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Chapter 7: Stress and Strain Analysis & Strength Theory

7.1 Introduction: Stress State

为什么要做应力应变分析？

同一个位置，作不同的截面，应力值不同。但应该是同一个状态！

Combined stresses

  • 主平面（Principal plane）：剪应力为0
  • 主应力（Principal stress）：主平面上的正应力
  • 单相应力状态（Uniaxial stress state）：只有一个主应力不为0
  • 二相应力状态（Biaxial stress state）：有两个主应力不为0
  • 三相应力状态（Triaxial stress state）：有三个主应力不为0

三者代数值关系（含正负号）： $$ \sigma_1 \geq \sigma_2 \geq \sigma_3 $$

7.2 Examples of State of Biaxial and Triaxial Stresses

7.3 Plane Stress State Analysis: Analytical Method

\(\tau_{xy}\)：

  • \(x\)（第一个下标）：截面的外法线方向
  • \(y\)（第二个下标）：应力方向
\[ \begin{aligned} & \Sigma F_x = 0, \quad \sigma_{\alpha} \mathrm{d}A \cos \alpha + \tau_{\alpha} \mathrm{d}A \sin \alpha - \sigma_x \mathrm{d}A \cos \alpha + \tau_{yx} \mathrm{d}A \sin \alpha = 0 \\ & \Sigma F_y = 0, \quad \sigma_{\alpha} \mathrm{d}A \sin \alpha - \tau_{\alpha} \mathrm{d}A \cos \alpha - \sigma_y \mathrm{d}A \sin \alpha + \tau_{xy} \mathrm{d}A \cos \alpha = 0 \\ \end{aligned} \]

化简得

\[ \begin{aligned} \sigma_{\alpha} &= \sigma_x \cos^2 \alpha + \sigma_y \sin^2 \alpha - 2 \tau_{xy} \sin \alpha \cos \alpha \\ \tau_{\alpha} &= \sigma_x \sin \alpha \cos \alpha - \sigma_y \sin \alpha \cos \alpha + \tau_{xy} (\cos^2 \alpha - \sin^2 \alpha) \end{aligned} \]

即

\[ \begin{equation} \tag{7-1} \boxed{ \begin{aligned} \sigma_{\alpha} &= \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2} \cos 2\alpha + \tau_{xy} \sin 2\alpha \\ \tau_{\alpha} &= \frac{\sigma_x - \sigma_y}{2} \sin 2\alpha + \tau_{xy} \cos 2\alpha \end{aligned} } \end{equation} \]

符号规定

  • 正应力 \(\sigma\)
    • \(\oplus\)：拉
    • \(\ominus\)：压
  • 切应力 \(\tau\)
    • \(\oplus\)：顺时针
    • \(\ominus\)：逆时针
  • 角度 \(\alpha\)（法线与 \(x\) 轴的夹角）
    • \(\oplus\)：逆时针
    • \(\ominus\)：顺时针

alt text

Extremum of \(\sigma_{\alpha}\)

\[ \frac{\mathrm{d} \sigma_{\alpha}}{\mathrm{d} \alpha} = -2\left[\frac{\sigma_x - \sigma_y}{2} \sin 2\alpha + \tau_{xy} \cos 2\alpha\right] = 0 \implies \tan 2\alpha = -\frac{2\tau_{xy}}{\sigma_x - \sigma_y} \]

7.4 Plane Stress State Analysis: Mohr's Circle

\[ \left(\sigma_{\alpha} - \frac{\sigma_x + \sigma_y}{2}\right)^2 + \tau_{\alpha}^2 = \left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2 \]
  • Radius \(R\)：
\[ R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2} \]
  • Center \(C\)：
\[ C = \left(\frac{\sigma_x + \sigma_y}{2}, 0\right) \]

7.5 State of Triaxial Stress

Stresses on the inclined plane

alt text

在微元体 \(\mathrm{d}x \mathrm{d}y \mathrm{d}z\) 内取一个斜截面。记 \(S\) 为合应力，斜截面法向量为 \(\vec{n} = (l, m, n) = (\cos \alpha, \cos \beta, \cos \gamma)\)，则有平衡方程

\[ \left \{ \begin{aligned} \Sigma F_x = 0, \quad S_x \mathrm{d}A - \sigma_x \mathrm{d}A \cdot l - \tau_{yx} \mathrm{d}A \cdot m - \tau_{zx} \mathrm{d}A \cdot n &= 0 \\ \Sigma F_y = 0, \quad S_y \mathrm{d}A - \tau_{xy} \mathrm{d}A \cdot l - \sigma_y \mathrm{d}A \cdot m - \tau_{zy} \mathrm{d}A \cdot n &= 0 \\ \Sigma F_z = 0, \quad S_z \mathrm{d}A - \tau_{xz} \mathrm{d}A \cdot l - \tau_{yz} \mathrm{d}A \cdot m - \sigma_z \mathrm{d}A \cdot n &= 0 \end{aligned} \right. \]

得到

\[ \begin{equation} \tag{7-2} \begin{aligned} S_x &= \sigma_x l + \tau_{yx} m + \tau_{zx} n \\ S_y &= \tau_{xy} l + \sigma_y m + \tau_{zy} n \\ S_z &= \tau_{xz} l + \tau_{yz} m + \sigma_z n \end{aligned} \end{equation} \]

合应力向法向投影，

The principal stresses

7.8 Generalized Hooke's Law

广义胡克定律

轴向拉伸/压缩：\(\sigma = E\varepsilon\)

泊松比 \(\varepsilon' = -\mu \varepsilon\)

纯剪切：\(\tau = G\gamma\)

叠加原理条件：

  1. \(\sigma \leq \sigma_p, \tau \leq \tau_p\)
  2. 小变形
  3. 各向同性材料
\[ \begin{equation} \tag{Hooke's Law} \label{Hooke's Law} \boxed{ \begin{aligned} \varepsilon_x &= \frac{1}{E} \left[\sigma_x - \mu (\sigma_y + \sigma_z)\right] \\ \varepsilon_y &= \frac{1}{E} \left[\sigma_y - \mu (\sigma_x + \sigma_z)\right] \\ \varepsilon_z &= \frac{1}{E} \left[\sigma_z - \mu (\sigma_x + \sigma_y)\right] \\ \gamma_{xy} =& \frac{\tau_{xy}}{G}, \, \gamma_{xz} = \frac{\tau_{xz}}{G}, \, \gamma_{yz} = \frac{\tau_{yz}}{G} \end{aligned} } \end{equation} \]

The relative change in volume

变形前：\(V_0 = \mathrm{d}x \mathrm{d}y \mathrm{d}z\)

变形后：

\[ \begin{aligned} V &= (\mathrm{d}x + \varepsilon_1 \mathrm{d}x)(\mathrm{d}y + \varepsilon_2 \mathrm{d}y)(\mathrm{d}z + \varepsilon_3 \mathrm{d}z) \\ &= (1 + \varepsilon_1)(1 + \varepsilon_2)(1 + \varepsilon_3) \mathrm{d}x \mathrm{d}y \mathrm{d}z \end{aligned} \]

忽略应变的乘积项，得到

\[ V = (1 + \varepsilon_1 + \varepsilon_2 + \varepsilon_3) \mathrm{d}x \mathrm{d}y \mathrm{d}z \]

体应变（Volume strain）:

\[ \theta = \frac{V - V_0}{V_0} = \varepsilon_1 + \varepsilon_2 + \varepsilon_3 \]

Discussion

  1. 结合 \eqref{Hooke's Law}，有
\[ \theta = \varepsilon_1 + \varepsilon_2 + \varepsilon_3 = \frac{1 - 2\mu}{E} \left(\sigma_1 + \sigma_2 + \sigma_3\right) \]

当 \(\mu = 0.5\) 时，\(\theta = 0\)，即不论外力如何变化，总体积不变。这种材料称为不可压缩材料（incompressible material），如橡胶。

  1. 令

7.10 Intro to Strength Theory

Two typical failure modes

  • 塑性：屈服
  • 脆性：断裂

Two types of resistance

  • Resistance to rupture（\(\sigma_{\text{rupt}}\)）

  • Resistance to shear（\(\tau_{\text{sh}}\)）

  • Ductile：\(\sigma_{\text{rupt}} > \tau_{\text{sh}}\)

  • Brittle：\(\sigma_{\text{rupt}} < \tau_{\text{sh}}\)

7.11 Four Practical Strength Theories

最大拉应力

最大正应变

\[ \varepsilon = \frac{1}{E} \left[\sigma - \mu (\sigma_1 + \sigma_2)\right] < \varepsilon_m \]

最大剪应力

单轴拉伸，\(\tau_{\max} = \frac{\sigma}{2}\)