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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.

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.

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.

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

Joint work with: C. Chatelain, P. Ciarletta and Julien Dervaux

Quasiconformal mappings u:Ω->Ω between open
domains in R^{n}, 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 C^{2} 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.

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)

http://www.physics.umass.edu/people/index.php?mode=full&id=198

http://www.physics.umass.edu/people/index.php?mode=full&id=198

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.

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.

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.

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.

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.

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.

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.”

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, The State University Of New Jersey )

http://www.math.pitt.edu/~lewicka/

Shankar Venkataramani (University of Arizona)

http://math.arizona.edu/~shankar

http://www.math.pitt.edu/~lewicka/

Shankar Venkataramani (University of Arizona)

http://math.arizona.edu/~shankar

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

I will propose techniques of persistence homology for use in analyzing growing cells.

Ronald Alan Siegel (University of Minnesota, Twin Cities)

http://www.pharmacy.umn.edu/faculty/siegel_ronald/home.html

http://www.pharmacy.umn.edu/faculty/siegel_ronald/home.html

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.

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.

May 19, 2011

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