The extension of Beltrami's ellipsoid to anisotropic bodies

Summary

The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. 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 efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, σ₁ and σ₂ which are shears (σ₂ being a simple shear and σ₁, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components σ₃, and σ₄, are the orthogonal supplements to the shear subspace of σ₁ and σ₂ and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle ω.

The spectral type of decomposition of the elastic stiffness or compliance tensors in elementary fourth-rank tensors thus serves as a means for the energy-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 property of defining energy-orthogonal stress states. That is, the stress-idempotent tensors are mutually orthogonal and at the same time collinear 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 respective σ_x, which are eigenstates of the compliance tensor S, this tensor also possesses the same remarkable property.

An interesting geometric interpretation arises for the energy-orthogonal stress states if we consider the “projections” of σ_x in the principal₃D stress space. Then, the characteristic state σ₂ vanishes, whereas stress states σ₁, σ₃ and σ₄ are represented by three mutually orthogonal vectors, oriented as follows: The ε₃ and ε₄ lie on the principal diagonal plane (σ₃δ₁₂) with subtending angles equaling (ω−π/2) and (π-ω), respectively. On the positive principal σ₃-axis, ω is the eigenangle of the orthotropic material, whereas the ε₁-vector is normal to the (σ₃δ₁₂)-plane and lies on the deviatoric π-plane. Vector ε₂ is equal to zero.

It was additionally conclusively proved that the four eigenvalues of the compliance, stiffness, and failure tensors for a transversely isotropic body, together with value of the eigenangle ω, constitute the five necessary and simplest parameters with which invariantly to describe either the elastic or the failure behavior of the body. The expressions for the σ_x-vector thus established represent an ellipsoid centered at the origin of the Cartesian frame, whose principal axes are the directions of the ε₁-, ε₃- and ε₄-vectors. This ellipsoid is a generalization of the Beltrami ellipsoid for isotropic materials.

Furthermore, in combination with extensive experimental evidence, this theory indicates that the eigenangle ω alone monoparametrically characterizes the degree of anisotropy for each transversely isotropic material. Thus, while the angle ω for isotropic materials is always equal to ω_i = 125.26° and constitutes a minimum, the angle |ω| progressively increases within the interval 90–180° as the anisotropy of the material is increased. The anisotropy of the various materials, exemplified by their ratiosE _L/2G_L of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tending asymptotically to very high values as the angle ω approaches its limits of 90 or 180°.