A thermovisco-hyperelastic constitutive model of HTPB propellant with damage at intermediate strain rates

The uniaxial compressive tests at different temperatures (223–298 K) and strain rates (\(0.40\mbox{--}63~\mbox{s}^{-1}\)) are reported to study the properties of hydroxyl-terminated polybutadiene (HTPB) propellant at intermediate strain rates, using a new INSTRON testing machine. The experimental results indicate that the compressive properties (mechanical properties and damage) of HTPB propellant are remarkably affected by temperature and strain rate and display significant nonlinear material behaviors at large strains under all the test conditions. Continuously decreasing temperature and increasing strain rate, the characteristics of stress-strain curves and damage for HTPB propellant are more complex and are significantly different from that at room temperature or at lower strain rates. A new constitutive model was developed to describe the compressive behaviors of HTPB propellant at room temperature and intermediate strain rates by simply coupling the effect of strain rate into the conventional hyperelastic model. Based on the compressive behaviors of HTPB propellant and the nonlinear viscoelastic constitutive theories, a new thermovisco-hyperelastic constitutive model with damage was proposed to predict the stress responses of the propellant at low temperatures and intermediate strain rates. In this new model, the damage is related to the viscoelastic properties of the propellant. Meanwhile, the effect of temperature on the hyperelastic properties, viscoelastic properties and damage are all considered by the macroscopical method. The constitutive parameters in the proposed constitutive models were identified by the genetic algorithm (GA)-based optimization method. By comparing the predicted and experimental results, it can be found that the developed constitutive models can correctly describe the uniaxial compressive behaviors of HTPB propellant at intermediate strain rates and different temperatures.     

Nonlinear behavior of bumper foams under uniaxial compressive cyclic loading

Theoretical and experimental studies on the nonlinear behaviors of bumper foams under cyclic loading are carried out in this paper. To study the compressible materials, the incompressible viscoelastic model proposed by Rajagopal and Srinivasa in 2000 is modified and expressed as a function of the principal stretches. The modified model is used to describe bumper foams for the first time. Besides, in order to better predict the nonlinear process of bumper foams under cyclic loading, a new compressible viscoplastic model is proposed, which is expressed separately as the invariants of stretches and the principal stretches. Then the compressible viscoelastic model and the compressible viscoplastic model are used to describe the response of bumper foams under cyclic loading with constant and variable amplitudes, respectively. The experimental results demonstrate that the compressible viscoelastic model and the compressible viscoplastic model are both suitable to describe the response of bumper foams under cyclic loading, the new proposed compressible viscoplastic model is more suitable to describe the deformation at the end of each cycle.     

On the nonlinear viscoelastic behaviour of nylon fiber

Nonlinear time-dependent stress relaxation was determined experimentally in nylon fiber in the small strain domain. This process is accounted for on the basis of a quasi-linear viscoelastic theory. A nonlinear modified constitutive equation for the viscoelastic medium is deduced. In processing the experimental data it turned out to be necessary to keep the nonlinear terms up to cubic in the constitutive equation for nylon fiber. The elastic and rheological constants in the constitutive equation for nylon fiber are evaluated.     

Characterization and modeling of stain-rate-dependent behavior of polymeric foams

A polymeric foam was characterized under quasi-static and dynamic loading and a constitutive model was proposed to describe its nonlinear behavior at varying strain rates. Four characteristic properties were identified in the compressive stress–strain curves: (1) yield stress, (2) peak or “critical” stress corresponding to collapse initiation of the cells, (3) plateau stress following the initial collapse of the cells, and (4) strain hardening stress at the end of the plateau region and before the onset of densification. All of the above characteristic stresses vary linearly with the logarithm of strain rate. A strain-based nonlinear constitutive model was proposed. A unified (master) constitutive model with built-in strain rate dependence was formulated and was shown to be in very good agreement with experimental results. The master stress–strain response was modeled in two parts, a power law and one consisting of two exponential terms.     

Branching of strain histories for nonlinear viscoelastic solids with a strain clock

Summary An important class of constitutive equations for nonlinear viscoelastic response utilizes the concept of a strain clock. The clock takes the form of a material time variable which is defined in terms of the strain history and which increases faster than physical time. Important consequences of the strain clock are that stress relaxation and creep occur faster as strain increases, and the stress may not increase monotonically with time. In this work, we discuss whether this non-monotonic response implies that strain histories may branch into multiple histories.     

高斯激励黏弹性夹层梁的非线性随机响应特性

黏弹性夹层梁的随机振动控制是一个重要的实际问题。基于性能可控黏弹性体的夹层梁具有无需改变结构设计的可优化性而倍受关注。虽然关于该可控黏弹性夹层梁的振动已有一定研究,但所用的动力学模型在几何或物理上是线性的,而对于较强激励情况则需要考虑非线性因素。首次考虑该黏弹性体的物理非线性,建立黏弹性夹层梁及其支承质量系统的非线性运动微分方程,并离散化为多模态耦合的非线性振动方程;对于平稳随机激励,运用统计线性化法推导等价拟线性系统,并计算系统的随机响应,得到黏弹性夹层梁非线性随机振动的均方位移,及等价的频响函数和功率谱,用以评价可控黏弹性夹层梁的响应抑制性能。     

Ratcheting in a nonlinear viscoelastic adhesive

Uniaxial time-dependent creep and cycled stress behavior of a standard and toughened film adhesive were studied experimentally. Both adhesives exhibited progressive accumulation of strain from an applied cycled stress. Creep tests were fit to a viscoelastic power law model at three different applied stresses which showed nonlinear response in both adhesives. A third order nonlinear power law model with a permanent strain component was used to describe the creep behavior of both adhesives and to predict creep recovery and the accumulation of strain due to cycled stress. Permanent strain was observed at high stress but only up to 3% of the maximum strain. Creep recovery was under predicted by the nonlinear model, while cycled stress showed less than 3% difference for the first cycle but then over predicted the response above 1000 cycles by 4–14% at high stress. The results demonstrate the complex response observed with structural adhesives, and the need for further analytical advancements to describe their behavior.     

An Assessment of Nonlinearly Viscoelastic Constitutive Models for Cyclic Loading: The Effect of a General Loading/Unloading Rule

Predicting the unloading and/or cyclic deformation behavior of polymers is a challenge for most nonlinear viscoelastic constitutive models. Experimental data of an epoxy polymer under uniaxial loading/unloading and two other types of cyclic loadings are used to assess the predictive capabilities of three types of nonlinear viscoelastic models. A general loading/unloading criterion and a switching rule, proposed recently by the authors, are further modified and incorporated into each of the three models. For each model, predictions by both the original formulations and that incorporating the proposed loading/unloading rule are compared with the test data. It is clearly shown that such a rule is essential to correctly simulate the unloading and cyclic loading behavior of polymers. By introducing such a rule to constitutive models, the quantitative predictions can be improved, to various degrees of success, with respect to cyclic deformation features such as ratcheting under cyclic loading with a mean stress and stress relaxation under cyclic straining with a mean strain.     

Analysis of nanocontact problems of layered viscoelastic solids with surface energy effects under different loading patterns

Surface stresses have a remarkable effect on nanocontact response of layered viscoelastic solids, especially under specific loading patterns. In the framework of nonlinear viscoelastic contact mechanics, a numerical model is developed to investigate the quasistatic nanocontact response of elastically layered viscoelastic solids under different loading patterns. The developed model accounts for surface energy effects by adopting the complete Gurtin–Murdoch surface elasticity model. The Schapery’s constitutive viscoelastic creep model is used for the stress, strain, and time relationships. The transient term in the creep compliance is expressed by Prony’s series. Frictionless contact condition is assumed throughout the contact interface. The equilibrium contact configuration, in which the contact constraints are exactly satisfied without any need for an appropriate value for the penalty parameter, is obtained by using the Lagrange multiplier method in the framework of the Newton–Raphson procedure. The developed model is applied to study and analyze the quasistatic nanocontact response of two different problems under different loading patterns. Results show the significant effect of the type of loading pattern and its rate on the nanocontact response of elastically layered viscoelastic solids.     

Single integral representations for nonlinear viscoelastic solids

The finite linear viscoelastic solid and single integral representation with nonlinear dependence on history are investigated in uni-axial stress. Both integral kernels in the stress formulation are determined by single-step constant strain tests, and both kernels in the strain formulation are determined by single-step constant stress tests. Single integral stress and strain formulations are not equivalent. The stress histories required to maintain constant strain-rate for both models are determined from the Volterra integral equations given by the strain formulations once their kernels are determined by constant stress tests. However, known constant strain-rate response does not determine the kernels. Examples are presented to show that variation of the kernel within a given qualitative shape can lead to different shapes of constant strain-rate response, so that both constant stress and constant strain-rate tests may be necessary to deduce the optimum single integral approximation, in preference to multi-step stress tests. It is shown that the apparently simpler finite linear viscoelastic model requires a far lengthier numerical algorithm to solve the Volterra equation, and leads to non-unique and physically unacceptable response, in comparison with the more flexible nonlinear history dependence which yields unique acceptable responses.     

Incremental forms of Schapery’s model: convergence and inversion to simulate strain controlled ramps

Schapery’s nonlinear viscoelastic model is written in incremental form, and three different approximations of nonlinearity functions in the time increment are systematically analysed with respect to the convergence rate. It is shown that secant slope is the best approximation of the time shift factor, leading to significantly higher convergence rate. This incremental form of the viscoelastic model, Zapas’ model for viscoplasticity, supplemented with terms accounting for damage effect is used to predict inelastic behaviour of material in stress controlled tests. Then the incremental formulation is inverted to simulate stress development in ramps where strain is the input parameter. A comparison with tests shows good ability of the model in inverted form to predict stress–strain response as long as the applied strain is increasing. However, in strain controlled ramps with unloading, the inverted model shows unrealistic hysteresis loops. This is believed to be a proof of the theoretically known incompatibility of the stress and strain controlled formulations for nonlinear materials. It also shows limitations of material models identified in stress controlled tests for use in strain controlled tests.     

A Viscoelastic Analysis of the Creep Behaviour of a Chopped Strand Mat E-Glass Fibre Reinforced Vinylester Resin

The viscoelastic response of a chopped strand mat E-glass fibre reinforced vinylester resin has been studied over a wide range of applied stress levels. At low applied stress levels, the material exhibited a linear viscoelastic response well represented by Schapery's power law model with constant C and n terms. At higher stresses nonlinear behaviour was observed which apparently is caused by the multiplicity of complex local phenomena associated with and preceding damage development (plastic deformation of the matrix, interfacial slippage, fiber-matrix debonding). The limits of linear viscoelastic behaviour and of damage initiation – about 0.48% strain or 42 MPa – coincide for this material. However, for successful modelisation of uniaxial creep strain in the nonlinear range a modified power law is proposed which uses stress-dependent creep parameters C and n.     

Description of depth-dependent nonlinear viscoelastic behavior for articular cartilage in unconfined compression

An optimized digital image correlation (DIC) technique was applied to investigate the depth-dependent nonlinear viscoelastic properties of articular cartilage and simultaneously the biphasic nonlinear viscoelastic relaxation model of cartilage was proposed and validated. The stress relaxation tests were performed with different strain levels and it is found that the initial stress and relaxed stress at any time increase with increasing strain levels. The depth-dependent strain of cartilage was obtained by analyzing the images acquired using the optimized DIC technique and moreover the inhomogeneous relaxation modulus distributions within the tissues were determined at different relaxation time points under strain of 11.35, 19.35 and 30% respectively. The strain rate dependent nonlinear stress and strain curves were obtained for articular cartilage through uniaxial compression tests. It is noted that the Young's modulus exhibits a slight increase near the cartilage surface, and then increases fast with depth and both the magnitude and the variation of the Young's modulus are affected by increasing strain rates. A biphasic nonlinear viscoelastic relaxation model was proposed to predict the depth-dependent relaxation behavior of cartilage under unconfined compression and the results show that there are good agreements between the experimental data and predictions.     

高分子材料非线性粘弹性问题的解法

尽管实验限制在小变形范围之内，高分子材料仍表现出明显的非线性粘弹性性能。目前由于分析上的困难，通常只好采用线粘弹性假设。基于前文，提出了一个求解高分子材料物理非线性（含线性）粘弹性问题的新原理，通过对改性聚丙烯材料的实验表明了该原理的正确性和对于此类材料的实用性。     

Numerical implementation of strain rate dependent thermo viscoelastic constitutive relation to simulate the mechanical behavior of PMMA

In this work, a nonlinear viscoelastic constitutive relation was implemented to describe the mechanical behavior of a transparent thermoplastic polymer polymethyl methacrylate (PMMA). The quasi-static and dynamic response of the polymer was studied under different temperatures and strain rates. The effect of temperature was incorporated in elastic and relaxation constants of the constitutive equation. The incremental form of constitutive model was developed by using Poila–Kirchhoff stress and Green strain tensors theory. The model was implemented numerically by establishing a user defined material subroutine in explicit finite element (FE) solver LS-DYNA. Finite element models for uniaxial quasi-static compressive test and high strain rate split Hopkinson pressure bar compression test were built to verify the accuracy of material subroutine. Numerical results were validated with experimental stress strain curves and the results showed that the model successfully predicted the mechanical behavior of PMMA at different temperatures for low and high strain rates. The material model was further engaged to ascertain the dynamic behavior of PMMA based aircraft windshield structure against bird impact. A good agreement between experimental and FE results showed that the suggested model can successfully be employed to assess the mechanical response of polymeric structures at different temperature and loading rates.     

A note on a correspondence principle in nonlinear viscoelastic materials

We show that models for the nonlinear viscoelastic response of solids generated on the basis of a correspondence principle developed by Schapery(1984) do not satisfy the balance of angular momentum for large deformations. This principle, which is valid if the displacement gradients are sufficiently small, has been used in several papers to develop models to describe the fracture of viscoelastic solids, and these studies need to be reexamined in the light of this note.     

On the influence of preloading in the nonlinear viscoelastic–viscoplastic response of carbon–epoxy composites

On an ongoing research for the nonlinear viscoelastic response of composites and polymers, a study of the influence of preloading applied to composite laminates subjected to creep–recovery loading is performed. In cases where high stress levels are applied, this response becomes highly nonlinear and has to be taken into account when designing composite parts. A major problem encountered in the experimental investigation of the nonlinear viscoelastic behaviour is the mode of the initial applied loading and its effect in the overall viscoelastic response of the test sample. The damage that occurs due to the instantaneous application of the load leads to an additional viscoelastic/viscoplastic strain component. In order to investigate this effect as well as to compare different preloading modes, as far as viscoelastic/viscoplastic response is concerned, a test program was initiated and the experimental data were investigated in the current study. A preloading mode is applied in each specimen prior to the creep–recovery testing at different applied stress levels. Useful results concerning the effect of preloading in the time dependent response of the material are concluded. Variation of the values of viscoplastic strain in respect to the preloading mode is also of great concern.     

The use of stress sweep tests for asphalt mixtures nonlinear viscoelastic and fatigue damage responses identification

Asphaltic materials are known to present a behavior that can be approximated by the theory of viscoelasticity. For these materials it is essential to characterize fatigue damage. An important aspect therein is the separation between nonlinear viscoelastic and fatigue damage responses. This is a complex issue, since both nonlinearity and damage have a similar effect on the overall material mechanical behavior, i.e. decrease in the stiffness and increase in the phase angle. This paper presents an experimental and a mathematical procedure to separate the nonlinear viscoelastic from the fatigue damage response for asphaltic materials. Stress sweep tests were used to characterize a hot mixture asphalt at nine conditions (three temperatures and three frequencies). Once all strain values were obtained in a stress controlled sweep test, a statistical analysis was used to find the maximum stress that can be applied to the material without invoking the damage response. The results showed that the transition stress value is directly associated with material properties, the stiffness being an important factor in this result. Consequently, stress, temperature and frequency determine together the mechanical response of the material (linear or nonlinear viscoelastic, fatigue damage and/or plastic deformation). Results from this study can be associated with other fatigue damage approaches in order to better select the stress or strain amplitude that should be used in fatigue tests, and to eliminate the amount of energy that is dissipated in the nonlinear viscoelastic region.     

基于非线性黏弹性本构模型的轮胎滚动和生热

轮胎的滚动阻力和自生热是造成轮胎失效的原因之一,轮胎内部能量的产生主要取决于轮胎中橡胶材料的黏弹性能量耗散。文中基于超弹性模型和并行流变模型(PRF)描述了橡胶材料的非线性黏弹性响应特征,提出了一种预测实心轮胎温度分布和滚动阻力的方法。首先将线性黏弹性Prony级数转化为PRF模型的初始参数,并利用Isight软件根据多应变工况加载得到的应力松弛测试数据来校准材料参数,然后采用显式-热力耦合分析方法分析基于Prony级数和PRF模型的实心轮胎滚动过程的差异。结果表明,Prony级数无法描述橡胶材料的非线性行为,在显式动力学下计算的轮胎生热结果为0;PRF模型可以表征橡胶材料的非线性行为,并且计算的轮胎模型在0.3s内温度上升了0.14℃。