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。     

炭黑填充橡胶材料改进Mooney模型

本文讨论了炭黑填充橡胶材料的唯象本构模型。考虑到Mooney模型无法表征橡胶类材料大变形阶段的力学特性,首先利用实验数据,对Mooney模型进行了分析,讨论了炭黑含量与Mooney模型准确表征橡胶材料应变区间大小的关系,Mooney模型对纯剪切和等比双向拉伸等复杂变形的预测能力,同时也分析了材料参数对Mooney模型的影响。最后在Mooney模型的基础上添加了一个修正项,且改进后的Mooney模型满足Treloar和Ogden六项假设。通过与实验数据对比分析,改进Mooney模型可以较好地描述橡胶材料大变形阶段的应力应变关系,同时提高了预测橡胶材料复杂变形的能力。     

炭黑颗粒对丁苯橡胶材料力学性能的影响

利用电子万能材料试验机,对不同体积分数的炭黑填充丁苯橡胶复合材料进行了单向拉伸和循环拉伸加卸载等准静态力学试验,研究了炭黑填充橡胶材料单向拉伸应力应变关系、Mullins效应、调制应变相关性和拉伸断裂力学行为等.试验结果表明:调制应变越大,橡胶材料的刚度越小,橡胶的非线性应力应变关系越明显;炭黑含量越高,橡胶材料的初始模量和刚度越大,应力反翘现象和Mullins效应越显著;同时,随着炭黑体积分数的增加,炭黑填充丁苯橡胶材料的拉伸强度和断裂伸长率将呈现先增大后减小的趋势.     

橡胶材料弹性的一种新的螺旋管模型

建立统计力学模型正确描述材料微观结构与宏观力学特性之间的关系是软物质类材料的最大挑战之一,已有的橡胶材料统计模型尚存在一些不足.文章根据橡胶类材料宏观各向同性、连续均匀和不可压缩特性,结合分子链的非高斯统计模型,提出一种橡胶材料网络结构的力学特性模型.该模型将代表体元上对应点之间的传力路径用一个类螺旋管区域约束的分子链子网络来描述,螺旋管的表面随材料的宏观变形做仿射变形,分子链子网络由方向和长度随机的分子链或链段首尾链接而成,在此基础上由分子链的熵推导出描述材料宏观力学特性的本构关系.通过大量的材料测试数据对本构模型进行拟合验证,拟合结果表明该模型具有非常好的精度,并且在采用两个参数时模型具有非常高的可靠性,仅用单轴拉伸实验数据拟合模型就能准确预测全部3类实验数据.该模型使用了仿射的弯曲管假设,能从微观结构尺度上说明材料的不可压缩特性,避免了直管模型的近似性,为微观尺度的随机性和宏观的均匀性的联系提出一个新的模型.     

炭黑增强橡胶复合材料力学行为的三维数值模拟

本文基于炭黑填充橡胶复合材料具有周期性细观结构的假设,采用一种新的、改进的随机序列吸附算法建立了三维多球颗粒随机分布式代表性体积单元,并通过细观力学有限元方法对炭黑颗粒填充橡胶复合材料的力学行为进行了模拟仿真。研究结果表明：采用改进的随机序列吸附算法所建立的模型更加便于有限元离散化;模拟中周期性边界条件的约束,使其更加符合实际约束的真实情况;炭黑填充橡胶复合材料的有效模量明显高于未填充橡胶材料,并随着炭黑颗粒所占体积分数的增加而增大;通过比较发现,本文提出的多球颗粒随机分布式三维数值模型对复合材料的应力-应变行为和有效弹性模量的预测结果与实验结果吻合良好,证实了该模型能够用于炭黑颗粒增强橡胶基复合材料有效性能的模拟分析。     

炭黑填充天然橡胶Mullins效应的仿真与计算分析

通过有限元分析软件Abaqus、Matlab数据拟合工具及单轴拉伸试验,研究了Yeoh本构模型、Ogden本构模型及伪弹性理论模型对炭黑填充天然橡胶的Mullins效应仿真效果及规律。结果表明,(1)对单轴拉伸主加载曲线的拟合,Ogden模型精度更高,而Yeoh模型随着炭黑含量的增加在小变形处的拟合效果变差;(2)提出了伪弹性参数与伸长比之间的关系,找到了一种快速计算伪弹性参数的方法,且精度良好,极大地提高了计算效率。     

橡胶弹性材料的一种混合本构模型

该文探讨不可压缩橡胶弹性材料的本构模型.考虑到小变形时分子链的端矢分布符合高斯函数,而大变形时符合非高斯函数,提出一个混合模型,用高斯链网络模型来描述小变形而用8链网络模型来描述大变形.引入权重函数,使小变形和大变形情况下混合模型分别退化或趋于高斯链网络模型和8链网络模型.由Treloar拉伸实验数据拟合得到模型参数,通过这些材料参数混合模型对等双轴拉伸和纯剪切变形模式的预测结果与Tre-loar实验数据基本吻合,说明混合模型具有同时描述不同变形模式的能力.通过比较分析,混合模型的总体预测精度均优于高斯链网络模型和8链网络模型,特别是对剪切变形.     

填充橡胶温度依赖的非弹性力学行为实验研究

填充橡胶具有复杂的非弹性力学行为，主要包括应变率依赖的粘弹性效应和变形历史依赖的Mullins效应。当前大多数对填充橡胶的实验研究集中于室温条件，针对以上问题，本文通过单轴压缩实验系统地研究了温度对氟橡胶粘弹性和Mullins效应这两种非弹性行为的影响。首先采用多次循环加载获得了完全消除了Mullins效应的预处理试样。通过对原试样和预处理试样的单轴加卸载实验应力响应进行对比，发现Mullins效应不受变形温度和应变率的影响。通过对消除Mullins效应橡胶材料应力松弛实验结果分析，发现粘弹性行为不仅与变形的温度、应变率相关，还受加载应变的影响，表现为较大的加载应变会抑制氟橡胶的应力松弛。     

一种新的橡胶材料弹性本构模型

橡胶类材料本构关系对于科学研究和工程应用具有重要意义,但已有的橡胶模型的拟合能力和可靠性需要进一步提高.为解决此问题,本文提出了一种新的橡胶材料的各向同性、不可压缩柯西弹性模型.研究了橡胶材料本构关系的模型形式,基于平面应力变形状态,提出了一种以较大的两个伸长率为自变量、适用于一般变形状态的橡胶材料弹性本构模型形式;研究了橡胶材料在侧面受约束条件下的变形规律,分析了橡胶材料本构关系需要满足的约束条件;在此基础上,结合一个可以通过实验确定的描述平面拉伸变形状态下的橡胶材料力学特性函数,提出一种将该函数拓展为平面应力状态一般模型的方法,并给出了一个具体的函数形式,形成了一个新的不可压缩、各向同性的橡胶材料弹性本构模型.使用5组包含3种类型实验的数据和一组较全面的双轴测试数据对该模型进行了参数拟合,结果表明:该模型具有很好的拟合精度和更高的可靠性,仅用一种类型实验数据,如单轴拉伸或者平面拉伸等,也能获得较好的拟合结果.      

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Langevin统计模型来表征熵弹性变形自由能, 通过管模型(tube model)引入缠结约束自由能, 并基于仿射假设, 建立微观变形与宏观变形之间的映射关系. 在宏观上, 所建立的超弹性模型的Helmholtz自由能同时包含熵弹性和缠结约束两部分, 其中熵弹性自由能与经典的Arruda-Boyce八链模型一致, 依赖于柯西-格林应变张量的第一不变量, 而缠结约束自由能依赖于柯西-格林应变张量的第二不变量. 与文献中的实验结果对比发现, 该三参数模型能准确地预测实验中所测得的橡胶材料在单轴拉伸、纯剪切和等双轴拉伸变形条件下的应力响应, 也能较好地描述不同预拉伸比条件下双轴拉伸实验结果. 最后, 本文比较了所建立的基于应变不变量的缠结约束模型与文献中相关的缠结约束模型在多种加载模式下自由能的异同. 总的来说, 本文所建立的本构理论能准确模拟橡胶等软材料的大变形力学行为, 对其工程应用有促进作用.      

A network evolution model for the anisotropic Mullins effect in carbon black filled rubbers

To the best of our knowledge, there are no constitutive models that properly describe experimental data on anisotropy of the Mullins effect. In this paper, such a micro-mechanical model is proposed for carbon black filled rubbers. The model describes the deformation induced anisotropy and permanent set as well. Damage of the polymer-filler network is considered as a consequence of chain sliding on or debonding from aggregates. In contrast to previous works on anisotropy of the Mullins effect we do not introduce any phenomenological damage function. Damage in different directions is governed by a network evolution concept which describes the changes in the inter-aggregate distribution of polymer chains. The model includes a few number of physically motivated material constants and demonstrates good agreement with own experimental data on subsequent uniaxial tensions in two orthogonal directions.     

A damage directional constitutive model for Mullins effect with permanent set and induced anisotropy

A stress softening, commonly known as the Mullins effect, is observed usually in filled rubberlike materials after the first load. Experimental observations have shown that the Mullins effect induces a permanent set and some anisotropy. In order to account for these two features, a strain energy density is proposed, which is based on material directions and coupled with a theory for network alteration. The constitutive law is based on a macromolecular approach, and the network alteration theory is motivated physically. The model has been compared to experimental data successfully.     

Cyclic stress-softening model for the Mullins effect in compression

This paper models the cyclic stress softening of an elastomer in compression. After the initial compression the material is described as being transversely isotropic. We derive non-linear transversely isotropic constitutive equations for the elastic response, stress relaxation, residual strain, and creep of residual strain in order to model accurately the inelastic features associated with cyclic stress softening. These equations are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic stress softening for a transversely isotropic, hyperelastic material, in particular a carbon-filled rubber vulcanizate. To establish the validity of the model we compare it with two test samples, one for filled vulcanized styrene–butadiene rubber and the other for filled vulcanized natural rubber. The model is found to fit this experimental data extremely well.     

A viscoelastic constitutive model of rubber under small oscillatory load superimposed on large static deformation

Summary  A viscoelastic constitutive equation of rubber that is under small oscillatory load superimposed on large static deformation is proposed. The model is derived through linearization of Simo's nonlinear viscoelastic constitutive model and reference configuration transformation. Most importantly, in this model, static deformation correction factor is introduced to consider the influence of pre-strain on the relaxation function. Natural statically pre-deformed state is served as reference configuration. The proposed constitutive equation is extended to a generalized viscoelastic constitutive equation that includes widely used Morman's model as a special case using objective stress increment. The proposed constitutive model is tested for dynamic behavior of rubber specimens with different carbon black content. It is concluded from the test that the assumption that the effects of static deformation can be separated from time effects, which is the basis of Morman's model, is only applicable to unfilled rubber. The viscoelastic constitutive equation for filled rubber must include, therefore, the influence of the static deformation on the time effects. The suggested constitutive equation with static deformation correction factor shows good agreement with test values. Received 4 January 2001; accepted for publication 13 June 2001     

A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage

A micromechanically based non-affine network model for finite rubber elasticity and viscoelasticity was discussed in Parts I and II [Miehe, C., Göktepe, S., Lulei, F., 2004. A micro-macro approach to rubber-like materials. Part I: The non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids 52, 2617-2660; Miehe, C., Göktepe, S., 2005. A micro-macro approach to rubber-like materials. Part II: Viscoelasticity model for polymer networks. J. Mech. Phys. Solids, published on-line, doi:10.1016/j.jmps.2005.04.006.] of this work. In this follow-up contribution, we further extend the micro-sphere network model such that it incorporates a deformation-induced softening commonly referred to as the Mullins effect. To this end, a continuum formulation is constructed by a superimposed modeling of a crosslink-to-crosslink (CC) and a particle-to-particle (PP) network. The former is described by the non-affine elastic network model proposed in Part I. The Mullins-type damage phenomenon is embedded into the PP network and micromechanically motivated by a breakdown of bonds between chains and filler particles. Key idea of the constitutive approach is a two-step procedure that includes (i) the set up of micromechanically based constitutive models for a single chain orientation and (ii) the definition of the macroscopic stress response by a directly evaluated homogenization of state variables defined on a micro-sphere of space orientations. In contrast to previous works on the Mullins effect, our formulation inherently describes a deformation-induced anisotropy of the damage as observed in experiments. We show that the experimentally observed permanent set in stress-strain diagrams is achieved by our model in a natural way as an anisotropy effect. The performance of the model is demonstrated by means of several numerical experiments including the solution of boundary-value problems.     

A constitutive model for carbon black filled rubber: Experimental investigations and mathematical representation

Abspract The stress-strain behavior of carbon black filled rubber is recognized to be nonlinearly elastic in its main part (see e.g. Gent [1]). In addition, inelastic effects occur under monotonic and cyclic processes. The inelastic behavior includes nonlinear rate dependence as well as equilibrium hysteresis. Moreover, the first periods of a stress-strain curve differ significantly from the shape of subsequent cycles; a characteristic feature, which is called the Mullins effect, because it has been pointed out by Mullins [2]. All inelastic phenomena are strongly influenced by the volume fraction of the filler particles (see e.g. Payne [3], So and Chen [4], Meinecke and Taftaf [5]).The aim of the present paper is to design a constitutive model, representing this kind of material behavior as a phenomenological theory of continuum mechanics. In order to motivate the basic structure of the constitutive theory, a series of uniaxial experiments between 100% in tension and 30% in compression are presented and analyzed. First of all, monotonic strain controlled experiments show the nonlinear rate dependence of the stress response. Then, a series of inserted relaxation periods at constant strain yields the monotonic equilibrium stress-strain curve, which is strongly nonlinear and unsymmetric with respect to the origin. Finally, cyclic experiments under strain control display pronounced hysteresis behavior. The hysteresis effects are mainly rate dependent, but there exists also a weak equilibrium hysteresis (compare to similar observations of Orschall and Peeken [6]). The Mullins effect corresponds to a softening phenomenon during the first few cycles. By means of an appropriate preprocess, this effect was excluded during the above experiments. Apart from the Mullins effect, neither hardening nor significant softening phenomena were observed in the context of cyclic loadings.These results motivate the structure of a constitutive model of finite strain viscoplasticity: The total stress is decomposed into an equilibrium stress and an overstress, where the overstress is a rate dependent functional of the strain history. The overstress represents the rate dependence of the material behavior and tends asymptotically to zero during relaxation processes. The nonlinearity of the rate dependence is incorporated by means of a stress dependent relaxation time. The equilibrium stress is assumed to be a rate independent functional of the strain history. For this quantity, we make use of an arclength representation, which was originally introduced by Valanis [7]. In case of vanishing equilibrium hysteresis and vanishing rate dependence our constitutive model reduces to finite strain hyperelasticity, which is the first approximation of the constitutive properties. In more general cases the main shape of a stress-strain curve is determined by hyperelasticity, superimposed by rate dependent and equilibrium hysteresis. The representation of the Mullins effect is incorporated by a continuum damage model.Some numerical simulations at the end of the paper demonstrate that the presented theory is able to represent the observed phenomena qualitatively and quantitatively with sufficient approximation.     

Anisotropic Mullins stress softening of a deformed silicone holey plate

Rubber like materials parts are designed using finite element code in which more and more precise and robust constitutive equations are implemented. In general, constitutive equations developed in literature to represent the anisotropy induced by the Mullins effect present analytical forms that are not adapted to finite element implementation. The present paper deals with the development of a constitutive equation that represents the anisotropy of the Mullins effect using only strain invariants. The efficiency of the modeling is first compared to classical homogeneous experimental tests on a filled silicone rubber. Second, the model is tested on a complex structure. In this aim, a silicone holey plate is molded and tested in tension, its local strain fields are evaluated by means of digital image correlation. The experimental results are compared to the simulations from the constitutive equation implemented in a finite element code. Global measurements (i.e. force and displacement) and local strain fields are successfully compared to experimental measurements to validate the model.