Maxwell Chain Research Articles

Experimentally informed modeling of the early‐age stress evolution in cementitious materials using exponential conversion from creep to relaxation

AbstractThis study presents comprehensive numerical modeling methods for simulating early‐age stress (EAS) relaxation in cementitious materials, based on the autogenous deformation (AD), elastic modulus, creep, and stress continuously tested by a mini temperature stress testing machine (Mini‐TSTM) and a mini AD testing machine from a very early age (i.e., from a few hours to a week). Four methods for converting creep compliance to relaxation modulus were discussed in detail and used for the one‐dimensional (1D) and three‐dimensional (3D) simulation of stress evolution in the Mini‐TSTM test. Furthermore, virtual creep and relaxation tests were conducted using an exponential algorithm with either the Kelvin or Maxwell chains to show their applicability in simulating the viscoelastic behavior of early‐age cementitious materials. The results showed that the exponential algorithm with the Maxwell chain using an exponential conversion function from creep to relaxation obtains good prediction accuracy of EAS in 3D analysis. The numerical solutions of the Volterra integral of creep compliance can lead to a negative relaxation modulus, thus introducing stress calculation errors in both 1D and 3D analysis.

A continuum model for the tensegrity Maxwell chain

<abstract><p>A recent study has presented a Maxwell mass–spring model for a chain formed by two different types of tensegrity prisms alternating with lumped masses. Such a model shows tensegrity theta prisms arranged in parallel with minimal regular prisms acting as resonant substructures. It features a tunable frequency bandgap response, due to the possibility of adjusting the width of the bandgap regions by playing with internal resonance effects in addition to mass and spring contrasts. This paper expands such research by presenting a continuum modeling of the tensegrity Maxwell chain, which is useful to conduct analytic studies and to develop finite element models of the plane wave dynamics of the investigated system. In correspondence to the high wave-length limit, i.e., in the low wave number regime, it is shown that the dispersion relations of the discrete and continuum models provide similar results. Analytic solutions to the wave dynamics of physical systems are presented, which validate the predictions of the bandgap response offered by the dispersion relation of the continuum model.</p></abstract>

Influence of microstructure on rate-dependent response of unidirectional fibrous composites

This paper outlines prediction of the macroscopic response of unidirectional fibrous composites made either from basalt or carbon fibers impregnated by a polymeric epoxy matrix. The viscoelastic response of the matrix phase was represented by the Maxwell chain model. A series of creep tests performed at several stress levels served as a stepping stone for the model calibration. The macroscopic behavior of both composites was first examined via computational homogenization. Attention was accorded to computational cells with variable size extracted from large representative images. We observed that selecting the computational model as a sufficiently large test window should be approached with caution. Because initial designs often need a large parametric study to test various material and geometrical patterns, this study was then supported by computationally much more effective Mori-Tanaka averaging scheme, clearly showing its potential even if loading the composite beyond its elastic limit.

NEWMARK ALGORITHM FOR DYNAMIC ANALYSIS WITH MAXWELL CHAIN MODEL

This paper investigates a time-stepping procedure of the Newmark type for dynamic analyses of viscoelastic structures characterized by a generalized Maxwell model. We depart from a scheme developed for a three-parameter model by Hatada et al. [1], which we extend to a generic Maxwell chain and demonstrate that the resulting algorithm can be derived from a suitably discretized Hamilton variational principle. This variational structure manifests itself in an excellent stability and a low artificial damping of the integrator, as we confirm with a mass-spring-dashpot example. After a straightforward generalization to distributed systems, the integrator may find use in, e.g., fracture simulations of laminated glass units, once combined with variationally-based fracture models.

Theoretical Investigation of Nonlinear Time-Dependent Behavior of Two-Way High-Strength Concrete Walls

High-strength concrete (HSC) walls have been increasingly used in the past decades. However, the time-dependent behavior of HSC wall panels in two-way action was not investigated, and the time effect of creep is not included in the design codes in most countries. For this purpose, the nonlinear long-term behavior of two-way HSC wall is investigated in this paper. A theoretical model is developed using time-stepping analysis considering geometric nonlinearity and creep of concrete. A rheological material model that is based on the generalized Maxwell chain is adopted to model the concrete creep. Von Karman plate theory is used to derive the incremental governing equations. The equations are solved numerically at each time step based on a Fourier series expansion of the deformations and loads and numerical multiple shooting method. It shows that the model can effectively predict the time-dependent behavior of two-way HSC panels, where the out-of-plane deflection and internal bending moments increase with time due to the combined effects of creep and geometric nonlinearity, which may ultimately lead to creep buckling failures. A parametric study shows that the long-term behavior of the panel is very sensitive to the in-plane load level and eccentricity, slenderness ratio, aspect ratio, and edge support conditions.

Experimental and Numerical Study of Viscoelastic Properties of Polymeric Interlayers Used for Laminated Glass: Determination of Material Parameters.

An accurate material representation of polymeric interlayers in laminated glass panes has proved fundamental for a reliable prediction of their response in both static and dynamic loading regimes. This issue is addressed in the present contribution by examining the time–temperature sensitivity of the shear stiffness of two widely used interlayers made of polyvinyl butyral (TROSIFOL BG R20) and ethylene-vinyl acetate (EVALAM 80-120). To that end, an experimental program has been executed to compare the applicability of two experimental techniques, (i) dynamic torsional tests and (ii) dynamic single-lap shear tests, in providing data needed in a subsequent calibration of a suitable material model. Herein, attention is limited to the identification of material parameters of the generalized Maxwell chain model through the combination of linear regression and the Nelder–Mead method. The choice of the viscoelastic material model has also been supported experimentally. The resulting model parameters confirmed a strong material variability of both interlayers with temperature and time. While higher initial shear stiffness was observed for the polyvinyl butyral interlayer in general, the ethylene-vinyl acetate interlayer exhibited a less pronounced decay of stiffness over time and a stiffer response in long-term loading.

Application of fractional time derivatives in modeling the finite deformation viscoelastic behavior of carbon-black filled NR and SBR

Mechanical behavior of elastomers includes nonlinear and time dependent phenomena such as creep and relaxation where due to low stiffness, these materials usually undergo large deformations. The classical approaches of viscoelasticity,i.e. differential constitutive models and hereditary integrals have been widely used to model the viscoelastic behavior of these materials even at finite deformations. However, from the viewpoint of computational analysis, the integral based methods are often time and memory consuming and the differential methods usually need a large number of material parameters whose identification is a real challenge. A potential solution is to introduce fractional order derivatives instead of regular differential operators in the standard spring-dashpot mechanical models which remarkably reduces the number of required material parameters. In the present paper, relaxation behavior of NR and SBR is investigated in finite strains. Due to the remarkable effects of filler particles and deformation rate on the mechanical properties of elastomers, compounds of each material with different amounts of filler content are investigated at two strain rates. A main interest is to examine efficiency of the fractional derivative models in modeling the finite deformation relaxation behavior of elastomers with different filler contents. The other goal is to investigate the effect of loading rate on the obtained model parameters. It is observed that, the fractional Maxwell chain is able to well fit the stress relaxation response of all compounds with only three parameters. Analyzing the obtained parameters and the experimental data, it is concluded that, order of the associated Maxwell element is not much affected with the filler content. Seemingly, it depends on the matrix material and more noticeably on the deformation rate. Moreover, as much as the deformation rate, the associated relaxation time is low. Decrease of the relaxation time with the deformation rate is more evident for compounds of SBR compared with NR compounds.

Microstructural Modeling of Early-Age Creep in Hydrating Cement Paste

AbstractThis paper presents a new approach to model the creep behavior of cement paste at early ages. The creep behavior is simulated by applying a time-varying generalized Maxwell model on the individual elements of a finite-element mesh of a simulated three-dimensional microstructure and compared with results in the literature. All mechanical properties of the constituent phases are taken from literature and Maxwell chain parameters are obtained by fitting the intrinsic creep of calcium silicate hydrate (C-S-H). A reasonable agreement between the simulations and the experimental results are obtained by assuming a constant C-S-H density of 2.0 g/cm3. It was found that better agreements could be obtained at low degree of hydrations, by assuming a loosely packed C-S-H growing in the microstructure. It was also found that the short-term creep characteristics of C-S-H from nanoindentation can be used to reproduce macroscopic creep at least over a few days. The results show how numerical models can be used t...

A tridimensional finite element approach to model a tunnel with shotcrete and precast concrete

ABSTRACT This paper describes a numerical simulation with 3D finite elements of a tunnel. The viscoplastic law of Perzyna represents the rockmass behavior. The concrete, shotcrete or precast, is modeled as a viscoelastic material through the Maxwell and Kelvin chain models. Finite element simulations are performed by incorporating subroutines for viscoelastic concrete models in the ANSYS code. The method to simulate tunnel excavations is by activating and deactivating elements in sequential steps. In the first part of the paper two validations are performed. The analytical solution and the deformation achieved on the stabilization in the ANSYS code are compared with an unlined tunnel. A lined tunnel, with an elastic and viscoplastic rockmass combined with an elastic lining, is compared with the results of the GEOMEC91 code. In the second part, it is compared the same tunnel with two different concrete lining for two chain models. Finally, it is modeled the Kielder experimental tunnel, which in situ measured data is available.

Brownian Motion in the Fluids with Complex Rheology

Theory of Brownian motion of a fine particle in a viscoelastic fluid continuum is developed. The rheology of the embedding medium is described in terms of classical structure (spring-and-damper) schemes. It is shown that a great variety of conceivable ramifications of such structures could be reduced to an effective Jeffreys model: a Maxwell chain shunted by a damper. This scheme comprises just three material parameters: two for viscosities (fast and slow) and one for elasticity. For the thermal motion of a particle in such a fluid, the set of Langevin equations is derived. In the cases of 1D translational and 2D orientational (rotation about fixed axis) motions, the time dependencies of mean-square displacements are found analytically. It is shown that in a Jeffreys fluid, the Brownian motion has three regimes: fast diffusion (short times), slow diffusion (long times) and a crossover of those resulting in virtual localization (dynamic confinement) of the particle. Experimental evidence for that taken from the literature is presented. The developed formalism is applied to the particles with “frozen-in” dipole moments, whose orientational motion is a combination of Brownian diffusion and regular excitation by an AC field. The dynamic susceptibilities are calculated for the ensembles of particles liable for either 2D or 3D rotations. Comparison shows that the out-of-phase part of the susceptibility χ′′(ω) in 3D case differs considerably from that of 2D case. Function χ(ω) for the forced 3D rotary diffusion in a Jeffreys fluid is found for the first time. In our view, it would be useful for realistic theoretical interpretation of microrheology data and the more so for estimating the particle-mediated AC field-induced heating (magnetic hyperthermia) in viscoelastic fluid media.

Constitutive modelling of the amplitude and frequency dependency of filled elastomers utilizing a modified Boundary Surface Model

A phenomenological uniaxial model is derived for implementation in the time domain, which captures the amplitude and frequency dependency of filled elastomers. Motivated by the experimental observation that the frequency dependency is stronger for smaller strain amplitudes than for large ones, a novel material model is presented. It utilizes a split of deformation between a generalized Maxwell chain in series with a bounding surface plasticity model with a vanishing elastic region. Many attempts to capture the behaviour of filled elastomers are found in the literature, which often utilize an additive split between an elastic and a history dependent element, in parallel. Even though some models capture the storage and loss modulus during sinusoidal excitations, they often fail to do so for more complex load histories. Simulations with the derived model are compared to measurements in simple shear on a compound of carbon black filled natural rubber used in driveline isolators in the heavy truck industry. The storage and loss modulus from simulations agree very well with measurements, using only 7 material parameters to capture 2 decades of strain (0.5–50% shear strain) and frequency (0.2–20Hz). More importantly, with material parameters extracted from the measured storage and loss modulus, measurements of a dual sine excitation are well replicated. This enables realistic operating conditions to be simulated early in the development process, before an actual prototype is available for testing, since the loads in real life operating conditions frequently are a combination of many harmonics.

Buckling of One-Way High-Strength Concrete Panels: Creep and Shrinkage Effects

A nonlinear theoretical model is developed in this paper for investigating the time-dependent behavior of one-way high-strength RC panels, with particular emphasis on the combined effects of creep and shrinkage on the buckling capacity and its degradation with time. A rheological generalized Maxwell chain model is used for modeling the creep of the concrete including its cracking, tension stiffening, and aging through strain- and time-dependent springs and dashpots. The incremental governing equations of the panel are derived and solved through a step-by-step time analysis that takes into account the variation of the internal stresses and deformations with time. A smeared cracking model is adopted, and an iterative procedure is conducted at each time step for the determination of the unknown rigidities of the cracked section, as well as the length of the cracked region. The capabilities of the proposed model are demonstrated through numerical and parametric studies, which show the important roles of creep and shrinkage in the buckling of high-strength concrete panels and which reveal the sensitivity of the nonlinear response to the magnitude and eccentricity of the sustained load and to the reinforcement ratio.

Viscoelasticity and Damage Model for Creep Behavior of Historical Masonry Structures

This paper presents a continuum model for the simulation of the viscous effects and the long-term damage ac-cumulation in masonry structures. The rheological model is based on a generalized Maxwell chain representation with a constitutive law utilizing a limited number of internal variables. Thanks to its computational efficiency, this approach is suitable for the analysis of large and complex structures. In the paper, the viscous and damage models are presented and their coupling is discussed. The FE simulation of the construction process of the representative bay of Mallorca Cathedral is presented, together with the analysis of the long-term effects. The parameters of the model are tentatively calibrated on the basis of the time-dependent viscous deformations detected during the cathedral monitoring.

Amplitude dependence of filler-reinforced rubber: Experiments, constitutive modelling and FEM – Implementation

The focus of the present paper is the experimental investigation, the constitutive representation and the numerical simulation of the amplitude dependence of filler-reinforced elastomers. A standard way to investigate the dynamic properties of viscoelastic materials is via the dynamic modulus which is obtained from stress signals due to harmonic strain excitations. Based on comprehensive experimental data, an amplitude-dependent constitutive model of finite viscoelasticity is developed. The model is based on a modified Maxwell chain with process-dependent viscosities which depend on additional internal state variables. The evaluation of this thermodynamically consistent model is possible in both the time domain, via stress-time signals, and in the frequency domain, via the dynamic modulus. This property is very profitable for the parameter identification process. The implementation of the constitutive model into the commercial finite element code ANSYS with the user-programmable feature (UPF) USERMAT for large deformations in updated Lagrange formulation is presented. This implementation allows simulating the time-dependent behaviour of rubber components under arbitrary transient loading histories. Due to physical and geometrical nonlinearities, these simulations are not possible in the frequency domain. But, transient FEM computations of large loading histories are sometimes not possible in an acceptable time. In the context of the parameter identification the fundamental ideas are presented, how this problem has been solved. Transient FEM simulations of real rubber components are also shown to visualize the properties of the model in the context of the transient material behaviour.

Material behaviour of filler‐reinforced rubber materials under cyclic loading processes, experiments and FEM‐simulations based on finite viscoelasticity.

AbstractA threedimensional constitutive model of finite viscoelasticity will be presented. It considers the amplitude dependence and is based on a Maxwell chain with intrinsic time scales. This type of viscoelasticity is thermodynamically consistent and provides a basis for modelling other effects (e.g. temperature dependence). The parameter identification for this model is possible in both the frequency and the time domain. Some finite element implementation and simulation results in the time domain are presented. The numerical simulation of multifrequent and other more complicated loading processes is possible only in the time domain. A very interesting loading process is a cyclic deformation with changing amplitudes in order to simulate the transient dynamic behavior of the Payne‐effect. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical Model for Time-Dependent Fracturing of Concrete

A finite-element formulation for the analysis of time-dependent failure of concrete is presented. The proposed formulation incorporates: (1) the viscoelastic behavior of uncracked concrete through a Maxwell chain model; and (2) the inelastic behavior of damaged concrete, characterized by a modified version of the microplane Model M4 which includes the rate dependence of fracturing. The proposed formulation is applied to the simulation of quasi-static concrete failure in the time domain. The different effects of creep and rate dependence of crack growth and their role in the lifetime of concrete structures are studied. The influence of different loading rates on the size effect is also analyzed with reference to single notched specimens, revealing the link between the size of the fracture process zone and the loading rate. The capability of the proposed numerical formulation is also verified for the case of sustained uniaxial compressive loads.

An incremental constitutive law for ageing viscoelastic materials: a three-dimensional approach

A new approach to a viscoelastic medium is presented and the constitutive equations of ageing viscoelastic materials are developed. The formulation is equivalent to ageing Kelvin or Maxwell chain models with age dependent elastic moduli and viscosities. It is shown that by using such models, the retaining of the complete past history of stress and strain in the memory of a digital computer is avoided. Finally, the formulation is generalized to deal with three-dimensional problems. To cite this article: C.F. Chazal, R. Moutou Pitti, C. R. Mecanique 337 (2009).

Mechanical behavior model for ASR-affected dam concrete under service load: formulation and verification

In this paper, a mechanical behavior model for concrete undergoing Alkali-Silica Reaction (ASR) expansions is presented. The relation between evolution of expansions and external pressure is specifically taken into account in the model, including a procedure to calculate inhibition pressure (defined as the compressive stress threshold beyond which no expansion can take place) from experimental data. Creep is represented by means of an eight element ageing Maxwell chain based on the shape proposed in the B3 model. An application of the model is provided for a case study using experimental data. Verification of the model is performed against data from the same batch of tests different from those used for calibration.

UHPFRC tensile creep at early age

Ultra high performance fibre reinforced concrete (UHPFRC) early age viscoelastic behaviour under tension was investigated. The tests results showed a high creep potential due to the high volume paste (88%). This result is of major importance because the viscoelastic properties contribute to mitigating the high early age stresses generated under restrained shrinkage. This beneficial effect was reflected by the increased linear-relationship between tensile creep and shrinkage. As expected, UHPFRC tensile creep behaviour was also sensitive to the loading level. Above 35% of the tensile strength at the loading age, the material exhibited viscoplastic behaviour. A Maxwell chain model was applied to predict the early age UHPFRC tensile creep and confirms the induced non-linear response.

Numerical Model for the Analysis of Unbonded Prestressed Members

A numerical model for the study of prestressed members with bonded and unbonded tendons has been implemented, considering the hybrid type finite element formulation for planar frames. With such an approach, accurate curvature distributions are obtained, allowing a good computational performance and an adequate unbonded tendon stresses evaluation. The computer program also includes geometric nonlinearity, composite construction, and time-dependent effects. Linear Maxwell chain models are used for both concrete and prestressed steel time-dependent behavior. This paper presents a general description of the numerical model proposed, including the structural idealization approach, a hybrid type finite element formulation review, the material constitutive models, and the main numerical procedures. A brief review of the behavior of unbonded prestressed members is also presented. Numerical and experimental results are compared, showing the good performance of the model for the analysis of unbonded prestressed elements. Unbonded tendon stresses and cracking pattern results are included, showing the capability of the numerical model. The model is proposed as an efficient analytical tool for further research on unbonded prestressed members.