https://www.zhuzhugst.com/ 2026-04-21T22:48:43+00:00 https://www.zhuzhugst.com/ https://www.zhuzhugst.com/lib/exe/fetch.php?media=logo.png text/html 2025-12-05T05:56:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.zhuzhugst.com/doku.php?id=%E5%BC%B9%E6%80%A7%E5%8A%9B%E5%AD%A6:%E5%BC%B9%E6%80%A7%E5%8A%9B%E5%AD%A6%E7%9A%84%E5%B9%B3%E9%9D%A2%E9%97%AE%E9%A2%98&rev=1764914182&do=diff 弹性力学平面问题 (Plane Problems in Elasticity) 在弹性力学中，许多实际工程问题（如水坝、厚壁圆筒、薄板受拉等）可以简化为二维平面问题。这不仅减少了未知量的个数，还大大降低了数学求解的难度。$t \ll L$$z = \pm t/2$$$ \sigma_z = 0, \quad \tau_{zx} = 0, \quad \tau_{zy} = 0 $$$\sigma_x, \sigma_y, \tau_{xy}$$$ \begin{cases} \varepsilon_x = \frac{1}{E}(\sigma_x - \mu \sigma_y) \\ \varepsilon_y = \frac{1}{E}(\sigma_y - \mu \sigma_x) \\ \gamma_{xy} = \frac{2(1+\mu)}{E} \tau_{xy} = \frac{1}{G} \tau_{xy} \end{cases} $$$\sigma_z=0$$\varepsilon_z \neq 0$$$ \varepsilon_z = -\frac{\mu}{E… text/html 2025-12-05T05:42:34+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.zhuzhugst.com/doku.php?id=%E5%BC%B9%E6%80%A7%E5%8A%9B%E5%AD%A6:%E5%BC%B9%E6%80%A7%E5%8A%9B%E5%AD%A6%E7%9A%84%E4%B8%80%E8%88%AC%E5%8E%9F%E7%90%86&rev=1764913354&do=diff 弹性力学基本原理 (Fundamental Principles of Elasticity) 在弹性力学问题的求解过程中，为了简化边界条件、确保解的可靠性以及处理复杂载荷，我们依赖于三大核心原理：圣维南原理、解的唯一性定理$\sigma_{ij}^{(1)}$$\sigma_{ij}^{(2)}$$u_i^{(1)}$$u_i^{(2)}$$$ \sigma_{ij}^* = \sigma_{ij}^{(1)} - \sigma_{ij}^{(2)} $$$$ u_i^* = u_i^{(1)} - u_i^{(2)} $$$\sigma_{ij}^{(1)}$$\sigma_{ij}^{(2)}$$F_{bi}$$\sigma_{ij,j} + F_{bi} = 0$$$ \sigma_{ij,j}^* = 0 $$$\sigma_{ij}^{(1)} n_j = P_i$$\sigma_{ij}^{(2)} n_j = P_i$$$ \sigma_{ij}^* n_j = 0 $$$\sigma_{ij}^*$$$ \sigma_{ij}^* = 0 \implies \sig… text/html 2025-12-05T05:35:52+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.zhuzhugst.com/doku.php?id=%E5%BC%B9%E6%80%A7%E5%8A%9B%E5%AD%A6:%E5%BC%B9%E6%80%A7%E4%BD%93%E5%8A%9B%E5%AD%A6%E9%97%AE%E9%A2%98%E7%9A%84%E5%BB%BA%E7%AB%8B&rev=1764912952&do=diff 弹性力学求解体系 (Solving System of Elasticity) 弹性力学的核心任务是在给定的边界条件下，求解弹性体内部的应力、应变和位移场。这是一个典型的偏微分方程组边值问题。 1. 基本架构：15个未知数与15个方程 $\sigma_x, \sigma_y, \sigma_z, \tau_{xy}, \tau_{yz}, \tau_{zx}$$\varepsilon_x, \varepsilon_y, \varepsilon_z, \gamma_{xy}, \gamma_{yz}, \gamma_{zx}$$u, v, w$$$ \begin{cases} P_x = \sigma_x l + \tau_{xy} m + \tau_{xz} n \\ P_y = \tau_{yx} l + \sigma_y m + \tau_{yz} n \\ P_z = \tau_{zx} l + \tau_{zy} m + \sigma_z n \end{cases} $$$P_i = \sigma_{ij} n_j$$l, m, n$$n_x, n_y, n_z$$$ u … text/html 2025-12-08T05:20:20+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.zhuzhugst.com/doku.php?id=%E5%BC%B9%E6%80%A7%E5%8A%9B%E5%AD%A6:%E8%83%BD%E9%87%8F%E6%B3%95&rev=1765171220&do=diff 弹性力学：能量法 (Energy Methods) 能量法是弹性力学中一种非常强大的求解方法。与直接求解微分方程（如平衡方程、几何方程、物理方程）不同，能量法通过研究物体变形过程中储存的能量（应变能）与外力做功之间的关系，将复杂的微分方程问题转化为积分问题或代数方程组求解，特别适合处理复杂边界条件和数值计算（如有限元法）。$U_0$$$ U_0 = \int_0^{\epsilon_x} \sigma_x \text{d}\epsilon_x = \frac{1}{2} E \epsilon_x^2 = \frac{1}{2} \sigma_x \epsilon_x $$$\sigma - \epsilon$$$ W = \int_0^{\epsilon_x} \sigma_x \left( \frac{\partial u}{\partial x} \text{d}x \right) \text{d}y\text{d}z = \int_0^{\epsilon_x} \sigma_x \text{d}\epsilon_x \text{d}V $$$U$$U_0$$V$$$ U = \… text/html 2025-12-05T05:12:08+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.zhuzhugst.com/doku.php?id=%E5%BC%B9%E6%80%A7%E5%8A%9B%E5%AD%A6:%E5%BA%94%E5%8F%98%E5%88%86%E6%9E%90&rev=1764911528&do=diff 弹性力学：应变与几何方程 本页面主要阐述弹性力学中关于物体变形的描述，包括位移、应变的概念、几何方程、变形协调方程以及主应变理论。 1. 物体的变形与位移 在弹性力学中，物体的变形通过空间中点的$P(x, y, z)$$P'(x', y', z')$$x$$u = u(x, y, z)$$y$$v = v(x, y, z)$$z$$w = w(x, y, z)$$\varepsilon$$\gamma$$$ \varepsilon_x = \frac{\partial u}{\partial x} $$$$ \varepsilon_y = \frac{\partial v}{\partial y} $$$$ \gamma_{xy} = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} $$$$ \begin{cases} \varepsilon_x = \frac{\partial u}{\partial x}, \quad \varepsilon_y = \frac{\partial v}{\partia… text/html 2025-12-04T05:48:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.zhuzhugst.com/doku.php?id=%E5%BC%B9%E6%80%A7%E5%8A%9B%E5%AD%A6:%E5%BA%94%E5%8A%9B%E5%88%86%E6%9E%90&rev=1764827290&do=diff 应力分析 (Stress Analysis) 本章主要探讨弹性体内部的受力状态，从一点的应力定义出发，扩展到三维空间的应力状态、坐标变换、主应力分析以及平衡方程。 1 应力的概念与定义 应力矢量 (Traction Vector) $P$$\Delta S$$n$$\Delta F$$$ \vec{T}_{(n)} = \lim_{\Delta S \to 0} \frac{\Delta \vec{F}}{\Delta S} $$$\Delta S$$\Delta S$$\Delta S$$\sigma_{xx} $$\sigma_{x} $$$ \sigma_{ij} = \left[ \begin{matrix} \sigma_x & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_y & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_z \end{matrix} \right] $$$$ \tau_{ij} = \tau_{ji} \quad (i \neq j) $$$\tau_{xy}… text/html 2025-12-05T05:20:13+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.zhuzhugst.com/doku.php?id=%E5%BC%B9%E6%80%A7%E5%8A%9B%E5%AD%A6:%E5%BA%94%E5%8A%9B%E5%92%8C%E5%BA%94%E5%8F%98%E7%9A%84%E5%85%B3%E7%B3%BB&rev=1764912013&do=diff 广义胡克定律 (Generalized Hooke's Law) 广义胡克定律是描述弹性体在比例极限内，应力与应变之间线性关系的物理方程。它是弹性力学中最基本的本构方程。 1. 应变形式 (Strain Form) 这是广义胡克定律最常见的形式，表达了应变分量如何由应力分量决定。$$ \begin{cases} \varepsilon_x = \frac{1}{E} [\sigma_x - \nu (\sigma_y + \sigma_z)] \\ \varepsilon_y = \frac{1}{E} [\sigma_y - \nu (\sigma_x + \sigma_z)] \\ \varepsilon_z = \frac{1}{E} [\sigma_z - \nu (\sigma_x + \sigma_y)] \end{cases} \quad \begin{cases} \gamma_{yz} = \frac{\tau_{yz}}{G} \\ \gamma_{zx} = \frac{\tau_{zx}}{G} \\ \gamma_{xy} = \frac{\tau_{x…