A field guide
Elasticity.
Suppose you pull on a strip of metal. Not enough to bend it forever. Just enough that it lengthens, fights your hand, and then returns when you let go. The metal remembered its shape because every small region inside it was storing a little elastic energy.
The whole subject is a bookkeeping system for that memory. Displacement makes strain. Strain makes stress. Stress has to balance the load.
A simple spring is the doorway. A solid is the same idea with direction, area, sideways contraction, shear, waves, and failure modes added back in.
II · Stress is force spread over area
Pull with the same force on a thick rod and a thin wire. The wire feels the larger stress because the force has less area to spread through. Divide by area, then compare extension with original length. That gives stress and strain.
For a one-dimensional tensile test, the linear elastic calculation is $\sigma=F/A$, $\varepsilon=\sigma/E$, and $\Delta L=\varepsilon L$. Move the sliders. Steel barely moves at these loads. Rubber-like material moves enough that you start to distrust the straight-line model.
III · Pull it long, watch it narrow
A one-dimensional spring law has no sideways direction. Real solids do. Pull in $x$ and most materials contract in $y$ and $z$. Poisson's ratio is the number that says how strongly that happens.
With uniaxial strain $\varepsilon_x$, the lateral strain is roughly $-\nu\varepsilon_x$. As $\nu$ approaches $0.5$, the volume change becomes small. That is the rubber-like, nearly incompressible limit.
IV · The area under the line is energy
When stress and strain are proportional, the stress-strain graph is a line. The energy density is the triangular area under it: $w=\frac12\sigma\varepsilon=\frac12E\varepsilon^2$.
This is why elasticity and finite elements talk so much about energy. The body chooses a displacement field that balances stored strain energy against external work.
V · A spring with inertia becomes a clock
Stretch a spring and hold it. That is static elasticity. Attach a mass, let go, and the stored energy starts moving. At the ends, energy is in the spring. At the center, energy is in the mass.
The simplest equation is $m\ddot{x}+c\dot{x}+kx=0$. With little damping, the natural frequency is close to $\omega_0=\sqrt{k/m}$. This is the same elastic law, now with inertia added.
VI · Bending hides tension and compression
A cantilever beam looks like one object, but a cross-section is doing opposite things at once. One side stretches. The other side compresses. The neutral axis in between barely changes length.
The small-deflection formula $\delta=FL^3/(3EI)$ is severe about geometry. Make the beam twice as thick and $I$ grows by eight.
VII · Stress depends on the cut
Cut an imaginary tiny square out of a loaded body. The traction on a face depends on the direction of that face. That is why stress is a tensor, not one number.
Rotate the little square. The normal and shear components change, even though the physical stress state is the same.
VIII · Elasticity can travel
If stress changes in one place, neighboring material accelerates. That makes an elastic wave. Shear waves care about $G$. Longitudinal waves care about both shear and bulk stiffness.
For an isotropic solid, $c_s=\sqrt{G/\rho}$ and $c_p=\sqrt{(K+4G/3)/\rho}$. Near incompressibility makes $K$ large.
IX · Materials do not share one curve
The phrase "elastic modulus" can make materials sound cleaner than they are. Steel has a long straight elastic region before yielding. Glass can stay nearly linear and then break. Tendon has a toe region where crimped collagen straightens before the slope stiffens. Rubber is elastic over large stretch but not linear.
The curves below are schematic teaching curves, not design data. Their job is to put four different kinds of "elastic" on the same page.
X · Time changes the answer
Hold a polymer at fixed strain and the force may relax. Hold a constant stress and the strain may creep upward. That is viscoelasticity: spring behavior mixed with dashpot behavior.
The controls compare two small models. Maxwell relaxation gives $\sigma(t)=\sigma_0 e^{-t/\tau}$. Kelvin-Voigt creep gives $\varepsilon(t)=\sigma_0(1-e^{-t/\tau})/E$.
XI · Contacts and cracks concentrate stress
Elasticity is often decided by a small region: the patch under a ball, the root of a notch, the tip of a crack. Geometry focuses the stress. A smooth contact grows a finite patch as load rises. A sharp crack is better described by stress intensity than by one maximum stress.
The notch model below uses an Inglis-style sharpness estimate, $K_t\approx1+2a/b$. It is a warning, not a fracture assessment.
XII · Elastic does not always mean linear
A rubber band can be elastic at large stretch. It comes back. But its stress is not a straight line in strain. Linear elasticity is a tangent near the reference shape.
The curve below compares a linear tangent with a simple incompressible neo-Hookean law, $P=\mu(\lambda-\lambda^{-2})$. It is a toy model, but it makes the warning visible.
XIII · Workshop: build the model until it complains
Each fragment has a job. First, one-dimensional Hooke law gets axial strain right and volume wrong. Then Poisson coupling repairs the missing sideways motion. Then local bar stiffness becomes a small truss solve. Finally, a rubber-like law bends away from the tangent line.
Fragment I
Scalar Hooke law
Step 1 of 4
XIV · Anatomy plates
Five drawings, same model family. First the stress cube: directions matter. Then the compliance matrix: normal stresses talk through Poisson's ratio, shear sits in its own slot. The beam shows bending as tension and compression arranged across a thickness. The oscillator and crack plates show what happens when inertia and geometry enter.