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IMA Newsletter #415

May 2011

2010-2011 Program

See http://www.ima.umn.edu/2010-2011/ for a full description of the 2010-2011 program on Simulating Our Complex World: Modeling, Computation and Analysis.

IMA Events

IMA Workshop

Strain Induced Shape Formation: Analysis, Geometry and Materials Science

May 16-20, 2011

Organizers: Marta Lewicka (Rutgers University), Shankar Venkataramani (University of Arizona)
Schedule

Monday, May 2

10:45am-11:15am Coffee breakLind Hall 400

Tuesday, May 3

10:45am-11:15am Coffee breakLind Hall 400

Wednesday, May 4

10:45am-11:15am Coffee breakLind Hall 400
2:30pm-3:30pm Math 8994: Discontinuous Galerkin methods: An introduction - Incompressible Navier-Stokes equationsBernardo Cockburn (University of Minnesota)Lind Hall 305

Thursday, May 5

10:45am-11:15am Coffee breakLind Hall 400

Friday, May 6

10:45am-11:15am Coffee breakLind Hall 400

Monday, May 9

10:45am-11:15am Coffee breakLind Hall 400

Tuesday, May 10

10:45am-11:15am Coffee breakLind Hall 400
11:15am-12:15pm Smoothness of Nonlinear Subdivision CurvesTom Duchamp (University of Washington)Lind Hall 305 PS

Wednesday, May 11

10:45am-11:15am Coffee breakLind Hall 400
2:30pm-3:30pm Math 8994: Discontinuous Galerkin methods: An introduction - Compressible Navier-Stokes equationsBernardo Cockburn (University of Minnesota)Lind Hall 305

Thursday, May 12

10:45am-11:15am Coffee breakLind Hall 400

Friday, May 13

10:45am-11:15am Coffee breakLind Hall 400

Monday, May 16

All Day Chair: Shankar Venkataramani (University of Arizona)

5-day Poster Session: Posters will be on display all day long in Lind Hall Monday through Friday.
SW5.16-20.11
8:15am-9:00am Registration and coffeeLind Hall 400 SW5.16-20.11
9:00am-9:15am Welcome to the IMAFadil Santosa (University of Minnesota)Lind Hall 305 SW5.16-20.11
9:15am-10:15am Direct and Inverse Problems in Shaping SheetsL. Mahadevan (Harvard University)Lind Hall 305 SW5.16-20.11
10:15am-10:45am Coffee breakLind Hall 400 SW5.16-20.11
10:45am-11:45am Wrinkling, microstructure, and energy scaling lawsRobert V. Kohn (New York University)Lind Hall 305 SW5.16-20.11
11:45am-1:30pm Lunch SW5.16-20.11
1:30pm-2:30pm Matching properties of isometries and elasticity of thin shells Marta Lewicka (Rutgers University)Lind Hall 305 SW5.16-20.11
2:30pm-2:40pm Group photo SW5.16-20.11
2:40pm-3:00pm Coffee breakLind Hall 400 SW5.16-20.11
3:00pm-3:30pm Wrinkles as a relaxation of compressive stresses in annular thin filmsPeter Bella (New York University)Lind Hall 305 SW5.16-20.11
3:30pm-4:00pm Effective Theories and Minimal Energy Configurations of Heterogeneous MultilayersBernd Schmidt (TU München)Lind Hall 305 SW5.16-20.11
4:00pm-4:30pm An Aronsson type approach to extremal quasiconformal mappings.Luca Capogna (University of Arkansas)Lind Hall 305 SW5.16-20.11
4:30pm-5:00pm The metric description of incompatible growth and irreversible deformationsEfi Efrati (University of Chicago)Lind Hall 305 SW5.16-20.11
5:00pm-6:00pm Reception and PostersLind Hall 400 SW5.16-20.11
Metric-induced wrinkling of an elastic thin filmPeter Bella (New York University)
Buckling of swelled ribbons: conical normals and minimal resonances AbstractBryan Gin-ge Chen (University of Pennsylvania)
Differential Growth and Ripples in Thin Elastic SheetsJohn Gemmer (University of Arizona)
Remote Control Jell-ODavid James Hamby (GELgothic)
Prestrained Kirchhoff shell theory: the derivation and the analysisHui Li (University of Minnesota)
Shaping via active deformation of thin elastic sheets Eran Sharon (Hebrew University)
Stable Phase-tip Splitting via Global Bifurcation Andras Arpad Sipos (Cornell University)

Tuesday, May 17

All Day Chair: Robert V. Kohn (New York University) SW5.16-20.11
8:30am-9:00am CoffeeLind Hall 400 SW5.16-20.11
9:00am-10:00am Tubes and CylindersAlain Goriely (University of Oxford)Lind Hall 305 SW5.16-20.11
10:00am-10:30am Coffee breakLind Hall 400 SW5.16-20.11
10:30am-11:30am Thin-limit behavior of engineered non-Euclidean plates, a theoretical analysisReza Pakzad (University of Pittsburgh)Lind Hall 305 SW5.16-20.11
11:30am-2:00pm Lunch SW5.16-20.11
2:00pm-3:00pm Geometrical Exactness and Nonlinear ResponseStuart S. Antman (University of Maryland)Lind Hall 305 SW5.16-20.11
3:00pm-3:30pm Coffee breakLind Hall 305 SW5.16-20.11
3:30pm-4:30pm Discussion sessionL. Mahadevan (Harvard University)Lind Hall 305 SW5.16-20.11

Wednesday, May 18

All Day Chair: Stuart Antman (University of Maryland) SW5.16-20.11
8:30am-9:00am Coffee Lind Hall 400 SW5.16-20.11
9:00am-10:00am Theoretical models for growth of melanomaMartine Ben Amar (École Normale Supérieure)Lind Hall 305 SW5.16-20.11
10:00am-10:30am Coffee breakLind Hall 400 SW5.16-20.11
10:30am-11:30am The Lame` problem: A prototypical model for wrinkling in thin sheetsBenny Davidovitch (University of Massachusetts)Lind Hall 305 SW5.16-20.11
11:30am-1:30pm Lunch SW5.16-20.11
1:30pm-2:30pm Soft machines and the mechanics of growing sheetsEran Sharon (Hebrew University)Lind Hall 305 SW5.16-20.11
2:30pm-3:00pm Coffee breakLind Hall 400 SW5.16-20.11
3:00pm-3:30pm Pattern Formation in Clamped Gel Membranes under Gradient StimulationRonald Alan Siegel (University of Minnesota)Lind Hall 305 SW5.16-20.11
3:30pm-4:00pm Shape Selection in Hyperbolic Non-Euclidean PlatesJohn Gemmer (University of Arizona)Lind Hall 305 SW5.16-20.11
4:00pm-4:30pm How to make a (designed) three-dimensional shape from a growing sheetChristian Santangelo (University of Massachusetts)Lind Hall 305 SW5.16-20.11
5:30pm-7:30pm Social hour at Stub and HerbsStub and Herbs
227 Oak St Minneapolis, MN 55414
(612) 379-0555
SW5.16-20.11

Thursday, May 19

All Day Chair: Martine Ben Amar (École Normale Supérieure) SW5.16-20.11
8:30am-9:00am CoffeeLind Hall 400 SW5.16-20.11
9:00am-10:00am And now for something (not quite) completely differentAlan C. Newell (University of Arizona)Lind Hall 305 SW5.16-20.11
10:00am-10:30am Coffee breakLind Hall 400 SW5.16-20.11
10:30am-11:30am Kinetics of lattice phase transitionsAnna Vainchtein (University of Pittsburgh)Lind Hall 305 SW5.16-20.11
11:30am-1:30pm Lunch SW5.16-20.11
1:30pm-2:00pm Persistence HomologyPatrick Daniel Shipman (Colorado State University)Lind Hall 305 SW5.16-20.11
2:00pm-2:30pm Geometric AnelasticityArash Yavari (Georgia Institute of Technology)Lind Hall 305 SW5.16-20.11
2:30pm-3:00pm Elastic Building Blocks in a Wrinkle CascadeRobert Schroll (University of Massachusetts)Lind Hall 305 SW5.16-20.11
3:00pm-3:30pm Coffee breakLind Hall 400 SW5.16-20.11
3:30pm-4:30pm Discussion sessionMarta Lewicka (Rutgers University)
Shankar Venkataramani (University of Arizona)
Lind Hall 305 SW5.16-20.11
6:30pm-8:30pm Social dinner (coordinated by the workshop organizers)Kafe 421 - 421 14th Avenue Southeast Minneapolis, MN 55414 SW5.16-20.11

Friday, May 20

All Day Chair: Marta Lewicka (University of Minnesota and Rutgers University) SW5.16-20.11
8:30am-9:00am CoffeeLind Hall 400 SW5.16-20.11
9:00am-9:30am Mathematical analysis of microstructures and low hysteresis shape memory alloysBarbara Zwicknagl (Carnegie Mellon University)Lind Hall 305 SW5.16-20.11
9:30am-10:00am The von Karman theory for incompressible elastic shellsHui Li (University of Minnesota)Lind Hall 305 SW5.16-20.11
10:00am-10:30am Curved Fold OrigamiMarcelo Azevedo Dias (University of Massachusetts)Lind Hall 305 SW5.16-20.11
10:30am-11:00am Coffee break Lind Hall 400 SW5.16-20.11
11:00am-12:00pm Isometric immersions, geometric incompatibility and strain induced shape formation.Shankar Venkataramani (University of Arizona)Lind Hall 305 SW5.16-20.11
12:00pm-12:10pm Closing remarks Lind Hall 305 SW5.16-20.11

Monday, May 23

2:30pm-3:00pm Coffee breakLind Hall 400

Tuesday, May 24

10:45am-11:15am Coffee breakLind Hall 400
11:15am-12:15pm Postdoc Seminar:An introduction to DPG methodsJay Gopalakrishnan (University of Florida)Lind Hall 305 PS

Wednesday, May 25

2:30pm-3:00pm Coffee breakLind Hall 400

Thursday, May 26

2:30pm-3:00pm Coffee breakLind Hall 400

Friday, May 27

2:30pm-3:00pm Coffee breakLind Hall 400

Saturday, May 28

2:30pm-3:00pm Coffee breakLind Hall 400

Monday, May 30

All Day Memorial Day. The IMA is closed.

Tuesday, May 31

2:30pm-3:00pm Coffee breakLind Hall 400
Abstracts
Stuart S. Antman (University of Maryland) Geometrical Exactness and Nonlinear Response
Abstract: The most accessible problems for the mechanics of deformable solid bodies are those for thin bodies, namely, rods and shells, because their equations respectively have but one and two independent spatial variables. There is a voluminous literature devoted to the derivations of various models for such bodies undergoing small deformations. On the other hand, geometrically exact theories are derived directly from fundamental principles. They readily accommodate general nonlinear material response. This lecture describes solutions of a variety of steady-state and dynamical geometrically exact problems, emphasizing the appearance of thresholds in constitutive response that separate qualitatively different behaviors.
Peter Bella (New York University) Wrinkles as a relaxation of compressive stresses in annular thin films
Abstract: It is well known that elastic sheets loaded in tension will wrinkle, with the length scale of wrinkles tending to zero with vanishing thickness of the sheet [Cerda and Mahadevan, Phys. Rev. Lett. 90, 074302 (2003)]. We give the first mathematically rigorous analysis of such a problem. Since our methods require an explicit understanding of the underlying (convex) relaxed problem, we focus on the wrinkling of an annular sheet loaded in the radial direction [Davidovitch et al, arxiv 2010]. While our analysis is for that particular problem, our variational viewpoint should be useful more generally. Our main achievement is identification of the scaling law of the minimum energy as the thickness of the sheet tends to zero. This requires proving an upper bound and a lower bound that scale the same way. We prove both bounds first in a simplified Kirchhoff-Love setting and then in the nonlinear three-dimensional setting. To obtain the optimal upper bound, we need to adjust a naive construction (one family of wrinkles superimposed on the planar deformation) by introducing cascades of wrinkles. The lower bound is more subtle, since it must be ansatz-free.
Peter Bella (New York University) Metric-induced wrinkling of an elastic thin film
Abstract: We study the wrinkling of a thin elastic sheet caused by a prescribed non-Euclidean metric. This is a model problem for the folding patterns seen, e.g., in torn plastic membranes and the leaves of plants. Following the lead of other authors we adopt a variational viewpoint, according to which the wrinkling is driven by minimization of an elastic energy subject to appropriate constraints and boundary conditions. Our main goal is to identify the scaling law of the minimum energy as the thickness of the sheet tends to zero. This requires proving an upper bound and a lower bound that scale the same way. The upper bound is relatively easy, since nature gives us a hint. The lower bound is more subtle, since it must be ansatz-free.
Martine Ben Amar (École Normale Supérieure) Theoretical models for growth of melanoma
Abstract: Like many biological systems, tumours undergo morphological changes during their evolution. For melanoma, these changes are the main diagnosis of the clinicians. In response to external (e.g. nutrient availability) or internal (e.g. genetic mutations) perturbations, a neoplasm may switch from an initially benign and highly localized symmetric state to an aggressive behaviour [1]. This rapid invasion of the surrounding tissues usually involves a morphological instability due to the heterogeneous nature of the growth process. This symmetry breaking is crucial in the clinical evaluation of the malignant character of a tumour. In order to describe this instability and highlight a fundamental process at work in morphogenesis, we first model the tumour as a ring of proliferative cells surrounding a core of quiescent cells. A biomimetic experiment of swelling gel with a similar geometry as avascular growth of melonama in the epidermis is presented to show that this instability has an elastic origin due to the growth process itself. Then I will present an adaptation of the mixture model to melanoma growth and show that this model exhibits travelling wave solutions in one dimension carrying a transverse perturbation of finite amplitude. In radial, we have found the same spatial instability for radially growing tumour with constant velocity. We establish a criterion for shape bifurcation in function of the biomechanical parameters of the skin and tumour cell properties that we compare to clinical data.

Joint work with: C. Chatelain, P. Ciarletta and Julien Dervaux
Luca Capogna (University of Arkansas) An Aronsson type approach to extremal quasiconformal mappings.
Abstract: Quasiconformal mappings u:Ω->Ω between open domains in Rn, are W{1,n} homeomorphisms whose dilation K=|du|/ (det du)1/n is in L∞. A classical problem in geometric function theory consists in finding QC minimizers for the dilation within a given homotopy class or with prescribed boundary data. In a joint work with A. Raich we study C2 extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a `localized' form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.
Bryan Gin-ge Chen (University of Pennsylvania) Buckling of swelled ribbons: conical normals and minimal resonances Abstract
Abstract: Differential growth processes play a prominent role in shaping leaves and biological tissues. Using both analytical and numerical calculations, we consider the shapes of closed, elastic strips which have been subjected to an inhomogeneous pattern of swelling. The stretching and bending energies of a closed strip are frustrated by compatibility constraints between the curvatures and metric of the strip. To analyze this frustration, we study the class of “conical” closed strips with a prescribed metric tensor on their center line. The resulting strip shapes can be classified according to their number of wrinkles and the prescribed pattern of swelling. We use this class of strips as a variational ansatz to obtain the minimal energy shapes of closed strips and find excellent agreement with the results of a numerical bead-spring model.
Benny Davidovitch (University of Massachusetts) The Lame` problem: A prototypical model for wrinkling in thin sheets
Abstract: Wrinkling is a fundamental mechanism for the relief of compressive stress in thin elastic sheets. It is natural to consider wrinkling as a (supercritical) instability of an appropriate flat, highly-symmetric state of the sheet. This talk will address the subtlety of this approach by considering wrinkling in the Lame` geometry: an annular sheet under radial tension. This axi-symmetric system seems to be the most elementary, yet nontrivial extension of Euler buckling (that emerges under uniaxial compression). Nevertheless, despite its apparent simplicity, the Lame` geometry exhibits a dramatic change of the wrinkling pattern beyond the instability threshold. I will address the distinct features of wrinkling patterns in the near-threshold (NT) and far-from-threshold (FFT) regimes, and will show how they emanate from different asymptotic expansions of Foppl-van-Karman (FvK) equations in these two limits. Our systematic theory of the FFT regime unifies the old “membrane limit” approach for the asymptotic stress field (Wagner, Stein&Hedgepeth, Pipkin) with more recent scaling ideas for the wavelength of wrinkles (Cerda&Mahadevan). Combining the analysis of these asymptotic regimes allows us to construct a complete “phase diagram” for wrinkling patterns in the Lame` geometry that sheds new light on experiments in this field. I will discuss general lessons that can be extracted from this analysis, and will conclude with some conjectures on possible universal aspects of this study.
Marcelo Azevedo Dias (University of Massachusetts) Curved Fold Origami
Abstract: Despite an almost two thousand year history, origami, the art of folding paper, remains a challenge both artistically and scientifically. Traditionally, origami is practiced by folding along straight creases. A whole new set of shapes can be explored, however, if, instead of straight creases, one folds along arbitrary curves. We present a mechanical model for curved fold origami in which the energy of a plastically-deformed crease is balanced by the bending energy of developable regions on either side of the crease.
Tom Duchamp (University of Washington) Smoothness of Nonlinear Subdivision Curves
Abstract: Subdivision Curves were discovered by Georges de Rham in 1947. He noticed that the sequence of polygonal lines obtained by a "corner cutting" (or "subdivision") process converges to a C1 curve. B-splines are obtained by a generalization of this procedure, which depends on the affine structure of Euclidean space. A "linear subdivision curve" is the limiting curve obtained by such a linear subdivision process. The theory of such curves, is well-understood. Because subdivision curves also support multiresolution structure, various wavelet based compression algorithms apply to them. Consequently, subdivision curves offer a way to compress data related to curves in Euclidean space.

But not all data live in Euclidean space (rotations, rigid motions, deformation tensors are examples), and the quantity of such data is proliferating. In "Multiscale representations of manifold-valued data", Rahman, Rori, Stodden, Donoho, and Schroder, therefore, considered manifold-valued subdivision curves, and their multiresolution structure. These curves are all based on corresponding linear subdivision curves, and Rahman-et-al. conjectured that they enjoy the same regularity properties.

In my talk, I will review linear subdivision curves, explain the generalization to manifold-valued curves, and close with some surprising results concerning their regularity properties.

The talk is based on joint work with Gang Xie and Thomas Yu.
Efi Efrati (University of Chicago) The metric description of incompatible growth and irreversible deformations
Abstract: The language of Riemannian geometry arises naturally in the elastic description of amorphous solids, yet in the long history of elasticity it was put to very little practical use as a computational tool. In recent years the usage of Riemannian terminology has been revived, mostly in the context of incompatible irreversible deformations. In this talk I will compare different approaches to the description of growth and irreversible deformations focusing on the metric description of incompatible growth. I will also discuss the appropriate reduced theories for slender bodies. Particularly, I will present a specific problem inspired by strictureplasty in which the metric approach elucidates the path to solution.
John Gemmer (University of Arizona) Shape Selection in Hyperbolic Non-Euclidean Plates
Abstract: We present a theoretical study of free non-Euclidean plates with a disc geometry and a prescribed metric that corresponds to a constant negative Gaussian curvature. We take the equilibrium configuration taken by the these sheets to be a minimum of a Foppl Von-Kàrmàn type functional in which configurations free of in plane stretching correspond to isometric immersions of the metric. We show for all radii there exists low bending energy configurations free of any in plane stretching that obtain a periodic profile. The number of periods in these configurations is set by the condition that the principle curvatures of the surface remain finite and grows approximately exponentially with the radius of the disc.
John Gemmer (University of Arizona) Differential Growth and Ripples in Thin Elastic Sheets
Abstract: In this poster we illustrate that isometric immersions of the hyperbolic plane into three dimensional Euclidean space with a periodic profile exist. These surfaces are piecewise smooth but have vastly lower bending energy then their smooth counterparts and could explain why periodic hyperbolic surfaces are proffered in nature.
Jay Gopalakrishnan (University of Florida) Postdoc Seminar:An introduction to DPG methods
Abstract: We will discuss a new class of discontinuous Petrov-Galerkin (DPG) methods that achieves stability by automatically finding stable pairs of trial and test spaces in a Petrov-Galerkin framework. The key is that the use of discontinuous finite element spaces (as in DG methods) allows local computation of the test spaces that guarantee stability. (This is joint work with L. Demkowicz.)
Alain Goriely (University of Oxford) Tubes and Cylinders
Abstract: Among the many typical biological structures, cylindrical and tubular structures such as hyphae, stems, roots, blood vessels, airways, oesophagus, and tree trunks abound in nature. Tubes are typically used for transport, mechanical support or both. Their morphogenesis usually involves complex genetic and biochemical processes mediated by mechanical forces. In many cases, tubes have (at least) two layers glued together. Each layer has different mechanical and geometric properties. Moreover, due to growth taking place in the layers, each tube may also develops residual stresses. In this talk, I will be discussing the range of mechanical properties and functions that can be obtained by tuning these different properties within the framework of nonlinear morphoelasticity. In particular, I will discuss how differential axial growth can be used to improved structural stiffness (with examples from plants and arteries), how relative radial growth of the tube can either induce hollowing (as found in plant aerenchyma), or generate mucosal folding (as found in oesophagus and airways), and how anisotropy can induce handedness reversal (as found in phycomyces). Given time, I will also discuss the inverse problem of designing a tube with desired mechanical properties through growth and remodelling.
David James Hamby (GELgothic) Remote Control Jell-O
Abstract: I propose Jell-O as a building material. The concept stems from a question in blobby architecture on transformable walls. Phase transition gels are known to expand and contract up to a thousand fold. Tiny wireless stimulators mixed inside these gels could direct local shape changes. A sum of small volume changes would, in theory, yield the overall shape desired. The immediate goal of creating a set of prototype gel models was to provide visual aids as a basis for discussion with other disciplines. Starting by experimenting with rigid and flexible molds, a series of 10-centimeter jiggly gel objects was formed and photographed. Next, as a proof of concept, a 1-meter pneumatic robot was designed and constructed to demonstrate motion via selective volume displacement. Following the successes of the gel mold objects and robot control experiments, the two components will now be mixed for preliminary tests of a “slosh-bot.”
Robert V. Kohn (New York University) Wrinkling, microstructure, and energy scaling laws
Abstract: The mechanics of a thin elastic sheet can be explored variationally, by minimizing the sum of "membrane" and "bending" energy. For some loading conditions, the minimizer develops increasingly fine-scale wrinkles as the sheet thickness tends to 0. While the optimal wrinkle pattern is probably available only numerically, the qualitative features of the pattern can be explored by examining how the minimum energy scales with the sheet thickness. I will introduce this viewpoint by discussing past work on simpler but related problems. Then I'll discuss recent work with Hoai-Minh Nguyen, concerning the cascade of wrinkles observed by J. Huang et al at the edge of a floating elastic film (Phys Rev Lett 105, 2010, 038302).
Marta Lewicka (Rutgers University) Matching properties of isometries and elasticity of thin shells
Abstract: In this talk we will discuss how the mechanical response of an elastic film is affected by subtle geometric properties of its mid-surface. The crucial role is played by spaces of weakly regular (Sobolev) isometries or infinitesimal isometries. These are the deformations of the mid-surface preserving its metric up to a certain prescribed order of magnitude, and hence contributing to the stretching energy of the film at a level corresponding to the magnitude of the given external force.

In this line, we will discuss results concerning the matching and density of infinitesimal isometries on convex, developable and axisymmetric surfaces. By a matching property, we refer to the possibility of modifying an infinitesimal isometry of a certain order to make it an infinitesimal isometry of a higher order. In particular, on a convex surface, any one parameter family of first order bendings generated by a first order isometry can be modified at a higher order of perturbation to a family of exact bendings (exact isometries).

We will show how this analysis can be combined with the tools of calculus of variations towards the rigorous derivation of a hierarchy of thin shell theories. The validity of each theory depends on the scaling of the applied force in terms of the vanishing thickness of the reference shell. The obtained hierarchy extends the seminal result of Friesecke, James and Muller valid for flat (plate-like) films, to shells whose mid-surface may have arbitrary geometry. When a matching property is established, the above-mentioned infinte hierarchy effectively collapses to a finite number of theories.
Hui Li (University of Minnesota) The von Karman theory for incompressible elastic shells
Abstract: We rigorously derive the von Karman shell theory (and the resulting von Karman equations) for incompressible materials. Our approach is variational and starts from the general nonlinear 3-dimensional elastic energy functional. Our only assumption is that the midsurface of the shell enjoys the following approximation property: C3 first order infinitesimal isometries are dense in the space of all W^{2, 2} infinitesimal isometries. The class of surfaces with this property includes: flat surfaces, convex surfaces, developable surfaces and rotationally invariant surfaces.
Hui Li (University of Minnesota) Prestrained Kirchhoff shell theory: the derivation and the analysis
Abstract: Applying methods of Calculus of Variations, we introduce and justify a variant of Kirchhoff theory for thin 3d shells, valid in presence of residual stresses. The effective 2d energy is the relative bending, appropriately modified while in the varying thickness and/or incompressible materials situation
L. Mahadevan (Harvard University) Direct and Inverse Problems in Shaping Sheets
Abstract: I will discuss some simple mathematical problems associated with the shaping of sheets inspired by the buckling of graphene, the rippling of leaf edges, the blooming of flowers, and the coiling of guts. One particular focus is the role of boundary conditions at a free edge, and a second is the question of inverse problems inspired by optimal design for tissue engineering.
Alan C. Newell (University of Arizona) And now for something (not quite) completely different
Abstract: Many of the challenges of finding the shapes of elastic surfaces have first cousins in the world of pattern formation. I will try to sketch out the connections and explain where there are similarities and where there are profound differences even though the equations and the free energies look much the same. If time permits, and with the indulgence of the audience, I shall also tell you how a three dimensional version of the ideas give rise to objects, "quarks and leptons," with spin and charge symmetries which arise because of symmetry breaking and which do not require to be put in by hand.
Reza Pakzad (University of Pittsburgh) Thin-limit behavior of engineered non-Euclidean plates, a theoretical analysis
Abstract: Non-Euclidean thin plates arise in different circumstances: differential growth, swelling, shrinking or plastic deformations can set the geometry of an elastic body to a preferred "target metric". In our model, the latter plays the main role in determining the shape of the plate. We use analytical techniques in the context of calculus of variations to predict the behavior of these structures for their very thin limits. We will moreover discuss a disparity between the theoretical analysis and experimental data, in which a sharp qualitative contrast between the negative and positive constant curvature cases has been observed.
Christian Santangelo (University of Massachusetts) How to make a (designed) three-dimensional shape from a growing sheet
Abstract: What three-dimensional shapes can be made with an elastic film of finite thickness upon which an isotropic, but inhomogeneous, pattern of growth has been prescribed? I will describe both theoretical progress in addressing this question, and an experimental realization in a swelling polymer film in which a metric is prescribed by modulating the local polymer cross-link density. By imposing a pattern of swelling dots, similar to half-toning in an inkjet printer, we can prescribe arbitrary swelling patterns. This system allows us to directly put mathematics to the experimental test. I will finally present a simple swelling geometry from which more complex shapes can be built, and rationalize some of the potentially counterintuitive behavior observed experimentally.
Bernd Schmidt (TU München) Effective Theories and Minimal Energy Configurations of Heterogeneous Multilayers
Abstract: We derive effective theories for heterogeneous multilayers from three-dimensional nonlinear elasticity by Gamma-convergence. Such materials have been used recently for a self-induced fabrication of nanotubes. The energy minimizers of the limiting functional turn out to be cylinders (scrolls) whose winding direction and radius depends on the equilibrium misfit of the specimen's layers. Taking a non-interpenetration condition into account we find spirals and double spirals as energetically optimal shapes.
Robert Schroll (University of Massachusetts) Elastic Building Blocks in a Wrinkle Cascade
Abstract: Several features, such as d-cones, minimal ridges, developable patches, and collapsed compressive stress, occur regularly in the the configuration of elastic sheets. We dub such features "building blocks." By understanding the shape of an elastic sheet as an amalgamation of these building blocks, we can understand its behavior without fully solving the governing equations. Here, we consider the building blocks that make up a wrinkle cascade. Such a cascade occurs when an elastic sheet is subject to confinement, so that it buckles at some optimal wavelength, but is required to have another wavelength at one end. The transition between imposed and optimal wavelength occurs in a cascade going through several intermediate wavelengths. We simulate a single generation of this cascade and demonstrate that it is composed of two different building blocks: a focused-stress feature reminiscent of a d-cone and a "diffuse-stress" feature. The former is characterized by a geometrical constraint (inextensibility), while the latter is governed by a mechanical constraint: the dominance of a single component of the stress tensor. We will discuss how boundary conditions affect which building blocks are chosen.
Eran Sharon (Hebrew University) Soft machines and the mechanics of growing sheets
Abstract: Many natural structures are made of soft tissue that undergoes complicated shape transformations as a result of the distribution of local active deformation of its "elements". Currently, the ability of mimicking this shaping mode in man-made structures is poor. I will present some results of our study of actively deforming thin sheets. We formulated a covariant elastic theory from which we derive an approximate 2D plate/shell theory for sheets with intrinsic incompatible metric and curvature tensors. With this theory we study selected cases of special interest. Experimentally, we use environmentally responsive gel sheets that adopt prescribed metrics upon induction by environmental conditions. With this system we study the shaping mechanism in different cases of imposed metrics and curvature. The generated sheets can be viewed as primitive soft machines. Finally, we study different cases of plant mechanics, connecting between the local growth tensor of the tissue and the evolution of the global shape of an organ.
Eran Sharon (Hebrew University) Shaping via active deformation of thin elastic sheets
Abstract: I will present our theoretical framework and experimental techniques, developed for constructing thin elastic sheets that undergo a known, nonuniform active deformation (or "growth") and calculating their equilibrium configurations. The poster includes two limit examples: 1) Non-Euclidean plates, in which the lateral growth is uniform along the thickness of the sheet, but varies across its surface. 2) An incompatible shell, in which the lateral growth is uniform across the surface, but varies along the sheet thickness, leading to double spontaneous curvature. "interesting" configurations and transitions, relevant to biological and chemical systems will be presented
Patrick Daniel Shipman (Colorado State University) Persistence Homology
Abstract: I will propose techniques of persistence homology for use in analyzing growing cells.
Ronald Alan Siegel (University of Minnesota) Pattern Formation in Clamped Gel Membranes under Gradient Stimulation
Abstract: A biochemomechanical oscillator has been developed in which a clamped, pH-sensitive hydrogel membrane containing N-isopropylacrylamide (NIPAAm) and methacrylic acid (MAA) separates a chamber containing glucose oxidase from a pH controlled external medium containing a constant concentration of glucose. This system undergoes oscillations in intrachamber pH and concomitant on/off switching of glucose permeation through the membrane due to a nonlinear feedback instability between the enzyme-mediated reaction, which converts glucose to hydrogen ion, and the swelling/glucose-permeability characteristic of the membrane. Oscillation period increases with time due to buildup in the chamber of a buffering product, gluconate ion, and eventually oscillations cease. During operation there is a fluctuating pH gradient between the chamber and the external medium. We have gathered experimental evidence that a sustained pH gradient leads to stress induced pattern formations in the hydrogel due to phase separation, which we believe may be responsible for cessation of oscillations. We have also shown pattern development in clamped thermally sensitive hydrogel membranes based on NIPAAm without MAA), with a temperature gradient applied across the membrane. A mathematical description of the observed phenomena is desirable.
Andras Arpad Sipos (Cornell University) Stable Phase-tip Splitting via Global Bifurcation
Abstract: We consider multi-phase equilibria of elastic solids under anti-plane shear.We use global bifurcation methods to determine paths of equilibria in the presence of small interfacial energy. In an earlier paper the rigorous existence of global bifurcating branches was established. The stability of the solutions along these branches are difficult to determine. By an appropriate numerical representation of the second variation we show, that phase-tip splitting at the boundary (which is typically observed in experiments with shape memory alloys) appears for stable solutions of our model.
Anna Vainchtein (University of Pittsburgh) Kinetics of lattice phase transitions
Abstract: Shape memory alloys and other active materials undergo displacive phase transitions which change the symmetry of the crystal lattice through a diffusionless coordinated motion of atoms. A signature feature of these materials is the hysteresis they exhibit in response to cyclic loading. The dissipation is due to propagating phase boundaries that can be represented at the continuum level as surfaces of discontinuity. Classical elastodynamics admits nonzero dissipation on moving discontinuities but provides no information about its origin and kinetics. In the presence of subsonic discontinuities, this results in an ill-posed initial-value problem. One can extract the missing information about phase boundary kinetics and regularize the continuum model by considering its natural discrete prototype. This leads to an incredibly rich lattice model that also describes other interesting phenomena such as phase nucleation and evolution of microstructures. In this talk I will describe some recent work in this direction.
Shankar Venkataramani (University of Arizona) Isometric immersions, geometric incompatibility and strain induced shape formation.
Abstract: Thin elastic sheets are usually modeled by variational problems for an energy with two scales, a (strong) stretching energy and a (weak) bending energy. A useful paradigm in understanding the behavior of theses sheets under various loadings/boundary conditions is the following -- Singularites/microstructure in the observed configurations of thin sheets reflect "geometric incompatibility", that is the non-existence of admissible, sufficiently smooth isometric immersions (zero stretching energy test functions). Using specific examples, and a combination of rigorous results and conjectures motivated by numerical computation, I will try to argue that the relation between the observed geometry of thin elastic sheets and the existence/nonexistence of isometric immersions is much more subtle.
Arash Yavari (Georgia Institute of Technology) Geometric Anelasticity
Abstract: There are different sources of residual stresses in solids. Nonuniform temperature distributions and defects (e.g. dislocations) have been of interest in mechanics in the last few decades (and of course bulk growth more recently). Distributed dislocations were geometrically studied by Kondo and Bilby (among others) in the 1950s. Dislocation density tensor has been identified with torsion of an affine connection (of a flat material manifold) in the literature. However, the successful phenomenological models in finite plasticity have been non-geometric and overwhelmingly based on a multiplicative decomposition of deformation gradient into elastic and plastic parts. This idea has been the main kinematic assumption in the early theories of bulk growth as well. In this talk I will outline a theory of anelasticity in which material manifold has an evolving geometry. In this framework elasticity is a very special case for which geometry of the material manifold is time independent. A connection will be made between Cartan's moving frames and decomposition of deformation gradient. The Riemannian manifold corresponding to a flat Riemann-Cartan material manifold will be defined. As an example of its application, I will show how to construct the material manifold of a single screw dislocation. Then the residual stress field of a single screw dislocation in an incompressible nonlinear elastic solid will be obtained.
Barbara Zwicknagl (Carnegie Mellon University) Mathematical analysis of microstructures and low hysteresis shape memory alloys
Abstract: For certain martensitic phase transformations, one observes a close relation between the width of the thermal hysteresis and the compatibility of two phases. The latter is in the context of geometrically non-linear elasticity measured by the deviation of the middle eigenvalue of the transformation stretch matrix from one. This observation forms the basis of a theory of hysteresis that assigns an important role to the energy of the transition layer (Zhang, James, Müller, Acta mat. 57(15):4332–4352, 2009). Following this ansatz, we study the energy barriers leading to hysteresis, and analyze the shapes of energetically optimal transition layers for low hysteresis alloys.
Visitors in Residence
Stuart S. Antman University of Maryland 5/16/2011 - 5/19/2011
Douglas N. Arnold University of Minnesota 9/1/2010 - 6/30/2011
Gerard Michel Awanou Northern Illinois University 9/1/2010 - 6/10/2011
Nusret Balci University of Minnesota 9/1/2009 - 8/31/2011
Pavel Belik Augsburg College 5/16/2011 - 5/20/2011
Peter Bella New York University 5/15/2011 - 5/20/2011
Martine Ben Amar École Normale Supérieure 5/15/2011 - 5/19/2011
Albert Boggess Texas A & M University 5/31/2011 - 6/2/2011
Joseph P. Brennan University of Central Florida 5/31/2011 - 6/2/2011
Susanne C. Brenner Louisiana State University 9/1/2010 - 6/10/2011
Leslie Button Corning Incorporated 5/31/2011 - 6/4/2011
Jeff Calder University of Michigan 5/15/2011 - 5/21/2011
Maria-Carme T. Calderer University of Minnesota 5/16/2011 - 5/20/2011
Luca Capogna University of Arkansas 5/15/2011 - 5/20/2011
Aycil Cesmelioglu University of Minnesota 9/30/2010 - 8/30/2012
Chi Hin Chan University of Minnesota 9/1/2009 - 8/31/2011
Bryan Gin-ge Chen University of Pennsylvania 5/15/2011 - 5/20/2011
Bernardo Cockburn University of Minnesota 9/1/2010 - 6/30/2011
Jintao Cui University of Minnesota 8/31/2010 - 8/30/2012
Vaishali J Damle Springer 5/16/2011 - 5/18/2011
Benny Davidovitch University of Massachusetts 5/15/2011 - 5/20/2011
Paul Davis Worcester Polytechnic Institute 5/31/2011 - 6/2/2011
Marcelo Azevedo Dias University of Massachusetts 5/15/2011 - 5/20/2011
Andrew Dienstfrey National Institute of Standards and Technology 5/31/2011 - 6/3/2011
David C. Dobson University of Utah 5/9/2011 - 5/12/2011
Achi Dosanjh Spring-Verlag 5/16/2011 - 5/18/2011
Tom Duchamp University of Washington 4/1/2011 - 6/15/2011
Efi Efrati University of Chicago 5/15/2011 - 5/20/2011
Matt Elsey University of Michigan 5/14/2011 - 5/21/2011
Selim Esedoglu University of Michigan 1/20/2011 - 6/10/2011
Randy H. Ewoldt University of Minnesota 9/1/2009 - 8/31/2011
Brittan Farmer University of Michigan 5/9/2011 - 5/14/2011
Oscar E. Fernandez University of Michigan 8/31/2010 - 8/30/2011
Daniel Frohardt Wayne State University 5/31/2011 - 6/2/2011
John Gemmer University of Arizona 5/15/2011 - 5/20/2011
Albert B. Gilg Siemens 5/31/2011 - 6/7/2011
Jay Gopalakrishnan University of Florida 9/1/2010 - 6/30/2011
Alain Goriely University of Oxford 5/15/2011 - 5/19/2011
Shiyuan Gu Louisiana State University 9/1/2010 - 6/30/2011
David James Hamby GELgothic 5/15/2011 - 5/21/2011
Christopher Heil Georgia Institute of Technology 5/31/2011 - 6/2/2011
Matthias Heinkenschloss Rice University 5/31/2011 - 6/1/2011
Yulia Hristova University of Minnesota 9/1/2010 - 8/31/2012
Farhad Jafari University of Wyoming 5/29/2011 - 6/2/2011
Michael S. Jolly Indiana University 5/31/2011 - 6/2/2011
Markus Keel University of Minnesota 7/21/2008 - 6/30/2011
Stephen Keeler Boeing 5/31/2011 - 6/2/2011
Erica Zimmer Klampfl Ford 5/31/2011 - 6/2/2011
Wolfgang Kliemann Iowa State University 5/31/2011 - 6/1/2011
Robert V. Kohn New York University 5/15/2011 - 5/20/2011
Pawel Konieczny University of Minnesota 9/1/2009 - 8/31/2011
Komandur R. Krishnan Telcordia 5/31/2011 - 6/2/2011
Suzanne Lenhart University of Tennessee 5/31/2011 - 6/2/2011
Gilad Lerman University of Minnesota 9/1/2010 - 6/30/2011
Mark Levi Pennsylvania State University 5/31/2011 - 6/2/2011
Marta Lewicka Rutgers University 5/15/2011 - 5/22/2011
Hengguang Li University of Minnesota 8/16/2010 - 8/15/2011
Hui Li University of Minnesota 5/15/2011 - 5/20/2011
Zhi (George) Lin University of Minnesota 9/1/2009 - 8/31/2011
Jiangguo (James) Liu Colorado State University 5/31/2011 - 6/2/2011
Jianfeng Lu New York University 5/16/2011 - 5/20/2011
Roger Lui Worcester Polytechnic Institute 5/31/2011 - 6/2/2011
Mitchell Luskin University of Minnesota 9/1/2010 - 6/30/2011
L. Mahadevan Harvard University 5/15/2011 - 5/18/2011
Kara Lee Maki University of Minnesota 9/1/2009 - 8/31/2011
Yu (David) Mao University of Minnesota 8/31/2010 - 8/30/2012
Irina Mitrea University of Minnesota 8/16/2010 - 6/14/2011
Dimitrios Mitsotakis University of Minnesota 10/27/2010 - 8/31/2012
Jeff Morgan University of Houston 5/31/2011 - 6/2/2011
Alan C. Newell University of Arizona 5/14/2011 - 5/20/2011
Alexandra Ortan University of Minnesota 9/16/2010 - 6/15/2011
Cecilia Ortiz-Duenas University of Minnesota 9/1/2009 - 8/31/2011
Reza Pakzad University of Pittsburgh 5/15/2011 - 5/22/2011
Petr Plechac University of Delaware 5/31/2011 - 6/4/2011
Serge Preston Portland State University 5/31/2011 - 6/2/2011
Sridevi Pudipeddi Waldorf College 5/1/2011 - 6/30/2011
Weifeng (Frederick) Qiu University of Minnesota 8/31/2010 - 8/30/2012
Vincent Quenneville-Belair University of Minnesota 9/16/2010 - 6/15/2011
Wayne Raskind Arizona State University 5/31/2011 - 6/2/2011
Fernando Reitich University of Minnesota 9/1/2010 - 6/30/2011
Christian Santangelo University of Massachusetts 5/15/2011 - 5/21/2011
Fadil Santosa University of Minnesota 7/1/2008 - 8/30/2011
Bernd Schmidt TU München 5/14/2011 - 5/21/2011
Robert Schroll University of Massachusetts 5/15/2011 - 5/20/2011
Shuanglin Shao University of Minnesota 9/1/2009 - 8/31/2011
Eran Sharon Hebrew University 5/15/2011 - 5/19/2011
Zhongwei Shen University of Kentucky 5/31/2011 - 6/2/2011
Patrick Daniel Shipman Colorado State University 5/15/2011 - 5/20/2011
Ratnasingham Shivaji Mississippi State University 5/31/2011 - 6/2/2011
Ronald Alan Siegel University of Minnesota 5/15/2011 - 5/20/2011
Andras Arpad Sipos Cornell University 5/15/2011 - 5/20/2011
Richard Sowers University of Illinois at Urbana-Champaign 5/31/2011 - 6/2/2011
Panagiotis Stinis University of Minnesota 9/1/2010 - 6/30/2011
Allan Struthers Michigan Technological University 5/31/2011 - 6/2/2011
Li-yeng Sung Louisiana State University 9/1/2010 - 6/15/2011
Nicolae Tarfulea Purdue University, Calumet 9/1/2010 - 6/15/2011
Luis Tenorio Colorado School of Mines 3/27/2011 - 6/10/2011
Dimitar Trenev University of Minnesota 9/1/2009 - 8/31/2011
Richard Tsai University of Texas at Austin 3/3/2011 - 5/1/2011
Anna Vainchtein University of Pittsburgh 5/17/2011 - 5/20/2011
Shankar Venkataramani University of Arizona 5/15/2011 - 5/20/2011
Chai Wah Wu IBM 5/31/2011 - 6/2/2011
Arash Yavari Georgia Institute of Technology 5/15/2011 - 5/21/2011
Yangbo Ye University of Iowa 5/31/2011 - 6/1/2011
Ganghua Yuan Northeast (Dongbei) Normal University 4/27/2011 - 7/27/2011
Zhengfang Zhou Michigan State University 5/31/2011 - 6/2/2011
Barbara Zwicknagl Carnegie Mellon University 5/15/2011 - 5/20/2011
Legend: Postdoc or Industrial Postdoc Long-term Visitor

IMA Affiliates:
Arizona State University, Boeing, Colorado State University, Corning Incorporated, ExxonMobil, Ford, General Motors, Georgia Institute of Technology, Honeywell, IBM, Indiana University, Iowa State University, Korea Advanced Institute of Science and Technology (KAIST), Lawrence Livermore National Laboratory, Lockheed Martin, Los Alamos National Laboratory, Medtronic, Michigan State University, Michigan Technological University, Mississippi State University, Northern Illinois University, Ohio State University, Pennsylvania State University, Portland State University, Purdue University, Rice University, Rutgers University, Sandia National Laboratories, Schlumberger Cambridge Research, Schlumberger-Doll, Seoul National University, Siemens, Telcordia, Texas A & M University, University of Central Florida, University of Chicago, University of Delaware, University of Houston, University of Illinois at Urbana-Champaign, University of Iowa, University of Kentucky, University of Maryland, University of Michigan, University of Minnesota, University of Notre Dame, University of Pennsylvania, University of Pittsburgh, University of Tennessee, University of Wisconsin-Madison, University of Wyoming, US Air Force Research Laboratory, Wayne State University, Worcester Polytechnic Institute