MECHANICAL BEHAVIOR OF PLANT TISSUES AS INFERRED FROM THE THEORY OF PRESSURIZED CELLULAR SOLIDS

The mechanical behavior of plant tissues and its dependency on tissue geometry and turgor pressure are analytically dealt with in terms of the theory of cellular solids. A cellular solid is any material whose matter is distributed in the form of beamlike struts or complete "cell" walls. Therefore, its relative density is less than one and typically less than 0.3. Relative density is the ratio of the density of the cellular solid to the density of its constitutive ("cell wall") material. Relative density depends upon cell shape and the density of cell wall material. It largely influences the mechanical behavior of cellular solids. Additional important parameters to mechanical behavior are the elastic modulus of "cell walls" and the magnitude of internal "cell" pressure. Analyses indicate that two "stiffening" agents operate in natural cellular solids (plant tissues): 1) cell wall infrastructure and 2) the hydrostatic influence of the protoplasm within each cellular compartment. The elastic modulus measured from a living tissue sample is the consequence of both agents. Therefore, the mechanical properties of living tissues are dependent upon the magnitude of turgor pressure. High turgor pressure places cell walls into axial tension, reduces the magnitude of cell wall deformations under an applied stress, and hence increases the apparent elastic modulus of the tissue. In the absence of turgid protoplasts or in the case of dead tissues, the cell wall infrastructure will respond as a linear elastic, nonlinear elastic, or "densifying" material (under compression) dependent upon the magnitude of externally applied stress. Accordingly, it is proposed that no single tangent (elastic) modulus from a stress-strain curve of a plant tissue is sufficient to characterize the material properties of a sample. It is also suggested that when a modulus is calculated that it be referred to as the tissue composite modulus to distinguish it from the elastic modulus of a noncellular solid material. THE MECHANICAL BEHAVIOR of plant tissues varies as a function of turgor pressure and the geometry of their constituent cells. Studies indicate that the elastic modulus of pith parenchyma increases monotonically as turgor pressure increases (Falk, Hertz, and Virgin, 1958; Lin and Pitt, 1986). (The elastic modulus is measured from the slope of the linear portion of a material's stress-strain curve. It measures a specimen's material properties.) A similar relationship has been reported for more anatomically complex structures, such as the leaves of Dubautia and Allium (Robichaux, Holsinger, and Morse, 1986; Niklas and O'Rourke, 1987). In addition to the influence of water content, the elastic modulus of a tissue increases as the ratio of cell wall to protoplasm increases (Niklas, 1989a), while the maximum elastic modulus of algal and higher plant cells decreases as cell size decreases but the cell wall fraction remains relatively constant (Steudle, Zimmermann, and Luttge, 1977; Robichaux I Received for publication 31 May 1988; revision accepted 7 February 1989. et al., 1986). Cell wall composition is another factor that can influence the mechanical properties of tissues. However, data relating to this issue are limited (see Mark, 1967). The influence of turgor pressure and the ratio of cell wall thickness to cell radius on the elastic modulus of pith parenchyma has been modelled by various authors (Nilsson, Hertz, and Falk, 1958; Pitt, 1982, 1984; Gatesetal., 1986; Lin and Pitt, 1986). In all cases, cells are approximated as thin-walled spheres, for which simple shell theory yields the relationship at = Pr/2, where a is the circumferential stress on a cell with a wall-thickness t, radius r, and a turgor pressure of P (see Lin and Pitt, 1986, p. 306). This relationship conforms well to empirical data from pith tissues, since t << r. An underlying assumption to these models is that tissue stiffness increases as the number of cells in a tissue sample increases because increasing cell numbers decrease the capacity for cell-tocell displacements, particularly as turgor pressure increases. This has been experimentally verified (Niklas, 1988). However, for plant tissues other than parenchyma, the assumptions of simple shell theory are violated (cell walls