Creep Function Research Articles

ХАРАКТЕРНЫЕ ОСОБЕННОСТИ СЕМЕЙСТВ КРИВЫХ ДЕФОРМИРОВАНИЯ ЛИНЕЙНЫХ МОДЕЛЕЙ ВЯЗКОУПРУГОСТИ

The qualitative analysis of linear integral constitutive equation of viscoelasticity with an arbitrary creep compliance function is continued. General equation and basic properties of quasi-static theoretic stress-strain curves (SSC) at constant stress rate are analytically studied herein and compared with properties of SSC at constant strain rate. It is proved for any increasing creep function that: 1) every SSC is an increasing and convex-up function of time (without extremum or inflection points); 2) SSC rise up as stress rate increase; 3) the instantaneous and longterm equilibrium moduli (i.e. the limit values of SSC derivative at zero and infinity points) don’t depend on a stress rate; 4) SSC family of any regular model converges uniformly to the linear function (instantaneous SSC of the model) as stress rate tends to infinity; SSC family of any non-regular model converges to vertical line; 5) if stress rate tends to zero, SSC family descends and converges uniformly to the linear function with the slope equal to equilibrium modulus (it may be zero if relaxation modulus tends to zero at infinity). All these properties are also proved to be valid for SSC at constant strain rate. Three main three classes of linear models (i.e. regular, unbounded and singular) are considered and distinctive features of their SSC are marked. It is proved for any regular model that SSC at a constant stress rate lies higher than its SSC at the corresponding constant strain rate (equal to the stress rate divided by the model instantaneous modulus). General properties and peculiarities of SSC families and their dependence on creep function parameters are illustrated by examination of classical rheological models. The specific features of SSC (and other basic curves) of the linear integral constitutive equation are highlighted that can indicate its applicability or non-applicability being compared with the list of typical properties of a certain material test curves. The list of indicators helps to check linear theory abilities to provide an adequate description of basic rheological phenomena and typical test curves of viscoelastoplastic materials. Keywords: creep compliance, relaxation modulus, integral constitutive equation, stress rate, strain rate, theoretic stress-strain curves, instantaneous modulus, regular and singular models, adequacy indicators for linear viscoelasticity.

A Study of Swelling Behaviour in a Tunnel Using Finite Element Methods

The aim of this research is to show swelling behaviour in a tunnel excavated through rocks by modeling them and using laboratory results. The engineering challenge is how to design a tunnel which contains swelling rocks such as marl. This aim is achieved through two methods. In the first method, the Field System Solution program (FISS) is calibrated using experimentally obtained laboratory graphs of the saturated rocks around the tunnel, and the parameters are applied to the geometry of the tunnel. Comparing a critical state model with stresses around the tunnel, stability of the rocks was examined. In the second method, stability of the rocks around the tunnel was investigated by using the Nisa-II program, adjusting the laboratory swelling graphs with the creep function to the program and, finally, drawing Von Mises stresses around the tunnel.

Rheology in Wood Engineering

Abstract The system strains under external loads in a certain amount of time and under the influence of the environmental factors define the rheological behavior. Rheological phenomena depend on many factors: temperature such as air humidity or moisture content of rheological system, radiations in term of intensity, duration, type - UV, IR, X, geometry of the parts; loadings in terms of intensity, variation, duration; defects; aggressive environment; composition, material properties; combinations of these factors. Rheology science is based on the theories of the strength of materials, thermodynamics, chemistry and materials science, but in terms of application, it provides a personalized analysis or diagnosis according to the condition of the structures/systems used. Wooden constructions are subjected to various loadings on both short and long durations. The joints can be elastic (flexible) if the failure occurs gradually, or plastic, if the failure occurs suddenly. Sudden failure of joints is caused by shear as predominant load because wood does not resist at shear stresses. In order to study the rheological behavior of the wood joints with metal rods under constant load, three types of joints in terms of diameters of bolts and stiffening systems were tested. They were stressed to traction force of 500 to 900 N for 200 days, in real conditions of temperature (-7 °C la +30 °C) and humidity (from 47.8% to 83.8%). The aim of the tests were to determine the rheological behavior of wooden joints; variation of deformations in relation to the relative humidity and temperature; rate of strain and connections in determining rheological model of wood with threaded rods. It was found that the low temperatures during winter (-7…0 °C) correlated with high relative humidity led to sudden changes in strain. It was observed that the high-speed deformation had a joint with the largest diameter rod (8 mm). The paper highlights the rheological analysis of joints in wooden rods in real conditions of temperature and humidity, with regards to applied tension and the determination of the creep function that characterizes these types of connections, establishing the optimum diameter rods.

A Modified Comprehensive Model for Piezoelectric Stack Actuators and Corresponding Parameter Identification Method

In order to accurately model the hysteresis and dynamic characteristics of piezoelectric stack actuators (PSAs), consider that a linear force and a hysteresis force will be generated by piezoelectric wafers under the voltage applied to a PSA, and the total force suffering from creep will result in the forced vibration of the two-degree-of-freedom mass-spring-damper system composed of the equivalent mass, stiffness, and damping of the piezoelectric wafers and the bonding layers. A modified comprehensive model for PSAs is put forward by using a linear function, an asymmetrical Bouc-Wen hysteresis operator, and a creep function to model the linear force, the hysteresis force, and the creep characteristics, respectively. In this way, the effect of the bonding layers on the hysteresis and dynamic characteristics of PSAs can be analyzed via the modified comprehensive model. The experimental results show that the modified comprehensive model for PSAs with the corresponding parameter identification method can accurately portray the hysteresis and dynamic characteristics of PSAs fabricated by different layering/stacking processes. Finally, the theoretical analyzing on utilizing the modified comprehensive model to linearize the hysteresis characteristics and design the dynamic characteristics of PSAs is given.

Time-Dependent Unified Hardening Model: Three-Dimensional Elastoviscoplastic Constitutive Model for Clays

AbstractA new three-dimensional (3D) elastoviscoplastic (EVP) constitutive model for both normally consolidated and overconsolidated clays is presented. In developing the new model, first the time effects on clays were connected with the change of overconsolidation degree according to the concepts of aging time and the instant normal compression line. This made it convenient to combine a logarithmic creep function with the reloading function of the unified hardening (UH) model (a model for overconsolidated clays), and, thus, an isotropic EVP relationship was built. Second, a time variable derived from the isotropic EVP relationship was embedded into the current yield function of the UH model, and, thereby, a time-dependent current yield function is proposed. Based on the time-dependent current yield function and the flow rule, an EVP model for triaxial compression stress states was built. The EVP model for triaxial compression stress states then was extended to a 3D EVP model for general stress states thr...

Assessment of the Elastic-Viscoplastic Behavior of Soft Soils Improved with Vertical Drains Capturing Reduced Shear Strength of a Disturbed Zone

AbstractSoil disturbance induced by the installation of vertical drains reduces the horizontal soil permeability and shear strength in the disturbed zone. Thus, the soil disturbance contributes to the reduced overconsolidation ratio (OCR) of the soil in the vicinity of drains, influencing soil deformation. Although a significant amount of research has been conducted on the effect of permeability variations in the smear zone, the influence of the reduced shear strength in the smear zone on the ground behavior has not been investigated. In this study, a numerical solution adopting an elastic-viscoplastic model with nonlinear creep function in combination with the consolidation equations has been developed. Moreover, the effects of shear strength variation in the disturbed zone on the time-dependent behavior of soft soil deposits improved with vertical drains and preloading have been studied. The applied elastic-viscoplastic model is based on the framework of the modified Cam-clay model, capturing soil creep...

Time-dependent analysis of composite and prestressed beams using the slope deflection method

The paper presents the viscoelastic analysis of composite and prestressed beams using the slope deflection method. For the fixed-end frame element, considering the viscoelastic behavior of concrete and relaxation of prestressing steel, the integral relation between the generalized element forces and the generalized element displacements, i.e., the element stiffness matrix, is derived and presented using the mathematical operators. From the element and system equilibrium equations, the integral equations of the problem with unknown displacements are formulated. Comparing to the analysis of homogeneous elastic structures, the governing equations of the problem are similar, but with integral equations instead of algebraic equations. In the presented method, the solution to the problem is derived using the linear integral operators without introducing any additional mathematical approximations, apart from the adopted rheological relations for constitute materials. In addition, the obtained expressions are general in a sense that any concrete creep functions can be used and the element can be with a variable cross section.

Generalized integral pulses of flows and forces and potentials of visco-elastic bodies of Boltzmann-Kelvin type

The paper considers full group of one-dimensional or multidimensional potentials of stress and strain of rheological materials, which are described by linear and quasilinear integral ratios of Boltzmann-Kelvin type, containing one pair of temporary relaxation and creep functions, interconnected together with linear integral equations of Maxwell reciprocity. The built group of potentials as a function of integral pulse flows and forces satisfies Legendre-Eiler transformations and forms a chain of Gibbs-Helmholtz in generalized Bourne diagram. Variants of potential relations are shown.

Response of frictional contact problems in thermo-rheologically complex structures

This paper presents a comprehensive computational model for predicting the nonlinear response of frictional viscoelastic contact systems under thermo-mechanical loading and experience geometrical nonlinearity. The nonlinear viscoelastic constitutive model is expressed by an integral form of a creep function, whose elastic and time-dependent properties change with stresses and temperatures. The thermo-viscoelastic behavior of the contacting bodies is assumed to follow a class of thermo-rheologically complex materials. An incremental-recursive formula for solving the nonlinear viscoelastic integral equation is derived. Such formula necessitates data storage only from the previous time step. The contact problem as a variational inequality constrained model is handled using the Lagrange multiplier method for exact satisfaction of the inequality contact constraints. A local nonlinear friction law is adopted to model friction at the contact interface. The material and geometrical nonlinearities are modeled in the framework of the total Lagrangian formulation. The developed model is verified using available benchmarks. The effectiveness and accuracy of the developed computational model is validated by solving two thermo-mechanical contact problems with different natures. Moreover, obtained results show that the mechanical properties and the class of thermo-rheological behavior of the contacting bodies as well as the coefficient of friction have significant effects on the contact response of nonlinear thermo-viscoelastic materials.

Straightforward Transient-Based Approach for the Creep Function Determination in Viscoelastic Pipes

This work introduces a simple and straightforward approach for the creep function determination in viscoelastic pipes on the basis of transient flow analysis. The governing equations are expanded and analytically solved for the first half period of the transient. This solution results in a direct formula for the viscoelastic Joukowsky pressure head as a function of the creep function coefficients which are a priori unknown. Utilizing the measured data, only in the first half water hammer period and the proposed viscoelastic Joukowsky formula, the problem unknowns are determined. To investigate the method’s merits and limitations, two experimental polyethylene pipes are taken into account from the literature. The results show that the proposed approach works very well for long enough pipelines which are completely crept in the half period of the transient flow. The method is found to be computationally efficient and easy to implement in comparison with traditional inverse transient analysis techniques.

NONLINEAR ANALYSIS OF FRICTIONAL THERMO-VISCOELASTIC CONTACT PROBLEMS USING FEM

This study presents a numerical finite element model to analyze the response of frictional thermo-viscoelastic contact systems, which experience material and geometrical nonlinearities. Thermo-rheologically complex behavior of the contacting bodies is assumed. The nonlinear viscoelastic constitutive model is expressed by an integral form of a creep function, whose elastic and time-dependent properties vary with stresses and temperatures. Adopting the assumption that the hydrostatic and deviatoric responses are uncoupled, the constitutive equation is expressed in an incremental form, with the hereditary integral updated at the end of each time increment by recursive computation. The Lagrange multiplier approach is applied to incorporate the inequality contact constraints, while friction effect along the contact interface is modeled using a local nonlinear friction law. The material and geometrical nonlinearities are modeled in the framework of the total Lagrangian formulation. The developed nonlinear viscoelastic model is verified using the available benchmarks. The applicability of the developed model is demonstrated by solving two thermo-viscoelastic frictional contact problems with different contact natures. Results show a distinct effect of the thermo-rheological behavior on viscoelastic contact status.

Finite element based dynamic analysis of viscoelastic solids using the approximation of Volterra integrals

Two new finite element based methods for time-domain dynamic analysis of viscoelastic solids are proposed in this research. The viscoelastic property is given by either the relaxation or creep functions and is simulated by the conventional generalized Kelvin–Voigt model. The viscoelastic behavior during the dynamic response is taken into account by the Volterra integral. This avoids the difficulties associated with the need for high order time-derivatives used in differential models of viscoelasticity. An accurate numerical approximation for the Volterra integrals is provided. It is used for implementation of the finite-element procedure in the time domain by the introduction of additional terms to the mass matrix (or the stiffness matrix) and the force vector. The additional terms are functions of calculations at the previous time step. In the two provided finite element formulations, one uses the relaxation and the other employs the creep compliance function. This makes it unnecessary to calculate the creep function from the given relaxation modulus or vice versa which are cumbersome operations. As a case study, the wave propagation in a one- and three-dimensional viscoelastic rod subject to a step and sinusoidal load is formulated and solved by the proposed two finite element methods. The computations were validated by direct solutions based on the Laplace transform method.

Determination of the Reduced Creep Function of Viscoelastic Compliant Materials Using Pipette Aspiration Method

Determining the mechanical properties of soft matter across different length scales is of great importance in understanding the deformation behavior of compliant materials under various stimuli. A pipette aspiration test is a promising tool for such a purpose. A key challenge in the use of this method is to develop explicit expressions of the relationship between experimental responses and material properties particularly when the tested sample has irregular geometry. A simple scaling relation between the reduced creep function and the aspiration length is revealed in this paper by performing a theoretical analysis on the aspiration creep tests of viscoelastic soft solids with arbitrary surface profile. Numerical experiments have been performed on the tested materials with different geometries to validate the theoretical solution. In order to incorporate the effects of the rise time of the creep pressure, an analytical solution is further derived based on the generalized Maxwell model, which relates the parameters in reduced creep function to the aspiration length. Its usefulness is demonstrated through a numerical example and the analysis of the experimental data from literature. The analytical solutions reported here proved to be independent of the geometric parameters of the system under described conditions. Therefore, they may not only provide insight into the deformation behavior of soft materials in aspiration creep tests but also facilitate the use of this testing method to deduce the intrinsic creep/relaxation properties of viscoelastic compliant materials.

Homogenization of periodic materials with viscoelastic phases using the generalized FVDAM theory

In the last decades, new generations of advanced materials have been designed and manufactured for specific applications. The micromechanics plays an important role in the development of heterogeneous materials, enabling efficient analyses of composite materials with complex geometries, circumventing the traditional trial-and-error approach, producing substantial cost savings. The unit cell problem to the analysis of periodic heterogeneous media can be solved by the well-established 0th order version of the finite-volume theory, named finite-volume direct averaging micromechanics (FVDAM) theory. This standard version of the FVDAM theory employs an incomplete second-order displacement field within individual subvolumes of a discretized analysis domain together with a surface-averaging framework, which does not enforce displacement or traction continuity in a point-wise manner. This, in turn, produces interfacial interpenetrations and non-traction stress discontinuities, thereby demanding very refined meshes in order to produce good interfacial conformability and pointwise stress continuity between adjacent subvolumes. To overcome these shortcomings, a generalized FVDAM theory has been proposed to enable analysis of periodic heterogeneous materials in the finite-deformation domain. The generalization is based on a higher-order displacement field representation and on the definition of elasticity-based surface-averaged kinematic and static variables related through a local stiffness matrix. Herein, we specialize the generalized FVDAM theory to the infinitesimal analysis of periodic materials with viscoelastic phases, where a total or secant formulation is employed, with the viscoelastic strains evaluated incrementally using an algorithm based on the concept of state variables. The generalized or 2nd order version considerably improves interfacial conformability and pointwise traction and non-traction stress continuity between adjacent subvolumes in comparison with the 0th order version, but with a higher computational cost. Furthermore, for the same mesh discretization, these two versions provide comparable macroscopic response. Considering these features, the 0th order version is recommended to evaluate the effective elastic properties and the homogenized creep and relaxation functions, while the 2nd order version is more efficient in the evaluation of the microscopic displacement and stress fields.

Analysis of nonlinear thermo-viscoelastic–viscoplastic contacts

This study introduces a numerical model for the analysis of contact problems of nonlinear thermo-viscoelastic–viscoplastic bodies, whose viscoelastic behavior belongs to a class of thermo-rheologically complex materials (TCM). The nonlinear viscoelastic behavior is expressed with an integral form of a creep function, whose properties change with stresses and temperatures, while the viscoplastic behavior follows the Perzyna model in the framework of associative viscoplasticity. To this end, a finite element model is derived based on uncoupled thermo-mechanical problems, which include both material and geometrical nonlinearities in the framework of the total Lagrangian description. An incremental-recursive form of the constitutive equations is derived. The Lagrange multiplier method is adopted to model the inequality contact constraints. The performance of the developed model is validated by analyzing three thermo-mechanical contact problems with different natures. Moreover, results show that the thermo-rheological behavior and the mechanical properties of the contacting bodies have distinct effects on the contact response of nonlinear thermo-viscoelastic–viscoplastic materials.

Long-term creep properties of cementitious materials: Comparing microindentation testing with macroscopic uniaxial compressive testing

This study is dedicated to comparing minutes-long microindentation creep experiments on cement paste with years-long macroscopic creep experiments on concrete and months-long macroscopic creep experiments on cement paste. For all experiments, after a transient period the creep function was well captured by a logarithmic function of time, the amplitude of which is governed by a so-called creep modulus. The non-logarithmic transient periods lasted for days at the macroscopic scale, but only for seconds at the scale of microindentation. The creep moduli (which thus govern the rate of the long-term logarithmic creep) of concrete samples were estimated from microindentations performed at the scale of cement pastes in combination with micro-mechanical models. Those estimates were proportional to the creep moduli measured on concrete samples by regular macroscopic uniaxial testing, thus proving that minutes-long microindentation can provide a measurement of the long-term creep properties of cementitious materials.

Gardner and Lockman Model in Creep Analysis of Composite Steel-Concrete Sections

The paper presents an analysis of the stress and deflections changes due to creep in statically determinate composite steel-concrete beams. The mathematical model involves the equation of equilibrium, compatibility, and constitutive relationship, that is, an elastic law for the steel part and an integral-type creep law of Boltzmann-Volterra for the concrete part. Based on the theory of the viscoelastic body of Arutyunian-Trost-Bažant for determining the redistribution of stresses in beam sections between a concrete plate and steel beam with respect to time t, two independent Volterra integral equations of the second kind have been derived. A numerical method based on linear approximation of the singular kernal function in the integral equation is presented. An example with the proposed model is investigated. The creep functions are suggested by the Gardner and Lockman model. The elastic modulus of concrete Ec(t) is assumed to be constant in time t.

Is the time-dependent behaviour of the aortic valve intrinsically quasi-linear?

The widely popular quasi-linear viscoelasticity (QLV) theory has been employed extensively in the literature for characterising the time-dependent behaviour of many biological tissues, including the aortic valve (AV). However, in contrast to other tissues, application of QLV to AV data has been met with varying success, with studies reporting discrepancies in the values of the associated quantified parameters for data collected from different timescales in experiments. Furthermore, some studies investigating the stress-relaxation phenomenon in valvular tissues have suggested discrete relaxation spectra, as an alternative to the continuous spectrum proposed by the QLV. These indications put forward a more fundamental question: Is the time-dependent behaviour of the aortic valve intrinsically quasi-linear? In other words, can the inherent characteristics of the tissue that govern its biomechanical behaviour facilitate a quasi-linear time-dependent behaviour? This paper attempts to address these questions by presenting a mathematical analysis to derive the expressions for the stress-relaxation G(t) and creep J(t) functions for the AV tissue within the QLV theory. The principal inherent characteristic of the tissue is incorporated into the QLV formulation in the form of the well-established gradual fibre recruitment model, and the corresponding expressions for G(t) and J(t) are derived. The outcomes indicate that the resulting stress-relaxation and creep functions do not appear to voluntarily follow the observed experimental trends reported in previous studies. These results highlight that the time-dependent behaviour of the AV may not be quasi-linear, and more suitable theoretical criteria and models may be required to explain the phenomenon based on tissue’s microstructure, and for more accurate estimation of the associated material parameters. In general, these results may further be applicable to other planar soft tissues of the same class, i.e. with the same representation for fibre recruitment mechanism and discrete time-dependent spectra.

Waterhammer modelling of viscoelastic pipes with a time-dependent Poisson's ratio

Poisson's ratio in viscoelastic materials is a function of time. However, recently developed waterhammer models of viscoelastic pipes consider it constant. This simplifying assumption avoids cumbersome calculations of double convolution integrals which appear if Poisson's ratio is time-dependent. The present research develops a mathematical model taking the time dependency of Poisson's ratio into account for linear viscoelastic pipes. Poisson's ratio is written in terms of relaxation function and bulk modulus which is assumed to be constant. The relaxation function is obtained from creep function given as the viscoelastic property data of pipe material. The results obtained from the present waterhammer model are compared with the experimental data for two different flow rates. The comparison reveals that with the application of the time-dependent Poisson's ratio and unsteady friction, the viscoelastic data of mechanical tests can directly be used for waterhammer analysis with less need for the calibration of the flow configuration. It was also shown that the creep curve calibrated based on the present model is closer to the actual creep curve than that calibrated based on previous models.

Time-Dependent Behavior of Diabase and a Nonlinear Creep Model

Triaxial creep tests were performed on diabase specimens from the dam foundation of the Dagangshan hydropower station, and the typical characteristics of creep curves were analyzed. Based on the test results under different stress levels, a new nonlinear visco-elasto-plastic creep model with creep threshold and long-term strength was proposed by connecting an instantaneous elastic Hooke body, a visco-elasto-plastic Schiffman body, and a nonlinear visco-plastic body in series mode. By introducing the nonlinear visco-plastic component, this creep model can describe the typical creep behavior, which includes the primary creep stage, the secondary creep stage, and the tertiary creep stage. Three-dimensional creep equations under constant stress conditions were deduced. The yield approach index (YAI) was used as the criterion for the piecewise creep function to resolve the difficulty in determining the creep threshold value and the long-term strength. The expression of the visco-plastic component was derived in detail and the three-dimensional central difference form was given. An example was used to verify the credibility of the model. The creep parameters were identified, and the calculated curves were in good agreement with the experimental curves, indicating that the model is capable of replicating the physical processes.