Eshelby's Inclusion Research Articles

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.

New solutions for an Eshelby inclusion of arbitrary shape in a plane or two jointed half‐planes

AbstractNew analytical solutions are derived for Eshelby's problem of an inclusion of arbitrary shape undergoing uniform in‐plane eigenstrains in a plane or in one of two bonded half‐planes. The simply‐connected domain occupied by the inclusion is described by a conformal mapping function that maps the interior of the inclusion onto the interior of the unit circle in the image plane. Elementary expressions of the elastic fields in the exterior of the inclusion and the mean stress inside the thermal inclusion are obtained. Typical examples of epitrochoidal, Booth's lemniscate, hypotrochoidal and generalized Booth's lemniscate thermal inclusions are presented to demonstrate the new solutions.

A Computational Scheme for Evaluating the Stress Field of Thermally and Pressure Induced Unconventional Reservoir

AbstractThe fluid flow connecting the hydraulic fracture and associated unconventional gas or oil reservoir is of great importance to explore such unconventional resource. The deformation of unconventional reservoir caused by heat transport and pore pressure fluctuation may change the stress field of surrounding layer. In this paper, the stress distribution around a penny-shaped reservoir, whose shape is more versatile to cover a wide variety of special case, is investigated via the numerical equivalent inclusion method. Fluid production or hydraulic injection in a subsurface resource caused by the change of pore pressure and temperature within the reservoir may be simulated with the help of the Eshelby inclusion model. By employing the approach of classical eigenstrain, a computational scheme for solving the disturbance produced by the thermally and pressure induced unconventional reservoir is coded to study the effect of Biot coefficient and some other important factors. Moreover, thermo-poro transformation strain and arbitrarily orientated reservoir existing within the surrounding layer are also considered.

Variant selection of primary–secondary extension twin pairs in magnesium: An analytical calculation study

Twining is an important deformation mode in magnesium. In a deformed magnesium sample, an extension twin crystal, i.e., {101¯2} twin, can form inside a {101¯2} primary twin, which is named {101¯2}–{101¯2} secondary twin. These secondary twins often appear at the intersection of two primary twins, and form primary–secondary twin pairs. Experimental observations show that the most frequently observed primary–secondary twin pairs have a unique misorientation, i.e., twin variant selection exists. Such variant selection of the primary–secondary twin pairs is studied in this work. The crystallographic analysis reveals that the twin planes of the primary and secondary twins that form a twin pair have coincident intersection lines with the boundary where the twin pair adjoins. An analytical calculation method based on Eshelby's inclusion theory is developed, and the calculation results show that only for this unique misorientation, the stress fields concentrated at the rims of the primary and the secondary twins are mutually favoured. The analysis is further extended to the incoming–outgoing twin pairs across ordinary grain boundaries, and compared with the commonly used geometrical compatibility factor m′. It is found that m′ only gives good prediction for twin transmission when the shear stress component on the twin plane along the twin shear direction of the incoming twin is the major contributor to the resolved shear stress of the outgoing twin. When other stress components play a dominant role, m′ becomes ineffective in prediction, which is the case for the primary–secondary twin pairs.

An Eshelby inclusion of parabolic shape in a Kirchhoff laminated anisotropic thin plate

An Eshelby inclusion of parabolic shape in a Kirchhoff laminated anisotropic thin plate

Tension-compression asymmetry of stress-relaxation ageing behavior of AA2219 alloy over a wide range of stress levels

The asymmetric stress-relaxation ageing behavior of AA2219 alloy has been experimentally investigated under both tension and compression loading conditions, over a wide range of stress levels at 165 °C for 11 h, using the stress-relaxation ageing tests, room-temperature tensile tests and transmission electron microscopy observations. The relaxed stresses in tension are much larger than those in compression and this deviation mostly originates from the initial rapid reduction stage of stress relaxation. With increasing initial stress, the yield strength in tension increases monotonously, while it remains basically unchanged with only a slight fluctuation of 8 MPa in compression. The mechanisms for asymmetric deformation and strengthening are unveiled on the basis of microstructural analysis. At low initial stress region, the yield strength under compression is higher as compared with that under tension, due to the GP zones and θ'' phases with larger size and weaker stress orientation effect, leading to more stronger hindering effects on the dislocation movement. At high initial stress region, massive dislocations caused by loading and main strengthening θ′ promoted by dislocations result in the larger yield strength under tension, exhibiting an unique reversed strength asymmetry in contrast to low stress level; meanwhile, these dislocations considerably accelerate the tensile stress relaxation process. In addition, the effect of tensile and compressive stresses on precipitate variants is quantitatively discussed based on the Eshelby inclusion theory. These findings benefit the deep understanding of tension-compression asymmetry and offer the theoretical basis for stress-relaxation ageing prediction models.

A viscoelastic Eshelby inclusion model and analysis of the Cell-in-Gel system

We develop a viscoelastic generalization of the elastic Eshelby inclusion solution, where the inclusion and surrounding matrix are two different viscoelastic solids and the inclusion’s eigenstrain is a time-periodic oscillatory input. The solution exploits the Correspondence Principle of Linear Viscoelasticity and a Discrete Fourier Transform to efficiently capture the steady-state oscillatory behavior of the 3-D mechanical fields. The approach is illustrated here in the context of the recently-developed in vitro Cell-in-Gel system, where an isolated live cardiomyocyte (the inclusion) is paced to contract periodically within a soft hydrogel (the matrix), for the purpose of studying the effect of mechanical load on biochemical signals that regulate contractility. The addition of viscoelasticity improves the fidelity of our previous elastic Eshelby inclusion analysis of the Cell-in-Gel system by accounting for the time-varying fields and the resulting hysteresis and dissipated mechanical energy. This mathematical model is used to study the parametric sensitivities of the relative stiffness of the inclusion, the inclusion’s aspect ratio (slenderness), and the cross-link density of the hydrogel matrix.

Shear Transformation Zone (STZ) plasticity analysis of constrained shear

The response in constrained shear of materials characterized by discrete Shear Transformation Zone (STZ) plasticity is investigated. A plane strain strip of finite thickness is analyzed within a small deformation framework and with the STZs modeled as Eshelby inclusions in an isotropic elastic solid. The strip is unbounded and consists of periodic unit cells in the shear direction. Within each unit cell, potential STZ sites are randomly distributed with a specified density and with randomly distributed critical values of shear strain energy density for STZ activation. For both a narrow and a broad distribution of values of critical shear strain energy density, the maximum shear stress is essentially independent of cell geometry but the post-maximum stress–strain response can vary with the geometry and size of the unit cell. The distribution of critical values of STZ shear strain energy density plays a key role in determining the post-maximum stress–strain response, with the post stress maximum shear stress dropping sharply for a narrow distribution of values of critical shear strain energy density and more gradually for a broader distribution of values of critical shear strain energy density. The deformation mode involves bands of enhanced shear strain that are nearly uniformly spaced for a narrow distribution of values of critical shear strain energy density, while for a broad distribution of values of critical shear strain energy density, the band spacing is more irregular. Within the bands the Mises effective stress magnitude can exceed the theoretical strength estimate E∕10.

A micromechanics-based model for visco-super-elastic hydrogel-based nanocomposites

This article presents a micromechanics-based model that constitutively relates internal network physics of hydrogel-based nanocomposites with their visco-super-elastic mechanics. The model is based on the Eshelby inclusion theory combined to the concept of cubic material volume to take into account the effective role of inorganic nanoparticles on the finite-strain response of hydrogels. Dynamic bonds between hydrogel chains and nanoparticles allow to describe the impressive time-dependent properties of hydrogel-based nanocomposites such as rate-dependent extreme stretchability, strong hysteresis upon stretching-retraction and room temperature self-healing facility. The model is compared to a few available experimental data of a variety of hydrogel-nanofiller material systems in terms of stress-strain response till failure, hysteresis, continuous relaxation and self-healing. The effects on the hydrogel behavior of loading conditions (strain rate and strain level) and internal network structures (due to variations in reinforcing elements and cross-linker amounts) are examined. The micromechanical model simulations are found in excellent agreement with experimental observations showing the relevance of the proposed approach. The mechanisms of nanofillers reinforcement and failure are discussed with respect to the model. The room temperature self-healing characteristics of hydrogel systems are discussed in connection to loading history and nanostructure. To further illustrate the model capabilities, the behavior of hydrogel systems is finally treated under different biaxial loading conditions.

Uniform stresses inside both a non-parabolic inhomogeneity and a non-elliptical inhomogeneity located in the vicinity of a circular Eshelby inclusion

We establish the uniformity of stresses inside both a non-parabolic open inhomogeneity and a non-elliptical closed inhomogeneity interacting with a nearby circular Eshelby inclusion undergoing uniform anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. Our procedure involves the introduction of a conformal mapping function for the doubly connected domain occupied by the matrix and the circular Eshelby inclusion. Two conditions are established in order to achieve the uniformity property inside each of the two inhomogeneities. Our results indicate that: (a) the internal uniform stresses are independent of the specific shapes of the two inhomogeneities and the existence of the nearby circular Eshelby inclusion; (b) the open and closed shapes of the respective inhomogeneities are significantly affected by the presence of the circular Eshelby inclusion. We also consider the two more complex cases involving: (a) an arbitrary number of circular Eshelby inclusions undergoing uniform eigenstrains; (b) a circular Eshelby inclusion undergoing linear eigenstrains. Detailed numerical results demonstrate the feasibility and effectiveness of the proposed theory.

The electroelastic fields inside and outside a piezoelectric parabolic inclusion with uniform eigenstrains and eigenelectric fields

AbstractUsing the Stroh octet formalism, we obtain the electroelastic fields in an infinite homogeneous piezoelectric medium containing a parabolic Eshelby inclusion undergoing uniform eigenstrains and eigenelectric fields. We prove that the internal electroelastic field defining stresses, total strains, electric displacements, total electric fields and rigid‐body rotation inside the parabolic inclusion is uniform. The internal uniform electroelastic field is expressed explicitly in terms of the reduced generalized compliances and the imposed eigenstrains and eigenelectric fields. The decaying non‐uniform electroelastic field in the exterior of the parabolic inclusion is also obtained.

“Volume collapse” instabilities in deep-focus earthquakes: A shear source nucleated and driven by pressure

Symmetry-breaking instabilities in high-pressure phase transformation produce the counterintuitive phenomenon of “volume collapse” producing only shear radiation, with little, or no, volumetric component, even under conditions of full isotropy, and explain the mystery of the long-standing observations in deep-focus earthquakes (400–700 km). Due to instability, at a critical “nucleation pressure”, an arbitrarily small densified region in the shape of a “pancake-like” flattened ellipsoidal Eshelby inclusion, grows self-similarly as a “lacuna” (zero particle velocity) with the phase transformation occurring under conditions of equilibrium in uniform strain/stress and at constant potential energy (at the vanishing of the M integral, when the radius-expanding driving force pεkk∗ overcomes the radius-shrinking self-force). The symmetry-breaking flattened shape favors minimization of the energy needed for the boundary to grow large, while for the accommodation of the large collapsing volume in the very thin inclusion deviatoric stresses are developed to avoid openings and overlaps. It is shown that, if an arbitrarily small flattened densified region is generated planarly, and the pressure exceeds the critical nucleation value, then it will necessarily produce a shear seismic source, with little or no, volumetric component, nucleated and driven to propagate by the pressure. The ellipsoid of phase change forms in the direction that minimizes the interaction energy with the pre-stress field and will be close to the direction of max shear pre-stress in the mantle. The obtained stress/deformation fields of a densified 2D flattened elliptical inclusion constitute a new defect that models the “anticrack” in geophysics and densified shear bands. The instability analysis can be extended to the nucleation and growth of the phase transition from water to a solid ice phase under high pressure, with the discovered instabilities providing insight to other phenomena of dynamic phase transformations, such as failure waves, amorphization, planetary impacts, etc.

A circular Eshelby inclusion with linear eigenstrains interacting with a coated non-parabolic inhomogeneity with internal uniform anti-plane stresses

We establish criteria which guarantee the uniformity of stresses inside a coated non-parabolic open inhomogeneity located in the vicinity of a circular Eshelby inclusion undergoing linear anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. The associated inverse problem in anti-plane elasticity which identifies the required geometry of the constituents of the composite is successfully solved using conformal mapping techniques and analytic continuation. Three conditions on the remote loading in the matrix, uniform eigenstrains imposed on the non-parabolic inhomogeneity and linear eigenstrains imposed on the circular Eshelby inclusion for given geometric and material parameters of the composite are found in order to ensure the uniformity of internal stresses. The internal uniform stress field is found to be independent of the specific open shapes of the inhomogeneity-coating and coating-matrix interfaces, the shear modulus of the coating and also of the existence of the nearby circular Eshelby inclusion. However, the open shapes of the two interfaces are significantly influenced by the presence of the circular Eshelby inclusion. We also discuss the general case in which arbitrary polynomial eigenstrains are imposed on the circular Eshelby inclusion.

An Eshelby inclusion of parabolic shape in an anisotropic elastic plane

An analytical solution is derived to the Eshelby's problem of a parabolic inclusion undergoing uniform in-plane and anti-plane eigenstrains in an anisotropic elastic plane. The stresses, total strains and rigid-body rotation are found to be uniform inside the parabolic inclusion. In addition, we obtain real-form expressions of these internal uniform physical quantities in terms of the reduced elastic compliances and the imposed eigenstrains. The constant Eshelby's tensor inside the parabolic inclusion can be completely determined by the reduced elastic compliances.

Massively parallel finite difference elasticity using block-structured adaptive mesh refinement with a geometric multigrid solver

Computationally solving the equations of elasticity is a key component in many materials science and mechanics simulations. Phenomena such as deformation-induced microstructure evolution, microfracture, and microvoid nucleation are examples of applications for which accurate stress and strain fields are required. A characteristic feature of these simulations is that the problem domain is simple (typically a rectilinear representative volume element (RVE)), but the evolution of internal topological features is extremely complex. Traditionally, the finite element method (FEM) is used for elasticity calculations; FEM is nearly ubiquitous due to (1) its ability to handle meshes of complex geometry using isoparametric elements, and (2) the weak formulation which eschews the need for computation of second derivatives. However, variable topology problems (e.g. microstructure evolution) require either remeshing, or adaptive mesh refinement (AMR) - both of which can cause extensive overhead and limited scaling. Block-structured AMR (BSAMR) is a method for adaptive mesh refinement that exhibits good scaling and is well-suited for many problems in materials science. Here, it is shown that the equations of elasticity can be efficiently solved using BSAMR using the finite difference method. The boundary operator method is used to treat different types of boundary conditions, and the “reflux-free” method is introduced to efficiently and easily treat the coarse-fine boundaries that arise in BSAMR. Examples are presented that demonstrate the use of this method in a variety of cases relevant to materials science: Eshelby inclusions, fracture, and microstructure evolution. Reasonable scaling is demonstrated up to ∼4000 processors with tens of millions of grid points, and good AMR efficiency is observed.

3D mechanical analysis of a self-contractile cell with stress fibers reorganization

Contractile stress generated by a cell itself is crucial in sensing its surrounding microenvironment and regulating cell adhesion, differentiation and cytokinesis. However, the precise mechanisms underlying mechanotransduction remain unknown. In this paper, based on the Eshelby inclusion problem, we develop a theoretical model to characterize quantitatively the mechanical environment around a self-contractile cell experiencing stress fibers reorganization. We divide the contractile stress into two parts: the constant contractile stress and the perturbed contractile stress due to stress fibers reorganization, for internal stress fibers have enough time to reorganize actively during long-term deformation, leading to changes of contractile stress in both magnitude and direction. Obtained results suggest that stress fibers reorganization may cause significant changes in the mechanical environment of the cell, helpful for exploring the mechanisms behind cell mechanotransduction.

Time-resolved shear transformations in the transient plastic regime of sheared amorphous silicon.

The accumulation of shear transformations (STs) in space and time is responsible for plastic deformation in amorphous solids. Here we study the effect of finite strain rates on STs during simulations of athermal shear deformation in an atomistic model of amorphous silicon. We present a time-resolved analysis of STs by mapping the plastic events identified in the atomistic simulations on a collection of Eshelby inclusions, which are characterized in terms of number, effective volume, lifetime, and orientation. Our analysis led us to distinguish between small and large events. We find that the main effect of a lower strain rate is to allow for a larger number of small events, roughly identified by an effective volume γ_{0}V_{0}<20 Å^{3}, while the number and characteristics of larger events are surprisingly independent of the strain rate. We show that at low strains, the decrease of the stress observed at lower strain rates is mainly due to the excess of small events, while at larger strains, when the glass approaches the yield point where a shear band forms, larger events start to play a role and organize due to their elastic interactions. This phenomenology is compared with the predictions of mesoscale elastoplastic models. The technique developed here can be used as a systematic tool to analyze plasticity during molecular dynamics simulations. It can also give valuable information to develop physically grounded mesoscale models of plasticity, providing quantitative predictions of the mechanical properties of amorphous materials.

Signature of local stress states in the deformation behavior of metallic glasses

The design of ductile heterogeneous metallic glasses (MGs) with enhanced deformability by purposely controlling the shear-band dynamics via modulation of the atomic-scale structures and local stress states remains a significant challenge. Here, we correlate the changes in the local atomic structure when cooling to cryogenic temperature with the observed improved shear stability. The enhanced atomic-level structural and elastic heterogeneities related to the nonaffine thermal contraction of the short-range order (SRO) and medium-range order (MRO) change the characteristics of the activation process of the shear transformation zones (STZs). The experimental observations corroborated by Eshelby inclusion analysis and molecular dynamics simulations disclose the correlation between the structural fluctuations and the change in the stress field around the STZ. The variations in the inclination axes of the STZs alter their percolation mechanism, affect the shear-band dynamics and kinetics, and consequently delay shear failure. These results expand the understanding of the correlation between the atomic-level structure and elementary plastic events in monolithic MGs and thereby pave the way for the design of new ductile metallic alloys.

Novel cloaking lamellar structures for a screw dislocation dipole, a circular Eshelby inclusion and a concentrated couple.

Using conformal mapping techniques, we design novel lamellar structures which cloak the influence of any one of a screw dislocation dipole, a circular Eshelby inclusion or a concentrated couple. The lamellar structure is composed of two half-planes bonded through a middle coating with a variable thickness within which is located either the dislocation dipole, the circular Eshelby inclusion or the concentrated couple. The Eshelby inclusion undergoes either uniform anti-plane eigenstrains or uniform in-plane volumetric eigenstrains. As a result, the influence of any one of the dislocation dipole, the circular Eshelby inclusion or the concentrated couple is cloaked in that their presence will not disturb the prescribed uniform stress fields in both surrounding half-planes.

Study on the Calculation Method of Stress in Strong Constraint Zones of the Concrete Structure on the Pile Foundation Based on Eshelby Equivalent Inclusion Theory.

In view of the strong constraint zones of the concrete structure on the pile foundation, there are some differences between the calculation results of the isotropic equivalent pile foundation by the volume replacement ratio method and the actual engineering. In this paper, referring to the relevant algorithm of rock mass with anchor, the anchor and rock mass are, respectively, compared to pile and surrounding soil foundation. Eshelby equivalent inclusion theory is introduced into the equivalent mechanical model of soil foundation with pile, and a new equivalent pile foundation algorithm considering anisotropic elastic constant is compiled by Fortran. Three kinds of calculation methods are used to calculate the stress field of the concrete structure of the large pump station on the pile foundation during the construction period, and the stress in the strong constraint zones of the concrete structure are mainly analyzed. It is found that the calculation accuracy of Algorithm 3 is the highest, and the calculation results of Algorithm 2 can be modified by the coefficients to achieve the calculation accuracy of Algorithm 3 and the calculation efficiency is actually improved. Finally, the accuracy of the proposed method is verified by the engineering measured data.