Most of our work is in the area of theoretical and applied elasticity, which forms the central core of the subject of solid mechanics. Elasticity is a broad fundamental science having applications in a diversity of other areas - for example, engineering structural mechanics (understanding the buckling and collapse of mechanical structures), materials science (modelling the mechanical properties of solids such as rubber), geophysics (interpretation of seismic data using elastic wave analysis), non-destructive evaluation of the integrity of materials using elastic waves, biomechanics (modelling the mechanical properties of soft tissue such as arteries so as to understand their mechanical performance and changes with age and disease).

The underlying mathematical theory of elasticity provides a rich framework for the study of such applications, and also offers many interesting and challenging mathematical questions in its own right relating to, for example, the governing partial differential equations and the qualitative properties of their solutions.

In the Department of Mathematics there is research in progress on theoretical and applied aspects related to the following topics:

Research areas

Nonlinear elasticity	Constitutive theory of anisotropic materials Residual stress effects Solution of boundary-value problems Stability, bifurcation and non-uniqueness analysis
Rubber elasticity and inelasticity	Characterization of inelastic properties of rubberlike materials, with particular reference to stress softening effects Influence of ageing and fatigue on the mechanical properties of rubber
Biomechanics of soft tissue	Mechanics of arterial walls and the influence of residual stress
Damage mechanics and pseudo-elasticity	The effect of damage on the evolution of nonlinear constitutive laws
Asymptotic methods	Boundary layers in solids WKB methods in the theory of elastic stability Amplitude equations and post-buckling analysis
Mechanics of thin-film surface-coated solids	The effect of surface coating on the stability of elastic structures. Wave propagation of surface-coated solids
Membrane theory	Stability of membrane structures Theory of wrinkling of membranes
Thermoelasticity	Thermoelastic constitutive laws and heat conduction effects
Elastic waves	Study of the effect of pre-stress on the propagation, reflection and transmission of elastic waves, and related questions of material stability
Nonlinear magnetoelasticity	Application to magneto-sensitive elastomers
Local instabilities	Stress concentration in inhomogeneous systems. Localised elastic instabilities in beams, plates, and shells.
Multi-scale material science	Quantum dots and nano-wires.
Thermo-viscoelasticity	Dynamic martensitic phase-transitions.

Ph.D. supervision is available in any of these areas. Interested students are encouraged to e-mail the members of the Solid Mechanics group for more information.

 

Members of the Solid Mechanics group

Academic staff

Raymond W. Ogden, David M. Haughton, Ciprian D. Coman, Stephen J. Watson, Steven M. Roper, Kenneth A. Lindsay , Xiaoyu Luo

 

Prof. Raymond W. Ogden

Research interests include the following. Fundamental aspects of the nonlinear theory of elasticity and thermoelasticity, and with the application of this theory to the study of stability and wave propagation phenomena in elastic bodies and to the modelling of the mechanical response of solids, including rubberlike solids. Inelastic behaviour of rubberlike solids. Modelling the mechanical behaviour of soft biological tissues such as artery walls, with particular reference to tissue structure and anisotropy. The interaction between growth and the development of residual stresses in soft tissue. Electromagnetic effects in solid continua: nonlinear magnetoelasticity of magneto-sensitive elastomers; nonlinear electroelasticity of electrorheological solids. Different aspects of the work can involve a wide variety of mathematical and computational methods, and opportunities exist for research students to work on problems that either straddle the theory/applications spectrum or are largely theoretically orientated or which emphasise the applications.

Sample of recent works:

1). Hyperelastic modelling of arterial layers with distributed collagen fibre orientations (with T. A. Gasser and G. A. Holzapfel), J. R. Soc. Interface3 (2006), 15-35.

2). Nonlinear magnetoelastic deformations (with A. Dorfmann), Quart. J. Mech. Appl. Math. 57 (2004), 599-622.

3). A theory of stress softening of elastomers based on finite chain extensibility (with C. O. Horgan and G. Saccomandi), Proc. R. Soc. Lond. A 460 (2004), 1737-1754.

Possible Ph.D. topics:

I). Modelling the electro-mechanical properties of electro-active polymers and applications.

II). Mathematical modelling of the interaction between mechanical stress and growth in soft biological tissues.

III). The influence of magnetic fields on the propagation of elastic waves in magneto-sensitive materials.

 

Dr. David M. Haughton

The main thrust of my work is bifurcation problems in nonlinear elasticity. For example, deciding when bulges will form in inflated balloons. This naturally leads to many related problems. Most recently, I have been looking at ways to decide if various solutions are stable or not. This requires calculus of variations and the numerical solution of nonlinear ordinary differential equations.

Other related work is the search for exact solutions to problems. This usually involves the (lucky, although I would claim judicious) choice of material model. But once a solution is found we can work backwards and find other (classes of) materials with the same solution.

Sample of recent works:

1). Stability and bifurcation of compressed elastic cylindrical tubes (with A. Dorfmann), Int. J. Engng. Sci. 44 (2006), 1353-1365.

2). A comparison of stability and bifurcation criteria for a compressible elastic cube, J.Engng. Math. 53 (2005), 79-98.

3). On nonlinear stability in unconstrained nonlinear elasticity, Int. J. Non-linear Mech.39 (2004), 1181-1192.

Possible Ph.D. topics:

I). Cusp formation in bending.

(This project will look at very severe deformations of solid slabs or inflated membranes where a sharp “corner” is formed along with some self-contact).

II). Bifurcation problems induced by growth.

(Usually a bifurcation problem involves a critical parameter which may be related to the loading conditions, geometry, material parameters, etc. Here we include the possibility of growth-as in tumour growth-changing the critical parameters).

 

Dr. Ciprian D. Coman

I am currently interested in a fundamental understanding of localised behaviour in solids, with a particular emphasis on the application of asymptotic techniques to this area. The motivation for this topic comes from a range of engineering applications. Many mechanical structures such as beams, plates, and shells, when subjected to various loading conditions experience stress concentrations that represent a common source of mechanical failure. In the most extreme cases such concentrations of stress lead to cracking and rupturing of the mechanical components, while in other instances they simply contribute to fatigue and a substantial reduction in the load-carrying capacity over prolonged periods of time. There is also an intermediate scenario, namely, when a structure subjected to global loading exhibits (undesirable) deformations which are concentrated at certain sites within the structure that experience high levels of stress/strain. This aspect is related to localised buckling, a phenomenon that still raises many unsolved questions.

I am also interested in the application of rigorous methods of bifurcation theory to more traditional problems of stability of solids and mechanical vibrations.

Sample of recent works:

1). Localised wrinkling instabilities in radially stretched annular thin films (with D. M. Haughton), Acta Mechanica 185 (2006), 179-200.

2). Secondary bifurcations and localisation in a three-dimensional buckling model, ZAMP 55 (2004), 1050-1064.

3). Solitary wave interaction phenomena in a strut buckling model incorporating restabilisation, (with M. K. Wadee and A. P. Bassom) Physica D163 (2002), 26-48.

Possible Ph.D. topics:

I). Stress concentration and local instabilities in thin elastic sheets.

(The aim of this project is to explore the applicability of WKB techniques to describing local instabilities caused by stress concentration in thin films, with a special emphasis on partial wrinkling).

II). Boundary layers, inhomogeneities and bifurcations in solids under finite deformations.

(The role of boundary layers in the stability of finitely strained solid will be assessed both analytically and numerically. Possible sources of inhomogeneities to be considered: small defects in elastic solids, anisotropic materials, fibre-reinforced materials, plasticity, electric or magnetic fields).

 

Dr. Stephen J. Watson

An area of recent focus has been the study of self-assembled faceted surfaces, whose characterization is a central theoretical challenge in surface science. This class of problems provide a mathematically rich and physically motivated paradigm wherein non-equilibrium phase ordering behaviour can be explored. My research in this field has led to a complete theoretical characterization of driven 1D facet ordering [2] in the long-wave approximation [3].This was extended to 2D in a recent publication [1] on self-assembled faceted crystal surfaces. The analytical aspects of these studies combines modern methods for partial differential equations with matched asymptotic analysis in novel ways, while the morphometric structure stems from the introduction of a unique computational geometry tool that utilizes the theory.

I also have a deep interest in the theory of PDE, and in particular in mixed hyperbolic-parabolic systems stemming from continuum physics [4].

Sample of recent works:

1).Scaling theory and morphometrics for a coarsening multiscale surface, via a principle of maximal dissipation (with S.A. Norris), Physical Review Letters 96 (17), Art. No. 176103 (2006).

2). Coarsening dynamics for the convective Cahn-Hilliard equation (with F. Otto, B. Rubinstein and S.H. Davies), Physica D 178 (2003), 127-148.

3). Crystal growth, coarsening and the convective Cahn-Hilliard equation, International Series of Numerical Mathematics, 147 (2003), Birkhauser.

4). Unique global solvability for initial-boundary value problems in one-dimensional nonlinear thermoviscoelasticity, Archive for Rational Mechanics and Analysis 153 (2000), 1-37.

Possible Ph.D. topics:

I). Modelling, analysis and simulation of faceted surfaces evolving by surface diffusion.

II). Thermo-viscoelastic materials with phase transitions: existence, uniqueness and temporal asymptotics.

Sample of recent works:

2). Buoyancy-driven crack propagation: the limit of large fracture toughness (with J. R. Lister). J. Fluid Mech. 580 (2007), 359-380.

3). Buoyancy-driven crack propagation from an over-pressured source (with J. R. Lister), J. Fluid Mech. 536 (2005), 79-98.

PhD students

Roger Bustamante, Fotios Kassianidis, Yunfei Zhu, Craig Chapman.

Former Students

Anna Guillou, Melanie Ottenio, Anthony Orr, Wasiq Hussain, Barry McKay, Xiamei-Jiang, Gillian Dryburgh.

Visitors (recent and current)

Dr. Michel Destrade, Universite Pierre et Marie Curie, Paris

Professor Jim Dunwoody, Queen's University, Belfast

Professor Tom Pence, Michigan State University

Professor David Steigmann, UC Berkeley

Professor Pham Chi Vinh, Hanoi National University

Dr. Wlodzimierz Domanski, IPPT, Warsaw

Alexander Ehret, University of Bayeuth

Dr. Baljeet Singh, Indian National Science Academy

Dr. Jose Merodio, University of Cantabria

Professor Gerhard Holzapfel

Dr. Luis Dorfmann