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Bulk Modulus and Dilation

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Quick
The bulk modulus is a constant also known as the modulus of compression. It has the same units as pressure and stress.
The dilation is the change in volume per unit volume.


Equation

(Eq1)     e = εx + εy + εz	Dilation, or change in volume per unit volume
(Eq2)     e = 1 − 2ν E (σx + σy + σz)	Dilation, or change in volume per unit volume
(Eq4)     k = E 3(1 − 2ν)  p	Bulk modulus, or modulus of compression
(Eq5)     e = −  p k	Dilation, or change in volume per unit volume
(Eq6)     B = − Δp ΔV/V0	Bulk modulus

Nomenclature

e	dilation
k	bulk modulus (mechanics of materials)
B	bulk modulus (physics)
ε	strain
E	modulus of elasticity
σ	stress
p	pressure
ν	Poisson's ratio
V	volume
Δp	change in pressure
ΔV	change in volume
V0	initial volume

Details

The effect of the normal stresses σ_x, σ_y, and σ_z on the volume of an element of isotropic material are examined. Consider the element shown:

Unstressed	Stressed

In its unstressed state, it is in the shape of a cube of unit volume; and under the stresses σ_x, σ_y, σ_z, it deforms into a rectangular parallelepiped of volume:

V = (1 + ε_x)(1 + ε_y)(1 + ε_z)

Since the strains ε_x, ε_y, ε_z, are much smaller than unity, their products will be even smaller and may be omitted in the expansion of the product. The following is therefore:

V = 1 + ε_x + ε_y + ε_z

Denoting by e the change in volume of the element:

e = V − 1 = 1 + ε_x + ε_y + ε_z − 1

or:

(Eq1)

e = εx + εy + εz

Since the element had originally a unit volume, the quantity e represents the change in volume per unit volume; it is referred to as the dilation of the material. Substituting for ε_x, ε_y, and ε_z from Eq1, Eq2, and Eq3 from the lesson Multiaxial Loading and Generalized Hooke's Law into Eq1, the following is:

e =

σx + σy + σz E

−

2ν(σx + σy + σz) E

(Eq2)

e = 1 − 2ν E (σx + σy + σz)

Since the dilation e represents a change in volume, it must be independent of the orientation of the element considered. It then follows from Eq1 and Eq2 that the quantities ε_x + ε_y + ε_z and σ_x + σ_y + σ_z are also independent of the orientation of the element.

A case of special interest is that of a body subjected to a uniform hydrostatic pressure p. Each of the stress components is then equal to −p and Eq2 yields:

(Eq3)

e = 3(1 − 2ν) E  p

Introducing the constant:

(Eq4)

k = E 3(1 − 2ν)  p

Eq3 is written in the form:

(Eq5)

e = −  p k

The constant k is known as the bulk modulus or modulus of compression of the material. It is expressed in the same units as the modulus of elasticity E, that is, in pascals or in psi.

Observation and common sense indicate that a stable material subjected to a hydrostatic pressure can only decrease in volume; thus the dilation e in Eq5 is negative, from which it follows that the bulk modulus k is a positive quantity. Referring to Eq4, it may be concluded that 1 − 2ν > 0, or ν < 1/2. On the other hand, recall from the lesson Poisson's ratio that ν is positive for all engineering materials. It may then be concluded that, for any engineering material:

0 < ν < 1/2

Note that an ideal material having a value of ν = 0 could be stretched in one direction without any lateral contraction. On the other hand, an ideal material for which ν = 1/2, and thus k = ∞, would be perfectly incompressible (e = 0). Referring to Eq2 it may also be noted that, since ν < 1/2 is in the elastic range, stretching an engineering material in one direction, for example in the x direction (σ_x > 0, σ_y = σ_z = 0), will result in an increase of its volume (e > 0). However, in the plastic range, the volume of the material remains nearly constant.

When Hooke's law is obeyed, an increase in pressure (bulk stress) produces a proportional bulk strain (fractional change in volume). The corresponding elastic modulus (ratio of stress to strain) is called the bulk modulus, denoted by B. When the pressure on a body changes by a small amount Δp, from p₀ to p₀ + Δp, and the resulting bulk strain is ΔV/V₀, Hooke's law takes the form:

(Eq6)

B = − Δp ΔV/V0

which is the bulk modulus. A minus sign is included in this equation because an increase of pressure always causes a decrease in volume. In other words, if Δp is positive, ΔV is negative. The bulk modulus B itself is a positive quantity.

For small pressure changes in a solid or a liquid, B is considered to be constant. The bulk modulus of a gas, however, depends on the initial pressure p₀. Its units, force per unit area, are the same as those of pressure (and of tensile or compressive stress).