Eshelby's Inclusion Method Research Articles

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.

Anisotropy profoundly alters stress fields within contractile cells and cell aggregates.

Many biological phenomena such as cell proliferation and death are correlated with stress fields within cells. Stress fields are quantified using computational methods which rely on fundamental assumptions about local mechanical properties. Most existing methods such as Monolayer Stress Microscopy assume isotropic properties, yet experimental observations strongly suggest anisotropy. We first model anisotropy in circular cells analytically using Eshelby's inclusion method. Our solution reveals that uniform anisotropy cannot exist in cells due to the occurrence of substantial stress concentration in the central region. A more realistic non-uniform anisotropy model is then introduced based on experimental observations and implemented numerically which interestingly clears out stress concentration. Stresses within the entire aggregate also drastically change compared to the isotropic case, resulting in better agreement with observed biomarkers. We provide a physics-based mechanism to explain the low alignment of stress fibers in the center of cells, which might explain certain biological phenomena e.g., existence of disrupted rounded cells, and higher apoptosis rate at the center of circular aggregates.

Internal Stress and Dislocation Interaction of Plate-Shaped Misfitting Precipitates in Aluminum Alloys.

The fine misfit precipitates in age-hardenable aluminum alloys have important roles due to their excellent age-hardening ability, by their interaction with dislocations. The present study focused on the internal stress field of plate-shaped misfitting precipitates to evaluate their roles in dislocation overcoming the precipitates by means of micromechanics based on Green’s function method. The stress field of misfit precipitates on {001} and {111} habit planes were reproduced by homogeneous misfit strain (eigenstrain) of the precipitate (Eshelby inclusion method), and the dislocation motion vector on the primary slip plane was predicted by the force acted on the dislocation by the Peach–Koehler formula. According to simulation results, the dislocation interaction strongly depends on the stress field and geometry of misfit precipitates; repulsive and attractive forces are operated on the dislocations lying on the primary slip plane when the dislocation approaches the misfit precipitates. The hardening ability of different orientations of precipitation variants was discussed in terms of interaction force acted on the dislocation.

Average Eshelby tensor and elastic field for helical inclusion problems

Elastic field of helical inclusion and average Eshelby tensor for helix, <Sh>, with various different helical parameters (p: pitch, d: core diameter, D: outer diameter, L: longitudinal length, n: number of turns) were predicted by two types of Green's function method, Eshelby inclusion method (EIM) and boundary integral equation method (BIE). Helical inclusion model revealed that the effects of helical pitch was found to have the most significant influence on 〈Sh〉, and resultant stress obtained by average Eshelby tensor model for helix (AETM) through EIM was comparable to that of spherical inclusion at the increased number of helical turn and aspect ratio, α = L/D. Local stress analysis by EIM and BIE under a given dilatational eigenstrain proved that the stress profile of helical cross-section shows large stress gradient in angular and radial stress components, and their equivalent stress on helical cross-section are largely different from the stress of entire helical volume computed by AETM, which reflect the influence of the rotational stress symmetry of helical inclusion on average stress. Finally, thermal residual stress of Fe0.7Pd0.3 nano-helix embedded in alumina template was predicted by equivalent inclusion method using AETM for nano-helix.

Evolution of Strain Energy During Recrystallization of Plated Cu Films

Microstructural evolution within thin films is dictated by energy minimization that can arise due to different mechanisms. Conventional treatments that estimate the driving force associated with elastic strain energy often employ a fiber-textured microstructure to arrive at an analytical solution. By approximating the case of a recrystallized grain in a randomly oriented film as an elastically anisotropic inclusion in an elastically isotropic matrix, we can apply Eshelby's inclusion method to calculate both the interaction strains and the strain energy density generated by this elastic incompatibility. A comparison of these two approaches for grains with cubic symmetry reveals that a lower amount of elastic strain energy is generated in the case of an elastically anisotropic grain in an isotropic matrix, suggesting that it is less energetically favorable for Cu (111) grains to recrystallize in films possessing strong (111) texture than in randomly textured films. High-resolution X-ray diffraction measurements are used to extract the difference in the elastic response between recrystallized grains and the untransformed matrix, providing experimental quantification of the induced interaction strains. In addition, the corresponding elastic strain energy values are compared with the appearance of recrystallized (100) grains as a function of film thickness, allowing for a threshold energy density to be extracted. This threshold energy density can be correlated to the difference in surface energy values between (100)- and (111)-oriented grains.

Study on crack-inclusion interaction using digital gradient sensing method

ABSTRACTThe interaction between matrix crack and a round inclusion was studied by the method of digital gradient sensing. First, the stress fields at the matrix crack tip in the neighbor of a round inclusion were derived based on transformation toughening theory and Eshelby inclusion method, and the effect of the inclusion on the stress intensity factor of the matrix crack was analyzed. Then, the non-contact optical measurement system of digital gradient sensing was built up, and a three-point bending test was carried out using a single-edge cracked specimen. The mode I stress intensity factor was extracted from the angular deflection of the light rays. Finally, the effect of the inclusion on the angular deflection fields and the stress intensity factor at the crack tip was analyzed experimentally. These results will play an important role for evaluating the fracture mechanism of crack-inclusion interaction in composites.

Crack–inclusion interaction due to mismatched thermal expansion under plane stress condition

As the temperature changes in composites molding, mico-cracks are observed due to the mismatch expansion for the reinforcement and medium. An approximate expression for the change of the stress intensity factor near a Mode I crack is proposed with the consideration of the reinforcement as an inclusion that is subjected to the coupled mechanical and thermal loads under the plane stress condition. The transformation toughening theory and Eshelby inclusion method is applied in the analysis, and the mismatched thermal expansion strain is employed as the additional eigen strain. The finite element analysis is then carried out to validate the developed formula. The effect of the temperature-dependence of material properties on the crack behavior is estimated and discussed. It is found that the mismatched expansion coefficient plays an important role on the crack–inclusion interaction under the free boundary condition.

Stress determination through diffraction: establishing the link between Kröner and Voigt/Reuss limits

The quantification of stress in polycrystalline materials by diffraction-based methods relies on the proper choice of grain interaction model that links the observed strain to the elastic stress state in the aggregate. X-ray elastic constants (XEC) relate the strain as measured using X-rays to the state of stress in a quasi-isotropic ensemble of grains. However, the corresponding interaction models (e.g., Voigt and Reuss limits) often possess unlikely assumptions as to mechanical response of the individual grains. The Kröner limit, which employs a self-consistent scheme based on the Eshelby inclusion method, is based on a more physical representation of isotropic grain interaction. For polycrystalline aggregates composed of crystals with cubic symmetry, Kröner limit XEC are equal to those calculated from a linear combination of Reuss and Voigt XEC, where the weighting fraction, xKr, is solely a function of the single-crystal elastic constants and scales with the material's elastic anisotropy. This weighting fraction can also be experimentally determined using a linear, least-squares regression of diffraction data from multiple reflections. Data on metallic thin films reveals that this optimal experimental weighting fraction, x*, can vary significantly from xKr, as well as that of the Neerfeld limit (x = 0.5).

The mode I crack–inclusion interaction in orthotropic medium

The interaction between a mode I crack and a symmetrical shape inclusion in an orthotropic medium subjected to the remote stress is investigated. Based on the transformation toughening theory and the Eshelby inclusion method, a closed-form formula for the change of stress intensity factor near the crack tip induced by the inclusion has been proposed. Both the stiffness and the shape for the inclusion are studied by finite element analysis to validate the developed formula. The elastic properties along the orientation perpendicular to the crack line play more influence on the change of stress intensity factor than those in the other directions.

Study on the plane stress mode II crack–inclusion interaction with coupled mechanical and thermal strains

The interaction between a mode II crack and an inclusion in an infinite medium is investigated under the coupled mechanical and thermal loads in plane stress condition. Based on the transformation toughening theory and the Eshelby inclusion method, a closed-form formula for the change of the stress intensity factor is obtained. The mismatched thermal expansion strains between the inclusion and the matrix are considered as part of eigenstrains. A composite of NiCrAlY alloy reinforced glass, as well as the inclusion with triangular shape is selected in finite element method to validate the accuracy for the stress intensity factor variation from the developed formula. The effects of temperature-dependent elastic properties on the inclusion–crack interaction are examined. It is found that the mode II crack is easily to propagate into the inclusion for the composite under the conditions of the free load and the decrease temperature.

Experimental study on interaction between matrix crack and fiber bundles using optical caustic method

The stress singularity of the matrix crack perpendicular to the fiber bundles was studied by the method of optical caustics. First, the strain fields at the matrix crack tip in the neighbor of the fiber bundles were derived based on transformation toughening theory and Eshelby inclusion method. Then the caustic equation at the matrix crack tip in the neighbor of the fiber bundles was established. Finally, an optical caustic experiment for the interaction between the matrix crack and the fiber bundles was conducted to visualize the stress singularity at the matrix crack tip in the form of caustic spots.

Study on the crack–inclusion interaction with coupled mechanical and thermal strains

The interaction between a Mode I crack and an inclusion in an infinite medium is examined with consideration of coupled mechanical and thermal loads under plane strain condition. A closed-form formula for the change of the stress intensity factor is obtained based on the transformation toughening theory and the Eshelby inclusion method. The mismatched thermal expansion strains between the inclusion and the matrix are conducted as eigen strains. Then finite element method is implemented to validate the accuracy for the stress intensity factor variation from the developed formula. Finally, the effects of temperature-dependent elastic properties on the inclusion–crack interaction are estimated. It is also found that the shielding or amplifying effect on the crack growth is the dependence on the mismatched expansion coefficient if only the temperature increase or decrease is considered.

A new approach to a bridging tensor

Bridging Model is an efficient micromechanical theory which can be used to predict the effective thermomechanical as well as strength properties of a fiber reinforced composite. The key to the Bridging Model is a bridging tensor, which was obtained on Eshelby's inclusion method [Huang and Zhou, Strength of Fibrous Composites, Vol. 53, Zhejiang University Press, Springer, Hangzhou, (2011)]. A different approach to the closed‐form expressions for the elements of a bridging tensor is presented in this work. The method is based on averaging the stress fields of a single‐fiber inclusion embedded within an infinite matrix subjected to various load conditions. By solving the bridging equations between the averaged stress components in the fiber and those in the matrix, the exact expressions for the bridging tensor elements are obtained. The Bridging Model can be established by neglecting some small quantities in the bridging tensor elements. Effective elastic moduli of a unidirectional (UD) composite are expressed as functions of a bridging tensor. Comparisons of predicted elastic properties of a total number of 18 UD composites on simplified and the exact bridging tensors are made. POLYM. COMPOS., 36:1417–1431, 2015. © 2014 Society of Plastics Engineers

Coherency Strain and Dislocations in Copper-Rich-Precipitate Embrittled A710 Ferritic Steels

The irradiation embrittlement of reactor pressure vessel steels was studied on surrogate samples of A710 ferritic steels, which is rich in copper. Dislocations and coherency strain caused by the copper-rich precipitates, as a function of aging time, were studied by small-angle neutron scattering, diffraction line-broadening analysis, transmission electron microscopy, and Eshelby inclusion method. Aging has induced copper-rich precipitate nucleation and growth, thus impeding the dislocation motion and hardening the steels. The precipitates are coherent with the matrix up to about 3.5 nm in radius, when they become semi coherent with the ferritic matrix, thus relaxing strains/stresses and softening it. There is good agreement between all experimental techniques used and the Eshelby inclusion model.

The interaction of an edge dislocation with an inclusion of arbitrary shape analyzed by the Eshelby inclusion method

The interaction of an edge dislocation with an inclusion of arbitrary shape is analyzed based on the Eshelby equivalent inclusion method. A general solution to determine the force on the dislocation is obtained, from which a set of simple approximate formulae is also suggested.

Fracture toughening mechanism of shape memory alloys under mixed-mode loading due to martensite transformation

This paper is devoted to the fracture toughening analysis of shape memory alloys (SMAs) with a macrocrack under mixed mode loads. The asymptotic analysis of the stress field and the derivation of the two-dimensional weight function for the semi-infinite crack in isotropic SMAs are investigated. The transformation boundaries for static crack and steady advanced crack are determined. The toughening mechanism due to martensite transformation of SMAs is also studied on the basis of the Eshelby inclusion method and weight function method. The results show that, for the mixed mode problem, the boundaries of the transformation zone is not symmetric and depend on the applied remote load field phase angle. Instead of stress intensity factors, the energy release rate and load phase angle are used for the analysis. It is found that the crack flanks may be closed owing to the martensite transformation near the crack tip when the remote load phase angle is small. The analytical results also show that the martensite transformation reduces the crack tip energy release rate and increases the toughness. The toughness of SMAs is enhanced by the transformation strain, which tends to limit or prevents crack advancing.

Fracture toughening mechanism of shape memory alloys due to martensite transformation

The fracture toughening mechanism of shape memory alloys is studied analytically. The asymptotic stress analysis of shape memory alloys under mode I loads is carried out using the Eshelby inclusion method and the weight function method. The toughening mechanism due to martensite transformation of shape memory alloys is investigated on the basis of the crack shielding theory of fracture mechanics. The transformation boundaries for static and steady advanced cracks are also determined. The analytic results show that the martensite transformation reduces the crack tip stress intensity factor and increases the toughness. The toughness of shape memory alloys is enhanced by the transformed strain fields which tend to limit or prevent crack opening and advancing.

The simulation of switching in polycrystalline ferroelectric ceramics

A polarization switching model for polycrystalline ferroelectric ceramics has been developed. It is assumed that a single ferroelectric crystallite in a ceramic, which is subjected to an electric field and/or a stress, undergoes a complete polarization change and a corresponding strain change if the resulting reduction in potential energy exceeds a critical value per unit volume of switching material. The crystallite’s switch causes a change in the interaction of its field and stress with the surrounding crystallites, which is modeled by the Eshelby inclusion method to provide a mean field estimate of the effect. Thus the model accounts for the effects of the mean electric and stress fields arising from the constraints presented by surrounding crystallites as well as the externally applied mechanical and electrical loads. The switching response of the ceramic polycrystal is obtained by averaging over the behavior of a large number of randomly oriented crystallites. The model, along with the linear dielectric, elastic, and piezoelectric behavior of the material, is implemented in a computer simulation. A fit to experimental electric displacement versus electric field, strain versus electric field, and strain versus stress curves of a ceramic lead lanthanum zirconate titanate PLZT at room temperature is used to obtain material parameters. The model then successfully predicts the electric displacement and strain hysteresis loops for the PLZT under varying electric fields with a constant applied stress.

Micromechanics of partially aligned short-fiber composites with reference to deformation processing

This paper presents a part of the continuing work toward a better modeling technique for the deformation of metal-matrix composites (MMCs) during mechanical deformation processing. The elastic/plastic behavior of partially aligned short-fiber composites loaded in the principal directions has been studied by using the generalized Eshelby inclusion method and incorporating the mean field theory. The plastic deformation of the matrix material is approximated by using the secant moduli for monotonic, proportional loading. Through a volume average procedure the basic formulation is found similar to that of a simple orientation average on the Eshelby tensor. The calculated Young's moduli are close to the lower bound for composites containing less aligned fibers. The influences of the distribution of fiber orientation on the plastic behavior of composites and the internal stress in fibers oriented at a particular angle are also studied. A case study of the extrusion of MMCs was conducted to obtain an insight into the fundamental behavior.