橡胶隔振器多轴疲劳寿命预测方法研究
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摘要
橡胶隔振器是汽车动力总成等系统中常见且重要的隔振元件，由金属和橡胶复合而成，其疲劳特性对汽车的安全舒适性、操作稳定性和其他零部件的疲劳特性有重要的影响。对橡胶隔振器的疲劳寿命预测，主要是对其主体隔振橡胶材料的疲劳特性进行试验和建模研究。本文基于疲劳设计的理论框架，完善和进一步发展了疲劳寿命计算的基本理论和方法，研究了大变形隔振器橡胶材料在单轴拉伸载荷、单轴拉伸/压缩载荷和多轴载荷作用下的疲劳寿命预测方法等一系列相关问题。现将这些工作归纳如下：
(1)研究了不同的疲劳损伤参量和橡胶试件的几何形状与橡胶材料疲劳寿命之间的关系。开展了隔振器橡胶材料的单轴拉伸疲劳试验，采用的试件是三种几何特征不同的橡胶试件（哑铃型试片、哑铃型试柱和圆环试件）；基于疲劳试验数据，对比了多种常用于橡胶疲劳寿命预测的损伤参量与橡胶拉伸疲劳寿命之间的关系；研究和讨论了基于这三种橡胶试件获取的疲劳寿命之间的相关性。结果表明，最大主应变(Green-Lagrange应变、Almansi-Euler应变、工程应变、对数应变和伸长率)峰值、八面体切应变峰值和应变能密度峰值为损伤参量的寿命预测模型均能很好地以幂法则来建立损伤参量与疲劳寿命之间的关系，预测寿命与实测寿命的相关系数均达到0.9以上。其中以最大主Green-Lagrange应变峰值为损伤参量的寿命模型预测效果最佳。橡胶材料拉伸疲劳寿命与所有橡胶试件的几何形状无关，是一种材料属性。在工程实践中可用试验成本较低的哑铃型试片来代表哑铃型试柱和圆环试件进行橡胶拉伸疲劳试验，可节省成本和缩短试验周期。
(2)研究应变比R对隔振器橡胶材料单轴疲劳寿命的影响，提出了考虑应变比R效应的疲劳损伤参量。对由汽车橡胶隔振器常见的橡胶材料制作的哑铃型试柱，进行不同应变比R下的单轴拉伸/压缩疲劳试验，并分析应变比R对疲劳寿命的影响。以应变峰值、应变幅值为损伤参量，建立相应的疲劳寿命预测模型，并对R≥0和R<0两种工况下模型的预测结果进行比较。结果表明：对于应变比R改变的疲劳载荷工况，以应变幅值为损伤参量的寿命预测模型、和以应变峰值为损伤参量的寿命预测模型这两种常规的模型都有一定的局限性。为了考虑应变比R对橡胶疲劳寿命的影响，提出了等效应变幅值。基于等效应变幅值的寿命模型可较好地预测R在较大范围内变化的载荷工况下的疲劳寿命(特别是对R≤0的载荷工况)，预测的疲劳寿命可为该橡胶材料在疲劳设计阶段提供有效的数据支持。
(3)开展了隔振器橡胶材料的裂纹扩展试验，并建立了橡胶材料裂纹扩展特性的统一模型。利用带单边缺口的纯剪试件，开展了隔振器橡胶材料的裂纹扩展试验，其中疲劳载荷是变幅加载的位移。对这种加载工况下的裂纹扩展数据，不宜采用传统的割线法或七点多项式法计算裂纹扩展速率。研究结果表明：采用插值函数来表征实测的裂纹扩展长度与循环次数的关系进而得到橡胶材料的裂纹扩展统一模型，这种方法是准确合理的。
(4)基于连续介质力学的理论，推导了有限变形和多轴应力状态下开裂能密度的求解方法，进而计算多轴载荷下橡胶材料的撕裂能。在多轴疲劳载荷下，有效用来驱动裂纹的扩展的那部分应变能密度称为开裂能密度。基于开裂能密度和橡胶材料的裂纹扩展特性预测橡胶材料的多轴疲劳寿命时，须计算在外载荷下橡胶隔振器的开裂能密度。为了由有限元软件(如ABAQUS)默认输出的应变计算在有限变形下的开裂能密度，推导了常见6种超弹性模型下开裂能密度的计算公式，并说明了所需的积分方法。结果表明：本文开裂能密度的计算方法是准确合理的，基于该方法计算的开裂能密度，可以将橡胶等双轴疲劳和单轴疲劳寿命较好地统一起来。
(5)基于开裂能密度和隔振器橡胶材料的裂纹扩展特性实测数据，预测了橡胶隔振器的多轴疲劳寿命。基于开裂能密度这一连续介质力学参数和隔振器橡胶材料裂纹扩展试验获得的裂纹扩展特性(裂纹扩展速率与撕裂能之间的关系)，得到了橡胶隔振器多轴疲劳寿命的计算公式，并计算了某一典型结构形式的橡胶隔振器的疲劳寿命。结果表明：采用上述方法对橡胶隔振器疲劳的预测结果(寿命、开裂位置和开裂方向)与实测结果较为一致。其中预测的疲劳寿命分布在实测疲劳寿命的2倍分散线之内，满足工程疲劳寿命预测的要求。本文橡胶隔振器的多轴疲劳寿命预测方法，用试验效率较高和投入较少的材料裂纹扩展试验代替需要耗费大量时间的材料疲劳破坏试验，可为橡胶隔振器前期的疲劳设计提供参照的同时，还可大幅地缩短产品疲劳设计的周期。
Rubber isolators are usually subjected to substantial static and dynamic loads, and oftenfail due to nucleation and growth of defects or cracks. Prevention of such mechanical fatiguefailures necessitates thorough understanding of the deformation mechanisms of the rubbermaterials during cyclic loading, so as to predict the fatigue life of rubber components moreaccurately. The investigation on the mechanical fatigue of rubbers has become one of thecurrent international hot issues. In this thesis，a series of fatigue behaviors for filled natrualrubbers under uniaxial tension, uniaxial tension/compression, and multiaxial loads, areinvestigated in the frame of mechanics of fatigue design. The major studies can besummarized as follows:
(i) Two factors that might influence the tension fatigue model of filled natural rubbersused in rubber isolators are investigated. One is the damage parameter and the other is thespecimen geometry used in the fatigue experiment. The uniaxial tension fatigue experimentsare carried out for three typical types of filled natural rubber specimens: a dumbbell simpletension specimen (STS), a dumbbell cylindrical specimen (DCS), and a hollow cylindricalspecimen (HCS). The commonly used damage parameters (Green-Lagrange strain,Almansi-Euler strain, engineering strain, logarithmic strain, stretch ratio, etc.) for fatigue lifeprediction are described and discussed. The fatigue life prediction models using the measuredtension fatigue life of the STS and different damage parameters are developed, and acorrelation coefficient is used to characterize the error between the measured fatigue life andthe estimated one using the developed fatigue life prediction model. It is concluded that alldamage parameters discussed in this paper can be used to estimate tension fatigue life withcorrelation coefficient greater than0.9. The fatigue life model with the STS is appropriate topredict the fatigue life of the DCS and the HCS, which shows that the relationship betweenthe tension fatigue life and the damage parameters is independent of the geometry of thespecimens. One can thus carry out tension fatigue test using only a STS to model the tensionfatigue of rubbers.
(ii) The effect of strain ratio R on the fatigue life of filled natural rubbers used in rubberisolators is investigated experimentally and numerically. A uniaxial tension/compressionfatigue experiment is conducted on DCS rubber specimens subjected to loads representingdifferent R-ratios. The experimental fatigue data are used to formulate two preliminary fatiguemodels based on peak strain and strain amplitude as the damage parameters. The deficienciesof these two models in predicting fatigue life over a wide range of R ratios are discussed, and an alternative life prediction model is proposed. The proposed model incorporates the effectof R-ratio using an equivalent strain amplitude. It is shown that the proposed model couldeffectively predict fatigue life over a wide range of R-ratios with an improved accuracy,praticually for loads of negative R ratio.
(iii) The fatigue crack growth (FCG) experiment and modeling method for filled naturalrubbers used in rubber isolators under variable amplitude loads are carried out using anedge-crack pure shear specimen. Variable amplitude loads are imposed on the edge-crack pureshear specimen, and such load provides a more effective way of obtaining the measured FCGdata under different load levels than the conventional constant amplitude load. The commonlyused data processing techniques for getting the crack growth rate (crank length versus numberof cycles) in metal materials, the secant method or the local incremental polynomial method,are not applicable for computing the crack growth rate of rubber material, since the FCG dataunder the variable amplitude loads embodies large fluctuating locally. Based on the knownFCG law and the measured FCG data under variable amplitude loads, a power function isproposed to fit the measured crack growth length and number of cycles using the least-squarestechnique. The crack growth rate is thus calculated from the determined power function, and aFCG prediction model for filled natural rubbers is established from the crack growth data andthe associated tearing energy. To validate the developed FCG model, a dumbbell specimenmade of the same rubber compound as the pure shear specimen is manufactured and is used tocarry out the tensional fatigue experiment. The comparsions between the measured tensionalfatigue life of the dumbbell specimen and that evaluated from the established FCG modelvalidates the proposed data processing method for FCG data of filled natural rubbers underthe variable loads.
(iv) A new method for calculation of cracking energy density (CED), which is related tothe tearing energy of rubbers under multiaxial loads, is proposed in the frame of continuummechanics. Using the measured fatigue crack growth characteristic of rubber materials and thecalculated CED of rubbers under external loads, one can predict fatigue life of rubbercomponents. To calculate CED using the output strain from the finite element software(ABAQUS) as inputs, the formula of CED under the principal coordinate system is derivedand the required integral technology is given. Six hyperelastic constitutive models (Ogden,Mooney-Rivlin, Neo-Hookean, Yeoh, Arruda-Boyce and Van der Waals) for rubber materialsare used in the method for calculating CED. It is shown that the CED calculated from theproposed method is valid and effective to unify the uniaxial and equiaxial fatigue life ofrubber materials.
(v) The multiaxial fatigue life of rubber isolators is predicted by combing the calcualtedCED and the FCG model of the studied rubber material. Using the measured FCGcharacteristic of rubber material and the calculated CED of rubbers under external loads, aformula for calculating the fatigue life of rubber components under multiaxial loads isestablished, and is applied for predicting a typical rubber isolator. It is shown that thepredicted fatigue life of the rubber isolator agree well with the measured fatigue life within afactor of two, and the predicted crack location and orientation are comparable with themeasured results. The fatigue life prediction method for rubber isolators under multiaxialloads can thus be used as an effective and low cost tool for up-front knowledge of rubbercomponents in the design stage.
(1)研究了不同的疲劳损伤参量和橡胶试件的几何形状与橡胶材料疲劳寿命之间的关系。开展了隔振器橡胶材料的单轴拉伸疲劳试验，采用的试件是三种几何特征不同的橡胶试件（哑铃型试片、哑铃型试柱和圆环试件）；基于疲劳试验数据，对比了多种常用于橡胶疲劳寿命预测的损伤参量与橡胶拉伸疲劳寿命之间的关系；研究和讨论了基于这三种橡胶试件获取的疲劳寿命之间的相关性。结果表明，最大主应变(Green-Lagrange应变、Almansi-Euler应变、工程应变、对数应变和伸长率)峰值、八面体切应变峰值和应变能密度峰值为损伤参量的寿命预测模型均能很好地以幂法则来建立损伤参量与疲劳寿命之间的关系，预测寿命与实测寿命的相关系数均达到0.9以上。其中以最大主Green-Lagrange应变峰值为损伤参量的寿命模型预测效果最佳。橡胶材料拉伸疲劳寿命与所有橡胶试件的几何形状无关，是一种材料属性。在工程实践中可用试验成本较低的哑铃型试片来代表哑铃型试柱和圆环试件进行橡胶拉伸疲劳试验，可节省成本和缩短试验周期。
(2)研究应变比R对隔振器橡胶材料单轴疲劳寿命的影响，提出了考虑应变比R效应的疲劳损伤参量。对由汽车橡胶隔振器常见的橡胶材料制作的哑铃型试柱，进行不同应变比R下的单轴拉伸/压缩疲劳试验，并分析应变比R对疲劳寿命的影响。以应变峰值、应变幅值为损伤参量，建立相应的疲劳寿命预测模型，并对R≥0和R<0两种工况下模型的预测结果进行比较。结果表明：对于应变比R改变的疲劳载荷工况，以应变幅值为损伤参量的寿命预测模型、和以应变峰值为损伤参量的寿命预测模型这两种常规的模型都有一定的局限性。为了考虑应变比R对橡胶疲劳寿命的影响，提出了等效应变幅值。基于等效应变幅值的寿命模型可较好地预测R在较大范围内变化的载荷工况下的疲劳寿命(特别是对R≤0的载荷工况)，预测的疲劳寿命可为该橡胶材料在疲劳设计阶段提供有效的数据支持。
(3)开展了隔振器橡胶材料的裂纹扩展试验，并建立了橡胶材料裂纹扩展特性的统一模型。利用带单边缺口的纯剪试件，开展了隔振器橡胶材料的裂纹扩展试验，其中疲劳载荷是变幅加载的位移。对这种加载工况下的裂纹扩展数据，不宜采用传统的割线法或七点多项式法计算裂纹扩展速率。研究结果表明：采用插值函数来表征实测的裂纹扩展长度与循环次数的关系进而得到橡胶材料的裂纹扩展统一模型，这种方法是准确合理的。
(4)基于连续介质力学的理论，推导了有限变形和多轴应力状态下开裂能密度的求解方法，进而计算多轴载荷下橡胶材料的撕裂能。在多轴疲劳载荷下，有效用来驱动裂纹的扩展的那部分应变能密度称为开裂能密度。基于开裂能密度和橡胶材料的裂纹扩展特性预测橡胶材料的多轴疲劳寿命时，须计算在外载荷下橡胶隔振器的开裂能密度。为了由有限元软件(如ABAQUS)默认输出的应变计算在有限变形下的开裂能密度，推导了常见6种超弹性模型下开裂能密度的计算公式，并说明了所需的积分方法。结果表明：本文开裂能密度的计算方法是准确合理的，基于该方法计算的开裂能密度，可以将橡胶等双轴疲劳和单轴疲劳寿命较好地统一起来。
(5)基于开裂能密度和隔振器橡胶材料的裂纹扩展特性实测数据，预测了橡胶隔振器的多轴疲劳寿命。基于开裂能密度这一连续介质力学参数和隔振器橡胶材料裂纹扩展试验获得的裂纹扩展特性(裂纹扩展速率与撕裂能之间的关系)，得到了橡胶隔振器多轴疲劳寿命的计算公式，并计算了某一典型结构形式的橡胶隔振器的疲劳寿命。结果表明：采用上述方法对橡胶隔振器疲劳的预测结果(寿命、开裂位置和开裂方向)与实测结果较为一致。其中预测的疲劳寿命分布在实测疲劳寿命的2倍分散线之内，满足工程疲劳寿命预测的要求。本文橡胶隔振器的多轴疲劳寿命预测方法，用试验效率较高和投入较少的材料裂纹扩展试验代替需要耗费大量时间的材料疲劳破坏试验，可为橡胶隔振器前期的疲劳设计提供参照的同时，还可大幅地缩短产品疲劳设计的周期。
Rubber isolators are usually subjected to substantial static and dynamic loads, and oftenfail due to nucleation and growth of defects or cracks. Prevention of such mechanical fatiguefailures necessitates thorough understanding of the deformation mechanisms of the rubbermaterials during cyclic loading, so as to predict the fatigue life of rubber components moreaccurately. The investigation on the mechanical fatigue of rubbers has become one of thecurrent international hot issues. In this thesis，a series of fatigue behaviors for filled natrualrubbers under uniaxial tension, uniaxial tension/compression, and multiaxial loads, areinvestigated in the frame of mechanics of fatigue design. The major studies can besummarized as follows:
(i) Two factors that might influence the tension fatigue model of filled natural rubbersused in rubber isolators are investigated. One is the damage parameter and the other is thespecimen geometry used in the fatigue experiment. The uniaxial tension fatigue experimentsare carried out for three typical types of filled natural rubber specimens: a dumbbell simpletension specimen (STS), a dumbbell cylindrical specimen (DCS), and a hollow cylindricalspecimen (HCS). The commonly used damage parameters (Green-Lagrange strain,Almansi-Euler strain, engineering strain, logarithmic strain, stretch ratio, etc.) for fatigue lifeprediction are described and discussed. The fatigue life prediction models using the measuredtension fatigue life of the STS and different damage parameters are developed, and acorrelation coefficient is used to characterize the error between the measured fatigue life andthe estimated one using the developed fatigue life prediction model. It is concluded that alldamage parameters discussed in this paper can be used to estimate tension fatigue life withcorrelation coefficient greater than0.9. The fatigue life model with the STS is appropriate topredict the fatigue life of the DCS and the HCS, which shows that the relationship betweenthe tension fatigue life and the damage parameters is independent of the geometry of thespecimens. One can thus carry out tension fatigue test using only a STS to model the tensionfatigue of rubbers.
(ii) The effect of strain ratio R on the fatigue life of filled natural rubbers used in rubberisolators is investigated experimentally and numerically. A uniaxial tension/compressionfatigue experiment is conducted on DCS rubber specimens subjected to loads representingdifferent R-ratios. The experimental fatigue data are used to formulate two preliminary fatiguemodels based on peak strain and strain amplitude as the damage parameters. The deficienciesof these two models in predicting fatigue life over a wide range of R ratios are discussed, and an alternative life prediction model is proposed. The proposed model incorporates the effectof R-ratio using an equivalent strain amplitude. It is shown that the proposed model couldeffectively predict fatigue life over a wide range of R-ratios with an improved accuracy,praticually for loads of negative R ratio.
(iii) The fatigue crack growth (FCG) experiment and modeling method for filled naturalrubbers used in rubber isolators under variable amplitude loads are carried out using anedge-crack pure shear specimen. Variable amplitude loads are imposed on the edge-crack pureshear specimen, and such load provides a more effective way of obtaining the measured FCGdata under different load levels than the conventional constant amplitude load. The commonlyused data processing techniques for getting the crack growth rate (crank length versus numberof cycles) in metal materials, the secant method or the local incremental polynomial method,are not applicable for computing the crack growth rate of rubber material, since the FCG dataunder the variable amplitude loads embodies large fluctuating locally. Based on the knownFCG law and the measured FCG data under variable amplitude loads, a power function isproposed to fit the measured crack growth length and number of cycles using the least-squarestechnique. The crack growth rate is thus calculated from the determined power function, and aFCG prediction model for filled natural rubbers is established from the crack growth data andthe associated tearing energy. To validate the developed FCG model, a dumbbell specimenmade of the same rubber compound as the pure shear specimen is manufactured and is used tocarry out the tensional fatigue experiment. The comparsions between the measured tensionalfatigue life of the dumbbell specimen and that evaluated from the established FCG modelvalidates the proposed data processing method for FCG data of filled natural rubbers underthe variable loads.
(iv) A new method for calculation of cracking energy density (CED), which is related tothe tearing energy of rubbers under multiaxial loads, is proposed in the frame of continuummechanics. Using the measured fatigue crack growth characteristic of rubber materials and thecalculated CED of rubbers under external loads, one can predict fatigue life of rubbercomponents. To calculate CED using the output strain from the finite element software(ABAQUS) as inputs, the formula of CED under the principal coordinate system is derivedand the required integral technology is given. Six hyperelastic constitutive models (Ogden,Mooney-Rivlin, Neo-Hookean, Yeoh, Arruda-Boyce and Van der Waals) for rubber materialsare used in the method for calculating CED. It is shown that the CED calculated from theproposed method is valid and effective to unify the uniaxial and equiaxial fatigue life ofrubber materials.
(v) The multiaxial fatigue life of rubber isolators is predicted by combing the calcualtedCED and the FCG model of the studied rubber material. Using the measured FCGcharacteristic of rubber material and the calculated CED of rubbers under external loads, aformula for calculating the fatigue life of rubber components under multiaxial loads isestablished, and is applied for predicting a typical rubber isolator. It is shown that thepredicted fatigue life of the rubber isolator agree well with the measured fatigue life within afactor of two, and the predicted crack location and orientation are comparable with themeasured results. The fatigue life prediction method for rubber isolators under multiaxialloads can thus be used as an effective and low cost tool for up-front knowledge of rubbercomponents in the design stage.
引文
[1] Lewitzke C. and Lee P. Application of elastomeric components for noise and vibrationisolation in the automotive industry [C]. SAE Technical Paper Series2001-01-1447.
[2] Shangguan W.B. Engine mounts and powertrain mounting systems: a review [J].International Journal of Vehicle Design,2009,49(4):237-258.
[3] Mars W.V. and Fatemi A. Factors that affect the fatigue life of rubber: A literature survey[J]. Rubber Chemistry and Technology,2004,76:391-412.
[4] Bannantine J.A., Comer J.J. and Handrock J.L. Fundamentals of metal fatigue analysis[M]. Englewood: Prentice Hall,1992.
[5] Mars W.V. and Fatemi A. A literature survey on fatigue analysis approaches for rubber [J].International Journal of Fatigue,2002,24:949-961.
[6] Verron E and Andriyana A. Definition of a new predictor for multiaxial fatigue cracknucleation in rubber [J]. Journal of the Mechanics and Physics of Solids,2008,56:417-443.
[7] Moon S.J., Cho I.J., Woo C.S. and Kim W.D. Study on determination of durabilityanalysis process and fatigue damage parameter for rubber component [J]. Journal ofMechanical Science and Technology,2011,25(5):1159-1165.
[8] Harbour R. J., Fatemi A. and Mars W.V. Fatigue life analysis and predictions for NR andSBR under variable amplitude and multiaxial loading conditions [J]. InternationalJournal of Fatigue,2008,30:1231-1247.
[9] Mars W.V. and Fatemi A. Multiaxial fatigue of rubber: Part I: equivalence criteria andtheoretical aspects [J]. Fatigue&Fracture of Engineering Materials&Structure,2005,28:515-522.
[10] Bonet J. and Wood R.D. Nonlinear continuum mechanics for Finite Element Analysis[M]. Cambridge: Cambridge University Press,1997.
[11] Lemaitre J. and Chaboche J.L. Mechanics of Solid Materials. Cambridge UniversityPress,1994.
[12] Griffith A.A. The phenomena of rupture and flow in solids [J]. PhilosophicalTransactions of the Royal Society of London Series A.1920,221:163–98.
[13] Rivlin R.S. and Thomas A.G. Rupture of rubbers.I. characteristic energy for tearing [J].Journal of Polymer Sciences,1953,10:291-318.
[14] Mirza S., Hansen P. and Harris J. Modelling and durability assessment for rubbercomponents in rail vehicles [J]. Plastics, Rubber and Composites,2011,40(4):185-193.
[15]Mars W.V. and Fatemi A. Fatigue crack nucleation and growth in filled natural rubber [J].Fatigue&Fracture of Engineering Materials&Structure,2003,26:779-789.
[16] Gent A.N., Lindley P.B. and Thomas A.G. Cut growth and fatigue of rubbers. I. Therelationship between cut growth and fatigue [J]. Journal of Applied Polymer Science,1964,8:455-466.
[17] Zine A., Benseddiq N. and Abdelaziz M.N. Rubber fatigue life under multiaxial loading:Numerical and experimental investigations [J]. International Journal of Fatigue,2011,33:1360-1368.
[18] Cadwell S.M., Merrill R.A., Sloman C. M., et al. Dynamic fatigue life of rubber [J].Industrial and Engineering Chemistry,1940,12:19-23.
[19] Fielding J.H. Flex life and crystallization of synthetic rubber [J].Industrial andEngineering Chemisty,1943,35(12):1259-1261.
[20] Ro H.S. Modeling and interpretation of fatigue failure initiation in rubber related topneumatic tires [D]. USA: Purdue University,1989.
[21] Roberts B.J. and Benzies J.B. The relationship between uniaxial and equibiaxial fatiguein gum and carbon black filled vulcanizates [J]. Proceedings of Rubbercon’77,1977,2.1:1–13.
[22] Roach J.F. Crack growth in elastomers under biaxial stresses [D]. USA: University ofAkron,1982.
[23]Zine A., Benseddiq N., Abdelaziz M.N., et al. Prediction of rubber fatigue life undermultiaxial loading [J]. Fatigue&Fracture of Engineering Materials&Structure,2006,29:267-278.
[24] Li Q., Zhao J. C. and Zhao B. Fatigue life prediction of a rubber mount based on test ofmaterial properties and finite element analysis [J]. Engineering Failure Analysis,2009,16:2304-2310.
[25] Kim W.D., Lee H.J., Kim J.Y., et al. Fatigue life estimation of an engine rubber mount[J]. International Journal of Fatigue,2004,26:553-560.
[26]Woo C. S., Kim W. D. and Kown J.D. A study on the material properties and fatigue lifeprediction of natural rubber component [J]. Materials Science and Engineering A,2008,483/484:376-381.
[27]Mars W.V. and Fatemi A. Multiaxial fatigue of rubber: Part II:experimental observationsand life predictions [J]. Fatigue&fracture of engineering materials&structure,2005,28:523-538.
[28]Hideki O., Yuuta A., Hiroshi N., et al. Fatigue characteristics of vulcanized natural rubberfor automotive engine mounting (Characteristics of composition and mechanicalproperties)[J]. Memoirs of the Faculty of Engineering,2007,67(2):75-83.
[29]André N., Cailletaud G. and Piques R. Haigh diagram for fatigue crack initiationprediction of natural rubber components [J]. Kautsch Gummi Kunstst1999,52:120-123.
[30]Poisson J.L., Lacroix F., Meo S., and et al. Biaxial fatigue behavior of a polychloroprenerubber [J]. International Journal of fatigue,2011,33:1151-1157.
[31]Abraham F., Alshuth T. and Jerrams S. The effect of minimum stress and stress amplitudeon the fatigue life of non strain crystallising elastomers [J]. Materials and Design,2005,26:239-245.
[32] Luo R.K. and Wu W.X. Fatigue failure analysis of anti-vibration rubber spring [J].Engineering Failure Analysis,2006,13:110-116.
[33]Luo R.K., Mortel W.J. and Wu X.P. Fatigue failure investigation on anti-vibration springs[J]. Engineering Failure Analysis,2009,16:1366-1378.
[34]Saintier N., Cailletaud G. and Piques R. Multiaxial fatigue life prediction for a naturalrubber [J]. International Journal of Fatigue,2006,28(5-6):530-539.
[35]余伟炜,陈旭,闫磊.多轴循环载荷下橡胶材料的力学行为[J].橡胶工业,2007,54:500-503.
[36]Mars, W.V. Multiaxial fatigue of rubber [D]. USA: The University of Toledo,2001.08.
[37]Mars W.V., Cracking energy density as a predictor of fatigue life under multiaxialconditions [J]. Rubber Chemistry and Technology,2002,75:1-17.
[38]Kachanov L.M. Time of the rupture process under creep conditions. Izv. Akad. Nauk.S.S.R.Otd. Tech. Nauk.1958,8:26-31.
[39]Wang B., Lu H.B. and Kim G.H. A damage model for the fatigue life of elastomericmaterials [J]. Mechanics of materials,2002,34:475-483.
[40]Ali A., Hosseini M. and Sahari B. Continuum damage mechanics modeling for fatiguelife of elastomeric materials [J]. International Journal of Structural Integrity,2010,1(1):63-72.
[41]Ayoub G., Na t-abdelaziz M., Za ria F., et al. Multiaxial fatigue life prediction ofrubber-like materials using the continuum damage mechanics approach [J]. ProcediaEngineering,2010,2:985-993.
[42] Ayoub G., Na t-abdelaziz M., Za ria F., et al. A continuum damage model for thehigh-cycle fatigue life prediction of styrene-butadiene rubber under multiaxial loading[J]. International Journal of Solids and Structures,2011,48:2458-2466.
[43]Hocine N.A., Abdelaziza M.N. and Imad A. Fracture problems of rubbers: J-integralestimation based upon η factors and an investigation on the strain energy densitydistribution as a local criterion [J]. International Journal of Fracture,2002,117:1–23.
[44]Thomas A.G. Rupture of rubbers: II the strain concentration at an incision [J]. Journal ofPolymer Sciences,1955,18:177-188.
[45]Rice J.R. A path independent integral and the approximate analysis of strainconcentration by notches and cracks [J]. Journal of Applied Mechanics,1968,35:379–386.
[46]Busfield J.J.C., Davies C.K.L. and Thomas A.G. Aspects of fracture in rubbercomponents [J]. Progress in Rubber and Plastics Technology,1996,12(3):191-207.
[47]Abdelaziza M.N., Za ri F., Qua Z., et al. J integral as a fracture criterion of rubber-likematerials using the intrinsic defect concept [J]. Mechanics of Materials,2012,53:80-90.
[48]Hamdi A., Abdelaziza M.N., Hocine N.A., et al. A fracture criterion of rubber-likematerials under plane stress conditions [J]. Polymer Testing25(2006)994-1005
[49]Hamdi A., Abdelaziza M.N., Hocine N.A., et al. A new generalized fracture criterion ofelastomers under quasi-static plane stress loadings [J]. Polymer Testing,2007,26:896-902.
[50]Hamdi A., Hocine N.A., Abdelaziza M.N., et al. Fracture of elastomers under staticmixed mode: the strain-energy-density factor [J]. International Journal of Fracture,2007,144:65–75.
[51]Lake G.J. and Lindley P.B. The mechanical fatigue limit for rubber [J]. Journal of AppliedPolymer Science,1965,9:1233-1251.
[52]Busfield J.J.C., Thomas A.G., Ngah M.F. Application of fracture mechanics for thefatigue life prediction of carbon black filled elastomers [C]. Constitutive Models forRubber, Edited by A. Dorfmann and A. Muhr, pp.249-256,1999.
[53]Busfield J.J.C., Jha V., Liang H., et al. Prediction of fatigue crack growth using finiteelement analysis techniques applied to three-dimensional elastomeric components [J].Plastics, Rubbers and Composites,2005,34(8):349-356.
[54]Asare S. and Busfield J.J.C. Fatigue life prediction of bonded rubber components atelevated temperature [J]. Plastics, Rubber and Composites,2011,40(4):194-200.
[55]汪艳萍.橡胶材料多轴疲劳寿命及微观结构研究[博士学位论文].天津：天津大学,2007.12.
[56]姜莞.商用车动力总成悬置性能模拟与疲劳寿命预测研究[博士学位论文].长春:吉林大学.2011.06.
[57]高勋朝.填充橡胶材料的疲劳性能研究[硕士学位论文].哈尔滨：哈尔滨工业大学.2006.06.
[58]丽平.发动机橡胶减振垫的疲劳性能研究[硕士学位论文].上海：上海交通大学.2008.02.
[59]丁智平,陈吉平,宋传江,等.橡胶弹性减振元件疲劳裂纹扩展寿命分析.机械工程学报,2010,46(22):58-64.
[60]姜纪鑫,靳晓雄,殷闻.发动机橡胶悬置元件的疲劳寿命分析与预测.佳木斯大学学报(自然科学版),2011,29(5):673-676.
[61]刘建勋,黄友剑,刘柏兵,等.一种橡胶弹性元件疲劳寿命预测方法的研究.电力机车与城轨车辆,2011,34(3):12-14.
[62]丁家松,张欢,童宗鹏,等.船用橡胶隔振器寿命评估方法研究.振动与冲击,2010,29(12):230-233.
[63]André N., Gailletaud G. and Piques R. Haigh diagram for fatigue crack initiationprediction of natural rubber components [J]. Kautschuk Gummi Kunstoffe,1999,52:120-123.
[64]Zarrin-Ghalami T. and Fatemi A. Cumulative fatigue damage and life prediction ofelastomeric components. Fatigue&Fracture of Engineering Materials&Structures,2013,36:270-279.
[65]Mars W.V. Fatigue life prediction for elastomeric structures [J]. Rubber Chemistry andTechnology,2007,80(3):481-503.
[66]ASTM Standard D4482-11. Standard test method for rubber property-extension cyclingfatigue, ASTM, West Conshohocken, PA.
[67]Mars W.V. and Fatemi A. A novel specimen for investigating the mechanical behavior ofelastomers under multiaxial loading conditions [J]. Experimental Mechanics,2004,44(2):136-146.
[68]Tao G. and Xia Z. A non-contact real-time strain measurement and control system formultiaxial cyclic/fatigue tests of polymer materials by digital image correlation method[J]. Polymer Testing,2005,24:844-855.
[69]Le Cam J.-B., Huneau B. and Verron E. Description of fatigue damage in carbon blackfilled natural rubber [J]. Fatigue&Fracture of Engineering Materials&Structures,2008,31:1031-1038.
[70]Andriyana A. and Verron E. Prediction of fatigue life improvement in natural rubberusing configurational stress [J]. International Journal of Solids and Structures,2007,44:2079-2092.
[71]陆明万,罗学富.弹性理论基础[M].北京:清华大学出版社,施普林格出版社,2001.
[72]徐秉业，黄炎，刘信声，等.弹塑性力学及其应用[M].北京：机械工业出版社,1984.
[73]McGarry M.D.J., Van Houten E.E.W., Perrinez P.R., et al. An octahedral shearstrain-based measure of SNR for3D MR elastography [J]. Physics in medicine andbiology,2011,56:153-164.
[74]Shangguan W.B., Wang X.L., Deng J.X., et al. Experiment and modeling of uniaxialtension fatigue performances for filled natural rubbers [J]. Materials and Design,2014,58:65-73.
[75]Dizon E.S., Hicks A.E. and Chirico V.E. The effect of carbon black parameters on thefatigue life of filled rubber compounds [J], Rubber Chemistry and Technology,1974,47:231-249.
[76]OriginLab Corporation. Origin8user guide [M].USA: OriginLab Corporation FirstEdition,2007.
[77]Sors L. Fatigue design of machine components [M]. Oxford: Pergamon Press,1971.
[78]Bannantine J.A., Comer J.J. and Handrock J.L. Fundamentals of metal fatigue analysis[M]. Englewood: Prentice Hall,1992.
[79]王小莉，上官文斌，刘泰凯，等.填充橡胶材料单轴拉伸疲劳试验及疲劳寿命模型研究[J].机械工程学报，2013，49(14)：65-73.
[80]Holzapfel G.A. Nonlinear solid mechanics. A continuum approach for engineering [M].Chichester：Wiley，2000.
[81]Kujawski D, Ellyin F. A unified approach to mean stress effect on fatigue thresholdconditions [J]. International Journal Fatigue,1995,17(2):101-106.
[82]Mars W.V. and Fatemi A. A phenomenological model for the effect of R Ratio on fatigueof strain crystallizing rubbers [J]. Rubber Chemistry and Technology,2003,76：1241-1258.
[83]Harbour J, Fatemi A and Mars W V. Fatigue crack growth of filled rubber under constantand variable amplitude loading conditions [J]. Fatigue&Fracture of EngineeringMaterials&Structures,2007,30:640-652.
[84]Mars W.V. and Fatemi A. Observations of the constitutive response and characterizationof filled natural rubber under monotonic and cyclic multiaxial stress states [J]. ASMEJournal of Engineering Materials and Technology,2004,126：19-28.
[85]Zarrin-Ghalami T. and Fatemi A. Material Deformation and Fatigue BehaviorCharacterization for Elastomeric Component Life Predictions [J]. Polym. Eng. Sci.,2012,52:1795-1805.
[86] Tao G. and Xia Z.H. Mean stress/strain effect on fatigue behavior of an epoxy resin [J].International Journal of Fatigue,2007,29：2180-2190.
[87]Wang X.L., Shangguan W. B., Rakhaja S., et al. A method to develop a unified fatiguelife prediction model for filled natural rubbers under uniaxial loads. Fatigue&Fractureof Engineering Materials&Structures,2014,37(1):50-61.
[88]上官文斌，王小莉，叶必军，等.应变比对填充天然橡胶疲劳特性影响的试验及其寿命预测方法研究[J].机械工程学报，2013，49(8)：49-56.
[89]Yeoh O. H．Analysis of deformation and fracture of pure shear rubber test piece[J]．Plastics，Rubber and Composites，2001，30(8)：389-397．
[90]贾法勇，霍立兴，张玉凤，等.疲劳裂纹扩展速率两种数据处理方法的比较[J].机械强度，2003，25(5)：568-571.
[91]孟宪红，张行.确定疲劳裂纹扩展速率的积分法[J].机械强度，2001，23(2)：213-215.
[92]American Society of Mechanical Engineers. ASME E647–08Standard test method formeasurement of fatigue crack growth rates [S].West Conshohocken：ASME，2008.
[93]郦正能.应用断裂力学[M].北京：北京航空航天大学出版社，2012.
[94]Kaang S.Y., Jin Y.W., Huh Y.I., et al. A test method to measure fatigue crack growth rateof rubbery materials [J]. Polymer Testing,2006,25:347-352.
[95]Thomas A.G. Rupture of rubber. V. Cut growth in natural rubber vulcanizates [J]. Journalof Polymer Science,1958,31:467-480.
[96]Lindley P.B. Relation between hysteresis and the dynamic crack growth resistance ofnatural rubber [J]. International Journal of Fracture，1973，9(4)：449-462.
[97]Lake G.J. Fatigue and fracture of elastomers [J]. Rubber Chemistry and Technology，1995，68(3)：435-459.
[98]Miller K. Testing elastomers for hyperelastic material models in finite element analysis[EB/OL].2004[2013-8]. http://www.axelproducts.com/pages/downloads.html.
[99]雷英杰，张善文，李续武，等. MATLAB遗传算法工具箱及应用[M].西安：西安电子科技大学出版社，2005.
[100] Mars, W.V. and Fatemi A. Analysis of fatigue life under complex loading: RevisitingCadwell, Merrill, Sloman, and Yost [J]. Rubber Chemistry and Technology,2006,79(4)：589-601.
[101] Greensmith H.W. Rupture of rubber. X. The change in stored energy on making asmall cut in a test piece held in simple extension [J]. Journal of Applied PolymerScience，1963，7：993-1002.
[102] At-Bachir M.，Mars W.V. and Verron E. Energy release rate of small cracks inhyperelastic materials [J]. International Journal of Non-Linear Mechanics，2012，47：22-29.
[103] ABAQUS6.10. Theory Manual,2010.
[104]上官文斌，吕振华.汽车动力总成橡胶隔振器弹性弹性的有限元分析[J].内燃机工程，2003，24(6)：50-55.
[105]王文涛，上官文斌，段小成，等.基于线性疲劳累计损伤橡胶悬置疲劳寿命预测研究[J].机械工程学报，2012，48(10)：56-65.
[106]谢新星.橡胶隔振器单轴疲劳寿命评价方法及预测研究[D].广州：华南理工大学，2011.
[107]王文涛.橡胶隔振器单轴疲劳特性试验与预测方法的研究[D].广州：华南理工大学，2012.
[108] Asare S. and Busfield J. J. C. Fatigue life prediction of bonded rubber components atelevated temperature [J]. Plastics, Rubber and Composites，2011，40(4)：194-200.
[2] Shangguan W.B. Engine mounts and powertrain mounting systems: a review [J].International Journal of Vehicle Design,2009,49(4):237-258.
[3] Mars W.V. and Fatemi A. Factors that affect the fatigue life of rubber: A literature survey[J]. Rubber Chemistry and Technology,2004,76:391-412.
[4] Bannantine J.A., Comer J.J. and Handrock J.L. Fundamentals of metal fatigue analysis[M]. Englewood: Prentice Hall,1992.
[5] Mars W.V. and Fatemi A. A literature survey on fatigue analysis approaches for rubber [J].International Journal of Fatigue,2002,24:949-961.
[6] Verron E and Andriyana A. Definition of a new predictor for multiaxial fatigue cracknucleation in rubber [J]. Journal of the Mechanics and Physics of Solids,2008,56:417-443.
[7] Moon S.J., Cho I.J., Woo C.S. and Kim W.D. Study on determination of durabilityanalysis process and fatigue damage parameter for rubber component [J]. Journal ofMechanical Science and Technology,2011,25(5):1159-1165.
[8] Harbour R. J., Fatemi A. and Mars W.V. Fatigue life analysis and predictions for NR andSBR under variable amplitude and multiaxial loading conditions [J]. InternationalJournal of Fatigue,2008,30:1231-1247.
[9] Mars W.V. and Fatemi A. Multiaxial fatigue of rubber: Part I: equivalence criteria andtheoretical aspects [J]. Fatigue&Fracture of Engineering Materials&Structure,2005,28:515-522.
[10] Bonet J. and Wood R.D. Nonlinear continuum mechanics for Finite Element Analysis[M]. Cambridge: Cambridge University Press,1997.
[11] Lemaitre J. and Chaboche J.L. Mechanics of Solid Materials. Cambridge UniversityPress,1994.
[12] Griffith A.A. The phenomena of rupture and flow in solids [J]. PhilosophicalTransactions of the Royal Society of London Series A.1920,221:163–98.
[13] Rivlin R.S. and Thomas A.G. Rupture of rubbers.I. characteristic energy for tearing [J].Journal of Polymer Sciences,1953,10:291-318.
[14] Mirza S., Hansen P. and Harris J. Modelling and durability assessment for rubbercomponents in rail vehicles [J]. Plastics, Rubber and Composites,2011,40(4):185-193.
[15]Mars W.V. and Fatemi A. Fatigue crack nucleation and growth in filled natural rubber [J].Fatigue&Fracture of Engineering Materials&Structure,2003,26:779-789.
[16] Gent A.N., Lindley P.B. and Thomas A.G. Cut growth and fatigue of rubbers. I. Therelationship between cut growth and fatigue [J]. Journal of Applied Polymer Science,1964,8:455-466.
[17] Zine A., Benseddiq N. and Abdelaziz M.N. Rubber fatigue life under multiaxial loading:Numerical and experimental investigations [J]. International Journal of Fatigue,2011,33:1360-1368.
[18] Cadwell S.M., Merrill R.A., Sloman C. M., et al. Dynamic fatigue life of rubber [J].Industrial and Engineering Chemistry,1940,12:19-23.
[19] Fielding J.H. Flex life and crystallization of synthetic rubber [J].Industrial andEngineering Chemisty,1943,35(12):1259-1261.
[20] Ro H.S. Modeling and interpretation of fatigue failure initiation in rubber related topneumatic tires [D]. USA: Purdue University,1989.
[21] Roberts B.J. and Benzies J.B. The relationship between uniaxial and equibiaxial fatiguein gum and carbon black filled vulcanizates [J]. Proceedings of Rubbercon’77,1977,2.1:1–13.
[22] Roach J.F. Crack growth in elastomers under biaxial stresses [D]. USA: University ofAkron,1982.
[23]Zine A., Benseddiq N., Abdelaziz M.N., et al. Prediction of rubber fatigue life undermultiaxial loading [J]. Fatigue&Fracture of Engineering Materials&Structure,2006,29:267-278.
[24] Li Q., Zhao J. C. and Zhao B. Fatigue life prediction of a rubber mount based on test ofmaterial properties and finite element analysis [J]. Engineering Failure Analysis,2009,16:2304-2310.
[25] Kim W.D., Lee H.J., Kim J.Y., et al. Fatigue life estimation of an engine rubber mount[J]. International Journal of Fatigue,2004,26:553-560.
[26]Woo C. S., Kim W. D. and Kown J.D. A study on the material properties and fatigue lifeprediction of natural rubber component [J]. Materials Science and Engineering A,2008,483/484:376-381.
[27]Mars W.V. and Fatemi A. Multiaxial fatigue of rubber: Part II:experimental observationsand life predictions [J]. Fatigue&fracture of engineering materials&structure,2005,28:523-538.
[28]Hideki O., Yuuta A., Hiroshi N., et al. Fatigue characteristics of vulcanized natural rubberfor automotive engine mounting (Characteristics of composition and mechanicalproperties)[J]. Memoirs of the Faculty of Engineering,2007,67(2):75-83.
[29]André N., Cailletaud G. and Piques R. Haigh diagram for fatigue crack initiationprediction of natural rubber components [J]. Kautsch Gummi Kunstst1999,52:120-123.
[30]Poisson J.L., Lacroix F., Meo S., and et al. Biaxial fatigue behavior of a polychloroprenerubber [J]. International Journal of fatigue,2011,33:1151-1157.
[31]Abraham F., Alshuth T. and Jerrams S. The effect of minimum stress and stress amplitudeon the fatigue life of non strain crystallising elastomers [J]. Materials and Design,2005,26:239-245.
[32] Luo R.K. and Wu W.X. Fatigue failure analysis of anti-vibration rubber spring [J].Engineering Failure Analysis,2006,13:110-116.
[33]Luo R.K., Mortel W.J. and Wu X.P. Fatigue failure investigation on anti-vibration springs[J]. Engineering Failure Analysis,2009,16:1366-1378.
[34]Saintier N., Cailletaud G. and Piques R. Multiaxial fatigue life prediction for a naturalrubber [J]. International Journal of Fatigue,2006,28(5-6):530-539.
[35]余伟炜,陈旭,闫磊.多轴循环载荷下橡胶材料的力学行为[J].橡胶工业,2007,54:500-503.
[36]Mars, W.V. Multiaxial fatigue of rubber [D]. USA: The University of Toledo,2001.08.
[37]Mars W.V., Cracking energy density as a predictor of fatigue life under multiaxialconditions [J]. Rubber Chemistry and Technology,2002,75:1-17.
[38]Kachanov L.M. Time of the rupture process under creep conditions. Izv. Akad. Nauk.S.S.R.Otd. Tech. Nauk.1958,8:26-31.
[39]Wang B., Lu H.B. and Kim G.H. A damage model for the fatigue life of elastomericmaterials [J]. Mechanics of materials,2002,34:475-483.
[40]Ali A., Hosseini M. and Sahari B. Continuum damage mechanics modeling for fatiguelife of elastomeric materials [J]. International Journal of Structural Integrity,2010,1(1):63-72.
[41]Ayoub G., Na t-abdelaziz M., Za ria F., et al. Multiaxial fatigue life prediction ofrubber-like materials using the continuum damage mechanics approach [J]. ProcediaEngineering,2010,2:985-993.
[42] Ayoub G., Na t-abdelaziz M., Za ria F., et al. A continuum damage model for thehigh-cycle fatigue life prediction of styrene-butadiene rubber under multiaxial loading[J]. International Journal of Solids and Structures,2011,48:2458-2466.
[43]Hocine N.A., Abdelaziza M.N. and Imad A. Fracture problems of rubbers: J-integralestimation based upon η factors and an investigation on the strain energy densitydistribution as a local criterion [J]. International Journal of Fracture,2002,117:1–23.
[44]Thomas A.G. Rupture of rubbers: II the strain concentration at an incision [J]. Journal ofPolymer Sciences,1955,18:177-188.
[45]Rice J.R. A path independent integral and the approximate analysis of strainconcentration by notches and cracks [J]. Journal of Applied Mechanics,1968,35:379–386.
[46]Busfield J.J.C., Davies C.K.L. and Thomas A.G. Aspects of fracture in rubbercomponents [J]. Progress in Rubber and Plastics Technology,1996,12(3):191-207.
[47]Abdelaziza M.N., Za ri F., Qua Z., et al. J integral as a fracture criterion of rubber-likematerials using the intrinsic defect concept [J]. Mechanics of Materials,2012,53:80-90.
[48]Hamdi A., Abdelaziza M.N., Hocine N.A., et al. A fracture criterion of rubber-likematerials under plane stress conditions [J]. Polymer Testing25(2006)994-1005
[49]Hamdi A., Abdelaziza M.N., Hocine N.A., et al. A new generalized fracture criterion ofelastomers under quasi-static plane stress loadings [J]. Polymer Testing,2007,26:896-902.
[50]Hamdi A., Hocine N.A., Abdelaziza M.N., et al. Fracture of elastomers under staticmixed mode: the strain-energy-density factor [J]. International Journal of Fracture,2007,144:65–75.
[51]Lake G.J. and Lindley P.B. The mechanical fatigue limit for rubber [J]. Journal of AppliedPolymer Science,1965,9:1233-1251.
[52]Busfield J.J.C., Thomas A.G., Ngah M.F. Application of fracture mechanics for thefatigue life prediction of carbon black filled elastomers [C]. Constitutive Models forRubber, Edited by A. Dorfmann and A. Muhr, pp.249-256,1999.
[53]Busfield J.J.C., Jha V., Liang H., et al. Prediction of fatigue crack growth using finiteelement analysis techniques applied to three-dimensional elastomeric components [J].Plastics, Rubbers and Composites,2005,34(8):349-356.
[54]Asare S. and Busfield J.J.C. Fatigue life prediction of bonded rubber components atelevated temperature [J]. Plastics, Rubber and Composites,2011,40(4):194-200.
[55]汪艳萍.橡胶材料多轴疲劳寿命及微观结构研究[博士学位论文].天津：天津大学,2007.12.
[56]姜莞.商用车动力总成悬置性能模拟与疲劳寿命预测研究[博士学位论文].长春:吉林大学.2011.06.
[57]高勋朝.填充橡胶材料的疲劳性能研究[硕士学位论文].哈尔滨：哈尔滨工业大学.2006.06.
[58]丽平.发动机橡胶减振垫的疲劳性能研究[硕士学位论文].上海：上海交通大学.2008.02.
[59]丁智平,陈吉平,宋传江,等.橡胶弹性减振元件疲劳裂纹扩展寿命分析.机械工程学报,2010,46(22):58-64.
[60]姜纪鑫,靳晓雄,殷闻.发动机橡胶悬置元件的疲劳寿命分析与预测.佳木斯大学学报(自然科学版),2011,29(5):673-676.
[61]刘建勋,黄友剑,刘柏兵,等.一种橡胶弹性元件疲劳寿命预测方法的研究.电力机车与城轨车辆,2011,34(3):12-14.
[62]丁家松,张欢,童宗鹏,等.船用橡胶隔振器寿命评估方法研究.振动与冲击,2010,29(12):230-233.
[63]André N., Gailletaud G. and Piques R. Haigh diagram for fatigue crack initiationprediction of natural rubber components [J]. Kautschuk Gummi Kunstoffe,1999,52:120-123.
[64]Zarrin-Ghalami T. and Fatemi A. Cumulative fatigue damage and life prediction ofelastomeric components. Fatigue&Fracture of Engineering Materials&Structures,2013,36:270-279.
[65]Mars W.V. Fatigue life prediction for elastomeric structures [J]. Rubber Chemistry andTechnology,2007,80(3):481-503.
[66]ASTM Standard D4482-11. Standard test method for rubber property-extension cyclingfatigue, ASTM, West Conshohocken, PA.
[67]Mars W.V. and Fatemi A. A novel specimen for investigating the mechanical behavior ofelastomers under multiaxial loading conditions [J]. Experimental Mechanics,2004,44(2):136-146.
[68]Tao G. and Xia Z. A non-contact real-time strain measurement and control system formultiaxial cyclic/fatigue tests of polymer materials by digital image correlation method[J]. Polymer Testing,2005,24:844-855.
[69]Le Cam J.-B., Huneau B. and Verron E. Description of fatigue damage in carbon blackfilled natural rubber [J]. Fatigue&Fracture of Engineering Materials&Structures,2008,31:1031-1038.
[70]Andriyana A. and Verron E. Prediction of fatigue life improvement in natural rubberusing configurational stress [J]. International Journal of Solids and Structures,2007,44:2079-2092.
[71]陆明万,罗学富.弹性理论基础[M].北京:清华大学出版社,施普林格出版社,2001.
[72]徐秉业，黄炎，刘信声，等.弹塑性力学及其应用[M].北京：机械工业出版社,1984.
[73]McGarry M.D.J., Van Houten E.E.W., Perrinez P.R., et al. An octahedral shearstrain-based measure of SNR for3D MR elastography [J]. Physics in medicine andbiology,2011,56:153-164.
[74]Shangguan W.B., Wang X.L., Deng J.X., et al. Experiment and modeling of uniaxialtension fatigue performances for filled natural rubbers [J]. Materials and Design,2014,58:65-73.
[75]Dizon E.S., Hicks A.E. and Chirico V.E. The effect of carbon black parameters on thefatigue life of filled rubber compounds [J], Rubber Chemistry and Technology,1974,47:231-249.
[76]OriginLab Corporation. Origin8user guide [M].USA: OriginLab Corporation FirstEdition,2007.
[77]Sors L. Fatigue design of machine components [M]. Oxford: Pergamon Press,1971.
[78]Bannantine J.A., Comer J.J. and Handrock J.L. Fundamentals of metal fatigue analysis[M]. Englewood: Prentice Hall,1992.
[79]王小莉，上官文斌，刘泰凯，等.填充橡胶材料单轴拉伸疲劳试验及疲劳寿命模型研究[J].机械工程学报，2013，49(14)：65-73.
[80]Holzapfel G.A. Nonlinear solid mechanics. A continuum approach for engineering [M].Chichester：Wiley，2000.
[81]Kujawski D, Ellyin F. A unified approach to mean stress effect on fatigue thresholdconditions [J]. International Journal Fatigue,1995,17(2):101-106.
[82]Mars W.V. and Fatemi A. A phenomenological model for the effect of R Ratio on fatigueof strain crystallizing rubbers [J]. Rubber Chemistry and Technology,2003,76：1241-1258.
[83]Harbour J, Fatemi A and Mars W V. Fatigue crack growth of filled rubber under constantand variable amplitude loading conditions [J]. Fatigue&Fracture of EngineeringMaterials&Structures,2007,30:640-652.
[84]Mars W.V. and Fatemi A. Observations of the constitutive response and characterizationof filled natural rubber under monotonic and cyclic multiaxial stress states [J]. ASMEJournal of Engineering Materials and Technology,2004,126：19-28.
[85]Zarrin-Ghalami T. and Fatemi A. Material Deformation and Fatigue BehaviorCharacterization for Elastomeric Component Life Predictions [J]. Polym. Eng. Sci.,2012,52:1795-1805.
[86] Tao G. and Xia Z.H. Mean stress/strain effect on fatigue behavior of an epoxy resin [J].International Journal of Fatigue,2007,29：2180-2190.
[87]Wang X.L., Shangguan W. B., Rakhaja S., et al. A method to develop a unified fatiguelife prediction model for filled natural rubbers under uniaxial loads. Fatigue&Fractureof Engineering Materials&Structures,2014,37(1):50-61.
[88]上官文斌，王小莉，叶必军，等.应变比对填充天然橡胶疲劳特性影响的试验及其寿命预测方法研究[J].机械工程学报，2013，49(8)：49-56.
[89]Yeoh O. H．Analysis of deformation and fracture of pure shear rubber test piece[J]．Plastics，Rubber and Composites，2001，30(8)：389-397．
[90]贾法勇，霍立兴，张玉凤，等.疲劳裂纹扩展速率两种数据处理方法的比较[J].机械强度，2003，25(5)：568-571.
[91]孟宪红，张行.确定疲劳裂纹扩展速率的积分法[J].机械强度，2001，23(2)：213-215.
[92]American Society of Mechanical Engineers. ASME E647–08Standard test method formeasurement of fatigue crack growth rates [S].West Conshohocken：ASME，2008.
[93]郦正能.应用断裂力学[M].北京：北京航空航天大学出版社，2012.
[94]Kaang S.Y., Jin Y.W., Huh Y.I., et al. A test method to measure fatigue crack growth rateof rubbery materials [J]. Polymer Testing,2006,25:347-352.
[95]Thomas A.G. Rupture of rubber. V. Cut growth in natural rubber vulcanizates [J]. Journalof Polymer Science,1958,31:467-480.
[96]Lindley P.B. Relation between hysteresis and the dynamic crack growth resistance ofnatural rubber [J]. International Journal of Fracture，1973，9(4)：449-462.
[97]Lake G.J. Fatigue and fracture of elastomers [J]. Rubber Chemistry and Technology，1995，68(3)：435-459.
[98]Miller K. Testing elastomers for hyperelastic material models in finite element analysis[EB/OL].2004[2013-8]. http://www.axelproducts.com/pages/downloads.html.
[99]雷英杰，张善文，李续武，等. MATLAB遗传算法工具箱及应用[M].西安：西安电子科技大学出版社，2005.
[100] Mars, W.V. and Fatemi A. Analysis of fatigue life under complex loading: RevisitingCadwell, Merrill, Sloman, and Yost [J]. Rubber Chemistry and Technology,2006,79(4)：589-601.
[101] Greensmith H.W. Rupture of rubber. X. The change in stored energy on making asmall cut in a test piece held in simple extension [J]. Journal of Applied PolymerScience，1963，7：993-1002.
[102] At-Bachir M.，Mars W.V. and Verron E. Energy release rate of small cracks inhyperelastic materials [J]. International Journal of Non-Linear Mechanics，2012，47：22-29.
[103] ABAQUS6.10. Theory Manual,2010.
[104]上官文斌，吕振华.汽车动力总成橡胶隔振器弹性弹性的有限元分析[J].内燃机工程，2003，24(6)：50-55.
[105]王文涛，上官文斌，段小成，等.基于线性疲劳累计损伤橡胶悬置疲劳寿命预测研究[J].机械工程学报，2012，48(10)：56-65.
[106]谢新星.橡胶隔振器单轴疲劳寿命评价方法及预测研究[D].广州：华南理工大学，2011.
[107]王文涛.橡胶隔振器单轴疲劳特性试验与预测方法的研究[D].广州：华南理工大学，2012.
[108] Asare S. and Busfield J. J. C. Fatigue life prediction of bonded rubber components atelevated temperature [J]. Plastics, Rubber and Composites，2011，40(4)：194-200.