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Growth of form in thin elastic structures.
Salem Al Mosleh1, Ajay Gopinathan2, Christian Santangelo1
1Department of Physics, University of Massachusetts Amherst, Amherst, MA 01003, USA. csantang@physics.umass.edu.
Soft Matter
|October 12, 2018
Summary
Heterogeneous growth in elastic structures is stabilized by coupling growth laws to local stress and curvature. Mean curvature flow is essential for stabilizing long-wavelength fluctuations in cylindrical shells.
Area of Science:
- Physics
- Materials Science
- Biophysics
Background:
- Heterogeneous growth is crucial for pattern formation in elastic structures like cell walls and flower petals.
- Understanding the stability and regulation of this growth is key to predicting and controlling shape evolution.
Purpose of the Study:
- To model and analyze the stability of heterogeneous growth in thin elastic structures.
- To investigate regulation mechanisms by coupling growth laws to local shape properties.
- To identify conditions for stable growth in cylindrical shells, relevant to bacteria.
Main Methods:
- Quasi-static time evolution model of a metric with rapid elastic shape relaxation.
- Coupling growth laws to local shape properties: curvature and stress.
- Analysis of stability for cylindrical shell models.
Main Results:
- Coupling growth to curvature alone leads to linear instability in cylindrical shells.
- Adding coupling to local stress stabilizes short-wavelength fluctuations.
- Mean curvature flow is required to stabilize long-wavelength fluctuations.
Conclusions:
- Local stress and curvature coupling offer pathways to stabilize heterogeneous growth.
- Mean curvature flow is a critical factor for overall stability in certain geometries.
- The framework provides insights into designing stable growth laws for diverse target shapes.
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