Elastic Modulii, Stiffness, and Thermal Fluctuations
In
a
post on Metamodern, I’ve written briefly about the
stiffness of various materials and why this property matters.
Here, I'd like to be a bit more quantitative, and to note some
of the simplifications made in the post.
Why stiffness matters to (nano)mechanical
engineers
Stiffness determines how much a component will
deform in response to force, or equivalently, how much elastic
energy is required to distort a component into a given shape.
In mechanical engineering for nanotechnology, this is of
special importance: Stiffness limits the amplitude of thermal
fluctuations, because the probability that a machine of a
distorted shape decreases exponentially as the elastic energy
increases. In a material, “stiffness” means elastic
modulus, and is measured in force per unit area. In a specific
structure, “stiffness” means spring constant, and
is measured in force per unit distance. In a structure of a
specific size and shape, the stiffness is proportional to the
elastic modulus of the material of which it is made. (As I note
below, however “elastic modulus” is not as simple
as this discussion suggests.)
Stiffness limits thermal fluctuations
To be more
specific, the amplitude of thermal fluctuations along a given
coordinate in a in a linear system has a Gaussian distribution
with an r.m.s. value of
(
kBT/
ks)
1/2,
where
kBT is the characteristic
thermal energy at a temperature
T, and
ks is the spring constant, which is
proportional to the elastic modulus of the materials of which
the machine is made. Quantum mechanics slightly increases the
fluctuation amplitude as a consequence of the zero-point
energy. For nanoscale components at ordinary temperatures, the
correction is a tiny fraction of a percent, and even measuring
it at temperatures near absolute zero is a challenge.
Nanosystems
discusses this topic in more detail, and I’m sure there
are ample discussions on the web.
There are several kinds of elastic modulus
Speaking as
if there were only one elastic modulus is a simplification. In
reality, there are several kinds of elastic modulus in
isotropic materials, and several more kinds in crystals, and in
inhomogeneous or nonlinear materials, the concept of
“elastic modulus” itself becomes ill-defined.
Granta Design has provided an
excellent tutorial that explains Young’s modulus,
shear modulus, bulk modulus, Poisson’s ratio, and how
they are related. When I speak of an “elastic
modulus” without qualification, I mean Young’s
modulus, the property that determines (for example) how much a
rod will stretch or bend in response to an applied force. Shear
modulus becomes important chiefly in thicker, more chunky
shapes. Fortunately, in typical
strong materials, the
values of the shear and Young’s modulus are roughly
proportional, hence Young’s modulus by itself is a good
basis for comparing materials for mechanical applications.
