The present research entirely relies on the Computer Algebric Systems (CAS) to develop techniques for the data analysis of the sets of elastic constant data measurements. In particular, this study deals with the development of some appropriate programming codes that favor the data analysis of known values of elastic constants for cancellous bone, hardwoods, and softwood species. More precisely, a “Mathematica” code, which has an ability to unfold a fourthorder elasticity tensor is discussed. Also, an effort towards the fabrication of an appropriate “MAPLE” code has been exposed, that can calculate not only the eigenvalues and eigenvectors for cancellous bone, hardwoods, and softwood species, but also computes the nominal average of eigenvectors, average eigenvectors, average eigenvalues, and the average elasticity matrices for these materials. Further, using such a MAPLE code, the histograms corresponding to average elasticity matrices of 15 hardwood species have been plotted and the graphs for I, II, III, IV, V, and VI eigenvalues of each hardwood species against their apparent densities are also drawn.
The study of material symmetry of 3dimensional space is of great interest due to having crucial theoretical as well as practical significance. This is because a symmetrical space includes crystals and all homogeneous fields without exceptions: electric, magnetic, gravitational, and so forth.
The variation of material properties with respect to direction at a stagnant point in a material is called material symmetry; for instance, if the material properties are same in all directions at some fixed point, they are called isotropic, whereas if the material properties show variation at the same point, they are called anisotropic [
Concerning computer assisted analysis of bone material, a few of the articles, like direct mechanical assessment of elastic symmetries and properties of trabecular bone architecture [
Now, as far as the present proposed research concerned, we turn our concentration towards cancellous bone and wood (soft and hard). According to [
Now, the question arises is what is the need of studying the mechanical properties of cancellous bone and wood? The rigorous answer is that the knowledge about mechanical properties of cancellous bone is essential for the determination of bone fracture risk in osteoporosis and other pathological conditions involving impaired bone strength [
Mechanically, clear wood obeys the law of elastic orthotropic material like the bone does. In general the woody substance also exposes the high toughness and stiffness properties and these properties vary according to the type of wood and the direction in which the woody substance is examined, as the woody substance shows a higher degree of anisotropy.
In addition, to deal with anisotropic Hooke’s law, one should be familiar with the algebra of the 4thorder tensor. We would like to emphasize that the 4thorder tensor algebra is not only involved in the study of material symmetry, but its crucial appearance has been set up in the field of diffusion tensor MRI [
Though an adequate amount of research work has been dedicated towards the algebra of 4thorder tensors, then also, in the following section, we shall present a brief digest regarding this issue, as this issue hits the present study up to some great level.
The splendid word “Tensor” has been derived from the Latin word “Tensus”, which is the past participle of “tenděre” and stands for “Stretch.” This word was used in anatomy in the early 1704 to represent muscle and stretches. However, in Mathematics, it existed in 1846 when William Rowan Hamilton has explored his quaternion Algebra. Even though the Hamilton sense regarding tensor did not survive. The recent meaning of tensor is due to Voigt, who used the term “Tensortriple” in crystal elasticity around 1899.
Likewise the secondorder tensor, a fourthorder tensor
The outer (or open) product of the two 4thorder tensors is given by the composition
In case of a secondorder tensor, the only known symmetry represents an invariance under the mutual rotation of two indices, while for the 4thorder tensor, there are several notions of symmetry that represent invariance under exchanging pair of indices. Thus, for the 4thorder tensor generally we have three types of symmetries, namely, major, minor, and total symmetries, and these notions of symmetries widely play an important role in the theory of elasticity [
For
Moreover, we have the decomposition
In the theory of elasticity, this symmetry is often evoked as “major symmetry.” Since the major symmetry represents invariance under exchanging the pair
Such a representation is called the spectral decomposition of
The second type of symmetry is called “minor symmetry” and is defined by
Now, the invariance under the exchange of first pair of the indices is called the “first minor symmetry” and the invariance under the exchange of second pair of indices is called “second minor symmetry.”
The set of all 4thorder tensors that satisfy the minor symmetry is denoted by
A fourthorder tensor
The set of the fourthrank tensors bearing total symmetry is denoted by
The components of totally symmetric and antisymmetric parts of a fourthorder tensor are described as [
Now, it is straightforward to show that if
Reference [
To meet the objectives of proposed research, let us turn our attention to the flattening (sometimes called unfolding) of the fourthorder 3dimensional tensor with minor symmetries, as it is customary in the 3dimensional elasticity theory.
In the following section, we discuss the notion of 9dimensional representation of a 3dimensional fourthorder tensor and then particularize this representation to a fourthorder elasticity tensor having minor symmetries due to the wellknown anisotropic Hooke’s law.
In accordance with the evolution of tensor theory, the algebra of the 3dimensional fourth order tensor is still not fully developed. For instance, the techniques for calculating the latent roots an latent tensors of a tensor like
In order to study material symmetries and anisotropic Hooke’s law, it is customary to transform the 3dimensional 4thorder tensor as a 9 × 9 matrix, that is, a 9dimensional representation of a 3dimensional fourthorder tensor. For this purpose, there is a basic notion of representing a fourthorder tensor as a secondorder tensor [
Therefore, any fourthrank tensor
Again, we have from (
Let us apply this concept to anisotropic Hooke’s law by assuming that the anisotropic Hooke’s law given below is generalized one and also valid in the case where stress and strain tensors are not necessarily symmetric.
The anisotropic Hooke’s law in abstract index notations is often depicted as
Thus, for Hooke’s law and for eigenstiffnesseigenstrain equations, one can have the following equivalences:
Now in the new basis
In generalized Hooke’s law, when the stressstrain symmetries do not affect the stiffness tensor
Moreover, imposing symmetrical connection, that is,
Here, we depict a figure (see Figure
Hooke’s law (reduction process of elastic coefficients). Here, still there are 36 components seen in the stiffness table, but due to the components, for example,
Now, with 21 significant independent components, the stiffness tensor
As the elasticity of a material is described by a fourthorder tensor with 21 independent components as shown in (Figure
We are fortunate to have a long series of research papers concerning this issue. For instance, [
The very first approach to map 21 significant components of the elasticity tensor on a symmetric 6 × 6 matrix was introduced by Voigt [
Let us briefly go through these three notions of tensor flattening one by one.
It is well known that the Voigt mapping preserves the elastic energy density of the material and elastic stiffness and is given by
This Voigt mapping can be visualized as shown in Table
Voigt’s mappings for stress (left) and strain tensors (right), respectively.


Thus, in accordance with Voigt’s mappings, Hooke’s law (
Index conversion rule of Voigt.

11  22  33  23 or 32  13 or 31  12 or 21 


1  2  3  4  5  6 
But in the Voigt notations, many disadvantages were noticed. For instance,
the
the norms of
the entries in all the three Voigt arrays (see (
Likewise Voigt’s mapping rules, Kelvin’s mapping rules also preserve the elastic energy density of material under the following methodology:
But the only disadvantage of this mapping is that the values of stiffness components are changed. However, there is a simple tool for conversion between Voigt’s and Kelvin’s notations. For this purpose, one just needed a single array (see Table
Voigt to Kelvin and vice versa.

1  1  1 



To preserve the tensor properties of Hooke’s law during the unfolding of fourthorder elasticity tensor, [
The stress, strain, elasticity, and compliance tensors components in Voigt’s, Kelvin’s, and MCN patterns.
Voigt's notations  Kelvin's notations  MCN  

Stress 





 


 


 


 


 


Strain 





 


 


 


 


 


Elasticity 





 


 


 


 


 






























































Compliance 





 


 


 


 


 



























































Even though the foregoing detailed description is interesting and important from the viewpoint of those who are beginners in the field of mathematical elasticity, the modern scenario of the literature of mathematical elasticity now involves the assistance of various computer algebraic systems (CAS), like MATLAB [
However, in the following section, we shall delineate a CAS approach for Section
“Mathematica” is a general computing environment intimated with organizing algorithmic, visualization, and userfriendly interface capabilities. Moreover, many mathematical algorithms encapsulated by “Mathematica” make the computation easy and fast [
A fabulous book entitled “Elasticity with Mathematica” has been produced by [
Overall the effort of [
We consider the following code that easily meets the purpose discussed in Section
Here, kindly note that only the input Mathematica codes are being exposed. For considering the entire processing, an Electronic (see Appendix A in Supplementary Material available online at
First of all install the “Tensor2Analysis” package in Mathematica using the following command
We have loaded the above package like Loading the package “Tensor2Analysis.m” by setting the following path:
SetDirectory[C:/Program Files/Wolfram/Research/Mathematica/8.0/AddOns/Packages].
Next is concerning the creation of tensor. The code in Algorithm
Use the MakeTensor command to create tensorial expressions having components with respect to symmetric conditions. This command combines all the previous commands to do so (see Algorithm
Now the next code deals with the index rules that is able to handle Hooke’s tensor conversion from the fourthorder to the secondorder tensor or secondorder Voigt form using indexrule (see Algorithm
Afterwards, we have a crucial stage that concerns with the transformation of Voigt’s notations into a fourthrank tensor and vice versa. This stage includes the codes like HookeVto4, Hooke4toV. Also the codes HookeVto2 and Hook2toV transform the Voigt notations into the secondorder tensor notations and vice versa. Finally the code in Algorithm
We have presented here a minimal code of mathematica developed by [
Let us now proceed to Section
Of course, “MAPLE” is a sophisticated CAS and produced under the results of over 30 years of cuttingedge research and development, which assists us in analyzing, exploring, visualizing, and manipulating almost every mathematical problem. Having more than 500 functions, this CAS offers broadness, depth, and performance to handle every phenomenon of mathematics. In a nutshell, it is a CAS that offers high performance mathematics capabilities with integrated numeric and symbolic computations.
However, in the present study, we attempt to explore how this CAS handles complicated analysis regarding elastic constant data.
The elastic constant data of our interest, for hardwoods and softwoods, as calculated by Hearmon [
The elastic constants data for hardwoods.
S. no.  Species 











1  Quipo  0.1  0.045  0.251  1.075  0.027  0.033  0.025  0.226  0.118  0.078 
2  Quipo  0.2  0.159  0.427  3.446  0.069  0.131  0.178  0.430  0.280  0.144 
3  White  0.38  0.547  1.192  10.041  0.399  0.360  0.555  1.442  1.344  0.022 
4  Khaya  0.44  0.631  1.381  10.725  0.389  0.520  0.662  1.800  1.196  0.420 
5  Mahogany  0.50  0.952  1.575  11.996  0.571  0.682  0.790  1.960  1.498  0.638 
6  Mahogany  0.53  0.765  1.538  13.010  0.655  0.631  0.841  1.218  0.938  0.300 
7  S. Germ  0.54  0.772  1.772  12.240  0.558  0.530  0.871  2.318  1.582  0.540 
8  Maple  0.58  1.451  2.565  11.492  1.197  1.267  1.818  2.460  2.194  0.584 
9  Walnut  0.59  0.927  1.760  12.432  0.707  0.936  1.312  1.922  1.400  0.460 
10  Birch  0.62  0.898  1.623  1.7173  0.671  0.714  1.075  2.346  1.816  0.372 
11  Y. Birch  0.64  1.084  1.697  15.288  0.777  0.883  1.191  2.120  1.942  0.480 
12  Oak  0.67  1.350  2.983  16.958  1.007  1.005  1.463  2.380  1.532  0.784 
13  Ash  0.68  1.135  2.142  16.958  0.827  0.917  1.427  2.684  1.784  0.540 
14  Ash  0.80  1.439  2.439  17.000  1.037  1.485  1.968  1.720  1.218  0.500 
15  Beech  0.74  1.659  3.301  15.437  1.279  1.433  2.142  3.216  2.112  0.912 
Source: this data is consulted from [
The elastic constants data for softwoods.
S. no.  Species 











1  Blasa  0.2  0.127  0.360  6.380  0.086  0.091  0.154  0.624  0.406  0.066 
2  Spruce  0.39  0.572  1.030  11.950  0.262  0.365  0.506  1.498  1.442  0.078 
3  Spruce  0.43  0.594  1.106  14.055  0.346  0.476  0.686  1.442  1.0  0.064 
4  Spruce  0.44  0.443  0.775  16.286  0.192  0.321  0.442  1.234  1.52  0.072 
5  Spruce  0.50  0.755  0.963  17.221  0.333  0.549  0.548  1.25  1.706  0.07 
6  Douglas Fir  0.45  0.929  1.173  16.095  0.409  0.539  0.539  1.767  1.766  0.176 
7  Douglas Fir  0.59  1.226  1.775  17.004  0.753  0.747  0.941  2.348  1.816  0.160 
8  Pine  0.54  0.721  1.405  16.929  0.454  0.535  0.857  3.484  1.344  0.132 
Source: this data is consulted from [
For cancellous bone, we have the elastic constants data available for three specimens [
The elastic constants data for the three specimens of cancellous bone.
Elastic constants  Specimen 1  Specimen 2  Specimen 3 


9.86  791.2  698.3 

2.147  360.9  281.6 

2.773  318.6  284.5 

−0.162  1.1  8.0 

−2.93  5.4  −0.7 

−0.821  −0.6  11.4 

9.35  852.9  896.7 

2.346  328.1  322.7 

−1.204  0.9  −162.3 

−0.936  −2.5  −3.2 

0.445  −4.2  16.0 

5.952  1473.0  916.0 

−1.309  −2.4  −162.2 

−1.791  16.9  −7.2 

0.54  0.6  2.4 

5.61  358.7  348.9 

0.798  0.5  10.5 

−2.159  5.1  −9.8 

6.032  344.8  242.6 

−0.335  −0.7  −64.4 

7.425  230.4  237.8 
Source: the data for cancellous bone has been taken from [
Precisely speaking, to meet the requirements of proposed research, a MAPLE code consisting of almost 81 steps has been developed. This MAPLE code (in its minimal form) is appended to Appendix A, while its entire working mechanism is included in an electronic appendix (Appendix B) and can be accessed from
Computating the eigenvalues and eigenvectors for all the 15 hardwood species, 8 softwood species, and 3 specimens of cancellous bone using MAPLE.
Computing the nominal averages of eigenvectors and the average eigenvectors for all the 15 hardwood species.
Computing the average eigenvalues for hardwood species.
Computing the average elasticity matrices for all the 15 hardwood species.
Plotting the histograms for elasticity matrices of all the 15 hardwood species.
Plotting the graphs for I, II, III, IV, V, and VI eigenvalues of 15 hardwood species against their apparent densities (see Figures
Here we present the outcomes of MAPLE code (which is mentioned in Appendix A) regarding the computation of eigenvalues and eigenvectors of the stiffness matrices
It is well known that the eigenvalues and eigenvectors of a stiffness matrix
A MAPLE code mentioned in Step 16 of Appendix A enables us to simultaneously compute the eigenvalues and eigenvectors for all the 15 hardwood species. Also, the results of this MAPLE code are summarized in Tables
The eigenvalues and eigenvectors for 15 hardwoods species calculated using MAPLE.
S. no.  Hardwood species  Eigenvalues 
Eigenvectors 

1  Quipo 




2  Quipo 




3  White 




4  Khaya 




5  Mahogany 




6  Mahogany 




7  S. Germ 




8  MAPLE 
