第12章静的つり合いと弾性

When you dive into water, you feel a force pressing on every part of your body from all directions. What you are experiencing then is bulk stress, or in other words, pressure. Bulk stress always tends to decrease the volume enclosed by the surface of a submerged object. The forces of this “squeezing” are always perpendicular to the submerged surface Figure 12.22. The effect of these forces is to decrease the volume of the submerged object by an amount $\text{Δ}V$ compared with the volume $V_0$ of the object in the absence of bulk stress. This kind of deformation is called bulk strain and is described by a change in volume relative to the original volume:

The bulk strain results from the bulk stress, which is a force $F_{\perp }$ normal to a surface that presses on the unit surface area A of a submerged object. This kind of physical quantity, or pressure p, is defined as

We will study pressure in fluids in greater detail in Fluid Mechanics. An important characteristic of pressure is that it is a scalar quantity and does not have any particular direction; that is, pressure acts equally in all possible directions. When you submerge your hand in water, you sense the same amount of pressure acting on the top surface of your hand as on the bottom surface, or on the side surface, or on the surface of the skin between your fingers. What you are perceiving in this case is an increase in pressure $\text{Δ}p$ over what you are used to feeling when your hand is not submerged in water. What you feel when your hand is not submerged in the water is the normal pressure $p_0$ of one atmosphere, which serves as a reference point. The bulk stress is this increase in pressure, or $\text{Δ}p,$ over the normal level, $p_0.$

When the bulk stress increases, the bulk strain increases in response, in accordance with Equation 12.33. The proportionality constant in this relation is called the bulk modulus, B, or

The minus sign that appears in Equation 12.39 is for consistency, to ensure that B is a positive quantity. Note that the minus sign $(–)$ is necessary because an increase $\text{Δ}p$ in pressure (a positive quantity) always causes a decrease $\text{Δ}V$ in volume, and decrease in volume is a negative quantity. The reciprocal of the bulk modulus is called compressibility $k,$ or

The term ‘compressibility’ is used in relation to fluids (gases and liquids). Compressibility describes the change in the volume of a fluid per unit increase in pressure. Fluids characterized by a large compressibility are relatively easy to compress. For example, the compressibility of water is $4.64\times {10}^{\mathrm{-5}}\text{/atm}$ and the compressibility of acetone is $1.45\times {10}^{\mathrm{-4}}\text{/atm}.$ This means that under a 1.0-atm increase in pressure, the relative decrease in volume is approximately three times as large for acetone as it is for water.