My work spans areas of mathematics including the Calculus of Variations, analysis, differential equations, topology, geometry, and nonlinear elasticity. Many of the problems are connected with the variational theory of nonlinear elasticity. A common feature of the problems studied is that there is an intriguing interplay between the constraints of the physical situation being modelled and the corresponding mathematics: with physical considerations yielding insight into the underlying mathematics and vice versa.
It is a typical feature of nonlinear problems that solutions may not exist, and that exact solutions cannot be obtained in general. Hence the necessity for abstract methods to study the existence and qualitative properties of solutions. Some of the topics I am interested in are listed below.
An exciting recent development (see , , , ,  below) is a new class of vector symmetrisation arguments for shells and bars. Given any deformation, these symmetrisation procedures can be applied to produce a symmetrised map with no greater energy than the original deformation. Consequently, these arguments give conditions on the stored energy function under which energy minimising configurations can be proven to be symmetric (in the case of shells) or homogeneous (in the case of the mixed problem for bars). To our knowledge, these are the first such arguments for nonlinear elasticity and the full potential of this approach is still to be investigated.
Nonlinear Elasticity and Fracture
In the variational approach to nonlinear elasticity one seeks equilibrium states of an elastic body by minimising the total energy that is stored in the deformed body over all possible configurations of the body. An intriguing aspect of the variational approach is that it is possible to start with an initially perfect body and to find (mathematically) that, if a sufficiently large boundary displacement or load is imposed, then the configurations which minimise the energy stored in the body must be discontinuous (see the seminal work on radial cavitation of J.M. Ball in Phil. Trans. R. Soc. A, 306, (1982), 557-610, the extension to nonsymmetric problems by Muller and Spector in Arch. Rational Mech. Anal., 131, (1995), 1-66). Work with S.J. Spector (see ) proposes a variational model for a nonlinear elastic material containing flaws in which the flaws are modelled by constraining the points of possible discontinuity of the admissible deformations .These discontinuous energy-minimising configurations can be interpreted as fractures such as cavities forming in the initially perfect elastic body and correspond to singular weak solutions of the equilibrium equations.
A main focus of my research to date has been to study the properties of such singular solutions (existence, uniqueness, analytic properties etc) and to incorporate these into a new theory of fracture.
In collaborative work with S.J. Spector, we now have a well-developed theory of the analytical properties of singular minimisers and a good understanding of the physical relevance of these singular solutions for general nonsymmetric 3D problems. Work in , , demonstrates how these singular solutions might model fracture initiation and shows how singular solutions of linear elasticity can be used to predict where the singularities will form. This shows that there are fundamental connections between variational problems with singular minimisers in nonlinear elasticity and classical engineering approaches to fracture mechanics, and should help in identifying limitations of the traditional methods (see ,  and the summary article pdf). For example,  proposes a variational model for crack initiation and relates this to the use of the energy-momentum tensor in engineering fracture mechanics.
Recent work with P. Negron-Marrero has been concerned with numerical methods to detect the onset of cavitation in radially symmetric [P2] (see also ) and non-symmetric problems [P3]. Our recent work [P3] uses the notion of a derivative of the energy functional with respect to hole-producing deformations and we conjecture that the zero set of this derivative represents the boundary, in strain space, of the set of general (possibly non-symmetric) boundary strains for which cavitation is energetically favoured.
I am interested in fundamental theoretical problems in the multidimensional Calculus of Variations: examples include the existence of multiple equilibria in elasticity under topological constraints (see ), extensions of the classical Weierstrass Field Theory to multidimensional problems of elasticity (see , , ), the derivation of conservation laws in weak form as necessary conditions for energy minimisers (see ), and the relationship between rank-one convexity and quasiconvexity (see ).
Other topics of interest include:
Existence and regularity of singular weak solutions to nonlinear elliptic systems arising in nonlinear elasticity. The paper  gives a direct proof of the existence of infinitely many weak solutions of the equations of nonlinear elasticity (for the same boundary value problem) each with infinitely many discontinuities.
Learned Societies: I am on the Editorial Board for the Journal of Elasticity and a member of the American Mathematical Society (AMS), London Mathematical Society (LMS), Society for Natural Philosophy (SNP), International Society for the Interaction of Mathematics and Mechanics (ISIMM), Society for Industrial and Applied Mathematics (SIAM).
[P4] "On the relative energies of the Rivlin and Cavitation instabilities for Compressible Materials", (2012), (Joint with J.G. Lloyd), Mathematics and Mechanics of Solids, vol. 17, no. 4, pp. 338–350.
 "On the stability of incompressible elastic cylinders in uniaxial tension", (2011), J. of Elasticity 105, 313-330 [pdf] (Joint with S.J. Spector.)
 "On the regularity of weak solutions of the energy-momentum equations", (2011), Proc. A Roy. Soc. Ed., [pdf] (Joint with S.J. Spector.)
 "On the symmetry of energy minimizing deformations in nonlinear elasticity I: incompressible materials", (2010), Arch. Rational Mech. Anal, 196, 363-394, [pdf] (Joint with S.J. Spector.)
 "On the symmetry of energy minimizing deformations in nonlinear elasticity II: compressible materials", (2010), Arch. Rational Mech. Anal., 196, 395-431, [pdf] (Joint with S.J. Spector.)
 "On the global stability of two-dimensional, incompressible elastic bars in uniaxial extension", (2010), Proceedings A of the Royal Society, 466, 1167-1176 [pdf], (Joint with S.J. Spector.)
 "Energy Minimization for an elastic fluid", (2010), Journal of Elasticity, 98, 189-203, [pdf] (Joint with R. Fosdick)
 "The numerical computation of the critical boundary displacement for radial cavitation", (2009), Math & Mech of Solids, 14, 696-726 (Joint with P.V. Negron-Marrero.) [pdf]
 "Energy minimizing properties of the radial cavitation solution in incompressible nonlinear elasticity ", (2008), Journal of Elasticity, 93, 177-187. [pdf] (Joint with S.J. Spector.)
 "Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity", Analyse Non-Lineaire (2008), AN25, 201-213 (Joint with S.J. Spector) [pdf]
 "The convergence of regularized minimizers for cavitation problems in nonlinear elasticity", SIAM J. Appl. Math (2006), 66, 736-757 . (Joint with S.J. Spector and V. Tilakraj) [pdf]
 "A variational approach to modelling initiation of fracture in nonlinear elasticity", IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics (A.B. Movchan ed), Springer, 2004, pp.295-306 (with S.J. Spector). [pdf]
 "Myriad radial cavitating equilibria in nonlinear elasticity", SIAM J. Appl. Math. (2003), 63, 1461-1473. (Joint with S.J. Spector) [pdf]
 "A construction of infinitely many singular weak solutions to the equations of nonlinear elasticity", Proc. R. Soc. Ed. (2002), 132A, 985-992. (Joint with S.J. Spector) [pdf]
 "On conservation laws and necessary conditions in the Calculus of Variations", Proc. R. Soc. Ed. (2002), 132A, 1361-1371. (Joint with G. Francfort.) [pdf]
 "On cavitation, configurational forces and implications for fracture in nonlinear elasticity", Journal of Elasticity (2002), 67, 25-49. (Joint with S.J. Spector.) [pdf]
 "On the optimal location of singularities arising in variational problems of elasticity", Journal of Elasticity (2000), 58, 191-224. (Joint with S.J. Spector.) [pdf]
 "On the existence of minimisers with prescribed singular points in nonlinear elasticity", Journal of Elasticity (2000), 59, 83-113. (Joint with S.J. Spector) [pdf]
 "On cavitation and degenerate cavitation under internal hydrostatic pressure", Proc. R. Soc. A (1999), 455, 3645-3664. [pdf]
 "The representation theorem for linear, isotropic tensor functions in even dimensions", Journal of Elasticity (1999), 57, 157-164. (Joint with K.A. Pericak-Spector and S.J. Spector.) [pdf]
 "On homotopy conditions and the existence of multiple equilibria in finite elasticity", Proc. R. Soc. Ed. (1997), 127A, 595-614. (Joint with K.D.E Post.) [pdf]
 "On the stability of cavitating equilibria", Q. of Appl. Math. (1995), LIII, 301-313. [pdf]
 "A pseudospectral three-dimensional boundary integral method applied to a nonlinear model problem from finite elasticity", SIAM J. Numer. Anal. (1994), 31, 1378-1414 (Joint with M. Ganesh and I.G. Graham)
 "Cavitation, the incompressible limit and material inhomogeneity", Q. of Appl. Math. (1991), 49, 521-541. [pdf]
 "The generalised Hamilton-Jacobi inequality and the stability of equilibria in nonlinear elasticity", Arch. Rational Mech. Anal. (1989), 107, 347-369. [pdf]
 "The structure of null lagrangians", Nonlinearity (1988), 1, 389-398. (Joint with P.J. Olver.) [pdf]
 "Implications of rank-one convexity", Analyse Non-Lineaire (1988), 5, 99-118. [pdf]
 "A field theory approach to stability of radial equilibria in nonlinear elasticity", Proc. Camb. Phil. Soc. (1986), 99, 589-604. [pdf]
 "Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity", Arch. Rational Mech. Anal. (1986) , 96, 97-136. [pdf]