics at short idle times, which progressively weakens to demonstrate a linear dependence in the limit of very long idle times. Remarkably, this behavior of nanoclay suspensions is akin to that observed for polymeric glasses. Consideration of aging as a first-order process suggests that continued hyper-aging dynamics causes cessation of aging. The dependence of relaxation time on aging time, therefore, must attenuate eventually producing linear or weaker dependence on time in order to approach a progressively low-energy state in the limit of very long times as observed experimentally. We also develop a simple scaling model based on a concept of aging of an energy well, which qualitatively captures various experimental observations very well, leading to profound insight into the hyper-aging dynamics of nanoclay suspensions.

The role of three-phonon Normal processes in the thermal conductivity of graphene

ABSTRACTWe have studied the thermal conductivity of graphene using Callaway’s effective relax-ation time theory and by employing analytical expressions for phonon dispersion relations and vibrational density of states based on the semicontinuum model by Nihira and Iwata. It is found that consideration of the momentum conserving nature of three-phonon Normal pro-cesses is very important for explaining the magnitude as well as the temperature dependence of the experimentally measured results. At room temperature, the N-drift contribution (the correction term in Callaway’s theory) provides 94% addition to the result obtained from the single-mode relaxation time theory, clearly suggesting that the single-mode relaxation time approach is inadequate for describing the phonon conductivity of graphene.

Delayed yielding in creep, time–stress superposition and effective time theory for a soft glass

In this work we investigate creep flow of aqueous suspension of Laponite, a model soft glassy material, at different aging times and stresses. We observe that this system shows time - aging time - stress superposition over a range of aging times and stresses when real time scale is transformed into effective time scale. Existence of superposition in an effective time domain facilitates prediction of long and very short time rheological behavior. Analysis of the observed behavior from effective time approach suggests that superposition is possible only when the shape of a relaxation time spectrum is preserved at various aging times and stresses. We also observe that creep curves at low aging times and greater stresses demonstrate delayed but sudden yielding. The critical time, at which material yields, increases with increase in aging time and decrease in stress. We argue that local rejuvenation of part of the glassy material causes variation in the rate of evolution of relaxation modes. The resulting interplay between aging and rejuvenating modes lead to delayed yielding as observed experimentally.

Simple method for obtaining viscoelastic parameters of polymeric materials by incorporating physical-aging effects

A simple method for obtaining viscoelastic parameters from the results of static tensile tests is presented herein. Viscoelastic parameters were obtained by fitting experimental results and calculated results based on the power law model and linear viscoelasticity. The static tensile tests were carried out at various pre-aging times and the effect of physical aging was determined. The data confirmed that the physical aging process has a significant effect on the viscoelastic behavior. A creep test was conducted in order to discuss the validity of the prediction using the results of the static tensile test. It was confirmed that the predictions based on the viscoelastic parameters obtained from static tensile tests cannot adequately model actual viscoelastic behavior. The effective time theory was incorporated into the prediction in order to account for the progress of physical aging. It was verified that incorporating effective time theory into the prediction allows for the precise prediction of the long-term viscoelastic behavior.

Long-term creep of polyphenylene sulfide (PPS) subjected to complex thermal histories: The effects of nonisothermal physical aging

The long-term viscoelastic behavior of polymeric materials used below the glass transition temperature ( T g ) is greatly affected by physical aging. In contrast to isothermal physical aging, long-term response under nonisothermal history has received far less attention. This paper reports experimental results and analytical methods of long-term creep behavior of polyphenylene sulfide (PPS) subjected to complex thermal histories in a temperature range below T g . To characterize the effects of aging, creep tests were performed using a dynamic mechanical analyzer (DMA). Besides the long-term data, short-term creep tests in identical thermal conditions were also analyzed; these were utilized with effective time theory to predict long-term response under both isothermal and nonisothermal temperature histories. The long-term compliance after a series of temperature changes was predicted by the effective time theory using the KAHR- a te model to obtain nonisothermal physical aging shift factors. Comparison of theoretical predictions with experimental data shows good agreement for various thermal histories.

Time-Temperature-Age Viscoelastic Behavior of Commercial Polymer Blends and Felt Filled Polymers

ABSTRACT This work presents an experimental investigation of creep behavior of polymer blends and felt-filled plastics used to rehabilitate deteriorated sewer pipelines, with emphasis on characterizing the effects of physical aging and temperature. The procedure for finding the aging shift rate μ is based on Struik's protocol and includes a novel method for rotating the data in addition to shifting. The master curve, obtained by Time Temperature Superposition (TTSP), of short-term data is shifted to the desired test temperature and initial age. A novel method is proposed to incorporate data from several specimens into a single averaged master curve and shift factor plot. Long-term behavior of the samples is predicted using the master curve and Effective Time Theory (ETT). The predicted long-term creep behavior is found to be close to the experimentally measured long-term creep behavior.

A new constitutive equation for the long-term creep of polymers based on physical aging

Based on the observation that during long-term creep the viscosity of polymers will continue to increase due to physical aging, a new constitutive equation is derived to describe the long-term creep behavior of polymers that are chrono-rheologically simple. The theory is developed using the concept of effective time for such materials whose long-term creep compliances with various aging times are characterized by a horizontal shift on the log(t)-scale. The derivation makes use of the basic mathematical structure for such a horizontal shift, with a result that is both sufficient and necessary. A linear viscosity function is found to be required for such a material, and the corresponding shift rate for both the long-term creep and the short-time creep is found to increase with aging time te, reaching an asymptotic value of unity. This theory improves Struik’s (1978) classic theory for the special class of chrono-rheologically simple materials, in that, when the aging time is sufficiently long, both theories are identical, but when it is short, the present one can account for the transition to the asymptotic state. The developed effectivetime theory is then extended to a polymer–matrix composite to predict the effect of physical aging on the long-term creep of a fiber-reinforced composite material.  2002 Editions scientifiques et medicales Elsevier SAS. All rights reserved.

Multi-Scale Modelling of Long-Term Mechanical Behaviour in Polymer Composite Laminates with Woven Fibre Architecture

A comprehensive analytical model for predicting the long-term durability of polymers and polymer matrix composites should in general take into account polymer viscoelastic/viscoplastic creep, hygrothermal effects, and the effects of physical and chemical ageing on material response. These effects, in turn are influenced by a multitude of factors such as polymer morphology, service temperature, ambient relative humidity, internal moisture concentrations, stacking sequence, fibre volume fraction, fibre architecture, applied stress level, degree of damage and ageing time. The primary objective of this paper is to present a multi-scale modelling methodology to simulate the long-term interlaminar properties in polymer matrix woven composites and then predict the critical regions where failure is most likely to occur. A micro-mechanics approach towards modelling the out-of-plane viscoelastic behaviour of a five-harness satin woven-fibre cross-ply composite laminate is presented, taking into consideration the weave architecture and time-dependent effects. In-plane properties are assumed to be dominated by the carbon fibres and are hence deemed elastic. The classical lamination theory model proposed by Raju and Wang is adapted to include the in-plane elastic behaviour of woven fibre composites. For the matrixdominated out-of-plane response, a viscoelastic creep model is employed to model the resin, based on Schapery's nonlinear viscoelastic constitutive law. In addition, physical ageing of the matrix has been included in the model, using the effective time theory proposed by Struik. Furthermore, the effect of large deflections and rotations on the time dependent out-of-plane behaviour is also investigated using the micro-mechanics model. The homogenized in-plane and out-of-plane compliance obtained using the proposed micro-mechanics methodology could be applied within the framework of a structural finite element code to model the macro-scale long-term behaviour of a woven fabric composite structure.

Long-term growth of core-set curl in poly(ethylene terephthalate) film

Solid polymeric films (plates) commonly exhibit only partial recovery after being subjected to a bending moment for some finite duration. This phenomenon, generally called bending recovery or core-set, has been described and successfully modeled in past studies using the viscoelastic plate bending theory. In this study, the viscoelastic theory of core-set curl was extended to account for physical aging effects during the bending recovery test. The contribution of physical aging was incorporated into the model through the effective time theory and assuming a thermorheologically simple behavior. The model was tested against long-term core-set data for a biaxially oriented poly(ethylene terephthalate) (PET) film under ambient conditions and was then used to map the effect of sample age on the evolution of core-set curl over long times. The predictions of the model agreed favorably with the data, and the results underscored the strong retarding effect of physical aging on the rate of acquisition of core-set curl in PET film at ambient temperature. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 1957–1967, 2001

Mechanical response of linear viscoelastic composite laminates incorporating non-isothermal physical aging effects

This paper presents a method of predicting the mechanical response of composite laminates including the effects of linear viscoelasticity and physical aging. Effective-time theory has been used to characterize the physical aging behavior of each linear viscoelastic lamina. In accordance with experimental findings, the aging behavior of each lamina is allowed to differ in the shear and transverse directions. The mechanical loading is restricted to the linear range, which decouples the aging and load behavior. A recursive algorithm has been used to solve the hereditary convolution integral that governs the response of each ply. Classical thin-laminate theory is then used to assemble the individual ply response equations and determine the overall laminate response to general in-plane force and moment loading. The method automatically recovers the ply-level stresses and strains, which are often critical to strength and durability predictions. The model can use either lamina compliance or modulus properties as its basis. Several illustrative examples of long-term laminate response to variable loading are presented and the impact of physical aging is explored. It is shown that for multidirectional laminates, the stiffness of the lamina in the fiber directions can allow simplifications of the model.

A continuous test data method to determine a reference curve and shift rate for isothermal physical aging

AbstractThe mechanical response of a material undergoing isothermal physical aging is governed by a reference curve (typically time‐dependent modulus or compliance) and a shift rate (which describes the rate of aging). In the standard analysis approach, these parameters are determined from a series of load/unload creep or relaxation tests, in which only the load portion data is used for property evaluation. In this paper, a new continuous test data (CTD) method is presented to determine these parameters using both the load and unload data from such a test series. This is accomplished by employing effective time theory to predict the full load/unload data set, with the reference curve and shift rate obtained as those which best fit the experimental data. It is demonstrated that the references curve obtained via the standard approach is not capable of predicting the full load/unload data set, while the CTD method using a Prony series fits this same data very well. Several concerns regarding the CTD method are addressed, including the differences in the CTD and standard method shift rate, the effect of vertical shifting or material nonlinearity on the results, and application of the method to other loading scenarios. Results are presented for long‐term and variable stress tests. The data used in the study was obtained from polyimide resin and polyimide/carbon fiber composite specimens at several temperatures below Tg.

Stress Relaxation in Semi-Crystalline PEN Films: Physical Aging Effects

The effect of physical aging on the stress relaxation response of a semicrystalline PEN film was studied at temperatures ranging from the secondaryB* transition (-70'C) up to its "amorphous" glass transition temperature (-120'C). Within this temperature range, the response is thermorheologically simple, i.e., the effect of physical aging is to shift the stress relaxation function along the log timescale to longer times following a simple powerlaw superposition rule with a characteristic temperaturedependent exponents (= aging rate parameter). They (T) function peaks at -80'C, and it follows a general pattern commonly observed in other polymer systems. Within the temperature range studied, the relaxation function also obeys timetemperature superposition and the corresponding shift factor appears to follow Arrhenius activation. The apparent activation energy is, however, strongly dependent on aging timeit increases nearly threefold over the range of aging times coveredwhich can be rationalized on the basis of the effective time theory.

Viscoelastic analysis of creep response for uniaxially-compressed laminates with initial deflection including the effect of physical aging

The viscoelastic behavior of simply-supported, cross-ply and angle-ply laminated plates with a physical aging phenomenon is examined. The laminates considered are under uniaxial inplane compression and have various amplitudes of initial imperfection. The analysis is based on a geometrically nonlinear viscoelastic formulation. The effective time theory is employed to construct the stress-strain relations. By assuming every component of relaxation moduli in each lamina has the same aging shift rate, the analysis is conducted on the effective time scale. The stress function is obtained by solving the compatibility equation by the use of the Laplace transform and a numerical integration scheme. Deflection is calculated from the moment equation by the Galerkin method in conjunction with the numerical integration. Numerical results for deflection history and edge shortening for the glass-reinforced vinyl ester laminates are presented. The effects of the physical aging, the pre-loading aging time, the amplitude of imperfection, the magnitude of loading, and the coupling relaxation moduli on the creep responses are illustrated in the numerical examples. Solutions based on the quasi-elastic approach are also presented for comparison.