Stretching Materials | 材料的拉伸

📚 Stretching Materials | 材料的拉伸

When a material is stretched, it experiences a force that tends to extend it. In A-Level Physics, you need to describe the relationship between load and extension, distinguish elastic and plastic behaviour, and use stress and strain to compare materials of different shapes. This topic links microscale ideas about atomic bonds to macroscale measurements such as the Young modulus and elastic strain energy.

当材料被拉伸时，它会受到使其伸长的力。在 A-Level 物理中，你需要描述载荷与伸长量之间的关系，区分弹性与塑性行为，并用应力和应变来比较不同形状的材料。这一主题将原子键的微观概念与杨氏模量和弹性应变能等宏观测量联系起来。

1. Hooke’s Law and the Force Constant | 胡克定律与力常数

Hooke’s law states that, provided the elastic limit is not exceeded, the extension of a spring or wire is directly proportional to the applied force. For a spring, we write this as F = kx, where F is the load, x is the extension, and k is the force constant or spring constant. The constant k has the SI unit newton per metre (N m⁻¹) and measures the stiffness of the spring.

胡克定律指出，只要不超过弹性极限，弹簧或金属丝的伸长量与施加的力成正比。对弹簧而言，我们可以写成 F = kx，其中 F 是载荷，x 是伸长量，k 是力常数或弹簧常数。常数 k 的 SI 单位是牛顿每米（N m⁻¹），它衡量弹簧的劲度。

F = kx

A graph of force against extension for a spring obeying Hooke’s law is a straight line through the origin. The gradient of this graph is equal to the spring constant k. A steeper line means a stiffer spring, because a larger force is needed to produce the same extension.

对于遵守胡克定律的弹簧，力-伸长量图是一条过原点的直线。该图线的斜率等于弹簧常数 k。斜率越大表示弹簧越硬，因为产生相同伸长量所需的力越大。

2. Force-Extension Graphs for Typical Materials | 典型材料的力-伸长图

Different materials produce very different force-extension graphs. A metal wire initially follows a straight line through the origin, showing Hooke’s law. Beyond the elastic limit, the graph curves and the wire shows a limit of proportionality, a yield point, and then plastic flow before breaking at the ultimate tensile force.

不同材料会产生截然不同的力-伸长图。金属丝最初沿过原点的直线变化，表现出胡克定律。超过弹性极限后，图线弯曲，金属丝依次出现比例极限、屈服点，然后在到达极限拉伸力之前发生塑性流动并最终断裂。

Rubber is an elastomer; its force-extension graph is not a straight line and shows hysteresis when loaded and unloaded. Glass and many ceramics are brittle: they obey Hooke’s law almost to fracture and show very little plastic deformation before breaking suddenly.

橡胶是一种弹性体，其力-伸长图不是直线，并且在加载和卸载时表现出滞后现象。玻璃和许多陶瓷是脆性材料：它们在几乎直到断裂时都近似遵守胡克定律，并在突然断裂前几乎没有塑性变形。

3. Elastic and Plastic Deformation | 弹性形变与塑性形变

Elastic deformation is reversible. When the load is removed, the material returns to its original length because the atoms are displaced only slightly from their equilibrium positions and can move back. Plastic deformation is permanent. In metals, plastic flow occurs when planes of atoms slip past each other, and the material does not return to its original shape when the load is removed.

弹性形变是可逆的。当载荷移除后，材料恢复到原始长度，因为原子只从其平衡位置发生微小偏移并能够回到原位。塑性形变是永久性的。在金属中，当原子平面彼此滑移时发生塑性流动，载荷移除后材料不会恢复原始形状。

The elastic limit is the maximum force before a material becomes permanently deformed. The limit of proportionality is the point where force and extension stop being directly proportional. For many materials these two points are very close, but they are distinct in principle.

弹性极限是材料发生永久形变前的最大力。比例极限是力与伸长量不再成正比的位置。对许多材料而言，这两点非常接近，但在原理上是不同的。

4. Stress and Strain | 应力与应变

To compare the strength and stiffness of materials with different lengths and cross-sectional areas, engineers use stress and strain. Stress is defined as the force applied per unit cross-sectional area. Strain is defined as the extension per unit original length. These definitions remove the effect of size and shape.

为了比较不同长度和截面积材料的强度和劲度，工程师使用应力和应变。应力定义为单位截面积上施加的力。应变定义为单位原始长度的伸长量。这些定义消除了尺寸和形状的影响。

σ = F / A

ε = ΔL / L₀

Stress has the SI unit pascal (Pa), where 1 Pa = 1 N m⁻². Strain is a ratio of two lengths, so it has no unit. It may be quoted as a decimal, a fraction, or a percentage.

应力的 SI 单位是帕斯卡（Pa），其中 1 Pa = 1 N m⁻²。应变是两个长度的比值，因此没有单位。它可以用小数、分数或百分比表示。

5. The Young Modulus | 杨氏模量

The Young modulus E of a material is the ratio of tensile stress to tensile strain within the proportional limit. It describes the stiffness of the material itself, independent of the dimensions of a particular sample. A large Young modulus means the material is difficult to stretch; a small value means it is easier to stretch.

材料的杨氏模量 E 是在比例极限内拉伸应力与拉伸应变的比值。它描述材料本身的劲度，与特定样品的尺寸无关。杨氏模量大表示材料难以拉伸；数值小表示材料较易拉伸。

E = σ / ε = (F L₀) / (A ΔL)

The SI unit of the Young modulus is the pascal. Typical values are about 2.0 × 10¹¹ Pa for steel, 1.2 × 10¹¹ Pa for copper, and 7.0 × 10¹⁰ Pa for aluminium. Since these values are very large, results are often quoted in GPa.

杨氏模量的 SI 单位是帕斯卡。典型值约为：钢 2.0 × 10¹¹ Pa，铜 1.2 × 10¹¹ Pa，铝 7.0 × 10¹⁰ Pa。由于这些数值很大，结果常用 GPa 表示。

6. Stress-Strain Graphs and Material Behaviour | 应力-应变图与材料行为

A stress-strain graph is obtained by dividing force by area and extension by original length. For a ductile metal such as copper or mild steel, the graph shows a straight-line region obeying Hooke’s law, then a yield point, a region of plastic flow, and finally fracture at the ultimate tensile stress.

应力-应变图是通过将力除以面积、伸长量除以原始长度得到的。对于铜或低碳钢等延性金属，该图先显示遵守胡克定律的直线区，然后出现屈服点、塑性流动区，最终在极限拉伸应力处断裂。

A brittle material such as glass shows a steep straight line and then sudden fracture with little plastic strain. A polymeric material such as rubber shows a large strain range and a curved, non-linear graph. The area under a stress-strain graph represents the strain energy stored per unit volume.

玻璃等脆性材料显示一条陡峭的直线，随后在几乎没有塑性应变的情况下突然断裂。橡胶等聚合物材料表现出很大的应变范围以及弯曲的非线性图线。应力-应变图下方的面积表示每单位体积储存的应变能。

7. Elastic Strain Energy | 弹性应变能

Work is done when a material is stretched, and this work is stored as elastic potential energy if the deformation is elastic. The work done is equal to the area under the force-extension graph. For a material obeying Hooke’s law, the graph is a triangle, so the energy stored is half the product of the final force and the total extension.

材料被拉伸时做功，如果形变是弹性的，这部分功以弹性势能的形式储存。所做的功等于力-伸长图下方的面积。对于遵守胡克定律的材料，图线为三角形，因此储存的能量等于最终力与总伸长量乘积的一半。

U = ½ F ΔL = ½ k (ΔL)²

The elastic strain energy has the SI unit joule (J). If the force is in newtons and the extension is in metres, the calculated energy is already in joules. This result is useful for designing springs, shock absorbers, and safety equipment.

弹性应变能的 SI 单位是焦耳（J）。如果力以牛顿为单位、伸长量以米为单位，计算出的能量就直接以焦耳为单位。这一结果对于设计弹簧、减震器和安全设备非常有用。

8. Measuring the Young Modulus | 杨氏模量的测量

In the laboratory, the Young modulus of a metal wire can be measured by hanging known masses from a long, thin wire. The original length L₀ is measured with a metre rule, and the diameter d is measured with a micrometer screw gauge at several positions to calculate the cross-sectional area A = πd²/4.

在实验室中，可以通过在长而细的金属丝上悬挂已知质量来测量其杨氏模量。原始长度 L₀ 用米尺测量，直径 d 用螺旋测微器在多个位置测量，以计算截面积 A = πd²/4。

The extension ΔL for each load is measured with a vernier scale or travelling microscope. Loads should be added gradually and removed gradually to check that the wire returns to its original length. A graph of stress against strain is plotted, and the gradient of the linear section gives the Young modulus.

每次加载后的伸长量 ΔL 用游标尺或移动显微镜测量。载荷应逐渐增加和逐渐减少，以检查金属丝是否恢复到原始长度。绘制应力-应变图，直线部分的斜率即为杨氏模量。

Keep the load below the elastic limit to avoid permanent stretching.
保持载荷低于弹性极限，以避免永久伸长。
Use a long, thin wire so the extension is easy to measure accurately.
使用长而细的金属丝，使伸长量易于精确测量。

9. Strain Energy Density and Toughness | 应变能密度与韧性

Strain energy density is the elastic energy stored per unit volume of a deformed material. For a material obeying Hooke’s law, this can be written as ½ × stress × strain, which equals ½ E ε². The area under a stress-strain graph gives the strain energy density even when the graph is not linear.

应变能密度是单位体积变形材料中储存的弹性势能。对于遵守胡克定律的材料，其值可写为 ½ × 应力 × 应变，即 ½ E ε²。即使应力-应变图不是线性的，图线下的面积仍给出应变能密度。

Toughness is the ability of a material to absorb energy before fracturing. A tough material has a large area under the whole stress-strain graph, combining both strength and ductility. A brittle material may be strong but not tough, because it fractures after only a small strain.

韧性是材料在断裂前吸收能量的能力。坚韧材料在整个应力-应变图下有较大的面积，兼具强度与延展性。脆性材料可能强度高但韧性差，因为它只经过很小的应变就会断裂。

10. Applications and Safety Considerations | 应用与安全考虑

Engineers use the concepts of Young modulus, yield stress, and strain energy density when selecting materials for construction, transport, and safety equipment. Steel is used in buildings and cables because it has a high Young modulus and good ductility. Rubber is used in tyres and engine mounts because it can absorb large strain energy and return to shape.

工程师在为建筑、交通和安全设备选择材料时，会使用杨氏模量、屈服应力和应变能密度等概念。钢因其高杨氏模量和良好的延展性而用于建筑和缆索。橡胶用于轮胎和发动机支架，因为它能吸收大量应变能并恢复形状。

Structures must be designed so that the maximum working stress remains well below the yield stress. A safety factor is the ratio of the yield stress or ultimate stress to the allowable working stress. This protects against unexpected loads, temperature changes, and material defects.

结构设计必须使最大工作应力远低于屈服应力。安全系数是屈服应力或极限应力与许用工作应力的比值。这可以防止意外载荷、温度变化和材料缺陷带来的破坏。

Published by TutorHao | Physics Revision Series | aleveler.com

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Stretching Materials | 材料的拉伸