On the frequency dependence of the modulus of elasticity of wood

Abstract This short note reviews the reasons for the frequency dependence of the Modulus of Elasticity, MOE, of wood. It has in fact been reported in several publications on wood that depending on the technique used in the test experiment, the value of the MOE depends to some degree on the frequency at which it is evaluated. The frequency ranges used are namely zero frequency in the case of static bending, audio frequencies when using mechanical vibrations or sound radiation and finally ultrasonics. The results from implementing these three different techniques show that the lowest value that may be obtained for the MOE occurs when using the static mode, and thereafter increases with increasing frequency. This property of increasing dynamic MOE with frequency is shared by all solid materials, and finds its theoretical explanation in the Kramers-Kronig relations. Dispersion in conjunction with the notion of complex MOE permit to establish the relation between the real and the imaginary components of the MOE, i.e. respectively the dynamic and loss moduli. Due to the mathematical difficulties encountered in using the exact expressions, approximations are necessary for applications in practical situations. Hence, an improved version of the Zener model for viscoelasticity, which has lately been proposed by Pritz (1999), is presented. With some assumptions, and under which excellent agreement has been obtained with the exact theory, this model is used for predicting the viscoelastic properties of wood.

[1]  Nicholas W. Tschoegl,et al.  The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction , 1989 .

[2]  E. Jaynes,et al.  Kramers–Kronig relationship between ultrasonic attenuation and phase velocity , 1981 .

[3]  J. Ferry.  Viscoelastic properties of polymers , 1961 .

[4]  H. C. Booij,et al.  Generalization of Kramers-Kronig transforms and some approximations of relations between viscoelastic quantities , 1982 .

[5]  Hota V. S. GangaRao,et al.  Assessment of defects and mechanical properties of wood members using ultrasonic frequency analysis , 1996 .

[6]  T. Pritz,et al.  VERIFICATION OF LOCAL KRAMERS–KRONIG RELATIONS FOR COMPLEX MODULUS BY MEANS OF FRACTIONAL DERIVATIVE MODEL , 1999 .

[7]  Eugen J. Skudrzyk,et al.  Simple and Complex Vibratory Systems , 1969 .

[8]  R. Bagley,et al.  On the Appearance of the Fractional Derivative in the Behavior of Real Materials , 1984 .

[9]  Voichita Bucur,et al.  Factors affecting ultrasonic measurements in solid wood , 1994 .

[10]  V. Bucur,et al.  Attenuation of ultrasound in solid wood , 1992 .

[11]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

[12]  W. Hume-rothery.  Elasticity and Anelasticity of Metals , 1949, Nature.

[13]  Eiichi Fukada,et al.  The Vibrational Properties of Wood. II. , 1950 .

[14]  T. Pritz,et al.  FREQUENCY DEPENDENCES OF COMPLEX MODULI AND COMPLEX POISSON'S RATIO OF REAL SOLID MATERIALS , 1998 .

[15]  Sven Ohlsson,et al.  Elastic Wood Properties from Dynamic Tests and Computer Modeling , 1992 .

[16]  D. W. Haines,et al.  Determination of Young's modulus for spruce, fir and isotropic materials by the resonance flexure method with comparisons to static flexure and other dynamic methods , 1996, Wood Science and Technology.

[17]  S. Hahn.  Hilbert Transforms in Signal Processing , 1996 .

[18]  C. Zener.  Elasticity and anelasticity of metals , 1948 .

[19]  R. Kronig.  On the Theory of Dispersion of X-Rays , 1926 .

[20]  Voichita Bucur,et al.  An ultrasonic method for measuring the elastic constants of wood increment cores bored from living trees , 1983 .