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 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 [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] "Infinite Energy Cavitation: a Variational Approach", (2022), (Joint with P. Negron-Marrero), to appear in SIAM J. App. Math. [pdf]

[P2] "On the convergence of a regularization scheme for approximating cavitation solutions with prescribed cavity volume size", (2020), SIAM J. App. Math, vol 80(1), (Joint with P. Negron - Marrero) [pdf].

[P3] "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 .

[P4] "On the Uniqueness of Energy Minimizers in Finite Elasticity", (2018), (Joint with S.J. Spector), J. of Elasticity vol 133, pp. 73 - 103. [pdf].

[P5] "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 [pdf].

[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]