📂Physics

Various Definitions of Modulus of Elasticity

Definition ¹

1. A force that causes a body’s deformation is called stress or stress.[../1673]
2. The quantity that measures the degree of deformation resulting from stress is called strain.
3. The ratio of stress to strain is called the elastic modulus. $$ \text{elastic modulus} = \frac{\text{stress}}{\text{strain}} $$

There are commonly three types of elastic moduli.

Elasticity of length

4. The ratio $F / A$ of an external force $F$ to the cross-sectional area $A$ perpendicular to that force is called tensile stress.
5. The ratio $\Delta L / L_{0}$ of the change in length $\Delta L$ to the original length $L_{0}$ is called tensile strain.
6. The ratio $Y$ of tensile stress to tensile strain is called Young’s modulus. $$ Y := \frac{\text{tensile stress}}{\text{tensile strain}} = \frac{F / A}{\Delta L / L_{0}} $$

Elasticity of shape

7. When a force $F$ is applied tangentially along a shear plane with area $A$, the ratio $F / A$ of force to area is called shear stress.
8. If the shear plane is displaced horizontally by $\Delta x$ and the height of the shear layer is $h$, then $\Delta x / h$ is called the shear strain.
9. The ratio $S$ of shear stress to shear strain is called the shear modulus. $$ S := \frac{\text{shear stress}}{\text{shear strain}} = \frac{F / A}{\Delta x / h} $$

Elasticity of volume

10. For a body with surface area $A$ subject to a force $F$, $P = F / A$ is called pressure. A change in pressure $\Delta P$ is called volume stress.
11. The ratio $\Delta V / V_{0}$ of the change in volume $\Delta V$ to the original volume $V_{0}$ is called volume strain.
12. The ratio $B$ of volume stress to volume strain is called the bulk modulus. $$ B := \frac{\text{volume stress}}{\text{volume strain}} = - \frac{\Delta F / A}{\Delta V / V_{0}} = - \frac{\Delta P}{\Delta V / V_{0}} $$
13. The reciprocal $B^{-1}$ of the bulk modulus is called compressibility.

Here, the minus sign in the definition of $B$ is necessary to make $B$ positive, reflecting that volume decreases when pressure increases.

Explanation

Putting other matters aside, shear stress can be hard to grasp intuitively; a common illustrative example is the deformation that occurs when one presses down on a thick book and slides it sideways, as in the figure above.

Definition of a fluid:

1. A collective term for liquids and gases
2. An assembly of molecules arranged randomly and interacting with one another²

3. A substance that, at rest, is subject to normal stress, and in a flowing state, deforms continuously under shear forces so that it flows¹

However, when actually dealing with shear stress one is usually not concerned with the sliding-book phenomenon per se; at least when working with substances like fluids, one will encounter shear effects more frequently.

At first glance the phrase in the fluid definition that “normal stress acts at rest and deformation occurs due to shear forces in the flowing state” may be hard to understand, but once you grasp the concept of shear stress as “being pushed sideways,” as in the book example, it becomes much clearer to visualize how a fluid moves.

That a normal stress acts in the static state literally means a tendency to collapse or flow vertically under normal forces. When stressed from above by gravity or atmospheric pressure, the material undergoes continuous deformation rather than catastrophic failure. In contrast, the statement that shear stress acts in the flowing state refers to deformation driven by horizontal (tangential) forces that lead to flow.