Creep Function Research Articles

A Generalized Maxwell Model for Creep Behavior of Artery Opening Angle

An artery ring springs open into a sector after a radial cut. The opening angle characterizes the residual strain in the unloaded state, which is fundamental in understanding stress and strain in the vessel wall. A recent study revealed that the opening angle decreases with time if the artery is cut from the loaded state, while it increases if the cut is made from the no-load state due to viscoelasticity. In both cases, the opening angle approaches the same value in 3 h. This implies that the characteristic relaxation time is about 10,000 s. Here, the creep function of a generalized Maxwell model (a spring in series with six Voigt bodies) is used to predict the temporal change of opening angle in multiple time scales. It is demonstrated that the theoretical model captures the salient features of the experimental results. The proposed creep function may be extended to study the viscoelastic response of blood vessels under various loading conditions.

Analysis of PVC Pipe-Wall Viscoelasticity during Water Hammer

This research work focuses on the analysis of hydraulic transients in polyvinyl chloride (PVC) pipes, which are characterized by a viscoelastic rheological behavior. Transient pressure data were collected in a pipe rig consisting of a set of PVC pipes. The creep function of the PVC pipes was determined by using an inverse transient model based on collected transient pressure data and compared with that obtained by carrying out mechanical tensile tests of PVC pipe specimens. The numerical results obtained from the transient solver have shown that the attenuation, dispersion, and shape of transient pressures were well described. The incorporation of the viscoelastic mechanical behavior in the hydraulic transient model has provided an excellent fitting between numerical results and observed data. Calibrated creep function based on inverse analysis fit the one determined by mechanical tests well, which emphasized the importance of pipe-wall viscoelasticity in hydraulic transients in PVC pipes.

Determining the parameters of the fractional exponential heredity kernels of linear viscoelastic materials

The parameters of the fractional exponential creep and relaxation kernels of linear viscoelastic materials are determined. Methods that approximate the kernel by using the Mittag-Leffler function, the Laplace-Carson transform, and direct approximation of the creep function by the original equation are analyzed. The parameters of fractional exponential kernels are determined for aramid fibers, parapolyamide fibers, glass-reinforced plastic, and polymer concrete. It is shown that the kernel parameters calculated through the direct approximation of the creep function provide the best agreement between theory and experiment. The methods are experimentally validated for constant-stress and variable-stress loading in the modes of additional loading and complete unloading

Some Applications of Linear Viscoelasticity to Problems of Consolidation under Variable Loading

The main purpose of this paper is to present a general method to derive closed form solutions for one-dimensional consolidation problems under time dependent loading using the Linear Viscoelasticity theory. A review of the basic concepts of this theory is initially presented and, mainly for illustration purposes, the method is applied to three consolidation problems, leading to relevant solutions for Geotechnical Engineering. In the first application, considering Terzaghi’s and Barron’s solutions, creep functions are determined for vertical and radial drainage, allowing derivation of expressions for one-dimensional consolidation under a number of linear loads for these drainage conditions. Using Carrillo’s equation, the creep function for combined vertical and radial drainage is obtained, leading to corresponding solution for linear variable loading. Partial submersion of embankments on soft soils is another consolidation problem under time dependent loading solved by means of Viscoelasticity. Classical approximate solutions are used in this second application to establish creep functions for vertical drainage and exact Barron’s solution to establish creep function for radial drainage. An expression for late stages of consolidation is also derived for combined vertical and radial drainage condition. The third application considers the problem of load transfer from a consolidating deposit of soft clay to a pattern of drain columns of finite stiffness. Diagrams concerning a case of consolidation under linear variable three-step loading and consolidation with partial submersion of the fill are provided to illustrate the solutions obtained.

Numerical Simulation and Characterization of Press Molding of Thermo-Plastic Resin Reinforced by Carbon Nanofiber

In the present paper, press molding for thermo-plastic resin toughened by carbon nanofiber were discussed. Polycarbonate (PC) and vapor grown carbon nanofiber (VGCF) are used for the resin matrix and the reinforcement. In order to investigate and simulate the process molding process of the PC/VGCF composite by finite element method, the thermo-viscoelastic property of the composites are estimated using unidirectional compression creep test. The creep functions were converted into relaxation function based on the convolution integral form of the basic equation using Laplace transform. Experimental testing for press molding of the PC/VGCF composites using V-shape die has been carried out. The results of the V-shape profiles and those obtained by FEM analysis were compared to confirm the validity of the present analysis method based on thermo-viscoelastic theory. The effect of the carbon nanofiber for the formability improvement of the PC/VGCF composites was discussed in detail.

Exact computation of creep compliance and relaxation modulus from complex modulus measurement data

It is well-known that for a linear viscoelastic material, there is a relationship between the complex modulus, the creep function and the relaxation modulus. It should be sufficient to collect one of these functions by a measurement to obtain the two other functions. However, this process seems to be rarely used. One reason for this may be the very large frequency intervals over which the Fourier transform should be computed. It is not unusual to find a ratio of 10 30 between the lowest and the highest frequencies. We propose a semi-analytical algorithm for computing the creep and the relaxation functions from complex modulus data. Exact in nature, it makes use of very basic assumptions. Good correlation with approximate alternative methods is found. In this paper, we illustrate our method using some examples of creep compliance and relaxation modulus synthesis from the master curves.

Creep and relaxation functions of a heterogeneous viscoelastic porous medium using the Mori-Tanaka homogenization scheme and a discrete microscopic retardation spectrum

In this paper the macroscopic creep and relaxation functions of a heterogeneous viscoelastic porous medium are derived by using Mori-Tanaka homogenization scheme. Analytical and semi-analytical solutions can then be determined with a parametric number of heterogeneous phases embedded in a viscoelastic matrix whose behavior is described with a parametric number of analogical units. Under some simplifying assumptions, a solution strategy is presented in order to make explicit how the microscopic retardation and relaxation times of the viscoelastic matrix control the distribution of the retardation and relaxation times of the homogenized medium.

Modelling the viscoelasticity of cell shape changes

AbstractIn the theory of linear viscoelasticity, creep and relaxation functions are the important delay kernels in the stress‐strain relations. Quite recently, a creep function power law has been obtained experimentally for an isolated living cell (on the scale of the cell). The corresponding form of the relaxation function is derived using the interconversion formula. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the sensitivity of interconversion between relaxation and creep

The interconversion equation of linear viscoelasticity defines implicitly the interrelations between the relaxation and creep functions G(t) and J(t). It is widely utilised in rheology to estimate J(t) from measurements of G(t) and conversely. Because different molecular details can be recovered from G(t) and J(t), it is necessary to work with both. This leads naturally to the need to identify whether it is better to first measure G(t) and then determine J(t) or conversely. This requires an understanding of the stability (sensitivity) of the recovery of J(t) from G(t) compared with that of G(t) from J(t). Although algorithms are available that work adequately in both directions, numerical experimentation strongly suggests that the recovery of J(t) from G(t) measurements is the more stable. An elementary theoretical rationale has been given recently by Anderssen et al. (ANZIAM J 48:C346–C363, 2007) for single exponential models of G(t) and J(t). It explicitly exploits the simple algebra of such functions. In this paper, corresponding bounds are derived that hold for arbitrary sums of exponentials. The paper concludes with a discussion, from a practical rheological perspective, about the implications and implementations of the results.

On the direct estimation of creep and relaxation functions

Two alternative approaches for estimating linear viscoelastic material functions from a single experiment under random excitation are derived and analyzed. First, Boltzmann’s superposition integral is discretized into a system of linear equations. Due to the ill-posedness of the resulting matrix equation, Tikhonov’s regularization is introduced. Second, the integral is transformed into a recursive formula, using a Prony series representation of viscoelastic material functions, in which gradient-based optimization is applied. Numerical results are provided to compare and verify the applicability of the presented numerical procedures.

Macrorheology and adaptive microrheology of endothelial cells subjected to fluid shear stress

Vascular endothelial cells (ECs) respond to temporal and spatial characteristics of hemodynamic forces by alterations in their adhesiveness to leukocytes, secretion of vasodilators, and permeability to blood-borne constituents. These physiological and pathophysiological changes are tied to adaptation of cell mechanics and mechanotransduction, the process by which cells convert forces to intracellular biochemical signals. The exact time scales of these mechanical adaptations, however, remain unknown. We used particle-tracking microrheology to study adaptive changes in intracellular mechanics in response to a step change in fluid shear stress, which simulates both rapid temporal and steady features of hemodynamic forces. Results indicate that ECs become significantly more compliant as early as 30 s after a step change in shear stress from 0 to 10 dyn/cm(2) followed by recovery of viscoelastic parameters within 4 min of shearing, even though shear stress was maintained. After ECs were sheared for 5 min, return of shear stress to 0 dyn/cm(2) in a stepwise manner did not result in any further rheological adaptation. Average vesicle displacements were used to determine time-dependent cell deformation and macrorheological parameters by fitting creep function to a linear viscoelastic liquid model. Characteristic time and magnitude for shear-induced deformation were 3 s and 50 nm, respectively. We conclude that ECs rapidly adapt their mechanical properties in response to shear stress, and we provide the first macrorheological parameters for time-dependent deformations of ECs to a physiological forcing function. Such studies provide insight into pathologies such as atherosclerosis, which may find their origins in EC mechanics.

A time-integration algorithm for thermo-rheologically complex polymers

Abstract This study presents nonlinear thermo-viscoelastic analyses of isotropic polymers that belong to a class of thermo-rheologically complex materials (TCM). The nonlinear viscoelastic constitutive model is expressed with an integral form of a creep function, whose initial, long-term, and transient properties change with stresses and temperatures. A combined recursive–iterative method is formulated to solve the viscoelastic integral equation. The recurrence formula allows bypassing the need to store entire strain histories. Two types of iterative procedures, fixed point (FP) and Newton–Raphson (NR), are examined within the recursive algorithm. Furthermore, a consistent tangent stiffness matrix is formulated to accelerate convergence and avoid divergence. The algorithm is compatible with displacement based finite element (FE) structural analyses for small deformation gradients and uncoupled thermo-mechanical problems. Verification of the numerical algorithm is performed using the thermo-viscoelastic experimental data available in the literature. The influences of temperature dependent material properties on the material’s viscoelastic responses are discussed for general thermo-mechanical loadings. The capability of the algorithm in predicting multi-axial viscoelastic responses is then verified using creep data of adhesive bonded joints at various temperatures. Finally, numerical simulations of time-dependent crack propagations in adhesive bonded joints are presented for the opening and shearing modes.

High-Temperature Mechanical Properties of Al<sub>2</sub>O<sub>3</sub>/Ti<sub>3</sub>SiC<sub>2</sub> Multilayer Ceramics

The high-temperature mechanical properties were very important to structural materials, especially structural ceramics. Hence, the strength, elastic modulus, stress relaxation and creep behavior of the multilayer materials at elevated temperature were studied in this paper. According to the curves of mechanical properties varieties with temperature risen from room-temperature to 1300°C, the multilayer materials could remain relatively high mechanical properties until 1150°C. Otherwise, the creep function of the multilayer composites was also determined, in which the stress exponent was 1.4 and activation energy was 204kJ/mol. By contrasting to the monolithic ceramics of Al2O3 and Ti3SiC2, the main creep mechanisms include: interface diffusion creep (in Al2O3 layers), dislocation movement creep, grain delamination and sliding (in Ti3SiC2 layers).

Indentation of Nonlinearly Viscoelastic Solids

AbstractMuch recent attention has been focused on the indentation of linearly viscoelastic solids, and analysis techniques have been developed for polymeric material characterization. However, there has been relatively little progress made in the development of analytical approaches for indentation of nonlinearly viscoelastic materials. Soft biological tissues tend to exhibit responses which are nonlinearly viscoelastic and are frequently modeled using a decomposition of the relaxation or creep function into a product of two functions, one time-dependent and the other stress- or strain-level dependent. Consideration here is for soft biological tissue-like responses, exhibiting approximately quadratic stress-strain behavior, which can be alternatively cast as linear dependence of elastic modulus on strain level. An analytical approach is considered in the context of indentation problems with flat punch, spherical and conical indenter shapes. Hereditary integral expressions are developed and solved for typical indentation experimental conditions including indentation creep, load-relaxation and monotonic constant load- or displacement-rate tests. Primary emphasis is on the deconvolution of material and geometrical nonlinearities during an indentation experiment. The simple analytical expressions that result from this analysis can be implemented for indentation characterization of soft biological tissues without the need for computationally- intensive inverse finite element approaches.

Suppression of elastically accommodated grain-boundary sliding in high-purity MgO

The first high-temperature mechanical spectroscopy experiments on high-purity polycrystalline MgO reveal a monotonically frequency- and temperature-dependent dissipation ‘background’—without any evidence of the superimposed dissipation peak observed in a previous study of a specimen of lower-purity [Webb, S. and Jackson, I. Phys. Chem. Min., 2003 30, 157]. The dissipation and associated relaxation of the shear modulus observed in both studies are well described in an internally consistent manner by a novel Burgers-type model based on a creep function incorporating suitable distributions of anelastic relaxation times. The contrasting patterns for the two materials reinforce an emerging generalisation concerning high-temperature viscoelastic behaviour, whereby the presence of a secondary intergranular phase of relatively low viscosity, and the associated rounding of grain edges, is apparently required to allow elastically accommodated grain-boundary sliding. The absence, in sufficiently pure polycrystalline materials, of a dissipation peak attributable to elastically accommodated grain-boundary sliding is in conflict with classical micromechanical models for grain-boundary sliding, which are therefore being revisited.

414 ガラスの熱粘弾性の評価とモールドプレスシミュレーション(OS4-2 先端材料システムの力学とメゾスケールモデリング,OS4 先端材料システムの力学とメゾスケールモデリング)

In the present paper, numerical simulation about press-molding of glass lens is investigated. Since constitutive equation of glass depend on time-history and temperature, stress relaxation is observed on the glass subjected to applied loading, and these behaviors vary drastically depending on the temperature. In the present study, creep test has been conducted to determine the thermo-viscoelastic properties of glass. The creep function obtained by the experiment has been transformed into relaxation function employing Laplace transform. Shift factor which gives the relation between the time and temperature can be determined by creep test under several temperature. In the present analysis for press-molding of glass lens, history of the temperature and its distribution is computed by unsteady heat conduction analysis prior to the analysis of cooling sequence. The total FEM analysis for press-molding are executed taking into account the nonuniformly temperature distribution in the glass lens. Residual stresses under some processing conditions are estimated, and the optimal conditions of forming process are discussed in detail.

Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions

Two higher-order fractional viscoelastic material models consisting of the fractional Voigt model (FVM) and the fractional Maxwell model (FMM) are considered. Their higher-order fractional constitutive equations are derived due to the models’ constructions. We call them the higher-order fractional constitutive equations because they contain three different fractional parameters and the maximum order of equations is more than one. The relaxation and creep functions of the higher-order fractional constitutive equations are obtained by Laplace transform method. As particular cases, the analytical solutions of standard (integer-order) quadratic constitutive equations are contained. The generalized Mittag–Leffler function and H-Fox function play an important role in the solutions of the higher-order fractional constitutive equations. Finally, experimental data of human cranial bone are used to fit with the models given by this paper. The fitting plots show that the models given in the paper are efficient in describing the property of viscoelastic materials.

Analytical techniques for indentation of viscoelastic materials

Indentation of linearly viscoelastic materials is explored using elastic–viscoelastic correspondence analysis for both conical-pyramidal and spherical indentation. Boltzmann hereditary integrals are used to generate displacement–time solutions for loading at constant rate and creep following ramp loading. Experimental data for triangle- and trapezoidal-loading are examined for commercially-available polymers and compared with analytical solutions. Emphasis is given to the use of multiple experiments to test the fidelity and predictive capability of the obtained material creep function. Plastic deformation occurs in sharp indentation of glassy polymers and is found to complicate the viscoelastic analysis. A new method is proposed for estimating a material time-constant from peak displacement or hardness data obtained in pyramidal indentation tests performed at different loading rates.

Computation and Analysis of the Creep Effect of Long-span Concrete-filled Steel Tubular Arch Bridge

There are a number of factors that influence the creep of concrete-filled steel tubular arch bridges, while the related research theory lags behind the state of practice. Based on the computing formula of creep coefficient in the code, the creep function is formulated. The computing method of creep in structural analysis is deduced, using the integral concepts of stress and time. Combined with the calculation of Meixi River Bridge it is noted out that creep affects the displacement of CFST arch bridge prominently, which will lead to redistribution of internal force. This should be recognized in related engineering circles.

Power laws in microrheology experiments on living cells: Comparative analysis and modeling

We compare and synthesize the results of two microrheological experiments on the cytoskeleton of single cells. In the first one, the creep function J(t) of a cell stretched between two glass plates is measured after applying a constant force step. In the second one, a microbead specifically bound to transmembrane receptors is driven by an oscillating optical trap, and the viscoelastic coefficient Ge(omega) is retrieved. Both J(t) and Ge(omega) exhibit power law behaviors: J(t) = A0(t/t0)alpha and absolute value (Ge(omega)) = G0(omega/omega0)alpha, with the same exponent alpha approximately 0.2. This power law behavior is very robust; alpha is distributed over a narrow range, and shows almost no dependence on the cell type, on the nature of the protein complex which transmits the mechanical stress, nor on the typical length scale of the experiment. On the contrary, the prefactors A0 and G0 appear very sensitive to these parameters. Whereas the exponents alpha are normally distributed over the cell population, the prefactors A0 and G0 follow a log-normal repartition. These results are compared with other data published in the literature. We propose a global interpretation, based on a semiphenomenological model, which involves a broad distribution of relaxation times in the system. The model predicts the power law behavior and the statistical repartition of the mechanical parameters, as experimentally observed for the cells. Moreover, it leads to an estimate of the largest response time in the cytoskeletal network: tau(m) approximately 1000 s.