📚 Stretching Materials: Hooke’s Law, Stress, Strain, and Young Modulus | 拉伸材料：胡克定律、应力、应变与杨氏模量
When forces are applied to solid objects, they can change shape – this is deformation. Understanding how materials respond to tensile forces is central to A‑level physics, linking everyday observations to fundamental mechanical properties. We explore Hooke’s law, elastic limits, stress and strain, the Young modulus, and the energy stored in stretched materials.
当力作用于固体时，固体会改变形状——这就是形变。理解材料如何响应拉伸力是A‑level物理的核心内容，它将日常观察与基本力学性质联系起来。我们将探讨胡克定律、弹性极限、应力和应变、杨氏模量以及储存在拉伸材料中的能量。
1. Deformation and Elasticity | 形变与弹性
Deformation is the change in shape or size of an object due to an applied force. Deformation can be elastic, meaning the object returns to its original dimensions when the force is removed, or plastic, where the deformation is permanent. Elasticity is the property that allows a material to regain its original form – understanding the boundary between elastic and plastic behaviour is essential for safe engineering design.
形变是指物体因受力而发生的形状或尺寸变化。形变可以是弹性的，即撤去力后物体恢复原始尺寸；也可以是塑性的，即形变永久保留。弹性是材料恢复原状的性质——理解弹性与塑性行为之间的界限对安全工程设计至关重要。
2. Hooke’s Law | 胡克定律
Hooke’s law states that, for many materials under modest loads, the extension ΔL produced is directly proportional to the applied force F, provided the elastic limit is not exceeded. Mathematically,
F = kΔL
where k is the spring constant (stiffness) of the sample. This linear relationship is the foundation for analysing springs and elastic materials, and it applies to any object that obeys Hooke’s law, not just coiled springs.
胡克定律指出，对于许多材料在适度载荷下，只要不超过弹性极限，产生的伸长量ΔL与施加的力F成正比。数学表达为
F = kΔL
其中k是样品的劲度系数（刚度）。这一线性关系是分析弹簧和弹性材料的基础，适用于任何遵循胡克定律的物体，而不仅仅是螺旋弹簧。
3. Force–Extension Graphs | 力–伸长量图像
A force–extension graph for a material usually shows a linear region where Hooke’s law holds, followed by a curve indicating the material no longer extends proportionally. The gradient of the linear part equals the spring constant k. Key points on the graph include the limit of proportionality (where linearity ends), the elastic limit (the maximum stress for purely elastic behaviour), and the yield point, beyond which plastic flow occurs.
材料的力–伸长量图像通常显示一个线性区域，在此区域内遵循胡克定律，随后出现曲线，表明材料不再按比例伸长。线性部分的斜率等于劲度系数k。图像上的关键点包括比例极限（线性终止处）、弹性极限（纯弹性行为的最大应力）以及屈服点，超过此点即发生塑性流动。
4. Stress and Strain | 应力与应变
To compare materials independently of shape and size, we define tensile stress σ as the force per unit cross‑sectional area:
σ = F / A
where F is the applied force and A is the original cross‑sectional area. Tensile strain ε is the fractional extension relative to the original length L₀:
ε = ΔL / L₀
Stress has units of pascals (Pa) and strain is dimensionless, often expressed as a percentage. Using stress and strain removes geometric effects, allowing a direct comparison of material stiffness.
为了在不受形状和尺寸影响的情况下比较材料，我们定义拉伸应力σ为单位横截面积上的力：
σ = F / A
其中F是施加的力，A是原始横截面积。拉伸应变ε是相对于原始长度L₀的伸长比率：
ε = ΔL / L₀
应力的单位是帕斯卡（Pa），应变是无量纲的，通常以百分比表示。使用应力与应变消除了几何因素的影响，可直接比较材料的刚度。
5. The Young Modulus | 杨氏模量
The Young modulus E characterises the stiffness of a material in tension. It is defined as the ratio of stress to strain within the limit of proportionality:
E = σ / ε = (F L₀) / (A ΔL)
he Young modulus is a fundamental property of the material and does not depend on the dimensions of a particular sample.
杨氏模量E表征材料在拉伸下的刚度。它被定义为在比例极限内应力与应变的比值：
E = σ / ε = (F L₀) / (A ΔL)
杨氏模量高意味着材料非常刚硬，在载荷下伸长很小（例如钢，E ≈ 200 GPa）。杨氏模量是材料的基本属性，不依赖于特定试样的尺寸。
6. Stress–Strain Graphs and Material Behaviour | 应力–应变图像与材料行为
A stress–strain graph captures the full mechanical fingerprint of a material. Below the limit of proportionality, the graph is a straight line whose gradient equals the Young modulus. For ductile materials such as copper, the curve levels off at the yield point and extends through a large plastic region before necking and fracture. Brittle materials like glass show little or no plastic region and fracture suddenly. Polymeric materials often display rubber‑like elasticity or viscoelasticity, with hysteresis loops on loading and unloading.
应力–应变图像捕捉了材料完整的力学指纹。在比例极限以下，图像是一条直线，其斜率等于杨氏模量。对于铜等韧性材料，曲线在屈服点趋于平缓，并经过大范围的塑性区域，然后颈缩并断裂。玻璃等脆性材料几乎或完全不显示塑性区域，会突然断裂。聚合物材料通常表现出类似橡胶的弹性或粘弹性，在加载和卸载时出现滞后回线。
7. Elastic Strain Energy | 弹性应变能
Work done to stretch an elastic material is stored as elastic potential energy. For an object obeying Hooke’s law, the energy stored is the area under the force–extension graph:
Eₚ = ½ F ΔL = ½ k (ΔL)²
s and strain, the elastic energy per unit volume is given by the area under the linear stress– seful when designing springs and safety components that must absorb energy without damage.
拉伸弹性材料所做的功以弹性势能的形式储存。对于遵循胡克定律的物体，储存的能量等于力–伸长量图像下的面积：
Eₚ = ½ F ΔL = ½ k (ΔL)²
释放力时，这部分能量可以恢复。用应力和应变表示，单位体积的弹性能量等于线性应力–应变图像下的面积：单位体积能量 = ½ σ ε。这一概念在设计弹簧和必须在不损坏的情况下吸收能量的安全部件时非常有用。
8. Plasticity, Ductility, and Brittleness | 塑性、韧性与脆性
Plastic deformation begins when the stress exceeds the elastic limit. In ductile materials, significant permanent extension occurs before fracture, which is accompanied by necking – a reduction in cross‑sectional area. Brittle materials, by contrast, fail with little prior plastic strain. The ultimate tensile stress (UTS) is the maximum stress a material can withstand before breaking; for brittle materials, UTS and fracture stress coincide. Understanding these differences helps engineers select materials for specific applications, such as ductile metals for cables and brittle ceramics for furnace linings.
当应力超过弹性极限时，塑性形变开始发生。在韧性材料中，断裂前会产生明显的永久伸长，并伴随颈缩——横截面积减小。相反，脆性材料在几乎没有塑性应变的情况下就会失效。极限拉伸应力（UTS）是材料在断裂前所能承受的最大应力；对于脆性材料，UTS与断裂应力重合。理解这些差异有助于工程师为特定应用选择材料，例如线缆用韧性金属，炉衬用脆性陶瓷。
9. Determining the Young Modulus Experimentally | 实验测定杨氏模量
length L₀ is measured with a metre rule, the diameter with a micrometer to calculate cross‑ ΔL yields a straight line whose gradient can be used to compute E = (gradient × L₀) / A. Care must be taken not to exceed the elastic limit and to account for any initial kinks in the wire.
经典的学校实验使用一根细长金属丝，一端固定，绕过滑轮，在另一端施加已知质量的重物拉伸它。用米尺测量原始长度L₀，用千分尺测量直径以计算横截面积A，每个载荷下的伸长量通过读取金属丝上的标记与游标尺的对应关系获得。绘制力F与伸长量ΔL的关系图，得到一条直线，利用其斜率可计算E = (斜率 × L₀) / A。必须注意不要超过弹性极限，并考虑金属丝中可能存在的初始弯折。
10. Summary of Key Relationships | 关键关系总结
A firm grasp of the following quantities and their links is vital for A‑level success:
- Hooke’s law: F = kΔL (valid only in the linear region)
- Stress: σ = F / A
- Strain: ε = ΔL / L₀
- Young modulus: E = σ / ε = (F L₀) / (A ΔL)
- Elastic energy: Eₚ = ½ F ΔL = ½ k (ΔL)²; energy per unit volume = ½ σ ε
- Limit of proportionality vs elastic limit: often close but conceptually distinct
- Ductile vs brittle: extensive plastic region vs sudden fracture
熟练掌握以下物理量及其关联对A‑level成功至关重要：
- 胡克定律：F = kΔL（仅在线性区域有效）
- 应力：σ = F / A
- 应变：ε = ΔL / L₀
- 杨氏模量：E = σ / ε = (F L₀) / (A ΔL)
- 弹性能量：Eₚ = ½ F ΔL = ½ k (ΔL)²；单位体积能量 = ½ σ ε
- 比例极限与弹性极限：往往接近但在概念上是不同的
- 韧性 vs 脆性：大范围塑性区域 vs 突然断裂
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图