Ultrasonic Determination of Elasticity Tensor
for Juvenile and Mature Ponderosa Pine
I.
Introduction
The objective of this work is to recover the full elasticity tensor for juvenile and
mature wood in ponderosa pine. The recovery of the elastic properties for the juvenile
wood is important because ponderosa pine has potential commercial value in a full round
configuration [1]. The importance of ponderosa pine is a result of the need for thinning
of the National Forests after many decades of fire suppression. Most of this material is of
a small diameter, less than 8 inches in diameter, and thus has a larger percentage of
juvenile wood.
The availability of elastic properties for this material can assist is
directing the material to its highest and best use which will enhance the commercial
viability of the mechanical thinning that may be required in many areas as an alternative
to controlled burns.
Previous work as been limited in the area of wood material characterization with
respect to developing an understanding of the full elastic properties of wood. In
recognition of the complexity of the wood as a material, bending modulus is often
reported for dimension lumber. This is of course a property that is of great interest in
many applications for wood and includes a significant effect of the shear stiffness in most
configurations for wood beams. A more complete description of the material properties
of small diameter timber is however useful since the loading of material in a number of
potential applications may be less straightforward depending on the connection details
and other application related details of the use of the material.
For these reasons as well as to support an improved understanding of the
properties of wood as a material, this work is focussed on the recovery of both shear and
normal properties of the material in all three material axes of the material. In much of the
literature of wood, it is assumed to be an orthotropic material [2]. This is a good
symmetry assumption for lumber that is derived from old growth timber. As the radius
from the center of the tree increases, the symmetry to the plane that has the radius of the
1
tree as the unit normal becomes increasingly reasonable. However, for small diameter
timber, the plane of the radius has a small radius of curvature (less than four inches) and
thus is less accurately considered to be a plane of symmetry (Figure 1). Because this
work is clearly focussed on the modulus of small diameter timber, the wood will thus be
assumed to be transversely isotropic like of a tree. The samples used for the
measurements are cut on the visible material axes and measurements are limited to the
material axes. The applicability of the material symmetry assumption is also explored in
a series of verification experiments to ensure that the assumptions in the testing
procedure do not influence the results of the study.
II.
Theoretical Background
A. Elastic Symmetry of Wood
It’s commonly accepted that for most of wood and wood-based composites,
orthotropic symmetry, which means material posses three mutually perpendicular planes
of symmetry, and transverse isotropic symmetries are observed. For transverse isotropy,
the material is characterized as having an axis of symmetry that all directions at right
angles to this axis are equivalent.
With the use of a Cartesian coordinate system, three-orientation directions are
defined. The direction along the wood fiber direction is called Longitudinal or X 1
direction, the radial orientation of wood rays is defined Radial or X 3 direction, and the
other one is defined as Tangential or X 2 direction. (Fig.1)
If the curved growth rings shown in Fig.1 can be modeled as planar growth layer,
the result is a clear wood which is modeled into an orthotropic material with geometric
axes( X 1 , X 2 , X 3 ) coincident with orthotropic axes ( L, T , R) . (Fig. 2)
B. Ultrasonic determination of elasticity tensor from wavespeed measurement
For an orthotropic material, there are nine independent elastic tensors that are
determined.
The equations of motion in a continuum are given by:
..
 ij , j   u i
2
Eq-1
where
 ij, j is stress components, including both normal and shear.
 is the density of the material.
u i is the components of displacement,
..
u i indicates the second partial derivative of the displacement with respect to time.
i, j , k , l are coordinate index, each can attain independently the values of 1, 2 or 3.
For an anisotropic linearly elastic solid the constitutive equation is
 ij  Cijkl kl
Eq-2
where
 kl is stain components, including both normal and shear.
C ijkl is the fourth order tensor for the material.
For an assumption of small displacement
..
Cijkl u k , jl   u i
Eq-3
where u k , jl is the second partial derivative of displacement with respect to coordinate
index j, and l .  is the density of the material.
Assuming a solution of wave propagation is of the form
ui  A0 i e i[ k (li xi t )]
Eq-4
where A0 is the incident aptitude which is divided into components that propagate in a
number of a direction determined by the material symmetrical axes.
Substituting Eq.4 into 3 gives an eigenvalue equation (Christoffel’s equation)
which is a cubic polynomial in V 2 .
(Cijkl nl n j   ik V 2 ) Pm  0
3
Eq-5
where
Pm is the polarization vector.
V is the phase velocity of ultrasonic wave.
 ij is the Kronecker delta symbol.
n is the wave propagation direction with components of n1 , n2 , n3 .
In order to have a nontrivial solution of the homogeneous system of equations, the
determinant of the coefficient matrix must vanish.
Or:
Det[ij   ik V 2 ]  0
Eq-6
where ij  Cijkl nl n j is called Christoffel’s tensor. For an orthotropic material it’s given
by
11  n12 C11  n22 C 66  n32 C 66
22  n12 C 66  n22 C 22  n32 C 44
33  n12 C55  n22 C 44  n32 C33
12  n1n2 (C12  C 66 )
Eq-7
23  n2 n3 (C 23  C 44 )
13  n1n3 (C13  C55 )
Therefore, for orthotropic and transverse isotropic materials, in any given
direction, there are three possible acoustic waves (a quasi-longitudinal wave, two
polarization of quasi-transverse waves) whose phase velocities are known functions of
the elasticity tensor and density.
As a result of material symmetry waves propagate along the longitudinal,
tangential or radial directions of the three material symmetry axes, in all three directions,
all three wave types can be generated.
4
A Cartesian coordinate system with axes X 1 , X 2 , X 3 parallel to the edges of the
cube is assumed. For wave motion in x1 (longitudinal direction), i.e., n1  1, n2  n3  0 ,
one longitudinal wave and two transverse wave. The two transverse waves are
propagated in orthogonal directions and are associated with wave velocities that
correspond to different terms of the elasticity tensor. (Fig. 3)
V112  C11 , longitudinal wave
V122  C66 , transverse wave
Eq-8
V132  C55 , transverse wave
Switching the wave motion in x2 or x 3 direction, all the six diagonal terms of
elasticity tensor can be obtained by a relation
Cii  V 2 
Eq-9
where i  1,2...6
C. Method of velocity measurement
The three longitudinal waves corresponding to the three components of the
elasticity tensor C11, C22 , C33 are obtained from the measurement of longitudinal wave
velocity using direct transmission technique (when the specimen is in contact with the
longitudinal transducer). In addition, a procedure is developed for the determination of
the three components of elasticity tensor ( C 44 ,C 55, C66 ) associated with transverse wave
propagation. The specimens used are the same for both the longitudinal and transverse
measurements. Fifteen specimens of juvenile wood are obtained and twenty of mature
wood, both from ponderosa pine specimens.
Again, in order to understand the measurement procedure; consider wave
propagation in the X 1 direction. The three waves propagate along the X 1 direction, which
is the material symmetry axis. A priori knowledge of the directions of particle oscillation
for the three waves is known from theory. The intent is to find the velocities
corresponding to wave motion for each wave using ultrasonic piezoelectric transducers.
5
X-Cut and Z-Cut transducers are available which are used to excite and measure
longitudinal and transverse waves.
The longitudinal wave is quite simple since only a single quasi-longitudinal wave
propagates. However, due to arbitrary particle displacement u (t ) of the transverse
components (Fig.5), it is necessary to rotate the transverse wave transducer through
180 0 to
find the largest amplitude of transverse component that should correspond to X 3 -
polarization, which in turn corresponds to the elasticity tensor component C 55 . From the
maximum amplitude, the transducer is rotated 90 0 to get another the orthogonal
component that corresponds to the X 2 -polarization, from which the C 66 component is
obtained.
Once the particle oscillations and wave velocities for the two models of transverse
wave propagation in the X 1 direction have been determined, the same testing procedure
is repeated in the X 2 direction and C 44 , C 66 components are obtained as well as C 44 and
C 55 components from switching to the X 3 direction.
III.
Specimen Preparation
A. CUBE HARVESTING
Using the logs from the log breaking experiment, cubes were harvested from the
area near the break. The middle section of the log was removed from the log using a
horizontal band saw. 1 ½” thick disks were cut from each end of the middle section, just
beyond the break. From these disks, 1” cubes were harvested from the juvenile and
mature portions of the log. It is assumed that the first 20 rings from the center of the disk
are juvenile wood. The outer portion of the disk is mature wood. Fig.6 shows a picture of
a cross section of a log.
The cubes were cut at different angles from the axis of the disk. Fig.7 show
typical samples of cube locations. The cubes were identified by writing the log name,
indicating which end the disk was taken from and the location of the cube in the disk. For
example, if the log name was CO 01 and the disk was taken from end B and the cube was
from the mature wood portion of the log, the cube name would thus be CO 01B S. For
6
cubes that are rotated at angles of 30, 45 and 60, the angle of rotation is added to the
name. For example, if the cube previously mentioned were rotated at an angle of 45,
then the cube name would be CO 01B 45 S.
For several of the log specimens, additional 3-5” cylinders were cut. These
sections were used to harvest cubes that are in a different orientation to the longitudinal
axis. The cylinders were divided into four parts, cutting a cross through the center of the
log section in the longitudinal axis. Cubes were then harvested from each quadrant. Fig.8
shows the orientation of the cubes that were cut. The cubes were named as described
above. However, the face the cubes were removed from was added to the designation.
The letters L and R indicate longitudinal and radial direction.
After cubes were removed from the log, the specimens were polished with a belt
sander. The belt sander was used to obtain flat and smooth faces and to ensure parallel
sides. The dimensions of the cubes were then taken and recorded. The cubes were stored
in a refrigerator held at approximately 48 F and 37% relative humidity overnight before
ultrasonic testing.
Tables 2 and 3 show data measurements of the mature wood and juvenile wood
cubes that were tested ultrasonically. From the samples, it was noted that the average
density of the juvenile wood samples was higher than the average density of the mature
wood samples. The highlighted portions of the table indicate a comparison between the
juvenile wood and mature wood samples that were taken from the same log.
D. Verification of the technique
The transducer used to generate transverse wave was very sensitive to the
relationship of the polarization of the wave to material axes. Verification of the shear
wave technique is necessary to ensure that the measurements of the elasticity tensor are
independent of the testing procedure. From the testing procedure outlined above, data for
C 44 , C 55 , C 66 is obtained twice. This data should be nearly identical for an orthotropic
material, the difference representing only the error in the measurement and deviation of
the material from strict orthotropic symmetry. Thus, the comparison of these two sets of
data can be used to estimate measurement uncertainty.
7
The comparison results are shown in Table.3.
For the juvenile wood, the values of the elasticity tensor C55 measured from
either the radial direction (R ) or tangential direction (T ) correspond well with those
obtained from longitudinal direction (L) . For example, the average value of C44
measured from T direction is 0.5282  0.2357(GPa) , the corresponding value measured
from R direction is 0.5244  0.2.23 (GPa) (Fig. 9(a))
However, for the mature wood, case is not the same. From Table.3, it is seen the
value of C44 measured from T direction, which is 0.3404  0.2530(GPa) , is significant
different with that measured from R direction, which is 0.5438  0.3871(GPa) . Indeed,
the error is introduced by modeling tangential direction (T ) of the wood block as an
orthotropic material. (Fig. 9(b))
E. Data and signal processing
In order to calculate the transverse wave velocity as well as the components of
elasticity tensor, data and signal processing must be done. This includes using crosscorrelation approach to obtain accurate relative time delays. The approach is used to
estimate the time difference ( ) between two signals, a reference signal for a known
material (in this data set, Aluminum was used) and an unknown sample, which are small
blocks of ponderosa pine.
For this data set, the technique of windowing of the signal was also required to
truncate the echoes from the small block samples.
1. Cross-correlation between signal of Aluminum and Cube
The cross-correlation approach is used to determine the relative time delay
( ) between two waveforms. A reference signal which was acquired from an aluminum
block x( t ) and wood block signal y( t ) are concerned, the cross-correlation between the
two signals is a sequence rxy ( t ) , which is defined as
N
rxy ( )   x(t ) y (t   )
  N
8
Eq-10
where  is time shift parameter and the subscript xy indicates the signals being
correlated. The sequence x( t ) is unshifted and y( t ) is shifted by  units in time. To the
right,  is positive, to the left,  is negative.
For the most general case, the delay is a function of wavelength or frequency. The
physical process that changes the delay is known as dispersion. The cross-correlation
rxy ( t ) provides a measure of similarity between the power in the two signals so that the
time delay is a delay between the peak power.
In this case,  for a narrow band signal is used to estimate the phase velocity
(Eq.11) of the ultrasonic waves propagating through the wood. From the velocities the
elasticity tensor can be obtained ( Eq.9).
Vii  (d wood  Val ) /( d al  ( ii  Val )
Eq-11
where
Val  3200(m / s) , which is the velocity of ultrasonic wave propagating in the reference
specimen (aluminum).
d wood and d al are the distances of wave propagating through Aluminum and cube.
 is the density of wood.
i  1,2,..6
Due to the size of the specimen used, the ultrasonic wavelength is on the order of
the dimensions of the cube. As a result, in addition to the material dispersion that may be
evident due to the visco-elasticity of the material, significant geometric dispersion should
result as well. However, the use of the cross-correlation to determine the velocity of the
ultrasonic wave results only in a measure of the relative difference in the group velocity
between the reference sample(the aluminum block) and the unknown sample (the
Ponderosa Pine) [3].
For the range of velocities considered, the only geometrical
dispersion that would impact the measurement is the different in the phase velocities
between the reference sample and the unknown sample. As a result, unlike absolute
9
measurements of velocity, the effect of dispersion even for this size sample is minimal
and may be neglected.
2. Signal windowing
If a long time window is used, an interference pattern is generated in the
waveform. The phase velocities obtained from the cross-correlation are not accurate
unless the signal is windowed. To ensure that multiple reflections are not included in the
time signal, the window is located at a point where the wave has traveled two complete
passes through the specimen. Since the time delay is not known a-priori, it is necessary to
use an adaptive calculation of the window position.
To make an initial estimate of the time of flight, a phase velocity is initially estimated
based on the reference velocity.
For all of the calculation the initial estimates used are shown in Table.2.
Simply truncating the signal using a rectangular window results in more energy in
the high frequencies. The increased energy in the high frequencies is a result of the abrupt
transition to zero energy. Instead, a triangular window is used as a compromise between
simplicity and minimization of the high-energy components. Specifically, if w( t ) is
defined as a window function, then
1
(t  0.8 p)


w(t )  ( p  t ) /(1  0.8)
(0.8  t  p)

0
(t  p)

Eq-12
where p is the time of flight when the wave has made two complete passes through the
specimen.
F. Result and discussion
Initial works have been done with the data and signal processing shown to
determine the elasticity tensor. The spatially averaged values of the elasticity tensor
measured from seventeen samples of juvenile wood and twenty samples of mature wood
are listed in Table 3. It should be mentioned again that the coordinate system is the one
that the longitudinal direction is labeled as X 1 , the tangential direction is labeled as X 2
10
along with radial direction labeled as X 3 . The comparison of elastic properties between
juvenile and mature wood is also presented.
IV.
Conclusion
1. A nondestructive testing method has been developed for determining the elasticity
tensor of wood or wood-based composites. The approach is based on the wave
velocity measurements and theory of polarization transformation of shear wave
propagation in a medium.
2. C11 and C22 show significant differences between juvenile and mature wood.
3. Shear modulus appears to be nearly the same for the juvenile and mature wood. The
approximate similarity of C55 and C 66 indicates that wood is at least orthotropic.
Further confirmation of its transversely isotropic property will be obtained after the
other three off-diagonal terms of elasticity tensor C12 , C13 , C 23 are determined.
4. Error introduced by modeling tangential direction (T) as a material symmetry should
be concerned.
V.
Future work
In order to get full elasticity tensor of wood, the components of C12 , C13 , C 23 need
to be determined from the wave velocity measurement when wave propagation is out of
the symmetry of the material. Further verification of its transversely isotropic property
can be available using the following relations:
2
C55  C 66  2 V23
/1
C 44  C 66  2 V132 / 1
C 44  C55  2 V122 / 1
C55  C 66
where Vij / k is the velocity of a transverse wave propagating out of the symmetry
axis of the material
11
12
X 3 ( R)
X 2 (T )
CROSS SECTION
X1 ( L)
Fig.2 Orthotropic model of a block of clear
wood
Fig.1 A typical cellular structure of a
softwood
13
V1 1  C1 1 / 
X1
V11  C11 / 
V1 3  C5 5 / 
X1
V1 2  C6 6 / 
f
V13  C55 / 
V12  C66 / 
Cross Section
Cross-section
X3
X3
X3
X3
V 33  C33 / 
V 31  C55 / 
V 31  C55 / 
X2
V2 2  C2 2 / 
V 33  C33 / 
X2
V32  C44 / 
V2 2  C2 2 / 
V32  C44 / 
V2 3  C6 6 / 
V2 2  C4 4 / 
V2 3  C6 6 / 
V2 2  C4 4 / 
(b). Ultrasonic wave propagation in the mature wood
(a).Ultrasonic wave propagation in juvenile wood
Fig.3 Ultrasonic wave velocity in an orthotropic wood
14
X1
X1
V13  C 55 / 
V12  C66 / 
X3
X2
(a)
X2
X 2 -polarized
X3
(b)
X 3 -polarized
X1
V11  C11 / 
X2
(c)
X 1 -polarized
X3
Fig 4. Wave propagation along the positive X direction of an
orthotropic solid
15
u(t )
X2
X1
X3
Fig 5.Polarized transverse wave to form arbitrary particle
displacement u (t )
16
Juvenile wood
Mature wood
Fig.6 Cross section of log showing portions of juvenile and mature wood
17
Mature wood
Mature 60 degree
Juvenile wood
Juvenile 45 degree
Mature 30 degree
Mature. 45 degree
(b). Rotation of cube in TR orthotropic plane
(a). No- rotation of cubic in TR orthotropic plane
Fig.7 Cube orientation for ultrasonic testing
18
Radial (X3)
Mature LR 45 degree
Mature LR 60 degree
Juvenile LR 30 degree
Longitudinal (X1)
Fig.8 Rotation of Cube Specimen in LR Orthotropic Plane
19
8
7
Frequency of cube number
6
5
Mature
Juvenile
4
3
2
1
0
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
More
Elasticity tensor(GaP)
20
Fig.10 Comparison of Elasticity Tensor C11 between Juvenile and Mature Wood
T
L
R
Transducer
Transducer
Cross Section
Cross Section
(b). Mature wood measured in T direction
(a).Juvenile wood measured in T direction
Fig.9 Wood block measured in T direction
21
6
Frequency of cube number
5
4
Mature
Juvenile
3
2
1
0
0.3
0.4
0.5
0.6
0.7
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6 More
Elasticity tensor (GPa)
Fig. 11 Comparison of elasticity tensor C22 between juvenile and mature wood
22
7
Frequency of cube number
6
5
Mature
4
Juvenile
3
2
1
0
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
More
Elasticity Tensor (GPa)
Fig. 12 Comparison of elasticity tensor C33 between juvenile and mature wood
23
6
Frequency of cube number
5
4
Mature
Frequency
3
2
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Elasticity Tensor (GPa)
Fig. 13 Comparison of elasticity tensor C44 between juvenile and mature wood
24
More
6
Frequency of cube number
5
4
Mature
Juvenile
3
2
1
0
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
2
2.2
2.4
Elasticity tensor (GPa)
Fig. 14 Comparison of elasticity tensor C55 between juvenile and mature wood
25
2.6
More
4.5
4
Frequency of cube number
3.5
3
Mature
Juvenile
2.5
2
1.5
1
0.5
0
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
2
2.2
2.4
2.6
Elasticity tensor (GPa)
Fig. 15 Comparison of elasticity tensor C66 between juvenile and mature wood
26
More
27
28
29
Table 1. Notation used to describe the direction and mode of vibration of the wave velocities.
Vii
velocity of a longitudinal wave traveling in the
Vij
velocity of a transverse wave traveling in the
xi direction
xi direction with particle motion in the
x j direction
Table 2. Initial estimated velocity for juvenile and mature wood
juvenile
wood
mature
wood
'
V11
'
V22
'
V33
V44 '
V 55 '
V66 '
'
V12
'
V13
'
V23
3800
1700
2300
600
1200
1200
1800
1800
1500
5000
1500
2300
600
1200
1200
2000
2000
1600
30
31
Table 3 Average elasticity tensor measured from juvenile and mature wood
Average elasticity tensor
Juvenile wood
Mature wood
(GPa)
(GPa)
C11
4.9095 (0.8425)
10.627 (2.9269)
C 22
2.2600 (0.7041)
0.6819 (0.2534)
C 33
2.2999 (0.6855)
1.9731 (0.5744)
measured in T direction
0.5282 (0.2357)
0.3404 (0.2530)
measured in R direction
0.5244 (0.2023)
0.5438 (0.3871)
measured in L direction
1.4792 (0.8350)
0.9927 (0.1650)
measured in R direction
1.4252 (0.7242)
1.0140 (0.3575)
measured in L direction
1.4793 (0.8351)
1.0418 (0.4209)
measured in T direction
1.5071 (0.7991)
0.9957 (0.3721)
C12
1.5834(0.6063)
1.5462(0.5646)
C13
2.0567(0.8657)
2.5312(0.6016)
C 23
1.3903(0.6063)
0.6482(0.2372)
C 44
C 55
C 66
standard deviations in parenthesis
32
Table 4 Elasticity tensor of juvenile wood (GPa)
C44
Sample
C11
C22
C33
1
3.9024
1.3142
2
4.1253
3
C55
C66
C12
C13
C23
2.6416
1.2343
2.7134
2.0505
2.5901
0.6847
1.5676
1.8666
1.7270
0.8827
3.5136
0.8259
2.239
3.0392
2.0209
0.8761
2.2656
0.8761
1.3896
1.1883
1.5794
0.8550
0.2408
1.2479
0.9320
1.2479
2.0492
0.8862
1.3073
0.9260
0.7883
0.6079
1.2029
1.8105
1.2029
2.1231
1.1608
0.9832
0.4594
2.5900
0.5932
0.3011
0.8640
2.3358
0.8640
0.6693
2.5821
2.3315
0.8760
2.1218
1.8555
0.6042
0.6483
0.8341
0.9131
0.8341
0.9738
1.3408
3.4849
2.139
5.1230
1.1567
3.0722
0.9881
0.5374
2.1530
0.8059
2.1530
2.1492
2.0279
0.7134
0.9188
10
4.6788
1.8735
2.5769
0.4671
0.3434
0.8398
1.9404
0.8398
2.8992
2.4172
2.7483
2.1740
11
5.8419
1.8121
1.8583
0.1126
0.6043
0.8023
0.6085
0.8023
1.3377
1.3405
0.7664
1.1322
12
4.8044
3.2974
2.9570
0.4467
0.4098
2.1221
0.8032
2.1221
0.6352
0.5305
1.9568
1.0900
13
5.0056
1.9788
1.1136
0.6132
0.5906
0.8768
0.9227
0.8768
1.1185
1.1839
2.3754
0.7411
14
3.9162
2.8974
1.3350
0.3576
0.1323
0.7104
1.0519
0.7104
0.6779
2.2563
1.9620
2.0991
15
4.9811
3.7965
3.6873
0.2162
0.6976
1.4269
2.4157
1.4269
2.4353
1.7963
3.024
1.6473
Average
4.9095
(0.8423)
2.2600
(0.7041)
2.2990
(0.6855)
0.52825
(0.2360)
0.5244
(0.2022)
1.4787
(0.8350)
1.4250
(0.7242)
1.4787
(0.8351)
1.5073
(0.7991)
1.5835
(0.6063)
2.0568
(0.8656)
1.3902
(0.6063)
measured
in T
measured
in R
measured
in L
measured
in R
measured
in L
measured
in T
2.0754
0.6515
0.6943
2.1203
2.6903
2.1203
1.7211
1.7934
0.7530
0.5990
2.5901
0.9971
5.1054
2.0759
2.9004
0.6807
0.9042
3.5136
4
6.0305
2.6297
2.0739
0.3109
0.5547
5
4.4166
2.3960
2.2129
0.3402
6
4.1776
2.4363
2.3834
7
4.5857
2.4824
8
6.9454
9
standard deviations in parenthesis
33
Table 5 Elasticity tensor of mature wood (GPa)
C44
C55
C66
Sample
C11
C22
C33
measured
in T
measured
in R
measured
in L
measured
in R
measured
in L
measured
in T
C12
C13
C23
1
7.2600
1.5632
3.5476
0.1961
0.1256
1.1022
1.1022
1.7789
0.9269
0.9493
2.4675
0.7306
2
12.022
0.7849
2.0936
0.1142
0.6356
1.0165
1.0027
0.6542
0.6542
1.4389
2.7182
0.6711
3
14.556
0.8061
2.4448
0.1792
0.5798
1.1530
1.1424
0.6784
1.6253
1.2497
2.9975
0.4373
4
10.641
0.6241
1.9367
0.2085
0.1024
0.9732
0.9642
0.955
0.6563
1.8850
2.9893
0.9039
5
13.935
0.5732
2.0481
0.5696
0.4622
1.0915
1.089
1.1381
1.7381
2.0674
3.3039
0.7517
6
8.1121
0.3905
0.8468
0.2267
0.7086
1.0819
0.5909
0.9345
1.1514
0.9859
3.2748
0.5290
7
7.1573
0.3881
1.1700
0.7177
0.2956
0.7337
0.7337
2.0494
0.7024
2.5215
1.9749
0.1426
8
11.482
0.6299
2.1940
0.5207
0.5316
1.0918
1.0841
1.1532
1.4484
1.8309
2.6640
0.9329
9
8.6521
0.6151
1.6882
0.3549
0.5488
0.9291
0.9314
0.7288
0.5720
1.8608
2.1586
0.3618
10
10.961
0.5732
1.9501
0.9152
1.0263
0.9934
0.9843
0.8885
0.8766
1.6027
2.2560
1.1337
11
16.045
0.8667
2.1844
0.1196
0.8664
1.0986
1.0986
1.0912
1.4473
0.9681
3.0855
0.7468
12
10.119
0.6678
1.9203
0.1027
0.4663
0.5773
0.5773
1.0930
0.9151
1.5305
1.2723
0.6667
13
10.965
0.5926
1.9689
0.0999
1.3409
1.0123
0.9984
0.6586
1.2393
1.1839
1.4498
0.3316
14
9.8500
0.9377
2.1854
0.6244
0.4256
0.9784
1.4971
0.6787
0.6787
2.8043
2.7227
0.7487
15
14.627
0.6844
2.1143
0.1003
0.2823
0.8113
0.9716
0.7815
0.7782
0.6623
3.1965
0.6841
6.4198
0.6097
2.0492
0.4171
0.5344
1.0715
0.8113
0.8883
0.4976
2.1369
2.7605
0.4886
14.646
0.5078
2.0337
0.1300
0.4080
0.8026
1.0715
0.6944
1.3317
1.6981
3.0269
0.8494
8.7644
0.7815
2.4190
0.4470
0.2981
1.3369
0.5065
0.7104
0.7118
0.8765
2.548
0.8340
9.6062
0.5793
1.7652
0.6728
0.6927
0.9689
0.9323
1.4832
1.1791
1.2258
1.9982
0.5888
6.7144
0.4618
0.9061
0.0911
1.6726
1.0272
2.1905
1.7923
0.7838
1.4469
1.7589
0.4310
10.6275
(2.9269
0.6819
(0.2534)
1.9733
(0.5744)
0.3404
(0.2536)
0.5438
(0.3871)
0.9926
(0.1650)
1.014
(0.3575)
1.04154
(0.4209)
0.9957
(0.3721)
1.5462
(0.5645)
2.5312
(0.6016)
0.6482
(0.2372)
Average
standard deviations in parenthesis
34
35
36
Table 5 Elasticity tensor of mature wood (GPa)
C 44
Sample
Number
C11
C 22
C55
C 66
C33
measured
in T
direction
Measured
in R
direction
measured
in L
direction
measured
in R
direction
measured
in L
direction
measured
in T
direction
1
7.2600
1.5632
3.5476
0.1961
0.1256
1.1022
1.1022
1.7789
0.9269
2
12.022
0.7849
2.0936
0.1142
0.6356
1.0165
1.0027
0.6542
0.6542
3
14.556
0.8061
2.4448
0.1792
0.5798
1.1530
1.1424
0.6784
1.6253
4
10.641
0.6241
1.9367
0.2085
0.1024
0.9732
0.9642
0.955
0.6563
37
5
13.935
0.5732
2.0481
0.5696
0.4622
1.0915
1.089
1.1381
1.7381
6
8.1121
0.3905
0.8468
0.2267
0.7086
1.0819
0.5909
0.9345
1.1514
7
7.1573
0.3881
1.1700
0.7177
0.2956
0.7337
0.7337
2.0494
0.7024
8
11.482
0.6299
2.1940
0.5207
0.5316
1.0918
1.0841
1.1532
1.4484
9
8.6521
0.6151
1.6882
0.3549
0.5488
0.9291
0.9314
0.7288
0.5720
10
10.961
0.5732
1.9501
0.9152
1.0263
0.9934
0.9843
0.8885
0.8766
11
16.045
0.8667
2.1844
0.1196
0.8664
1.0986
1.0986
1.0912
1.4473
12
10.119
0.6678
1.9203
0.1027
0.4663
0.5773
0.5773
1.0930
0.9151
13
10.965
0.5926
1.9689
0.0999
1.3409
1.0123
0.9984
0.6586
1.2393
14
9.8500
0.9377
2.1854
0.6244
0.4256
0.9784
1.4971
0.6787
0.6787
15
14.627
0.6844
2.1143
0.1003
0.2823
0.8113
0.9716
0.7815
0.7782
16
6.4198
0.6097
2.0492
0.4171
0.5344
1.0715
0.8113
0.8883
0.4976
17
14.646
0.5078
2.0337
0.1300
0.4080
0.8026
1.0715
0.6944
1.3317
18
8.7644
0.7815
2.4190
0.4470
0.2981
1.3369
0.5065
0.7104
0.7118
19
9.6062
0.5793
1.7652
0.6728
0.6927
0.9689
0.9323
1.4832
1.1791
20
6.7144
0.4618
0.9061
0.0911
1.6726
1.0272
2.1905
1.7923
0.7838
Average
10.6275
(2.9269
0.6819
(0.2534)
1.9733
(0.5744)
0.3404
(0.2536)
0.5438
(0.3871)
0.9926
(0.1650)
1.014
(0.3575)
1.04154
(0.4209)
0.9957
(0.3721)
standard deviations in parenthesis
38
39
40
41
1
I will get a reference from Forest Products Lab
2
Bodig, J. and Jayne, 1993, B. A., Mechanics of Wood and Wood Composites Krieger Publishing, Malabar, Florida.
3 Peterson Relative Velocity Paper
42