Elasticity

Instructor: Zhang Jie
中文版：弹性力学
Back to: Soft Matter Physics Course Notes

Core Picture

Elasticity describes reversible deformation of solids. In linear elasticity, stress is proportional to strain. Full 3D deformation requires strain and stress tensors.

Stretching and Poisson Ratio

stress=$F/A$

strain=$\Delta l/l$

Hooke=$\frac{F}{A}=Y\frac{\Delta l}{l}$

Poisson=$\frac{\Delta w}{w}=-\sigma\frac{\Delta l}{l}$

Y measures resistance to stretching or compression. sigma measures transverse contraction during longitudinal extension.

hydro=$\frac{\Delta l}{l}=-\frac{p}{Y}(1-2\sigma)$

range=$-1<\sigma<\frac{1}{2}$

mu=$\mu=\frac{Y}{2(1+\sigma)}$

K=$K=\frac{Y}{3(1-2\sigma)}$

Hydrostatic compression requires volume to decrease, giving sigma < 1/2. Positive shear modulus gives sigma > -1.

Shear, Confined Compression, and Torsion

shear=$g=\mu\theta$

confined=$Y'=\frac{Y(1-\sigma)}{(1+\sigma)(1-2\sigma)}$

order=$\mu<Y<Y'$

torsion=$\tau=\frac{\mu\pi a^4}{2L}\phi$

The key torsion result is the a^4 scaling: doubling radius increases torsional stiffness by 16 times.

Beam Bending and Buckling

bending=$M=YI/R$

moment=$I=\int y^2\,dA$

cantilever=$z(L)=\frac{WL^3}{3YI}$

buckling=$F_c=\frac{\pi^2YI}{L^2}$

Buckling is a structural instability. A slender rod can bend sideways before the material is crushed.

Tensor Form and Elastic Waves

strain=$e_{ij}=\frac{1}{2}(\partial_i u_j+\partial_j u_i)$

Hooke 3D=$S_{ij}=2\mu e_{ij}+\lambda e_{kk}\delta_{ij}$

motion=$\rho\frac{d^2u_i}{dt^2}=\partial_j S_{ij}$

shear wave=$c_s=\sqrt{\mu/\rho}$

long wave=$c_l=\sqrt{(\lambda+2\mu)/\rho}$

Exam Cue

Identify whether the problem is stretching, shear, bending, torsion, buckling, or full tensor elasticity. Boundary conditions such as free expansion, confinement, fixed end, or free end directly determine the effective modulus and deformation.