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The objective of this paper is to validate a measurement-driven, model-based approach to estimate residual stress (RS) in samples machined from quenched aluminum stock. Model input is derived from measurement of RS in the parent stock. Validation is performed for prismatic T-sections removed from bars at different locations. We find RS predicted agrees with RS measured, by contour and neutron diffraction methods, with root-mean-square model-measurement difference of 22 MPa. Follow-on work (in Part 2) applies the RS estimation to samples representative of aircraft structures and examines the effects of RS on fatigue crack growth in the RS-bearing samples.

This paper assesses the accuracy of fatigue crack growth (FCG) predictions for high-strength aluminum samples containing residual stress (RS) and complex two-dimensional cracks subjected to constant amplitude load. FCG predictions use linear-elastic, multi-point fracture mechanics. A first prediction includes RS estimated by the model described in Part 1; a second prediction includes RS measured by the contour method. FCG test data show a significant influence of RS. Ignoring the RS results in a +60% error in predicted FCG life (non-conservative). Including RS improves predictions of crack growth significantly (errors better than +26% (estimated RS) and −14% (measured RS)).

Laser peen forming (LPF) is promising to form large-scale metal sheets in various applications. Understanding the bending deformation of the LPF process is of particular importance for accurate shape prediction and process planning elaboration. This study established a simplified eigenstrain-based finite element (FEM) model by incorporating loading sequence and geometric non-linearity to explore the bending deformation and shape prediction of the LPF process. The simplified eigenstrain was inversely determined based on the arc heights of plates subjected to full-covered LPF. The above model incorporating loading sequence and geometric non-linearity was verified by the comparison of experimental results, and it improved shape prediction accuracy considerably compared to the common beam bending analytical model. Moreover, the bending deformation processes and mechanisms of LPF plates were elucidated. It was found that biaxially curved surfaces were formed for thick plates with 5 mm or larger thickness, in which the geometric non-linearity and flattening effects played significant roles on their formation. On the other side, monoaxially curved surfaces were obtained for thin plates with 4 mm or smaller thickness after the LPF process. Three stages, i.e., forward bending, reverse flattening, and buckling, occurred and contributed to the formation of the monoaxially curved surface. With the increase in the eigenstrain amplitude and side length, and the decrease in plate thickness, the LPF plates transit from a biaxially curved shape to a monoaxially curved one. [ABSTRACT FROM AUTHOR]

The compressive residual stress distribution in shot peening can be simulated using the nonlinear elastic–plastic finite element method (FEM) or obtained using various measurement methods; however, a simple and efficient prediction is desirable for practical applications. The residual plastic strain component produced by shot peening, known as the inherent strain or Eigenstrain, has a simple distribution and can be used to predict the residual stress via linear elastic analysis. In this study, the residual stress distributions in aluminum–magnesium alloy plates due to various shot velocities and shot diameters were simulated using the nonlinear FEM; further, the residual stress distributions were verified using the X-ray diffraction method. The inherent strain was identified using these results as datasets, and the strain distribution was quantitatively expressed using a simple equation. Finally, an analytical method for shot peening residual stress prediction using inherent strain and plate bending theory was developed. The residual stress distributions predicted from the inherent strain determined for 5 -mm-thick plate were consistent with the experimentally measured and numerical simulated results for 2-, and 1 -mm-thick plates with different shot blasting machines and shot diameters. [ABSTRACT FROM AUTHOR]

• A novel homogenization model for elastic periodic multiphase composites is presented. • The model construction is based on the eigenstrain concept. • Composites with arbitrary number of phases and shapes of inclusions can be considered. • The model performance is evaluated by comparison with other homogenization techniques. This paper presents a novel micromechanical procedure for the linear elastic homogenization of composites with periodic microstructures. The procedure is developed for composites with arbitrary number of phases and geometric shapes of the inhomogeneities, in contrast with most existing homogenization approaches. Also, no restriction is made in relation to the mismatch between the properties of the phases and volume fractions of the inhomogeneities. The proposed procedure is based on the Eshelby equivalent inclusion approach and extends a model originally derived for evaluating the effective elastic moduli of periodic two-phase composites. The procedure represents the fluctuating elastic fields within each multiphase repeating unit cell (RUC) using Fourier series, resulting in Lippmann-Schwinger integral equations governing the unknown eigenstrain fields of the inclusions. Unlike traditional iterative algorithms used in Fast Fourier Transform (FFT)-based approaches, the procedure solves the integral equations straightforwardly from a scheme of partition of the domain of each inclusion. The efficiency of the proposal procedure is demonstrated through applications to composites with different arrays of coated fibers and constituent materials. [ABSTRACT FROM AUTHOR]

Cold expansion (CX) is a material processing technique that has been widely used in the aircraft industry to enhance fatigue life of structural components containing holes. CX introduces compressive hoop residual stresses that slow crack growth near the hole edge. The objective of this paper is to predict residual stresses arising from cold expansion using two different finite element (FE) approaches, and compare the results to measurement data obtained by the contour method. The paper considers single-hole, double-hole, and triple-hole configurations with three different edge margins. The first FE approach considers process modeling, and includes elastic-plastic behavior, while the second approach is based on the eigenstrain method, and includes only elastic behavior. The results obtained from the FE models are in good agreement with one another, and with measurement data, especially close to the holes, and with respect to the effect of edge margin on the residual stress distributions. The distribution of the residual stress and equivalent plastic strain around the holes is also explored, and the results are discussed in detail. The eigenstrain method was found to be very useful, providing generally accurate predictions of residual stress.

In linear elasticity, universal displacements for a given symmetry class are those displacements that can be maintained by only applying boundary tractions (no body forces) and for arbitrary elastic constants in the symmetry class. In a previous work, we showed that the larger the symmetry group, the larger the space of universal displacements. Here, we generalize these ideas to the case of anelasticity. In linear anelasticity, the total strain is additively decomposed into elastic strain and anelastic strain, often referred to as an eigenstrain. We show that the universality constraints (equilibrium equations and arbitrariness of the elastic constants) completely specify the universal elastic strains for each of the eight anisotropy symmetry classes. The corresponding universal eigenstrains are the set of solutions to a system of second-order linear PDEs that ensure compatibility of the total strains. We show that for three symmetry classes, namely triclinic, monoclinic, and trigonal, only compatible (impotent) eigenstrains are universal. For the remaining five classes universal eigenstrains (up to the impotent ones) are the set of solutions to a system of linear second-order PDEs with certain arbitrary forcing terms that depend on the symmetry class. [ABSTRACT FROM AUTHOR]

Eigenstrain is a distributed strain field considered in mechanics that is particularly helpful in evaluating residual stress fields in the finite element method, and estimating the stress intensity factor due to residual stress in cracked components. The objective of this paper is to provide a solution for a simple eigenstrain problem in a two-dimensional rectangular domain that can serve as a benchmark for validation of fracture mechanics analysis methods. The solution provides residual stress fields and the stress intensity factor for a single edge crack as a function of crack size. Documenting the benchmark provides opportunities to demonstrate the correlation of different means to determine the stress intensity factor and to highlight details in implementing stress intensity factor calculations.

The paper presents a general solution of the isotropic elasticity problem for the sandwiched inclusion in an elastic cylinder and having axially varying eigenstrain. This problem is relevant in considerations of a cylinder with axially varying thermal expansion coefficient or a nanowire having an axial inhomogeneity of the crystal lattice parameter due to growth and processing conditions. The technique of the solution and the results for the elastic fields and energies of dilatational inclusions with different distribution of eigenstrain, namely, constant, trapezoidal, and diffusion-like, along the axis of the cylinder, are given in full details. The technique explores the axial superpositions of infinitely thin dilatation disks inserted in the cylinder, the elastic fields of which are found in analytical form. In addition, the energies of interfaces separated domains with constant eigenstrains in the cylinder are given. It is demonstrated that the blurring of the eigenstrain in the transition region leads to a decrease in interface energy. [ABSTRACT FROM AUTHOR]

This work presents force and shape control strategies for adaptive structures subjected to quasi-static loading. The adaptive structures are designed using an integrated structure-control optimization method developed in previous work, which produces minimum “whole-life energy” configurations through element sizing and actuator placement optimization. The whole-life energy consists of an embodied part in the material and an operational part for structural adaptation during service. Depending on the layout, actuators are placed in series with the structural elements (internal) and/or at the supports (external). The effect of actuation is to modify the element forces and node positions through length changes of the internal actuators and/or displacements of the active supports. Through active control, the stress is homogenized and the displacements are kept within required limits so that the design is not governed by peak demands. Actuation has been modeled as a controlled non-elastic strain distribution, here referred to as eigenstrain. Any eigenstrain can be decomposed into two parts: an impotent eigenstrain only causes a change of geometry without altering element forces while a nilpotent eigenstrain modify element forces without causing displacements. Four control strategies are formulated: (C1) force and shape control to obtain prescribed changes of forces and node positions; (C2) shape control through impotent eigenstrain when only displacement compensation is required without affecting the forces; (C3) force control through nilpotent eigenstrain when displacement compensation is not required; and (C4) force and shape control through operational energy minimization. Closed-form solutions to decouple force and shape control through nilpotent and impotent eigenstrain are given. Simulations on a slender high-rise structure and an arch bridge are carried out to benchmark accuracy and energy requirements for each control strategy and for different actuator configurations that include active elements, active supports and a combination of both.

Evolution equations, or equations of motion, of moving defects are the balance of the “driving forces”, in the presence of external loading. The “driving forces” are defined as the configurational forces on the basis of Noether’s theorem, which governs the invariance of the variation of the Lagrangean of the mechanical system under infinitesimal transformations. For infinitesimal translations, the ensuing dynamic J integral equals the change in the Lagrangean if and only if the linear momentum is preserved. Dislocations and inclusions are “defects” that possess self-stresses, and the total driving force for these defects consists only of two terms, one expressing the “ self-force” due to the self-stresses, and the other the effect of the external loading on the change of configuration (Peach–Koehler force). For a spherically expanding (including inertia effects) Eshelby (constrained) inclusion with dilatational eigenstrain (or transformation strain) in general subsonic motion, the dynamic J integral, which equals the energy-release rate, was calculated. By a limiting process as the radius tends to infinity, the driving force (energy-release rate) of a moving half-space plane inclusion boundary was obtained which is the rate of the mechanical work required to create an incremental region of eigenstrain (or transformation strain) of a dynamic phase boundary. The total driving force (due to external loading and due to self-forces) must be equal to zero, in the absence of dissipation, and the evolution equation for a plane boundary with eigenstrain is presented. The equation applied to many strips of eigenstrain provides a system to solve for the position/ evolution of strips of eigenstrain.

Investigation of residual stresses is of crucial importance due to their effect on the performance of engineering components. Recently, inverse methods have been developed for determination of the residual stresses. Inverse eigenstrain method is one of the mentioned inverse methods. The inverse eigenstrain method, which is based on the eigenstrain theory, uses limited measurements of residual elastic strains obtained from the experimental tests. In this study, effective parameters on result accuracy obtained from the 2D inverse eigenstrain method in residual stresses measurement were investigated using numerical experiment. The results indicated that in the inverse eigenstrain method the accuracy of the results increases with increasing the basis functions order and the number of the points where displacement is measured. Additionally, the result accuracy increases selecting the appropriate basis functions. Moreover, in this paper the inverse eigenstrain method was applied for an actual part. The results showed that in the real conditions too, accurate results can be obtained by selecting the appropriate parameters of the inverse eigenstrain method.

Ericksen's problem consists of determining all equilibrium deformations that can be sustained solely by the application of boundary tractions for an arbitrary incompressible isotropic hyperelastic material whose stress-free configuration is geometrically flat. We generalize this by first, using a geometric formulation of this problem to show that all the known universal solutions are symmetric with respect to Lie subgroups of the special Euclidean group. Second, we extend this problem to its anelastic version, where the stress-free configuration of the body is a Riemannian manifold. Physically, this situation corresponds to the case where nontrivial finite eigenstrains are present. We characterize explicitly the universal eigenstrains that share the symmetries present in the classical problem, and show that in the presence of eigenstrains, the six known classical families of universal solutions merge into three distinct anelastic families, distinguished by their particular symmetry group. Some generic solutions of these families correspond to well-known cases of anelastic eigenstrains. Additionally, we show that some of these families possess a branch of anomalous solutions, and demonstrate the unique features of these solutions and the equilibrium stress they generate. [ABSTRACT FROM AUTHOR]

Background: The integral method of incremental hole-drilling is used extensively to determine the residual stress distribution in isotropic materials. When used with Tikhonov regularization, the method is robust and produces accurate results with minimal uncertainty. Alternatively, an optimal hole depth distribution can be found using the method of Zuccarello to improve the conditioning of the calibration matrices. If substantial measurement noise or a steep variation in stress exists, however, considerable uncertainty in, or distortion of, the calculated residual stress distribution can occur. Series expansion offers an alternative solution, but it has been reported to become unstable before meaningful accuracy can be achieved. Objective: Investigate the use of series expansion to determine a rapidly changing throughthickness residual stress distribution in an aluminium alloy 7075 plate subjected to laser shock peening treatment. Methods: Power series expansion of eigenstrains is used in finite element modelling to calculate the calibration coefficients. Monte Carlo simulation is used to determine robust uncertainties in the residual stress distributions. This allows the series order with the lowest RMS uncertainty in stress to be selected from those series orders that have converged. The best estimate of the residual stress distribution is thereby obtained. Results: Series expansion is shown to be stable up to 8th order and convergence to a stress solution can be found before instability dominates. The method is insensitive to measurement errors due to the least-squares approach employed by the inverse solution. Conclusions: The use of series expansion reduces the RMS uncertainty in stress when compared to the regularized integral and Zuccarello methods. [ABSTRACT FROM AUTHOR]

This paper describes an improved data reduction scheme for the deep-hole method of residual stress measurement. The deep-hole method uses the changes in diameter of a reference hole, drilled through the thickness of a component, to determine residual stress. The diameter changes result from the removal of a cylindrical core from the component, where the core is larger than and concentric with the reference hole. The new data reduction seeks to determine the unknown eigenstrain distribution that gives rise to the residual stress state and to the reference hole deformations; once the eigenstrain distribution is found, it is input to an elastic finite element analysis to provide the residual stress distribution in the original component. The new data reduction relies on expressing the unknown eigenstrain field in a polynomial basis, and finding the unknown basis function amplitudes from the measured reference hole diameter changes. The new data reduction is compared with the current technique, and it is shown that the proposed scheme offers several advantages to the current method of data reduction.

• Repeated slitting safe distance (RSSD) which determines the region where the material undergoes residual stress relief is defined. • The introduced RSSD parameter determines the minimum safe distance between consecutive slitting experiments to get accurate results. • The presented method reduces the run-time and slitting experiment expenses for low thickness specimens significantly. • The proposed method to repeat the slitting method on a single part was validated for an isotropic aluminium beam and for a transversely isotropic laminated composite. The slitting method is a destructive procedure whereby the residual stresses are determined throughout the thickness of the specimen. One should repeat this experiment to ascertain the uncertainties of the obtained results. However, performing the slitting experiment relieves the residual stresses around the cut which influences the results of the next experiments. In this paper the repeated slitting safe distance (RSSD) parameter was introduced which determines the proper distance between slitting experiments to guarantee the validity of the next ones. This method also contributes to reducing the calculation process and experiment costs especially for specimens with low thicknesses. The RSSD was obtained by two different methods including the eigenstrain based method and supplemental stress analysis. First, this distance was verified numerically for both isotropic and composite materials. Next, the slitting experiment was conducted for these materials and it was shown that the proposed method was successful to determine a proper distance to exclude the effects of the previous cuts. Image, graphical abstract [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the numerical simulation of coil sagging. This problem arises within the framework of the steel making industry where strips are wound on themselves for storage. Coil sagging is a major defect that can occur for recent grades undergoing phase transitions during the coiling process. The detailed mechanisms leading to coil sagging are still not well understood, making this phenomenon very difficult to prevent. The coil is a multilayer hollow cylinder where sliding takes place at each interface and significantly contributes to the overall deformation. However, a detailed numerical simulation addressing the contact problem, considering both pressure and sliding is difficult to perform under non-axisymmetric conditions. This paper presents a simplified approach considering an orthotropic hollow cylinder instead of a multilayer coil. The anisotropy is due to contact roughness that tends to decrease the radial stiffness. The hollow cylinder is subjected to gravity and an eigenstrain representing thermal expansion, phase transitions and transformation induced plasticity. Sliding at each interface is taken into account through a continuous plastic-like shear strain that is determined through an energetic principle. The proposed solution relies on analytical developments so that computation time is compatible with parametric studies. Results are addressed in order to give a better understanding of mechanisms and conditions under which coil sagging occur. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a fast method of solving 3D contact problems when one of the mating bodies has an elastic-damageable behavior. The damage model is implemented in a semi-analytical model using Eshelby' s equivalent inclusion method in the contact solver. The proposed technique can be seen as an enrichment technique for which the effect of heterogeneous inclusions is surimposed on the homogeneous solution in the contact algorithm. Contact pressure and subsurface stress field computation time is kept small due to a massive use of 3D and 2D Fast Fourier Transforms. Cuboidal inclusions with the same size as the discretization of the half-space and with the same elastic properties are surimposed. The damage model affects the elastic properties of the cuboidal inclusions. The emphasis is put on the effects of the fretting regimes on the contact pressure and damage evolution. [ABSTRACT FROM AUTHOR]

Highlights • The least squares method for implementing the unit pulse functions and the Tikhonov regularization technique were developed for determining the residual stresses using the slitting method. • The developed least squares and Tikhonov regularization techniques were validated by reproducing a known residual stress field inside an aluminum beam using the four-point bending experiment. • Laminated composite specimens were subjected to thermal loading and the induced residual stresses were determined by the slitting method and utilizing the developed least squares and Tikhonov regularization techniques. • The present methods significantly decreased the noise and uncertainty in the experimental measurements of both metals and composites. Abstract One of the challenges of measuring the residual stresses in metals and composites is the adverse effect of the noise and error on final results. The least squares and the Tikhonov regularization methods are the two available techniques for the noise reduction of the residual stress measurement in materials. In the present paper, the least squares method was developed for the residual stress measurement in metallic and laminated composites. Moreover, the application of the Tikhonov regularization method was extended for the eigenstrain based method and used for the measurement of residual stresses in laminated composites. The proposed methods were validated by performing experiments on an aluminum beam and a laminated composite plate using the slitting method. It was illustrated that the present methods diminished the uncertainty and sensitivity to noise of the final results successfully. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

Abstract Tungsten is the main candidate material for plasma-facing armour components in future fusion reactors. Bombardment with energetic fusion neutrons causes collision cascade damage and defect formation. Interaction of defects with helium, produced by transmutation and injected from the plasma, modifies defect retention and behaviour. Here we investigate the residual lattice strains caused by different doses of helium-ion-implantation into tungsten and tungsten‑rhenium alloys. Energy and depth-resolved synchrotron X-ray micro-diffraction uniquely permits the measurement of lattice strain with sub-micron 3D spatial resolution and ~10−4 strain sensitivity. Increase of helium dose from 300 appm to 3000 appm increases volumetric strain by only ~2.4 times, indicating that defect retention per injected helium is ~3 times higher at low helium doses. This suggests defect retention is not a simple function of implanted helium dose, but strongly depends on material composition and presence of impurities. Conversely, analysis of W-1 wt% Re alloy samples and of different crystal orientations shows that both the presence of rhenium, and crystal orientation, have a comparatively small effect on defect retention. These insights are key for the design of armour components in future reactors where it will be essential to account for irradiation-induced dimensional change when predicting component lifetime and performance. Graphical abstract Unlabelled Image Highlights • Helium-implantation causes large strains likely to reduce longevity of plasma-facing tungsten armour in fusion reactors. • Non-linear defect retention (ten times dose increase only increases strain three times) suggests saturation regime onset. • Little grain-orientation dependence means texture of tungsten armour can be optimised for other performance aspects. • Not all transmutation elements strongly affect defect retention, e.g. 1% Re causes little increase in defect retention. [ABSTRACT FROM AUTHOR]

Current methods for incremental hole-drilling in composite laminates have not been successfully applied in laminates of arbitrary construction or where significant variation of residual stress exists within a single ply. This work presents a method to overcome these limitations. Series expansion is applied to each ply orientation separately so that the discontinuities in the residual stresses at ply interfaces can be correctly captured. Temperature variations described by power series are used to set up eigenstrains and consequent stresses which vary in the through-thickness direction. The calibration coefficients at each incremental hole depth are calculated through the use of finite element modelling. The inverse solution employs a least-squares approach which makes the resulting solution insensitive to measurement uncertainty. Robust uncertainties in the residual stress distributions are determined using Monte Carlo simulation. The residual stress distribution is found from that combination of series orders in the different ply orientations that has the lowest RMS uncertainty, selected only from those combinations that have converged. The method is demonstrated on a GFRP laminate of [02/+45/−45]s construction where it is found that transverse cracking of the plies at the inner surface of the hole may have impacted on the accuracy of the results. [ABSTRACT FROM AUTHOR]

A novel approach to separating macroscopic (Type I) from microscopic (Type II + III) residual stress is presented, based on Focused Ion Beam – Digital Image Correlation (FIB-DIC) ring-core stress evaluation and eigenstrain modelling. This approach was applied to study the residual stresses for a titanium alloy bar following plastic four-point bending. It was found that electrochemical polishing is a surface preparation technique that is very well suited for FIB-DIC ring-core measurements, in the sense that it removes the influence of prior sample grinding and polishing, leads to a stress profile that satisfies force and moment equilibrium, and thus enables the evaluation of absolute values of total residual stress. The obtained relief strain profile across the bar width is asymmetric, highlighting the difference in the alloy's response to tension and compression. Total experimental residual stress values were calculated using (i) the assumption of material elastic isotropy, with an average Young's modulus, and (ii) under the assumption of elastic anisotropy, taking into account the crystallographic orientation of each investigated grain. Based on the measured relief strain values, the eigenstrain distribution in the bar was reconstructed and used to obtain the macroscopic (Type I) residual stress profile. The differences in the residual stress between the eigenstrain reconstruction values and the individual experimental results were ascribed to the local microscopic (Type II + III) residual stresses. This conclusion was substantiated by revealing the correlation between the residual stress values in individual grains in the elastic zone and their respective Young's moduli in the loading direction, as well as the correlation between the residual stress values in grains located in the plastic zone and their respective Schmid factors for basal slip. [ABSTRACT FROM AUTHOR]

The main source of welding residual stress is identified by investigating the distribution of permanent plastic strains in as-welded and post-weld heat treated specimens of Inconel 740H. For this purpose, inverse eigenstrain problem is solved using experimental displacement data obtained by high precision coordinate measuring machine measurements from the surface of transversal wire-cut of electric discharge machining. A new multi-component iterative inverse eigenstrain model is developed and, in total, three different inverse eigenstrain models are analysed to get a high-quality fit with experimental data and have a reasonable prediction of residual stresses in the whole body of nonuniform bead-on-plate specimen design. In order to determine the boundaries of permanent plastic strains, which are the main source of welding residual stress, eigenstrain distribution size is analysed in terms of mean squared error using three models. The distribution size that provides the best match between the calculated displacements and the measured deplanation on the surface of cut is determined. The multi-component iterative eigenstrain reconstruction model validated itself by providing a good agreement with experimentally determined displacement and residual stress profiles, and this model is used to predict residual stress distribution in the whole body. [ABSTRACT FROM AUTHOR]

The development of harder and thinner new steel grades requires computationally efficient numerical simulations of forming processes in order to optimize industrial conditions through parametric studies. Within this general framework, the present contribution deals with one particular process, namely the run out table. Thus, this paper focuses on the evolution of residual stresses of thin strips during cooling on the run out table. Due to the fact that the complete problem is a nonlinear multiphysics process, it is known that simulating such processes with fully coupled numerical procedures leads to high computational costs. Therefore, a simplified numerical strategy has been developed. This procedure consists of three steps: (i) solving the thermal problem coupled with multiphase transitions; (ii) computing thermal expansion, metallurgical deformation and transformation induced plasticity and (iii) solving the associated mechanical problem. Residual stress profiles through the strip thickness are also computed in order to evaluate classic flatness defects such as crossbow and longbow . A post-processing is also included in order to quantify out of plane displacements that would take place if the strip was cut off the production line. The post-processing consists in computing at finite strain the relaxation of residual stresses when the tension applied by the coiler is released. The proposed numerical strategy has been tested on common industrial conditions. [ABSTRACT FROM AUTHOR]

We present a comprehensive study of the effects of internal boundaries on the accuracy of residual stress values obtained from the eigenstrain method. In the experimental part of this effort, a composite specimen, consisting of an aluminum cylinder sandwiched between steel cylinders of the same diameter, was uniformly heated under axial displacement constraint. During the experiment, the sample temperature and the reaction stresses in the load frame in response to changes in sample temperature were monitored. In addition, the local (elastic) lattice strain distribution within the specimen was measured using neutron diffraction. The eigenstrain method, utilizing finite element modeling, was then used to predict the stress field existing within the sample in response to the constraint imposed by the load frame against axial thermal expansion. Our comparison of the computed and measured stress distributions showed that, while the eigenstrain method predicted acceptable stress values away from the cylinder interfaces, its predictions did not match experimentally measured values near them. These observations indicate that the eigenstrain method is not valid for sample geometries with this type of internal boundaries. [ABSTRACT FROM AUTHOR]

The eigenstrain tensor is defined in terms of the unit vector normal to the fracture plane and the dislocation vector to characterize the fracture geometry and to determine the fracture induced deformation field. Compared to the conventional nonlinear fracture model, the forward fracture model provides a linear relationship between fracture geometry parameters (eigenstrain) and the induced deformation. Therefore, a linear inverse problem needs to be tackled in mapping the fracture geometry by inversion of tilt observations. The fracture orientation and opening distribution of a tensile-dominated hydraulic fracture can be mapped iteratively in a coupled way by using the forward model defined in terms of the dislocation vector. The proposed analysis method is applied to field tilt data collected for monitoring the growth of a hydraulic fracture placed in the preconditioning of a roof rock over a coal longwall panel at a test site. Inversion of the tilt data produces a low fracture dip angle of 6.8°, which agrees with a nearly horizontal fracture orientation confirmed by intersection data collected in offset monitoring boreholes. Tikhonov regularization technique is applied to stabilize the discrete ill-posed inverse problem. The regularized solution provides information on the fracture opening distribution and asymmetric fracture growth. The inversion shows that the fracture grows uniformly and roughly in a radial pattern at early injection times, but grows asymmetrically, preferentially, and consistently to the northwest during later times. The predicted fracture growth agrees with the temperature logging to measure fracture intersections with the offset monitoring boreholes. • Linear relationship between fracture geometry and induced deformation is defined. • Tiltmeter array design is quantitively assessed by studying resolution matrices. • Linear inverse problem is solved to map fracture by inversion of tilt data. • Mapped asymmetric fracture growth agrees with logging in monitoring boreholes. [ABSTRACT FROM AUTHOR]

Statistical predictions of crack propagation are requested to evaluate remaining lifetime of operating welded structures. Today, crack growth rate for each observed crack cannot be evaluated accurately without neutron diffraction and synchrotron X-ray diffraction due to the difficulty of nondestructive measurements of welding residual stresses in the thickness direction. However, it is difficult to apply those nondestructive diffraction methods as on-site measurement techniques because the higher energy diffraction methods are available only in special irradiation facilities. To make things worse, measured results by diffraction methods cannot be directly applied to the FEM (finite element method) model for crack propagation prediction. From this view point, the methods based on the eigenstrain methodology have been proposed. In the bead flush method, for example, three-dimensional welding residual stresses are calculated by an elastic FEM analysis from eigenstrains which can be estimated by the inverse analysis from released strains during the removal of the weld reinforcement. Here, the removal of the excess metal is nondestructive treatment essentially because it is effective to eliminate stress concentration zone. In this study, numerical simulations for a welded pipe under SCC (stress corrosion cracking) were carried out to evaluate crack propagation statistically. As well, estimation accuracies of crack propagation using residual stresses estimated by the bead flush method were compared with the accuracy using residual stresses assumed to be measured by diffraction methods. Prediction accuracies of crack propagation estimated by this method were higher than that by diffraction methods. It is because estimated results base on the eigenstrain methodology satisfy the self-equilibrium condition of residual stress.

Elastostatic measurements of deformations by using tiltmeters located at moderate to large distances from a hydraulic fracture have often been used to map fracture geometry. The fracture is generally modeled as a quasi-static elastic dislocation, and the analytical solution for the induced deformation by the dislocation in a homogeneous elastic material is commonly used as a forward model to map the fracture geometry by fitting the deformation measurements. However, the mechanical properties of the reservoir in which a hydraulic fracture is placed are often very different from those of the surrounding rocks. In order to obtain a correct interpretation of the created fracture geometry, the mechanical property contrasts need to be recognized and taken into account. In this paper, a mechanical model is presented and applied to the problem of calculating the far-field deformation and stresses induced by a fracture in a reservoir with mechanical properties different from those of the otherwise homogeneous elastic half-space. The fracture is modeled as a constant displacement discontinuity over the fracture plane, and the corresponding eigenstrain associated with the fracture is defined based on the geometries of the fracture and reservoir. Analytical expressions connecting the induced far-field deformation and stresses with the geometrical and mechanical properties of the inhomogeneous system have been established based on the equivalent inclusion method. The proposed model offers a simple and computationally efficient method to account for the effect of inhomogeneous mechanical properties on the far-field deformation induced by the fracture, which can then be used to undertake tiltmeter mapping of the fracture geometry. The mechanical model is verified by comparing with finite element results and analytical solutions available in the literature. [ABSTRACT FROM AUTHOR]

Understanding the fields that are set up in and around inhomogeneities is of great importance in order to predict the manner in which heterogeneous media behave when subjected to applied loads or other fields, e.g., magnetic, electric, thermal, etc. The classical inhomogeneity problem of an ellipsoid embedded in an unbounded host or matrix medium has long been studied but is perhaps most associated with the name of Eshelby due to his seminal work in 1957, where in the context of the linear elasticity problem, he showed that for imposed far fields that correspond to uniform strains, the strain field induced inside the ellipsoid is also uniform. In Eshelby's language, this corresponds to requiring a uniform eigenstrain in order to account for the presence of the ellipsoidal inhomogeneity, and the so-called Eshelby tensor arises, which is also uniform for ellipsoids. Since then, the Eshelby tensor has been determined by many authors for inhomogeneities of various shapes, but almost always for the case of uniform eigenstrains. In many application areas in fact, the case of non-uniform eigenstrains is of more physical significance, particularly when the inhomogeneity is non-ellipsoidal. In this article, a method is introduced, which approximates the Eshelby tensor for a variety of shaped inhomogeneities in the case of more complex eigenstrains by employing local polynomial expansions of both the eigenstrain and the resulting Eshelby tensor, in the case of the potential problem in two dimensions. [ABSTRACT FROM AUTHOR]

During solid-solid phase transitions, the eigenstrain introduced by the geometrical transformation in the newly formed phase is a significant issue. Indeed, it is responsible for very large elastic energy and dissipation at the continuum scale that have to be added to the total energy in order to determine if a phase transition can occur. The eigenstrain can cause sliding of the newly formed grain. In this paper, an analytical method coupled with numerical energetic optimization is derived to solve the problem of a two-dimensional circular elastic sliding inclusion accounting for plastic dissipation at the interface. Numerical calculations under plane stress assumption show that dissipation enables an effective decrease in the energy needed for the phase transformation to occur. [ABSTRACT FROM AUTHOR]

Numerical modelling of the residual stresses and strains within mechanical components is of great importance for improving the quality and reliability of design for structural integrity. A particularly versatile and powerful approach is offered by direct and inverse eigenstrain modelling. The nature of the eigenstrain modelling approach is that it not only generates an efficient parametric representation of the residual stress field, but also ensures consistency by enforcing stress equilibrium and strain compatibility. In the present study we propose a particular way of prescribing the eigenstrain field due to surface treatment such as shot peening. Eigenstrain variation is described by a continuous function of the distance from the boundary of the object in a two-dimensional model of its cross-section. The procedure is compatible with the use of commercial numerical simulation software, and allows correct assignment of all eigenstrain components. We apply the technique to the evaluation of residual strain within an additively manufactured nickel superalloy compressor blade that was subsequently subjected to shot peening treatment. Two experimental techniques are used to validate the model, namely, Focused Ion Beam ring core milling (FIB-DIC) and synchrotron X-ray Powder Diffraction (SXRPD). Consistency between model prediction and experimental measurements provides verification of the suitability of eigenstrain modelling as consistent basis for the incorporation of residual stress effects on the deformation behaviour of manufactured components. [ABSTRACT FROM AUTHOR]

In this study, the residual stress relaxation behaviour occurring during low-cycle fatigue in shot-peened specimens with either a flat or a notched geometry has been studied. A representative low-pressure steam turbine material, FV448, was used. The residual stress and strain hardening profiles caused by shot peening were measured experimentally and were then incorporated into a finite element model. By allowing for both effects of shot peening, the residual stress relaxation behaviour was successfully simulated using this model and correlated well with the experimental data. Although more modelling work may be required to simulate the interaction between shot peening effects and external loads in a range of notched geometries, the model predictions are consistent with the specimens tested in the current study. The novelty of this study lies in the development of such a modelling approach which can be used to effectively simulate the complex interaction between shot peening effects and external loads in notched regions. Compared with the un-notched geometry, the notched geometry was found to be more effective in retaining the improvement in fatigue life resulting from shot peening, by restricting the compressive residual stress relaxation during fatigue loading. [ABSTRACT FROM AUTHOR]