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Notes: 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-reinf orced 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, σ 1 and σ 2, which are shears (σ 2 being a simple shear and σ 1, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components σ 3 and σ 4, are the orthogonal supplements to the shear subspace of σ 1 and σ 2 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-tensors, 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 principal 3D stress space. Then, the characteristic state σ 2 vanishes, whereas stress states σ 1, σ 3 and σ 4 are represented by three mutually orthogonal vectors, oriented as follows: The ε 3- and ε 4-vectors lie on the principal diagonal plane (σ3∂12) 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 ε 1-vector is normal to the (σ3∂12)-plane and lies on the deviatoric π-plane. Vector ε 2 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 ε 1, ε 3- and ε 4-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 ratios E 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°.

Notes: 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 σx, 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 σx 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 σx-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 ωi = 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/2GL 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°.