Mean and axial green and Eshelby tensors for an inclusion with finite cylindrical 3D shape

• Inclusions of general shapes in composites are widely approximated by ellipsoids.
• The reason of this (ab)use is to benefit of simpler Eshelby and Green operators.
• The mean tensors calculated on the real shape are a more accurate approximation.
• We provide them for cylindrical inclusions with finite length for elastic isotropy.
• This new result under fully analytical form opens on many applications of interest.

Inclusions with the shape of a finite cylinder are generally approximated by a spheroid of same aspect ratio in order to take benefit of the simple and often analytical form of the Eshelby tensor for ellipsoids. Although mathematically advantageous, such an approximation is worse than considering the mean tensor related to the true inclusion shape. Using the Radon transform method, we give an exact formal expression of the mean shape function for a finite cylinder of general aspect ratio in any media and of the related mean Green and Eshelby tensors in isotropic elastic media, a 3D case which to the author knowledge has no reported solution in the literature so far. Thanks to this mean shape function we provide analytical forms for the fundamental integrals to compare with those of the spheroidal approximation. Noticeable differences show up which are discussed in terms of best spheroidal approximation. Since these tensors, as their fundamental integrals, are not uniform in non ellipsoidal inclusions, we also provide a simple exact solution for the fundamental integrals at any point of the finite fibre axis, so allowing analytical check of property variations along a long cylinder, what is not possible in the ellipsoidal approximation context. Case examinations and comparisons will be the purpose of a forthcoming paper.