A CAS Approach to Handle the Anisotropic Hooke’s Law for Cancellous Bone and Wood

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 fourth-order 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. 1. Introduction The study of material symmetry of 3-dimensional 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 [1]. Of course, the familiarity with material symmetries is the best way to categorize the materials. However, according to [2], many materials are anisotropic and inhomogeneous due to the varying composition of their constituents. In such materials, it becomes ticklish to identify the symmetries or more particularly, the elastic symmetries. The variable composition method to identify material’s elastic symmetry becomes complicated and hence to overcome this difficulty, an approach is developed by [3], called “averaging anisotropic elastic constant data.” In this approach, the identification of elastic symmetries and method of variable composition are analyzed separately. With the aid of this fabulous approach, a sheer volume of research towards material symmetry has been put forward by various researchers, for