The Continuum: Strain, Stress and Hooke's Law in 3D

From atoms to a smooth field

A real solid is a lattice of 10^{23} jostling atoms. The continuum hypothesis replaces that swarm with smooth fields — a density \rho(\mathbf{x}) and, crucially, a displacement \mathbf{u}(\mathbf{x}) at every point — valid whenever we look at scales far larger than an atom yet far smaller than the whole body. It is the same leap that turns individual molecules into a fluid with a smooth pressure and velocity field.

The key variable is the displacement field \mathbf{u}(\mathbf{x}): how far the material point that was at \mathbf{x} has moved. But \mathbf{u} itself is not what builds up internal force — pick up the whole body and translate it and nothing is stressed. What matters is how \mathbf{u} varies from point to point.

Strain: measuring deformation

Deformation lives in the gradients \partial u_i/\partial x_j. But not all of that gradient is real strain: its antisymmetric part is just a local rigid rotation, which stores no energy. Keeping only the symmetric part isolates genuine stretching and shearing — the strain tensor:

\varepsilon_{ij} = \tfrac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)

The strain tensor — the symmetric gradient of the displacement field; dimensionless, and small for ordinary elastic deformation.

Its diagonal entries \varepsilon_{11},\varepsilon_{22},\varepsilon_{33} are the normal strains — the fractional stretch along each axis. Its off-diagonal entries are the shear strains — the change in angle between axes as the material skews. And the trace \varepsilon_{kk}=\varepsilon_{11}+\varepsilon_{22}+\varepsilon_{33} is exactly the fractional change in volume (the dilatation).

Stress: internal force across a surface

Now the forces. Slice the material with an imaginary plane; the two sides push and pull on each other with a force per unit area — a traction. That traction depends on how the cut is oriented. Cauchy's insight is that the traction is linear in the surface's unit normal \mathbf{n}, mediated by the stress tensor:

t_i = \sigma_{ij}\,n_j

Cauchy's stress principle — the traction (force per area) on a surface with unit normal \mathbf{n}; \sigma_{ij} is the i-th force component on a face whose normal points along the j-th axis.

The diagonal \sigma_{ii} are normal stresses — pressure when negative, tension when positive. The off-diagonal are shear stresses. And \sigma_{ij}=\sigma_{ji} is not an assumption: it follows from the balance of angular momentum on a shrinking element — an unsymmetric stress would spin a tiny cube up without limit. So stress, like strain and inertia, is a symmetric rank-2 tensor with six independent components.

Hooke's law in three dimensions

Strain says how the body has deformed; stress says what forces it carries. The bridge between them is the material's constitutive relation. In Volume I Hooke's law was F=-kx, a single spring. Its continuum descendant is a linear relation between the two tensors, with a rank-4 elasticity tensor C_{ijkl} holding up to 21 independent elastic constants:

\sigma_{ij} = C_{ijkl}\,\varepsilon_{kl}

The generalized Hooke's law; C_{ijkl} is the elasticity (stiffness) tensor encoding all the material's elastic properties.

Twenty-one constants sounds forbidding, but symmetry rescues us. For an isotropic material — the same in every direction, like glass or unstructured steel — the whole tensor collapses to just two numbers, the Lamé parameters \lambda and \mu:

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

Isotropic Hooke's law: \mu (the shear modulus) resists change of shape, while \lambda ties stress to the change of volume \varepsilon_{kk}.

The engineering moduli you know are just combinations of these two. Young's modulus E (stiffness under a simple pull) and Poisson's ratio \nu (how much it thins sideways as you stretch it) are:

E = \frac{\mu(3\lambda + 2\mu)}{\lambda + \mu}, \qquad \nu = \frac{\lambda}{2(\lambda + \mu)}

The everyday elastic moduli recovered from the two Lamé parameters.

The equation of motion for a solid

Finally, Newton's second law for a tiny material element: its mass times acceleration equals the net internal force (the divergence of the stress tensor) plus any body force f_i such as gravity. This is Cauchy's momentum equation:

\rho\,\frac{\partial^2 u_i}{\partial t^2} = \frac{\partial \sigma_{ij}}{\partial x_j} + f_i

Cauchy's equation of motion — ma=F for a continuum, with the divergence of the stress tensor as the net internal force per unit volume.

This is the master equation of the continuum. Drop the acceleration and it governs statics — a loaded bridge, a bent beam, a stressed dam. Keep the acceleration and substitute Hooke's law, and — as the next guide shows — it becomes a wave equation, and sound is born.