Dynamics of a Viscous Liquid within an Elastic Shell with Application to Soft Robotics

1. Dynamics of a Viscous Liquid within an Elastic Shell with Application to Soft Robotics Shai B. Elbaz and Amir D. Gat Technion - Israel Institute of Technology Faculty of Mechanical Engineering 12/01/2013

2. Background (1/2) – Soft Robotics • Emerging field of experimental soft robotics. (Stokes et al. ,2013, Shepherd et al. 2013, others) • Embededfluidic networks • Constant spatial pressure - inflation/deflation • Essentially: creating a solid deformations field by a fluidic stress field.

3. Background (2/2) - Biological Flows Interaction between fluid and solid dynamics involving viscous flow through elastic cylinders extensively studied. • Heil & Pedley 1996,1997 studied the stability of cylindrical shells conveying viscous flow and stokes flow in collapsible tubes. • Paidoussis (1998) extensively studied fluid-structure Interactions for the case of axial flow in slender structures. • Canic & Mikelic 2003 studied viscous incomp. flow through a long elastic tube in the context of arterial blood flow. • Many others.

4. Our Goal • Apply models and methods used in biological flows to study time varying deformation patterns in soft-robotics. • Add a new level of control to soft-robotics • Introduce visco-elastic motion to traditional mechanical eng. applications. (Math. Overview)

5. Problem Definition • Fluid-structure interaction between: • Viscous, Newtonian, incompressible flow. • Slender, linearly elastic cylindrical shell closed at one end. • Assume negligible inertia in liquid and solid. • External stress and pressure

6. Elastic Medium (1/2) – Governing Eq. • Conservation of Momentum, • Strain Displacement Relations, • Hook’s Law, , ,

7. Elastic Medium (2/2) – Final Formulation • Follow elastic axi-symmetric shell theory. • Boundary conditions imposed on stress field at fluidic and external interface. • We relate the deformations to fluidic pressure and stress.

8. Fluidic medium (1/2) – Governing Eq. • Conservation of Momentum, • Conservation of Mass, • Velocity boundary conditions - no-slip and no-penetration imposed at solid-liquid interface.

9. Fluidic Medium (2/2) – Final Formulation • Axial velocity profile, • Non-zero velocity boundary conditions yield char. time scale, • Integrating continuity Eq.,

10. Coupled Fluidic-Elastic System • Subs. elastic relations into integrated continuity relation, • are known rational functions of . • For incompressible materials, , effect of . • We may formulate an IBVP on the pressure field,

11. Results Overview • Solid-liquid material properties, • Slenderness ratio – • Wall thickness ratio – • Liquid - Silicone oil at • Shell material - rubber • Examine response to, • Constant Pressure inflation – step response • Oscillating pressure at inlet of the form - , • Response to an external sudden force

12. Results (1/2) Constant pressure inflation of a slender elastic cylinder with internal viscous flow

13. Results (2/2) Quasi-steady diffusion of a slender elastic cylinder with internal viscous flow

14. Oscillatory Time Response

15. Oscillatory Frequency Response

16. Response to External Obstacle (1/2)

17. Response to External Obstacle (2/2)

18. Concluding Remarks • Closed analytic solution for pressure, velocity and deformation fields. • Characteristic time scale of the visco-elatic interaction. • Analysis of governed inlet pressure and external domain on the deformation field of the shell. • Elastic material compressibility. • Inducing the flow off the base frequency. • Phase reversal. • Boundary pressure feedback – movement detection.

19. Future Research • The current study lays the foundation for treatment of an external fluidic domain. • Control based on plant-model and boundary feedback to navigate/propel the vehicle. • Multi-channel networks – complex deformations. • A new breed of visco-elastic robots?