Eshelby's Theory Research Articles

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

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

Effect of ferromagnetic particles on the elastic and magnetostriction parameters of bulk superconductors

Based on the magnetic dipole theory and the Eshelby's equivalent inclusion theory, a theoretical model of ferromagnetic particle inclusion-superconducting matrix is established in this paper, intrinsic strain is introduced and uniform stress boundary conditions are used to calculate the mechanical properties of the composite material. The effects of changes in parameters such as the elastic modulus, volume fraction ratio, and magnetic field of ferromagnetic particle inclusions on the effective mechanical properties of bulk superconductors such as the effective elastic modulus, effective Poisson's ratio, and effective magnetostriction were compared in detail, providing a theoretical reference for the preparation of superconducting materials containing ferromagnetic particle inclusions.

Precipitate sheared by low-angle grain boundary in Mg-Al alloys

The interaction between Mg17Al12 precipitate and low-angle grain boundaries (LAGBs) in Mg-Al alloys was studied using molecular dynamics (MD) simulations. The precipitate was observed to be not sheared by the LAGBs with a small rotation angle < 8°, but sheared by the LAGBs with a large rotation angle ≥ 9° To explain these two different interaction styles, a critical GB shear was proposed analytically using the Eshelby theory, which indicates that the critical GB rotation angle is 8.4° and agrees well with the current MD simulations. The critical GB shear was found to be related to the aspect ratio of precipitate.

Superelasticity degradation of NiTi shape memory alloy in wide ranges of temperature and loading level: Experimental observation and micromechanical constitutive model

In this work, cyclic tension-unloading tests with different peak strains (10%-18%) in the temperature range from 313 K to 393 K are performed to investigate the effects of temperature and loading level on the cyclic deformation behavior of NiTi shape memory alloy (SMA) wires. Experimental results demonstrate that the superelasticity degradation occurs during the cyclic deformation process, and such phenomenon aggravates with the rise of ambient temperature and loading level, which can be attributed to the complex interactions among the martensite transformation (MT), austenite plasticity (AP), martensite plasticity (MP) and the transformation-induced plasticity (TRIP). Then, a micromechanical cyclic constitutive model is proposed based on the framework of irreversible thermodynamics and Eshelby's inclusion theory. The martensite (M) phase is treated as mobile inclusions with alterable eigenstrain embedded in the austenite (A) phase matrix. The volume fractions of the M-phase, A-phase, and A-M interface-phase are introduced. The non-uniform stress fields in the three phases are estimated by employing the Mori-Tanaka's homogenization scheme and interfacial operator. The driving forces of the four types of the inelastic deformation mechanisms, MT, AP, MP, and TRIP are derived based on the proposed new Helmholtz free energy, instantaneous growth hypothesis of M domains, and the energy dissipation inequality. The inheritances of the plastic deformation induced by the movement of A-M interfaces during the repeated MT and its reverse are incorporated. Finally, to validate the predictive capability of the proposed model, the predicted results for the superelasticity degradation of NiTi SMA with various peak strains at different ambient temperatures are compared with the experimental ones. Moreover, the dominant plastic deformation mechanisms and the effects of AP, MP, TRIP on the cyclic deformation of NiTi SMA are discussed.

An LCF lifetime model for PM superalloy considering equivalent ellipsoidal inclusion

The non-metallic inclusions are one of the main defects in powder metallurgy (PM) superalloy, which may promote crack initiation under the cyclic loading and seriously affect the safety of aero engines. To quantify the effects of inclusions to the low cycle fatigue (LCF) lifetime, the LCF tests on PM FGH96 superalloy are firstly carried out to observe the inclusions on fracture. By equating the inclusion to an ellipsoid, the relationship between the inclusion's feature and the stress field is established based on Eshelby's theory and finite element method (FEM). Then, the ellipse-equated fracture is determined according to the maximum shear stress's location. Subsequently, key inclusion-related parameters influencing the LCF lifetime are introduced to construct a modified SWT model. Finally, the model's accuracy is validated by comparing with the LCF experimental data. It is demonstrated that the modified SWT model has a good accuracy, with an improvement of scattering band from ±15 to ±4 compared with the original SWT model.

Influence of stress redistribution and fracture orientation on fracture permeability under consideration of surrounding rock in underground gas storage

UGS (underground gas storage) is of great significance for ensuring emergent supply and CCUS (Carbon Capture, Utilization, and Storage). However, very different from gas reservoir development, some stress changes in reservoir, caprock, and bottom support rock may affect its operation. Sichuan Basin is rich in natural-gas resources and geological targets for the UGS rebuilt from fractured carbonate reservoirs. Moreover, gas storage is featured by rapid injection and production, and the change of in-situ stress affects intensively the permeability of reservoir fracture. However, most existing models on the fracture permeability assume that the surrounding-rock stress is constant. So, a new model has been established for the fracture permeability based on the Eshelby theory, considering the effect of both direct caprock and bottom support rock (also called surrounding rock) and the stress redistribution mechanism of the entire geological body during UGS injection and production. Results show that for the fracture, the permeability is related to not only its compressibility and orientation, but also reservoir shape and elastic property of the entire geological body including reservoir and surrounding rock. In most disk-shaped or elliptical reservoirs, vertical fracture is less susceptible to pore pressure than horizontal one. The surrounding-rock elastic property is also an important factor in the fracture-permeability model. Hard surrounding rock can reduce the stress sensitivity in fracture. When the elastic modulus ratio (elastic modulus of reservoir vs. that of surrounding rock) is 0.1, the fracture permeability decreases by only 15%. This model is also applicable to multiple fractures. Using the superposition principle, it is found that the main direction of permeability and the degree of permeability anisotropy may change when there are multiple fractures in the reservoir in spite of the same fracture compressibility.

Mesomechanical damage of fiber mortar under temperature difference

The present study aims to establish a mesomechanical damage model for the description of anisotropic damage by mesocracking under temperature difference condition. With the basic solution of Eshelby theory and the Clapeyron equation, a mechanics framework is developed for cement materials, which are considered to be heterogeneous materials consisting of a cement stone matrix and imbedded penny-shaped microcracks. From the mesomechanical analysis, the critical conditions corresponding to the failure of fiber mortar under different initial crack density conditions are obtained. Furthermore, the Clapeyron equation is used to describe the relationship between the stress and the initiation or expansion of the fiber mortar crack and the damage evolution rule is expressed by the energy equation of the stiffness coefficient. The proposed model can accurately quantify the factors such as fiber distribution, microcrack density, temperature differences and the relationship between directional microcracks and crack strength in mortar. The results verified by laboratory tests and numerical simulations show that as the temperature difference increases from 0 °C to 75 °C, the cracking strength of the fiber mortar decreases and the number of cracks increases in the form of a power exponential function, which proves the micromechanical damage the validity of the model.

Multiscale simulation of elastic response and residual stress for ceramic particle reinforced composites

In this study, the elastic response and residual stress of the ceramic matrix composites were simulated in multiscale view for exploring the elastic modulus of the two-phase composites reinforced with ceramic particles by diffraction method. Combined with the macroscopic stress-strain relationship of two-phase composites, the effective elastic response model under uniaxial loading was theoretically predicted by introducing the effective representative volume element (RVE). Subsequently, the effective elastic properties of the material and its constituents were calculated, along with comparison with the experimental values reported in the literature. On the basis of Eshelby's inclusion theory, a micromechanical elastic response model for the two-phase composites was established, and a series of micromechanical field equations related to the crystal planes diffraction elastic constants were derived. A comparison with the different micromechanical models for predicting the crystal planes diffraction elastic constants of the ceramic matrix and particle reinforcement was carried out, so as to qualitatively and quantitatively analyze the residual stress. It was revealed that the effective elastic parameters of the ZrB2 matrix and SiC particle reinforcement were very close to the experimental values reported in literature, and the calculated values of the elastic modulus in seven crystallographic diffraction directions were similar to the experimental values. In addition, the average residual stress of the SiC reinforcement and the ZrB2 ceramic matrix predicted by the developed model were consistent with the measured, which proved the reliability and accuracy of the theoretical model for multiscale simulation of the residual stress of two-phase composites.

Stress and Strain Rate Evolution During the Finite Strain Deformation of a Weak Ferropericlase Grain by Diffusion Creep: Implications for Shear Localization in the Lower Mantle

AbstractWe investigate the finite deformation of a weak inclusion embedded in a strong matrix. The goal is to understand the possible causes for shear localization that would explain the presence of geochemical heterogeneity in the Earth's lower mantle. The weak phase in the lower mantle is ferropericlase (Fp), and the strong phase is bridgmanite (Br). We herein consider deformation by diffusion creep for which some supporting evidence is available. When an inclusion deforms by diffusion creep, deformation itself modifies its geometry and stress distribution. The modified stress distribution in turn leads to strain‐dependent rheological properties due to the changes in diffusion path length and the stress gradient that drives diffusion flux. We investigate the evolution of deformation of a weak inclusion (i.e., Fp‐like weak grain embedded in Br‐like strong matrix) using the Eshelby theory (for stress and strain rate fields) combined with a theory of diffusional mass transport caused by the gradient in the normal stress within the inclusion. We found that finite strain leads to a significant strain weakening under simple shear deformation but not under axial deformation. Since diffusion creep is thought to be a dominant mechanism of the lower mantle rheology, the results of the present study provide a basis for investigating the nature of shear localization and its important implication for the preservation of geochemical heterogeneity and the distribution of seismic anisotropy in the lower mantle.

A coupled fictitious stress method and Eshelby inclusions as a meshless technique for inhomogeneity problems

In this paper, the fictitious stress method (FSM) as a meshless technique is extended to solve problems with inhomogeneities. The Eshelby's equivalent inclusion theory is coupled with the FSM as a set of particular solutions. There is no domain discretization as analytical solutions for inclusions are employed. The coupling procedure is presented, and the solution is carried out either in a direct or in an iterative approach. Numerical examples are presented to verify accuracy of the developed formulations.

Damage model of unsaturated frozen soil while considering the influence of temperature rise under impact loading

Constitutive models and damage laws of frozen soil under high strain rates are crucial for the design of underground soil structures, and they lay an essential foundation for the research of instability phenomena in cold region engineering. The expansion and evolution of micro-defects and temperature damage are two basic mechanisms that control the macro and micro damage behavior of frozen soil. Considering frozen soil as a composite material, this study analyzes the damage mechanism of brittle materials subjected to impact loading. Finally, a dynamic model of unsaturated frozen soil under high pressure is established based on the complementary energy equivalence principle and Eshelby's equivalent inclusion theory. To analyze the effect of high-speed compression on the meso behavior of frozen soil, according to the Hugoniot energy theory, the evolution of temperature rise in the impact process is formulated. The results for frozen sandy soil confirm that the rate of temperature rise increases with the strain rate and has a significant weakening effect on the bearing capacity. Thus, the damage behavior of frozen soil changes according to different loading histories. Finally, based on the corresponding theoretical results, the mechanical response of frozen soil under different loading histories and the corresponding performance indicators are compared and analyzed.

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.

Explicit boundary element modeling of nonlocal damage with Eshelby theory

Modeling nonlinearities, including damage, in the boundary element method (BEM) is usually carried out in implicit way, or in other words via applying initial stresses or strains over a discretized domain part. Such initial values have no physical meaning. They are only used to compensate the stress level due to the occurred nonlinearity. In this paper explicit implementation of nonlocal damage is proposed. The damaged points inside the domain is physically weakened by decreasing their modulus of elasticity. With the help of Eshelby's equivalent inclusion theory, this idea is developed and implemented in this work. Load control solution algorithm is used. Both average strain and average damage nonlocal models are considered. Numerical examples are presented to verify the developed formulation. Factors that affect the solution accuracy are studied in details.

Geometry and energy barrier of martensite in the initial stage martensitic transformation in B19’ TiNi shape memory alloy

This study addresses the selectivity of self-accommodation microstructures in the initial stage of martensitic transformation in B19’ TiNi. It is known that specific habit plane variant (HPV) pairs, including {111¯}mtype I twin planes that are called initial stage microstructure (ISM) in this study, are preferentially formed in the initial stage of martensitic transformations. To elucidate on the selectivity of the ISM, the energy barrier at the formation of the twin plane, which is the candidate for the starting point of HPV pair formation, and the geometry of the HPV pair are theoretically analyzed in this paper. Using geometrical non-linear theory analysis, the compatible twin plane which has exact twin orientation relationship can be formed in the specific five groups of HPV pair. Using ab-initio simulations and Eshelby theory analysis results, {111¯}mtype I twin is determined to have a relatively small energy barrier among the twins in TiNi. Therefore, this paper shows that the HPV pair having a twin plane with a small energy barrier of formation and a structure that can effectively reduce the elastic strain is preferentially formed as the ISM.

A new fracture permeability model: Influence of surrounding rocks and matrix pressure

An appropriate fracture permeability model needs to include terms that characterize the effects of surrounding rocks and varying matrix pore pressure. This paper describes the use of Eshelby's inhomogeneity theory to derive equations that can be used to predict fracture permeability behavior for different shapes of reservoirs whose elastic properties differ from the surrounding rocks. In the equations, permeability is affected by two factors: reservoir volumetric strain and matrix volumetric strain. Reservoir volumetric strain is determined by properties of both the reservoir and surrounding rock. Matrix volumetric strain is controlled by fracture pressure as well as matrix pressure. When fracture permeability is significantly greater than matrix permeability, pressure drop first occurs in fractures and then matrix pressure decreases. In this case, fracture permeability first decreases and then increases. The increase of permeability is related to the matrix pressure drop and the bulk modulus ratio between the matrix and the grains. Results show that the smaller the initial fracture porosity, matrix modulus, shear modulus ratio between the reservoir and surrounding rocks, Poisson's ratio of the reservoir and shape factor of the reservoir, the stronger the stress-sensitivity of fracture permeability during hydrocarbon production. It is also found that when shear modulus of the reservoir is the same as that of surrounding rocks, fracture permeability is independent of reservoir shapes. The model is suitable for heterogeneous reservoirs and can be embedded into the reservoir numerical software to more accurately simulate the development of fractured reservoirs.

Stress distribution analysis and interface influence on fiber reinforced composites

In this paper, Eshelby's equivalent inclusion theory, Mori-Tanaka method and shear lag theory are used to derive the analytical formula of stress distribution on the fiber. The effective elastic modulus of the equivalent fiber combining the fiber and the interface layer is predicted and analyzed. Moreover, the theoretical expression of stress field on fibers under three-phase structure is proposed, and the factors affecting the stress distribution on the fiber are also discussed. In addition, because the mechanical properties of different interface layers will have an impact on the parameters of the equivalent fiber phase, this study considers the influence of different interface properties on the interface stress ability, and the effect of interface properties on stress transfer is also investigated. It is proved that the effect of interface layer on stress transfer between matrix and fiber can not be neglected.

A numerical application of the Eshelby theory for geobarometry of non-ideal host-inclusion systems

In the complex geodynamic processes occurring at convergent plate margins, rocks can be subducted at depth into the Earth experiencing metamorphism. A mineral inhomogeneity entrapped into another mineral, after exhumation to the Earth surface, will exhibit stress and strain fields different from those of the host because of the different thermoelastic properties. In the present paper, we propose a finite-element-based approach to determine the Eshelby and the relaxations tensors for any morphology of the inhomogeneity and for any crystallographic symmetry of the host. The proposed procedure can be directly applied in the framework of elastic geobarometry to estimate, on the basis of the Eshelby theory, the entrapment conditions (pressure and temperature) from the residual strain field measured in the inhomogeneity. This aspect represents a step forward to currently available models for geobarometry allowing the investigation of complex morphologies of the inhomogeneity in systems with general material anisotropy. We validate the proposed approach versus Eshelby analytical solutions available for spherical and ellipsoidal inclusions and we show the application to a real geological case of high pressure metamorphic rocks.

Predicting the fatigue life of an AlSi10Mg alloy manufactured via laser powder bed fusion by using data from computed tomography

A modelling strategy is proposed to evaluate the influence of defect morphology on the fatigue limit of additively manufactured Al alloys by: (i) obtaining an x-ray micro-Computed Tomography (μ-CT) 3D image of the material, (ii) computing the Equivalent Inertia Ellipsoid of each individual pore, (iii) modelling the influence of the defect on the fatigue limit through the Defect Stress Gradient (DSG) approach coupled to the Eshelby theory and, (iv) 3D mapping the criticality of each individual defect. For this fatigue study, an AlSi10Mg alloy was manufactured by laser powder bed fusion using sub-optimal deposition parameters in order to produce large lack-of-fusion defects. After a T6 heat treatment, tension-compression fatigue tests, with R = −1, were conducted on specimens oriented with their loading axis either parallel or normal to the Z-axis of the additive manufacturing equipment. Two samples were characterised before μ-CT testing in order to characterise the initial 3D defect population. Each sample was fatigued step by step in order to determine the fatigue limit. The fracture surface was observed in order to identify the critical defect in the initial μ-CT image. A comparison with the fatigue results led to the following conclusions: (i) when the longest axis of the defect is perpendicular to the loading axis, modelling the defect as an equivalent inertia prolate ellipsoid gives better results (5 % error on the fatigue limit) than modelling it as a simple equivalent sphere (22 % error on the fatigue limit), (ii) the prolate ellipsoid is not relevant when the longest axis of the defect is oriented along the loading axis; in this case an oblate equivalent ellipsoid should be used, (iii) the concept of ‘size’ for a complex 3D shaped defect should be linked to the inertia and the loading, (iv) with this approach, surface defects are shown to be more critical than internal ones for fatigue life and, (v) a 3D defect criticality map of the entire sample can be plotted to provide visual feedback on which defects are the most critical for fatigue life.

Simulation of salt-cavity healing based on a micro–macro model of pressure solution

CO 2 storage in salt rock is simulated with the finite element method (FEM), assuming constant gas pressure. The initial state is determined by simulating cavity excavation with a continuum damage mechanics (CDM) model. A micro–macro healing mechanics model is proposed to understand the time-dependent behaviour of halite during the storage phase. Salt is viewed as an assembly of porous spherical inclusions that contain three orthogonal planes of discontinuity. Eshelby's self-consistent theory is employed to homogenize the distribution of stresses and strains of the inclusions at the scale of a representative elementary volume (REV). Pressure solution results in inclusion deformation, considered as eigenstrain, and in inclusion stiffness changes. The micro–macro healing model is calibrated against Spiers’ oedometer test results, with uniformly distributed contact plane orientations. FEM simulations show that independent of salt diffusion properties, healing is limited by stress redistributions that occur around the cavity during pressure solution. In standard geological storage conditions, the displacements at the cavity wall occur within the first 5 days of storage and the damage is reduced by only 2%. These conclusions still need to be confirmed by simulations that account for changes in gas temperature and pressure over time. For now, the proposed modelling framework can be applied to optimize crushed salt back-filling materials and can be extended to other self-healing materials.

An anisotropic elastic–plastic model for predicting the rafting behavior in Ni-based single crystal superalloys

An anisotropic elastic–plastic model was developed to investigate the rafting behavior of Ni-based single crystal superalloys. This model was developed by introducing the plastic constitutive equation of face-centered cubic (FCC) single crystal in the framework of Eshelby's equivalent inclusion theory. The Hill's equivalent stress was calculated when applying a tensile or compressive loading along the [001] direction. The calculated results successfully predict the rafting direction for alloys with a positive or negative mismatch, consistent with pervious experimental and theoretical studies. Moreover, based on this model, the mismatch degree and the elastic-constant differences of the matrix and precipitate phases and their effects on the speed of rafting are carefully discussed. Regardless of a positive or negative mismatch alloy, the larger absolute value of mismatch degree can more effectively accelerate the process of rafting in Ni-based single crystal superalloys. However, when the alloys enter the plastic deformation stage, the variation in mismatch degree slightly affects the speed of rafting. For the elastic-constant differences (C11r′−C11r,C12r′−C12r,and C44r′−C44r), the smaller the value of C11r′−C11r, or the greater the value of C12r′−C12r, the more effective the acceleration of γ′ rafting; whereas the value of C44r′−C44r has no effect on the rafting of alloys. The research results provide a new theory and method for studying the rafting behavior and its influencing factors for Ni-based single crystal superalloys.