DETERMINATION OF CONSTANT ELASTIC OF PEQUI WOOD USING ULTRASOUND DETERMINAÇÃO DAS CONSTANTES ELÁSTICAS DA MADEIRA PEQUI UTILIZANDO ULTRASSOM

Wood is a building material with high level of sustainability and renewable that captures and retains CO2 throughout its life cycle. Due to this peculiarity, it is important that the design when used as a structural materialbe as accurate as possible. Thus, it will be possible to use this material with greater performance. For this, it is necessary to know, with the highest accuracy, the mechanical characteristics of the wood. Mechanical tests, due to being an orthotropic material, are difficult and expensive. It is necessary to determine three modules of elasticity, three modules of transversal deformation and six Poisson's ratio. So the objective of this paper is to determine the elastic constants using the ultrasonic wave propagation method in Pequi wood (Caryocar Villosum). The device used was the Panametrics NDT EPOCH4 with longitudinal and shear wave transducers and a frequency of 1MHz. Specimens with dimensions of 5x5x15 cm were used, requiring six types of specimens and seven repetitions. Each specimen was prepared, with the necessary fiber direction, to determine the stiffness constants. With the application of Christoffel's equations, the 12 elastic constants were determined. It is concluded that it is possible to determine the values of the elastic constants using the ultrasonic wave propagation method, with good precision, fast and with reduced cost.


INTRODUCTION
With the devastation of forests raised the concern about the use of wood properly characterized and methods development and faster analysis, which generate fewer losses. In this context, are introduced non-destructive methods for evaluation of thematerial. The study of wave propagation principle applies to moving orthotropic materials began in 1880 with the development of the Christoffel equation (BUCUR, 1984). In addition, about a century later, initiated studies aimed at physical and mechanical characterization of wood (STANGERLIN et al., 2010). Authors like Bolza and Kloot (1963), Kennedy (1965), Bodig and Godmann (1972), Curry and Tory (1976), Warren (1979), Atherton (1980), Oberhofnerová et al. (2016) and Haseli, et al. (2020) have contributed to the perception of the importance of acoustic wave propagation technique to estimate the properties of the wood.
To the present day, there are differences of opinions regarding the evaluation of the modulus of elasticity of wood through wave propagation. Some authors, such as Steiger (1997), Ross et al. (1998), Gautam and Bartholomeu (2000), and Bjelanović et al. (2019), used the Edin (dynamic modulus of elasticity) of the relationship between the speed of wave propagation and wood density (EQ. Christoffel), doing an approximation of equation, without taking into account the various influences of the Poisson coefficient and of transverse elasticity modules. Other authors such as Bucur (1983), Sandoz (1989, Bartholomeu et al. (2003), Gonçalves et al. (2011) andHaseli et al. (2020) called the Christoffel equation of elastic constant. They claim to be considered like and din is required to be taken into account a number of other factors, such as the Poisson coefficients. The purpose of this study is to estimate the elastic properties of wood with the use of non-destructive method of ultrasonic wave propagation considering orthotropic material.

MATERIALS AND METHODS
The non-destructive tests were performed in the laboratory of construction materials and structures, Lab END-UNICAMP. Woods tested were in the form of polyhedra (Caryocar villosum), pequi. In all, were used in the six directions, oriented CPs the three major,longitudinal (I), radial (II), tangential (III) and three intermediate, longitudinal -radial (IV), tangential -radial (V), longitudinal -tangential (VI), figure 1. It is worth mentioning that most of the worksin Brazilian woods is carried out only with wood oriented in three main directions. Took care to use seven repetitions in order to determine the characteristic values for the calculation of these properties, these data were not available in the literature. The device used was the Panametrics-NDT EPOCH4 (Olympus Panametrics NDT/Inc, San Diego, CA), and use of longitudinal and transverse transducers, with frequency of 1 MHz.
For each wood pequi were used seven beams of 2.5 m length and cross section of 150 x 150 mm. They were air dried and subsequently deployed in smaller pieces. Then were stored for stabilization of humidity, and subsequently made--of-proof bodies (CPs) in accordance with Brazilian standard NBR 7190 (1997), for determination of the moisture content.

Wave propagation
The wave propagation in wood is described by movement equations established for an anisotropic solid, which can be found through a combination of Newton's law and the generalized Hooke's law (Bucur (1984), Carrasco and Azevedo Junior (2003), Equation (1).
Assuming that plane harmonic waves are spreading in the material, the solution of Equation (3) is the Equation (2).
(ρ ω² δ im -C iklm k k k l )u m = 0 (2) Where u 0i are the amplitudes of the components of the displacement vector and k j are the components of the wave vector. The value u 0i can be written as u 0αi , where u 0 is the amplitude of the displacement and α i are the direction cosines of the displacement vector of the particle.
Substituting the value expressed by Equation (2) into Equation (1) yields to Equation (3). This equation can be written in a more homogeneous by making u i =u mδim , where the tensor δ im is the unit tensor or Kronecker delta, Equation (4).
However, for certain directions of propagation in a given medium material, in which k  is an eigenvector of λ im , a wave is strictly longitudinal and the other two are purely transverse. For a pure longitudinal wave the displacement vector of the particle u  is parallel to the unit vector perpendicular to the wave fronts n  . Therefore, the vectorial product x is zero. On the other hand for a pure transverse wave, the same vectors are perpendicular to each other and, consequently, the scalar product . is zero.
λ mi α i = ρ v² δ mi α i The determination of elastic constants of wood can be simplifi ed when considering it, as a fi rst approximation, as a solid orthogonally isotropic or simply orthotropic. The matrix of elastic coeffi cients of an orthotropic solid is given by equation (13). Thus, it is possible to distinguish, in a wooden piece, three structural planes of symmetry that are both elastic planes of symmetry, as illustrated in fi gure 1. With the velocity of wave propagation in the ultrasonic tests and wood apparent density, using Christoff el's equation, the wood stiff ness constant values were estimated to be approximately the dynamic moduli of elasticity. This equation (14) used by Gonçalves and Bartholomeu (2000) and Bucur (2006), obtained good results. c = ρV 2 g (14) Where: C = stiff ness constant, MPa; V= velocity of wave propagation, m.s -1 ; ρ = wood apparent density, at 12% moisture content, kg.m -3 and g= acceleration of gravity, 10 m.s -2 .

RESULTS AND CONSIDERATIONS
In the table 1 is presented the numerical values of speed of propagation of the wave and their respective constant elastics (CE). It is worth emphasizing that, the direction 1 refers to the longitudinal direction, the 2 to the radial one, and the 3 to the tangential one (BUCUR, 2006) Figure 2 is shown the relation between the static values and the appreciated ones by the non-destructive method of ultrasonic wave's propagation, for the 3 directions (L, T and R).

CONCLUSIONS
It is possible to determine the values of the elastic constants using ultrasonic wave propagation. The theory C11> C22> C33, C44 <C55 <C66 and C12> C13> C23 was attended.
It is possible to estimate with precision the value of Estatic based on the propagation of ultrasonic waves.