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铝合金随机荷载作用下的疲劳性能确定疲劳寿命的 2024-T3 和铝合金 6061-T6是由作者G.M.Brown和R.Ikegami进行实验调查摘要-本文介绍了实验的进行，以确定两个铝合金材料的疲劳寿命 (2024-T3 及 6061-T6)。它们受到这两个等应变幅正弦和窄带随机应变幅疲劳载荷。从窄带随机测试获得的疲劳寿命值基于迈纳的线性积累的损伤假说的理论预测进行比较。铝合金材料制作的悬臂梁测试标本通过电磁振动器励磁的方式受到等应变幅正弦或窄带随机基础激励。发现两种合金的 e N 曲线可以非常接近三直线段的低、 中和高周疲劳寿命范围内。迈纳的假说用于此类型的预测材料的窄带随机疲劳周期-N 的性能。这些疲劳寿命预测发现了一贯高估实际疲劳值的 2 或 3 倍。但是，发现预测的疲劳寿命曲线的形状和高周疲劳性能的两种材料都与实验的结果相吻合。简介 受到随机载荷的金属结构的疲劳寿命预测的问题，一般是先制定一个损害标准的积累，然后应用此标准规定条件的不同循环载荷幅值解决。第一，仍然预测，在疲劳损伤积累的最常用的标准是由A.Palmgr-N提出并由M.A.迈纳申请。这一标准假定积累损害的问题可被视为一个其中疲劳寿命的分数从振幅 -N 曲线确定不同的负载级别用完可能只需添加到提供一个索引的疲劳损伤，一般称为迈纳的线性-积累-损失的条件。 本文介绍了部分研究结果，以确定受到窄带随机载荷-铝合金结构疲劳寿命预测方法进行随机载荷。实验方案进行了确定悬臂束测试标本，必须等应变振幅的正弦或窄带随机应变振幅疲劳载荷的周期。试样在随机载荷作用下的疲劳寿命与基于预测的迈纳标准应用进行对比。实验方案 两种常用的铝合金2024-T3和6061-T6进行疲劳试验。这两种合金的力学性能列于表1，真应力和真应变曲线图1和图2。tupian应变硬化指数，表现了塑性范围中的应力应变关系。Gongshi 这些属性是根据确定的单轴拉伸试验用两种合金制成的张力试样。在这两种情况下，张力标本和疲劳标本是从加工平行的滚动方向的标本的纵向轴与铝、 同一负债表。这样做是为了确保测试这两种类型之间的一致性。做疲劳试验的大多数是对电磁振动器励磁，与正在执行 Instron 测试仪的低周期等应变幅测试的一小部分。为进行测试，已在振动励磁机上执行，疲劳试验模型是基础的激励作用下的悬臂梁显示于图 3 (a)。 窄带随机疲劳测试中，激励指定的频率带宽较窄带信号与高斯基地加速振幅和均匀频谱。激发带为中心的根本束共振。正弦疲劳测试是略高于基本梁共振励磁频率进行的。因此循环加载的类型被完全扭转弯曲。悬臂标本异形沿梁长度最大弯曲应力，远离固定端将在第一振动模式。草绘的测试样本配置如图 4 所示。试验样品被这直接相连的电枢的振动励磁机安装夹具的夹在中间。从图中可以看出，每个疲劳试验标本所载两悬臂梁标本的同时都感到兴奋。结束质量、 形式的恩德夫科公司模型 2216年水晶加速度计，被附加在一端的悬臂梁标本。在图所示的模式形状和相应的弯曲应力分布的前两个振动模式。3（b)和图3（c）。从梁共振，最大弯曲应力发生在7/8英寸，从梁的固定端的距离。第一束共振的频率是大约 115 cps。标本被仔细手抛光之前，删除任何尖角，消除所有可见表面的划痕，该地区的最大应力测试发生。要测量应变级别疲劳测试期间，应变片被装在每个标本在发生弯曲应力的最大值的位置点。发现疲劳寿命应变仪安装的是一般的试件疲劳寿命比小得多。为此，从安装在梁的自由端的加速度计的信号用于确定疲劳标本的故障时间。从加速度计的信号用于触发的停用一个计时器时加速级别降至 50%的名义 RMS 加速级别的继电器。据指出出现故障，加速级别删除非常迅速，这样，计时器表示非常密切的标本的失效的总时间。图 5 中的图片显示装载的振动励磁机只是之前，测试上的疲劳试验标本。从图中可以看出，由拼接两个更小、 更灵活，铅电线电缆标准来说加速度计建造特别加速度传感器电缆。这样做是为了尽量减少对标本的加速度计电缆振动的影响。虽然接头增加噪音污染，信号水平是如此之大这种增加噪音不是明显的。来自应变计和加速度计的信号是在测试期间监测，并记录在磁带上。每个测试之后, 录制的信号被播放到波分析仪系统以确定 RMS 级别。数字计算机被用于执行时间序列从窄带随机疲劳测试获得的随机信号分析。典型的窄带随机测试期间的应变响应的应变规范扫雷具密度情节如图 6 所示。如所料，这张图指示可以认为疲劳标本是一个很轻阻尼的单自由度体系。最可能的这种窄带反应的振动频率可以显示系统的共振频率。因此，周期失效 （即零与正斜率的过路处的总人数） 的总数被假定为失效的总时间，以秒为单位乘以每秒周期的共振频率。在等幅、 正弦测试周期失效的总数是只是总时间失效以秒为单位乘以励磁频率每秒周期。对于振动激振器上进行的测试，失效的总周期的范围是从2) 10 5 10 周期。相应的应变水平从600到3500不等显微窄带随机疲劳试验RMS，从1700年到7000微应变恒定振幅测试。由于循环对振动激励器的快速率,它无法可能获得的生命值低于约 210疲劳数据周期内，使用上面描述的测试设置。为此，英斯特龙测试仪进行了一些低周期、 等应变幅疲劳测试。迈纳的标准的低周期的了解中，恒定振幅两种材料的疲劳性能必要关联恒定振幅和窄带随机疲劳的结果。（图表）为瑞利分布，RMS值的3.72倍，超过峰值的概率是0.1%。可以预期RMS的应变窄带随机试验以来的最高水平是3500微应变，峰值应变水平约14,000微应变。振动激振器能够产生恒定振幅只有7000微应变峰值应变，因此，其他手段都必须以产生一个连接图高达14000微应变迈纳的故障预测所需的水平。英斯特郎测试率约5 cpm在编程周期与应变振幅高达40000微应变是用来获取低周期数据。振动励磁机测试中使用标本再次加载不完全弯曲基本上相同。 图7中提供登记表疲劳测试的测试设置的照片。试样标本振动激振器安装时举行以同样的方式举行试样安装夹具安装在横梁英斯特郎测试。标本的一角举行了两个队的成员连接的尖端试验机的框架固定。基地或横梁，然后通过不断地偏转循环。从登记表测试仪计数器确定失效的周期数。振动励磁机上执行的疲劳测试的结果如图8和图9所示，RMS应变级别已被绘制与周期来的悬臂梁标本失效的总数。（图表）等幅和窄带随机结果已在上绘制在同一图表强调疲劳寿命的差异获得两个不同类型的加载同一RMS应变级别。 可以注意到这两种材料，当等应变幅正弦和窄带随机应变变幅疲劳荷载，展示耐力极限现象在很低的应变级别。大多数的恒定振幅的数据似乎在RMS应变振幅的周期总数的积失效记录档记录的直线下降。6061-T6在高应变水平的数据表明有一些偏差。这些点对应的应变以及成塑性的地区是可以预料的。相信2024-T3的材料会高于获得能够与现有的振动励磁测试设置显示在较高的应变水平的同一类的偏差。窄带随机疲劳试验结果表明：双对数坐标上的同一个斜坡，这是为相应的恒定振动振幅相同的线性关系数据RMS应变水平较低。这些数据，然后开始偏离较多的应变峰值预计将超过屈服应变在较高的应变水平。英斯特朗测试仪上执行的固定应变振幅疲劳试验的结果显示如图10和图11所示，峰值应变幅度已与绘制的周期总数的悬臂梁试样的失效。斯特朗测试数据绘制恒定振幅与振动励磁测试结果。由于材料表现出低频效果一般显示较低的疲劳寿命值作为循环频率降低，他认为在循环频率的差异并没有很大影响的两个试验材料的疲劳性能，因此两套数据可以直接比较。但一般认为，几乎没有显着影响时，工作频率高达CPS约1000。如图10和图11所显示的拉伸试验的结果可以看出经历过大的塑性应变时的应力幅与失效的周期数积将是没有多大用处。由于塑性的范围类一个大的增量应力应变关系的性质提高应变对应一个非常小的压力逐步增加。因此在本文中无论是RMS应变振幅或者振幅的峰值应变绘制对比周期数。符号是峰值应变，而不是“塑性变形”。如图10和图11显示，低循环疲劳数据似乎在一条直线上的峰值应变振幅与失效的周期数的log-log图。在这两种情况下，这条线的斜率是远远高于高周期数据通过及格线的斜率更大。这些日志记录图上的直线关系表示，为常数，应变振幅循环，失效和峰值应变振幅周期总数之间的近似关系可以在指数形式获得。在高周期范围，这种关系可以表示为 在低周期范围，这种关系可以表示为 k1和k2是常数。N是在峰值应变振幅和 周期失效下b1和b2两种材料的值在图10和图11和表2中。这种斜率在低周期的变化，高应变振幅地区是在与许多其他试验者的意见一致。L.F.Coffin观察，许多材料，如果绘制周期数与失效的塑性应变振幅，b1是约等于2。使用的固定应变振幅疲劳试验的结果，窄带随机疲劳试验的结果可以与迈纳的线性积累损伤的假设为基础的预测。迈纳的线性累积损伤准则正如前面所讨论的，迈纳的故障预测假设失败是由于一个损坏的线性积累，可以确定其外的恒定振幅测试结果。（表12）测试材料10和11完整的恒定振幅的结果，包括低收入和高循环疲劳寿命范围内，已在图绘制。这两种材料的-N曲线从这些数字可以看出近似由三个直线段时，日志记录的规模绘制，可以表明这三个范围的原理如图12所示。在范围2内，它会假设，-N图是一个直线上的log-log图与1/b2负斜率从零到N1周期。在范围1内，它会假设从N直线延迟300到N周期具有负斜率的1/b1。在水平对应的耐力应变。认为它低于这个水平的应变没有最终的疲劳失效的影响。因此-N曲线可以表示为和 在这种情况下，线性积累的损伤在峰值应变方程可以写成振幅Pp(p)的概率密度函数的峰值应变振幅，T表示要失效的总时间和Mo是正峰值每秒数。因此，MoT代表正峰失效总数。由于应变响应窄带过程中，峰值应变激发加强高斯分布的概率密度函数是一个雷射线分布，可表示为利用（3），（4）和（5）累计可得公式假定发生时，损伤的积累是同等统一，使正逢失效总数最终失效。这个方程的解为CDC6400数字电脑编程。在迈纳的两个材料进行预测计算中使用的参数列于表2。在图8和图9虚线表示对迈纳假设为基础的预测。正如从这些数字可以看出，虽然按照数据的趋势形式出现，迈纳的预测始终高估疲劳寿命的2到3倍。结论共振电磁激振器振动疲劳试验似乎是一个非常快速，经济的方式获得固定应变振幅的正弦和随机应变振幅疲劳载荷数据。然而，由于高循环率，低周疲劳的数据就无法获得这种方式了。测试的铝合金，似乎有三个不同的疲劳范围对应三种不同的线路段须符合的-N数据绘制在一个日志记录的规模。窄带随机载荷条件下地疲劳寿命预测适用于迈纳标准，-N的数据必须是完整的了至少四次RMS应变水平的随机载荷的应变水平。对迈纳标准为基础的预测似乎始终高估了实际的疲劳寿命，然而预测曲线的形式似乎是正确的，并按照数据的趋势非常接近。参考文献 1. Miner.M.A横贯“疲劳，累积损伤”。对ASME.Jnl.APPL.应用力学,87（2），159（1945）。2. Brown.G.W和Ikegami研究“铝合金疲劳受随机载荷，”众议员MD-69-1号，加州工程学院，加州大学伯克利分校（1969年1月）。3. Harris.W.J.Jr.“金属疲劳”。专着航空航天系列，Pergamon出版社，纽约（1961年）。4.Kennedy.A.J研究金属徐变和疲劳，“奥利弗”过程，英国爱丁堡手臂博伊德有限公司（1962）。5. Coffin.L.F.Jr“低循环疲劳：回顾与展望”，APPL.Matter.Res，1（3），129（1962年10月）。6. Bendat.J.S“原则和随机噪音理论的应用”，纽约约翰.威利父子公司（1958年）。7. Robson.J.D“随机振动概论”英国爱丁堡,爱丁堡大学出版社(1964年）。The Fatigue of Aluminum Alloys Subjected to Random Loading Experimental investigation is undertaken by the authors to determine the fatigue life of 2024-T3 and 6061-T6 aluminum alloys G. W. Brown and R. Ikegami ABSTRACT-This paper describes an experimental investi- gation which was carried out to determine the fatigue life of two aluminum alloys (2024-T3 and 6061-T6). They were subjected to both constant-strain-amplitude sinus- oidal and narrow-band random-strain-amplitude fatigue loadings. The fatigue-life values obtained from the nar- row-band random testing were compared with theoretical predictions based on Miners linear accumulation of dam- age hypothesis. Cantilever-beam-test specimens fabricated from the aluminum alloys were subjected to either a constant- strain-amplitude sinusoidal or a narrow-band random base excitation by means of an electromagnetic vibrations exciter. It was found that the e-N curves for both alloys could be approximated by three straight-line segments in the low-, intermediate- and high-cycle fatigue-life ranges. Miners hypothesis was used to predict the narrow-band random fatigue lives of materials with this type of -N behavior. These fatigue-life predictions were found to consistently overestimate the actual fatigue lives by a factor of 2 or 3. However, the shape of the predicted fatigue-life curves and the high-cycle fatigue behavior of both materials were found to be in good agreement with the experimental results. Symbols bl, b = constants kl, k2 = constants N = number of cycles to failure S = stress S = yield stress (1,N1) = reference point on e-N diagram e = strain e = yield strain ep = peak strain e = endurance-limit strain = strain-hardening exponent G. W. Brown is Professor of Mechanical Engineering, University of Cali- fornia, Berkeley, Calif. R. Ikegami is Research Engineer, Structural Dynamics Group, Boeing Aircraft, Seattle, Wash. Paper was presented at 1970 SESA Spring Meeting held in Huntsville, Ala. on May 19-22. r = strain variance Symbols not shown here are defined in the text. Introduction The problem of predicting the fatigue lives of metal structures which are subjected to random loadings is generally solved by first formulating an accumula- tion of damage criteria, then applying this criteria to the specified conditions of varying cyclic-load amplitude. The first and still the most commonly used criterion for predicting the accumulation of damage in fatigue was proposed by A. Palmgren and applied by M. A. Miner. 1 This criterion as- sumes that the problem of accumulation of damage may be treated as one in which the fractions of fa- tigue life used up at different load levels as deter- mined from the constant amplitude e-N curve may be simply added to give an index of the fatigue damage and is generally known as Miners linear- accumulation-of-damage criteria. This paper describes a portion of the results of a study 2 which was conducted to determine a method of predicting the fatigue lives of alumimum-alloy structures which were subjected to narrow-band random loadings. An experimental program was carried out to determine the lives of cantilever-beam test specimens which were subjected to either con- stant-strain-amplitude sinusoidal or narrow-band random strain-amplitude fatigue loadings. The fatigue lives of the test specimens subjected to the random loadings were then compared to predictions based on the application of Miners criteria. Experimental Program The fatigue tests were performed on two com- monly used aluminum alloys, 2024-T3 and 6061-T6. The mechanical properties of these two alloys are Experimental Mechanics 321 u 10 .3 10 .2 TRUE STRAIN L in/in ) 10 1 Fig. 1-True stress vs. true strain for 2024-T3 aluminum 0.5 a. Beam Configuration b. Normalized Mode Shape -0.5 O 0.5 STRAIN G A G E ACCEL. : ! I I I 100 0-3 _ i i rTr 10-2 10 -I TRUE STRAIN (in/in I Fig. 2-True stress vs. true strain for 6061-T6 aluminum 0.5 Normalized Bending Stress 1 /st Mode -0.5 -1 Fig. 3-Vibration characteristics of fatigue specimen given in Table 1, and the true-stress vs. true-strain curves are shown in Figs. 1 and 2. The strain-hardening exponent, , characterizes the stress-strain relationship in the plastic range. These properties were determined from uniaxial tensile tests using tension specimens made from the two alloys. In both cases, the tension speci- mens and the fatigue specimens were machined from the same sheets of aluminum, with the longitudinal axes of the specimens parallel to the direction of roll- ing. This was done to insure uniformity between these two types of tests. The majority of the fatigue tests was done on an electromagnetic vibrations exciter, with a small portion of the low-cycle constant-strain-amplitude tests being performed on an Instron tester. For the testing which was performed on the vibra- tions exciter, the fatigue-test model was a cantilever TABLE 1-MATERIAL PROPERTIES Material True True Strain- Elastic Yield fracture fracture harden- modulus, stress, stress, strain, ing ex- psi psi psi in./in, ponent 2024-T3 i0.6 X 106 51,000 90,700 0.240 0.147 6061-T6 10.6 X 108 40,500 62,000 0.440 0.0875 beam subjected to a base excitation, as shown schematically in Fig. 3(a). For the narrow- band random fatigue tests, the excitation was a nar- row-band signal with a Gaussian base-acceleration amplitude and uniform spectrum over a specified frequency bandwidth. The excitation band was centered at the fundamental beam resonance. The sinusoidal fatigue tests were performed with the ex- citation frequency slightly above the fundamental beam resonance. The type of cyclic loading was therefore completely reversed bending. The cantilever specimens were profiled along the length of the beam to move the maximum bending stress in the first mode of vibration away from the fixed end. A sketch of the test-specimen configura- tion is shown in Fig. 4. The test specimens were clamped in the middle by a mounting fixture which was attached directly to the armature of the vibra- tions exciter. As can be seen from the figure, each fatigue-test specimen contained two cantilever- beam specimens which were excited simultaneously. An end mass, in the form of an Endevco Model 2216 crystal accelerometer, was attached at the free end of the cantilever-beam specimens. The mode shapes and corresponding bending-stress distribu- tions for the first two modes of vibration are shown in Figs. 3(b) and 3(c). At the fundamental beam resonance, the maximum bending stress occurred at 322 I August 1970 i -CLAMPED- o e-j +- -1- I -zt - =! r- Fig. 4-Fatigue-test specimen p,f c .=_ E z r Fig. 5-Fatigue specimen on vibration exciter a distance of 7/8 in. from the fixed end of the beam. The frequency of the first beam resonance was ap- proximately 115 cps. The specimens were care- fully hand polished prior to testing to remove any sharp corners and to eliminate all visible surface scratches in the region in which the maximum stress occurred. To measure the strain level during the fatigue tests, strain gages were mounted on every specimen at the point where the maximum bending stress oc- curred. It was found that the fatigue life of the strain-gage installation was generally much smaller than the fatigue life of the specimen. For this reason, the signal from the accelerometer mounted at the free end of the beam was used to determine the time to failure of the fatigue specimens. The signal from the accelerometer was used to trigger a relay which deactivated a timer when the accelera- tion level dropped to 50 percent of the nominal RMS acceleration level. It was observed that, at failure, the acceleration level dropped very rapidly so that the timers indicated very closely the total time to failure of the specimen. The picture in Fig. 5 shows a fatigue-test speci- men mounted on the vibrations exciter just prior to testing. As can be seen from this figure, special accelerometer cables were constructed by splicing standard Microdot accelerometer cables with two smaller, more flexible, lead wires. This was done to minimize the effect of the vibrations of the ac- celerometer cable on the specimen. Although the splice increased the noise pickup, the signal level was so large that this increase in noise was not noticeable. 60 80 100 FREQUENCY ( Hz ) Fig. 6-Strain spectral density 120 140 160 Fig. 7-Fatigue specimen in Instron tester The signals from the strain gages and the ac- celerometers were monitored during the tests and recorded on magnetic tape. After each test, the recorded signals were played back into a wave- Experimental Mechanics I 323 analyzer system to determine the RMS levels. A digital computer was used to perform a time-series analysis of the random signals obtained from the narrow-band random fatigue tests. A strain spec: tral-density plot of the strain-gage response during a typical narrow-band random test is shown in Fig. 6. As expected, this plot indicates that the fatigue specimen can be considered to be a very lightly damped, single-degree-of-freedom system. The most probable frequency of vibration of this nar- row-band response can be shown to be the resonance frequency of the system. Therefore, the total num- ber of cycles to failure (i.e., the total number of zero crossings with positive slope) was assumed to be the total time to failure in seconds multiplied by the resonance frequency in cycles per second. In the constant-amplitude, sinusoidal tests the total num- ber of cycles to failure was merely the total time to failure in seconds multiplied by the excitation fre- quency in cycles per second. For the tests performed on the vibrations exciter, the range of the total cycles to failure was from 2 ) 10 a to 5 10 cycles. The corresponding strain levels ranged from 600 to 3500 microstrain RMS for the narrow-band random fatigue tests, and from 1700 to 7000 microstrain for the constant-amplitude testing. Due to the fast rate of cycling on the vibrations exciter, it was not possible to obtain fatigue data for life values below approximately 2 103 cycles using the test setup described above. For this reason, some low-cycle, constant-strain-amplitude fatigue testing was performed on an Instron tester. A knowledge of the low-cycle, constant-amplitude . - _ , 9 8 7 10 4 I0 s CYCLES TO FAILURE Constant Amplitude Results Narrow Band Results S-N Diagram - Miners Prediction _ L_i. lO s ID 7 1o Fig. 3-Fatigue-test results for 2024-T3 aluminum o E .J n ?9 . -_ -_ _ - ?9 Constant Amplitude Results . . . . . S-N Diagram 10 4 10 5 l0 s 10 7 1D s CYCLES TO FAILURE Fig. 9-Fatigue-test results for 6061-T6 aluminum 324 August 1970 fatigue behavior of the two materials is necessary to correlate the constant-amplitude and narrow- band random fatigue results using Miners criteria. For a Rayleigh distribution, the probability of the peaks exceeding 3.72 times the RMS value is 0.1 percent. Since the highest RMS-strain level for the narrow-band random tests was 3500 micro- strain, peak strain levels of approximately 14,000 microstrain could be expected. The vibrations ex- citer was capable of producing constant-amplitude peak strains of only 7000 microstrain; conse- quently, other means had to be employed to produce an e-N diagram up to the 14,000-microstrain level required for Miners prediction of failure. The Instron tester programmed to cycle at a rate of approximately 5 cpm with strain amplitudes of up to 40,000 microstrain was used to obtain the low- cycle data. The specimens again were loaded in completely reversed bending and were essentially the same as those used in the vibrations-exciter tests. A photograph of the test setup for the Instron fatigue testing is given in Fig. 7. The test speci- men was mounted on the crosshead of the Instron tester by means of a mounting fixture which held the specimen in the same manner that the specimens were held when mounted on the vibrations exciter. The tip of the specimen was held stationary by means of the two-force member which connected the tip to the frame of the testing machine. The base or crosshead was then cycled through a constant deflection. The number of cycles to failure was determined from a counter on the Instron tester. The results of the fatigue tests performed on the vibrations exciter are shown in Figs. 8 and 9, where the RMS strain level has been plotted vs. the total number of cycles to failure of the cantilever-beam specimens. Both the constant-amplitude and the Fig. lO-Constant-amplitude fatigue.test results for 2024-T3 aluminum so I ! II I I Ill t I111 i. Vibr ._-_ jl i i iil I ltll . ?9 , 1 i 1o I 10 1o lo LO IO 10 lO CYCLES TO FAILURE Fig. ll-Constant-amplitude fatigue-test results for 6061-T6 aluminum x m 50 .o .- 10 B 1 lO 1 b2 2.35 II I lO 2 I ! 10 3 n io 4 iij!: Ill i ii(l=- t 1 ! IIj l0 s iO s lO 7 l0 s CYCLES TO FAILURE Experimental Mechanics 325 narrow-band random results have been plotted on the same graph to emphasize the difference in fa- tigue life obtained with the two different types of loading at the same RMS strain level. It can be noticed that both materials, when sub- jected to constant-strain-amplitude sinusoidal and narrow-band random strain-amplitude fatigue loads, exhibited an endurance-limit phenomenon at the very low strain levels. Most of the constant- amplitude data seems to fal on a straight line in the log-log plot of RMS strain amplitude vs. the total number of cycles to failure. Some deviation is indicated in the data for 6061-T6 at the high strain levels. This is to be expected since the strains corresponding to these points are well into the plastic region. It is believed that the 2024-T3 material would show the same type of deviation at higher strain levels than could be obtained with the existing vibrations-exciter test setup. The narrow-band random fatigue-test results show the same linear behavior on a log-log plot with a slope which is the same as the corresponding constant- amplitude data at the lower RMS strain levels. The data then begin to deviate at the higher strain levels where a greater number of strain peaks would be expected to exceed the yield strain. The results of the constant-strain-amplitude fa- tigue tests performed on the Instron tester are shown in Figs. 10 and 11, where the amplitude of peak strain, %, has been plotted vs. the total num- ber of cycles to failure of the cantilever-beam speci- mens. The Instron test data are plotted with the constant-amplitude results from the vibrations-ex- citer tests. Since materials which exhibit a fre- quency effect generally show lower fatigue-life val- ues as the cycling frequency is lowered, 3 it is be- lieved that the difference in cycling frequency did not greatly affect the fatigue behavior of the two materials tested; consequently the two sets of data can be directly compared. It is generally believed that there is little significant effect when working with frequencies of up to about 1000 cps. 4 From the results shown in Figs. 10 and 11 and the results of the tensile tests, it can be seen that a plot of stress amplitude vs. the number of cycles to fail- ure would be of little use when large plastic strains are experienced. Due to the nature of the stress- strain relations in the plastic range, a large incre- mental increase in strain corresponds to a very small incremental increase in stress. Therefore, in this paper, either the RMS strain amplitude or else the amplitude of peak strain ep is plotted vs. number of cycles. The symbol % is for peak strain, rather than for plastic strain. Figures 10 and 11 show that the low-cycle-fatigue data seem to fall on a straight line in a log-log plot of peak-strain amplitude vs. the number of cycles to failure. In both cases, the slope of this line is much greater than the slope of the line passing through the high-cycle data. These straight-line relation- ships on the log-log plots indicate that, for con- TABLE 2-PARAMETERS USED IN MINERS CALCULATIONS N1 el ee Material b b2 cycles in./in, in./in. 2024-T3 5.80 3.22 1.35 X 104 0.00510 0.00198 6061-T6 6.50 2.35 4.20 X 103 0.00545 0.00180 stant-strain-amplitude cycling, approximate re- lationships between the total number of cycles to failure and the peak-strain amplitude can be ob- tained in exponential form. In the high-cycle range, this relationship can be expressed as Nf = klep -I (1) and in the low-cycle range as N = k2ep -b (2) where kl and ks are constants. N s is the number of cycles to failure at the peak strain amplitude ep. The values of b and b2 for both materials are given in Figs. 10 and 11 and in Table 2. This change of slope in the low-cycle, high-strain- amplitude region is in agreement with the observa- tions of numerous other experimenters. L. F. Coffin 5 has observed that for many materials, if the plastic-strain amplitude* is plotted vs. the number of cycles to failure, b2 is approximately equal to 2. Using the results of the constant-strain-amplitude fatigue tests, the results of the narrow-band random fatigue tests can be compared with a prediction based on Miners linear-accumulation-of-damage hypothesis. Miners Linear-accumulation-of-damage Criteria As was discussed previously, Miners prediction of failure assumes that failure is due to a linear accu- mulation of damage which can be determined from the results of the constant-amplitude testing. The complete constant-amplitude results, including both * For this case, the plastic-strain amplitude would be the peak-strain amplitude m inus the yield strain. N1 Ne log N Fig. 12-Idealized e-N diagram 326 I August 1970 the low- and high-cycle-fatigue life ranges, have been plotted in Figs. 10 and 11 for both of the ma- terials tested. As can be seen from the figures, the Final failure is assumed to occur when the accumu- lation of damage is equal to unity so that the total number of positive peaks to failure is MoT = Nae2 (7) E Y, s 1/el bl ep bi+l e -p/2 d% + 1/el b ep b2+l e-/2g d% e e-N curves for both materials can be approximated by three straight-line segments when plotted on a log-log scale. These three ranges can be indicated schematically as shown in Fig. 12. In range 2, it will be assumed that the e-N dia- gram is a straight line on a log-log plot from zero to N cycles with a negative slope of 1/b. In range 1, it will be assumed that the straight line extending from N to N cycles has a negative slope of 1/b. The horizontal line at e corresponds to the endur- ance strain. It will be assumed that strains below this level have no influence on the final fatigue fail- ure. Thus, the e-N curve can be expressed as = for el ep oo, 0 N N (3) and = for ee_ %_ e, N1 N N s For this case, the linear-accumulation-of-damage equation can be written in terms of the peak-strain amplitude as Ace. Dam. = MeTIf N() + i = P(e)d%l N(%)J (4) where p(e) is the probability density function of the peak-strain amplitude, T is total time to failure and Mo is the number of positive peaks per second. Thus MoT represents the total number of positive peaks to failure. Since the strain response is an extremely narrow-band process, the peak-strain probability density function resulting from the im- posed Gaussian distri