Can a continuous mineral foam explain the stiffening of aged bone tissue? A micromechanical approach to mineral fusion in musculoskeletal tissues

Recent experimental data revealed a stiffening of aged cortical bone tissue, which could not be explained by common multiscale elastic material models. We explain this data by incorporating the role of mineral fusion via a new hierarchical modeling approach exploiting the asymptotic (periodic) homogenization (AH) technique for three-dimensional linear elastic composites. We quantify for the first time the stiffening that is obtained by considering a fused mineral structure in a softer matrix in comparison with a composite having non-fused cubic mineral inclusions. We integrate the AH approach in the Eshelby-based hierarchical mineralized turkey leg tendon model (Tiburtius et al 2014 Biomech. Model. Mechanobiol. 13 1003–23), which can be considered as a base for musculoskeletal mineralized tissue modeling. We model the finest scale compartments, i.e. the extrafibrillar space and the mineralized collagen fibril, by replacing the self-consistent scheme with our AH approach. This way, we perform a parametric analysis at increasing mineral volume fraction, by varying the amount of mineral that is fusing in the axial and transverse tissue directions in both compartments. Our effective stiffness results are in good agreement with those reported for aged human radius and support the argument that the axial stiffening in aged bone tissue is caused by the formation of a continuous mineral foam. Moreover, the proposed theoretical and computational approach supports the design of biomimetic materials which require an overall composite stiffening without increasing the amount of the reinforcing material

The asymptotic homogenization elasticity tensor properties for composites with material discontinuities

The classical asymptotic homogenization approach for linear elastic composites with discontinuous material properties is considered as a starting point. The sharp length scale separation between the fine periodic structure and the whole material formally leads to anisotropic elastic-type balance equations on the coarse scale, where the arising fourth rank operator is to be computed solving single periodic cell problems on the fine scale. After revisiting the derivation of the problem, which here explicitly points out how the discontinuity in the individual constituents’ elastic coefficients translates into stress jump interface conditions for the cell problems, we prove that the gradient of the cell problem solution is minor symmetric and that its cell average is zero. This property holds for perfect interfaces only (i.e., when the elastic displacement is continuous across the composite’s interface) and can be used to assess the accuracy of the computed numerical solutions. These facts are further exploited, together with the individual constituents’ elastic coefficients and the specific form of the cell problems, to prove a theorem that characterizes the fourth rank operator appearing in the coarse-scale elastic-type balance equations as a composite material effective elasticity tensor. We both recover known facts, such as minor and major symmetries and positive definiteness, and establish new facts concerning the Voigt and Reuss bounds. The latter are shown for the first time without assuming any equivalence between coarse and fine-scale energies (Hill’s condition), which, in contrast to the case of representative volume elements, does not identically hold in the context of asymptotic homogenization. We conclude with instructive three-dimensional numerical simulations of a soft elastic matrix with an embedded cubic stiffer inclusion to show the profile of the physically relevant elastic moduli (Young’s and shear moduli) and Poisson’s ratio at increasing (up to 100 %) inclusion’s volume fraction, thus providing a proxy for the design of artificial elastic composites

Homogenized out-of-plane shear response three-scale fiber-reinforced composites

In the present work we embrace a three scales asymptotic homogenization approach to investigate the effective behavior of hierarchical linear elastic composites reinforced by cylindrical, uniaxially aligned fibers and possessing a periodic structure at each hierarchical level of organization. We present our novel results assuming isotropy of the constituents and focusing on the effective out-of-plane shear modulus, which is computed exploiting the solution of the arising anti-plane problems. The latter are solved semi-analytically by means of complex variables and successfully benchmarked against the results obtained by finite elements. Our findings can pave the way for multiscale modeling of complex hierarchical materials (such as bone and tendons) at a negligible computational cost

Three scales asymptotic homogenization and its application to layered hierarchical hard tissues

In the present work a novel multiple scales asymptotic homogenization approach is proposed to study the effective properties of hierarchical composites with periodic structure at different length scales. The method is exemplified by solving a linear elastic problem for a composite material with layered hierarchical structure. We recover classical results of two-scale and reiterated homogenization as particular cases of our formulation. The analytical effective coefficients for two phase layered composites with two structural levels of hierarchy are also derived. The method is finally applied to investigate the effective mechanical properties of a single osteon, revealing its practical applicability in the context of biomechanical and engineering applications

Microstructural modeling and computational homogenization of the physically linear and nonlinear constitutive behavior of micro-heterogeneous materials

Engineering materials show a pronounced heterogeneity on a smaller scale that influences the macroscopic constitutive behavior. Algorithms for the periodic discretization of microstructures are presented. These are used within the Nonuniform Transformation Field Analysis (NTFA) which is an order reduction based nonlinear homogenization method with micro-mechanical background. Theoretical and numerical aspects of the method are discussed and its computational efficiency is validated

Homogenization Methods for Problems with Multiphysics, Temporal and Spatial Coupling

There are many natural and man-made materials with heterogeneous micro- or nanostructure (fine-scale structure) which represent a great interest for industry. Therefore there is a great demand for computational methods capable to model mechanical behavior of such materials. Direct numerical simulation resolving all fine-scale details using very fine mesh often becomes very expensive. One of alternative effective group of methods is the homogenization methods allowing to model behavior of materials with heterogeneous fine-scale structure. The essence of homogenization is to replace heterogeneous material with some equivalent effectively homogeneous material. The homogenization methods are proven to be effective in certain classes of problems while there is need to improve their performance, which includes extension of the range of applicability, simplification, usage with conventional FE software and reducing computational cost. In this dissertation methods extending the range of applicability of homogenization are developed. Firstly, homogenization was extended of the case of full nonlinear electromechanical coupling with large deformations, which allows simulating effectively behavior of electroactive materials such as composites made of electroactive polymers. Secondly, homogenization was extended on wave problems where dispersion is significant and should be accounted for. Finally, the homogenization was extended on the case where the size of microstructure. The distinctive feature of the methods introduced in this dissertation is that they don't require higher order derivatives and can be implemented with conventional FE codes. The performance of methods is tested on various examples using Abaqus

Effective properties of hierarchical fiber-reinforced composites via a three-scale asymptotic homogenization approach

The study of the properties of multiscale composites is of great interest in engineering and biology. Particularly, hierarchical composite structures can be found in nature and in engineering. During the past decades, the multiscale asymptotic homogenization technique has shown its potential in the description of such composites by taking advantage of their characteristics at the smaller scales, ciphered in the so-called effective coefficients. Here, we extend previous works by studying the in-plane and out-of-plane effective properties of hierarchical linear elastic solid composites via a three-scale asymptotic homogenization technique. In particular, the approach is adjusted for a multiscale composite with a square-symmetric arrangement of uniaxially aligned cylindrical fibers, and the formulae for computing its effective properties are provided. Finally, we show the potential of the proposed asymptotic homogenization procedure by modeling the effective properties of musculoskeletal mineralized tissues, and we compare the results with theoretical and experimental data for bone and tendon tissues

Thermo-mechanical properties prediction of Ni-reinforced Al2_2O3_3 composites using micro-mechanics based representative volume elements

For effective cutting tool inserts that absorb thermal shock at varying temperature gradients, improved thermal conductivity and toughness are required. In addition, parameters such as the coefficient of thermal expansion must be kept within a reasonable range. This work presents a novel material design framework based on a multi-scale modeling approach that proposes nickel (Ni)-reinforced alumina (Al$_2$O$_3$) composites to tailor the mechanical and thermal properties required for ceramic cutting tools by considering numerous composite parameters. The representative volume elements (RVEs) are generated using the DREAM.3D software program and the output is imported into a commercial finite element software ABAQUS. The RVEs which contain multiple Ni particles with varying porosity and volume fractions are used to predict the effective thermal and mechanical properties using the computational homogenization methods under appropriate boundary conditions (BCs). The RVE framework is validated by the sintering of Al$_2$O$_3$-Ni composites in various compositions. The predicted numerical results agree well with the measured thermal and structural properties. The properties predicted by the numerical model are comparable with those obtained using the rules of mixtures and SwiftComp, as well as the Fast Fourier Transform (FFT) based computational homogenization method. The results show that the ABAQUS, SwiftComp and FFT results are fairly close to each other. The effects of porosity and Ni volume fraction on the mechanical and thermal properties are also investigated. It is observed that the mechanical properties and thermal conductivities decrease with the porosity, while the thermal expansion remains unaffected. The proposed integrated modeling and empirical approach could facilitate the development of unique Al$_2$O$_3$-metal composites with the desired thermal and mechanical properties for ceramic cutting inserts

A phase field approach for damage propagation in periodic microstructured materials

In the present work, the evolution of damage in periodic composite materials is investigated through a novel finite element-based multiscale computational approach. The proposed methodology is developed by means of the original combination of asymptotic homogenization with the phase field approach to nonlocal damage. This last is applied at the macroscale level on the equivalent homogeneous continuum, whose constitutive properties are obtained in closed form via a two-scale asymptotic homogenization scheme. The formulation considers different assumptions on the evolution of damage at the microscale (e.g., damage in the matrix and not in the inclusion/fiber), as well as the role played by the microstructural reinforcement, i.e. its volumetric content and shape. Numerical results show that the proposed formulation leads to an apparent tensile strength and a post-peak branch of unnotched and notched specimens dependent not only on the internal length scale of the phase field approach, as for homogeneous materials, but also on microstructural features. Down-scaling relations provide the full reconstruction of the microscopic fields at any point of the macroscopic model, as a simple post-processing operation