FLEXURAL STIFFNESS AND MODULUS OF ELASTICITY OF FLOWER STALKS FROM ALLIUM SATIVUM AS MEASURED BY MULTIPLE RESONANCE FREQUENCY SPECTRA

Multiple resonance frequency spectra (MRFS) provide a rapid and repeatable method for determining the flexural stiffness and modulus of elasticity, E, of segments of plant stems and leaves. Each resonance frequency in a spectrum can be used to compute E, and removal of the distal portion of an organ produces characteristic shifts in spectra dependent upon the geometry of an organ. Hence, MRFS can be used to quantitatively determine the extent to which a particular leaf or stem morphology can be modelled according to beam theory. MRFS of flower stalks of Allium sativum L. are presented to illustrate the technique. The fundamental, f, and higher resonance frequencies, f2 . .. fn, of stems and the ratios of f2/f,, f3/f, and f3/f2 increase as stalk length is reduced by clipping. The magnitudes of these shifts conform to those predicted from the MRFS of a linearly tapered beam. Morphometric data confirm this geometry in 21 flower stalks. Based on this model, the average modulus equals 3.71 x 108 ? 0.32 x 108 N/M2, which compares favorably with values of E determined by static loading (3.55 x 108 ? 0.22 x 108 N/M2) and is in general agreement with ultrasonic measurements (3.8 x 108 to 4.4 x 108 N/M2). Data indicate that determinations of E from a single resonance frequency are suspect, since each resonance frequency yields slightly different values for E. Statistical evaluations from all the frequencies within a MRFS are more reliable for determining E and testing the appropriateness of beam theory to evaluate the biomechanical properties of plants. THE EXTENT to which a stem can support a weight at its tip or continue to grow vertically before buckling under its own weight depends upon its flexural stiffness and the modulus of elasticity of constituent tissues (McMahon, 1975; King, 1981; Givnish, 1982). Provided it can be modelled according to some beamlike geometry, the critical buckling weight and critical buckling length of a stem can be calculated from empirically determined values of the elastic modulus, E, which measures the proportionality between stress and strain for a structure (Silk, Wang, and Cleland, 1982; Niklas and O'Rourke, 1982, 1987). Consequently, a number of workers have devised methods to determine E. Among the most common is the Instron method of testing specimens under a ' Received for publication 1 October 1987; revision accepted 25 January 1988. The authors wish to thank Mr. William Holmes and Scott Copeland (Department of Theoretical and Applied Mechanics, Cornell University) for their technical assistance, Ms. Barbara Bernstein (Section of Plant Biology, Cornell University) for rendering figures from computer hard-copy printouts, and Professor Wolfgang H. Sachse and Mr. Howard J. Susskind (Theoretical and Applied Mechanics, Cornell University) for providing preliminary data from a longitudinal ultrasonic examination of flower stalks. Support from a National Science Foundation grant BSR 8320272 (KJN) is gratefully acknowledged. uniaxial, constant strain rate (Cleland, 1967, 1971, 1984). Although this technique provides rapid and repeatable measurements of E, the appropriateness of a beam geometry to model a particular stem morphology often remains conjectural. In addition, plant organs capable of supporting their own static weight can undergo dynamic mechanical failure. We present a technique for measuring the flexural stiffness and modulus of elasticity of plant stems which can also be used to evaluate the extent to which a particular stem morphology conforms to each of a variety of beam models. The technique is based on the mathematical relationship between the elastic properties and the multiple resonance frequencies of vibration of tapered or untapered beams with various transverse geometries (Timoshenko and Gere, 1961; Gorman, 1975; Blevins, 1979). A large body of empirical and theoretical studies underpin this approach, and to a limited extent it has been applied to examining the turgor pressure and rigidity of tissues (Virgin, 1955; Falk, Hertz, and Virgin, 1958). However, to our knowledge, multiple resonance frequency patterns have not been used to study plant organs in the method presented here. We have selected the flower stalk of Allium sativum L., to illustrate this method. Garlic