![]() Where \(\mu\) is the viscosity, which we also call the dynamic viscosity. Similarly, a linear viscous fluid (or simply, a “viscous fluid”) has a linear relationship between stress, \(\sigma\), and strain rate, \(\dot\), We often just call this an “elastic material”, with the “linear” part implied. If you graphed their relationship, it would be a straight line with a slope given by Young’s Modulus, $E$. The above example is of a linear elastic material because there is a direct proportionality between stress and strain. A force per unit area is defined as a stress, $\sigma$:Ī change in distance per distance is a unitless quantity called a strain, $\varepsilon$:Īt this point, it is straightforward to rewrite the above equation with Young’s modulus to one that links stress and strain. Now, let’s apply a couple of definitions. It is like a spring constant, except it depends only only the properties of the material – not on its geometry. What is this $E$ term? This is Young’s Modulus. To address (2), we normalize the force by the cross-sectional area of the spring material, $A$. To address (1), we reframe $x$ as a change in length of the material $\Delta l$ divided by its starting length, $l$. This removes the effect of the wire thickness. ![]() We normalize by the cross-sectional area of the material.This removes the effect of the spring length. We normalize deformations to the length, $l$, of the spring.But in order to understand the material properties, we want to strip away these questions of length and wire width. One of the reasons that we study springs in physics – arguably, the main reason – is because they are such a good analogy to elastic materials in general. What is the material? Different materials have different rigidities.How thick is the wire? Thicker wires are stiffer.How long is the spring? Longer springs have more coils that can deform, and therefore become more easily stretched or compressed.The spring constant wraps (heh!) together many different variables. Here, \(F\) is a force, \(k\) is the spring constnt, and \(x\) is some amount of displacement. Linear relationship between stress and strain with a spring: image from Wikimedia Commons.įirst-semester physics courses typically introduce the linear relationship between an applied force and the lengthening or shortening of a spring. You may recall the spring constant from an introductory physics course, which relates to Young’s modulus and is helpful to recall when learning about viscosity. ![]() In the abstract, it is a fluid-mechanics analogy to Young’s Modulus in Hooke’s Law for elastic materials. Viscosity is a measure of resistance to flow.
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