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 interesting development (see [P3], [5], [7], [8], [9], [12] 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

**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 [22]) 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 [19], [21], 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 [19], [21] and the summary article pdf). For example, [19] proposes a variational model for crack initiation and relates this to the use of the energy-momentum tensor in engineering fracture mechanics.

Work with P. Negron-Marrero has been concerned with numerical methods to detect the onset of cavitation in radially symmetric [4] (see also [11]) and non-symmetric problems [1]. Our work [1] 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 [27]), extensions of the classical Weierstrass Field Theory to multidimensional problems of elasticity (see [32], [33], [35]), the derivation of conservation laws in weak form as necessary conditions for energy minimisers (see [18]), and the relationship between rank-one convexity and quasiconvexity (see [34]).

Other topics of interest include:

Existence
and regularity of singular weak solutions to nonlinear elliptic systems arising
in nonlinear elasticity. The paper [17] 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:** Editorial Board for the Journal of Elasticity (2012 - present) and Proceedings A of the Royal Society (2005-2012), 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).

[P1] "On the Structure of Linear Dislocation Field Theory", (2019), (Joint with A.Acharya, R.J. Knops), Journal of the Mechanics and Physics of Solids 130, pp. 216–244 .

[P2] "On the Uniqueness of Energy Minimizers in Finite Elasticity", (2018), (Joint with S.J. Spector), J. of Elasticity vol 133, pp. 73 - 103. [pdf].

[P3] "On the convergence of a regularization scheme for approximating cavitation solutions with prescribed cavity volume size", (2019), to appear in SIAM Journal on Applied Mathematics, (Joint with P. Negron - Marrero) [pdf].

[P4] "On the Global Stability of Compressible Elastic Cylinders in Tension", (2015), (Joint with S.J. Spector), J. of Elasticity, vol 120, pp. 161-195 [pdf].

[1] " A characterisation of the boundary conditions which induce cavitation in an elastic body, (2012), (Joint with P. Negron-Marrero), J. of Elasticity, vol. 109, pp. 1–33.

[2] "Aspects of the nonlinear theory of Type II thermoelastostatics", (2012), (Joint with R. Quintanilla), European Journal of Mechanics - A/Solids, vol. 32, pp. 109–117.

[3] "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.

[4] "The volume derivative and its approximation in the case of radial cavitation", (2011), (Joint with P. Negron -Marrero), SIAM Journal on Applied Mathematics, vol. 71 (6), pp. 2185-2204.

[5] "On the stability of incompressible elastic cylinders in uniaxial tension", (2011), J. of Elasticity **105**, 313-330 [pdf] (Joint with S.J. Spector.)

[6] "On the regularity of weak solutions of the energy-momentum equations", (2011), Proc. A Roy. Soc. Ed., [pdf] (Joint with S.J. Spector.)

[7] "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.)

[8] "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.)

[9] "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.)

[10] "Energy Minimization for an elastic fluid", (2010), Journal of Elasticity, **98**, 189-203, [pdf]
(Joint with R. Fosdick)

[11] "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]

[12] "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.)

[13] "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]

[14] "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]

[15] "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]

[16]
"Myriad radial cavitating equilibria in nonlinear elasticity",
SIAM J. Appl. Math. (2003), **63, **1461-1473. (Joint with S.J. Spector) [pdf]

[17] "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]

[18] "On conservation
laws and necessary conditions in the Calculus of Variations", Proc. R.
Soc. Ed. (2002), **132A**, 1361-1371.
(Joint with G. Francfort.) [pdf]

[19] "On cavitation,
configurational forces and implications for fracture in nonlinear
elasticity", Journal of Elasticity (2002), **67**, 25-49. (Joint with S.J. Spector.) [pdf]

[20] "An explicit
radial cavitation solution in nonlinear elasticity", Maths. and Mech. of
Solids (2002), **7**, 87-93. (Joint with
K. Pericak-Spector and S.J. Spector.)

[21] "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]

[22] "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]

[23] "An
isoperimetric estimate and W ^{1,p}-quasiconvexity in nonlinear elasticity", Calculus of Variations
(1999), **8**, 159-176. (Joint with S. Muller and S.J. Spector.)

[24] "On cavitation and degenerate cavitation
under internal hydrostatic pressure", Proc. R. Soc. A (1999), **455**,
3645-3664. [pdf]

[25] "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]

[26] "A new
spectral boundary integral collocation method for three-dimensional potential
problems", SIAM J. Num. Anal. (1998), 35, 778-805. (Joint with M. Ganesh
and I.G. Graham.)

[27] "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]

[28] "On the stability
of cavitating equilibria", Q. of Appl. Math. (1995), LIII,
301-313. [pdf]

[29] "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)

[30] "Singular minimisers in the Calculus of
Variations: a degenerate form of cavitation", Analyse Non-Lineaire
(1992), **9**, 657-681.[pdf]

[31] "Cavitation, the incompressible limit and
material inhomogeneity", Q. of Appl. Math. (1991), **49**, 521-541. [pdf]

[32] "The generalised Hamilton-Jacobi inequality and
the stability of equilibria in nonlinear elasticity", Arch. Rational Mech.
Anal. (1989), **107**, 347-369. [pdf]

[33] "The structure of null lagrangians",
Nonlinearity (1988), **1**, 389-398. (Joint with P.J. Olver.) [pdf]

[34] "Implications of rank-one convexity",
Analyse Non-Lineaire (1988), **5**, 99-118. [pdf]

[35] "A field theory approach to stability of
radial equilibria in nonlinear elasticity", Proc. Camb. Phil. Soc. (1986),
**99**, 589-604. [pdf]

[36] "Uniqueness of regular and singular equilibria for spherically
symmetric problems of nonlinear elasticity", Arch. Rational Mech. Anal.
(1986) , **96**, 97-136. [pdf]