Graduate Thesis Or Dissertation
 

Continuum mechanics methods applied to root growth and the penetration of soil by roots

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  • The principles of continuum mechanics provide a consistent framework for the derivation of mathematical statements describing transport of water and solute, and growth in plant tissue. These derivations are based upon the explicit consideration of the tissue as a mathematical continuum composed of a cell wall matrix, water, and solute phases. The differential equations for water and solute transport in growing plant tissue are derived from the continuity equations for each phase, and the constitutive relations for water and solute flow with respect to the cell wall matrix. The differential equations of axial plant growth are derived from a statement of the conservation of thermodynamic potential energy and kinetic energy of deformation. The axial growth model is given by a system of coupled, first-order, partial differential equations for the local tissue displacement velocities and the longitudinal tissue strain. The coefficient of the growth model describes the change of tissue thermodynamic potential energy with respect to the longitudinal strain and is termed the specific energy capacity for tissue growth. The results of this analysis indicate that axial growth rates are controlled by the derivatives with respect to the tissue strain of the turgor, osmotic potential, extent of the tissue biosynthetic reaction, and the tissue stress from an external force. The axial plant growth equations are solved by the method of characteristics. The solution is applied to measurements of steady root growth given in Erickson and Sax (1956, Proc. Am. Phili. Soc. 100:499) and Goodwin and Avers (1956, Am. J. Bot. 43:479). The theoretical, spatial displacement velocities accurately regenerate the steady state velocity measurements and a comparison of the theoretical cell lengths with the experimental cell length measurements yields a justification of the model. The concept that the thermodynamic potential energy performs the work of deformation is more general and applicable to biological systems than the force balance approach. The overall mathematical expression of transport and deformation in plant tissue is given by four differential equations coupled through the gradient of the tissue displacement velocities. This coupling provides the explicit connection between the local tissue deformation and the instantaneous water potential and solute concentration of the tissue.
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