ARTICULO
CONCERNING THE ELASTIC ORTHOTROPIC MODEL APPLIED TO WOOD ELASTIC PROPERTIES
MODELO ELÁSTICO ORTOTRÓPICO APLICADO A LA MADERA
Nilson Tadeu Mascia^{1}
^{1}Associate Professor. State University of Campinas- UNICAMP-BRAZIL
Correspondig Author: nilson@fec.unicamp.br
ABSTRACT
Among the construction materials, wood reveals an orthotropic pattern, because of unique characteristics in its internal structure with three axes of wood biological directions (longitudinal, tangential and radial). elastic symmetry: longitudinal, tangential and radial, reveals an orthotropic pattern. The effect of grain angle orientation onin the elastic modulus constitutes the fundamental cause forof wood anisotropy. It is responsible for the greatest changes in the values of the constitutive tensor components, and, thus in the elastic constant values of wood. The goal of this article is to verify the adequacy of the orthotropic model for wood, basically expressed by the modulus of elasticity E_{i} related to a determined direction. It was considered a cCoordinate transformation between the materials axes (L, R and T) and Euler′s angles is considered to use a constitutive equation for orthotropic materials. The main purpose of this analysis is to theoretically explore the coordinate transformation in the three - dimensional point of view and also to statistically compare the results of modulus of elasticity from compression test in a Brazilian wood species, Guapuruvú (Schizolobium parahyba), with values obtained from a theoretical expression. The results from the analysis, in which the coefficient of determination (R-sq) was equal to 0.965 for a linear least squares analysis, showed that the orthotropic model is adequate to be applied.
Keywords: Modulus of Elasticity of Wood, Compression Test, Orthotropic Material.
RESUMEN ]]> La madera, entre los materiales de construcción, presenta un comportamiento ortotrópico, esto debido a su estructura interna con tres ejes elásticos de simetría: longitudinal, tangencial y radial. El efecto de la orientación angular de las fibras en el módulo elástico constituye la causa fundamental de la anisotropía en la madera. Este efecto es responsable por los grandes cambios en los valores de los componentes del tensor constitutivo, y, consecuentemente en los valores de las constantes elásticas de la madera. El objeto de este artículo es verificar la adecuación del modelo ortotrópico para la madera, expresado básicamente por el módulo de elasticidad E_{i }referido a una dirección determinada. Se consideró una transformación de coordenadas entre los ejes del material (L, R y T) y los ángulos de Euler para utilizar una ecuación constitutiva para materiales ortotrópicos. El propósito principal de este análisis es explorar teóricamente la transformación de coordenadas, desde el punto de vista tridimensional y también, comparar estadísticamente los resultados del módulo de elasticidad obtenido a través del ensayo de compresión en la madera brasileña de la especie, Guapuruvú, con los valores obtenidos por medio de una expresión teórica. Los resultados de este análisis, en el que el coeficiente de determinación (R-sq) fue igual a 0.965 en un análisis de mínimos cuadrados lineal, mostró que el modelo ortotrópico es válido para ser aplicado.
Palabras Claves: Módulo de Elasticidad de la Madera, Ensayo de Compresión, Material Ortrotópico.
INTRODUCTION
The most general elastic constitutive model formulated to describe the mechanical behavior of material is the anisotropic model. This kind of model implies that there is no material symmetry, and mechanical properties in each certain directions are different. On the other hand, if there is material symmetry, the material can be denominated, for example, orthotropic or isotropic. In this context, the adequacy of a determined material for a certain elastic model is based on the existence of elastic symmetry axes. In these axes, denominated elastic principal axes, there is invariance of the constitutive relations under a group of transformations of coordinate axes.
In fact, the study of anisotropy is to develop implies knowing the constitutive law that governs the elastic behavior of the material and, consequently, to determineing the constitutive tensor, S_{ijkl}, and its components. In a completely elastic and anisotropic model this tensor has 81 unknown coefficients. By using adequate simplifications, this number can be reduced to 9 coefficients, which is denominated orthotropric model, or to 2 independent constants, which is the isotropic model.
Among the construction materials, wood follows an orthotropic pattern, due to its unique internal structure along the where there are three axes of wood biological directions (longitudinal, tangential and radial). Thus, there are 9 coefficients to be determined.
Because of the nature of wood, there are some parameters that can interfere within these elastic constants and such parameters include as moisture content, specific gravity and the grain angle orientation.
On the other hand, focusing on the wood anisotropy, the variation of grain angle constitutes its fundamental cause. It is responsible for the greatest changes in the values of the constitutive tensor components, and, consequently in the wood elastic constants.
Many procedures have been used to analyze the wood behavior under a uniaxial direction and, determined the elastic constants of wood using uniaxial compression or There are some tensile tests published!) tests tensile) tests as well. Conversely, wood under a three-dimensional field view is less studied and consequently the constants of wood and their relationship are not fully quantified.
In this way, the aim of this paper is to verify the adequacy of the orthotropic model for wood, analyzing the results of compression tests in Brazilian wood species, Guapuruvú (Schizolobium parahyba) and then comparing them with the theoretical values obtained from a specific expression. In order to carry out this analysis, it is necessary to transform the angles measured on the surface of the specimen to Euler’s angles and to apply the correct coordinate transformation tensor. Thus, this study addresses an investigation, on the basis of both theoretical and experimental aspects, in which it is focuses on a three-dimensional coordinate transformation, which is usually less investigated in this area.
MATERIALS AND METHODS
]]> Elastic Properties Of An Anisotropic MaterialsAccording to Love (1944) and Chen and Saleeb (1982), among others, the laws and equations that govern engineering problems are related to the stored energy in a solid. So, an elastic solid is capable of storing the energy developed by the external work and transforms it into potential elastic energy that is denoted as strain energy. During this process, the body is deformed, but recovers its original shape and size once the external force is removed.
In this condition, if no energy is dissipated during the process of deformation, under adiabatic and isothermal conditions, the derived equations from this supposition are termed elastic models of Green and the material that makes the body as hyperelastic material. Thus, a hyperelastic material is the one that has a strain energy function, denoted by U_{0.}
The elastic material of Green is, in fact, a special case of the most general elastic material called elastic material of Cauchy, but considering the existence of the U_{0}, in order to maintain unaltered the laws of thermodynamics. These laws say that an elastic material produces no work in a closed loading cycle. Using the strain energy function and considering the Green elastic model, formulations of the constitutive laws for different classes of elastic materials can be established. So, consider a strain energy function given by:
(1) |
where C_{0} , ρ _{ij} ,β _{ijkl} , α _{ij} are constants and ε _{kl} is the strain tensor. In view of the strain energy formulation where the strain energy has a stationary value in relation to the strain tensor, it is possible to set C_{0} = 0. From equation (1) and considering that: , the stresses can be expressed by:
(2) |
For an elastic body, the current state of stress depends only on the current state of strain. It may also be taken into the account the fact that α _{ij} =0, since that the initial strain field corresponds to an initial stress free state, and (β _{ijkl} +β _{klij}) can be taken as C_{ijkl} , so, mathematically, the constitutive laws can be written as:
]]> |
(3) |
in which: ρ _{ij} is the stress tensor, ε _{kl} is the strain tensor, C_{ijkl} is the tensor of material elastic constants.
Agreeing that | C_{ijkl} | ≠ 0 , the equation (3) can be expressed as:
(4) |
where: S_{ijkl} is the compliance tensor.
The constitutive laws may also be written in matrix form as:
(5) |
(6) |
On the other hand, according to Lekhnitskii (1981), all bodies, on the whole, can be divided into homogeneous and non-homogeneous bodies, and isotropic and anisotropic as well.
When a body is considered to be homogeneous, its physical properties, such as density, remain invariant in all directions, in any of its points. For non-homogeneous body its properties are not constants.
If the elastic properties of the material are the same in certain directions at a point, then the material exhibits symmetry with respect to these directions. If symmetry exists, the material is generally said to be isotropic. Otherwise, if there is no symmetry at all, the material is said to be anisotropic.
Another interesting issue to be pointed out is that when a body presents certain kinds of symmetry, the constitutive relations are simplified. These simplifications can be done in different ways just as those used by Love (19442), where the strain energy function remains unaltered by all symmetrical coordinate system substitutions. Thus, for example, a corresponding substitution given by three axes of elastic symmetry, () does not change the value of U_{o}. Lekhnitskii (1981), on the other hand, performs these simplifications by in developing in two different coordinate systems, symmetrical one to other. The author compareds the obtained constitutive relations and , identifiedying, in this way, the existence of the elastic symmetry.
A material with elastic symmetry under the linear transformation , with being the coordinate transformation tensor, requires that the constitutive tensor, either C_{rspq} or S_{rspq} , be comply with submitted to the following condition:
(7) |
Thus, a body referred to a coordinate system χ_{i} is defined as orthotropic material if through each point there are three mutually perpendicular axes of elastic symmetry. Then, using the coordinate system χ_{1}, χ_{2} and χ_{3} (or χ, γ, and Ζ), perpendicular to the three planes of material symmetry and considering the elastic properties to be invariant under counterclockwise rotation 180° of about three axes, and using one at time as showed in Figure 13, it is possible to determine the constitutive tensor for orthotropic materials.
Figure 1-180° - Rotation about χ_{3}. |
Consequently, we obtain that:
(8) |
And we find, either C_{rspq} or S_{rspq} can be wtitten by:
(9) |
(10) |
where: E_{i} is the modulus of elasticity related to i direction, G_{ij} is the shear modulus related to ij -plane and ν _{ij} is the Poisson’s ratio in ij - plane.
In this way, arbitrating for wood the orthotropic model, with the three elastic principal axes denoted L, R and T , which can be seen in Figure 2, the components of the any tensor are determined by replacing the indices 1,2 and 3 by L, T and R.
Figure 2- Material Axes and Board Axes for wood. |
The effect of Grain Angle
In general, as already pointed out the variation of grain angle causes greatest changes in the constitutive tensor components, and, obviously in the wood elastic constant values.
Many researchers, in the theoretical and experimental point of view have long studied the effect of grain angle. One of the most important procedures was formulated by Hearmon (1948), who reported the effect of grain angles onin all the components of S_{ijkl }, showing that for wood it is possible to obtain negative values of Poisson’s ratio, which emphasized the wood anisotropy.
]]> Goodman and Bodig (1970), presented the following coordinate transformation matrix in order to determine the wood elastic properties with respect to rotation Θ about the L axis and Φ about R - axis:
(11) |
in order to determine the wood elastic properties with respect to rotation Θ about the L axis and Φ about R - axis. The material axes and the board axes are: R , T ,and L , χ_{i} (χ,γ, Ζ), respectively. Figure 3 shows these axes.
Figure 3- Goodman and Bodig’s coordinate transformation for wood. |
Equation (11) can be generalized found using for two coordinate transformations that can be written by:
(12) |
In terms of tensor notation, where represent the set of direction cosines and 1 and 2 are, respectively, the first and the second rotations. Since no rotation about T was considered, this coordinate transformation is limited to cases where the L material axis lies in the χ_{1}- χ_{3} plane.
]]> Bindzi and Samson (1995) derived another carried out the following coordinate transformation relation with rotation Φ about L-axis and Ψ about R-axis as follows:
(13) |
It can be noticed that the R-axis lies in the χ-γ plane. This equation can be got using the Equation (12).
Both this and Goodman and Bodig’s transformations are considered limited since it is not possible to obtain all relations between the board and material axes.
Hermanson (1996),studying the transformation of elastic properties for lumber to align these axes χ_{i }(χ,γ,z) with the material axes χ’_{i} (L, R, T), used three rotations Θ, ρ and φ (denoted Euler’s angles) about χ, γ and Ζ axes, as can be seen in Figure 4.
Thus, we can write that:
(14) |
or in terms of matrix, that:
]]> |
(15) |
with A being a product of three matrices as defined as followsequals to:
(16) |
Observe that the eEquation (14) is similar to Equation (12), where the superscript 3 represents the third rotation.
Figure 4 - Euler’s angles λ , ρ and φ . |
The final relation between among these systems leads to:
]]> (17) |
or:
(18) |
where A is determined byleads to:
(19) |
After that, the three Euler’s angles were related to the surface angles α, β and γ , through the following relations:
(20) |
Figure 5- Surface angles α , β and γ . |
In this way, it was possible to find the Euler’ angles by knowing the surface angles and evaluating all wood elastic constants by using the complete coordinate transformation to elastic properties, described by eEquation (7) or using the Equation (21) given by:
(21) |
or in terms of matrices by:
(22) |
where: , K and [K]^{T}^{ }, the transpose of K, are function of components of the matrix A. The matrix K is the following matrixone:
]]> |
(23) |
and:
(24) |
Now, from Equation (21), we can determine, for example, S_{1111 } , S_{2222 },_{ }S_{3333}_{ }or, simply, the board elastic moduli (see Equation (10)), by:
(25) |
where: i= x, y, z .
It is noted that the terms: are obtained from Equations (18) and (19).
]]> Experimental MethodsThe experimental data used to analyze the adequacy of the wood elastic behavior for orthotropic elastic model were obtained in Lamem- São Carlos-Brazil, following the procedure used by Mascia (1993). The specimens which were used consisted of wood blocks of 6 cm x 6 cm x 18cm of Guapuruvú (Schizolobium parahyba) species obtained from tree stem, according to the following schematic. The moisture content was around 12% at ambient temperature ( 25º C) and the specific gravity around 0.40 g/cm^{3}.
Figure 6- Scheme of the specimens used in compression test |
The goal of this procedure is to determine the modulus of elasticity in some grain fiber orientations determined with respect to the R-T, the R-L and T-L plane.
To achieve this, firstly, it was cut lumbers were cut in such a way so that in one axe direction in the R-T plane varying in the following angles: 0° ,20° ,45° ,70° and 90° and the length axe being parallel to the L direction. After this, blocks were obtained from these lumbers following the same direction but varying the angles over the lumber axe in the R-T plane by 0° , 3° ,5° ,7° and 10° . In this way, 25 specimens were obtained for the compression tests. We have to emphasize that this procedure was very carefully designed and needs great care to accomplish.
It was used the AMSLER test machine with 250 kN load capacity and strain gages to measure the strains.
RESULTS AND DISCUSSIONS
In this work, only the modulus of elasticity in z-axis by comparing obtained from the experimental data, denominated E_{exp} with and the theoretical predictions from Equation (25), E_{theo}. To use Equation (25), it is necessary to know we the following elastic constants:
The above values have the unit of MPa. The values of Poisson’s ratios are given as follows (Mascia (1993)):
Modulus of Elasticity: Experimental And Theoretical Data Comparison
In the range covered by the 25 sets of data, it was considered to establish relations between theoretical and experimental values, through the statistical regression analysis.
This analysis provides the following regression:
(26) |
Detailed data both for the theoretical calculations and experimental measurements, are given in Table 1. whereas Table 2 shows the Analysis of Variance. The analysis was based on the methods of Montgomery and Peck (1992) and Ryan (1994), and Minitab software.
Table 1- Modulus of Elasticity E_{i} (MPa) | |||||
Specimen | E_{exp} | E_{theo} | Specimen | E_{exp } | ]]> E_{theo} |
0 – 0 – 0 | 3500 | 3495 | 45 –70 | 311 | 494 |
0 – 20 | 1884 | 2102 | ]]> 45 – 90 | 250 | 519 |
0 – 45 | 876 | 694 | 70 –0– 7 | 3391 | 3245 |
0 –70 | ]]> 413 | 341 | 70 – 20 | 1500 | 1963 |
0 – 90 | 519 | 519 | 70 –45 | 656 | ]]> 858 |
20 – 0 – 3 | 3666 | 3442 | 70 – 70 | 379 | 561 |
20 – 20 | 1489 | 2045 | ]]> 70 –90 | 211 | 519 |
20 – 45 | 489 | 674 | 90 – 0–10 | 3500 | 3112 |
20 – 70 | ]]> 379 | 363 | 90 – 20 | 1400 | 1900 |
20 – 90 | 338 | 519 | 90 – 45 | 677 | ]]> 864 |
45 – 0 –5 | 3700 | 3353 | 90 – 70 | 486 | 572 |
45 – 20 | 1692 | 1987 | ]]> 90 - 90 | 287 | 519 |
45 – 45 | 588 | 746 |
]]> Table 2- Analysis of the Variance for Significance of Regression | |||||
Source of Variation | Sum of Squares | Degrees of Freedom | Mean Square | F_{0} | P(%) |
Regression | 29810815 | ]]> 1 | 29810815 | 663.61 | 0.0001 |
Residual | 1033210 | 23 | 44922 | ||
]]> Total | 30844025 | 24 |
From this analysis of the variance we can conclude that the regression between the theoretical values and the experimental data is significant. This means that we can reject the null hypothesis with a high level of significance. We can observe that if the observed value of F_{0}_{ } is large, the parameter E_{exp }is non- zero for a reached probability p . In other words, the agreement among the theoretical values of the elastic modulus and the experimental values described by Equation (26) is satisfactory.
Figure 7 shows the regression plot, in which, by analyzing the prediction interval, with 0.95 confidence coefficient, we can observe that some results in the 25-data set did not adequately fit in the confidential interval of the linear model but all of them fitted adequately in the predicted interval. This statistically reflects the capacity of predicting of this model.
]]> Figure 7- Regression Plot Of Experimental and Theoretical Modulus of Elasticity .(In MPa) |
To better illustrate this item argument we presented the Figure 8 and Figure 9 showing the agreement between the experimental and theoretical results as a function of grain angles. To construct these three-dimension diagrams of Elastic Modulus and Euler’s angles, and the lateral view of this figure, Figure 10, it was used the Matlab software and the reference: Hanselman and Littlefield (1998).
Figure 8- Theoretical Curve and Experimental Values of the Modulus of Elasticity. |
Figure 9- Theoretical and Experimental Values of the Modulus of Elasticity. |
Figure 10- Lateral View Of The Three -Dimension Diagram Of Modulus of elasticity From The Theoretical And Experimental Data. |
In summary, we can point out that there are some values that perturb the linear model, but, in general, the regression analysis provides results that isare considered satisfactory for wood.
CONCLUSIONS
In this paper, it was described the general concepts of the orthotropic elastic model, particularly the rectilinear model, in order to verify the adequacy of this model for wood, by analyzing experimental data obtained from compression tests in Guapuruvú and theoretical data from a specific expression resulted from this model.
We have already commented that the variation of grain angle constitutes the main reason for wood anisotropy, and strongly affects the values of the constitutive tensor components.
We have also observed that the use of the test device and the specimen configuration are important to avoid perturbation in stress and strain fields. It is convenient to measure the strains as far as possible from the contact between specimen and test device surface.
In general, the most important conclusion that was drawn from this study can be summarized as follows:
-The agreement between the rectilinear orthotropic model, described by the theoretical values and the experimental values, can be considered satisfactory. The present statistical analyses indicated that only some results of the data did not adequately fit in the model especially because wood to be a non-homogeneous and an anisotropic material.
-It is important to notice that this conclusion is restricted to the current experimental data. In order to make generalizations about these results, it is necessary to perform develop more tests taking into account other species of wood, and in other different physical situations of moisture content, specific gravity and temperature as well.
REFERENCES
]]>Bindzi, I.; Samson, M. 1995. New Formula for Influence of Spiral Grain on Bending Stiffness of Wooden Beams. J. of Struct. Div. 121(11): 1541-1546. [ Links ]
Chen, E.F.; Saleeb, A. 1982. Constitutive Equations For Engineering Materials. New York: John Wiley & Sons, Inc, New York, N.Y. [ Links ]
Goodman, J.R.; Bodig, J. 1970. Orthotopic Elastic Properties of Wood. J. of Struct. Div. 96 (11): 2301-2319. [ Links ]
Hanselman, D.; Littlefield, B. 1998. Mastering Matlab 5: A Comprehensive Tutorial and Reference. Prentice Hall, Inc, Upper Saddle River, New Jersey. [ Links ]
Hearmon, R.F.S. 1948. The Elasticity of Wood and Plywood. Forest Products Res. Special Rep. No. 7, Dep. of Scientific and Industrial Res., London, U.K. [ Links ]
Hearmonson, J.C. 1996. The Triaxial Behavior Of Redwood Using A New Confined Compression Device. PhD thesis, Dept. of Civil Engr.,Univ. of Wisconsin, Wisc. [ Links ]
Lekhnitskii, S.G. 1981. Theory of Elasticity of an Anisotropic Body. Mir, Moscow, Russia. [ Links ]
Love, A. E. 1944. A Treatise On The Theory Of Elasticity. Dover Publications, New York, N.Y. [ Links ]
Mascia, N.T. 1993. Considerações a respeito da anisotropia na madeira. PhD Thesis. EESC-USP- São Carlos-Brasil. [ Links ] ( In Portuguese)
]]>Montgomery, D.C.; Peck, E.A. 1992. Introduction To Linear Regression Analysis. John Wiley & Sons, Inc, New York, N.Y. [ Links ]
Ryan, B.F.; Joiner, B.L. 1994. Minitab Handbook. Duxbury Press. Belmont, California. [ Links ]
NOTATION
C_{0 }, δ _{ij } , β _{ijkl} , α _{ij} | : | constants |
i,j,k | : | indices |
ε _{kl} | : | strain tensor, strain |
σ _{ij} | : | stress tensor, stress |
x_{i} , x_{1}, x_{2} and x_{3} (or x , y, and z): | : | coordinate system |
U_{o} | : | strain energy function |
C_{ijkl} | : | tensor of material elastic constants |
S_{ijk}_{l} | ]]> : | compliance tensor |
E_{i} | : | modulus of elasticity related to i direction |
G_{ij} | : | shear modulus related to ij -plane |
ν_{ij} | : | Poisson’s ratio in ij - plane |
E_{exp} | : | modulus of elasticity from the experimental data |
E_{theo} | : | modulus of elasticity in z-axis from theoretical data |
A, K | : | matrix, |
α_{ij} | : | elements of tensor or matrix |
l_{ij} | : | coordinate transformation tensor, coordinate |
x'_{i} (L, R , T ) | : | material axes |
L | ]]> : | Longitudinal direction |
T | : | Tangential direction |
R | : | Radial direction |
λ , ρ , φ | : | Euler’s angles |
R-sq | : | Coefficient of Determination |
p | : | probability |
F_{0}_{} | : | parameter of analysis of the variance |
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