When osteoporosis occurs, the long cortical bone shows increased porosity. The effect of porosity on the ultrasonic body wave and axial ultrasonic guided wave propagation in the long cortical bone has not been fully investigated. In this article, in order to evaluate the effects of partial porosity on the elastic properties and the ultrasonic wave propagation in the dry long cortical bone, the relation between the partial porosity and the effective anisotropic elastic modulus of the dry cortical bone has been established using the stiffness contribution tensor and the Maxwell homogenization scheme. The anisotropic porous long cortical bone model has been constructed, the acoustic field is analytically solved, and the effects of different levels of partial porosities on the elastic properties of long cortical bone are studied. The effects of porosity on the ultrasonic body wave propagation velocity and guided wave dispersion characteristics are carefully investigated. The results show that the elastic moduli C3(3)(d) , C-44(d) , the body waves (including longitudinal wave, SH wave (shear wave motion in a vertical plane), and SV wave (shear wave motion in a horizontal plane) propagating along the vertical direction (perpendicular to the isotropic plane of long cortical bone), longitudinal guided wave modes L-1 and L-2 in the low-frequency plateau region are more sensitive to the canalicular porosity; However, the elastic moduli C-11(d) , C-66(d) , the body waves (including longitudinal wave, SH wave) propagating along the horizontal direction (along the isotropic plane of long cortical bone) are more sensitive to the Haversian porosity. Flexural guided wave modes are sensitive to both types of porosity. This article provides a theoretical basis for nondestructive evaluation of osteoporosis using ultrasonic waves.

This paper focuses on the electrical conductivity and thermal expansion coefficient of copper–graphite composites in the range of 50–90 vol.% graphite. In the previous authors’ work, elastic and conductive properties for the range of 0–50 vol% graphite were studied and cross-property connections were established between (1) electrical and thermal conductivities and thermal expansion coefficient and (2) electrical resistivity and shear modulus. Copper–graphite composites are prepared using hot isostatic pressing from the mixture of copper and graphite powders. The microstructure of the composites consists of copper particles that are homogeneously distributed within the dominant graphite phase (50–90 vol% graphite). The density, electrical conductivity, and thermal expansion coefficients of copper–graphite composites for the range of 0–90 vol% graphite are measured. Cross-property connections between electrical resistivity and thermal expansion coefficients are modeled using various techniques of micromechanics that account for the volume fraction, shape, size, and orientation of composites’ constituents. Cross-property connections between the said properties are verified using the obtained experimental data.

This paper is devoted to the analytical calculation of the contribution and opening displacement tensors of an arbitrarily oriented crack in a transformed transversely isotropic (TraTI) matrix. It generalizes a recent work considering elliptically orthotropic (EO) symmetry, as EO is a particular case of TraTI. The latter allows to explore a wider range of symmetries, including non-orthotropic symmetries. The approach is based on the linear transformation between boundary value problems with TraTI and transversely isotropic (TI) bodies. A detailed analysis shows that TraTI fourth-order tensors are described by a set of 11 parameters (5 material parameters and 6 angles defining the transformation). This is a significant enrichment beyond the set of EO tensors which depends on 7 parameters (4 material parameters and 3 angles). New analytical results are obtained for elliptical and circular cracks embedded, respectively, in TraTI, orthotropic TraTI and monoclinic TraTI matrices. It is shown that the most general case leading to analytical derivations of crack contribution and opening displacement tensors is that of a TraTI matrix with a specific restriction: the initial crack must be aligned along the isotropy plane of the TI matrix. To the best of our knowledge, this corresponds to the current largest space of matrix anisotropy allowing analytical derivation of the compliance of a single elliptical crack embedded in an infinite matrix as well as an extension of analytical 3D results showing a coupling between opening and shear modes. Numerical results are presented to illustrate potential applications of the method for the case of an arbitrarily oriented elliptical crack in an infinite uniform matrix of stiffness of arbitrary anisotropy. The best-fit problem investigated in previous papers is revisited and a new algorithm providing the closest stiffness tensor for which there exists an analytical solution to the crack opening displacement tensor is developed. Numerical applications to real TI materials are finally presented.

The research focuses on the evaluation of the mechanical properties of osteonal cortical bone at the lamellar level. Elastic properties of the mid-diaphysis region of the bovine tibia are investigated via cantilever-based nanoindentation at the submicron length scale utilizing Atomic Force Microscopy, where the force-displacement curves are used for the elastic assessment using the Derjaguin-Muller-Toropov model to calculate indentation modulus. Variations of the modulus and the directional mechanical response of the osteonal bone at different distances from the Haversian canal are investigated. Additionally, the effects of demineralization on the indentation modulus are discussed. It was found that in the axial direction, the first and last untreated thick lamella layers show a significant indentation modulus difference compared to all other layers (4.26 ± 0.4 and 4.6 ± 0.3 GPa vs ∼3.5 GPa). On the other hand, the indentation modulus of transverse thick lamella layers shows a periodic variation between ∼3 ± 0.7 GPa and ∼4 ± 0.3 GPa from near the Haversian canal to near the interstitial bone. A periodic variation in the anisotropy ratio was found. Mineral content was quantified via energy-dispersive X-ray microanalysis at different levels of mineralization and shows a positive correlation with the indentation modulus.

The problem of reconstruction and quantitative characterization of the microstructure of random composites, as a fundamental problem of material sciences, has been a subject of a considerable amount of literature. Thus far, previous studies used for the reconstruction either statistical microstructure descriptors or the overall property of real material. This paper makes a major contribution to research on reconstruction by formulating a procedure to recover the micro-structure that produces the same effective thermal conductivity as the real composite material. In particular, our goal is to find a binary representation of 'replacement' microstructure that, being a two-phase statistically isotropic medium, produces minimal disagreement with the experimental data. Such a binary microstructure is invariant with respect to the conductivity of fluid occupying the porous space. Thus, in some sense, the paper is an extension of the concept proposed by Lydzba et al. (2018), who showed, in the framework of analytical homogenization, that any isotropic microstructure can be represented by randomly oriented spheroids of certain distribu-tion over the aspect ratios. The efficiency of our methodology was illustrated by examples including Wiener and Hashin-Shtrikman bounds as well as the microstructure created by the system of non-overlapping disks. Finally, we use our algorithm to construct the 'replacement' microstructure for the real porous medium, i.e., medium sand. The main advantage of the digital representation of 'replacement' microstructure over the analytical one, is that it can be further used in computational modeling as well as in 3D printing applications.

The aim of this paper is to extend recent elastic work to thermal problem. In the first part of the paper, approximate relations for the resistivity contribution tensor of pores of two reference shapes, supersphere and axisymmetrical superspheroid, are developed on the basis of 3D Finite Element Modelling, presented in the companion paper, and known exact solutions for the limiting cases of spherical pores. In the second part application to effective elastic coefficients of transversely isotropic materials such as clay rocks, in the frame of homogenization theory, is presented to illustrate the impact of concavity parameter on overall properties.

This paper focuses on the modeling of the effect of different porosity on the overall elastic properties of saturated cortical bone and ultrasound wave propagation in such bone. We first utilize micromechanical model of Zhou, Cui and Sevostianov (2020) to model anisotropic effective elastic stiffness of saturated cortical bone and evaluate the effect of partial porosities. It is shown that the moduli Csat33 and Csat 44 of saturated bone are more sensitive to canalicular porosity, while the moduli Csat11and Csat 66 of saturated bone are more sensitive to Haversian porosity. This model has been validated by experimental data. We compare the extents of anisotropy of the saturated bones for different levels of partial porosity. Last but not the least, we apply this micromechanical model to the elastic field of saturated cortical bone to evaluate the effect of partial porosity on the velocity of different bulk waves and guided waves. Results show that the velocity of the fast longitudinal wave and the slower SH wave along the horizontal direction are more sensitive to Haversian porosity phi Hav, but the velocity of the fast longitudinal wave and SH wave along the vertical direction and the velocity of the SV wave along the horizontal direction are more sensitive to the canalicular porosity phi Can. Further, it is found that, the plateau region of the axisymmetric longitudinal guided waves mode L(0, 1) and L(0, 2) phase velocity are more sensitive to canalicular porosity phi Can, and the phase velocity of the lowest nonaxisymmetric flexural mode F(1, 1) in low frequencie is more sensitive to Haversian porosity phi Hav.

We consider here the problem of a three-dimensional (3D) body subjected to an arbitrarily oriented and remotely applied stationary heat flux. The body includes a non-conductive inhomogeneity (or pore) having the shape of two intersecting spheres with different radii. Using toroidal coordinates, the steady-state temperature field and the heat flux have been expressed in terms of Mehler–Fock transforms. Then, by imposing Neumann BCs at the surface of the spheres, a system of two Fredholm integral equations is obtained and solved based on Gauss–Laguerre quadrature rule. It is shown that the components of the resistivity contribution tensor exhibit a non-monotonic trend with the distance between sphere centers. In particular, if the inhomogeneity has a symmetric dumbbell-shape, then the extrema of the resistivity contribution tensor components occur when the two overlapping spheres have the same size. Differently, when the inhomogeneity has a lenticular shape, then these extrema are attained for a non-symmetric configuration, namely, for different radii of the intersecting spheres.

In general, traditionally manufactured steel alloys exhibit a typical elastic plastic deformation distinguished by linear and nonlinear stress–strain curves. However, an unconventional elastic–plastic behavior was observed from the 3D-printed 316L stainless steel specimens in this study, which showed no noticeable hardening in their elastic–near perfect plastic behavior. For comparison, similar tests were also performed on conventionally processed (hot rolled) 316L stainless steel samples. Microhardness indentation, resonant frequency, and tensile test have been performed on both types of samples and the results were compared with the ones available in the literature. Microstructural characteristics of the specimens produced by the two technologies were compared to explain the unconventional elastic–plastic behaviors of the 3D-printed specimens. Optical microscopy, scanning electron microscopy (SEM), and transmission electron microscopy (TEM) were performed for the analysis of microstructural characteristics of the specimens. High concentrations of dislocations along the grain boundaries were observed in the 3D-printed specimens that explains the increase of yield strength and exceptional plastic behavior of the material.

We evaluate thermal conductivity of the skeleton of porous sandstone from the measurements of the effective thermal conductivities of dry and saturated specimens provided by Woodside and Messmer (Woodside and Messmer in J Appl Phys 32:1688, 1961). Six types of the sandstone of different porosity levels are evaluated—Berkeley (porosity 0.03), St. Peters (porosity 0.11), Tensleep (porosity 0.155), Berea (porosity 0.22), Teapot (porosity 0.29), and Tripolite (porosity 0.59). The approach is based on recently developed concept of equivalent microstructure and solution of the inverse homogenization problem in the framework of Mori–Tanaka–Benveniste micromechanical model. We obtained that skeleton conductivities of all the sandstones except of Tripolite are smaller than the value of λ_s = 8.40 W/mK typically used in the literature. The equivalent microstructures that lead to the experimentally measured overall conductivity are obtained as combinations of almost spherical pores and strongly oblate, crack-like pores. The obtained results can be used for modeling of thermal conductivity of saturated sandstone.

The micromechanical behavior of an annealed Ti-6Al-4V material produced by Laser Powder Bed Fusion was characterized by means of in-situ synchrotron X-ray diffraction during a tensile test. The lattice strain evolution was obtained parallel and transversal to the loading direction. The elastic constants were determined and compared with the conventionally manufactured alloy. In the plastic regime, a lower plastic anisotropy exhibited by the lattice planes was observed along the load axis (parallel to the building direction) than in the transverse direction. Also, the load transfer from α to β phase was observed, increasing global ductility of the material. The material seems to accumulate a significant amount of intergranular strain in the transverse direction.

The full set of transversely isotropic elastic stiffness constants of inorganic shale (mudrock with total organic carbon less than 1.5%) can be successfully modeled and, therefore, predicted based on the mineral composition, mineral stiffnesses, clay platelet orientation distribution function, and microgeometry of the pore space. A fundamentally novel concept drawing from the Maxwell homogenization scheme allows a zero-porosity mineral matrix of the mudrock to be expressed as a polycrystal of variable composition and clay mineral alignment. Introduction of the brine-saturated pore space allows us to account for realistic 3D pore types and their combinations as well as elastic interactions, opening the way for better integration of rock physics and geomechanics with modern petrographic investigations and better shale velocity/anisotropy prediction as a function of diagenetic porosity reduction. We were able to calibrate the model using a limited subset of high-quality ultrasonic measurements on shale and constrain main pore geometries such as tetrahedra and irregular spheroids, often reported in modern scanning electron microscopy images. The model is then used to constrain the anisotropy tensor elements of illite-dominated clay, impossible to measure directly, and explore the main compositional and microstructural controls on the anisotropic elasticity of inorganic shale, including the most troublesome [Formula: see text] stiffness and its derivative — the anisotropy parameter [Formula: see text], which is of paramount importance in quantitative seismic interpretation.

This paper focuses on the analysis and quantitative characterization of the effect of saturation on the viscoelastic properties of human root dentin. Uniaxial compression tests under creep conditions have been performed on root molar dentin with tubules fully saturated with a viscous physiological fluid, as well as samples with non-saturated tubules (dry dentin samples). Blair-Rabotnov (BR) fraction-exponential model is used to characterize the overall viscoelastic properties of dentin and correlate them to the level of saturation. Experimental data are compared with theoretical predictions that interrelate the viscoelastic properties of saturated and dry specimens. The results show that saturation increases the viscous creep strains of dentin, which indicates a reduced capacity for stress relief. The uniaxial compression test under creep conditions, in combination with the BR kernel model, allows us to analyze the creep-relaxation behavior of dentin.

The mechanism of the effect of stresses on wave propagation in a fluid-saturated porous media has not been well understood. The goal of this paper is to fill this gap. First we formulated the general equations of motion in a homogeneously pre-stressed fluid-saturated medium and use them to derive explicit expressions of velocity dependence on stresses. Equations for fast and slow longitudinal waves allow substantial simplifications. The feasibility of fast longitudinal wave simplification is verified by comparison with experimental data available in literature.

The paper focuses on the problem of evaluating analytically the electric and elastic fields in a piezoelectric material of the symmetry class 6 containing a penny-shaped crack loaded by both normal and tangential tractions and by electric charges. A solution of this kind has been developed for symmetry class 6 mm by Karapetian et al. (Arch. Appl. Mech. 70: 201–229, 2000). To the best of our knowledge, it has never been derived for a material of class 6. The research is motivated by needs in evaluation of damage accumulated in bone tissue, which possesses such a symmetry. The problem is solved by the method of potential functions developed by Fabrikant (Applications of Potential Theory in Mechanics: Selection of New Results. Kluwer, Dordrecht,1989). Closed form solutions are obtained for the full electric and elastic fields. The approach is first verified by comparison with the results of Karapetian et al. (Arch. Appl. Mech. 70: 201–229, 2000) for the limiting case of piezoceramic of class 6 mm (PZT-6B) and then applied to calculation of the electroelastic fields associated with a penny-shaped crack in cortical bone (material of class 6).

In literature on the effective properties of composites, one-particle homogenization schemes have been developed alongside with solutions for periodic arrangements of inhomogeneities (starting with the pioneering work of Lord Rayleigh, 1892). It has often been claimed that the two approaches are, to some extent, equivalent. These claims have been based, mostly, on the fact that in the case of cubic arrangement of spherical particles the two approaches, indeed, yield close results. The present work shows that the mentioned case (that is the simplest, and very special one) constitutes an exception: generally, application of one-particle homogenization schemes to periodic composites may generate very large errors.

This work provides comparative analysis of the effect of manufacturing processes and process parameters on mechanical and thermal properties of stainless steel 316L. The properties are compared for the parts produced by additive manufacturing (Selected Laser Melting), thermal spraying, casting, hot/cold rolling, hot-isostatic pressing, and forging technologies. Each of these manufacturing processes leads to formation of its own microstructure formed by pores and cracks of different shape and orientation, as well as by dislocations and grain boundaries. A comprehensive analysis of the data available in literature is supplemented by our measurements of the mechanical properties of hot rolled and 3D printed samples.

The overall properties—thermal, electric, elastic, etc.—of any porous material are strongly dependent on their three-dimensional (3D) microstructures, which include the porosity, pore sizes, and shapes and connectivity of the porous space. These microstructural parameters can be collectively described as the “tortuosity of the porous space” (see, e.g., Chen et al., J Power Sources 273:486–494 2013). We propose to use the tortuosity parameter to characterize the overall material properties of a porous material with interconnected porous space. In the text to follow, we discuss the concept of this parameter and its application to characterize, elastic, electric, and mass transport properties as well as the cross-property connections.

In this paper, we derive the second-order crack opening displacement tensor for an arbitrarily oriented elliptical crack in an elliptically orthotropic (EO) matrix. This result is obtained in explicit closed form. The approach is based on the Saint-Venant’s idea of linear transformation between boundary value problems for elliptically orthotropic and isotropic bodies. The solution utilizes the classical representation of an ellipsoid crack where the smallest aspect ratio approaches zero and the transformation of the Taylor expansion of the corresponding Hill tensor. It is shown, in particular, that transformed cracks have neither the same in-plane aspect ratio nor the same vanishing aspect ratio. It requires a correction factor in the crack opening displacement tensor. Some specific relative orientations of the crack with respect to the symmetry planes of the EO matrix are considered in detail and effective properties are calculated in the case of randomly distributed cracks. The result is also extended to the case of a cylindrical (plane strain) crack.