Continuous Retardation Spectrum Research Articles

An improved approach for the continuous retardation spectra of concrete creep and applications

Creep is an important physical property of concrete and can lead to additional displacement, stress redistribution, and even cracking in concrete structures, inducing prestress loss of large-scale prestressed concrete structures. When an exponential algorithm is used to calculate the long-term creep of concrete, it is usually necessary to apply the continuous retardation spectra of the material. In the improved approach proposed here, the continuous retardation spectra can be obtained by the Weeks inverse Laplace transform. The CEB MC90 creep model is taken as an example to analyze the computational process, efficiency, and error of the approach. The improved approach is further applied to the ACI 209R-92, JSCE, and GL2000 concrete creep models. Through comparison with other methods, the advantages of the improved approach are illustrated, and some useful conclusions are drawn.

Continuous spectrum approach to evaluate the effect of binder modification and aging

ABSTRACT The present work proposes a novel mathematical approach to directly obtain the continuous spectrum of asphalt binder from frequency domain data. Using the asymmetric sigmoidal function to obtain continuous spectrum offers advantages of flexibility and improvement in describing the spectrum well. Initially, complex modulus, storage modulus, and storage compliance master curves were constructed using experimental data. In the next stage, the continuous relaxation and retardation spectrum were constructed directly from the storage modulus and storage compliance master curves, respectively. Statistical indicators like mean, variance, and skewness were used to describe the shape of these finalised continuous spectra. To check the efficacy of the proposed approach, asphalt binders altered with modifiers (at different dosages) were aged under laboratory conditions. Critical evaluation of the results indicated that the underlying changes could be captured using the continuous spectra (which were not obvious with respective master curves or black space diagrams). Continuous spectra of binders constructed (with different dosages of modifier and under different aging conditions) were compared using the statistical indicators. Thus, the spectral shape parameters could capture the effect of aging and modifier dosage. These changes can be attributed to molecular changes, chemical reactions occurring with the addition of modifiers and aging.

Predicting the Time‐dependent Mechanics of Concrete Based on a Multiscale Model

To accurately predict the time‐dependent deformation of concrete, a multiscale model with its focus pinned on mesoscale is proposed here to break down the constitutive law of concrete to the mechanics of its different constituent phases. A three‐phase unit cell, consisting of one coarse aggregate, mortar matrix, and the interfacial transition zone (ITZ), is employed to represent the basic structural element of concrete on mesoscale. Following Eshelby’s inclusion theory, the Mori‐Tanaka homogenization, continuous retardation spectrum method, and isotropic continuum damage model are applied to capture the time‐dependent behavior of the unit cell. To take into account the shape effect of aggregate, the explicit Eshelby’s tensor of polygonal inclusion is obtained based on an enhanced approach. The proposed multiscale material model is incorporated into ABAQUS, and its effectiveness and robustness are documented by the simulations of unit cells containing aggregates of different shapes.

The Discrete and Continuous Retardation and Relaxation Spectrum Method for Viscoelastic Characterization of Warm Mix Crumb Rubber-Modified Asphalt Mixtures.

To study the linear viscoelastic (LVE) of crumb rubber-modified asphalt mixtures before and after the warm mix additive was added methods of obtaining the discrete and continuous spectrum are presented. Besides, the relaxation modulus and creep compliance are constructed from the discrete and continuous spectrum, respectively. The discrete spectrum of asphalt mixtures can be obtained from dynamic modulus test results according to the generalized Maxwell model (GMM) and the generalized Kelvin model (GKM). Similarly, the continuous spectrum of asphalt mixtures can be obtained from the dynamic modulus test data via the inverse integral transformation. In this paper, the test procedure for all specimens was ensured to be completed in the LVE range. The results show that the discrete spectrum and the continuous spectrum have similar shapes, but the magnitude and position of the spectrum peaks is different. The continuous spectrum can be considered as the limiting case of the discrete spectrum. The relaxation modulus and creep compliance constructed by the discrete and continuous spectrum are almost indistinguishable in the reduced time range of 10−5 s–103 s. However, there are more significant errors outside the time range, and the maximum error is up to 55%.

Tridimensional Long-Term Finite Element Analysis of Reinforced Concrete Structures with Rate-Type Creep Approach

This paper presents a general procedure for a rate-type creep analysis (based on the use of the continuous retardation spectrum) which avoids the need of recalculating the Kelvin chain stiffness elements at each time step. In this procedure are incorporated three different creep constitutive relations, two recommended by national codes such as the ACI (North-American) and EC2 (European) building codes and one by the RILEM research association. The approximate expressions of the different creep functions with the corresponding Dirichlet series are generated using the continuous retardation spectrum approach based on the Post–Widder formula. The proposed rate-type formulation is implemented into a 3D finite element code and applied to study the long-term deflections of a prestressed concrete bridge built in Romania, which crosses a wide artificial channel that connects the Danube river to the port of Constanta in the Black Sea.

A framework for linear viscoelastic characterization of asphalt mixtures

The master curve of a viscoelastic variable is of significance due to its capability of characterizing the linear viscoelastic (LVE) property in an extended time or frequency range. However, the master curves constructed using the traditional approach fail to strictly comply with the LVE theory, leading to inaccurate predictions in the extended range. In order to address this issue, a framework was developed for the LVE characterization of asphalt mixtures. The generalized logistic sigmoidal model was adopted as the master curve model of storage modulus. A numerical model of loss modulus was established in relation to the continuous relaxation spectrum, whose mathematical model was derived in light of its relationship with the storage modulus. The model parameters determined using the storage modulus and loss modulus test data were employed to construct the master curves of storage modulus, loss modulus, dynamic modulus and phase angle. Then the relaxation modulus master curve was generated by establishing a numerical model. Afterwards, the continuous retardation spectrum was solved numerically based on its relationship with the continuous relaxation spectrum. The master curves of storage compliance, loss compliance and creep compliance were obtained using the corresponding numerical models that were established with respect to the continuous retardation spectrum. The interrelationship among the viscoelastic variables was then employed to obtain the dynamic compliance and phase angle master curves. It was demonstrated that the developed framework ensured the master curves of all viscoelastic variables complied with the LVE theory.

Anisotropic microplane constitutive model for coupling creep and damage in layered geomaterials such as gas or oil shale

An anisotropic constitutive model for coupling creep with damage of shale and other anisotropic geomaterials under complex loading paths is formulated. The material damage is described by a previously developed spherocylindrical microplane model, which can simulate not only the stress-induced incremental anisotropy but also the inherent material anisotropy. Like material damage, the creep must be expected to be anisotropic as well. The creep responses of shale with inherent transverse isotropy are first separately constructed for the loading by compressive principal stress oriented either (i) parallel or (ii) normal to the bedding planes, and (iii) for shear stress along the bedding planes produced by the differences of the bedding plane inclination angle. This leads to three elementary linear creep models, which are mathematically formulated based on a continuous retardation spectrum of generalized Kelvin chain. These three creep responses are combined with the spherocylindrical microplane model. This leads to a general anisotropic constitutive creep model for arbitrary three-dimensional applied loading and arbitrary loading path. To achieve high numerical efficiency, a fully explicit numerical algorithm for the microplane constitutive model and finite element analysis is then formulated. Short-time (two-day) creep tests of shale cylinders with multilevel deviatoric stress and various inclinations of bedding planes relative to the axial compressive load are conducted. The numerical simulations by the microplane model are compared with the results of laboratory creep tests. The comparisons demonstrate a linear dependence of creep on the stress and verify the applicability of the anisotropic microplane model for creep and damage. Extensions to multi-year time ranges are in principle possible but will necessitate further calibration and verification.

Characterizing rheological behavior of asphalt binder over a complete range of pavement service loading frequency and temperature

The present study presented a new procedure for characterizing the rheological properties of asphalt binders over the complete range of pavement service frequency and temperature. The proposed procedure involved two primary steps. In the first step, a modified windowing method (MWM) with a pre-smoothing operation and a modified windowing inter-conversion method (MWIM) with an accurate distribution of the time constants were developed to convert the flexural creep compliance observations from the bending beam rheometer (BBR) test into the corresponding complex shear moduli, so that these measurements could be well combined with the DSR test results. In the second step, the 2S2P1D model was adopted to construct the full master curves of the asphalt binders in frequency domain. To extend the application of the procedure for simulating the master curves in time domain, an existing continuous relaxation spectrum model and a derived continuous retardation spectrum model for the 2S2P1D model were employed. The results showed that the combination of the BBR and DSR data substantially expanded the scale of the complex shear modulus. The corresponding linear viscoelastic (LVE) master curves could be incorporated into the existing asphalt concrete complex modulus predictive equations for better pavement response analysis and performance assessment.

Characterization of asphalt concrete linear viscoelastic behavior utilizing Havriliak–Negami complex modulus model

This study proposed a new procedure for characterizing linear viscoelastic (LVE) behavior of asphalt concrete using the Havriliak–Negami (HN) complex modulus model and corresponding continuous relaxation spectrum model. To extend its application to modeling retardation behavior of asphalt concrete, the continuous retardation spectrum was analytically derived from the HN model by performing an inverse Fourier–Laplace transform followed by a variable substitution. The two continuous spectra of the HN model allow the accurate construction of all the modulus and compliance function master curves in time and frequency domains as well as the efficient determination of the high-quality discrete relaxation and retardation spectra. The advantages of the proposed approach over existing ones were demonstrated through two different complex modulus test data sets. Also, a simple and practical numerical interconversion method was presented based on the LVE relations between the continuous relaxation and retardation spectra, which can effectively compute one continuous spectrum from the other for any characterization model. Further, for practical considerations, a two-step method for determining reduced master curves with fewer discrete spectrum lines was suggested.

Accurate approximations of concrete creep compliance functions based on continuous retardation spectra

This paper presents a detailed analysis of the continuous retardation spectra corresponding to a number of concrete creep models stipulated by various codes and recommendations. Approximations of various orders based on the Post–Widder formula are constructed, and the accuracy of the corresponding Dirichlet series approximating the compliance function is assessed. It is shown that the accuracy can often be substantially increased by appropriate modifications of the discrete retardation times used by the Dirichlet series. Practical hints regarding the choice of the discrete retardation times and the order of integration used in the evaluation of the compliance coefficients are provided.

Excessive Long-Time Deflections of Prestressed Box Girders. II: Numerical Analysis and Lessons Learned

As a sequel to Part I, which clarified the causes of the unexpectedly large deflections of the Koror-Babeldaob Bridge in the Pacific island nation of Palau, Part II presents the numerical procedure and reviews the lessons learned. The box girder represents a thick shell that is discretized by eight-node, three-dimensional (3D) finite elements. Except for corrections due to cracking, concrete creep is assumed to follow aging linear viscoelasticity and is modeled by a rate-type law based on the Kelvin chain, the properties of which are adjusted for humidity conditions and temperature. In each time step and at each integration point, Widder’s formula is used to convert the aging compliance function to a continuous retardation spectrum for the current age of concrete, and discretization of the spectrum yields the current elastic moduli of the Kelvin units. The shrinkage strains depend on the environmental humidity and the thickness of each plate in the cross section. The computations proceed according to Bazant’s exponential algorithm, which is unconditionally stable and reduces the problem to a sequence of elasticity problems with an orthotropic effective stiffness of material and nonisotropic inelastic strains, different for each integration point in each time step. These problems are solved by commercial software ABAQUS. The segmental construction sequence is also modeled. The computer results reported in Part I explain the excessive deflections and compare the performance of various material models for creep and shrinkage. Part II formulates the lessons learned and makes recommendations for implementation.

Improved Algorithm for Efficient and Realistic Creep Analysis of Large Creep-Sensitive Concrete Structures

Recent compilation of data on numerous large-span prestressed segmentally erected box girder bridges revealed gross underestimation of their multi-decade deflections. The main cause has been identified as incorrect and obsolete creep prediction models in various existing standard recommendations and is being addressed in a separate study. However, previous analyses of the excessive deflections of the Koror-Babeldaob (KB) Bridge in Palau and of four Japanese bridges have shown that a more accurate method of multi-decade creep analysis is required. The objective of this paper is to provide a systematic and comprehensive presentation, appropriate not only for bridges but also for any large creep-sensitive structure. For each time step, the solution is reduced to an elastic structural analysis with generally orthotropic elastic moduli and eigenstrains. This analysis should normally be three-dimensional (3-D). It can be accomplished with a commercial finite element code such as ABAQUS. Based on the Kelvin chain model, the integral-type creep law is converted to a rate-type form with internal variables, which account for the previous history. For time steps short enough to render aging during each step to be negligible, a unique continuous retardation spectrum for each step is obtained by Laplace transform inversion using simple Widder's formula. Discretization of the spectrum then yields the current Kelvin chain moduli. The rate-type creep analysis is computationally more efficient than the classical integral-type analysis. More importantly, though, it makes it possible to take into account the evolution of various inelastic and nonlinear phenomena such as tensile cracking, cyclic creep, and stress relaxation in prestressing tendons at variable strain, as well as the effects of humidity and temperature variations, and the effect of wall thickness variation on drying creep and shrinkage. Finally, the advantages compared to the existing commercial programs, based on step-by-step integration of memory integrals, are pointed out and illustrated by a simple example.

Viscoelastic repetitive creep and recovery in bituminous materials

Repetitive creep and recovery tests in bituminous materials- asphalt and asphalt paving mix are studied at high service temperatures. The non-exponential- stretched exponential type of continuous retardation spectrum is defined and used for the calculation of creep compliance function in repetitive shear and tensile creep and recovery experiments and also in the wheel tracking test on asphalt paving mix. It is shown that the used model, with only five adjustable parameters, can describe the test quite well. The model is also adequate for the description of the composite compliance function constructed from the linear viscoelastic data and the apparently nonlinear compliance function obtained in the wheel tracking test, for asphalt paving mix.

Accumulated strain in polymer-modified asphalts

One of the distress modes of asphalt pavements is the rutting that occurs at high operating temperatures, and it is believed that the accumulated strain in asphalt binder is mainly responsible for the rutting. The asphalt industry has tried to use the parameter 1/J″ as the specification parameter for the determination of the susceptibility of asphalt pavements to rutting. It was, however, recognized that this parameter is not suitable for polymer-modified asphalts, and currently, the repeated creep and recovery test (dynamic creep test) is studied as a new specification tool. In this paper, we study the general response of binders to a train of shear-stress pulses with a constant rest time between the pulses. We have found that the viscous approximation of the creep compliance function yields a suitable description of the test when the creep recovery is very weak. To extend the description to cases with significant recovery, an alternative model is proposed. Using the simple approximation of the continuous retardation spectrum, L, by the reduced loss compliance, L is modeled by a gamma-type distribution function. With this form of the shear compliance function, one can successfully describe the dynamic creep test in various polymer-modified asphalts.

Broad frequency range characterization of molten polymers

We report a technique for obtaining the continuous retardation spectrum or relaxation spectrum over a very broad range of relaxation times. The major challenge is to obtain reliable data at the long-time end of the spectrum. It is often not practical to use low-frequency oscillatory shear to obtain the necessary data because of the very long time required for the measurements. Polyolefins pose special problems in long-time characterization, because time-temperature superposition is not very useful. Information about the long-time behavior can be obtained by means of creep experiments, but now another problem arises. If we use a stress sufficiently large to yield precise data, the strain may go beyond the limit of linear viscoelastic behavior before steady state is achieved. This problem can be addressed by eliminating the stress well before nonlinearity becomes a problem and tracking the resulting recovery. Using a method proposed by Meissner, the combined creep and recovery data can be used to construct the entire creep curve. The final challenge is to combine the data from the oscillatory shear and creep experiments to obtain a composite linear spectrum. We propose a new method for accomplishing this. Retardation spectra are inferred from both sets of data and compared to reveal the range of times over which both techniques provide reliable data and the range in which each technique is uniquely reliable. The two spectra can then be combined to form a composite spectrum, which can be used to calculate any linear material function. This procedure is demonstrated for two branched polypropylenes whose terminal zones could not be accessed by means of oscillatory shear. For each of these materials, we found that the spectra obtained by the two experimental techniques superposed perfectly within the region of overlapping experimental windows, making it possible to construct a reliable composite spectrum valid over a wide range of times. The composite continuous spectrum is more directly related to molecular structure than rheological properties, such as the complex modulus, which involve integrals of the spectrum.

Continuous retardation spectrum for solidification theory of concrete creep : Z. P. Bazant & Yunping Xi, Journal of Engineering Mechanics — ASCE, 121(2), 1995, pp 281–288

Continuous retardation spectrum for solidification theory of concrete creep : Z. P. Bazant & Yunping Xi, Journal of Engineering Mechanics — ASCE, 121(2), 1995, pp 281–288

Continuous Retardation Spectrum for Solidification Theory of Concrete Creep

The basic creep of concrete is the time-dependent strain caused by a sustained stress in absence of moisture movements. It is the strain observed on sealed specimens. Similar to other properties of concrete, it is dependent on the age of concrete, as a consequence of long-time chemical reactions associated with the hydration of cement. This paper formulates the solidification theory with a continuous retardation spectrum, and shows how this spectrum can be readily and unambiguously identified from arbitrary measured creep curves and how it then can be easily converted to a discrete spectrum for numerical purposes. The identification of the continuous spectrum is based on Tschoegl's work on viscoelasticity of polymers. Attention is limited to basic creep.