# Elastic Modulii and Poisson Ratio

What are the constants and ?

First, we’ll define the Bulk Modulus “” by considering a block oriented along the principal strains:

Initially,

After deformation, ; ;

where terms have been eliminated.

Now, from eq. 4 in Section 8: Linear infinitesimal Elasticity, we know that can be expressed as follows:

Summing both sides on (and on , as always) and noting that (pressure – e.x. hydrostatic)

or

, where = “bulk modulus”

note: is sometimes called the “compressibility”, since “” relates hydrostatic pressure to volumetric strain. Really though, compressibility is determined from , where “” is the “shear modulus”. , where “” is the Poisson Ratio. “” = incompressible material.

Next, we’ll define the Young’s Modulus, “”, the Poisson Ratio, “”, and the shear modulus, “”. Any two modulii are needed to define an isotropic material ( or , etc.)

We need in terms of ;

We’ll start with eq. 4 in Section 8: Linear infinitesimal Elasticity, which is re-written, below:

(1)

Take the trace:

(2)

(2) (1) and invert:

(1)

Now we’re ready to find , , :

Consider the case of uniaxial tension:

* – must be since those of us that have done laboratory testing of linear, isotropic, materials, know that for a simple uniaxial test

(2)

Now, ,

where is the lateral strain that occurs from the uniaxial (longitudinal) stress.

By definition (i.e. as defined in undergraduate mechanics of materials),

(3)

Lastly, consider pure shear:

Since and (by definition),

(4)

It can also be easily shown that