Constitutive modeling of visco-hyperelastic behavior of double-network hydrogels using long-term memory theory

Double-network hydrogels with viscoelastic behavior are appropriate materials for biomechanical applications. In this article, the standard linear solid (SLS) rheological model for the linear viscoelastic materials is generalized to the viscoelastic materials with large nonlinear deformations. Based on this viewpoint, the constitutive equation is proposed as sum of two parts including the strain-dependent elastic stress, and the viscous stress, which depends on the strain and strain rate. The elastic part of the stress is modeled via considering a hyperelastic strain energy function, while the main core of the viscous stress part requires a time-dependent weight function to satisfy the long-term memory fading principle. In addition, the weight function is proposed such that it can capture the mechanical behavior trend corresponding to the strain and strain rate for a double-network hydrogel in the relaxation test. Finally, to evaluate the performance of the proposed constitutive equation for the mechanical behavior modeling of double-network hydrogels, the tests on these materials have been used, and the material parameters are determined from fitting the experimental results to the theory. The agreement of test and theory results showed that the proposed model is capable to model the mechanical behavior of double-network hydrogels.     

Extensive validation of a thermodynamically consistent, nonlinear viscoelastic model for glassy polymers

The nonlinear thermoviscoelastic formalism presented in the preceding paper is validated with four amorphous polymer systems. Validation is performed over a broad range of relaxation phenomena in the glass transition region, including the temperature and rate-dependence of the stress-strain behavior through yield, volume and enthalpy relaxation, and stress relaxation during multi-step loading histories. The objective is to obtain quantitative agreement between the constitutive theory and all experimental results using one set of model parameters for each material system. The nonlinear viscoelastic formalism is shown to predict the wide range of behavior observed experimentally, indicating that the formalism does capture the essential physics of glassy polymers. Moreover, the material parameters required in the constitutive formalism can be readily obtained from independent experiments and are relatively insensitive to how these parameters are determined experimentally from the various characterization techniques.     

Prediction of the relationship between the rate of deformation and the rate of stress relaxation in glassy polymers

Kim, et al. (Polymer, 54(15), 3949, 2013) recently reported on the unexpected relaxation behavior of an amorphous polymer in the T_g-region, where the rate of stress relaxation increased with deformation at a strain rate of 1.5 × 10⁻⁴ s⁻¹ but decreased at a strain rate of 1.2 × 10⁻⁵ s⁻¹. This inversion in the ordering with strain rate challenges the underlying structure of the existing nonlinear viscoelastic and viscoplastic constitutive models, where the key nonlinearity is a deformation dependent material clock. The nonlinear stress relaxation predictions of a recently developed stochastic constitutive model, SCM, (Medvedev, et al., J. Rheology, 57(3), 949, 2013) that acknowledge dynamic heterogeneity of the glass have been investigated. The SCM predicts the inversion in the ordering of the mobility with the loading strain rate as reported by the stress relaxation response. The change in perspective on the nonlinear viscoelastic behavior of glassy polymers engendered by the SCM is discussed.     

有限变形下的橡胶材料非线性高弹-粘弹性本构模型

本文基于唯象学原理提出了一种适用于橡胶材料在有限应变条件下的非线性高弹-粘弹性本构模型,该模型将应力拆分为两部分：第一部分基于Yeoh模型计算得到的高弹性应力;第二部分由Boltzmann叠加原理计算得到的带有率依赖性的粘弹性应力。该本构模型中应变依赖性的高弹性应力通过修正Zener模型中的非线性弹簧表示,时间及应变依赖性的粘弹性应力通过修正Zener模型中的非线性Maxwell模型表示。利用提出的高弹-粘弹性本构模型模拟不同应变条件下的拉伸、回复、应力松弛及多步松弛在内的复杂加载过程,并与实验值对比,结果表现出良好的吻合性,说明该本构模型的合理、可靠。     

橡胶材料非线性高弹-粘弹性本构模型

本文提出的修正Zener模型,即以非线性弹簧与粘壶代替原来的线性弹簧与粘壶,得到一种适合橡胶材料在单轴作用下的非线性高弹-粘弹性本构模型。该模型可同时描述加载、卸载及应力松弛行为。模型将应力分解为弹性应力与粘性应力。弹性应力由Yeoh高弹性模型得到;粘性应力由加载及卸载过程的瞬时应力通过积分变换得到。此外考虑到应力松弛的作用,因此将粘性应力分解成两部分：一是在加载、卸载过程所用时间内由应变的改变所导致的粘性力;二是在该时间内由应力松弛所产生的反作用力。最后,将实验结果与该模型的计算结果进行对比,结果表明计算结果与实验结果具有良好的一致性,说明该本构模型可靠、合理。     

On Mooney and Mooney stress relaxation. II. A nonlinear viscoelastic description: Experimental and numerical results

It has been investigated whether the stress build‐up and the stress relaxation involved in a Mooney test, with subsequent Mooney stress relaxation, can be described by nonlinear viscoelastic theory, more particularly the Wagner constitutive model. For this purpose, the viscoelastic behavior of three nonvulcanized EPDM materials, with similar Mooney viscosity but varying elasticity, has been studied. Relaxation time spectra were obtained from dynamic mechanical experiments, from which the step‐strain stress‐relaxation modulus was calculated. Stress build‐up experiments were performed with a cone and plate system in order to obtain the so‐called damping function (a measure for the deformation sensitivity) of the materials. Using these material functions, the Mooney test was successfully described with the Wagner constitutive model. Experimental and theoretical Mooney stress‐relaxation rates are in close agreement. The predicted Mooney viscosity is up to 25% lower than the measured value. This may be due to nonideal conditions during the Mooney test, such as inhomogeneous heating and secondary flows, and to inaccuracy of the damping function. The model calculations confirm the strong experimental dependence of Mooney measurements on small variations in instrumental conditions such as geometry, rotation speed, and so forth. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 1220–1233, 1999     

A practical approach to modeling time‐dependent nonlinear creep behavior of polyethylene for structural applications

The purpose of this work is to develop a practical method for constitutive modeling of polyethylene, based on a phenomenological approach, which can be applied for structural analysis. Polyethylene is increasingly used as a structural material, for example, in pipes installed by trenchless methods, where the relatively low stiffness of polyethylene reduces the required installation forces, chemical inertness makes it applicable for corrosive environments, and adequate strength allows its use in sewer, gas, and water lines. Polyethylene exhibits time‐dependent constitutive behavior which is also dependent on the applied stress level resulting in nonlinear stress–strain relationships. Nonlinear viscoelastic theory has been well established and a variety of modeling approaches have been derived from it. To realistically utilize the nonlinear modeling approaches in design, a simple method is needed for finding a constitutive formulation for a specific polyethylene type. This paper presents such a practical approach to nonlinear viscoelastic modeling utilizing both the multi‐Kelvin element theory and the power law functions to model creep compliance. Creep tests are used to determine material parameters and models are generated for four different polyethylene materials. The corroboration of the models is completed by comparisons with results from different tensile creep, step‐loading creep, and load‐rate tests. POLYM. ENG. SCI., 48:159–167, 2008. © 2007 Society of Plastics Engineers     

PE100燃气管材的非线性蠕变行为及其本构模型研究

对PE100燃气管材开展不同应力水平（2.4~9.6 MPa）下的常温蠕变测试,结果显示应力不超过5.4 MPa时,材料的蠕变柔量与应力水平无关,呈现出线性黏弹性行为,当应力高于5.4 MPa时,材料发生非线性黏弹性蠕变。基于单积分型非线性黏弹性本构理论,采用Findley模型对蠕变行为进行分析,并与Struik经验模型进行比较。结果表明,Findley模型和Struik模型均能较好地描述PE100管材的蠕变行为,但对高应力下的非线性蠕变行为,Findley模型比Struik模型描述得更准确。     

Nonlinear mechanical response of high density polyethylene. Part I: Experimental investigation and model evaluation

The nonlinear behavior of high density polyethylene (HDPE) is investigated for samples cut from thick-walled HDPE pipe. Extensive experimental work has been performed to characterize the non-linear time-dependent response of the material tested under uniaxial compression. Tests were conducted under conditions of constant strain rate, creep, stress relaxation, constant loading rate, abrupt change of strain rate, creep-recovery, cyclic strain rate, and various combinations of these loading conditions. Creep and stress relaxation response after strain reversal and the effect of the transient response on the following stress-strain behavior is examined. Permanent strains for the test specimens and their dependence on loading histories are investigated. Specimens cut at various orientations from the pipe are used to quantify the small amounts of local anisotropy in the pipe specimen. The experimental work has been used to develop both nonlinear viscoelastic (NVE) and viscoplastic (VP) constitutive models in a companion paper. Both the test results and the corresponding model predictions are reported in this paper. It is found that the VP model reproduces the nonlinear viscoelastic-viscoplastic behavior of HDPE very well provided that the current strain is not below the maximum strain imposed (there is no strain reversal). The NVE model predicts the material behavior reasonably well for some loading conditions, but inadequately for others.     

Constitutive modeling for mechanical behavior of PMMA microcellular foams

Constitutive equations for nonlinear tensile behavior of PMMA foams were studied. Five viscoelastic models composed of elastic and viscous components were accounted for the modeling of the constitutive equations. The developed constitutive equations are expressed in terms of material properties and foam properties such as strain, strain rate, elastic modulus, relative density of foam, and relaxation time constant. It was found that the stress-strain behaviors by Generalized Maxwell model, Three Element model and Burgers model could be described by the constitutive equation obtained from the Maxwell model. For the verification of the constitutive model, poly(methyl methacrylate) (PMMA) microcellular foams were manufactured using batch process method, and then uniaxial tensile tests were performed. The stress-strain curves by experiment were compared with the theoretical results by the constitutive equation. It was demonstrated that nonlinear tensile stress-strain behaviors of PMMA foams were well described by the constitutive equation.     

PE80燃气管道的应力松弛模型与实验验证

采用实验方法研究聚乙烯(PE)管道在机械载荷条件下的应力松弛行为,通过卷积分解析法和基于Maxwell模型的有限单元数值模拟法对其进行验证。结果表明,在小应变下采用Boltzmann叠加原理推导出的PE黏弹性本构模型可用于预测PE管道的应力松弛行为;基于实验参数,利用有限单元法能有效地对PE的黏弹性行为进行预测。     

A rate‐type nonlinear viscoelastic–viscoplastic cyclic constitutive model for polymers: Theory and application

A thermodynamically consistent rate‐type viscoelastic–viscoplastic constitutive model is developed in the framework of isothermal and small deformation to describe the nonlinear and time‐dependent deformation behaviors of polymers, e.g., ratchetting, creep, and stress relaxation. The model is proposed on the base of a one‐dimensional rheological model with several springs and dashpot elements. The strain is divided into viscoelastic and viscoplastic parts, and the stress is also decomposed into two components. Each stress component is further divided into elastic and viscoelastic sub‐components. The viscoelasticity is described by introducing pseudo potentials, and the ratchetting is considered by the viscoplastic flow which is derived by the codirectionality hypotheses. The capability of the proposed model to describe the nonlinear and time‐dependent deformation of polymers is then verified by comparing the simulations with the corresponding experimental results of polycarbonate (PC) polymer. It is shown that the nonlinear and time‐dependent stress–strain responses of the PC can be reasonably predicted by the proposed model. POLYM. ENG. SCI., 56:1375–1381, 2016. © 2016 Society of Plastics Engineers     

Transient modeling of viscosity

Capillary and parallel plate rheological characterization was conducted for a low‐density polyethylene. In contrast with conventional rheological analysis, steady conditions were not assumed. Transient data, with time steps between 0.0001 and 0.2 s, were analyzed with a nonlinear, viscoelastic constitutive model in which the relaxation time was modeled as a function of the applied stress. The fit model explained more than 99% of the observed transient variation in the capillary and parallel plate rheometers. The model coefficients for the capillary and parallel plate were compared directly to conventional linear viscoelastic analysis of the same parallel plate data. The results indicate that the described constitutive model closely predicts the observed viscoelastic behavior of the polymer melt tested in the parallel plate rheometer. Furthermore, the results indicate that the relaxation spectrum modeled with the transient analysis of the capillary rheological data correlate closely to the results predicted by the same transient analysis of parallel plate rheological data. The conclusion is that described constitutive modeling describes the viscoelastic behavior in both capillary and parallel plate rheometers. Moreover, the analysis and results suggest that the viscoelastic behavior of the polymer melt is a significant factor during the rheological characterization and the modeling of the transient response should be taken into consideration during rheological analysis to provide high fidelity models. POLYM. ENG. SCI., 57:1110–1118, 2017. © 2017 Society of Plastics Engineers     

Deformation behavior of an epoxy resin subject to multiaxial loadings. Part II: Constitutive modeling and predictions

Based on the experimental data presented in Part I, a nonlinear viscoelastic constitutive model, in differential form, is presented here. A distinctive feature of this model is the inclusion of a criterion to delineate loading and unloading in multiaxial stress states, and different moduli for loading and unloading behaviors. In addition, the model contains only five material constants and one modulus function, which can be calibrated in accordance with a well‐defined procedure. A comparison with the experimental data shows that the current differential model is capable of predicting the nonlinear viscoelastic behavior of the epoxy polymer qualitatively and quantitatively, including both the loading and unloading behavior. The predictions of an integral form of constitutive model are also included for comparative purposes.     

Nonlinear viscoelastic modeling of soft polymers

A one‐dimensional phenomenological constitutive model, representing the nonlinear viscoelastic behavior of polymers is developed in this study. The proposed model is based on a modification of the well‐known three element standard solid model. The linear dashpot is replaced by an Eyring type one, while the nonlinearity is enhanced by a nonlinear, strain dependent spring constant. The new constitutive model was proved to be capable of capturing the main aspects of nonlinear viscoelastic response, namely, monotonic and cyclic loading, creep and stress relaxation, with the same parameter values. Model validation was tested on the experimental results at various modes of deformation for two elastomeric type materials, performed elsewhere. A very good agreement between model simulations and experimental data was obtained in all cases. © 2015 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2015 , 132, 42141.     

On the nonlinear characterization of the long term behavior of polymeric materials

In the linear viscoelastic range the long term behavior of viscoelastic materials—such as polymers—can be described by using exponential series with a limited number of terms for the approximation of the relaxation modulus or of the creep compliance. This procedure can be extended to the nonlinear viscoelastic range by multiplying the linear parameters of the material by certain nonlinearity factors, which depend upon the level of the applied loading. Application of this method to stress relaxation data of several polymers has shown that nonlinearity factors can be approximated as linear functions of the applied constant strain. From creep tests, on the other hand, one can observe that the immediate strain response to the suddenly applied stress is linear elastic even in the nonlinear viscoelastic range of the investigated polymer. The computation of the linear viscoelastic material parameters as well as of the nonlinearity factors is conducted numerically by using least squares techniques. Good agreement between computed results and experimental data can be observed in the presented examples.     

Mechanical response of beams of a nonlinear viscoelastic material

A constitutive equation for nonlinear viscoelasticity is used to model the mechanical response of solid polymers such as polycarbonate. The nonlinearity arises from a reduced time which causes stress relaxation to accelerate with increasing strain. The constitutive equation can account for the occurrence of yield in a homogeneous uniaxial constant strain rate test. The constitutive equation is used in a study of the pure bending of beams. It is assumed that the classical assumption of beam theory is valid, i.e., plane sections remains plane. At each fixed time, the strains vary linearly through the depth of the beam. At a fixed material element the strain varies in time with the curvature. This spatial variation of the strains combined with the nonlinear dependence of the reduced time on strain leads to a significantly different response from that given by traditional beam theory. The implications of this for the bending moment history, stress distributions, and other factors that relate to beam design are discussed.     

Viscoelastic-plastic deformation behavior of polypropylene after cyclic preloadings

Stress-relaxation behavior is studied in polypropylene samples subjected to different cyclic preloadings and to simple tension. The relaxation tests are performed under different sets of strain amplitude, number of cycles, and strain rate, using a closed-loop, electrohydraulic, servocontrolled testing machine. The calculated stress-strain curves are determined from a constitutive equation based on an overstress theory in which an equilibrium stress and a viscosity function are treated. The calculated results agree well with the experimental ones. It is concluded that the overstress theory explains the nonlinear viscoelastic-plastic behavior of polypropylene.     

Modeling the uniaxial rate and temperature dependent behavior of amorphous and semicrystalline polymers

Strain rate and temperature dependent constitutive equations are proposed for polymer materials based on existing isotropic formulations of viscoplasticity. The proposed formulations are capable of simulating some of the important features of deformation behavior of amorphous and semicrystalline polymers. The materials model is based on the assumption that the evolution of flow stress is dependent on the rate of deformation, temperature, and an appropriate set of internal variables. The proposed theory is capable of modeling yielding, strain softening, and the orientation hardening exhibited by amorphous polymers. It is also possible to model the initial viscoplastic and subsequent nonlinear hardening behavior shown by semicrystalline polymers at large strains. Uniaxial tensile tests with uniform and hourglass specimens are made at temperatures ranging from 23 to 100°C and under various crosshead speeds. Both amorphous polycarbonate and semicrystalline polypropylene sheet materials are tested to characterize the stress and strain behavior of these materials and to determine their appropriate material constants. Load relaxation experiments are also conducted to obtain the necessary material constants describing the rate and temperature dependent flow stress behavior of polypropylene. Simulation results compare favorably against experimental data for these polymer materials.     

Constitutive model and failure criterion for orthotropic ceramic matrix composites under macroscopic plane stress

To predict the nonlinear stress-strain behavior and the rupture strength of orthotropic ceramic matrix composites (CMCs) under macroscopic plane stress, a concise damage-based mechanical theory including a new constitutive model and two kinds of failure criteria was developed in the framework of continuum damage mechanics (CDM). The damage constitutive model was established using strain partitioning and damage decoupling methods. Meanwhile, the failure criteria were formulated in terms of damage energy release rate (DERR) in order to correlate the failure property of CMCs with damage driving forces, and the maximum DERR criterion and the interactive DERR criterion were suggested simultaneously. For the sake of model evaluation, the theory was applied to a typical CMC with damageable and nonlinear behavior, that is, 2D-C/SiC. The damage evolution law, strain response and rupture strength under incremental cyclic tension along both on-axis and off-axis directions were completely investigated. Comparison between theoretical predictions and experimental data illustrates that the newly developed mechanical theory is potential to give reasonable and accurate results of both stress-strain response and failure property for orthotropic CMCs.