Fluids Deforming Solids - What About Geometry

What about geometry? Mechanics of Thin Dancing Disks Axisymmetric Disks Dynamic Plate Shape Diffusive dynamics: Bending dynamics: 0 Dynamics: Twisting Dynamics: Twisting How do thin structures grow? Permanent shape change and mass increase. • • Radial growth Through-thickness growth Shaping Sheets Shaping elastic sheets by prescribing non-Euclidean metrics • Prepare gels that undergo nonuniform shrinkage. • Buckling thin films based on chosen metrics. Shaping Sheets Shaping elastic sheets by halftone gel lithography • Photopattern thin films. • Thermal-actuated shape change. • Swell to embedding based on prescribed metric. Geometric Composite Goal: Use swelling to predictably & permanently morph plates into shells Geometric Composite Stretching Dominated Will bend as much as possible while minimizing stretching. Stretching Energy of the Plate (incompressible) Realized metric Target metric Stretching Energy First Fundamental Form Gaussian curvature Minimize Stretching Energy (Constant K metric) Geometric Composite Minimize Stretching Energy (Constant K metric) Taylor Expand Flat metric Kind of non-Euclidean metric Experiments: Mechanical Strain Residual Swelling The elastomer contains free, uncrosslinked polymer chains. Residual Swelling Residual Swelling Residual Swelling Stretching Energy Modulus difference Stretching ratio • • • • Depends on chemical and material properties. Should vary with R/Re Will depend on concentration gradient of free chains. from conservation of mass & proportional to mass uptake in annulus. Residual Swelling Approximate Analytical Solution • • Taylor expand about and 2D analog to Timoshenko’s model for thermal beam bending of bimetallic strips. Swelling Dynamics Diffusion of free chains Characteristic length: Growing Sheets Consider a thin structure with a growing top layer. • Caused by swelling, growth, heating, etc. Elastic energy density depends on material properties & metric tensors Lateral distances ( ) and curvatures ( ) that make the sheet stress free. Metric tensor Curvature tensor Isometric Limit Isometric Limit Numerical Simulations using COMSOL Multiphysics • • • Residual Swelling Experiments • • • • PVS Bilayer (Total thickness ~ 600um) Axisymmetric, circular plate. Pink (top) source of swelling for green (bot). Real time = 75 minutes (Video: 640x RT) Finite, incompatible tridimensional elasticity with a Neo-Hookean incompressible material. Distortions used to simulate prestretch. Top layer subjected to distortion field: Isometric Limit Ellipse Rectangle Cross Leaf Isometric Limit Isometric Limit In the isometric limit, the stretching energy is zero. • i.e. Curvature tensor (Cartesian) • Second fundamental form: Minimize bending energy • • Constrain the mid-surface to be flat Impose Lagrange multiplier enforcing Minimization yields: v Bifurcation Rectangular Sheet Stretching Energy Density In the limit of large stretching, the sheet adopts an isometry. For small stretching, the sheet is initially spherical curved. • Bifurcation from spherical to cylindrical. Classical problem (limited to circular and elliptical disks) • • Stoney formula relating stress to curvature. Strain mismatch work (Hyer, Freund, Seffen, etc.) Bifurcation Stretching Energy Assuming a metric with constant K (Gauss normal coords) Bending Energy In the spherical shape: Shape factor: • • Assume metric is axisymmetric. Assume K is homogenous. Shape factor: Structural Slenderness: Bifurcation Energetic cost to continue bending into a spherical cap: Energy to bend as a cylinder: v Energy balance: Bifurcation curvature: Shell Growth Swelling Structures Funding NSF CMMI – Mechanics of Materials (#1300860) Acknowledgements Matteo Pezzulla Anupam Pandey Suzie Protiere Pierre-Thomas Brun Paola Nardinocci (Sapienza) (Virginia Tech) (UPMC) (MIT) (Sapienza) Publications 1. 2. 3. A. Pandey and D.P. Holmes. “Swelling Induced Deformations: A Materials-Defined Transition from Structural Instability to Surface Instability.” Soft Matter, 9, 7049, (2013.) D.P. Holmes, M. Roché, T. Sinha, and H.A. Stone. “Bending and Twisting of Soft Materials by Non-homogenous Swelling” Soft Matter, 7, 5188, (2011). M. Pezzulla, S. Shillig, P. Nardinocchi, and D.P. Holmes. “Morphing of Geometric Composites via Residual Swelling,” Soft Matter, 11, 5812-5820, (2015).