Eshelby Model Research Articles

Effect of a micro-scale dislocation pileup on the atomic-scale multi-variant phase transformation and twinning

In this paper, we perform concurrent atomistic–continuum (CAC) simulations to assess the contribution of the internal stress induced by the microscale dislocation pileup at an atomically structured interface to the atomic-scale phase transformations (PTs), reverse PTs, and twinning. The main novelty of this work is to unify the atomistic description of the interface and the coarse-grained (CG) description of the lagging dislocations away from the interface within one single framework. Our major findings are: (a) the interface dynamically responds to a pileup by forming steps/ledges, the height of which is proportional to the number of dislocations arriving at the interface; (b) the pileup-induced internal stress concentration profile follows neither the classical Eshelby model nor the super-dislocation model alone, but a combination of them; (c) when the pre-sheared sample is compressed, a direct square-to-hexagonal PT occurs ahead of the pileup tip and eventually grows into a wedge shape. The two variants of the hexagonal phases form a twin with respect to each other; (d) upon a further increase of the loading, part of the newly formed hexagonal phase transforms back to the square phase. The square product phase resulting from this reverse PT forms a twin with respect to the initial square phase. All phase boundaries (PBs) and twin boundaries (TBs) are stationary and correspond to zero thermodynamic Eshelby driving forces; and (e) the microscale dislocation pileup-induced internal shear stress and the structural change at the atomic-scale interface reduces the stress required for initiating a PT by a factor of 5.5, comparing with that in the sample containing no dislocations. This work is the first characterization of the behavior of PTs/twinning resulting from the reaction between a microscale dislocation slip and an atomically structured interface. The gained knowledge will advance our understanding of how the multi-phase material behaves in many complex physical processes, such as the synthesis of multi-phase high-entropy alloys or superhard ceramics under high-pressure torsion, deep mantle earthquakes in geophysics, and so on, which all involve dislocation slip, PTs, twinning, and their interactions across from the atomistic to the microscale and beyond.

Buckling and Free Vibration Analyses of Various Nanoparticle Reinforced Concrete Beams Resting on Multi-Parameter Elastic Foundations.

Given their considerable specific surface area and amorphous characteristics, nanoparticles exhibit excellent pozzolanic activity, and when undergoing a reaction with calcium hydroxide, this leads to the generation of a denser matrix by promoting the formation of a greater amount of C-S-H gel, thereby enhancing the strength and durability of the concrete and fortifying the overall structure. Indeed, the present study investigates a comparative study of the buckling and free vibration analyses of concrete beams reinforced with various types of nanoparticles. For its simplicity and accuracy, a higher-order shear deformation theory will be used to analytically model the reinforced concrete beam. Furthermore, the powerful Eshelby's model is used to derive the equivalent nanocomposite properties. The soil medium is simulated with Pasternak elastic foundation, including a shear layer, and Winkler's spring, interlinked with a Kerr foundation. The motion equations are derived using Hamilton's principle. Moreover, based on Navier's analytical methods, the closed-form solutions of simply supported beams have been obtained. Different parameters, such as type and volume percent of nanoparticles, geometrical parameters, choice of theory and soil medium, on the buckling and dynamic behavior of the beam, are exercised and shown. The major findings of this work indicate that the use of nanoparticles in concretes increases better mechanical resistance and amplifies the natural frequencies. In addition, the elastic foundation has a significant impact on the buckling and vibration performances of concrete beams.

Thermomechanical Analysis of Glass Powder Based Eco-concrete Panels: Limitations and Performance Evaluation

This article presents a comprehensive investigation into the thermomechanical analysis of glass powder as an additive in concrete. The efficient Eshelby's model is utilized to determine the relevant composite properties, considering the spherical shape of the glass powder. A higher-order shear deformation plate theory is employed to theoretically simulate the reinforced concrete panel, ensuring accuracy and simplicity. Equilibrium equations are derived using the virtual work concept, and energy equations are derived using Hamilton's principle. Navier's analytical techniques are applied to obtain closed-form solutions for simply supported plates. A comprehensive parametric study is conducted, analyzing the impact of factors such as glass powder volume, geometric parameters, and thermal loading on the thermomechanical behavior of the panel. The findings highlight the challenges associated with using glass powder in concrete for thermomechanical applications and suggest the need for alternative approaches to optimize its effectiveness in such scenarios, also the study provides first-time numerical results that serve as guidelines for further research on reinforced concrete.

The mechanism for an orientation dependence of grain boundary strengthening in pure titanium

The Hall-Petch slope k represents the magnitude of grain boundary strengthening. For the first time, a strong orientation dependence of the k for pure titanium (Ti) is reported in the present study, namely, a much lower k for the TD-tension of a Ti plate (188 MPa μm1/2) than the k values for the RD-tension of the Ti plate (358 MPa μm1/2), SD-tension (369 MPa μm1/2) and SD-compression (397 MPa μm1/2) of a Ti rod. Here, the RD and TD are respectively the rolling direction and transverse direction of the plate, and SD is the axial direction of the rod, while tension and compression stand for uniaxial tension and compression, respectively. It is found that the mechanisms reported previously cannot explain the orientation dependence of k experimentally observed in the present study. A new mechanism is proposed by combining crystal plasticity finite element modeling and the Eshelby model for the stress concentration at grain boundaries. The results indicate that an orientation-mediated deformation-transfer is the main contributor to this orientation effect on k. More specifically, for the RD-tension, SD-tension and SD-compression, a single slip system is predominant in most grains when plastic deformation-transfer occurs. In contrast, multiple slip systems are activated in the majority of grains in the TD-tension. The activation of additional slip systems remarkably reduces the stress concentration at grain boundaries, leading to a lower k. Afterward, the reasons for the orientation-mediated deformation transfer behavior are discussed.

Creep of particle and short fibre reinforced polyurethane rubber

Tensile stress–strain testing and creep testing have been carried out on a polyurethane rubber, at three temperatures, with and without either particulate or short fibre alumina reinforcement. A previous paper reported concerning composites with particulate reinforcement and the present work is focused on the effect of the fibres. The samples were made via a blending and extrusion process that produced a certain degree of fibre alignment (along the direction of loading). Prior milling procedures were used to produce fibres with two different ranges of aspect ratio (with averages about 10 and 16). When expressed as true stress–strain relationships, all materials exhibit approximately linear responses. The dependence of stiffness on the volume fraction and aspect ratio of the reinforcement was found to conform well to the Eshelby model predictions. Moreover, the creep behaviour of all of the materials can be captured well by a Miller–Norton formulation, using the average matrix stress predicted by the Eshelby model. A striking conclusion is that it is both predicted and observed that short fibres are much more effective in reducing the creep rate than is the case with particles.

Hybrid model of nonlinear homogenisation of anisotropic composites with ellipsoidal inclusions

A hybrid model of nonlinear homogenisation of anisotropic composites was developed, based on the secant Eshelby's model of the second order. At first a scaling factor was determined between constrained strains of matrices of finite and infinite dimensions for an isotropic composite. The scaling factor was then used to calculate the constrained strain of the matrix and the macrostress of the anisotropic composite. If the calculation results and reference values differ greatly, the scaling factor serves as a fitting parameter. A parametric hybrid model has been developed to predict the properties of composites with different content of inclusions. Good agreement of the calculation results with the numerical solutions of the FEM and experimental data was obtained.

Residual stresses in additively manufactured AlSi10Mg: Raman spectroscopy and X-ray diffraction analysis

Metal parts produced by Additive Manufacturing, and in particular Laser Powder Bed Fusion (LPBF), suffer from residual stresses due to high thermal gradients causing cyclic expansion and contraction of the alloy.This work deals with the determination of residual stress in rapidly solidified AlSi10Mg samples using two non-destructive techniques: Raman spectroscopy, rapid, unconventional but applicable to AlSi alloys, and XRD (ω-method), used as benchmark, being a classical method for determining residual stresses. Al stress level was studied by XRD both on the surface of LPBF samples and in the interior, after in-depth sectioning. Raman was employed to assess the stress on Si. The effect of particle size and stress on the Raman was separated determining the size distribution of Si particles, making Raman suitable to study residual stresses in alloys containing free Si.Al and Si stresses were evaluated also by means of the Williamson-Hall method: stresses are of tensile type with agreement among all methods. Considering the alloy as a composite, stress on Si was estimated using the Eshelby's model, showing that larger eutectic particles undergo lower stress with respect to nanometric precipitates.

Modeling for onset strain of deformation-induced martensite transformation in Q&P steel by a mean-field Eshelby approach

Based on Eshelby model, a multi-domain incremental mean-field method was revised to investigate the microstructure effect on the onset strain of deformation-induced martensite transformation (DIMT) in Q&P980 steel, which is designed to show good energy absorbability when the car meets with an accident. Inputting elastoplastic numerical constitutive relationships of both particulate retained austenite (RA) and primary martensite innovatively, this model predicts the overall steel stress–strain curve successfully so that it can calculate stress evolution in any constituent phase in the steel during straining. The model is used to eventually examine the onset strain of DIMT determined by a stress criterion of martensite transformation as a function of far-field strain and a numerical relationship between transformation fraction of new-born martensite from DIMT and the applied strain is fitted out to quantify the amount of DIMT in the steel. The effects of volume fraction and aspect ratio of the original RA and primary martensite on the onset strain of DIMT are quantitatively distinguished. The microstructure effect on the rest fraction of the RA during straining is also examined. These results are to supply guidance by varying microstructure of the Q&P steel to achieve high onset strain and low speed in DIMT to protect the RA during automobile manufacture.

Viscoelastic characteristics of short fiber reinforced polybutylene terephthalate

The mechanical characteristics of short-fiber-reinforced thermoplastics should be evaluated based on viscoelastic theory because of the time and temperature dependent characteristics of the matrix resin. In this study, the viscoelastic characteristics of short-fiber-reinforced polybutylene terephthalate, PBT composite, are evaluated by material testing and micromechanics model. Firstly, tensile tests are carried out PBT composite with three fiber volume ratios under various temperature and strain rate conditions. The results show that the viscoelastic characteristics of the PBT composites can be obtained by the time-temperature superposition principle using shift factor of the PBT resin. Next, the Eshelby model combined with the Mori–Tanaka theory is used to predict to the viscoelastic characteristics of the PBT composites based on the measurement of fiber length and fiber orientation angle. The predicted viscoelastic characteristics of the PBT composites are in good agreement with the tensile test.

Implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for any crystal symmetry

In this paper, we will report about the implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for general, triclinic crystal symmetry. With applying appropriate symmetry relations, the point groups of higher crystal symmetries are covered as well. This simplifies the implementation effort to cover the calculations for any crystal symmetry. In the literature, several models can be found to estimate the polycrystalline elastic properties from single crystal elastic constants. In general, this is an intermediate step toward the calculation of the polycrystalline response to different techniques using X-rays, neutrons, or ultrasonic waves. In the case of X-ray residual stress analysis, the final goal is the calculation of X-ray Elastic constants. Contrary to the models of Reuss, Voigt, and Hill, the Kröner–Eshelby model has the benefit that, because of the implementation of the Eshelby inclusion model, it can be expanded to cover more complicated systems that exhibit multiple phases, inclusions or pores and that these can be optionally combined with a polycrystalline matrix that is anisotropic, i.e., contains texture. We will discuss a recent theoretical development where the approaches of calculating bounds of Reuss and Voigt, the tighter bounds of Hashin–Shtrikman and Dederichs–Zeller are brought together in one unifying model that converges to the self-consistent solution of Kröner–Eshelby. For the implementation of the Kröner–Eshelby model the well-known Voigt notation is adopted. The 4-rank tensor operations have been rewritten into 2-rank matrix operations. The practical difficulties of the Voigt notation, as usually concealed in the scientific literature, will be discussed. Last, we will show a practical X-ray example in which the various models are applied and compared.

Simulation of stress distribution and evolution in SiC particle reinforced iron matrix composites

An Eshelby particle-compounded model was employed to investigate the stress distribution in the SiCp/Fe composites during straining in order to predict the stress-strain curves of the composites. Effect of the SiC volume content and aspect ratios on yield strength of the SiCp/Fe composites was studied by the simulation. The modeling results indicate that the stress in the iron matrix is much smaller than that in SiC particles as straining the composites, which means the strengthening mechanisms of the SiCp/Fe composites is determined by the load transfer mechanisms. The aspect ratios of the particles play a significant role in yield strength of the SiCp/Fe composites especially as the particle volume content is large. The stress in SiC particles has a stronger dependence on the aspect ratios of the particles only if the reinforcement volume content is small.

Predicting the mechanical properties of ordinary concrete and nano-silica concrete using micromechanical methods

By combining several materials with specific mechanical properties, new materials with unknown mechanical properties are obtained. Various experiments are required to determine the mechanical properties of the produced composite materials. Since conducting experiment processes is costly and time-consuming, comprehensive studies have been conducted in recent years to solve the problem. Fortunately, it is possible to easily predict the mechanical properties of composite materials without the need to construct them, by inspecting their constituent’s properties using micromechanical methods. Although various micromechanical methods have been presented so far, few of them yielded precise predictions of the properties of composite materials. Therefore, selecting a method suitable to predict the properties of composite materials is of much importance. In this study, some micromechanical approaches, including Hirsch, Hansen, Bache, Cavento, Mori–Tanaka, Eshelby, self-consistent, effective interface and double-inclusion models, were employed for the estimation of elasticity modulus and Poisson’s ratio of ordinary and nanomaterial concretes. The results obtained from the micromechanical methods were compared to those obtained from experimental tests. The obtained numerical results showed that Bache’s model is the most accurate micromechanics model for predicting the elastic mechanical properties of ordinary and nanomaterial concretes.

Micromechanical Constitutive Equations for the Effective Thermoelastic Properties of Carbon Nanotube-Reinforced Composites

Predicting thermomechanical properties of composites containing carbon nanotubes (CNTs) is significantly depending on the assumed microstructural parameters (MSPs) of CNTs and CNT/matrix morphology. These MSPs include geometry, dispersion and orientation. On the other hand, CNT/matrix morphology refers to two microstructural observations. The first is whether or not an interphase exists between CNTs and matrix, whereas the second is whether or not voids exist due to, for example, debonding of CNTs. In this work, the aim is to propose micromechanical constitutive equations, which are based on the micromechanics principles of Eshelby and Mori-Tanaka models, for considering all of these MSPs altogether in addition to the other well-known MSPs. Accordingly, these equations can be used for modeling realistic nanocomposites to predict their effective thermomechanical properties in different directions. The obtained computational results are compared with other results of both experimental and theoretical investigations found in the literature, and good agreement is obtained.

Uncertainty quantification of fiber orientation distribution measurements for long-fiber-reinforced thermoplastic composites

We present a detailed methodology for experimental measurement of fiber orientation distribution in injection-molded discontinuous fiber composites using the method of ellipses on two-dimensional cross sections. Best practices to avoid biases occurring during surface preparation and optical imaging of carbon-fiber-reinforced thermoplastics are discussed. A marker-based watershed transform routine for efficient image segmentation and the separation of touching fiber ellipses is developed. The sensitivity of the averaged orientation tensor to the image sample size is studied for the case of long-fiber thermoplastics. A Mori–Tanaka implementation of the Eshelby model is then employed to quantify the sensitivity of elastic stiffness predictions to biases in the fiber orientation distribution measurements.

Reliability in the characterization of fiber length distributions of injection molded long carbon fiber composites

The mechanical properties of a discontinuous fiber reinforced composite material are dependent on the length of the reinforcing fibers. Long‐fiber thermoplastic (LFT) composites offer a compromise between ease of manufacture and mechanical properties for structural applications and are conventionally made using fibers that are initially 10–14 mm long. However, fiber attrition during the injection molding process significantly reduces the final fiber lengths in molded composite and produces a fiber length distribution (FLD), which depends on the processing parameters. Thus, accurate FLD measurements are important for reliable predictions of the composite properties. Here, we present a detailed methodology for experimentally measuring the FLD in LFT composites based on best practices. Possible biases occurring during the measurement process are discussed and their effect on the measured FLD and averaged fiber length values are evaluated. We then employ a Mori–Tanaka implementation of the Eshelby model to predict the elastic stiffness of the LFT composite and analyze the sensitivity of the stiffness values to the biases in the FLD measurements. POLYM. COMPOS., 39:4594–4604, 2018. © 2017 Society of Plastics Engineers

Finite element estimates of viscoelastic stiffness of short glass fiber reinforced composites

An unstructured mesh Galerkin finite element method is used to obtain estimates of the viscoelastic moduli of short glass fiber reinforced polymer composites. Periodic Monte Carlo models with 125 identical inclusions are studied. Both spheroidal and spherocylindrical inclusions are considered. The estimates are compared against predictions of the dilute approximation, Mori-Tanaka (MT) and self-consistent (SC) models. It is shown that at small fiber fractions, the dilute approximation (Eshelby) model is exact. However, for the axial stiffness the dilute regime is limited to fiber volume loadings of a few tens of a percent while typical short glass fiber polymer composites have fiber loadings from 10 to 20 percent. It is found that in this concentrated regime, both MT and SC models give excellent predictions for all but the axial stiffness modulus. To assess the feasibility of reliable stiffness and vibration damping design of composite structures from short fiber reinforced polymers, Monte Carlo models with various fiber orientation distribution (FOD) states are studied. It is shown that the quick Voigt (constant strain) orientation averaging procedure gives excellent viscoelastic stiffness predictions provided that the finite element estimates are used for the required moduli of the basis FOD state with fully aligned fibers.

Morphological explanation of high tear resistance of EPDM/NR rubber blends.

The fatigue properties of cross-linked blends of ethylene propylene diene rubber (EPDM) with low natural rubber (NR) content and reinforced with carbon black (CB) are studied. It is found that such EPDM/NR compounds have superior crack growth resistance and fatigue lifetime. For low NR contents, transmission electron microscopy reveals that the NR phase forms small droplets of 20-50 nm. Remarkably, these droplets are even smaller than the primary CB particles. Atomic force microscopy shows that the the NR phase droplets have a higher loss factor and a smaller elastic modulus than the surrounding EPDM matrix. Rheometer measurements are used to study the effect of the phase morphology on the rubber mechanical properties. These rheological data are compared with the prediction of the Eshelby model describing the effect of elastic inclusions on solids. A complex interplay between the rubber phase morphology and the solubility of both the sulfur cross-linking system and CB is observed, which cannot be predicted theoretically. It is proposed that the soft NR droplets effectively inhibit the crack propagation in the EPDM matrix.

The effect of internal stresses due to precipitates on twin growth in magnesium

Twinning is an important deformation mode in hexagonal close packed (HCP) metals, including magnesium alloys. Precipitates are used to provide strengthening in many of these alloys. The effect of precipitates in strengthening against deformation by slip is well understood, but this is not the case for twinning. Recent studies have indicated that precipitates are usually not sheared by twins, but the Orowan law for strengthening against slip by dislocation bowing does not give a good prediction when applied to twinning. It has therefore been proposed that the dominant strengthening effect inhibiting thickening of a twin arises from an additional back-stress that results from embedding a unsheared precipitate in twinned matrix. The present paper uses an Eshelby model to assess the influence of precipitate shape and habit on the internal stresses that arise from embedding a non-shearing precipitate in a {101¯2} twin (the dominant twin type). It is demonstrated that the elastic stresses generated easily exceed the critical resolved shear stress for activation of slip and therefore plastic relaxation is to be expected. In all cases, the predicted plastic zone is confined to a region local to the particle. The implications of these predictions for design of precipitation strengthened HCP alloys are discussed.

Load partitioning between the bcc-iron matrix and NiAl-type precipitates in a ferritic alloy on multiple length scales.

An understanding of load sharing among constituent phases aids in designing mechanical properties of multiphase materials. Here we investigate load partitioning between the body-centered-cubic iron matrix and NiAl-type precipitates in a ferritic alloy during uniaxial tensile tests at 364 and 506 °C on multiple length scales by in situ neutron diffraction and crystal plasticity finite element modeling. Our findings show that the macroscopic load-transfer efficiency is not as high as that predicted by the Eshelby model; moreover, it depends on the matrix strain-hardening behavior. We explain the grain-level anisotropic load-partitioning behavior by considering the plastic anisotropy of the matrix and elastic anisotropy of precipitates. We further demonstrate that the partitioned load on NiAl-type precipitates relaxes at 506 °C, most likely through thermally-activated dislocation rearrangement on the microscopic scale. The study contributes to further understanding of load-partitioning characteristics in multiphase materials.

Hierarchical modelling of elastic behaviour of human enamel based on synchrotron diffraction characterisation

Human enamel is a hierarchical mineralized tissue with a two-level composite structure. Few studies have focused on the structure-mechanical property relationship and its link to the multi-scale architecture of human enamel, whereby the response to mechanical loading is affected not only by the rod distribution at micro-scale, but also strongly influenced by the mineral crystallite shape, and spatial arrangement and orientation. In this study, two complementary synchrotron X-ray diffraction techniques, wide and small angle X-ray scattering (WAXS/SAXS) were used to obtain multi-scale quantitative information about the structure and deformation response of human enamel to in situ uniaxial compressive loading. The apparent modulus was determined linking the external load and the internal strain in hydroxyapatite (HAp) crystallites. An improved multi-scale Eshelby model is proposed taking into account the two-level hierarchical structure of enamel. This framework has been used to analyse the experimental data for the elastic lattice strain evolution within the HAp crystals. The achieved agreement between the model prediction and experiment along the loading direction validates the model and suggests that the new multi-scale approach reasonably captures the structure-property relationship for the human enamel. The ability of the model to predict multi-directional strain components is also evaluated by comparison with the measurements. The results are useful for understanding the intricate relationship between the hierarchical structure and the mechanical properties of enamel, and for making predictions of the effect of structural alterations that may occur due to the disease or treatment on the performance of dental tissues and their artificial replacements.