Tensor-structural model relating x-ray computed tomography data to the elastic properties of anisotropic materials

Аuthors

Sachenkov O. A.

Kazan Federal University, 18, Kremlyovskaya St., 420008, Russia

e-mail: 4works@bk.ru

Abstract

This paper presents a theoretical justification of the relationship between X-ray computed tomography data and the mechanical properties of structural materials. Based on the introduction of an orientation distribution function on the unit sphere and its expansion in generalized spherical harmonics, a method for reconstructing the structure tensor from tomographic data is developed. By approximating the point cloud obtained from segmented CT data with a second-order surface, the quadratic form is determined, from which a symmetric positive-definite structure tensor is recovered. Using a polynomial expansion of the stress tensor in terms of the strain tensor and the structure tensor, analytical relations linking the elastic constants tensor components to the eigenvalues of the structure tensor are obtained. It is shown that the resulting elastic constants tensor describes the orthotropic symmetry family. A dimensionless fractional anisotropy index is introduced to quantitatively assess the degree of anisotropy from tomography data. The model is validated through numerical simulation on 10 test samples with porous structures ranging from isotropic to pronounced orthotropic cases. It is established that fractional anisotropy increases from 0.03 to 0.97 with changing pore geometry, while stiffness reduction in the direction of the minimum eigenvalue reaches 70%. The approach opens new perspectives for predicting elastic properties and diagnosing hidden damage in aerospace and mechanical engineering components without destroying the samples.

Keywords:

X-ray computed tomography, mechanical properties, elastic properties, structure tensor, fractional anisotropy, orientation distribution function

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