# 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 (*k*

_{B}

*T*/

*k*

_{s})

^{1/2}, where

*k*

_{B}

*T*is the characteristic thermal energy at a temperature

*T*, and

*k*

_{s}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.