从形变约束到材料应力From deformation constraints to material stress
材料题在近年的 Section 1 中多次出现，所选题每题 4–5 分。关键不是多记公式，而是先判定什么被固定：力、伸长、温度还是曲率。常见失分点是混淆应力与应变、漏掉平方单位换算，以及把总伸长当成伸长变化。Materials questions recur in recent Section 1 papers; the selected questions carry 4–5 marks each. First identify what is constrained: force, extension, temperature or curvature. Common losses come from confusing stress with strain, area conversion, and total extension with its change.
1应力、应变与弹性能Stress, strain and elastic energy
应力是单位面积承受的力，应变是伸长量与原长之比。在线弹性范围内，杨氏模量是应力–应变图的斜率，图下三角形面积是单位体积储能。Stress is force per area; strain is extension divided by original length. In the linear range, Young modulus is the stress–strain slope and the triangular area is stored energy per volume.
σ = F/A; ε = x/L; E = σ/ε; k = EA/LU = ½Fx = ½kx²; U/(AL) = ½σε
应变无单位；1 cm² = 10⁻⁴ m²。Strain is dimensionless; 1 cm² = 10⁻⁴ m².
2串联看力，并联看伸长Series: common force; parallel: common extension
串联的线材受同样的拉力，各段伸长相加。并联并固定在共同端板上的线材伸长相同，力相加；原长相同的牢固连接复合管还具有相同应变。Wires in series carry the same force and their extensions add. Parallel wires attached to common end plates share extension and their forces add; bonded tubes of equal initial length also share strain.
1/kseries = Σ(1/kᵢ); kparallel = ΣkᵢFbonded = εΣ(EᵢAᵢ)
同应变不等于同应力：较大的 E 对应较大的 σ。Equal strain does not mean equal stress: larger E gives larger σ.
3热约束与弯曲几何Thermal restraint and bending geometry
热膨胀若被完全阻止，机械压缩应变抵消自由热应变。弯曲时，中性层不伸长；离中性层越远，应变绝对值越大。Under complete thermal restraint, mechanical compression cancels the free thermal strain. In bending the neutral layer does not extend; strain magnitude grows with distance from it.
εth = αΔT; |σthermal| = EαΔTεbending = y/R; |σsurface| = Ea/(2R)
两者都先从约束求应变，再用 σ = Eε。In both cases, obtain strain from the constraint before applying σ = Eε.
按特征选方法Choose a method by the constraint
| 子题型Subtype | 识别特征Recognise | 核心步骤Core step | 真题Question |
|---|---|---|---|
| 热冲击Thermal shock | 内外温差Temperature difference | 用强度反求温差Solve for ΔT at the strength | 2025 Q1(i) |
| 复合管并联Parallel tubes | 牢固连接Firmly bonded | 共同应变，分力相加Equal strain; add forces | 2023 Q1(n) |
| 弯曲Bending | 圆弧与截面厚度Circular arc and thickness | 最大应变 a/(2R)Maximum strain a/(2R) | 2021 Q1(o) |
| 固定端热膨胀Fixed-end heating | 两端不动Immovable ends | F = EAαΔTF = EAαΔT | 2018 Q1(n) |
| 出水伸长变化Emerging load | 浮力消失Buoyancy is lost | Δx = ΔFL/(EA)Δx = ΔFL/(EA) | 2017 Q1(p) |