A numerical pathology in discretization methods (commonly finite elements) where the approximation becomes overly stiff and fails to reproduce the correct deformation or response, typically occurring for thin structures or nearly incompressible materials.

Principle

When the discrete function space or integration scheme cannot represent key deformation modes, constraints act artificially stiff and the numerical solution is dominated by spurious constraint energy rather than true deformation energy.

Demonstration

In low-order finite elements applied to thin bending beams, the method predicts vanishing deflections as thickness decreases (shear locking); similarly, standard displacement elements applied to nearly incompressible elasticity overpredict stiffness (volumetric locking).

Misapplication

Treating locking as merely a mesh-refinement issue and refining uniformly without changing element formulation or stabilization will not remove the pathology and leads to wasted computation with persistent bias.

Consequence

When unaddressed, locking yields qualitatively wrong stiffness and loads, can prevent convergence under mesh refinement, and may mislead design or physical interpretation of simulation results.

Reversal

Using mixed formulations, reduced/selective integration, higher-order or specially designed elements removes locking by restoring correct representability of constrained modes at the discrete level.

Boundary

Concerns discretization-induced stiffness in numerical approximations; does not refer to physical material hardening, nor to solver-level ill-conditioning unrelated to representability of deformation modes.

Semantic Tension

Tension between element simplicity and fidelity: cheap low-order formulations are attractive but compete with locking-robust mixed or enriched formulations that are more complex but preserve correct kinematics.

Synthesis

Locking is the emergence of artificial stiffness from an incompatible discrete representation; recognizing it directs the practitioner to reformulate, enrich, or stabilize the discretization so numerical compliance matches physical compliance.