Eshelby Tensor Research Articles

Hierarchical elastoplasticity of cortical bone: Observations, mathematical modeling, validation

Motivated by the water layer-coated nanoscale bone mineral crystals and the elastoplastic behavior seen at the extracellular scale, we develop a six-step hierarchical micromechanics model for the elastoplasticity of cortical bone. For that purpose, the Eshelby problem-based concentration-influence tensor concept is generalized for a multi-scale situation, quantifying the mechanical interaction between elastic and plastic strains between material phases across six orders of magnitude in observation scale. This hierarchical interaction scheme is complemented by non-associated Mohr–Coulomb plasticity assigned to the mineral crystal phases, and a return-mapping algorithm which adapts classical computational mechanics approaches for the realm of semi-analytical continuum micromechanics. Founded on elastic and strength properties of molecular collagen and hydroxyapatite, the model passes experimental validation against ultrasonic and quasi-static tests at the extrafibrillar, extracellular, extravascular, and cortical observation scales, across different tissues and species. It reveals cortical bone strength to increase nonlinearly with the vascular porosity, and to depend bi-linearly on the extracellular mass density, while elucidating plastic spreading events at the nanocrystal scale, which are fundamentally different in tensile and compressive loading.

Strain concentration factor of heterogeneous materials and analytical influence functions based on Eshelby tensor

In this manuscript, Eshelby tensor is employed to assess the strain concentration that arises in the matrix phase at the interface, offering precise values and locations of maximum strain under specific loading conditions for both spherical and cylindrical inclusions. When compared to numerical simulation results, the analytical predictions grounded in the Eshelby tensor exhibit satisfactory accuracy. Then an analytical calculation method based on Eshelby tensor for the elastic strain influence functions of reduced-order homogenization (ROH) method is developed and adopted on particle-reinforced and fibrous composites, presenting its feasibility and advantage on off-line stage calculation of ROH method. The error analyses between analytical and numerical results are conducted. The numerical results also exhibit the necessity of finer interface partitioning to obtain the response on micro-scale with higher resolution.

Fast Fourier transform method in peridynamic micromechanics of composites

We consider a static linear bond–based peridynamic (proposed by Silling, see J. Mech. Phys. Solids 2000; 48:175–209) composite materials (CMs) of a periodic structure. In the framework of the second background of micromechanics (also called computational analytical micromechanics), one proved that local micromechanics (LM) and peridynamic micromechanics (PM) are formally analogous to each other for CM of both random and periodic structures. It allows a straightforward generalization of LM methods (including fast Fourier transform, FFT) to their PM counterparts. So, in the PM counterpart of the implicit periodic Lippmann–Schwinger (L-S) equation in LM, we have three convolution kernels corresponding to the properties of the matrix, inclusions, and interactive interface. Eshelby tensor in LM, depending on the inclusion shape, is replaced by PM counterparts depending on the shapes of inclusions, and the interaction interface (although the Eshelby concept of homogeneous eigenfields does not work in PM). The mentioned tensors are estimated once (as in LM) in a frequency domain (also by the FFT method). The possible incorrectness of FFT applications to PM is analyzed and corrected. The polarization schemes of the solution of the L-S equation in the Fourier space have one primary unknown variable (polarization), whereas the PM counterpart contains three primary ones estimated at each step, which are formally similar to the LM case. A description of the generalized basic scheme and the Krylov approach is presented. Computational complexities O(N log2 N) of the FFT methods are the same in both LM and PM.

A microstructure-based parameterization of the effective anisotropic elasticity tensor of snow, firn, and bubbly ice

Abstract. Quantifying the link between microstructure and effective elastic properties of snow, firn, and bubbly ice is essential for many applications in cryospheric sciences. The microstructure of snow and ice can be characterized by different types of fabrics (crystallographic and geometrical), which give rise to macroscopically anisotropic elastic behavior. While the impact of the crystallographic fabric has been extensively studied in deep firn, the present work investigates the influence of the geometrical fabric over the entire range of possible volume fractions. To this end, we have computed the effective elasticity tensor of snow, firn, and ice by finite-element simulations based on 391 X-ray tomography images comprising samples from the laboratory, the Alps, Greenland, and Antarctica. We employed a variant of Eshelby's tensor that has been previously utilized for the parameterization of thermal and dielectric properties of snow and utilized Hashin–Shtrikman bounds to capture the nonlinear interplay between density and geometrical anisotropy. From that we derive a closed-form parameterization for all components of the (transverse isotropic) elasticity tensor for all volume fractions using two fit parameters per tensor component. Finally, we used the Thomsen parameter to compare the geometrical anisotropy to the maximal theoretical crystallographic anisotropy in bubbly ice. While the geometrical anisotropy clearly dominates up to ice volume fractions of ϕ≈0.7, a thorough understanding of elasticity in bubbly ice may require a coupled elastic theory that includes geometrical and crystallographic anisotropy.

Two-dimensional nonlocal Eshelby’s inclusion theory: eigenstress-driven formulation and applications

The classical Eshelby’s theory, developed based on local linear elasticity, cannot be applied to inclusion problems that involve nonlocal (long range) elastic effects often observed in micromechanical systems. In this study, we introduce the extension of Eshelby’s inclusion theory to nonlocal elasticity. Starting from Eringen’s integral formulation of nonlocal elasticity, an eigenstress-driven nonlocal Eshelby’s inclusion theory is presented. The eigenstress-driven approach is shown to be a valid mathematical extension of the classical eigenstrain-driven approach in the context of nonlocal inclusion problems. Two individual numerical approaches are developed and applied to simulate inclusion problems and numerically extract the corresponding nonlocal Eshelby tensor. The numerical results obtained from both approaches confirm the validity of the derived nonlocal Eshelby tensor and its ability to capture the non-uniform eigenstress distribution within an elliptic inclusion. These results also help reveal the fundamental difference between the mechanical behaviour of the classical local and the nonlocal inclusion problems. The eigenstress-driven nonlocal inclusion theory could provide the necessary theoretical foundation for the development of homogenization methods of nonlocal heterogeneous media.

Energy variation and stress fields of spherical inclusions with eigenstrain in three-dimensional anisotropic bi-materials

In this study, expressions for the interior stress fields of a spherical inclusion with uniform eigenstrain embedded in an anisotropic bi-material featuring a planar interface are derived. These expressions involve surface integrals of the imaginary term of the first derivative of the Green tensor in an anisotropic bi-material which are well-suited for standard numerical integrations. Specific formulations are provided for cases where the inclusion belongs to either the same material or both materials. Additionally, expressions are presented for the equivalent Eshelby tensor. The interior stress fields and variations in elastic strain energy are computed using cubic elastic constants of Cu. Various inclusion positions relative to the interface, eigenstrain forms, and crystallographic orientations are considered. For instances involving dilatational eigenstrain, the elastic strain energy variation with the inclusion’s position may exhibit multiple extrema. The global minimum and maximum consistently occur when the inclusion spans both materials. In the context of symmetrical tilt boundaries, energy variations are perfectly symmetric, with a global minimum on the interface that decreases with the tilt angle. The significance of the observed energy variations for defects segregation is quantitatively assessed by comparing them with the interaction energy between an eigenstrain and a grain boundary stress field.

Non-Invasive Imaging of Mechanical Properties of Cancers In Vivo Based on Transformations of the Eshelby's Tensor Using Compression Elastography.

Knowledge of the mechanical properties is of great clinical significance for diagnosis, prognosis and treatment of cancers. Recently, a new method based on Eshelby's theory to simultaneously assess Young's modulus (YM) and Poisson's ratio (PR) in tissues has been proposed. A significant limitation of this method is that accuracy of the reconstructed YM and PR is affected by the orientation/alignment of the tumor with the applied stress. In this paper, we propose a new method to reconstruct YM and PR in cancers that is invariant to the 3D orientation of the tumor with respect to the axis of applied stress. The novelty of the proposed method resides on the use of a tensor transformation to improve the robustness of Eshelby's theory and reconstruct YM and PR of tumors with high accuracy and in realistic experimental conditions. The method is validated using finite element simulations and controlled experiments using phantoms with known mechanical properties. The in vivo feasibility of the developed method is demonstrated in an orthotopic mouse model of breast cancer. Our results show that the proposed technique can estimate the YM and PR with overall accuracy of (97.06 ± 2.42) % under all tested tumor orientations. Animal experimental data demonstrate the potential of the proposed methodology in vivo. The proposed method can significantly expand the range of applicability of the Eshelby's theory to tumors and provide new means to accurately image and quantify mechanical parameters of cancers in clinical conditions.

FAST FOURIER TRANSFORM METHOD FOR PERIDYNAMIC BAR OF PERIODIC STRUCTURE

The basic feature of the peridynamics [introduced by Silling (2000)] considered is a continuum description of material behavior as the integrated nonlocal force interactions between infinitesimal material points. A heterogeneous bar of the periodic structure of constituents with peridynamic mechanical properties is analyzed. One introduces the volumetric periodic boundary conditions (PBCs) at the interaction boundary of a representative unit cell (UC), whose local limit implies the known locally elastic PBCs. This permits us to generalize the classical computational homogenization approach to its counterpart in peridynamic micromechanics (PM). Alternative to the finite element methods (FEM) for solving computational homogenization problems are the fast Fourier transforms (FFTs) methods developed in local micromechanics (LM). The Lippmann-Schwinger (L-S) equation-based approach of the FFT method in the LM is generalized to the PM counterpart. Instead of one convolution kernel in the L-S equation, we use three convolution kernels corresponding to the properties of the matrix, inclusions, and interaction interface. The Eshelby tensor in LM depending on the inclusion shape is replaced by PM counterparts depending on the inclusion size and interaction interface (although the Eshelby concept of homogeneous eigenfields does no work in PM). The mentioned tensors are estimated one time (as in LM) in a frequency domain (also by the FFT method). Numerical examples for 1-D peridynamic inhomogeneous bar are considered. Computational complexities O (N <i>log</i><sub>2</sub> N) of the FFT methods are the same in both LM and PM.

A novel method for depolarization tensor and average form of an arbitrarily shaped inclusion: Extension to different physical fields and their effective transport properties of composites

Depolarization tensor (DT), exclusively controlled by the inclusion shape, is crucial in assessing composite transport behavior. However, the solution of DT for a non-ellipsoidal inclusion exhibits nonuniformity and lacks a closed-form expression. Average DT is then introduced to ensure uniformity and seamlessly integrate the evaluation of effective transport properties of composites. In this work, we devise a novel method to obtain DT and average DT of an arbitrarily shaped inclusion by leveraging mathematical similarity from Greenʼs function to degenerate the Eshelby tensor and average Eshelby tensor in an elastic field. The method circumvents singularity and complexity in direct integration and successfully extracts all tensor components. As natural extensions, we also derive the Eshelby conduction tensor, polarizability tensor, and their averages. The accuracy of the method is validated by comparing the average polarizabilities of Platonic solids and stem cells with existing literature data. Furthermore, we incorporate average DT into a developed homogenization model to predict effective transport properties of two-phase composites comprising randomly distributed congruent irregular inclusions. Comparisons with finite element simulations demonstrate accurate predictions of the effective transport properties like effective permeability, diffusivity, conductivity, and permittivity. The proposed framework offers valuable guidance for tailoring inclusion shapes to design particulate/fibrous/porous/biological composites.

Investigation of non-uniform oxidation based on a mechanochemical phase field model with nonlinear reaction kinetics and large inelastic deformation

A mechanochemical model is proposed to investigate the non-uniform oxidation of thermal barrier coatings (TBCs) that involves large inelastic deformation and nonlinear reaction kinetics. The large-deformation theory incorporates the higher-order term of geometric nonlinearity for a more precise description of the deformation and stress evolution in an oxide layer. The effect of stresses on the reaction kinetics is considered, which is expressed as the Eshelby stress tensor to account for the conformational volume change and deformation energy. A nonlinear reaction kinetics is adopted for a more accurate description of the nonequilibrium thermodynamic processes. The 2D simulations reveal a non-uniform oxide growth, three modes of oxide-metal interfacial morphology evolution, and tensile stress concentrations in the oxide scale. These simulation results agree with the experimental observations that cannot be described by the previous models. With the model, it is further demonstrated that a stable interfacial morphology and a significantly reduced tensile stress can be achieved by increasing the creep rate of the oxide and the flatness of the oxide-metal interface. This model thus provides an approach to extend the service time of TBCs.

Computation of minimal covariants bases for 2D coupled constitutive laws

We produce minimal integrity bases for both isotropic and hemitropic invariant algebras (and more generally covariant algebras) of most common bidimensional constitutive tensors and — possibly coupled — laws, including piezoelectricity law, photoelasticity, Eshelby and elasticity tensors, complex viscoelasticity tensor, Hill elasto-plasticity, and (totally symmetric) fabric tensors up to twelfth-order. The concept of covariant, which extends that of invariant is explained and motivated. It appears to be much more useful for applications. All the tools required to obtain these results are explained in detail and a cleaning algorithm is formulated to achieve minimality in the isotropic case. The invariants and covariants are first expressed in complex forms and then in tensorial forms, thanks to explicit translation formulas which are provided. The proposed approach also applies to any n-uplet of bidimensional constitutive tensors.

Eshelby tensors and effective stiffness of one-dimensional orthorhombic quasicrystal composite materials containing ellipsoidal particles

Eshelby tensors and effective stiffness of one-dimensional orthorhombic quasicrystal composite materials containing ellipsoidal particles

Thermal Conductivity Prediction of Metal Matrix Particulate Composites: Theoretical Methodology and Application

To make more accurate predictions of the effective thermal conductivity (ETC) of the composites, a systematic method for predicting the effective thermal conductivity of metal matrix particle composites with arbitrarily shaped particles was proposed, and the geometry of random particles with controlled shape characteristics is reconstructed. In addition, the geometric vertices of the reconstructed particles are used to characterize the morphology of inclusions with complex profile in two-dimensional isotropic elasticity, and its explicit expression for the Eshelby tensor are explored. Moreover, the material mismatch between the particles and the matrix phase is simulate using a continuously distributed source field based on the Eshelby's equivalent inclusion method. The relationship between micro-structure and effective performance is established. Finally, the effective thermal conductivity of CuCr alloys was predicted using the ETC prediction model. Through the comparison of the numerical simulations, experiments, and calculations, the results show that the ETC model has reliable predictive capability.

An analysis of composites with two-dimensional decagonal quasicrystal matrix and spheroidal inclusions

Eshelby tensors in elasticity are obtained for two-dimensional (2D) decagonal quasicrystal containing an ellipsoidal inclusion. Based on the intrinsic strain formula and Mori–Tanaka theory, the effective elastic properties of 2D decagonal composites are obtained. The effects of inclusion volume fraction and inclusion aspect ratio on elastic properties of composites with 2D decagonal quasicrystal are discussed.

Study on effective electroelastic properties of one-dimensional hexagonal piezoelectric quasicrystal containing randomly oriented inclusions

In this paper, the effective electroelastic properties of one-dimensional (1D) hexagonal piezoelectric quasicrystal containing randomly oriented inclusions are considered. The explicit expressions are obtained for the Eshelby tensors for 1D hexagonal piezoelectric quasicrystals containing rod-shaped and penny-shaped inclusions. The closed forms of the electroelastic constants are acquired for four special cases of random orientations of inclusions. Numerical results are given for the 1D hexagonal piezoelectric quasicrystal containing randomly oriented ellipsoidal inclusions. The results indicate that the effective electroelastic properties of 1D hexagonal piezoelectric quasicrystal composites are strongly affected by both the aspect ratio and the orientation of inclusions.

Average Eshelby tensor of an arbitrarily shaped inclusion from convexity to non-convexity: Effective elastic properties of composites

The effective mechanical properties of composites are strongly influenced by the morphology, volume fraction, and properties of the inclusion phase. Extensive theoretical models for the evaluation of effective properties are mainly based on the continuum micromechanics derived from the well-known Eshelby’s solution of an ellipsoidal inclusion problem, which are subjected to the limitations on the Eshelby tensor for a non-ellipsoidal inclusion. This work devises a three-step method to derive the average Eshelby tensor of an arbitrarily shaped inclusion from convexity to non-convexity in an elastic field. The proposed three-step method is user-friendly and avoids the singularity that arises from solving the complicated integrals for an arbitrarily shaped inclusion. The calculated average Eshelby tensors of inclusions with different morphologies, including Platonic solids, superballs, superellipsoids, doughnuts, helices, and non-convex particles, are thereafter compared against the existing data reported in the literature, in order to verify the reliability of the present method. Moreover, we also incorporate the proposed method into a micromechanical model to predict the effective elastic moduli of two-phase composites, in which congruent non-ellipsoidal inclusions of a specific volume fraction are randomly distributed in a homogeneous matrix. Comparisons with the direct finite element method (FEM) simulations indicate that the developed micromechanical model is a reliable means to evaluate the elastic properties of composites. The proposed average Eshelby tensor of an arbitrarily shaped inclusion can also be generalized to the theoretical predictions of the effective transport properties of composites with inclusions of diverse types and shapes, such as particles, rocks, granules, pores, fibers, and cracks.

Effective thermal conductivity of composites with anisotropic particles of various shapes embedded in an isotropic matrix

The paper demonstrates a method to calculate effective thermal conductivity in a two-phase medium with an isotropic host phase and embedded anisotropic particles. The presented scheme enables easy incorporation of various regular shapes of filler particles if the principal values of their Eshelby tensor are supplied. Basic shapes, such as spheroids, may approximate a wide range of fillers used to enhance thermal conduction, such as polymers filled with highly conductive metal and ceramic powders, graphite and graphene flakes, and nanotubes. Different spatial orientations and sizes of particles may be supplied to the model in the form of probability distributions. The method is well suited to numerical implementation, and the resulting computations are less demanding than popular mesh-based methods, i.e., Finite Volume and Finite Element Methods. It does not include interfacial thermal resistance between particles and matrix, but the addition of this component is planned in future papers.

The Derivation of Elastic Fields of a Curvilinear Inclusion

The disturbed elastic fields of a curvilinear inclusion in an isotropic elastic plane are investigated analytically by a newly proposed technique. The boundary of the inclusion is characterized by arbitrary Laurent polynomials in the 2D Cartesian coordinate system, and constant eigenstrains are considered to occur in the inclusion. Based on the irreducible decomposition of an arbitrary tensor, the Eshelby tensor is attributed to two integrals on the curved boundary of the inclusion. The analytical solutions for the induced stress and displacement fields outside the inclusion domain are explicitly derived by utilizing the newly developed technique, including the salient features of the Faber polynomials. Examples show the efficiency of the technique in this paper.

Eshelby Tensors for Two-Dimensional Decagonal Piezoelectric Quasicrystal Composites

The Eshelby tensor for two-dimensional (2D) piezoelectric quasicrystal composites (QCs) is considered. The explicit expressions of Eshelby tensors for 2D piezoelectric QCs are given using the Green’s function method and the interior polarization tensor method, respectively. On this basis, numerical examples of the Eshelby tensor for 2D piezoelectric QCs with ellipsoidal inclusions are discussed in detail.

A computationally efficient coupled multi-scale model for short fiber reinforced composites

A coupled multi-scale (macro–micro) model is developed to predict non-linear elasto-plastic behavior of short fiber reinforced composites. At the microscopic level, a recently proposed micro-mechanical model, developed based on a two-step orientation averaging approach, is used. A wide range of micro-structural parameters, including matrix and fiber constitutive parameters, fiber volume fraction and fiber aspect ratio, can be accommodated in the model. Different interactions including Voigt, Reuss and a self-consistent assumption are considered in the model. This micro-mechanical model is then incorporated in a Finite Element model of the macro-scale problem, enabling coupled macro–micro simulations of real-life structures/specimens. Numerical examples and comparisons with experimental data, taken from literature, show that the model gives good predictions. Besides, several strategies and techniques are employed to improve the computational efficiency of the model. These techniques include replacing originally utilized trapezoidal integration (for fiber orientations and calculation of the Eshelby tensor) with more efficient integration schemes, and using a more efficient method for data storage. Comparisons of the computational efforts shows that these improvements substantially decreased the computational cost of the model.