THE EXTENSION OF BELTRAMIS ELLIPSOID TO ANISOTROPIC BODIES

The spectral decomposition of the compliance, stiffness, and failure t ensors for transversely isotropic materials was studied and their char acteristic values were calculated using the components of these fourth -rank tensors in a Cartesian frame defining the principal material dir ections. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite ), and the eigenvalues derived from them define in a simple and effici ent way the respective elastic eigenstates of the loading of the mater ial. It has been shown that, for the general orthotropic or transverse ly isotropic body, these eigenstates consist of two double components, sigma1 and sigma2, which are shears (sigma2 being a simple shear and sigma1, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two e igenstates, with stress components sigma3 and sigma4, are the orthogon al supplements to the shear subspace of sigma1 and sigma2 and consist of an equilateral stress in the plane of isotropy, on which is superim posed a prescribed tension or compression along the symmetry axis of t he material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle omega. The spect ral type of decomposition of the elastic stiffness or compliance tenso rs in elementary fourth-rank tensors thus serves as a means for the en ergy-orthogonal decomposition of the energy function. The advantage of this type of decomposition is that the elementary idempotent tensors to which the fourth-rank tensors are decomposed have the interesting p roperty of defining energy-orthogonal stress states. That is, the stre ss-idempotent tensors are mutually orthogonal and at the same time col linear with their respective strain tensors, and therefore correspond to energy-orthogonal stress states, which are therefore independent of each other. Since the failure tensor is the limiting case for the res pective sigma(x)-tensors, which are eigenstates of the compliance tens or S, this tensor also possesses the same remarkable property. An inte resting geometric interpretation arises for the energy-orthogonal stre ss states if we consider the ''projections'' of sigma(x) in the princi pal 3D stress space. Then, the characteristic state sigma2 vanishes, w hereas stress states sigma1, sigma3 and sigma4 are represented by thre e mutually orthogonal vectors, oriented as follows: The epsilon3- and epsilon4-vectors lie on the principal diagonal plane (sigma3delta12) w ith subtending angles equaling (omega - pi/2) and (pi - omega), respec tively. On the positive principal sigma3-axis, omega is the eigenangle of the orthotropic material, whereas the epsilon1-vector is normal to the (sigma3delta12)-plane and lies on the deviatoric pi-plane. Vector epsilon2 is equal to zero. It was additionally conclusively proved th at the four eigenvalues of the compliance, stiffness, and failure tens ors for a transversely isotropic body, together with value of the eige nangle omega, constitute the five necessary and simplest parameters wi th which invariantly to describe either the elastic or the failure beh avior of the body. The expressions for the sigma(x)-vector thus establ ished represent an ellipsoid centered at the origin of the Cartesian f rame, whose principal axes are the directions of the epsilon1-, epsilo n3- and epsilon4-vectors. This ellipsoid is a generalization of the Be ltrami ellipsoid for isotropic materials. Furthermore, in combination with extensive experimental evidence, this theory indicates that the e igenangle omega alone monoparametrically characterizes the degree of a nisotropy for each transversely isotropic material. Thus, while the an gle omega for isotropic materials is always equal to omega(i) = 125.26 -degrees and constitutes a minimum, the angle \omega\ progressively in creases within the interval 90-180-degrees as the anisotropy of the ma terial is increased. The anisotropy of the various materials, exemplif ied by their ratios E(L)/2G(L) of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tendin g asymptotically to very high values as the angle omega approaches its limits of 90 or 180-degrees.