elastic

sources:

• wikipedia
• claude

materials deform following a stress-strain curve.

for many materials, this curve starts out approximately linear, in what is called the region of linear elasticity. in this range you can think of the material as being a spring, or a chain or lattice of beads connected by springs. (molecular bonds can often be well approximated as springs, ie systems with "harmonic potentials")

elastic deformation is reversible; it does not break any bonds. every material has a yield point beyond which bonds are broken and the deformation is irreversible. This permanent deformation is called plastic deformation.

linear elastic deformation is modeled by Hooke's law, $\sigma = E \epsilon$, where

• $\sigma$, the 'stress', is the force per unit area applied to the material
• $\epsilon$, the 'strain', is the proportional deformation of the material, e.g., $\epsilon = 0.1$ if the material stretches by 10%. negative values indicate contraction.
• $E$ is Young's modulus, the slope of the stress-strain curve in the linear elastic region for this material. Like $\sigma$ this has units of Pascals or $N/m^2$.

Equivalently, Young's modulus is the "spring constant" of a unit cylinder of the given material. It measures the stiffness of the material to compression or stretching.

As we pull on a rod or wire, stretching it out, it will also grow thinner. This is measured by Poisson's ratio, $\nu = -\frac{d\epsilon_y}{d\epsilon_x} = -\frac{d\epsilon_z}{d\epsilon_x}$ , the ratio between the deformation in the axial dimension $x$ (the direction of the stretching force) and the two transverse dimensions $y, z$. The maximum possible ratio is 0.5, for a perfectly incompressible material where volume is preserved at each instant of this stretching. ![Note that the relationship between axial and transverse deformation is only *locally* linear --- in general you have to integrate across the range of deformation which gives a nonlinear relationship at the macro scale. for example stretching a unit cube by a factor of 2 in one axis, to preserve volume the other two axes must then each be $1/\sqrt{2}$! ]

Shear modulus

Bulk modulus

Generalized Hooke's Law

Todo: understand the elasticity tensor https://en.wikipedia.org/wiki/Elasticity_tensor