Thermovisco-hyperelastic behavior of elastomeric materials modified by filler nanoparticles

We present the results of experimental studies of hyperelastic and relaxation properties of polymer composites with elastomeric matrix made of hydrogenated nitrile butadiene rubber filled with nanoparticles of technical carbon in the temperature range 19–150° C. We present typical experimental diagrams of deformation of the material with constant strain rate and the stress relaxation curves at different strain levels under tension and compression conditions. We consider a possible version of constitutive relations for describing some singularities of the behavior of the material under study. We developed a method for determining all the parameters of the accepted relations on the basis of the results of uniaxial tests. We found a nonmonotone dependence of the relaxation modulus on the temperature and proposed a formula for describing this dependence in the temperature range under study. To justify the possible use of the considered constitutive relations to perform calculations under conditions of arbitrary compound stress state, we performed numerical modeling of the compression experiment for cylindrical samples. A rather satisfactory agreement between the computational results and experimental data was obtained.     

基于横观各向同性超弹性理论的短纤维增强橡胶本构模型的建立与应用Establishment and application of short fiber reinforced rubber constitutive model based on transversely isotropic hyperelastic theory

提出了一种能够表征短纤维增强橡胶的横观各向同性超弹性本构模型,并结合试验体系,对其在数值分析中的应用方法和效果进行了研究。基于连续介质力学理论,建立了横观各向同性材料的应变能函数,推导得到不同变形形式下的应力应变关系,给出材料参数辨识试验方法,并成功应用于某短纤维增强橡胶测试中,得到表征其超弹性特性的相关材料参数。利用有限元软件ANSYS对不同纤维排布方向的单轴拉伸和平行纤维方向的平面拉伸进行仿真计算,并对比相应试验数据,以验证材料参数的可靠性。最后基于已验证的本构模型,建立了某铣槽装备减振环仿真模型,并对其进行了校核计算。研究结果表明,本文提出的本构模型能够有效表征短纤维增强橡胶的静态力学特性并且方便嵌入现有的有限元软件中,具有材料参数少、测试简便和结果准确等特点,工程实用性强。     

Understanding the need of the compression branch to characterize hyperelastic materials

Soft biological tissues are frequently modeled as hyperelastic materials. Hyperelastic behavior is typically ensured by the assumption of a stored energy function with a pre-determined shape. This function depends on some material parameters which are obtained through an optimization algorithm in order to fit experimental data from different tests. For example, when obtaining the material parameters of isotropic, incompressible models, only the extension part of a uniaxial test is frequently taken into consideration. In contrast, spline-based models do not require material parameters to exactly fit the experimental data, but need the compression branch of the curve. This is not a disadvantage because as we explain herein, to properly characterize hyperelastic materials, the compression branch of the uniaxial tests (or valid alternative tests) is also needed, in general. Then, unless we know beforehand the tendency of the compression branch, a material model should not be characterized only with tensile tests. For simplicity, here we address isotropic, incompressible materials which use the Valanis-Landel decomposition. However, the concepts are also applicable to compressible isotropic materials and are specially relevant to compressible and incompressible anisotropic materials, because in biomechanics, materials are frequently characterized only by tensile tests.     

Biaxial deformation of dough using the bubble inflation technique. II. Numerical modelling

In Part I the bubble inflation test was used to measure the stress-strain relationship of dough. A large disagreement was found between the stress-strain curve based on experimental data and the curve derived from Bloksma's analytical model. In Part II, a numerical simulation of the bubble inflation test is performed using Finite Element Analysis, in order to obtain further information regarding the accuracy of the analytical predictions. A hyperelastic model is assumed for the dough, with a strain energy potential described by the compressible form of the Mooney-Rivlin model. Four cases were investigated, corresponding to various combinations of material parameters of the Mooney-Rivlin model. The numerical results reinforce the conclusions drawn in Part I of the study, specifically that Bloksma's analysis of the bubble inflation could lead to large errors in the stress-strain curve. It was further concluded that the accuracy of the analysis was dependent on the material properties. For a neo-Hookeian material, the analysis leads to accurate results. This is because, for this material, all the assumptions made in the analysis regarding the bubble shape, the material's incompressibility and the bubble wall thickness distribution are accurate.     

Finite dynamic deformations of a hyperelastic,anisotropic, incompressible and prestressed tube. Applications to in vivo arteries

In this paper, we study the mechanical behavior of a prestressed tube subjected to finite dynamic deformations. The tube is assumed to be made of a hyperelastic, anisotropic and incompressible material. The analysis is carried out by using a Mooney–Rivlin stored energy function augmented with fiber reinforcements in four unidimensional orientations. A semi-analytical solution is proposed to study the radial dynamic mechanical response of an artery by using in vivo data. The optimal model parameters describing the mechanical characteristics of arterial wall microconstituents are obtained by minimizing the difference between computed and measured inner pressures over the cardiac cycle using a nonlinear regression. Theoretical and experimental results on rat carotid elastic arteries are compared in order to assess the validity of the approach by estimating differences of the model parameters and wall stresses with aging.     

Finite element analysis of V-ribbed belts using neural network based hyperelastic material model

A three-dimensional finite element model was built to study V-ribbed belt pulley contact mechanics. The model consists of a pulley and a segment of V-ribbed belt in contact with the pulley. A material model for the belt, including the rubber compound and the reinforcing cord is developed. Rubber is modeled as hyperelastic material. The hyperelastic strain energy function is approximated by neural network trained by rubber test data. Reinforcing cord is modeled as elastic rebar. The material model developed is implemented in the commercial finite element code ABAQUS to simulate the V-ribbed belt-pulley system. A study is then conducted to investigate the effect of belt pulley system parameters on the contact mechanics. The effects of temperature and aging on belt materials are also investigated. The information gained from the analysis can be applied to optimize V-ribbed belt and pulley design.     

一种考虑剪切作用的各向异性超弹性本构模型

基于纤维增强复合材料连续介质力学理论,提出了一种轮胎帘线/橡胶复合材料的各向异性超弹性本构模型. 应变能被分解为分别代表橡胶、帘线、帘线/橡胶之间的角剪切和正剪切应变能4个部分. 给出了模型参数的简单确定方法,通过拟合文献中的实验数据,得到了本构模型参数,并用该模型预测了其他变形条件下的力学行为,得到了和实验数据较一致的结果,验证了该模型的有效性,为整体轮胎的有限元分析打下了基础.     

Time-dependent behavior of passive skeletal muscle

An isotropic three-dimensional nonlinear viscoelastic model is developed to simulate the time-dependent behavior of passive skeletal muscle. The development of the model is stimulated by experimental data that characterize the response during simple uniaxial stress cyclic loading and unloading. Of particular interest is the rate-dependent response, the recovery of muscle properties from the preconditioned to the unconditioned state and stress relaxation at constant stretch during loading and unloading. The model considers the material to be a composite of a nonlinear hyperelastic component in parallel with a nonlinear dissipative component. The strain energy and the corresponding stress measures are separated additively into hyperelastic and dissipative parts. In contrast to standard nonlinear inelastic models, here the dissipative component is modeled using an evolution equation that combines rate-independent and rate-dependent responses smoothly with no finite elastic range. Large deformation evolution equations for the distortional deformations in the elastic and in the dissipative component are presented. A robust, strongly objective numerical integration algorithm is used to model rate-dependent and rate-independent inelastic responses. The constitutive formulation is specialized to simulate the experimental data. The nonlinear viscoelastic model accurately represents the time-dependent passive response of skeletal muscle.     

Constitutive modeling of particle reinforced rubber-like materials

The present study is focused on the constitutive modeling for the mechanical behavior of rubber reinforced with filler particles. A filler-dependent energy density function is proposed with all the continuum mechanics-based necessities of an effective hyperelastic material model. The proposed invariant-based energy function comprises a single set of material parameters for a material subjected to several modes of loading conditions. The model solution agrees well with existing experimental results. Later, the effect of varying concentrations of filler particles in the rubber matrix is also studied.     

Elastic Cherenkov effects in transversely isotropic soft materials-I: Theoretical analysis,simulations and inverse method

A body force concentrated at a point and moving at a high speed can induce shear-wave Mach cones in dusty-plasma crystals or soft materials, as observed experimentally and named the elastic Cherenkov effect (ECE). The ECE in soft materials forms the basis of the supersonic shear imaging (SSI) technique, an ultrasound-based dynamic elastography method applied in clinics in recent years. Previous studies on the ECE in soft materials have focused on isotropic material models. In this paper, we investigate the existence and key features of the ECE in anisotropic soft media, by using both theoretical analysis and finite element (FE) simulations, and we apply the results to the non-invasive and non-destructive characterization of biological soft tissues. We also theoretically study the characteristics of the shear waves induced in a deformed hyperelastic anisotropic soft material by a source moving with high speed, considering that contact between the ultrasound probe and the soft tissue may lead to finite deformation. On the basis of our theoretical analysis and numerical simulations, we propose an inverse approach to infer both the anisotropic and hyperelastic parameters of incompressible transversely isotropic (TI) soft materials. Finally, we investigate the properties of the solutions to the inverse problem by deriving the condition numbers in analytical form and performing numerical experiments. In Part II of the paper, both ex vivo and in vivo experiments are conducted to demonstrate the applicability of the inverse method in practical use.     

Rheological model optimization using advanced evolutionary computation for the analysis of the influence of recycled rubber on rubber blend dynamical behaviour

Rheological models of tested rubber blend samples, with different ratios of smut/recycled material, were developed, analyzed and discussed according to experimental data. The experimental measurements show the dynamical behavior of these samples after a short-term axial load. The rheological model of samples was assumed as a viscoelastic Voigt-Maxwell and hyperelastic Mooney-Rivlin model. Real coded genetic algorithms were implemented for optimal model parameters identification in order to minimize the difference between the real sample behavior and dynamical model output. Numerical results obtained for analyzed mixture were compared with the original sample to verify the presented method. In purpose of predicting the dynamical characteristics for desirable rubber blends, an interrelation was formed based on the concentration of recycled material in sample blends and dynamical model parameters dependence.     

Combining the logarithmic strain and the full-network model for a better understanding of the hyperelastic behavior of rubber-like materials

Suitably defined invariants of the logarithmic strain are shown to be more adequate than the usual invariants of the left Cauchy-Green tensor to define the type and intensity of a strain applied to hyperelastic rubber-like materials. Coupling these invariants with the macromolecular full-network model clarifies some features of the state of strain dependence of these materials. Finally, comparisons of the model with experimental data illustrate the efficiency of the full-network model and the dependence of the material parameters on the applied loading history.     

基于仿真和智能算法骨骼肌超弹性本构参数的反演方法研究

从事高强度的体力工作者经常会发生肌肉软组织的损伤,因此对骨骼肌的变形特性和应力分布的研究受到了越来越多的重视.获取正确的本构参数对于生物软组织的力学行为的研究至关重要,而本构参数的确定本质上是一个逆过程,具有很大的挑战性.本文分别采用K近邻(K-nearest neighbor,KNN)模型和支持向量机回归(suppo...     

On some connections between equivalent single material and mixture theory models for fiber reinforced hyperelastic materials

There are two approaches that can be used to model the large strain mechanical response of material systems in which elastic fibers are embedded in an elastic matrix. In the first approach, a fiber reinforced material undergoing large deformation is homogenized in the sense that it is assumed to act as an equivalent single material that is transversely isotropic and hyperelastic. Both constituents then share a common reference configuration, which is typically assumed to be a natural or stress-free configuration for the equivalent single material. The stress depends on a single deformation gradient defined with respect to the natural configuration.In the second approach, the fiber/matrix system is treated as a mixture, with the matrix and the fibrous constituents having their own reference configurations and material symmetries. The total stress depends on the deformation gradients and material symmetries for both constituents, defined with respect to their reference configurations.Under appropriate assumptions, the constitutive theory developed using mixture theory can coincide with the constitutive theory assuming an equivalent single material that is transversely isotropic and hyperelastic. This paper explores the connection between the two approaches by considering the various reference configurations and material symmetries.     

Hyperelastic constitutive model for rubber-like materials based on the first Seth strain measures invariant

The mechanical behaviour of isotropic and incompressible vulcanized natural rubbers (NR's) and that of quasi-incompressible carbon black filled vulcanized natural rubbers (NR 70) are considered both theoretically and experimentally. We start by generalising the neo-Hookean model to derive an original form of the strain energy density function (W). W satisfies the hypothesis of the Valanis–Landel function, which allows reducing the number of needed experimental tests to identify the parameters of the model. In the present study, in the order to identity the analytical form of W, we undertake only simple tension tests. The two-dimensional field of in-plane homogeneous displacements is determined here using a home-developed image analysis cross-correlation technique. Our model is also identified using results taken from the literature in the case of (NR's) for different types of solicitations, including simple tension, equibiaxial tension and pure shear deformation. Comparison of numerical results with the experimental data indicates that the present model can characterise the hyperelastic behaviour of NR's and that of NR 70 for all the tested modes of deformation. Moreover, it seems to be valid over a wide range of deformation intervals.     

NEPE推进剂低高应变率下改进的黏-超弹本构模型

为研究低高应变率条件下NEPE推进剂的力学特性,通过电子万能试验机和分离式霍普金森杆装置,对NEPE推进剂进行了准静态和冲击实验,得到了不同应变率下（1.667×10?4～4 500 s?1）的应力-应变曲线。实验结果表明NEPE推进剂具有明显的非线性弹性和应变率敏感性,随着应变率的增加,材料的强度、屈服应力和弹性模量显著增加,与低应变率相比,高应变率条件下材料的应变率敏感性更高。在高速冲击下材料内部瞬间产生大量热量无法及时散发出去,使得材料内部温度升高,导致材料出现软化效应,力学性能降低。本文建立了一个非线性黏超弹本构模型,其中采用Rivlin应变能函数来描述稳态超弹响应部分,采用积分型本构模型来描述材料的动态黏弹性响应部分,考虑到松弛时间具有应变率相关性,本文采用了一个率相关松弛函数来替代传统的Prony级数形式。使用极慢速压缩实验数据对本构模型中的超弹部分进行拟合获得超弹参数,然后用准静态和动态实验数据对本构模型进行拟合得出其他参数。不同应变率下的预测曲线与实验曲线具有较好的重合度,证明了该模型可以很好地描述低高应变率下NEPE推进剂的力学特性。     

Modelling cell wall growth using a fibre-reinforced hyperelastic–viscoplastic constitutive law

A fibre-reinforced hyperelastic–viscoplastic model using a finite strain Finite Element (FE) analysis is presented to study the expansive growth of cell walls. Based on the connections between biological concepts and plasticity theory, e.g. wall-loosening and plastic yield, wall-stiffening and plastic hardening, the modelling of cell wall growth is established within a framework of anisotropic viscoplasticity aiming to represent the corresponding biology-controlled behaviour of a cell wall. In order to model in vivo growth, special attention is paid to the differences between a living cell and an isolated wall. The proposed hyperelastic–viscoplastic theory provides a unique framework to clarify the interplay between cellulose microfibrils and cell wall matrix and how this interplay regulates sustainable growth in a particular direction while maintaining the mechanical strength of the cell walls by new material deposition. Moreover, the effect of temperature is taken into account. A numerical scheme is suggested and FE case studies are presented and compared with experimental data.     

A model for the nonisothermal behavior of viscoelastic media

Summary A new constitutive model is derived for the viscoelastic behavior of polymers under non-isothermal loading. The model extends the concept of adaptive links (entanglements) between polymeric molecules to thermoviscoelastic media. By using experimental data for Nylon-6 and polyisobutylene in the vicinity of the glass-transition temperature, we find parameters of the model and study their dependence on temperature. The model is employed for the numerical analysis of the material response to time-periodic loads under isothermal conditions and to time-varying loads under heating. The results of numerical simulation demonstrate fair agreement with experimental data. Accepted for publication 23 May 1996     

Orthotropic elastic–plastic material model for paper materials

Paper and paperboard generally exhibit anisotropic and non-linear mechanical material behaviour. In this work, the development of an orthotropic elastic–plastic constitutive model, suitable for modelling of the material behaviour of paper is presented. The anisotropic material behaviour is introduced into the model by orthotropic elasticity and an isotropic plasticity equivalent transformation tensor. A parabolic stress–strain relation is adopted to describe the hardening of the material. The experimental and numerical procedures for evaluation of the required material parameters for the model are described. Uniaxial tensile testing in three different inplane material directions provides the calibration of the material parameters under plane stress conditions. The numerical implementation of the material model is presented and the model is shown to perform well in agreement with experimentally observed mechanical behaviour of paper.