In this blog post, we’ll use a rubber band as an example to examine the relationship between stress and strain, and explore the process by which an object deforms and eventually fails when a force is applied to it.
Many of you have probably played with a rubber band before. If you stretch a rubber band and then let go, it may return to its original shape, or it may remain stretched beyond its original length. Furthermore, if you pull it too hard, it will eventually snap. It is well known that applying sufficient force to an object causes it to change in length. So, how much force must be applied to deform an object to the desired length? The answer to this question can be found by understanding the relationship between stress and strain.
First, we need to examine what stress and strain are. Stress (σ) refers to the force acting per unit area; it is the force (P) applied to an object divided by its cross-sectional area (A). Mathematically, this is expressed as σ = P/A. For example, if a force of 50 N is applied to an object with a cross-sectional area of 10, the stress is 50 N ÷ 10 = 5 Pa. When stress is applied to an object in this way, its length changes; the ratio of the deformed length to the original length is called strain (ε). For example, if a metal rod with a length of 1.00 cm is subjected to stress and its length becomes 1.01 cm, the strain is ε = (1.01 – 1.00) / 1.00 = 0.01.
So, what is the relationship between stress and strain? There is a standard experimental method to determine this: a rod specimen with a constant cross-sectional area is secured, then pulled to apply stress, and the resulting strain is measured. Using the values measured in this way, a curve representing the relationship between stress and strain can be obtained.
If you stretch a rubber band and then release it, it returns to its original length. Similarly, the range in which an object returns to its original length after the stress is removed is called the elastic region (elastic behavior). In this region, stress and strain are linearly proportional to each other, and this relationship is known as Hooke’s Law. In this case, the relationship between stress and strain can be expressed as σ = Eε. Here, E is Young’s modulus, an intrinsic material property that varies depending on the type of material. The maximum stress within the range where Hooke’s Law applies is called the proportional limit. Once the stress slightly exceeds the proportional limit, the relationship between stress and strain is no longer linear. The object then reaches the elastic limit; up to this point, if the stress is removed, the object’s length returns to its original state.
If a rubber band is stretched too far and then released, it may not return to its original length but remain stretched. The phenomenon in which the length changes permanently due to stress beyond the elastic range is called yielding, and the range in which this deformation occurs is referred to as the plastic range. The stress that causes yielding is called the yielding stress. Once the yielding stress is reached, the object continues to deform even without a significant increase in stress. Thereafter, a region appears where the strain rate increases only when the stress is raised further; this is called strain hardening. In this region, the stress continues to increase until it reaches a peak; the stress at this point is called the ultimate tensile strength. This represents the maximum stress the material can withstand. Beyond the ultimate tensile strength, the object continues to stretch, but the cross-sectional area of a specific section begins to decrease rapidly.
This is similar to the phenomenon where the middle section of a rubber band turns white and becomes thinner just before it snaps. This phenomenon is called necking. Once necking begins, the engineering stress applied to the object gradually decreases below the ultimate tensile strength. Eventually, the object can no longer withstand the load and fails; the stress at this point is called the fracture stress.
However, while some objects break only after being significantly stretched, others fail almost immediately without much elongation. What causes this difference? It can be distinguished by examining the relationship between stress and strain. A ductile material is one that exhibits a large amount of strain before failure. It also has the property of absorbing a great deal of impact or energy. Typical examples of ductile materials include mild steel, brass, and zinc. Conversely, materials that exhibit almost no yield and very small strain are called brittle materials. Brittle materials are weak under tensile stress but strong under compressive stress. Typical examples include concrete and cast iron. However, it is more appropriate to understand a rubber band not as a ductile material in the same sense as metal, but as an elastic material that exhibits very large elastic deformation.
In this way, we have examined the process by which an object deforms and eventually fails when subjected to tensile stress. The relationship between the force pulling on an object and the resulting deformation may seem somewhat difficult to grasp, but by considering the familiar example of a rubber band, we can understand the relationship between stress and strain much more easily.
