Eigenstrain Method Research Articles

Warpage analysis of multilayer thin film/substrate systems using the Eigenstrain method

Warpage resulting from the deposition of films on substrates remains a persistent challenge that significantly impacts the application of numerous advanced devices. To precisely evaluate warpage in multilayer thin film/substrate systems, we employ the eigenstrain method in the warpage analysis. An analytical model is developed to predict warpage and residual stress in multilayer film/substrate system based on classical laminate theory. We introduce a novel concept termed “Eigenstress”, which serves as the fundamental cause of warpage and characterizes the influence of the manufacturing process on warpage. Theoretical analysis and simulation demonstrate that warpage is entirely determined by eigenstress and other process-independent parameters. This model also explains that there is little interaction between thin films deposited on the same substrate. An experimental method is studied for measuring eigenstress using a standard plate. The measured eigenstress differs significantly from the corresponding thermal stress. We also propose a critical condition and its governing equation for warpage in multilayer film/substrate systems. Experimental results confirm that adding a compensation layer on a warped substrate can considerably reduce warpage. The warpage model based on “Eigenstress” provides a theoretical foundation for accurately predicting and controlling warpage in multilayer thin film/substrate systems.

Residual stress prediction across dimensions using improved radial basis function based eigenstrain reconstruction

Accurate prediction of residual stress has significant impacts on structural fatigue and long-term performance. Eigenstrain reconstruction emerges as a competitive method for residual stress prediction, due to the appealing engineering adaptability and cost effectiveness. However, traditional eigenstrain methods, by polynomial forms, have difficulties in reconstructing residual stress with complex contour. In such cases, high-order polynomials have to be used to enhance reproduction capability within the domain. Unfortunately, polynomial interpolation with high degree over a set of measurements often encounters unexpected numerical oscillation, known as the Runge's phenomenon. This numerical limitation will misinterpret the residual stress with abrupt change or near the boundary of structures. To tackle these limitations, this paper investigates the residual stress prediction across dimensions. An eigenstrain reconstruction method based on radial basis function (RBF) is proposed in this work. By virtue of least squares method, full-field residual stress can be reproduced by solving an inverse eigenstrain problem through minimizing the residual errors between numerical predictions and experimental measurements. The radial basis function is used as the basis function space to reconstruct the full scale eigenstrain, and then the residual stress in three different dimensions. More importantly, a novel elliptical radial basis function has been proposed for predicting welding residual stress. Thanks to the unique scatter interpolation feature of radial basis function with limited experimental data, the distinctive advantage of the proposed method lies in the excellence of accurately predicting complex residual stresses in one dimension, the whole two dimensions or even three-dimensional components. The eigenstrain reconstruction method based on radial basis function enriches residual stress prediction with complex profiles in two-dimensional and three-dimensional space.

Investigating the applicability of the layered average eigenstrain method in laser shock peening

The eigenstrain model is widely used to predict residual stress and deformation resulting from laser shock peening (LSP). However, most existing eigenstrain models rely on the input of layered average eigenstrain (LAE), which has not been fully verified, particularly considering the non-uniform eigenstrain distribution in the workpiece. To address this issue, a modified eigenstrain model is proposed that uses periodic array eigenstrain (PAE) as input to verify the accuracy of the LAE method. First, a dynamic LSP model is developed to obtain the initial eigenstrain, which is then used in both the LAE and PAE models. The effects of overlapping rate and laser energy of LSP are analyzed by comparing the results of residual stress and displacement predictions. The results show that although the LAE model accurately predicts the displacement induced by LSP, it has errors in predicting residual stress. Although the LAE model has errors in predicting residual stress, its results can still be used to predict the stress intensity factor (SIF) of laser shock peening with some fluctuations. Therefore, we conclude that the LAE method is applicable under high overlapping conditions in LSP, while the proposed PAE-based eigenstrain model provides a more accurate approach for predicting residual stress and deformation during LSP. This has practical implications for the design and optimization of LSP processes.

Voxel‐Based Full‐Field Eigenstrain Reconstruction of Residual Stresses

Inverse eigenstrain (inherent strain) analysis methods are shown to be effective for the reconstruction of residual stresses in plane eigenstrain problems (continuously processed bodies) while conversely residual stress reconstruction in discontinuously processed bodies is extremely challenging and necessitates the use of complex regularizing assumptions. Herein, a new generic inverse eigenstrain method suitable for the reconstruction of residual stresses along with residual elastic strains and displacements in discontinuously processed bodies is introduced. The proposed method uses the superposition of eigenstrain radial basis functions together with a set of limited experimental data for model‐free (unconstrained) determination of unknown eigenstrain fields. This approach eliminates the limitations introduced by global basis functions such as polynomials. The novel point of this method is the ability to account for all six components of strain in an isotropic body without using regularizing assumptions. By lifting complex guiding formulation, the fidelity of full‐field eigenstrain reconstruction becomes directly related to the quality of experimental data and proper discretisation of the model domain. The FEniCS implementation has been validated using the experimental data of pointwise high‐energy synchrotron X‐ray diffraction measurements from a bent titanium alloy bar. A hybrid high throughput computing approach is also introduced for effective parallel computing.

A novel analysis of stress intensity factors for distributed non-planar surface cracks and its application to material quality control

A novel method for the numerical analysis of stress intensity factors (SIFs) was developed for multiple, 3-d-distributed, semi-elliptical cracks on the surface of a semi-infinite elastic body under arbitrary loading conditions. Emphasis was placed on the nature of interactions between cracks, with examination by the distributed infinitesimal dislocation loop technique, an eigenstrain method. SIFs of 100 distributed surface cracks were acquired in approximately 15 min when calculated on a standard personal computer, owing to the numerical integration approaches and meshing procedure adopted. The validity of the obtained results is verified and its applicability to some engineering problems is discussed.

Numerical Investigation of Influence of Spot Geometry in Laser Peen Forming of Thin-Walled Ti-6Al-4V Specimens

The aviation industry demands thin-walled structures of high dimensional accuracy. Varying radii and individual use-cases, e.g. for repair purpose, require flexible forming techniques. Laser peen forming (LPF) represents such a forming process providing precise energy input by a pulsed laser over a wide energy range. Among adjustable parameters such as laser intensity and focus size, the spot shape, i.e. square and circular, is usually fixed for a specific laser system. As the spot shape is a crucial parameter, this work focuses on the effect of the spot shape on structural deformation after LPF application. Therefore, models for laser peen forming of thin-walled Ti-6Al-4V strips for LPF systems with circular and square focus shapes are set up. Geometric conditions on both focus shapes ensure equal energy input during the laser processing. The numerical simulation relies on the so called eigenstrain method, leading to a cost-efficient calculation of resulting deformation after the dynamic LPF process. Square-based peening pattern exhibit higher deflection. For increasing spot size, the deflection difference between square and circle-based patterns increase slightly.

On the elastic modulus of rock-filled concrete

Rock-filled concrete (RFC) is constructed by placing large rocks into the prescribed formwork then pouring self-compacting concrete (SCC) to fill interstitial voids. The prelaid rock skeleton transfers forces and provides a certain degree of stiffness at the beginning of the hardening process, which makes the elastic modulus evolution of RFC different from that of conventional concrete. This study aims to reveal the elastic modulus evolution of RFC at early ages, bridging the gaps of RFC research in the numerical computational field. Firstly, RFC is modeled using a mesoscopic finite element approach in which prelaid rocks, SCC, and interfacial transition zone are represented explicitly. This model is validated by onsite full-scale test results. Secondly, the hardening process of RFC is simulated using the validated model, and a qualitative evolution function is obtained. Finally, the advanced Dual Eigenstrain method is employed and calibrated based on the simulation results to predict the elastic modulus of RFC at any curing age.

A fuzzy finite element model based on the eigenstrain method to evaluate the welding distortion of T-joint fillet welded structures

T-joint fillet welded structure is one of the most commonly welded structures in the assembly of vessels, so it is crucial to accurately predict the welding distortion of T-joint fillet as fast as possible for in-process quality controls in ship building industry. Eigenstrain distribution which forms after welding accounts for the appearance of welding distortion, the results obtained from experiment and thermal elastoplastic finite element (FE) analysis indicate that it is closely related with the formation of heat-affected zone (HAZ), however, a quantitative description needs to be determined based on limited experimental data. To deal with this problem, a fuzzy finite element model (fFEM) is proposed to evaluate welding distortion in T-joint fillet welding using the eigenstrain method. A set of coefficients which determines the eigenstrain distribution is introduced as the optimization parameters in fFEM. The optimized coefficients of eigenstrains can be found by introducing the genetic algorithm (GA) and combining it with the commercial FE software ANSYS. The eigenstrain distribution calculated by the optimized coefficients can predict the welding distortion with high accuracy and efficiency, the same optimized coefficients and corresponding eigenstrain distribution can be applied to various dimensions of T-joint welded structures if the welding parameters remain unchanged.

An iterative approach combined with multi-dimensional fitting of limited measured stress points to reconstruct residual stress field generated by laser shock peening

In this paper, an iterative approach combined with least squares fitting of limited measured points is introduced to reconstruct residual stress distribution after laser shock peening (LSP) treatment. The iterative approach is started by introducing prescribed stress components in the data measurement region (the mapped zone) in the finite element (FE) model, then a static equilibrium analysis is conducted, which will cause the stress distribution in the mapped zone and unmapped zone both redistribute if they don't converge to the target stresses. By repeatedly introducing the input stress in the mapped zone to converge to the prescribed stress components, the complete residual stress distribution in the whole region can be obtained. Firstly, to validate the viability of this method, based on the full stress components information of ‘measured’ sample points extracted from FE models, residual stress reconstructions of single-pass welding of a thick plate and a single shot of LSP have been carried out, respectively. After that, to examine the robustness of the method, the iterative approach combined with multi-dimensional least squares fitting is tested using a limited set of ‘measured’ sample points extracted from the FE model of single shot LSP treatment. It is found that besides the commonly studied depth direction, the in-plane stress components are also varied with its in-plane locations, which are due to “edge effects”. The residual stress fields for the whole region reconstructed by the proposed approach agree well with the actual stress fields even if some of the non-critical stress information are missing. Finally, the results are compared with those reconstructed by the eigenstrain method. It is shown that the proposed iteration approach can reconstruct the residual stress distribution in the whole region at a satisfactory level by solving a direct linear elastic static equilibrium problem, while the eigenstrain method needs to solve an inverse problem to obtain the appropriate eigenstrain distribution.

A novel approach to reconstruct residual stress fields induced by surface treatments in arbitrary 3D geometries using the eigenstrain method

A novel approach has been proposed to reconstruct residual stress fields after surface treatments in arbitrary 3D geometries using the eigenstrain method based on bicubic spline interpolation surface and finite element (FE) method. Firstly, the complex curved surface is represented by piecewise interpolation of the smooth surface using explicit bicubic functions, then the computation of the distance and corresponding orientation are calculated by solving a constrained nonlinear multivariable problem using MATLAB, which is insensitive to the complexity of the curved treated surface and reduces the errors induced by differentiation operations when using Wall distance module in COMSOL in previous works. Three instances of 3D geometries after surface treatment studied previously in literature and finally an instance of a real turbine blade after LSP treatment conducted by ourselves are analyzed, which validate the proposed approach and illustrate its effectiveness in the reconstruction of residual stress fields for complex 3D geometries in various cases. The proposed approach provides a general and flexible way to the implementation of eigenstrain method in arbitrary 3D geometries using data interchange between the mathematical calculation software and FE analysis software for the design in industrial applications.

The use of profilometry techniques and eigenstrain theory for the analysis of creep behavior in nickel superalloy welds

This paper presents a summary of recently developed experimental and computational tools to reconstruct residual stress fields and analyze creep in nickel superalloy welds used in aerospace engineering components. This approach combines experimental data with eigenstrain theory to reconstruct stress fields at the macroscopic scale and provided reliable means for numerical prediction of creep behavior of welded components under complex loading conditions. Experimental data in the form of profilometry scans was interpreted using a range of iterative eigenstrain methods that included the adaptation of the contour method and artificial intelligence models for eigenstrain-creep analysis. The integration of principles of artificial intelligence with eigenstrain models allowed highly accurate results to be obtained which were validated by comparison with experimental data obtained using independent techniques such as neutron diffraction. The use of artificial intelligence models is discussed for residual stress reconstruction and creep behavior prediction in annular aeroengine parts manufactured using inertia friction welding. To extend the range of experimental data taken into consideration, the height Digital Image Correlation (hDIC) technique was introduced that utilizes information regarding triaxial displacements obtained from profilometry, allowing deeper and more reliable analysis to be conducted. The hDIC technique was validated using operando tensile testing data.

Effect of thermal residual stresses on ductile-to-brittle transition of a bi-material specimen by using the CAFE method

The effect of residual stress on fracture of materials or structures has been widely studied. However, its influence on ductile-to-brittle transition (DBT), a crucial phenomenon of structural materials, has rarely been investigated so far. In the present study, employing the eigenstrain method residual stresses are introduced into a bi-material specimen, where two configurations of crack and interface, e.g., one with interface perpendicular and one parallel to the crack extension, are designed to study the influence of residual stress. The DBT of the bi-material specimen in the presence of residual stresses is numerically studied by using the CAFE method where temperature dependent surface energy is implemented to calculate absorbed energy of Charpy impact testing specimen. It is found that residual stress generated in the two configurations affect the DBT in a similar manner. The DBT curves generally shift to higher temperature due to the decrease of absorbed energy with the increase of residual stress. It is found that the decrease of absorbed energy in both configurations is caused by the additional constraint on the notch root, which is induced by the residual stress and can facilitate the fracture.

Eigenstrain Modeling of Laser–Shock Processing of Materials

Finite-element modeling of classical one-time and repeated laser–shock processing of materials by the eigenstrain method is considered. The distribution of the residual compressive stress in the surface layers is analyzed.

Reconstruction of residual stress and work hardening and their effects on the mechanical behaviour of a shot peened structure

Generation of residual stress through inhomogeneous plastic deformation is inevitably coupled with work hardening/softening activities of materials. Being able to precisely model the residual stress and the work hardening is of great importance for structure optimization and design. Eigenstrain is an effective method to reconstruct residual stress field in a structure with the consistency of stress equilibrium and strain compatibility. However, the hardening behaviour of materials is generally not considered in the eigenstrain method. This paper presents a reconstruction approach of residual stress and work hardening generated by shot peening in a cylindrical structure. This approach takes into account both the stress equilibrium and the elastic-plastic equilibrium conditions. In order to experimentally validate this reconstruction approach, a cylindrical sample in 316 L stainless steel is shot peened. The in-depth residual stress and work hardening variations are measured and evaluated using X-ray diffraction. Self-equilibrium analyses based on Finite Element (FE) modelling indicate that the approach developed in this work is efficient to reconstruct equilibrated residual stress and work hardening fields. Moreover, the effects of the initial residual stress and the work hardening on the mechanical behaviour of the structure are investigated through simulation of tensile tests using the FE model. It is demonstrated that an accurate consideration of work hardening during reconstruction, which is strongly deformation history-dependent, is essential to precisely describe the mechanical behaviour of a shot peened material.

Finite Element Modeling of the Technology of Multiple Laser Shock Processing of Materials Using the Eigenstrain Method

The laser shock processing (LSP) of material is an efficient modern technology of processing of metal materials, during which significant compressive residual stresses contributing to an increase in their strength and tribological and operational characteristics are generated in the subsurface area. The finite element modeling of the technology of multiple laser shock processing is carried out using the eigenstrain method. The level of the compressive residual stresses arising under LSP is determined. It is shown that the residual stresses on the surface of the VT-6 alloy grow from 510 to 830MPa with an increase in the number of pulses from 1 to 4, and the depth of the zone of the compressive residual stresses increases respectively from 1.26 mm after the first pulse to 1.60 mm after the fourth pulse.

Boundary Effects in the Eigenstrain Method

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.

A simplified FEM eigenstrain residual stress reconstruction for surface treatments in arbitrary 3D geometries

Residual stress is of great significance for the structural integrity of components and assemblies. Its proper evaluation and integration into failure criteria may considerably enhance the engineering parts performance. Although experimental measurements of residual stress provide important input, the crucial next step is the incorporation of the information obtained into numerical modelling. Eigenstrain method is a powerful way of specifying residual stress in structures in a self-consistent way that satisfies the mechanics requirements of stress equilibrium and strain compatibility. This approach also enables the verification of experimental data consistency. In combination with the principle of transferability of eigenstrain, it allows the application of well-characterised material processing conditions to different solid geometries with the purpose of predicting residual stress. In the present work, a novel user-friendly implementation framework is presented that is able to predict the residual stress fields arising due to material surface treatment of arbitrary three-dimensional geometries. In the first instance, detailed analysis of potential sources of error was conducted, followed by verification of the validity of the procedure conducted by analysing relevant cases from the literature: laser shock peening and carburising combined with quenching. The work presented here opens new ways to use the eigenstrain method for real industrial applications where the mechanical components geometry is generally complex. Particularly, it enables the integration of residual stress into well-established FEM fatigue prediction codes when the experimental data available is limited.

An Experimentally Validated Combined Stiffness Formulation for a Finite Domain Considering Volume Fraction, Shape, Orientation, and Location of a Single Inclusion

This work characterizes the stiffness of a finite domain containing one (biaxial ellipsoidal) void due to the combined effect of inclusion’s attributes: (1) size or volume fraction, VF, (2) shape or aspect ratio, AR, (3) angular orientation, and (4) location (position) within the matrix. The values and ranges of these ellipsoidal inclusion attributes are varied according to a matrix developed using design of experiments (DOE). Modified Mori–Tanaka method combined with dual-eigenstrain method (interior and exterior eigenstrain methods) is used to determine the effective stiffness tensor of the composite domain. Employing the numerically calculated normalized axial modulus [Formula: see text] values in SAS/STAT®, a nonlinear mathematical expression of [Formula: see text] as function of the void’s variables is arrived at Stiffness values found from the numerical homogenization scheme are experimentally corroborated using compression tests conducted on 3D-printed ABS cubes having a single ellipsoidal inclusion of various geometric attributes. In addition, finite element simulations were run of said uniaxial compression test cases to further validate the numerical homogenization results. Corroborated findings suggest that while the location of the inclusions in the matrix have no significant effect on normalized modulus [Formula: see text], the void’s volume fraction has the largest effect where it decreases with VF. The effect of the void’s orientation and elliptical aspect ratio are significant. [Formula: see text] increases with AR at angles ranging from 0–[Formula: see text]; at [Formula: see text][Formula: see text] are almost constant with AR, at angles of 60–[Formula: see text] values of [Formula: see text] decrease with AR. As AR approaches unity, the effect of orientation decreases significantly.

Residual Stress From Cold Expansion of Fastener Holes: Measurement, Eigenstrain, and Process Finite Element Modeling

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.

固有ひずみ法による厚肉溶接平板に対する溶接残留応力推定のためのX線回折法の導入

固有ひずみ法による厚肉溶接平板に対する溶接残留応力推定のためのX線回折法の導入