Workshop on Elastic Stability

3rd October 2008
Mathematical Institute, University of Oxford

The workshop will address current issues related to the stability of solutions in nonlinear elasticity, including local energy minimizers, the stability of growing bodies, global existence for small data, bifurcation and continuation of solutions, and Saint-Venant’s principle.

Registration

The workshop will take place on Friday, 3rd October 2008 at the Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford. Registration with coffee will be at 10.00–10.30 am. Lunch will be at St Anne’s College and is free if the box is ticked on the registration page. You can register online or by contacting OxMOS. There is no registration fee. Limited funding is available to cover expenses. Applications will be treated on a case by case basis. Students and researchers at the beginning of their careers are particularly encouraged to apply.

Accommodation

If you need to stay overnight, you will need to arrange your own accommodation (unless you are speaking). The Travelodge at Oxford Pear Tree is on an easy Park and Ride bus route which will take you very close to the Maths Institute, and their rooms start at £19 if booked and paid for 21 days in advance. If you would like something more central in Oxford, prices will reflect this. The Tower House Hotel is very central and is quite reasonably priced but the rooms are rather quirky and individual. The Linton Lodge Hotel is well placed for the Maths Institute. Oxford City Council has an accommodation search which may also be useful. Please note that to be close to the Maths Institute or on an easy bus route you will want to search the Central and North areas. Please contact oxmos@maths.ox.ac.uk if you need further advice or help.

Abstracts

Local-in-time persistence of dynamic phase boundaries in elastic media
Heinrich Freistühler (Universität Leipzig, Germany)

Sufficient conditions for strong local minima in vectorial variational problems
Yury Grabovsky (Temple University, Philadelphia, USA)

The vectorial variational problem refers to the variational functional involving multiple integrals, where the unknown is a vector field. Such problems arise naturally in the context of non-linear elasticity and modeling of shape memory materials. We prove a conjecture of John Ball that the strengthened version of the set of well-known necessary conditions is sufficient for strong local minima. We achieve the goal by studying the effect of an arbitrary strong variation on the value of the functional. The key tool is the recently developed Decomposition Lemma, first proposed by Jan Kristensen (1994), that permits us to split the variation into the purely strong and weak part. We show that the two parts of the variation act on the functional independently (orthogonality principle). Positivity of second variation ensures that the weak part cannot decrease the functional, while the quasiconvexity conditions (the vectorial analog of the Weierstrass convexity condition) ensure that the strong part is unable to decrease the functional either. The latter part is accomplished by means of the localization principle. The use of the three key components of our analysis: the Decomposition Lemma, the orthogonality and the localization principles were inspired by the work of Fonseca, Mueller and Pedregal (1998).

Energy Minimizers as Weak Solutions in Second-Gradient Nonlinear Elastostatics
Tim Healey (Cornell University, USA)

We briefly recall the setting of classical nonlinear elastostatics and discuss the apparent incompleteness of all known approaches to existence theory. One possible remedy is to incorporate a second-gradient term representing “interfacial energy” into the model. In this talk we consider such a class of models in which the internal potential energy density is taken as the sum of a convex function of the second gradient and a general function of the gradient. However, in consonance with the classical theory, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of an energy minimizer is now routine, the existence of a weak solution is not, and we focus our efforts on that question. In particular, in the presence of reasonable growth conditions, we demonstrate that the determinant of the gradient of any admissible deformation corresponding to finite total energy is uniformly positive on the closure of the domain. With this in hand, Gateaux differentiability of the potential energy functional at a minimizer is clear, yielding the existence of a weak solution. We indicate how our results hold for a general class of boundary value problems, including “mixed” boundary conditions.

Spatial Elastic Stability
Robin Knops (Heriot-Watt University, UK)

Progress in dynamical elastic stability is severely hampered by lack of an appropriate existence theory. Obviously, this is much less the case in elastostatics. In particular, this enables the quasi-Hamiltonian structure of problems possessing a preferred spatial direction to be exploited by the application of techniques from dynamics. A review of this approach, principally to the estimation of edge effects (Saint-Venant’s principle), will be presented that includes the application of Liapunov stability theory, the more recent use of dual extremum principles in both linear and nonlinear theories, and the determination of growth and decay rates mainly in cylindrical bodies.

Simple aspects of growth and form
L. Mahadevan (Harvard University, USA)

The growth and form of a soft solid, such as what we are, poses a range of problems in mathematical physics, that focus on the stability of an inhomogeneously growing body as it leads to the evolution of form and its feedback on the growth process itself. I will discuss three examples of growth and form in the plant and animal world:

at the microscopic scale, I will discuss the growth of an actin network, and use it to look at the mechanics of polymerization engines
at the mesoscopic scale, I will discuss the dynamics of pollen tube growth, and use it to look at the mechanics of turgor engines
at the macroscopic scale, I will discuss the shape of a leaf or petal whose morphology is directly influenced by external mechanical forces.

Time permitting, I will close with some observations of the growth of a soft sheet in a confined space, using the mammalian brain for motivation.

Perturbative Methods in Nonlinear Elasticity
Becca Thomases (University of California, Davis, USA)

Global existence for nonlinear elasticity can be proven for small data perturbations via energy methods using weighted L² and Sobolev estimates. In this talk I will discuss these analytical tools which can be applied to compressible and incompressible isotropic elastic materials in three space dimensions. These methods can also be applied to problems in elasticity in exterior domains.