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L7 — 2D Problems（二维问题）

Q34 ⭐ 平面应力 vs 平面应变

EN: Plane stress applies to thin structures (plates with in-plane loading):

  • Condition: σz=τxz=τyz=0\sigma_z = \tau_{xz} = \tau_{yz} = 0
  • BUT εz≠0\varepsilon_z \neq 0 (the plate is free to contract/expand in thickness)
  • Typical application: thin plates with in-plane forces, thin-walled structures

Plane strain applies to thick/long structures (cross-section of long bodies):

  • Condition: εz=γxz=γyz=0\varepsilon_z = \gamma_{xz} = \gamma_{yz} = 0
  • BUT σz≠0\sigma_z \neq 0 (constrained in the longitudinal direction)
  • Typical application: long dams, tunnels, retaining walls, shafts

中文： ⚠️ 高频考点——平面应力（薄板）：σz=0\sigma_z=0 但 εz≠0\varepsilon_z\neq0，板厚方向自由变形，用于薄板面内载荷；平面应变（长厚体）：εz=0\varepsilon_z=0 但 σz≠0\sigma_z\neq0，纵向受约束，用于长坝、隧道、挡土墙等。两个的D矩阵不同！

Q35 🟡 平面应力D矩阵

EN: Plane stress constitutive matrix (3×3):

D=E1−ν2[1ν0ν10001−ν2]\boxed{\mathbf{D} = \frac{E}{1-\nu^2}\begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1-\nu}{2} \end{bmatrix}}

Stress and strain vectors: σ={σx,σy,τxy}T\boldsymbol{\sigma} = \{\sigma_x, \sigma_y, \tau_{xy}\}^T, ε={εx,εy,γxy}T\boldsymbol{\varepsilon} = \{\varepsilon_x, \varepsilon_y, \gamma_{xy}\}^T.

中文： 平面应力D矩阵（3×3），因子为 E/(1−ν2)E/(1-\nu^2)。对应应力/应变分量各3个（σz=0\sigma_z=0所以不出现）。

Q36 🟡 平面应变D矩阵

EN: Plane strain constitutive matrix (3×3):

D=E(1+ν)(1−2ν)[1−νν0ν1−ν0001−2ν2]\boxed{\mathbf{D} = \frac{E}{(1+\nu)(1-2\nu)}\begin{bmatrix} 1-\nu & \nu & 0 \\ \nu & 1-\nu & 0 \\ 0 & 0 & \frac{1-2\nu}{2} \end{bmatrix}}

Alternatively: use the plane stress D-matrix but replace EE with E′=E/(1−ν2)E' = E/(1-\nu^2) and ν\nu with ν′=ν/(1−ν)\nu' = \nu/(1-\nu).

中文： 平面应变D矩阵（3×3），因子为 E/[(1+ν)(1−2ν)]E/[(1+\nu)(1-2\nu)]。对角项为 1−ν1-\nu 而非1。等价变换：E→E/(1−ν2)E\to E/(1-\nu^2), ν→ν/(1−ν)\nu\to\nu/(1-\nu)。

Q37 🟡 二维应变-位移关系

EN: In 2D (plane problems), the three strain components are:

εx=∂u∂x,εy=∂v∂y,γxy=∂u∂y+∂v∂x\varepsilon_x = \frac{\partial u}{\partial x}, \quad \varepsilon_y = \frac{\partial v}{\partial y}, \quad \gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}

In matrix form:

{εxεyγxy}=[∂∂x00∂∂y∂∂y∂∂x]{uv}\begin{Bmatrix} \varepsilon_x \\ \varepsilon_y \\ \gamma_{xy} \end{Bmatrix} = \begin{bmatrix} \frac{\partial}{\partial x} & 0 \\ 0 & \frac{\partial}{\partial y} \\ \frac{\partial}{\partial y} & \frac{\partial}{\partial x} \end{bmatrix} \begin{Bmatrix} u \\ v \end{Bmatrix}

中文： 二维应变——正应变 εx=∂u/∂x\varepsilon_x=\partial u/\partial x, εy=∂v/∂y\varepsilon_y=\partial v/\partial y；剪应变 γxy=∂u/∂y+∂v/∂x\gamma_{xy}=\partial u/\partial y+\partial v/\partial x（工程剪应变，非张量分量）。

Q38 ⭐ CST常应变三角形

EN: The Constant Strain Triangle (CST, T3) is a 3-node triangular element with linear displacement functions:

u=a1+a2x+a3y,v=a4+a5x+a6yu = a_1 + a_2 x + a_3 y, \quad v = a_4 + a_5 x + a_6 y

Since displacements are linear, their derivatives (strains) are constant throughout the element. The B-matrix is constant:

ke=tA BTDB\mathbf{k}_e = tA\,\mathbf{B}^T\mathbf{D}\mathbf{B}

Advantages: Simple derivation; good for complex geometries.
Disadvantages: Low accuracy (constant stress); overly stiff in bending; requires dense mesh for good results.

中文： CST（常应变三角形，T3）：3节点，位移线性 → 应变为常数（故名"常应变"）。优点：简单、几何适应性强；缺点：精度低、弯曲问题中偏硬、需密网格。刚度矩阵 ke=tABTDB\mathbf{k}_e=tA\mathbf{B}^T\mathbf{D}\mathbf{B}，无需数值积分。

Q39 ⭐ CST形函数的一阶导数

EN: For a CST element, shape functions are:

Ni=αi+βix+γiy2AN_i = \frac{\alpha_i + \beta_i x + \gamma_i y}{2A}

where AA is the element area, and βi=yj−ym\beta_i = y_j - y_m, γi=−(xj−xm)\gamma_i = -(x_j - x_m) (cyclic permutation).

The first derivatives ∂Ni/∂x=βi/(2A)\partial N_i/\partial x = \beta_i/(2A) and ∂Ni/∂y=γi/(2A)\partial N_i/\partial y = \gamma_i/(2A) are constants. Therefore, the B-matrix is constant.

For a quadrilateral element, the derivatives ∂Ni/∂x\partial N_i/\partial x and ∂Ni/∂y\partial N_i/\partial y involve J−1\mathbf{J}^{-1} (Jacobian inverse), which generally varies with position → derivatives are not constant.

中文： CST的形函数导数是常数（因位移线性）；四边形单元的形函数导数通过Jacobian逆矩阵变换得到，随位置变化，不是常数。这就是"三角形常应变，四边形非常应变"的原因。

Q40 🔴 CST的优缺点总结

EN: CST advantages: (1) Simple formulation — easy to program; (2) Good for complex geometries — triangles can mesh any shape; (3) No numerical integration needed; (4) Good for preliminary/quick analysis.

CST disadvantages: (1) Constant stress/strain — poor accuracy in regions with high gradients; (2) Overly stiff in bending — requires very fine mesh for beam/plate problems; (3) Lower accuracy per DOF compared to quadrilateral or higher-order elements; (4) Not suitable for stress concentration regions (holes, corners).

Recommendation: Use CST in areas with small strain gradients, mesh transition zones, or quick preliminary analysis. Avoid CST at stress concentrations — use higher-order elements or very fine mesh.

中文： 优点：简单、几何适应性好、无需数值积分。缺点：常应力精度低、弯曲问题偏硬、应力集中处表现差。建议：应变梯度小处用CST，应力集中处用高阶单元或加密网格。

Q41 🔴 不同二维单元的比较

EN: Comparison of 2D element types:

Element Nodes Displacement Strain Stress
T3 (CST) 3 Linear Constant Constant
Q4 (Bilinear Quad) 4 Bilinear Linear (varies) Linear (varies)
T6 (LST) 6 Quadratic Linear (varies) Linear (varies)
Q8 (Quadratic Quad) 8 Quadratic/cubic Quadratic Quadratic

Rule of thumb: For the same number of DOFs, quadrilaterals generally perform better than triangles. Higher-order elements give better accuracy per DOF but have higher bandwidth.

中文： T3（常应变）、Q4（双线性，应变线性变化）、T6（位移二次，应变线性变化）、Q8（位移三次，应变二次变化）。经验法则：同样DOF下，四边形优于三角形；高阶单元精度更高但带宽更大。

Q42 🔴 平面应力和平面应变的适用范围

EN: Plane stress is appropriate for:

  • Thin plates with in-plane loading (thickness ≪ in-plane dimensions)
  • Thin-walled pressure vessels (membrane stresses)
  • Sheet metal forming analysis
  • Web of an I-beam (local analysis)

Plane strain is appropriate for:

  • Very long/thick structures with uniform cross-section
  • Dams, tunnels, retaining walls
  • Cylindrical rollers under compression
  • Soil/geotechnical cross-sections

中文： 平面应力——薄板面内加载、薄壁容器、板料成形；平面应变——长坝、隧道、挡土墙、轧辊、岩土截面。判断标准：厚度方向的变形是否被约束。

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