What are the constants and ?
First, we’ll define the Bulk Modulus “” by considering a block oriented along the principal strains:
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)
, 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:
Take the trace:
(2) (1) and invert:
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
where is the lateral strain that occurs from the uniaxial (longitudinal) stress.
By definition (i.e. as defined in undergraduate mechanics of materials),
Lastly, consider pure shear:
Since and (by definition),
It can also be easily shown that