Study on Nonlinear Viscoelastic Constitutive Equations for Rock

A new constitutive equation has been developed for describing the nonlinear viscoelastic properties, including yield, of polymer solids. The nonlinear viscoelastic constitutive equation is the logical extension of the standard three‐dimensional linear viscoelastic constitutive equation to include the effects of deformation‐induced changes in the thermodynamic state of the polymer on the rate of viscoelastic relaxation. The rate of viscoelastic relaxation depends upon the fraction of unoccupied lattice sites (i.e., holes) in the Simha‐Somcynsky statistical thermodynamic equation of state, and the hole fraction is a known function of the temperature, pressure, and specific volume. The nonlinear viscoelastic yield and postyield behavior are a direct consequence of the deformation‐induced dilation. All material constants contained in the constitutive equation can be determined from independent elastic, linear viscoelastic, and pressure‐volume‐temperature measurements—there are no adjustable constants. The constitutive equation has been solved for model viscoelastic materials in uniaxial extension and shear deformations where the applied strain rate is constant. It is shown that an experimentally realizable “shear” deformation is only isochoric in the linear viscoelastic limit. Yield and postyield softening are predicted in both uniaxial extension and experimental shear, and the predicted yield stress exhibits the appropriate strain rate and temperature dependence. The effects of using various finite strains measures in the nonlinear viscoelastic constitutive equation are analyzed.

This chapter deals with how to measure nonlinear viscoelastic functions and various nonlinear viscoelastic constitutive equations. Since there have been developed a number of constitutive equations which cannot be included in a single chapter, this chapter considers only a few popularly used ones, which are classified to four groups: the models based on mathematical approximations; the models obtained from the generalization of linear viscoelastic models; the models based on the simplification of polymer structure; and the Leonov model which is a class of constitutive equations based on irreversible thermodynamics.

For the class of all discrete two-velocity models of the nonlinear Boltzmann equation consistent with theH-theorem an exact representation of the corresponding nonlinear constitutive equation will be derived. Applying the fractional calculus we derive, in addition, the nonlinear fractional Boltzmann equation as the starting point for the formulation of a nonlinear fractional constitutive equation which relates the densities and fluxes through an integro-differential equation. Linearization leads to the fractional version of Cattaneo's constitutive law. The model presented here may be regarded as a first step, towards the formulation of a fractional extended irreversible thermodynamics.

Considering the influence of temperature and strain variables on materials. According to the relationship of conjugate stress-strain, a complete and irreducible non-linear constitutive equation of isotropic hyperelastic materials is derived and the constitutive equations of 16 types of isotropic hyperelastic materials are given we study the transformation methods and routes of 16 kinds of constitutive equations and the study proves that transformation of two forms of constitutive equation. As an example of application, the non-linear thermo-elastic constitutive equation of isotropic hyperelastic materials is combined with the natural vulcanized rubber experimental data in the existing literature base on MATLAB, The results show that the fitting accuracy is satisfactory.

A comparison of second-order constitutive theories for hyperelastic materials

Анализируется скоростная чувствительность семейства диаграмм деформирования, порождаемых физически линейным определяющим соотношением вязкоупругости Больцмана-Вольтерры с произвольной функцией релаксации в одноосных испытаниях с постоянными скоростями деформации. Выведено общее выражение для показателя скоростной чувствительности (скоростного упрочнения) и аналитически исследованы его общие качественные свойства: зависимость от деформации, скорости деформации и характеристик функции релаксации, диапазон значений, интервалы монотонности и существование точек экстремума, предельные значения при стремлении скорости деформации к нулю или бесконечности, способы определения по диаграммам деформирования или по кривым релаксации. Установлено, что (в рамках линейной теории вязкоупругости) этот показатель зависит не от двух независимых аргументов (деформации и скорости деформации), а только от их отношения, что он выражается через отношение касательного модуля к секущему и может быть вычислен по одной диаграмме деформирования с произвольной скоростью деформации, и что по заданной (или измеренной в испытаниях) функции скоростной чувствительности можно однозначно восстановить функцию релаксации. Доказано, что значения показателя скоростной чувствительности всегда лежат в интервале от нуля до единицы (т.е. линейное определяющее соотношение описывает только псевдопластические среды и не может описывать дилатантные) и могут быть сколь угодно близки к единице (верхней границе для псевдопластических сред), что как функция скорости он не только может монотонно возрастать или убывать, но может иметь точки экстремума, в частности точку максимума (при малообременительных ограничениях на функцию релаксации). Тем самым обнаружена неожиданная способность линейной теории вязкоупругости не только порождать семейство диаграмм деформирования с выраженными участками течения при практически постоянном напряжении, но и качественно описывать «сигмоидальную» форму зависимости напряжения от скорости деформации (в логарифмических осях) и очень высокую скоростную чувствительность, характерные для режима сверхпластического деформирования материалов. Установленные свойства показателя скоростной чувствительности и его характерные особенности проиллюстрированы на примерах классических регулярных, сингулярных и фрактальных моделей вязкоупругости (Максвелла, Фойгта, Кельвина, Зенера, Бюргерса, Скотт-Блэра) и их параллельных соединений.

High-resolution laser Doppler velocimetry and flow-induced birefringence measurements were used to determine velocity and stress fields and to calculate a transient planar elongation viscosity profile for viscoelastic flow on the shear-free centerplane of a planar contraction geometry. A novel, two-component Boger fluid was developed that enabled Weissenberg numbers, We, as high as 2.9 to be attained and allowed the stress-optical coefficients of the fluid components to be measured. The experimental apparatus was designed to subject the fluid traveling on the centerplane to Hencky strains, ε, as great as 3.5. Despite the high We and ε, the flow induced only a weakly nonlinear viscoelastic stress response in the fluid. A spectral decomposition of predictions of linear and nonlinear constitutive equations was used to demonstrate that the weakly nonlinear response was a result of the nonhomogeneous form of the centerplane strain-rate profile in conjunction with the finite response time of the individual relaxation modes. The degree of nonlinear stress response in this experiment cannot be increased by increasing the contraction ratio to increase the total Hencky strain experienced by a fluid element traveling along the centerplane, increasing the flow rate to increase the strain rate at a given point along the centerplane, or introducing a taper to the contraction geometry to change the strain rate profile. Thus, the experimental configuration of flow through a contraction appears to be of limited value in evaluating the accuracy of nonlinear constitutive models in predicting the stress response of a viscoelastic polymer solution to a generalized elongational flow.

A simplified three‐dimensional viscoelastic equation, based on the Green–Rivlin theory, was used to characterize the nonlinear creep behavior of two food products, i.e., apple and potato. Uniaxial and triaxial creep tests were conducted to evaluate the nonlinear behavior of apple and potato flesh. Four material property functions in the nonlinear constitutive equation were determined using experimental results. The nonlinear constitutive equation was then used to predict the load response of both apple and potato flesh under various loading conditions. The nonlinear constitutive model calculated values compared favorably with the experimental data. Results show that the nonlinearity of apple flesh under the uniaxial creep loading was significant especially for extended loading time periods. The initial load response has a pronounced effect on the nonlinearity of potato flesh under the uniaxial creep loading.

The Rabotnov physically non-linear constitutive equation with two arbitrary (increasing) material functions for elasto-viscoplastic materials is studied analytically in order to outline the set of basic rheological effects it can simulate, to clarify the material functions governing abilities, to indicate application field of the relation and to develop identification techniques.Under minimal primary restrictions on two material functions, the general equation of the stress-strain curves family produced by the model at constant strain rates is derived and analyzed in uniaxial case. The main qualitative properties of the stress-strain curves and their dependence on a strain rate and material functions are examined and compared to typical properties of test stress-strain curves of elasto-viscoplastic materials and to the properties of stress-strain curves generated by the Boltzmann - Volterra linear viscoelasticity theory (with arbitrary creep compliance). The last one have been generalized to formulate the Rabotnov relation and so the inherited properties and the new properties acquired due to the introduction of the second material function governing non-linearity are in the focus of attention. In particular, it is proved that the stress-strain curves family increases monotonously as strain rate parameter grow and converges to limit curve (instantaneous or equilibrium) as strain rate tends to zero or infinity. Two-sided bounds for stress-strain curves and conditions for their monotonicity with respect to strain or for existence of maximum are obtained. It is shown that the Rabotnov constitutive equation is able to simulate (qualitatively) deformation softening of materials, i.e. a non-monotone behavior and existence of a decreasing segment of stress-strain curves in tensile tests under constant strain rate condition. The linear viscoelasticity theory fails to do it as it generates only increasing and convex-up stress-strain curves.

This paper is concerned with the elasto-viscoplastic constitutive equation of normally consolidated clay. A three dimensional constitutive equation is derived based on the Perzyna’s elasto-viscoplastic theory and Cambridge theory. Comparing the theoretical results with the experimental results by triaxial compression test, it becomes evident that the derived constitutive equation can explain the behavior not only such time-dependent behavior as creep, stress relaxation and strain rate effect, but also as secondary compression, aging and isotropic stress relaxation. The equation has eight material parameters which can be determined by two undrained strain rate constant triaxial compression tests with different strain rates, usual consolidation and swelling tests. The proposed theory has a feature to be able to determine the secondary consolidation rate from the results of strain rate controlled undrained triaxial compression test.

Products bear a lot of risks of getting damaged because of falling, getting compressed, etc. during transportation of logistics. In order to preserve products shape and quality, we can reduce the impact force during the logistics using packaging system. In general, we consider that packaging materials would not exhibit the strain rate effect while we predict free drop test results. In other words, the quasi-static compression stress-strain curve is used to simulate the results. But some packaging materials probably have the strain rate effect. Therefore, the foam packaging system of free drop test in this research considers the strain rate effect, that is the packaging materials are strain rate dependent. The energy loss is also considered into account when using numerical simulation methods to predict the maximum acceleration. The aim of this research is to understand the behavior of foam packaging materials during free drop by numerical simulation methods. Most of the literature on relevant research of foam packaging system neglects the strain rate effect to simplify the numerical simulation. But in this research it is shown that the strain rate effect would influence the maximum strain captured by high speed camera, even if there is no difference between free drop test and numerical simulation on maximum acceleration if the strain rate effect is not considered. So this research generalizes some experimental results to coordinate the relationship between the strain rate effect and the energy loss. Later, both energy method and single degree-of-freedom of nonlinear spring system are used to predict the results from free drop test. The constitutive law of foam packaging materials used in this research, for example expanded polypropylene (EPP), is constructed by series of quasi-static compression test and high-rate dynamic impact test. A numerical material constitutive law model considering the strain rate effect is also published to simulate the strain rate dependent property about EPP. We can accomplish the aim, considering the strain rate effect is used in this model when we are simulating the results, and to develop the numerical simulation which can predict maximum strain more accurately. In addition, for verifying that EPP has the strain rate effect, the high speed camera cooperated with the accelerometer through free drop test is used to derive the actual stress-strain curve of EPP packaging system. Finally, the fact that EPP is strain rate depenedent material has been proved. The microstructure photos of EPP by both optical electron microscope (OEM) and scanning electron microscope (SEM) were captured. This research also attempts to understand the relationship between strain rate effect and the microstructure of EPP. Furthermore, for designing EPP packaging system of real products, the appropriate range of strain energy density of EPP was chosen. The maximum acceleration, maximum strain and actual stress-strain curve of free drop test are predicted by numerical simulation methods considering the strain rate effect or not. To conclude, the results by several numerical simulation methods and free drop test are compared to explain that these methods which considers strain rate effect are feasible and accurate.

The changes of mechanical properties of filled polymers and their dependence on deformation history are the subject of this paper. For most of filled polymers in practical use, the theory of linear viscoelasticity cannot be applied, even at small deformations. In this work, samples of glass bead filled polybutadiene rubber with different filler levels were investigated at small strains (e<10%). The evolution of the relaxation modulus and Poisson's ratio was observed in cyclic experiments, which were also applied in inducing a defined deformation history for the succeeding relaxation experiments. In these experiments, the relaxation modulus and Poisson's ratio were measured as functions of time, with strain, strain rate, filler level and the preceeding deformation history as parameters. The results indicate dewetting as the main reason for the changes of the mechanical properties of the filled materials.

Material properties are represented by constitutive equations, which complete the set of balance equations for momentum and energy. According to experimental observations, material behavior may be idealized to be rate independent or rate dependent. Moreover, rate dependent or rate independent hysteresis effects may occur: This implies four different categories of constitutive models, namely the theories of elasticity, viscoelasticity, plasticity and viscoplasticity. In case of elasticity the stress depends only on the present strain. For inelastic materials the stress is related to the present strain and its past history. This relationship can be represented explicitly as a functional of the strain process. An implicit representation of a functional relation is obtained from the formulation of a system of ordinary differential equations for a set of additional internal variables. These evolution equations specify the rate of change of internal variables depending on their present values and the strain (or stress) input. Different constitutive models imply different mathematical characteristics of the evolution equations. These concern the existence of equilibrium solutions and their stability properties. Some principal ideas for a modelling of material properties are outlined for the purely mechanical case and then generalized to thermodynamics. In the thermodynamic context the existence of rate independent hysteresis effects, encountered in plasticity and viscoplasticity, leads to a basic problem: It is impossible to approximate reversible processes through slow deformation and temperature histories. In the theory of viscoplasticity the evolution equations have equilibrium solutions, which depend on the process history and represent equilibrium hysteresis behavior.

Knowledge regarding the time-dependent behavior of rocks is essential to estimate the long-term deformation and stability of underground structures. The rock strength increases with the increasing loading rate. However, the loading rate dependence in postfailure regions under confining pressure remains unclear. In this study, triaxial compression tests were conducted on four types of rocks to examine the loading rate dependence in both peak and postfailure regions. Results demonstrate that an increase in residual strength with a tenfold increase in the strain rate was approximately proportional to 4% of the residual strength. Furthermore, the increase in peak strength with a tenfold increase in the strain rate increased at a rate of approximately 4% of peak strength. The obtained results were applicable to all the sample rocks and can be easily employed for improving the constitutive equations. Finally, the effect mechanism of the confining pressure on the loading rate dependence of rock strength is discussed.

Constitutive Model for the Subsequent Time-Dependent Deformations of Type 304 Stainless Steel at Room Temperature