平均应变效应 橡胶疲劳和金属疲劳的区别第1部分 ➥ 橡胶和金属的力学行为有非常大的差异，我们首先可以从平均应变或应力对材料疲劳性能影响的角度来分析这种差异。Rubber and metal are very different materials that exhibit very different behaviors. Consider the effect of mean strain or stress on the fatigue performance of these materials. 图1显示了几个典型的等幅应变循环，每个循环都处于不同的平均应变水平。在循环疲劳试验中，如果施加的应力幅度等于平均应力，我们把这种情况称为脉冲载荷循环或全松弛载荷循环。如果平均应力为零，我们把这种情况称为完全反转的拉伸/压缩加载循环。如果最小应力总是正的，则称为非全松弛载荷循环（即试样总是处于加载状态）。非全松弛载荷循环在应用中很常见，例如：在安装过程中对产品施加了预载荷；衬套在模压过程中产生的压缩预应力、过盈配合、由于热膨胀/收缩而产生的内应力；以及在轮胎中，帘线的形状记忆效应。Figure 1 illustrates a few typical constant amplitude strain cycles, each at a different level of mean strain. If the stress amplitude is equal to the mean stress, we say that we have pulsating tension or fully relaxing tension. If the mean stress is zero, we say that we have fully reversed tension/compression. If the minimum stress is always positive, then we have nonrelaxing tension (i.e. always under load). Nonrelaxing cycles are quite common in applications. Examples include: pre-loads applied during installation; swaging of a bushing to induce compressive pre-stresses, interference fits, self-stresses occurring due to thermal expansion/contraction; and in tires, shape-memory effects of textile cords. 图1. 在三种不同的平均应变下的恒定振幅加载循环Figure 1. Constant amplitude cycles at three different mean strains. 在金属疲劳分析方法中，通常以应力幅度σa和平均应力σm相对于屈服应力σy和极限应力σu的大小来定义应力均值效应对金属材料疲劳行为的影响。如图2所示。当加载应力处于疲劳阈值应力σ0以下时，材料具有无限寿命。Haigh图（或Goodman图）（图2左）将疲劳寿命绘制为这些变量的函数[1]。Wohler曲线（图2右）提供了类似的信息。对于金属材料，有一个普遍适用的简单规则：增加平均应变将降低疲劳寿命。通常还假设金属的潜在疲劳开裂面垂直于最大主应力方向。In metal fatigue an alysis, it is customary to define the effect in terms of stress amplitude σa and mean stress σm, relative to the yield stress σy and the ultimate stress σu, as shown in Figure 2. Below the fatigue threshold stress σ0, indefinite life is predicted. The Haigh (or Goodman) diagram (left) maps fatigue life as a function of these parameters [1]. Wohler curves (right) provide similar information. For metals, a simple rule may be applied universally: increasing mean strain is detrimental fatigue life. It is also commonly assumed for metals that the critical plane is perpendicular to maximum principal stress direction.图2. 显示平均应变对金属疲劳寿命影响的Haigh图（左）和Wohler曲线（右）Figure 2. Haigh diagram (left) and Wohler curves (right) showing mean strain effects on fatigue life for a metal. 橡胶材料与金属材料有许多不同之处 There are many ways that rubber materials differ from metallic materials. 01 在原子尺度上 At the atomic scale橡胶由长链分子组成，这些分子经历恒定的热运动，同时以永久的网络拓扑结构相互连接。这种结构允许发生大的弹性/可逆应变。而金属则完全不同，它们以单个原子的形式存在于有序的晶体中，偶尔会出现位错或晶格空位。在非弹性变形发生之前，这种结构仅允许极小的应变。Rubber is composed of long chain molecules experiencing constant thermal motion while interlinked with a permanent network topology. This structure permits large, elastic/reversible straining to occur. Metals could not be more different, existing as individual atoms packed into well-ordered crystals with occasional dislocations or lattice vacancies. This structure permits only vanishingly s mall strains before inelastic deformation occurs.02 在介观尺度上 At the meso scale橡胶通常是一种含有填料（如炭黑、二氧化硅或粘土）以及其它化学成分的复合材料。而金属的介观尺度通常用晶粒边界和夹杂物或空隙来描述。橡胶表现出许多在金属中看不到的“特殊效应”，例如：速率和温度依赖性、老化特性、循环软化特性。基于这些因素，橡胶的分析方法与金属的分析方法有很大不同，这并不奇怪。Rubber is typically a composite material containing fillers such as carbon black, silica or clay, as well as other chemical agents. The mesoscale of a metal is generally described in terms of crystalline grain boundaries and inclusions or voids. Rubber exhibits many “special effects” that are not seen in metals: rate and temperature dependence, ageing, cyclic softening. It is unsurprising that an alysis methods for rubber differ substantially from those applied for metals. 橡胶的疲劳性能与平均应变的关系更为复杂。对于无定形（即非结晶）橡胶，与金属材料一样，增加平均应变会降低疲劳寿命。但是，对于表现出应变诱导结晶的橡胶，平均应变的存在可能会大大增加疲劳寿命，如图3所示。因此，橡胶的疲劳仿真必须考虑材料的应变结晶效应。Rubber’s fatigue performance has a more complex dependence on mean strain. For amorphous (ie non-crystallizing) rubbers, increasing mean strain reduces the fatigue life, as with metals. But for rubbers that exhibit strain-induced crystallization, mean strain can greatly increase fatigue life, as illustrated in Figure 3. Fatigue simulations therefore must take account of the strain crystallization effect.图3. 在恒定振幅下进行的简单拉伸疲劳试验表明，增加载荷均值后，有应变结晶的天然橡胶（NR）的寿命显著增加，而无定形的丁苯橡胶（SBR）的寿命减少[2]。Figure 3. Fatigue tests run in simple tension under constant amplitude show a significant increase in life for Natural Rubber (NR), which strain crystallizes, and a decrease of life for Styrene Butadiene Rubber (SBR) which is amorphous [2]. 在Endurica疲劳仿真分析软件中，基于橡胶材料的断裂力学行为，通过定义等效全松弛撕裂能Teq来描述材料的平均应变效应。当全松弛载荷对应的撕裂能产生与非松弛条件下撕裂能相同的裂纹扩展速率时，则称此时的全松弛撕裂能为等效撕裂能。Mean strain effects are specified in the Endurica fatigue code in terms of fracture mechanical behavior, using the concept of an equivalent fully relaxing tearing energy Teq. The tearing energy for fully relaxing conditions is said to be equivalent when it produces the same rate of crack growth as the nonrelaxing condition. 对于无定形橡胶，等效的全松弛撕裂能Teq等于撕裂能幅值ΔT，ΔT=Tmax-Tmin，也可以用R=Tmin/Tmax来描述平均应变效应。将该规则代入到幂律裂纹扩展速率函数中，即可得到众所周知的Paris模型，由该模型可以预测出，当平均应变增加时，材料的裂纹扩展速率会加快。For amorphous rubbers, the equivalent R=0 tearing energy Teq is simply the range ΔT of the tearing energy cycle, which can be expressed in terms of the min and max tearing energies Tmin and Tmax, or in terms of R= Tmin / Tmax. Plugging this rule into the power law crack growth rate function yields the well known Paris law, which predicts faster crack growth for increasing mean strain.对于应变结晶橡胶，可以使用Mars-Fatemi模型来定义等效全松弛撕裂能。在这种情况下，等效全松弛撕裂能取决于函数F(R)，该函数通过定义裂纹扩展速率模型的幂律斜率，来描述应变结晶效应对材料疲劳的影响。无定形橡胶和结晶橡胶的关系总结在表1中[3，4]。For a strain crystallizing rubber, the equivalent fully relaxing tearing energy can be specified using the Mars-Fatemi law. In this case, the equivalent fully relaxing tearingenergy depends on a function F(R), which specifies the crystallization effect in terms of its influence on the powerlaw slope of the crack growth rate law. The relationship for amorphous and crystallizing rubbers are summarized in Table 1 [3,4].表1. 计算无定形和应变结晶橡胶中裂纹扩展速率的模型Table 1. Models for computing crack growth rate in amorphous and strain-crystallizing rubbers. 橡胶的疲劳特性可以绘制在Haigh图中，但橡胶的等寿命曲线与金属有很大不同。在金属疲劳分析中，假设裂纹总是垂直于最大主应力方向扩展，而这对于橡胶来说并不总是正确的，特别是在涉及应变结晶和非松弛载荷的情况下。因此，对于橡胶疲劳分析，需要使用临界平面分析方法[5]，通过计算材料单元在多个潜在疲劳开裂面上的疲劳寿命，找出其中具有最短寿命的裂纹平面，将其确定为最危险的开裂面。Rubber’s fatigue behavior may be plotted in a Haigh diagram, but the contours can be quite different than for metals. In metal fatigue an alysis, it is assumed that cracks always develop perpendicular to the max principal stress direction. This is not always true for rubber, especially in cases involving strain crystallization and nonrelaxing loads. For rubber fatigue an alysis it is therefore required to use critical plane a nalysis [5], in which fatigue life is computed for many potential crack orientations, and in which the crack plane with the shortest life is identified as the most critical plane. 图4显示了疲劳寿命和临界平面方向对应变幅度和平均应变的依赖性。为每对应变振幅和平均应变坐标绘制一个球体，其上的颜色表示疲劳寿命，单位法向量表示临界平面方向。可以看出，平均应变和应变幅值的不同组合可以产生一定范围的裂纹平面取向。Figure 4 shows the dependence of the fatigue life and the critical plane orientation on strain amplitude and mean strain. A sphere is plotted for each pair of strain amplitude and mean strain coordinates, on which the colors represent fatigue life, and unit normal vectors indicate critical plane orientations. It can be seen that different combinations of mean strain and strain amplitude can produce a range of crack plane orientations.图4. 临界平面分析包括整合每个可能裂纹方向的裂纹扩展速率定律，并确定产生最短寿命的方向（图4左）。Haigh图（图4右）中的每个点都与其自身的临界平面方向相关联。Figure 4. Critical plane an alysis consists in integrating the crack growth rate law for every possible crack orientation, and identifying the orientation that produces the shortest life (left). Each point in the Haigh diagram (right) is associated with its own critical plane orientation. 天然橡胶（NR）和丁苯橡胶（SBR）的Haigh图如图5所示。在这些图像中，红色表示疲劳寿命短，蓝色表示疲劳寿命长。对于天然橡胶（图5左），Haigh图的长寿命区域呈现出显著的圆顶状形状，表明在应变诱导结晶的影响下平均应变对提高寿命的有益效果。相反，SBR的疲劳寿命总是随着平均应变的增加而降低。即使如此，SBR的Haigh图具有与材料的超弹性相关的非线性特征，这也不同于金属。The Haigh diagrams for natural rubber (NR) and for styrene butadiene rubber (SBR) are shown in Figure 5. In these images, red represents short fatigue life, and blue long life. For natural rubber (on the left), the long-life region of the Haigh diagram exhibits a notable dome-like shape, indicative of a beneficial effect of mean strain under the influence of strain-induced crystallization. In contrast, SBR always exhibits decreased fatigue life as mean strain increases. Even so, the Haigh diagram for SBR has a nonlinear character associated with the material’s hyperelasticity that is also distinct from a metal.图5. 为NR（左）和SBR（右）橡胶计算的图表。Figure 5. Haigh diagrams computed for NR (left) and for SBR (right) rubbers.需要注意的是，橡胶中的应变结晶效应与温度相关。在较冷的温度下，这种影响较强，而在较高温度下，这种影响较弱。图6比较了三种不同温度下结晶橡胶的实验Haigh图[6]（顶部）和计算结果（底部）。It should be noted that the strain crystallization effect in rubber depends on temperature. At colder temperatures, the effect is stronger, and at higher temperatures it is weaker. Figure 6 compares experimental Haigh diagrams [6] (top) for a crystallizing rubber to computed results (bottom) for three temperatures. 图6. 天然橡胶在3个温度下的实验Haigh图【6】（顶部），与计算的Haigh图（底部）进行比较。温度升高往往会降低应变结晶的有益效果。Figure 6. Experimental Haigh diagram [6] for natural rubber at 3 temperatures (top), compared to computed Haigh diagram (bottom). Increasing temperature tends to reduce the beneficial effect of strain crystallization. 总之，拉伸平均应力对于金属材料总是有害的，但对于橡胶材料中，它可能是有益的，也可能是有害的，这取决于橡胶是否具有应变结晶效应。给橡胶施加平均应力的好处可能十分明显——有时会让疲劳寿命提高几个数量级。尤其是在较低的环境温度下，对寿命的增强效果更强，而在较高的温度下，这种有益效果会减弱。临界平面分析对于准确预测橡胶中的应变结晶效应至关重要。In summary, while tensile mean stresses are always detrimental in metals, in rubber they may be either beneficial or harmful, depending on whether the rubber can strain crystallize. The benefits of mean stresses in rubber can be quite strong – sometimes amounting to more than several orders of magnitude. The beneficial effect is stronger at colder temperatures and is reduced at higher temperatures. Critical Plane An alysis is essential for accurately predicting the effects of strain crystallization in rubber. 如果将金属疲劳分析的Wohler曲线（S-N曲线）理论用于橡胶的疲劳分析，会错误地认为材料断裂平面总是垂直于最大主应力方向。对于受到非松弛加载作用的应变结晶橡胶而言，这一结论是不对的。Wohler curves, commonly used for metal fatigue an alysis, incorrectly assume that the worst-case plane is always normal to the max principal stress direction. This is not an accurate approach for strain crystallizing rubber under mean strain. 推荐使用Endurica疲劳求解器来准确地分析疲劳耐久性问题！Use the Endurica fatigue solvers to accurately capture these effects when its important to get durability right!References[1] Stephens, R. I., Fatemi, A., Stephens, R. R., & Fuchs, H. O. (2000). Metal fatigue in engineering. John Wiley & Sons.[2] Ramachandran, Anantharaman, Ross P. Wietharn, Sunil I. Mathew, W. V. Mars, and M. A.Bauman.(2017) “Critical plane selection under nonrelaxing simple tension with strain crystallization.” In Fall 192nd technical meeting of the ACS Rubber Division, pp. 10-12.[3]Mars,W. V. (2009). Computed dependence of rubber’s fatigue behavior on strain crystallization.Rubber Chemistry and Technology, 82(1), 51-61.[4] Harbour, Ryan J., Ali Fatemi, and Will V. Mars. “Fatigue crack growth of filled rubber under constant and variable amplitude loading conditions.” Fatigue & Fracture of Engineering Materials & Structures 30, no. 7 (2007): 640-652.[5] Mars, W. V. (2021). Critical Plane An alysis of Rubber. Fatigue Crack Growth in Rubber Materials: Experiments and Modelling, 85-107.[6]Ruellan, Benoît, J-B. Le Cam, I. Jeanneau, F. Canévet, F. Mortier,and Eric Robin. “Fatigue of natural rubber under different temperatures.” International Journal of Fatigue 124 (2019): 544-557. 来源：Endurica
