📚 Compressive and Tensile Forces | 压缩力与拉伸力
When external forces act on a solid object, they tend to change its shape or size. Depending on the direction of the applied force relative to the object, materials may experience tension (being pulled apart) or compression (being pushed together). Understanding how materials respond to tensile and compressive forces is central to solid mechanics, and it forms the basis of many engineering and structural applications. In this article, we explore the concepts of stress, strain, elasticity, plasticity, and the characteristic behaviours of ductile and brittle materials under both types of loading.
当外力作用于固体时，它们会使物体的形状或大小发生变化。根据作用力方向与物体取向的关系，材料可能受到拉伸（被拉开）或压缩（被压紧）作用。理解材料在拉伸力与压缩力下的响应是固体力学的核心，也是许多工程与结构应用的基础。本文将探讨应力、应变、弹性、塑性等概念，以及延性材料和脆性材料在这两类载荷下的典型行为。
1. Defining Tensile and Compressive Forces | 定义拉伸力与压缩力
A tensile force is one that tends to stretch a material. It acts along the length of an object, pulling neighbouring particles away from each other, which increases the object's length if it is free to deform. Common examples include the tension in a cable supporting a lift or the stretching of a rubber band. Tensile forces are usually considered positive in sign when calculating stress.
拉伸力是一种趋向于拉长材料的力。它沿物体的长度方向作用，将相邻粒子拉开，如果物体能够自由变形，其长度就会增加。常见的例子有支撑电梯的缆绳中的张力，或橡皮筋的拉长。在计算应力时，拉伸力通常取为正值。
A compressive force, on the other hand, tends to squash or shorten a material. It pushes particles together, reducing the distance between them and decreasing the length of the object. Columns supporting a building, the legs of a chair, and the piston rod in an engine all experience compressive forces. In many contexts, compressive stress is considered negative, although magnitudes are often used when comparing tensile and compressive strengths.
相反，压缩力趋向于压扁或缩短材料。它将粒子推向一起，减小它们之间的距离，从而缩短物体的长度。支撑建筑物的柱子、椅子的腿、发动机中的活塞杆都承受压缩力。在许多场合下，压应力被视作负值，但在比较拉伸强度与压缩强度时，通常只使用其绝对值。
2. Stress and Strain | 应力与应变
To compare the effect of forces on materials of different sizes, engineers use the notion of stress (σ). Tensile or compressive stress is defined as the force F applied perpendicularly to a cross-section, divided by the original cross-sectional area A₀. The SI unit of stress is the pascal (Pa) or N m⁻².
为了比较力对不同尺寸材料的影响，工程师引入了应力（σ）的概念。拉伸应力或压缩应力定义为垂直施加在横截面上的力 F 除以原始横截面积 A₀。应力的国际单位是帕斯卡（Pa），即 N m⁻²。
σ = F / A₀
Strain (ε) is the measure of deformation representing the relative change in length. For both tension and compression, strain is usually calculated as the change in length ΔL divided by the original length L₀. Because strain is a ratio of two lengths, it has no units; it is often expressed as a percentage.
应变（ε）是衡量形变的量，表示长度的相对变化。无论是拉伸还是压缩，应变通常用长度变化量 ΔL 除以原始长度 L₀ 来计算。由于应变是长度之比，因此无单位；常用百分数表示。
ε = ΔL / L₀
In tension, ΔL is positive and the strain is tensile; in compression, ΔL is negative and the strain is compressive. The stress–strain relationship for small strains is the foundation of elasticity.
在拉伸中，ΔL 为正，应变是拉伸应变；在压缩中，ΔL 为负，应变是压缩应变。小应变下的应力–应变关系是弹性的基础。
3. Hooke's Law and the Spring Constant | 胡克定律与弹簧常数
Many materials obey Hooke' e spring constant (stiffness) of the sample. For a wire under tension, k depends on the material, its original length, and its cross-sectional area. Within the limit of proportionality, a graph of force against extension is a straight line through the origin, and the material returns to its original length when the load is removed.
许多材料在某个限度内遵守胡克定律：伸长量 x（或压缩量）与所施加的力 F 成正比。这可以写成 F = k x，其中 k 是试样的弹簧常数（劲度系数）。对于承受拉伸的金属丝，k 取决于材料、原始长度和横截面积。在比例极限内，力–伸长图是一条通过原点的直线，撤去载荷后材料会恢复原始长度。
It is important to appreciate that Hooke's law applies equally to compressive forces as long as buckling is avoided and the strain is small. For a helical spring, compression simply shortens the coils, and the same linear relationship holds until the coils touch one another.
重要的是要认识到，只要不发生屈曲且应变很小，胡克定律同样适用于压缩力。对于螺旋弹簧，压缩只是缩短了线圈，同样的线性关系保持到线圈相互接触为止。
4. The Force–Extension Graph for a Metal Wire | 金属丝的力–伸长图
A force–extension (or stress–strain) graph for a typical ductile metal, such as annealed copper or mild steel, reveals several distinct regions. Starting from the origin, a straight-line portion corresponds to Hookean behaviour. The end of this straight line marks the limit of proportionality. Slightly beyond it lies the elastic limit: up to this point, the wire will return to its original length when unloaded, although the relationship may no longer be linear.
典型延性金属（如退火铜或低碳钢）的力–伸长（或应力–应变）图上显示出几个不同的区域。从原点开始，直线段对应胡克行为。直线的终点标志着比例极限。再稍远处是弹性极限：在此点之前，虽然关系可能不再呈线性，但卸去载荷后金属丝仍能恢复原始长度。
Beyond the elastic limit, the material undergoes plastic deformation – permanent changes that remain after the force is removed. In mild steel, a clear yield point is often observed, where the strain increases with little or no increase in stress. Further loading leads to work hardening, and the force reaches a maximum, which defines the ultimate tensile strength. Afterwards, a neck begins to form, the load drops, and the wire eventually fractures.
超出弹性极限后，材料发生塑性形变——撤去力后仍保留永久变化。在低碳钢中，通常可观察到一个明显的屈服点，此时应力增加很小而应变却大幅增加。继续加载导致加工硬化，力达到最大值，该最大值定义了极限拉伸强度。随后开始形成颈缩，载荷下降，最终金属丝断裂。
5. Elastic Limit and Yield Point | 弹性极限与屈服点
The elastic limit is the maximum stress a material can withstand without any permanent deformation. In practice, it is very close to the limit of proportionality for many metals. If the stress never exceeds the elastic limit, the material behaves elastically, and the loading–unloading cycle follows the same path on the force–extension graph.
弹性极限是材料在完全不产生永久形变的条件下所能承受的最大应力。实际上，对许多金属来说，弹性极限与比例极限非常接近。如果应力从未超过弹性极限，材料就表现为弹性行为，加载–卸载循环在力–伸长图上沿着同一条路径返回。
The yield point is the stress at which a material begins to deform plastically. Some materials, like mild steel, show a distinct upper yield point followed by a lower yield point and a period of large extension at nearly constant load. For materials without a sharp yield point, such as aluminium, a proof stress is adopted – typically the stress that produces a 0.2% permanent strain.
屈服点是材料开始发生塑性形变的应力。一些材料，如低碳钢，会显示出明显的上屈服点，然后是下屈服点，并且在近乎恒定的载荷下延伸较长距离。对于没有明显屈服点的材料（如铝），则采用规定非比例延伸强度——通常是产生 0.2% 永久应变的应力。
6. Plastic Deformation and Necking | 塑性形变与颈缩
Plastic deformation occurs when atomic planes slide over one another and do not return to their original positions. In ductile materials, this sliding is facilitated by the movement of dislocations. Once plastic strain has occurred, the shape change is permanent, and the work done in producing it is dissipated as heat or stored as internal energy.
当原子层之间发生滑移且不能回到初始位置时，就发生塑性形变。在延性材料中，这种滑移通过位错运动来实现。一旦发生塑性应变，形状变化就是永久的，产生这一变化所做的功以热的形式耗散或以内能形式储存。
After the ultimate tensile strength is reached, a local reduction in cross-sectional area, called necking, begins. Because the true stress continues to increase within the neck, the load-carrying capacity declines, and the engineering stress–strain curve drops. Necking concentrates the deformation in a small region, leading rapidly to fracture in tension. Compressive loading does not produce a neck in the same way; instead, ductile cylinders tend to bulge uniformly.
达到极限拉伸强度后，局部横截面积的减小——即颈缩——开始出现。由于颈内真应力继续增加，承载能力下降，工程应力–应变曲线随之下降。颈缩使形变集中在一个小区域，导致拉伸试样快速断裂。压缩载荷不会以相同方式产生颈缩；相反，延性圆柱体会均匀地向外鼓出。
7. Ultimate Tensile Strength and Fracture | 极限拉伸强度与断裂
The ultimate tensile strength (UTS) is the maximum engineering stress a material can withstand while being stretched before necking becomes significant. It is calculated by dividing the maximum applied force Fₘₐₓ by the original cross-sectional area A₀. The UTS is a key design parameter, indicating the maximum load a component can carry under simple tension.
极限拉伸强度（UTS）是材料在发生显著颈缩前所能承受的最大工程应力。它通过最大作用力 Fₘₐₓ 除以原始横截面积 A₀ 来计算。UTS 是一个关键的设计参数，表示构件在简单拉伸下所能承受的最大载荷。
UTS = Fₘₐₓ / A₀
Fracture occurs when the material can no longer sustain the load. In tension, the breaking stress is often lower than the UTS because of the necking effect. Brittle materials, on the other hand, fracture at the maximum load without appreciable necking, so their breaking strength is essentially equal to their UTS. Compressive fracture in brittle materials often involves shear cracking at an angle of about 45° to the load direction.
当材料无法再承受载荷时，断裂便发生了。在拉伸中，由于颈缩效应，断裂应力通常低于 UTS。相反，脆性材料在最大载荷处断裂而无明显的颈缩，因此其断裂强度基本等于 UTS。脆性材料的压缩断裂往往会产生与载荷方向约成 45° 角的剪切裂纹。
8. Compressive Behaviour of Materials | 材料的压缩行为
Under compressive loading, the cross-sectional area increases and the length decreases. For ductile metals, the stress–strain curve in compression is initially identical to that in tension; the Young modulus and yield strength are roughly the same. However, instead of necking, a cylinder of ductile material will bulge outwards as its height is reduced, and it will not fracture unless extremely deformed.
在压缩载荷下，横截面积增大而长度减小。对于延性金属，压缩下的应力–应变曲线初始部分与拉伸相同；杨氏模量和屈服强度也大致相等。然而，随着高度的减少，延性材料圆柱体不会发生颈缩，而是向外鼓出，除非极度变形，否则不会断裂。
Brittle materials such as concrete, glass and cast iron behave quite differently. They are usually much stronger in compression than in tension. A concrete column, for instance, can support a very large compressive load but will crack under relatively small tensile stresses. This is why steel reinforcement is added to concrete in order to carry tensile forces in reinforced-concrete structures.
脆性材料（如混凝土、玻璃和铸铁）的行为则大不相同。它们的抗压强度通常远高于抗拉强度。例如，混凝土柱可承受非常大的压缩载荷，但在较小的拉应力下就会开裂。这就是钢筋混凝土结构中添加钢筋来承受拉伸力的原因。
9. Young Modulus: Tensile and Compressive | 杨氏模量：拉伸与压缩
The Young modulus E is a measure of the stiffness of a material. It is defined as the ratio of tensile (or compressive) stress to tensile (or compressive) strain, provided the material is
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