John Willis has made highly original contributions to many branches of the mechanics of solids, and especially to the theory of anisotropic and heterogeneous media. For example, in classical anisotropic elasticity (both static and dynamic), he is responsible for solutions to important three-dimensional problems; much of this was achieved by his perceptive and original application of Radon transforms.

In fracture mechanics, he made notable contributions to the understanding of both brittle and ductile fracture; his solution for the perturbation of a dynamically growing cracks permitted mathematical verification of the existence of ‘crack front waves. He has also contributed extensively to the theory of dislocations and of radiation damage.

In the theory of heterogeneous media, he was among the first to employ statistical correlations in the formulation of variational principles, which have led to improved bounds on the overall properties of composites. More recently, he has successfully extended this approach to composites with nonlinear response, for dynamic as well as static problems, including the effective response of metamaterials.