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Chapter 11: Alternating Stress

11.1 Introduction

Practical examples

  1. 齿轮
  2. 只有啮合的时候有应力，周期性脉冲
  3. 火车车轴
  4. 盯住某一点看，\(y = r \sin(\omega t)\)，弯曲应力是周期变化的
  5. 简支梁上放着旋转的偏心轮

Concepts

  1. Alternating stress
  2. Fatigue failure
  3. Mechanism of fatigue failure
graph LR
    A[Crack nucleation] --> B[Crack extension]
    B --> C[Fatigue crack]
    C --> D[Fracture]

11.2 Alternating Stress

  • The cycle characteristic
\[ r = \frac{\sigma_{\min}}{\sigma_{\max}} \]
  • The mean stress \(\sigma_m\)
  • The stress amplitude \(\sigma_a\)
  • Symmetrical cycle
\[ r = -1, \sigma_m = 0, \sigma_a = \sigma_{\max} \]
  • Pulsant cycle
\[ r = 0, \sigma_m = \sigma_a = \frac{1}{2}\sigma_{\max} \]
  • Static stress
\[ r = 1, \sigma_a = 0, \sigma_m = \sigma \]
  • Asymmetrical cycle = Symmetrical cycle + Static stress

11.3 Determination of Fatigue Strength Limit

Fatigue Strength Limit（持久极限）：The index of resistance to failure under symmetrical cyclic loading.

\(S - N\) Curve

  • \(S\): Stress
  • \(N\): Number of cycles to failure

\(N > 10^7 \sim 10^8\) 时，认为不会发生疲劳断裂。此时的应力记作 \(\sigma_{-1}\)

11.4 Fatigue Strength Limit

Effect of stress concentration

Effect of size

\[ \varepsilon_{\sigma} = \]

Effect of surface finish

\[ \beta = \frac{(\sigma_{-1})_{\beta}}{\sigma_{-1}} \]
  • \(\beta\): surface quality coefficient
  • \(\sigma_{-1}\): endurance limit for ground specimen
  • \((\sigma_{-1})_{\beta}\): endurance limit for other specimens

11.5 Fatigue Strength Calculation Under Symmetrical Cycle

  • Fatigue strength condition
\[ \sigma_{\max} \leq \left[\sigma_{-1}\right] = \frac{\sigma_{-1}^0}{n} = \frac{\varepsilon_{\sigma} \beta}{nK_{\sigma}}\sigma_{-1} \]
  • \(\varepsilon_{\sigma}\): size coefficient 尺寸因数
  • \(K_{\sigma}\): stress concentration factor 有效应力集中因数
  • \(\beta\): surface quality coefficient 表面质量因数
  • \(n\): safety factor

Another form:

\[ n_{\sigma} = \frac{\sigma_{-1}^0}{\sigma_{\max}} = \frac{\varepsilon_{\sigma} \beta}{K_{\sigma}\sigma_{\max}}\sigma_{-1} \geq n \]
  • \(n_{\sigma}\): 实际有效的安全因数

11.6 Endurance Limit Diagram

  • \(\sigma_r\): endurance limit of unsymmetrical cycle
    • \(\sigma_{-1}\): symmetrical cycle
    • \(\sigma_0\): pulsant cycle
    • \(\sigma_{0.3}\)
\[ \left. \begin{aligned} \sigma_m &= \frac{\sigma_{\max} + \sigma_{\min}}{2} = \frac{1 + r}{2}\sigma_{\max} \\ \sigma_a &= \frac{\sigma_{\max} - \sigma_{\min}}{2} = \frac{1 - r}{2}\sigma_{\max} \end{aligned} \right\} \implies \tan \alpha = \frac{\sigma_a}{\sigma_m} = \frac{1 +r}{1 - r} = \text{const.} \]

Endurance limit diagram \(\sigma_a\) vs. \(\sigma_m\)

11.7 Fatigue Strength Calculation Under Asymmetrical Cycle

Tests show that stress concentration, size and surface finish of members have influence only on stress amplitude \(\sigma_a\), not on mean stress \(\sigma_m\).

In the endurance limit diagram, \(\alpha_1 \geq \alpha\)

\[ \begin{aligned} \tan \alpha_1 &= \frac{\frac{\varepsilon_{\sigma} \beta}{K_{\sigma}}\sigma_{-1} - n\sigma_a}{n\sigma_m} \\ \tan \alpha &= \frac{\frac{\varepsilon_{\sigma} \beta}{K_{\sigma}}\left(\sigma_{-1} - \frac{\sigma_0}{2}\right)}{\frac{\sigma_0}{2}} \end{aligned} \]

Let \(\psi_{\sigma} = \frac{\sigma_{-1} - \sigma_0 / 2}{\sigma_0 / 2}\), then the actual safety factor is

\[ n_{\sigma} = \frac{\sigma_{-1}}{\frac{K_{\sigma}}{\varepsilon_{\sigma} \beta}\sigma_a + \psi_{\sigma} \sigma_m} \geq n \]

which is the fatigue strength condition under asymmetrical cycle.

11.8 Fatigue Strength Calculation Under Combined Bending and Torsion

第四强度理论

\[ \sqrt{\sigma^2 + 3\tau^2} \leq \sigma_s \overset{\tau_s = \sigma_s / \sqrt{3}}{\longrightarrow} \left(\frac{\sigma}{\sigma_s}\right)^2 + \left(\frac{\tau}{\tau_s}\right)^2 \leq 1 \]

At critical point, provided safety factor \(n\):

\[ \left(\frac{n\sigma}{\frac{\varepsilon_{\sigma} \beta}{K_{\sigma}}\sigma_{-1}}\right)^2 + \left(\frac{n\tau}{\frac{\varepsilon_{\tau} \beta}{K_{\tau}}\tau_{-1}}\right)^2 \leq 1 \implies \left(\frac{n}{n_{\sigma}}\right)^2 + \left(\frac{n}{n_{\tau}}\right)^2 \leq 1 \]

i.e.

\[ n \leq \frac{n_{\sigma} n_{\tau}}{\sqrt{n_{\sigma}^2 + n_{\tau}^2}} = n_{\sigma \tau} \]

11.9 Practical Measurements for Preventing Fatigue Failure

  1. 倒角尽可能大，避免应力集中
  2. Improve the quality of machining. 提高表面加工质量，避免磕碰
  3. Increasing the strength of surface. 提高表面强度