Constitutive modeling of strain-induced softening in swollen elastomers

Under cyclic loading, elastomeric material exhibits strong inelastic responses such as stress-softening due to Mullins effect, hysteresis and permanent set. The corresponding inelastic responses are observed in both dry and swollen rubbers. Moreover, it is observed that inelastic responses depend strongly on the swelling level. For engineering applications involving the interaction and contact between rubber components and solvent, the understanding and consideration of swelling are essential pre-requisites for durability analysis. In this paper, a simple phenomenological model describing Mullins effect in swollen rubbers under cyclic loading is proposed. More precisely, the proposed model adopts the concept of evolution of soft domain microstructure with deformation originally proposed by Mullins and Tobin. The swollen rubbers are obtained by immersing dry ones in solvent until desired degrees of swelling are achieved. Subsequently, their mechanical responses, in particular Mullins effect, under cyclic loading are investigated. These experimental data are used to assess the efficiency of the proposed model. Results show that the model agrees qualitatively well with experiments. Furthermore, the model captures well the fundamental features of strain-induced softening.     

On hysteretic response and stationary phase fronts in rubber

We consider the dynamic response of natural, latex and synthetic, nitrile rubbers under non-monotonic dynamic loading conditions; in particular, we recreate an experiment first considered by Kolsky (Nature 224:1301, 1969) in which two segments of a long rubber specimen are initially maintained at different strain levels by external force and then allowed to evolve dynamically towards equilibrium. We show that as a result of the hysteretic behavior, a phase boundary that is stationary with respect to the material points can be established in both these materials. We also show that this phase boundary persists indefinitely in strain-crystallizing natural, latex rubber, but disappears quickly in the non-crystallizing nitrile rubber.     

TEST AND CHARACTERIZATION FOR THE INCOMPRESSIBLE HYPERELASTIC PROPERTIES OF CONDITIONED RUBBERS UNDER MODERATE FINITE DEFORMATION

In this paper, the automated grid method is applied to test for the mechanical properties of conditioned rubbers under the moderate ?nite deformation (not exceeding 100%). More accurate stress-strain curves of conditioned rubber specimens unde…     

Some specific features and consequences of the thermal response of rubber under cyclic mechanical loading

The present paper deals with the specificities of the thermal response of rubber under cyclic mechanical loading at constant ambient temperature. This question is important, since the stabilized thermal response is used in fatigue life criteria, especially for the fast evaluation of fatigue life. For this purpose, entropic coupling in a thermo-hyperelastic framework is first used to predict the variation in the heat source produced or absorbed by the material during cyclic loading. The heat diffusion equation is then used to deduce temperature variations under adiabatic and non-adiabatic conditions. The influence of several parameters on the stabilized thermal response is studied: signal shape, frequency, minimum and maximum stretch levels, multiaxiality of the mechanical state. The results show that, in the steady-state regime, the mean value between the maximum and minimum temperature variations over a mechanical cycle is different from zero. This is due to the specific variation in the heat source, which depends on both the stretch rate and the stretch level. This result has numerous consequences, in particular for fatigue. Indeed, the stabilized mean value between the maximum and minimum temperature variations during fatigue tests does not reflect only fatigue damage, since the entropic coupling also leads to a value different from zero. This is a major difference with respect to materials exhibiting only isentropic coupling, such as metallic materials.     

考虑缠结效应的超弹性本构模型

经典熵弹性模型,如Neo-Hookean模型和Arruda-Boyce八链模型,被广泛应用于预测橡胶等软材料的超弹性力学行为.然而,大量实验结果也显示仅采用一套模型参数,这类模型不能同时准确地描述橡胶在多种加载模式下的应力响应.为了克服上述模型的不足,本文在熵弹性的模型基础上引入缠结约束效应.微观上,采用Langevi...     

Experiments and modeling of iron-particle-filled magnetorheological elastomers

Magnetorheological elastomers (MREs) are ferromagnetic particle impregnated rubbers whose mechanical properties are altered by the application of external magnetic fields. Due to their coupled magnetoelastic response, MREs are finding an increasing number of engineering applications. In this work, we present a combined experimental and theoretical study of the macroscopic response of a particular MRE consisting of a rubber matrix phase with spherical carbonyl iron particles. The MRE specimens used in this work are cured in the presence of strong magnetic fields leading to the formation of particle chain structures and thus to an overall transversely isotropic composite. The MRE samples are tested experimentally under uniaxial stresses as well as under simple shear in the absence or in the presence of magnetic fields and for different initial orientations of their particle chains with respect to the mechanical and magnetic loading direction.Using the theoretical framework for finitely strained MREs introduced by Kankanala and Triantafyllidis (2004), we propose a transversely isotropic energy density function that is able to reproduce the experimentally measured magnetization, magnetostriction and simple shear curves under different prestresses, initial particle chain orientations and magnetic fields. Microscopic mechanisms are also proposed to explain (i) the counterintuitive effect of dilation under zero or compressive applied mechanical loads for the magnetostriction experiments and (ii) the importance of a finite strain constitutive formulation even at small magnetostrictive strains. The model gives an excellent agreement with experiments for relatively moderate magnetic fields but has also been satisfactorily extended to include magnetic fields near saturation.     

温度对炭黑填充橡胶复合材料力学性能影响的实验研究

采用岛津万能试验机对天然与丁苯共混橡胶(NSBR)试件进行不同温度下多步松弛循环加卸载实验和单向拉伸加卸载循环实验，对比两者的宏观力学响应，并研究温度对炭黑填充复合材料力学性能的影响．结果表明：(1)随着温度的升高，未填充橡胶逐渐变“硬”，而炭黑填充橡胶是“先变软后变硬”．(2)随着温度的升高，橡胶的迟滞损耗呈线性减小．(3)比较单向拉伸和应力松弛平衡点的应力应变曲线，填充橡胶“先变软后变硬”的温度转折点不同，这是由于应力松弛过程基本消除了粘弹性效应．     

Computational modeling of porous shape memory alloys

In this study the mesoscopic behavior of porous shape memory alloys has been simulated with particular attention to the mechanical response under cyclic loading conditions. A recently developed constitutive law, accounting for full martensite reorientation as well as phase transformation, was implemented into the commercial finite element code ABAQUS. Due to stress concentrations in a porous microstructure, the constitutive law was enhanced to account for the development of permanent inelasticity in the shape memory matrix. With this simulation method, the complex interaction between porosity, local phase transformation and macroscale response has been evaluated. The results have implications for use of porous SMAs in biomedical and structural applications.     

Fluid pressure response in poroelastic materials subjected to cyclic loading

When cyclic loading is applied to poroelastic materials, a transient stage of interstitial fluid pressure occurs, preceding a steady state. In each stage, the fluid pressure exhibits a characteristic mechanical behavior. In this study, an analytical solution for fluid pressure in two-dimensional poroelastic materials, which is assumed to be isotropic, under cyclic axial and bending loading is presented, based on poroelasticity. The obtained analytical solution contains transient and steady-state responses. Both of these depend on three dimensionless parameters: the dimensionless stress coefficient; the dimensionless frequency; and, the axial-bending loading ratio. We focus particularly on the transient behavior of interstitial fluid pressure with changes in the dimensionless frequency and the axial-bending loading ratio. The transient properties, such as half-value period and contribution factor, depend largely on the dimensionless frequency and have peak values when its value is about 10. This suggests that, under these conditions, the transient response can significantly affect the mechanical behavior of poroelastic materials.     

Constitutive modeling of isotropic hyperelastic materials in an exponential framework using a self-contained approach

In this paper, an exponential framework for strain energy density functions of elastomers and soft biological tissues is proposed. Based on this framework and using a self-contained approach that is different from a guesswork or combination viewpoint, a set strain energy density functions in terms of the first and second strain invariants is rebuilt. Among the constructed options for strain energy density, a new exponential and mathematically justified model is examined. This model benefits from the existence of second strain invariant, simplicity, stability of parameters, and the state of being accurate. This model can capture strain softening, strain hardening and is able to differentiate between various deformation-state dependent responses of elastomers and soft tissues undergoing finite deformation. The model has two material parameters and the mathematical formulation is simple to render the possibility of numerical implementations. In order to investigate the appropriateness of the proposed model in comparison to other hyperelastic models, several experimental data for incompressible isotropic materials (elastomers) such as VHB 4905 (polyacrylate rubber), two various silicone rubbers, synthetic rubber neoprene, two different natural rubbers, b186 rubber (a carbon black-filled rubber), Yeoh vulcanizate rubber, and finally porcine liver tissue (a very soft biological tissue) are examined. The results demonstrate that the proposed model provides an acceptable prediction of the behavior of elastomers and soft tissues under large deformation for different applied loading states.     

A rheology model of high damping rubber bearings for seismic analysis: Identification of nonlinear viscosity

The mechanical behavior of high damping rubber bearings (HDRBs) is investigated under horizontal cyclic shear deformation with a constant vertical compressive load. On the basis of experimental observations, an elasto-viscoplastic rheology model of HDRBs for seismic analysis is developed. In this model, the Maxwell model is extended by adding a nonlinear elastic spring and an elasto-plastic model (spring-slider) in parallel. In order to identify constitutive relations of each element in the rheology model, an experimental scheme comprised of three types of tests, namely a cyclic shear (CS) test, a multi-step relaxation (MSR) test, and a simple relaxation (SR) test, are carried out at room temperature. HDRB specimens with the standard ISO geometry and three different high damping rubber materials are employed in these tests. A nonlinear viscosity law of the dashpot in the Maxwell model is deduced from the experimental scheme, and incorporated into the rheology model to reproduce the nonlinear rate dependent behavior of HDRBs. Finally, numerical simulation results for sinusoidal loading are presented to illustrate capability of the proposed rheology model in reproducing the mechanical behavior of HDRBs.     

Determination of material response functions for prestrained rubbers

The mechanical response of two natural rubber compounds is examined in order to determine relevant material parameters by non-linear finite element analysis. The materials are subjected to (a) combined static torsion and extension, and (b) small, steady-state torsional oscillations superposed on a large static simple extension. The materials are assumed to be incompressible and isotropic in their undeformed state and a time-strain separable relaxation modulus tensor is employed in order to characterize the steady-state harmonic viscoelastic response. The combined static torsion and extension experiments are used to determine the basic delayed elastic response functions. A Rivlin-type strain energy expression of third-order accuracy is used for the purpose. The two-constant, Mooney-Rivlin form is found to be adequate for both materials in the relatively limited range of deformation magnitudes considered.The torsional storage and loss moduli are determined under quasistatic conditions as functions of frequency and axial static pre-strain. The time-strain separability is found to be a resonable approximation in a relatively limited range of static prestrain magnitudes and frequencies considered for the natural gum rubbers investigated. The experimental methodology is discussed in some detail.     

N330炭黑增强天然橡胶材料力学性能的实验研究

为了研究炭黑对橡胶材料力学性能的影响,对9种不同体积含量的炭黑填充橡胶材料进行了准静态力学实验研究。利用循环拉伸加卸载实验分析了炭黑对橡胶Mullins效应及能量损耗的影响,通过单轴拉伸实验研究了炭黑对橡胶材料刚度和起始模量的影响,采用多步松弛拉伸加卸载实验研究了炭黑对橡胶材料应力行为的应变历史相关性的影响。实验结果表明:炭黑填充量越高,橡胶材料的刚度越大,初始模量越大,Mullins效应也越明显;随着炭黑填充量的增加,橡胶在加卸载循环中所产生的迟滞损耗、Mullins效应相对能量损耗以及残余应变都呈现出非线性增长趋势;随着炭黑填充量的升高,橡胶在加卸载过程中的应力松弛现象越明显,其平衡态迟滞损耗以及与时间相关部分的迟滞损耗也越大。     

A Study on the Evaluation of the Fracture Process Zone in CCNBD Rock Samples

The safety of many civil and mining concrete and rock structures including pre-existing crack networks is fundamentally affected by the mechanical behaviour of the material under static and cyclic loading. In cyclic loading case, cracks can grow at a lower load level compared to the monotonic case. This phenomenon is called fatigue due to subcritical crack propagation and depends on the behaviour of the fracture process zone (FPZ). This study presents the results of laboratory diametrical compression tests performed on Brisbane tuff disc specimens to investigate their mode-I (tensile) fracture toughness response to static and cyclic loading and relevant FPZ. The FPZ and fracture toughness response to cyclic loading was found to be different from that under static loading in terms of the ultimate load and the damage mechanisms in front of the chevron crack. A maximum reduction of the static fracture toughness (K _IC ) of 42 % was obtained for the highest amplitude increasing cyclic loading test. Detailed scanning electron microscope (SEM) examinations were performed on the surfaces of the tips of the chevron notch cracks, revealing that both loading methods cause FPZ development in the CCNBD specimens. When compared with monotonic FPZ development, the main difference with the cyclically loaded specimens was that intergranular cracks were formed due to particle breakage under cyclic loading, while smooth and bright cracks along cleavage planes were formed under static loading. Further, the SEM images showed that fatigue damage in Brisbane tuff is strongly influenced by the failure of the matrix because of both intergranular and transgranular subcritical fracturing.     

On the response of rubbers at high strain rates—I. Simple waves

In this series of papers, we examine the propagation of waves of finite deformation in rubbers through experiments and analysis; in the present paper, Part I, attention is focused on the propagation of one-dimensional waves in strips of natural, latex and synthetic, nitrile rubber. Tensile wave propagation experiments were conducted at high strain rates by holding one end fixed and displacing the other end at a constant velocity. A high-speed video camera was used to monitor the motion and to determine the evolution of strain and particle velocity in rubber strips. Analysis of the response through the theory of finite waves indicated a need for an appropriate constitutive model for rubber; by quantitative matching between the experimental observations and analytical predictions, an appropriate instantaneous elastic response for the rubbers was obtained. This matching process suggested that a simple power-law constitutive model was capable of representing the high strain-rate response for both rubbers used.     

Ultrasonic devulcanization of waste rubbers: Experimentation and modeling

Continuous ultrasonic devulcanization of ground tire rubber (GRT) and styrene-butadiene rubber (SBR) is considered. Experiments are performed under various processing conditions. Two recipes of SBR with different amounts of polysulfidic linkages are utilized. Gel fraction and crosslink density of devulcanized rubbers are measured and a unique relationship between them is established. Die characteristics with and without imposition of ultrasonic waves are determined. Devulcanized samples are revulcanized and mechanical properties are measured. In some cases, properties of revulcanized SBR samples exceeded those of virgin vulcanizates. This is explained based on the presence of a double network in the revulcanized rubber. A modification of acoustic cavitation and flow modeling of ultrasonic devulcanization of SBR and GRT is proposed using a concept of effective viscosity characterizing the flow of vulcanized particles before devulcanization combined with a shear rate, temperature and gel fraction-dependent viscosity of devulcanized rubber. Velocity, shear rate, pressure, and temperature field along with gel fraction, crosslink density and number of bonds broken are simulated. Predicted data on gel fraction, crosslink density, and pressure using the present modification of the model are found to be closer to experimental data then previously reported.Dedicated to the memory of Professor Tasos C. Papanastasiou     

Transient response of fluid pressure in a poroelastic material under uniaxial cyclic loading

Poroelasticity is a theory that quantifies the time-dependent mechanical behavior of a fluid-saturated porous medium induced by the interaction between matrix deformation and interstitial fluid flow. Based on this theory, we present an analytical solution of interstitial fluid pressure in poroelastic materials under uniaxial cyclic loading. The solution contains transient and steady-state responses. Both responses depend on two dimensionless parameters: the dimensionless frequency Ω that stands for the ratio of the characteristic time of the fluid pressure relaxation to that of applied forces, and the dimensionless stress coefficient H governing the solid-fluid coupling behavior in poroelastic materials. When the phase shift between the applied cyclic loading and the corresponding fluid pressure evolution in steady-state is pronounced, the transient response is comparable in magnitude to the steady-state one and an increase in the rate of change of fluid pressure is observed immediately after loading. The transient response of fluid pressure may have a significant effect on the mechanical behavior of poroelastic materials in various fields.     

Prediction of fatigue life improvement in natural rubber using configurational stress

To some extent, continua can no longer be considered as free of defects. Experimental observations on natural rubber revealed the existence of distributed microscopic defects which grow upon cyclic loading. However, these observations are not incorporated in the classical fatigue life predictors for rubber, i.e. the maximum principal stretch, the maximum principal stress and the strain energy. Recently, Verron et al. [Verron, E., Le Cam, J.B., Gornet, L., 2006. A multiaxial criterion for crack nucleation in rubber. Mech. Res. Commun. 33, 493–498] considered the configurational stress tensor to propose a fatigue life predictor for rubber which takes into account the presence of microscopic defects by considering that macroscopic crack nucleation can be seen as the result of the propagation of microscopic defects. For elastic materials, it predicts privileged regions of rubber parts in which macroscopic fatigue crack might appear. Here, we will address our interest to a broader context. Rubber is assumed to exhibit inelastic behavior, characterized by hysteresis, under fatigue loading conditions. The configurational mechanics-based predictor is modified to incorporate inelastic constitutive equations. Afterwards, it is used to predict fatigue life. The emphasis of the present work is laid on the prediction of the well-known fatigue life improvement in natural rubber under tension–tension cyclic loading.     

Mechanical annealing under low-amplitude cyclic loading in micropillars

Mechanical annealing has been demonstrated to be an effective method for decreasing the overall dislocation density in submicron single crystal. However, simultaneously significant shape change always unexpectedly happens under extremely high monotonic loading to drive the pre-existing dislocations out of the free surfaces. In the present work, through in situ TEM experiments it is found that cyclic loading with low stress amplitude can drive most dislocations out of the submicron sample with virtually little change of the shape. The underlying dislocation mechanism is revealed by carrying out discrete dislocation dynamic (DDD) simulations. The simulation results indicate that the dislocation density decreases within cycles, while the accumulated plastic strain is small. By comparing the evolution of dislocation junction under monotonic, cyclic and relaxation deformation, the cumulative irreversible slip is found to be the key factor of promoting junction destruction and dislocation annihilation at free surface under low-amplitude cyclic loading condition. By introducing this mechanics into dislocation density evolution equations, the critical conditions for mechanical annealing under cyclic and monotonic loadings are discussed. Low-amplitude cyclic loading which strengthens the single crystal without seriously disturbing the structure has the potential applications in the manufacture of defect-free nano-devices.     

On the response of rubbers at high strain rates—II. Shock waves

In this series of papers, we examine the propagation of waves of finite deformation in rubbers through experiments and analysis; in the present paper, Part II, attention is focused on the propagation of one-dimensional tensile shock waves in strips of latex and nitrile rubber. Tensile wave propagation experiments were conducted at high strain rates by holding one end fixed and displacing the other end at a constant velocity. A high-speed video camera was used to monitor the motion and to determine the evolution of strain and particle velocity in rubber strips. Shock waves have been generated under tensile impact in prestretched rubber strips; analysis of the response yields the tensile shock adiabat for rubbers. The propagation of shocks is analyzed by developing an analogy with the theory of detonation; it is shown that the condition for shock propagation can be determined using the Chapman-Jouguet shock condition.