Eshelby's Inclusion Research Articles

An additive Mori–Tanaka scheme for elastic–viscoplastic composites based on a modified tangent linearization

Mean-field modeling based on the Eshelby inclusion problem poses some difficulties when the non-linear Maxwell-type constitutive law is used for elasto–viscoplasticity. One difficulty is that this behavior involves different orders of time differentiation, which leads a long-term memory effect. One of the possible solutions to this problem is the additive interaction law. Generally, mean field models solely use the mean values of stress and strain fields per phase, while variational approaches consider the second moments of stresses and strains. It is seen that the latter approach improves model predictions allowing to account for stress fluctuation within the phases. However, the complexity of the variational formulations still makes them difficult to apply in the large scale finite element calculations and for non-proportional loadings. Thus, there is a need to include the second moments within homogenization models based on the additive interaction law. In the present study, the incorporation of the second moments of stresses into the formulation of the additive Mori–Tanaka model of two-phase elastic–viscoplastic material is discussed. A modified tangent linearization of the viscoplastic law is proposed, while the Hill–Mandel’s lemma is used to track the evolution of second moments of stresses. To study the model performance and efficiency, the results are compared to the full-field numerical calculations and predictions of other models available in the literature. Very good performance of the modified tangent linearization is demonstrated from these benchmarks for both monotonic and non monotonic loading responses.

Optimal Arrangements and Local Anisotropy of {100} Guinier-Preston (GP) Zones by Parametric Dislocation Dynamics (PDD) Simulations.

Stress-oriented precipitation and the resulting mechanical anisotropy have been widely studied over the decades. However, the local anisotropy of precipitates with specific orientations has been less thoroughly investigated. This study models the interaction between an edge dislocation source and {100} variants of Guinier-Preston (GP) zones in Al-Cu alloys using the parametric dislocation dynamics (PDD) method. Concentric geometrically necessary dislocation (GND) loops were employed to construct a line integral model for thin platelets. The simulations, conducted with our self-developed code based on Green's function method and Eshelby inclusion theory revealed distinct strengthening behavior along the strong and weak directions for 60° GP zones, demonstrating anisotropic strengthening from the perspective of elastic interactions. Furthermore, the optimal inclined arrangement of the GP zone array was determined through elastic energy calculations, and these results were corroborated by TEM observations.

Mean-field homogenization of liquid metal elastomer composites: comparative study and exact dilute solution of core–shell inclusion

Liquid metal-elastomer composites (LMECs) have gathered significant attention for their potential applications in various functional stretchable devices, with inclusion sizes ranging from micrometers to nanometers. These composites exhibit exceptional properties, such as high electric permittivity and thermal conductivity, surpassing those of the elastomer matrix, thus enabling a broader range of applications without compromising the material's stretchability. To investigate the diverse effective elastic and functional properties of LMECs, micromechanics-based homogenization method based on Eshelby's inclusion solution are invaluable. However, the extreme contrast in elastic constants among the phases in LMECs, particularly for nanosized inclusions where a considerable amount of stiff metal oxide forms around the inclusions, can lead to critical failure in predicting effective properties if inadequate homogenization approach is employed. In this study, we present multiple mean-field homogenization approaches applicable to LMECs with core-shell morphology, namely: (i) multi-phase, (ii) sequential, (iii) pseudo-grain, and (iv) direct approaches. We compare the accuracy of the models concerning effective elastic, thermal, and dielectric properties, evaluated against numerical homogenization results and compared with reported experimental data. Specifically, we highlight homogenization scheme utilizing exact field solutions of dilute core-shell inclusion, emphasizing the importance of accurately capturing the field in the micromechanics of LMECs. Furthermore, we demonstrate that widely utilized interphase model could not properly resolve the core-shell morphology and thus should be avoided. This comprehensive assessment provides critical insights into the proper homogenization strategies for designing advanced LMECs with precise prediction of effective properties.

Synchrotron X-ray diffraction studies of the internal load transfer in Ni–CrC metal matrix composites

Strengthening in metal matrix composites (MMCs) is primarily due to the load transfer from the compliant matrix to the stiff reinforcement. While internal load transfer has been studied for conventionally manufactured MMCs, the extent to which it may be affected by the preexisting defects in additively manufactured MMCs remain elusive. In this study, we performed uniaxial compression loading on cold sprayed Ni–CrC particulate-reinforced MMC. We observe significantly enhanced strength and ductility compared to the brittle behavior previously reported in tension for the same MMC. In contrast to the absence of load transfer in tension, the presence of initial defects did not preclude the internal load transfer in the composites under compressive loading. In-situ high-energy X-ray diffraction analysis revealed a relatively constant load partitioning in the elastic regime, which aligned well with predictions by the Eshelby's inclusion model. Furthermore, an internal load transfer was observed from the Ni matrix to the CrC reinforcement upon plastic deformation of the Ni matrix. Finite element modeling further confirmed this and demonstrated that localized tensile stress at the interface in the transverse direction resulted in the interfacial debonding and partial relaxation of the matrix. We also report reinforcing particle fracture upon further compression which in turn triggered the eventual failure of the MMC.

Topological characterization of rearrangements in amorphous solids.

In amorphous materials, plasticity is localized and occurs as shear transformations. It was recently shown by Wu etal. that these shear transformations can be predicted by applying topological defect concepts developed for liquid crystals to an analysis of vibrational eigenmodes [Z. W. Wu et al., Nat. Commun. 14, 2955 (2023)10.1038/s41467-023-38547-w]. This study relates the -1 topological defects to the displacement fields expected of an Eshelby inclusion, which are characterized by an orientation and the magnitude of the eigenstrain. A corresponding orientation and magnitude can be defined for each defect using the local displacement field around each defect. These parameters characterize the plastic stress relaxation associated with the local structural rearrangement and can be extracted using the fit to either the global displacement field or the local field. Both methods provide a reasonable estimation of the molecular-dynamics-measured stress drop, confirming the localized nature of the displacements that control both long-range deformation and stress relaxation.

A dislocation density-based crystal plasticity constitutive model: comparison of VPSC effective medium predictions with ρ-CP finite element predictions

This work presents a dislocation density-based crystal plasticity constitutive model for glide kinetics, strengthening and dislocation density evolution, implemented in the effective medium-based visco-plastic self consistent (VPSC) framework and the spatially resolved, ρ-CP crystal plasticity finite element framework. Additionally, a distribution of intragranular stresses is introduced in the VPSC framework, instead of the conventionally used mean value of grain stress for effective medium calculations. The ρ-CP model is first calibrated to predict the mechanical response of a bcc ferritic steel with an initial rolled texture. The same set of constitutive model parameters are then used in VPSC to predict the aggregate stress–strain response and total dislocation densities. For these VPSC simulations, the interaction parameter governing the interaction between the grain and the effective medium in the Eshelby inclusion formalism, and a scalar parameter representative of the distribution of intragranular stresses within a grain, are used to calibrate the VPSC predictions in order to match the predictions of the ρ-CP model. A parametric study is performed to understand the effect of these two parameters on the VPSC predictions. Further, simulations are also performed for a random untextured polycrystal to identify the corresponding VPSC simulation parameters for predicting a similar response as the ρ-CP model. The novelty of the work is in the same set of constitutive models and associated parameters have been implemented in VPSC and ρ-CP to predict similar aggregate stress–strain response and total dislocation densities. This finite element-calibrated effective medium crystal plasticity approach reduces the computational time by at least two orders of magnitude and represents an advance towards the development of multiscale crystal plasticity modeling tools.

Unexpected nucleation mechanism of T1 precipitates by Eshelby inclusion with unstable stacking faults

AbstractAluminum‐lithium (Al‐Li) alloy is one of the most promising lightweight structural materials in the aeronautic and aerospace industries. The key to achieving their excellent mechanical properties lies in tailoring T1 strengthening precipitates; however, the nucleation of such nanoparticles remains unknown. Combining atomic resolution HAADF‐STEM with first‐principles calculations based on the density functional theory (DFT), here, we report a counterintuitive nucleation mechanism of the T1 that evolves from an Eshelby inclusion with unstable stacking faults. This precursor is accelerated by Ag‐Mg clusters to reduce the barrier, forming the structural framework. In addition, these Ag‐Mg clusters trap the free Cu and Li to prepare the chemical compositions for T1. Our findings provide a new perspective on the phase transformations of complex precipitates through solute clusters in terms of geometric structure and chemical bonding functions.

Numerical modeling of edge inhomogeneity in a macro cantilever beam

This study explores the inhomogeneity problem near the edge of a cantilever beam. Common in gears and turbine blades, inhomogeneities under local edge contact, coupled with macroscale bending effects, result in extreme and complex stress concentration. This model employs the Conjugate Gradient Method to combines the macroscopic beam theory and microscopic elasticity. The Eshelby inclusion method for simulating inhomogeneities. The Hetényi image method is used for address edge contacts in the beam. Enhanced by the Discrete Convolution fast Fourier Transform algorithm, the model’s accuracy and efficiency are validated with the Finite element method. Parametric analysis highlights the influence of distance between indenter and edge, stress components distribution and potential failure in cross-scale challenges.

Void nucleation and growth behavior of TIG welded AA2219 deformed at cryogenic temperatures

This work investigates the failure behaviour of tungsten inert gas (TIG) welded 2219 aluminum alloy butt welds deformed over a range of cryogenic temperatures relevant to liquid hydrogen/oxygen propellent storage tanks. A novel in situ tensile testing machine, able to deform samples at temperatures down to 83 K, was designed to be accommodated on synchrotron computed tomography beamlines. In this way we were able to observe the accumulation of damage from void nucleation through growth to coalescence and ultimate failure by time-lapse X-ray microtomography during tensile straining at 293 K and 123 K. The results reveal that micro-void nucleation is triggered by θ (Al2Cu) precipitates and α-Al + θ (Al2Cu) eutectics in the form of particle fracture and interface decohesion. After nucleation, these freshly nucleated voids, along with pre-existing welding defects, extend with straining along the loading direction. This leads to void linkage, a very short void coalescence stage and subsequent catastrophic failure. The probability distribution function that describes the rate at which voids nucleate by interface decohesion considering the particle orientation was determined through the Eshelby inclusion theory. The evolution of void aspect ratio and void growth were modelled using the Gurson-Tvergaard-Needleman model and Le Roy model, respectively. This suggests that pre-existing defects grow faster, but that the numerous closely spaced nucleated voids play a significant role in failure.

Physics-driven neural networks for nonlinear micromechanics

Micromechanical problems seek full-field solutions in response to external and/or internal thermal-mechanical loads, which have been increasingly encountered in material property and process assessments. Here we present a machine learning based strategy of solving micromechanics at finite strains. Deep neural networks (DNNs) are trained by employing the elastic energy together with other physical constraints to formulate the loss function. In particular, the crossover of material points, which can be a common issue for similar DNNs when applied to compression-dominant problems, is shown to be effectively addressed by including a kinematic penalty term that forces the DNN to maintain structural integrity and stability and avoid unphysical behavior. The generality and accuracy of the proposed physics-driven neural networks (PDNNs) are demonstrated through various micromechanical problems, including single crystal anisotropy elasticity, Eshelby's inclusion problem, buckling, and elastic homogenization of polycrystals. The effect of the choice of optimizers and hyperparameters on the PDNN training are discussed and the computational efficiency is also analyzed. It is shown that this novel PDNN framework can completely remove the need of labeled training data and exhibit improved performances as compared to the conventional physics-informed neural networks (PINNs) in terms of avoiding unphysical solutions and attaining higher computational efficiency. PDNNs can thus serve as a promising tool to solve some critical challenges associated with a wide range of nonlinear micromechanical problems.

Two ways to estimate precipitate elastic fields through Eshelby inclusion theory

The stress-free misfit strain obtained from simulation, neglecting the constraint from the matrix phase, serves as the inhomogeneity eigenstrain. On the other hand, the experimentally measured misfit strain under the constraint of the matrix phase serves as the constrained strain. This study demonstrates two ways to calculate the equivalent eigenstrain for precipitates through Eshelby inclusion theory: the inhomogeneity eigenstrain way (IEW) and the constrained strain way (CSW).

Energy release rate for steady-state fiber debonding in structural battery composites

Structural battery composites are multifunctional materials intended to provide energy storage capacity while maintaining their strength and load bearing capacity under significant mechanical loads. In this study, we investigate the mechanics of carbon fiber debonding, a critical failure mechanism in structural battery composites. The carbon fibers are intended to serve both as an electrochemically active material and to bear mechanical load in the composite system. We derive an analytical solution to the energy release rate for steady-state fiber debonding for different electrochemical and mechanical loading cases with the aid of the classical solution for the Eshelby inclusion problem. The analytical solutions are validated with finite element simulations. We find a higher energy release rate and thus a greater driving force for fiber debonding is caused either by a lower lithium concentration in the fiber and/or by greater transverse mechanical loads such as biaxial tension or shear applied at the far field in the matrix. We find that the model is predictive even for transversely isotropic fibers despite the assumption that the fiber is elastically isotropic in the model. This work can provide guidance for the design of mechanically robust structural batteries.

Modeling a sample size-dependency of martensitic phase transformation using a mesoscale framework

Predicting a sample size-dependency of the constitutive response of any material system when it is being used in nano- and micro-scale devices, such as NEMS and MEMS, is very crucial for their design process. This also includes the shape memory alloys (SMAs), where predicting a sample size-dependency of the phase transformation process is the key aspect. Many experimental studies with micropillar compression tests showed an increase in critical stress to transformation and an increase in transformation hardening as the sample size decreases below a critical size. In this work, we have employed and extended a recently developed mesoscale framework to model such size effects in SMAs. A plane strain tensile SMA strip of finite width is analyzed within a small deformation framework. The martensitic transformation regions are modeled as Eshelby inclusions in an austenite phase matrix. Circular potential nucleation sites are considered with a square packing for their spatial locations and with nucleation stress assigned to each site from a Gaussian distribution. The combination of the Gaussian distribution of nucleation stress and the size of the nucleation site represents the underlying SMA material microstructure. Our numerical analysis, with varying values of the aspect ratio between the strip width and the nucleation site size, provides a critical value of the aspect ratio characterizing a transitioning length scale between a bulk vs. a sample size-dependent phase transformation behavior. At the transition, the critical stress to phase transformation and the transformation hardening increase drastically as the aspect ratio decreases below its critical value. The limitations of these predictions are discussed with existing literature on experimental observations, and some perspectives to improve the model are stated.

Polynomial inclusions: Definitions, applications, and open problems

Predictive modeling in physical science and engineering is mostly based on solving certain partial differential equations where the complexity of solutions is dictated by the geometry of the domain. Motivated by the broad applications of explicit solutions for spherical and ellipsoidal domains, in particular, the Eshelby’s solution in elasticity, we propose a generalization of ellipsoidal shapes called polynomial inclusions. A polynomial inclusion (or p-inclusion for brevity) of degree k is defined as a smooth, connected and bounded body whose Newtonian potential is a polynomial of degree k inside the body. From this viewpoint, ellipsoids are identified as the only p-inclusions of degree two; many fundamental problems in various physical settings admit simple closed-form solutions for general p-inclusions as for ellipsoids. Therefore, we anticipate that p-inclusions will be useful for applications including predictive materials models, optimal designs, and inverse problems. However, the existence of p-inclusions beyond degree two is not obvious, not to mention their explicit algebraic parameterizations.In this work, we explore alternative definitions and properties of p-inclusions in the context of potential theory. Based on the theory of variational inequalities, we show that p-inclusions do exist for certain polynomials, though a complete characterization remains open. We reformulate the determination of surfaces of p-inclusions as nonlocal geometric flows which are convenient for numerical simulations and studying geometric properties of p-inclusions. In two dimensions, by the method of conformal mapping we find an explicit algebraic parameterization of p-inclusions. We also propose a few open problems whose solution will deepen our understanding of relations between domain geometry, Newtonian potentials, and solutions to general partial differential equations. We conclude by presenting examples of applications of p-inclusions in the context of Eshelby inclusion problems and magnet designs.

A Hall-Petch-like relationship linking nanoscale heterogeneity to yield stress of heterogeneous metallic glasses

Metallic glasses (MGs) undergo plastic deformation through shear transformation zones (STZs) at small deformations and shear banding at large deformations. The STZ mechanism is linked to microstructural heterogeneities, including nano-heterogeneities associated with atom clustering and their spatial distribution. Shear banding occurs due to the coalescence of STZs. In this study, we investigate the yielding behavior of heterogeneous MGs by varying the spatial correlation and standard deviation of local shear moduli associated with clustering atoms on the nanoscale. Using a mesoscale shear transformation zone (STZ) dynamics model, we compute the yielding strength and observe deformation behaviors associated with the formation and propagation of STZs. We propose a Hall-Petch-like relationship where the yield stress of the MG scales inversely with the square root of the spatial correlation length. Our results show that the yielding of MGs corresponds to the percolation of STZs from soft to hard regions. Prior to macroscopic yielding, the activated STZs tend to accumulate inside soft regions, forming a Super-STZ array, which is similar to the formation of dislocation pile-up at grain boundaries. We derive the stress concentration in the front of the Super-STZ using the Eshelby inclusion model and formulate a Hall-Petch-like equation to quantify the yield of MGs with the spatial correlation length and standard deviation of the nanoscale heterogeneity. Our results provide insights into the structure-property relationship of MGs and have important implications for the design of nanoscale MGs with tunable properties.

Effect of a Long-Range Dislocation Pileup on the Atomic-Scale Hydrogen Diffusion near a Grain Boundary in Plastically Deformed bcc Iron

In this paper, we present concurrent atomistic-continuum (CAC) simulations of the hydrogen (H) diffusion along a grain boundary (GB), nearby which a large population of dislocations are piled up, in a plastically deformed bi-crystalline bcc iron sample. With the microscale dislocation slip and the atomic structure evolution at the GB being simultaneously retained, our main findings are: (i) the accumulation of tens of dislocations near the H-charged GB can induce a local internal stress as high as 3 GPa; (ii) the more dislocations piled up at the GB, the slower the H diffusion ahead of the slip–GB intersection; and (iii) H atoms diffuse fast behind the pileup tip, get trapped within the GB, and diffuse slowly ahead of the pileup tip. The CAC simulation-predicted local H diffusivity, Dpileup−tip, and local stresses, σ, are correlated with each other. We then consolidate such correlations into a mechanics model by considering the dislocation pileup as an Eshelby inclusion. These findings will provide researchers with opportunities to: (a) characterize the interplay between plasticity, H diffusion, and crack initiation underlying H-induced cracking (HIC); (b) develop mechanism-based constitutive rules to be used in diffusion–plasticity coupling models for understanding the interplay between mechanical and mass transport in materials at the continuum level; and (c) connect the atomistic deformation physics of polycrystalline materials with their performance in aqueous environments, which is currently difficult to achieve in experiments.

Elastic modelling of lattice distortions in concentrated random alloys

Lattice distortions, i.e. small atomic displacements away from the average lattice, are linked to a number of functional and mechanical properties of concentrated metallic alloys, particularly their yield strength. Here we develop an elastic model of lattice distortions where every atom is modeled as an Eshelby inclusion in a homogeneous elastic matrix. Local environment effects are included by considering fluctuating anisotropic eigenstrain tensors associated to the inclusions. The model is tested on several concentrated alloys, face-centered cubic (FCC) AlMg and FeNiCr alloys, as well as a fictitious body-centered cubic (BCC) binary alloy to study systematically and independently the effects of a size and an elastic modulus mismatch between the constituents. The elastic model predicts lattice distortions well when the size and elastic modulus mismatches are typically less than 5% and 25%, respectively. Interestingly, we find that when the size mismatch is small, as in FeNiCr alloys, the lattice distortion is dominated by the fluctuations of the dilatational eigenstrains rather than their average value as often assumed in elastic models of concentrated alloys. Moreover, models usually assume homogeneous elastic constants, while the limit obtained here of 25% is often exceeded in concentrated alloys, particularly with a BCC structure.

Analytic solutions for displacements in quantum-wire structures

Quantum wires (QWs) and quantum dots (QDs) have been widely applied in semiconductor devices due to their excellent mechanical, electronic, and optical properties. Faux and Downes [J. Appl. Phys. 82 (1997) 3754–3762] have obtained the closed-form solutions for strain distributions produced by QWs, whose cross section is composed of any combination of line elements and circular arcs. In this paper, Eshelby's inclusion model is established to simulate QWs and the closed-form solutions for the resultant displacements are obtained. By employing the method of Green's function, the displacement solutions may be formulated as area integrals and then converted into contour integrals along the boundary of the QW. The present study complements Faux and Downes' work and provides an efficient shortcut for analyzing the displacements of a QW, whose boundary may be discretized into line segments and circular arcs.

Direct observation of quadrupolar strain fields surrounding Eshelby inclusions in metallic glasses.

For decades, scanning/transmission electron microscopy (S/TEM) techniques have been employed to analyze shear bands in metallic glasses and understand their formation in order to improve the mechanical properties of metallic glasses. However, due to a lack of direct information in reciprocal space, conventional S/TEM cannot characterize the local strain and atomic structure of amorphous materials, which are key to describe the deformation of glasses. For this work, we applied 4-dimensional STEM to map and directly correlate the local strain and the atomic structure at the nanometer scale in deformed metallic glasses. We observe residual strain fields with quadrupolar symmetry concentrated at dilated Eshelby inclusions. The strain fields percolate in a vortex-like manner building up the shear band. This provides a new understanding of the formation of shear bands in metallic glass. This article is protected by copyright. All rights reserved.

A mesoscopic damage model for the low-cycle fatigue of an extruded magnesium alloy

This paper presents a novel mesoscopic damage model to characterize the low-cycle fatigue damage evolution of an extruded AZ31 magnesium (Mg) alloy, taking into account the effect of twinning. The damage caused by the slip bands (SBs)–twin boundaries (TBs) and SBs–grain boundaries (GBs) interactions is treated based on the Tanaka–Mura model and the Eshelby inclusion theory. Strain energy values at the TBs and GBs are defined as the TB and GB damage variables, respectively. Explicit formulae for the TB and GB damage evolution are derived from the characteristics of the {101¯2} extension twin and the basal texture. A fracture energy-based crack initiation criterion is established, and the proposed damage model is validated against the existing experimental results. The model demonstrates its ability to reproduce the damage evolution processes, and the predictions of the crack initiation life are within the twice error band.