publications

2025

1. NeuberNet: a neural operator solving elastic-plastic PDEs at V-notches from low-fidelity elastic simulations

   T. Grossi , M. Beghini , and M. Benedetti

   Research Square Preprint, Mar 2025

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   We present NeuberNet, a nonlinear manifold decoder that learns a family of operator mappings on the domain of reentrant corners between far-field displacement boundary conditions obtained with low-fidelity elastic simulations and the high-resolution stress and strain fields that stem from the elastic-plastic axisymmetric solid mechanics equations, under the only assumptions of small-scale plasticity and bilinear isotropic hardening. We envision NeuberNet as a data-driven application of the substructuring principle in solid mechanics, engineered to simulate complex geometries by employing plastic material behavior only in the vicinity of stress raisers where nonlinearities are most likely to occur. We provide practical guidelines for mesh resolution in the initial low-fidelity elastic simulations; we show how NeuberNet can detect violations of the small-scale plasticity assumption, signaling the need for full-scale nonlinear models when required; finally, we show that NeuberNet can perform zero-shot inference on 3D problems with axisymmetric geometries and non-symmetric boundary conditions.

2024

1. Towards a Reliable Uncertainty Quantification in Residual Stress Measurements with Relaxation Methods: Finding Average Residual Stresses is a Well-Posed Problem

   M. Beghini , and T. Grossi

   Experimental Mechanics, Apr 2024

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   Background  In a previous work, the problem of identifying residual stresses through relaxation methods was demonstrated to be mathematically ill-posed. In practice, it means that the solution process is affected by a bias-variance tradeoff, where some theoretically uncomputable bias has to be introduced in order to obtain a solution with a manageable signal-to-noise ratio. Objective  As a consequence, an important question arises: how can the solution uncertainty be quantified if a part of it is inaccessible? Additional physical knowledge could—in theory—provide a characterization of bias, but this process is practically impossible with presently available techniques. Methods  A brief review of biases in established methods is provided, showing that ruling them out would require a piece of knowledge that is never available in practice. Then, the concept of average stresses over a distance is introduced, and it is shown that finding them generates a well-posed problem. A numerical example illustrates the theoretical discussion Results  Since finding average stresses is a well-posed problem, the bias-variance tradeoff disappears. The uncertainties of the results can be estimated with the usual methods, and exact confidence intervals can be obtained. Conclusions  On a broader scope, we argue that residual stresses and relaxation methods expose the limits of the concept of point-wise stress values, which instead works almost flawlessly when a natural unstressed state can be assumed, as in classical continuum mechanics (for instance, in the theory of elasticity). As a consequence, we are forced to focus on the effects of stress rather than on its point-wise evaluation.

2. Numerical Analysis of a Nozzle Guided Vane Filled With Lattice Structures

   I. Senegaglia , T. Grossi , G. Macoretta , and 5 more authors

   In ASME Turbo Expo 2024: Turbomachinery Technical Conference and Exposition , Aug 2024

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   Abstract. Gas turbines play a critical role in industries such as power generation and aviation. Additive manufacturing has emerged as a game-changing technology for gas turbine components, offering superior design flexibility and performance enhancements. The present work provides an overview of a multistep approach for integrating lattice structures into a specific gas turbine component, the Nozzle Guide Vane (NGV), using additive manufacturing technology. The first step involves a comprehensive assessment of lattice structures’ influence on the mechanical and thermal properties of the exposed part of NGV. Through computational simulations and experiments, an ideal lattice geometry is determined, optimizing structural integrity and heat transfer properties while minimizing volume usage. The second step sets the baseline performances of the current NGV system components, which were investigated and selected for additive manufacturing analysis. The third step focuses on the overall effect of additive manufacturing capabilities in the NGV system. The fourth and final step optimizes the additive manufacturing process for fabricating gas turbine components with lattice structures. Laser Powder Bed Fusion (L-PBF) technology, united with advanced Topological Optimization analyses, and high-temperature alloys were selected to withstand the demanding gas turbine operating conditions. This multistep approach represents a significant step forward in gas turbine technology, capitalizing the advanced mechanical applications as lattice designs and additive manufacturing, aiming in enhanced performance, reduced weight, and improved efficiency. These developments hold the potential to achieve more sustainable and cost-effective energy generation and transportation systems.

3. Regularization of Hole-Drilling Residual Stress Measurements with Eccentric Holes: An Approach with Influence Functions

   M. Beghini , L. Bertini , M. Cococcioni , and 3 more authors

   Journal of Materials Engineering and Performance, Apr 2024

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   The hole-drilling method is one of the most widespread techniques to measure residual stresses. Since the introduction of the Integral Method to evaluate non-uniform stress distributions, there has been a considerable improvement in the instrumentation technology, as step increments of about 10 microns are now achievable. However, that spatial resolution makes the ill-posedness of the problem stand out among other sources of uncertainty. As the solution becomes totally dominated by noise, an additional regularization of the problem is needed to obtain meaningful results. Tikhonov regularization is the most common option, as it is also prescribed by the hole-drilling ASTM E837 standard, but it has only been studied in the reference case of a hole with no eccentricity with respect to the strain rosette. A recent work by Schajer addresses the eccentricity problem by defining a correction strategy that transforms strain measurements, allowing one to obtain the solution with the usual decoupled equations. In this work, Tikhonov regularization is applied to the eccentric hole case through the influence functions approach, in order to avoid the introduction of new error-compensating functions and bias-prone interpolations. Some useful general considerations for a practical implementation of the procedure and an experimental test case on an aluminum specimen are presented.

4. Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel

   M. Beghini , and T. Grossi

   Applied Mechanics, Jul 2024

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   By means of relaxation methods, residual stresses can be obtained by introducing a progressive cut or a hole in a specimen and by measuring and elaborating the strains or displacements that are consequently produced. If the cut can be considered a controlled crack-like defect, by leveraging Bueckner’s superposition principle, the relaxed strains can be modeled through a weighted integral of the residual stress relieved by the cut. To evaluate residual stresses, an integral equation must be solved. From a practical point of view, the solution is usually based on a discretization technique that transforms the integral equation into a linear system of algebraic equations, whose solutions can be easily obtained, at least from a computational point of view. However, the linear system is often significantly ill-conditioned. In this paper, it is shown that its ill-conditioning is actually a consequence of a much deeper property of the underlying integral equation, which is reflected also in the discretized setting. In fact, the original problem is ill-posed. The ill-posedness is anything but a mathematical sophistry; indeed, it profoundly affects the properties of the discretized system too. In particular, it induces the so-called bias–variance tradeoff, a property that affects many experimental procedures, in which the analyst is forced to introduce some bias in order to obtain a solution that is not overwhelmed by measurement noise. In turn, unless it is backed up by sound and reasonable physical assumptions on some properties of the solution, the introduced bias is potentially infinite and impairs every uncertainty quantification technique. To support these topics, an illustrative numerical example using the crack compliance (also known as slitting) method is presented. The availability of the Linear Elastic Fracture Mechanics Weight Function for the problem allows for a completely analytical formulation of the original integral equation by which bias due to the numerical approximation of the physical model is prevented.

5. Determination of Chaboche and Bouc-Wen parameters for quenched and tempered steel

   C. Santus , L. Romanelli , T. Grossi , and 1 more author

   Journal of Theoretical and Applied Mechanics, Jul 2024

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   During cyclic loadings, metal alloys can undergo cyclic plasticity, for example, at notches. The Chaboche kinematic hardening model provides a versatile and realistic description of the material stress-strain behaviour under multiaxial cyclic loadings. In this work, the global properties, extracted from stabilized cycles of strain-controlled tests and from a force-controlled test, are employed to calculate the parameters. Alternatively, the Bouc-Wen model can provide a reliable representation of nonlinear hysteretic phenomena, and the classic nonlinear least squares approach is employed to tune its constants. The performances of the two proposed techniques are compared, and a final discussion is provided.

2023

1. Ill-Posedness and the Bias-Variance Tradeoff in Residual Stress Measurement Inverse Solutions

   M. Beghini , T. Grossi , M.B. Prime , and 1 more author

   Experimental Mechanics, Mar 2023

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   Background  Relaxation methods determine residual stresses by measuring the deformations produced by incremental removal of a subdomain of the specimen. Measured strains at any given increment, determined by the cumulative effect of the relieved stresses, appear as an integral equation, which must be inverted to obtain residual stresses. In practice, stress distributions are discretized by a finite-dimensional basis, to transform the integral equations into a linear system of equations, which is often ill-conditioned. Objective  This article demonstrates that the problem is actually ill-posed and comes with an inherent bias-variance tradeoff. Methods  The hole drilling method is used as an example application, and the practical effects of ill-posedness are illustrated. Results  Traditional regularization of the solution by limiting the resolution of the discretization reduces solution variance (noise) at the expense of increased bias and often results in the ultimately harmful practice of taking fewer data points. A careful analysis including the alternate Tikhonov regularization approach shows that the highest number of measurements should always be taken to reduce the variance for a given regularization scheme. Unfortunately, the variability of a regularized solution cannot be used to build a valid confidence interval, since an unknown bias term is always present in the true overall error. Conclusions  The mathematical theory of ill-posed problems provides tools to manage the bias-variance tradeoff on a reasonable statistical basis, especially when the statistical properties of measurement errors are known. In the long run, physical arguments that provide constraints on the true solution would be of utmost importance, as they could regularize the problem without introducing an otherwise unknown bias. Constraining the minimum length scale to some physically meaningful value is one promising possibility.

2. Elastic–plastic analysis of high load ratio fatigue tests on a shot-peened quenched and tempered steel, combining the Chaboche model and the Theory of Critical Distances

   C. Santus , L. Romanelli , T. Grossi , and 4 more authors

   International Journal of Fatigue, Sep 2023

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   This study aimed to predict the uniaxial fatigue strength of shot peened notched specimens made of 42CrMo4 quenched and tempered steel. The Chaboche kinematic hardening rule was employed to model the cyclic plastic behaviour of the tested samples and the evolution of the residual stresses, which were measured with X-ray diffraction and then reproduced through a finite element software with a novel proposed procedure. This latter introduced a distribution of eigenstrains to reproduce the initial residual stresses and a hardening of the material to replicate the residual stresses measured after the loading for a run-out specimen. The Theory of Critical Distances combined with the Smith-Watson and Topper multiaxial fatigue criterion were then employed to predict the fatigue strength resulting in a good level of accuracy.

3. Validation of a strain gauge rosette setup on a cantilever specimen: Application to a calibration bench for residual stresses

   M. Beghini , T. Grossi , and C. Santus

   Materials Today: Proceedings, Jun 2023

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   It is commonly known that the most difficult part of measuring residual stresses through diffraction or relaxation methods is the high sensitivity of the results to input errors, such as noise in the strain data. Then, quantifying and minimizing stress uncertainties is at least as important as the residual stress results themselves. Results are often validated by leveraging different measurement techniques, although each method is somehow specialized at detecting residual stresses at different locations and length scales. This leads to a fundamental lack of ground truth data and an inherent difficulty in detecting biases. The authors have introduced a calibration bench that facilitates the application of a well-known bending stress distribution on a specimen while conducting residual stress measurements using either the Hole-Drilling Method (HDM) or X-ray Diffraction (XRD). By leveraging Bueckner’s superposition principle, the bench allows for determination of both the residual stress distribution and the reference stress distribution through a single experimental setup. This approach not only enables direct evaluation of accuracy but also identification of any procedural systematic errors, as the reference stress distribution is known with a high degree of certainty.

4. A computationally fast and accurate procedure for the identification of the Chaboche isotropic-kinematic hardening model parameters based on strain-controlled cycles and asymptotic ratcheting rate

   C. Santus , T. Grossi , L. Romanelli , and 2 more authors

   International Journal of Plasticity, Jan 2023

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   The Chaboche isotropic-kinematic hardening (CIKH) model provides a versatile and realistic description of the material stress–strain behavior under generic multiaxial cyclic loadings. However, identifying the backstress parameters is challenging, and can be formulated as an optimization problem using different approaches. Instead of a computationally expensive pointwise search, in this paper the global properties of the cyclic curves are fitted to the experimental data. The conditions introduced are the hysteresis areas, peak stress values and tangent moduli at the extreme points, however the framework can be easily adapted to other target quantities. One linear and two non-linear backstress components of the kinematic hardening model are introduced, although the analytical equations developed can be used to refine the model further, with more components. Two stabilized cycles are required to identify the main kinematic parameters. New analytical expressions for asymptotic ratcheting rates in uniaxial tests are developed and then used to tune the dynamics of the slightly non-linear (hence, slowest) backstress component. After obtaining the kinematic parameters, isotropic hardening laws can also be identified, by considering the evolution of the extreme points of the strain-controlled cycles before stabilization. Practical demonstrations of the procedure are provided by experimental tests carried out on a 7075-T6 aluminum alloy, 42CrMo4+QT steel, and a high-silicon ferritic ductile cast iron. An accurate reproduction of the material behavior is achieved, at a negligible computational cost.

5. Investigation of Chaboche and Bouc–Wen Parameters of Quenched and Tempered Steel and Comparison of Model Predictive Capabilities

   C. Santus , L. Romanelli , T. Grossi , and 5 more authors

   Applied Sciences, Feb 2023

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   The aim of this paper is to model the elastic–plastic uniaxial behaviour of a quenched and tempered steel. The common Chaboche isotropic kinematic hardening model (CIKH) is introduced, and a physics-based procedure is proposed to determine its parameters. This procedure is based on strain- and stress-controlled tests and is focused on the stabilized cycles. The imposed cycle properties are the hysteresis area, the stress range, the slope at the inversion points, obtained from the stabilized cycles of strain-controlled tests, and the ratcheting rate extracted from a stress-controlled test. The novelty of the algorithm is to determine the hardening parameters from the global properties of the cycle rather than imposing a pointwise fitting, which is also implemented to calculate the parameters for a comparison. The Bouc–Wen model showed great flexibility in describing nonlinear behaviours, corresponding to different physical phenomena, through an appropriate tuning of its parameter values. In this paper, another optimization approach is developed to estimate the Bouc–Wen coefficients and accurately describe the same experimental cycles. The performances of the Bouc–Wen model are compared with the predictions of the Chaboche model, and a discussion comparing the techniques used to reproduce cyclic plastic behaviour is provided.

6. Residual stress measurements on a deep rolled aluminum specimen through X-Ray Diffraction and Hole-Drilling, validated on a calibration bench

   M. Beghini , T. Grossi , C. Santus , and 2 more authors

   IOP Conference Series: Materials Science and Engineering, Feb 2023

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   Residual stress measurements are notably affected by a high sensitivity to errors in input data. Measurements should then be presented together with an estimation of their accuracy. A common strategy is to carry out more measurements and/or to compare the results of different techniques. However, error contributions due to biases could be dangerously left unseen. In a previous work, the authors presented a calibration bench which can impose a known bending stress distribution on a specimen while simultaneously performing X-Ray Diffraction (XRD) or Hole-Drilling Method (HDM) residual stress measurements. Since the external load can freely be applied and removed, the superposition principle can be exploited to simultaneously identify either the reference bending stress distribution or the actual residual stress distribution, with the same experimental setup. A deep rolling treatment was measured and analyzed on the calibration bench with both XRD and HDM. First, residual stresses on the surface were evaluated with XRD measurements, then electrochemical material removal was performed to investigate stresses at higher depths. After that, HDM measurements were carried out and compared with the results of XRD. Both methods were also used to identify the known bending stresses, providing an additional validation of the residual stress results.

7. Revealing systematic errors in hole drilling measurements through a calibration bench: the case of zero-depth data

   M. Beghini , T. Grossi , C. Santus , and 1 more author

   Journal of Theoretical, Computational and Applied Mechanics, Oct 2023

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   An accurate estimation of the measurement error in the hole drilling method is needed to choose an appropriate level of regularization and to perform a sensitivity analysis on the stress results. The latest release of ASTM E837 standard for the hole drilling method includes a procedure aimed at estimating the standard deviation of the random error component on strain measurements, proposed by Schajer. Nevertheless, strain measurements are also affected to some extent by systematic errors which are not included in the estimation and need to be compensated. For example, an error in the rosette gage factor or in the identification of the zero-depth datum systematically affects all strain measurements in a strongly correlated fashion. This paper describes a calibration bench, designed to superimpose a reference bending stress distribution on a given specimen while simultaneously performing a hole drilling measurement. Since the reference solution is known a priori and shares the measurement instrumentation, the hole geometry and the stepping process with the actual residual stress distribution, the bench provides the user with a direct validation of the obtained accuracy. In addition, strategies aimed at compensating systematic errors can be tested on the reference solution and then applied on the residual stress evaluation. The imperfect hole geometry and drilling alignment are proven to cause a significant underestimation of stresses near the surface, as they lead to an incorrect identification of the zero-depth datum. It is shown that this effect can be corrected through the proposed calibration bench.

8. Tuning Modal Behavior Of Additively Manufactured Lattice Structures

   M. Beghini , T. Grossi , G. Macoretta , and 5 more authors

   Journal of Engineering for Gas Turbines and Power, Dec 2023

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   Abstract Thanks to the increasingly widespread additive manufacturing technology and promising properties, the use of Lattice Structures (LS) is becoming increasingly frequent. LS allows the components to be designed with tunable stiffness, which can unlock the control of natural frequencies. However, crucial challenges must be faced to integrate LS into the typical design process. In the present work, an experimental and numerical study of LS-enabled tuning of natural frequencies in mechanical components is proposed. In a first step, the difficulties arising with the large amount of FEM nodes, that are required to predict LS complex shapes in detail, are overcome by modeling LS with an elastic metamaterial whose stiffness properties are determined through ad hoc finite element analyses. After that, a simplified investigation can be conducted on the modal properties of components with fixed external shape and variable internal LS filling, based on Triply Periodic Minimal Surfaces (TPMS) lattices. In those conditions, the parameters of the LS core can be tuned to control and optimize the global modal frequencies of the entire geometry. In addition, the admissible range of frequencies can be estimated. Optimized plates results are validated through an experimental test campaign on additively manufactured specimens made with Laser Powder Bed Fusion (L-PBF) technology. The samples are hammer-tested with various boundary conditions while laser sensors measure the oscillation data of selected points. Finally, estimated and identified natural frequencies were compared. The described model is suitable to be implemented in an automated tool for designers.

2022

1. X-Ray Diffraction and Hole-Drilling residual stress measurements of shot peening treatments validated on a calibration bench

   M. Beghini , T. Grossi , C. Santus , and 3 more authors

   In ICSP14–14th International Conference on Shot Peening , Dec 2022

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   The inverse problem of determining residual stresses from diffraction or relaxation methods is notoriously affected by a high sensitivity to errors in input data. A particular care must be devoted to ensuring that their input errors are minimized, and results shall come with a quantification of the corresponding uncertainties.

2. Procedura di determinazione dei parametri di chaboche di un acciaio con comportamento elasto-plastico in regime di fatica ad alto numero di cicli

   C. Santus , L. Romanelli , T. Grossi , and 3 more authors

   In Atti della conferenza AIAS2022 , Dec 2022

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   High-strength metal alloys usually have a purely elastic behaviour during high-cycle fatigue regime. Nevertheless, some low alloy steels undergo plasticity even in high-cycle fatigue regime and thus by considering a purely elastic behaviour can produce significant prediction errors. Chaboche Kinematic Hardening model, combined with Voce Isotropic Hardening model, provides generally a reliable modelling of the cyclic-plastic condition of a component under cyclic loading. The purpose of this work is to determine the Chaboche and Voce parameters just by imposing the global properties, such as stress range, slope at the extremes and hysteresis area of the stabilized cycles of strain-controlled tests. This procedure is therefore easier and more efficient instead of using complex algorithms of fitting.

3. A calibration bench to validate systematic error compensation strategies in hole drilling measurements

   M. Beghini , T. Grossi , C. Santus , and 1 more author

   In ICRS 11–11th International Conference on Residual Stresses , Dec 2022

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   An accurate estimation of the measurement error in the hole drilling method is needed to choose an appropriate level of regularization and to perform a sensitivity analysis on the stress results. Latest release of ASTM E837 standard for the hole drilling method includes a procedure aimed at estimating the standard deviation of the random error component on strain measurements, proposed by Schajer. Nevertheless, strain measurements are also affected to some extent by systematic errors which are not included in the estimation and need to be compensated. For example, an error in the rosette gage factor or in the identification of the zero-depth point systematically affects all strain measurements in a strongly correlated fashion.

4. Torsional-loaded notched specimen fatigue strength prediction based on mode I and mode III critical distances and fracture surface investigations with a 3D optical profilometer

   C. Santus , L. Romanelli , T. Grossi , and 5 more authors

   International Journal of Fatigue, Aug 2022

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   Torsional and axial fatigue tests were performed on steel 42CrMo4+QT and aluminium alloy 7075-T6, and a 3D optical profilometer was used for the investigations of the fracture surfaces. The mode I and mode III critical distances of the two materials were regarded as reference scales for evaluating the initial orientation of the cracks and then identifying the tests valid for the calibration of the normal and those for the shear-fatigue models. The torsional strength of blunt notched specimens was then predicted with the Smith-Watson-Topper and the FatemiSocie criteria, combined with the critical distances and the Chaboche model for the steel.