7.3 Hooke’s law in 3-D and the elastic moduli

Stress and strain are linked by a constitutive law: how much stress does a given strain produce? For a linear, isotropic, elastic solid the answer is the three-dimensional generalisation of Hooke’s ut tensio sic vis — and it requires exactly two independent constants.

The constitutive law

For small strains the stress is a linear function of the strain. Isotropy — no preferred direction in the material — restricts that linear relation to just two coefficients, the Lamé parameters $\lambda$ and $\mu$ :

\sigma_{ij} \;=\; \lambda\,\varepsilon_{kk}\,\delta_{ij} \;+\; 2\mu\,\varepsilon_{ij}.

where

$\sigma_{ij}$: stress tensor Pa
$\varepsilon_{ij}$: strain tensor —
$\varepsilon_{kk}$: volumetric strain $\nabla\cdot\mathbf{u}$ —
$\lambda$: first Lamé parameter Pa
$\mu$: shear modulus (second Lamé parameter, also $G$) Pa

The structure mirrors the strain decomposition of the previous lesson. The first term responds to the volumetric strain $\varepsilon_{kk}$ and acts equally in all directions (through $\delta_{ij}$ ); the second responds to the full strain, shape changes included. Two constants, one for changing volume and one for changing shape, are all an isotropic elastic solid needs.

The engineering moduli

The Lamé pair is rarely how a material is quoted. Four other constants, each an easily-measured ratio, package the same information:

Young’s modulus $E$ — uniaxial stress divided by the resulting uniaxial strain, with the sides free to contract. The stiffness felt when you stretch a rod.
Shear modulus $G = \mu$ — shear stress divided by shear strain.
Bulk modulus $K$ — pressure divided by the fractional volume decrease. The resistance to uniform compression.
Poisson’s ratio $\nu$ — the lateral contraction divided by the axial extension in a uniaxial pull.

Because an isotropic solid has only two independent constants, any two of these fix all the rest. The most useful conversions are

G \;=\; \frac{E}{2(1+\nu)}, \qquad K \;=\; \frac{E}{3(1-2\nu)}, \qquad \lambda \;=\; K - \tfrac23 G.

σ_xx = 0.050 σ_yy = 0.000 τ_xy = 0.000 Young modulus E = 1.00 Poisson ratio ν = 0.30

Tension in x alone (σ_xx > 0, σ_yy = 0) elongates the block in x and *contracts* it in y by Poisson's ratio ν. A pure shear stress τ_xy tilts the block without changing its volume. At ν = 0.5 the material is incompressible (volumetric strain = 0 for any deviatoric stress), and the bulk modulus K diverges — the limit relevant for water and most biological tissues.

Drive the stress components and watch the unit block respond. Pulling along one axis narrows the perpendicular directions in proportion to Poisson’s ratio — the block conserves volume better as $\nu \to \tfrac12$ . Shear tilts the block while leaving its volume unchanged.

The physical range of Poisson’s ratio

For most solids $\nu$ lies between $0.2$ and $0.35$ : stretch a steel bar and it thins by about a third of its elongation, in relative terms. Two limits bound the range. As $\nu\to\tfrac12$ the bulk modulus $K = E/[3(1-2\nu)]$ diverges: the material becomes incompressible, resisting any volume change absolutely while still yielding to shear — the limit that rubbers and water-saturated tissues approach. As $\nu\to -1$ the material is auxetic, thickening laterally when stretched; ordinary matter never does this, but engineered metamaterials with re-entrant microstructure can. The thermodynamic requirement that elastic energy be positive confines $\nu$ to $(-1, \tfrac12)$ .

⏳ The history — Robert Hooke's anagram and the slow disclosure of linear elasticity

Robert Hooke published his law of elasticity in 1660 as an anagram: ceiiinosssttuv. Publishing a scrambled result was a way to establish priority without disclosing the discovery to rivals; Hooke gave the solution only in 1678, in De Potentia Restitutiva (“Of Spring”). Unscrambled it reads ut tensio sic vis — “as the extension, so the force” — the first quantitative statement of a constitutive law.

The three-dimensional generalisation took another 150 years. Augustin-Louis Cauchy in 1822 introduced the stress tensor and founded the systematic theory of continuum mechanics; Gabriel Lamé developed the modern modulus algebra in the 1850s. The compact two-constant $(E,\nu)$ description of an isotropic solid settled into the engineering literature only in the early twentieth century.

What endures is that the linearity Hooke posited for a single spring — extension proportional to force — survives intact to the full tensor law, because $\sigma_{ij}(\varepsilon)$ is just the first-order Taylor expansion of any smooth stress–strain relation about the unstressed state. Hooke’s law is the statement that, for small enough deformation, the leading term is all that matters.

Read the original: De Potentia Restitutiva, or Of Spring (Robert Hooke, 1678)