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Notizen: 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°.

Notizen: 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°.

Notizen: Abstract In this paper sufficient conditions are derived to ensure the convergence of the Elliott and Hunter types of quadrature rules for the evaluation of weighted Cauchy principal-value integrals of the form: The simultaneous convergence in the interval (−1, 1) of both quadratures was established for a class of Hölder-continuous functionsf(f∈H μ ). Corrections of some previous statements on the subject of convergence of such quadratures are also included. Moreover, a simple derivation of the Hunter and Elliott types of quadrature rules for the evaluation of the derivative of thep-th-order of the abovestated integral was given and sufficient conditions for the convergence of the Hunter-type quadrature were obtained. Thus, the convergence of this integral was ensured for functionsf such thatf (p) ∈H μ .

Notizen: Abstract An accurate relationship for the shear modulus of particulates is derived based on the Kerner model, but not using its approximate relations. Furthermore, the model takes into account the existence of the mesophase layer between the inclusions and the matrix, which acts as a smooth transition boundary layer between constituent materials. By applying Christensen's field to the Kerner model, modified by introducing the mesophase, the new model is liberated from any inconsistencies. Experimental evidence and application to a glass particle-epoxy resin-matrix composite indicated the superiority of the model over previous ones.

Notizen: Abstract A theoretical model, consisting of a series of infinite concentric cylinders surrounding a fiber in a composite material, was introduced in this paper to give a quantitative account of interface phenomena, already experimentally observed. A series of specimens, conveniently designed to represent the theoretical model, were subjected to dynamic modes of loading to measure the amount of adhesion between fibers and matrices by means of an adhesion coefficient developed in the theory. It was found that theoretical results for the adhesion between matrix and filler were compatible with the structural characteristics of the specimens tested.

Notizen: Abstract Water absorption in particulate composites at ambient temperature influences their thermomechanical properties. Second Fick's law of diffusion was used in this paper to predict the diffusion coefficient of the composite materials tested. In all cases the matrix material was a diglycidyl ether of bisphenol-A polymer cured with 8 phr triethylene tetramine and filled with iron particles with an average diameter 150 μm at five distinct volume fractionsv f =0, 0.05, 0.10, 0.16 and 0.20. The modification of the modulus of elasticity, ultimate stress, breaking strain and breaking energy due to moisture absorption was examined. Moreover, differential scanning calorimetry was used to study the influence of the time exposure into water and the filler concentration of the particulates on their glass transition temperature. Finally, the void occupancy in the composite was evaluated from free volume considerations.