von Mises 应力
许多实验室疲劳试验是在单轴加载下进行的。然而，超声疲劳可能涉及沿多个方向作用的应力（多轴应力）；一个简单的例子是径向谐振的圆盘，它同时存在径向应力和环向应力。von Mises 应力 \( \sigma_v \) 是一种（基于理论的）人为定义的应力，试图将这种多轴疲劳应力与更简单的实验室单轴疲劳应力 σu 关联起来。因此，如果实验室单轴试样在疲劳应力 \( \sigma_{uf} \) 下（平均而言）发生失效，那么同种材料、承受多轴应力的谐振器应当在 \( \sigma_v \) 等于 \( \sigma_{uf} \) 时（平均而言）失效。von Mises 应力适用于各向同性的延性材料。
对于沿三个互相垂直方向（x、y 和 z）作用的应力，von Mises 应力由下式给出 —
\begin{align} \label{eq:13801a} \sigma_v = \left[ \frac{(\sigma_x - \sigma_y)^2 + (\sigma_y - \sigma_z)^2 + (\sigma_z - \sigma_x)^2 + 6 \, (\tau_{xy}^2 + \tau_{yz}^2 + \tau_{zx}^2)} {2} \right]^{1/2} \end{align}
其中 —
| \( \sigma \) | = 正应力 |
| \( \tau \) | = 剪应力 |
如果局部应力立方体的取向使其坐标轴与主轴 1、2、3 重合（即不存在剪应力，只有拉应力或压应力），则上式变为 —
\begin{align} \label{eq:13802a} \sigma_v = \left[ \frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2} {2} \right]^{1/2} \end{align}
注：
- 也称为 "von Mises - Hencky 应力"、"Hencky von Mises 应力"、"八面体剪应力"或"畸变能应力"。
- 有限元分析（FEA）软件通常会计算并显示 von Mises 应力。
- 除非另有说明，本站所有应力值均为 von Mises 应力。
参见 Juvinall，第 86 - 89 页、第 230 页、第 316 - 317 页，以及 Shigley，第 152 - 154 页。另见Frequently Asked Questions on Von Mises Stress Explained。
von Mises stress
Many laboratory fatigue tests are conducted under uniaxial loading. However, ultrasonic fatigue may involve stresses that act in several directions (multiaxial stress); a simple example is a radially resonant disk which would have both radial stress and hoop stress. The von Mises stress \( \sigma_v \) is an artificial stress (based on theory) that attempts to correlate such a multiaxial fatigue stress to the simpler uniaxial laboratory fatigue stress σu. Thus, if uniaxial laboratory samples fail (on average) at a fatigue stress \( \sigma_{uf} \), then resonators of the same material with multiaxial stresses should fail (on average) when \( \sigma_v \) equals \( \sigma_{uf} \). The von Mises stress is applicable to isotropic materials that are ductile.
For stresses acting in three perpendicular directions (x, y, and z), the von Mises stress is given by —
\begin{align} \label{eq:13801a} \sigma_v = \left[ \frac{(\sigma_x - \sigma_y)^2 + (\sigma_y - \sigma_z)^2 + (\sigma_z - \sigma_x)^2 + 6 \, (\tau_{xy}^2 + \tau_{yz}^2 + \tau_{zx}^2)} {2} \right]^{1/2} \end{align}
where —
| \( \sigma \) | = normal stress |
| \( \tau \) | = shear stress |
If the local stress cube is oriented such that its axes align with the principal axes 1, 2, and 3 (i.e., there are no shear stresses, only tensile or compressive stresses), then the above equation becomes —
\begin{align} \label{eq:13802a} \sigma_v = \left[ \frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2} {2} \right]^{1/2} \end{align}
Notes:
- Also called "von Mises - Hencky stress", "Hencky von Mises stress", "octahedral shear stress", or "distortion energy stress".
- Finite element analysis (FEA) software generally will calculate and display the von Mises stress.
- All stress values on this site are von Mises unless otherwise stated.
See Juvinall, pp. 86 - 89, p. 230, pp. 316 - 317 and Shigley, pp. 152 - 154. Also see Frequently Asked Questions on Von Mises Stress Explained.