Does Stress Cause Strain — or the Other Way Around?
In mechanics, the relationship between stress and strain is fundamental — but understanding which causes which is more nuanced and requires careful consideration.
The standard view is that stress causes strain. Apply a force to a material (which creates stress — force per unit area), and the material deforms (which produces strain — deformation per unit length). This relationship is formalized by Hooke’s Law for linear elastic materials:
\[\epsilon = \frac{\sigma}{E}\]Here, \(\epsilon\) is strain, \(\sigma\) is stress, and \(E\) is Young’s modulus — a material constant that quantifies stiffness. This equation treats stress as the input and strain as the output.
However, in practical applications — especially in lab settings or engineering design — this relationship often seems reversed. We might impose a known strain by stretching or compressing a material by a fixed amount, and then measure the resulting stress. In this context, strain appears to come first, and stress becomes the system’s response.
So, which one actually causes the other?
Causality Depends on the Perspective
Physically, external forces — such as loads, pressures, or constraints — generate internal stress within a material. The material then responds to that stress by deforming. In this framework, stress is the cause, and strain is the effect. This view aligns with how materials behave in the natural world.
Experimentally, however, the setup is often reversed. We control how much we stretch or compress a material (strain) and then observe how much resistance (stress) it produces. This doesn’t contradict the physical model — it simply shifts the practical focus to what’s being controlled versus what’s being measured.
Mathematically, the stress-strain relationship is linear (at least in the elastic range), so either stress or strain can be treated as the input, depending on the situation. The equation works in both directions.
But There Are Exceptions
In many real-world cases, strain is the starting point, and the material reacts with internal stress.
Take thermal expansion as an example: heating a material causes it to expand — that’s strain. If the material is free to expand, no stress develops. However, if it’s constrained, the expansion is resisted, and stress arises as a response to the imposed strain.
A similar concept applies in shrink-fit assemblies, where a component is intentionally strained to fit tightly inside another. The strain is applied during assembly, and the resulting stress locks the parts together.
In plastic deformation, once a material yields, it undergoes permanent strain even if the applied stress doesn’t increase. After unloading, residual strain remains — and in some cases, internal stress can also persist.
These examples show that strain can sometimes be directly imposed, with the material’s internal stress developing as a response. In these cases, stress is not the original cause; rather, it’s the system’s reaction to a constraint or deformation.
Conclusion
In classical mechanics, stress is typically considered the cause of strain, as external forces induce deformation. But in experimental settings and real-world applications, we also impose strain and measure the resulting stress. The key is context: what is being controlled and what is being observed.