Additional: Relation between Elastic Constants

Relationships between Y, B, G, and $\nu$

For a perfectly elastic, homogeneous, and isotropic material, the three elastic moduli (Young's Modulus $Y$, Bulk Modulus $B$, and Shear Modulus $G$) and Poisson's ratio ($\nu$) are not independent of each other. They are related by specific mathematical equations. Isotropic means the material properties are the same in all directions, and homogeneous means the properties are uniform throughout the material.

Deriving these relationships from first principles involves considering the strain components resulting from various stress states in a three-dimensional elastic solid and applying Hooke's Law in its generalized form. These derivations are typically covered in more advanced courses on mechanics of materials or theory of elasticity.

However, the resulting relationships are important for understanding how these properties are interconnected and for calculating one property if others are known. Here are the key relationships:

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Relationships Formulas

The relationships between $Y$, $B$, $G$, and $\nu$ for an isotropic solid are:

$ Y = 2G(1+\nu) $

$ Y = 3B(1-2\nu) $

From these two fundamental relationships, we can derive other relations:

Equating the expressions for Y:

$ 2G(1+\nu) = 3B(1-2\nu) $

$ 2G + 2G\nu = 3B - 6B\nu $

$ 2G\nu + 6B\nu = 3B - 2G $

$ \nu (2G + 6B) = 3B - 2G $

$ \nu = \frac{3B - 2G}{6B + 2G} $

Also, we can express Y in terms of B and G:

From $Y = 2G(1+\nu)$, $\nu = \frac{Y}{2G} - 1$.

Substitute this into $Y = 3B(1-2\nu)$:

$ Y = 3B \left(1 - 2\left(\frac{Y}{2G} - 1\right)\right) $

$ Y = 3B \left(1 - \frac{Y}{G} + 2\right) = 3B \left(3 - \frac{Y}{G}\right) $

$ Y = 9B - \frac{3BY}{G} $

$ Y + \frac{3BY}{G} = 9B $

$ Y \left(1 + \frac{3B}{G}\right) = 9B $

$ Y \left(\frac{G + 3B}{G}\right) = 9B $

$ Y = \frac{9BG}{3B+G} $

So, the key relationships are:

$ \mathbf{Y = 2G(1+\nu)} $

$ \mathbf{Y = 3B(1-2\nu)} $

$ \mathbf{\nu = \frac{3B - 2G}{6B + 2G}} $

$ \mathbf{Y = \frac{9BG}{3B+G}} $

These equations show that if any two of the elastic constants (Y, B, G, $\nu$) for an isotropic material are known, the other two can be calculated.

For most materials, $\nu$ lies between 0 and 0.5. This implies constraints on the relative values of $Y$, $B$, and $G$. For example, since $1-2\nu \ge 0$, $Y = 3B(1-2\nu) \le 3B$. Also, since $1+\nu \ge 1$, $Y = 2G(1+\nu) \ge 2G$.

Answer:

Compressibility (Inverse of Bulk Modulus)

The Bulk modulus ($B$) quantifies a material's resistance to compression under uniform pressure. A high Bulk modulus means the material is hard to compress. The opposite concept, how easily a material can be compressed, is described by compressibility.

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Definition of Compressibility ($k$)

Compressibility ($k$) is defined as the reciprocal of the Bulk modulus ($B$).

$ k = \frac{1}{B} $

Since $B = \frac{-\Delta P}{\Delta V/V}$, the units of $B$ are N/m$^2$ or Pa. Therefore, the units of compressibility are the reciprocal of pressure units, which are m$^2$/N or Pa$^{-1}$.

Compressibility represents the relative change in volume per unit change in pressure. From the definition $B = -V \frac{\Delta P}{\Delta V}$, we can write $\frac{\Delta V}{V} = -\frac{1}{B} \Delta P = -k \Delta P$. This shows that for a given change in pressure, the change in volume per unit volume is proportional to the compressibility.

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Physical Significance

A material with high compressibility undergoes a large fractional change in volume for a given change in pressure. A material with low compressibility undergoes a small fractional change in volume for the same pressure change.

• Gases: Gases are highly compressible. Their volume changes significantly with pressure. This means gases have a low Bulk modulus and therefore high compressibility.
• Liquids: Liquids are much less compressible than gases, but more compressible than solids. Their volume changes noticeable under very high pressures, but for typical pressure changes, the volume change is very small. Liquids have a higher Bulk modulus and lower compressibility than gases.
• Solids: Solids are generally the least compressible state of matter. Their volume changes very little even under very large pressures. Solids have the highest Bulk moduli and lowest compressibilities among the common states of matter.

For example, the Bulk modulus of water is about 2.2 GPa, while the Bulk modulus of steel is around 160 GPa. The compressibility of water is $1/2.2 \text{ GPa} \approx 0.45 \times 10^{-9} \text{ Pa}^{-1}$, while the compressibility of steel is $1/160 \text{ GPa} \approx 0.00625 \times 10^{-9} \text{ Pa}^{-1}$. This clearly shows steel is much less compressible than water.

Compressibility is an important property in fields like fluid dynamics (where the compressibility of liquids and gases affects flow behaviour), acoustics (where the speed of sound depends on bulk modulus and density), and material science.