Evaluation of the stress–strain curve of metallic materials by spherical indentation

AbstractA method for deducing the stress–strain uniaxial properties of metallic materials from instrumented spherical indentation is presented along with an experimental verification.An extensive finite element parametric analysis of the spherical indentation was performed in order to generate a database of load vs. depth of penetration curves for classes of materials selected in order to represent the metals commonly employed in structural applications. The stress–strain curves of the materials were represented with three parameters: the Young modulus for the elastic regime, the stress of proportionality limit and the strain-hardening coefficient for the elastic–plastic regime.The indentation curves simulated by the finite element analyses were fitted in order to obtain a continuous function which can produce accurate load vs. depth curves for any combination of the constitutive elastic–plastic parameters. On the basis of this continuous function, an optimization algorithm was then employed to deduce the material elastic–plastic parameters and the related stress–strain curve when the measured load vs. depth curve is available by an instrumented spherical indentation test.The proposed method was verified by comparing the predicted stress–strain curves with those directly measured for several metallic alloys having different mechanical properties.This result confirms the possibility to deduce the complete stress–strain curve of a metal alloy with good accuracy by a properly conducted instrumented spherical indentation test and a suitable interpretation technique of the measured quantities

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

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. In this work, a detailed characterization of the stress and strain fields generated by the externally applied load was pursued. Then, the calibration bench was used to perform a validated characterization of residual stresses produced by two shot peening treatments, through both XRD and HDM. Additionally, both techniques were employed to verify the recognized bending stresses, thereby validating the findings of the residual stress measurements

Surface and subsurface rolling contact fatigue characteristic depths and proposal of stress indexes

The rolling contact fatigue is distinguished into subsurface initiated (spalling and case crushing) and surface initiated (pitting and micropitting). A characteristic depth was identified for each of these mechanism. The characteristic depth of the case crushing is the hardening depth, while for the spalling it is the maximum cyclic shear stress depth. The pitting depth is the size of the crack for which the mode I stress intensity factor range, due to the fluid pressurization, is higher than the threshold. This depth can be similar to the micropitting depth, in the order of 10 µm, for heavily loaded small radius contacts. Rolling contact fatigue cyclic shear stress indexes are then defined on the basis of the characteristic depths, and they identify the load intensity of each rolling contact fatigue mechanism. The characteristic depths and the stress index approach can be used to relate specific tests to component design, without any size effect misinterpretation

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

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

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

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. 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. Two bias correction strategies are discussed and validated on a 7075-T651 aluminum specimen. It is observed that the imperfect hole geometry and drilling alignment lead to a significant underestimation of stresses near the surface. With the proposed bench, it is shown that this effect can be corrected

Integral method coefficients for the ring-core technique to evaluate non-uniform residual stresses

The ring-core technique allows for the determination of non-uniform residual stresses from the surface up to relatively higher depths as compared to the hole-drilling technique. The integral method, which is usually applied to hole-drilling, can also be used for elaborating the results of the ring-core test since these two experimental techniques share the axisymmetric geometry and the 0°–45°–90° layout of the strain gage rosette. The aim of this article is to provide accurate coefficients which can be used for evaluating the residual stress distribution by the ring-core integral method. The coefficients have been obtained by elaborating the results of a very refined plane harmonic axisymmetric finite element model and verified with an independent three-dimensional model. The coefficients for small depth steps were initially provided, and then the values for multiple integer step depths were also derived by manipulating the high-resolution coefficient matrices, thus showing how the present results can be practically used for obtaining the residual stresses according to different depth sequences, even non-uniform. This analysis also allowed the evaluation of the eccentricity effect which turned out to be negligible due to the symmetry of the problem. An applicative example was reported in which the input of the experimentally measured relaxed strains was elaborated with different depth resolutions, and the obtained residual stress distributions were compared

Comportamento a fatica di strutture meccaniche in piena scala: risultati sperimentali e previsioni

Il lavoro si propone di presentare le principali attività di ricerca svolte, negli ultimi anni, presso il Dipartimento di Ingegneria Meccanica, Nucleare e della Produzione (DIMNP) dell’Università di Pisa, anche in collaborazione con l’Università di Trento, nel campo della resistenza a fatica delle strutture meccaniche, in particolare per quanto riguarda la conduzione di “test” su componenti in piena scala e la loro interpretazione. Viene quindi condotta un’illustrazione di alcune recenti campagne sperimentali (Es.: giunzioni filettate in acciaio, elementi di sospensione in alluminio, ingranaggi ad elevate prestazioni), alla quale segue una descrizione delle attività di caratterizzazione di base e di modellazione condotte al fine di costituire una adeguata base di conoscenze per la interpretazione. Infine, vengono analizzati i risultati ottenuti, evidenziando alcuni problemi aperti, sia sul piano concettuale che su quello applicativo

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

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. Residual stress measurements are often validated by comparing the results of different techniques. Although this approach can strengthen the measurement confidence, it does not highlight potential biases of the methods. The authors presented a calibration bench [1, 2] which can impose a known bending distribution on a specimen while simultaneously performing an X-Ray Diffraction (XRD) or Hole-Drilling Method (HDM) residual stress measurement. Since the external load can freely be applied and removed, Bueckner’s superposition principle [3] can be exploited to simultaneously identify both the reference bending distribution and the actual residual stress distribution with the same experimental setup. As the first is accurately known, the bench provides a direct estimation of the achieved accuracy. Moreover, it can reveal systematic errors in the chosen procedures. Two shot peening treatments were 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: that provided an additional validation of the residual stress results

Modello di tenuta della flangia bullonata, senza guarnizione, mediante l'analogia della meccanica della frattura di una fessura parzialmente aperta

I compressori centrifughi di elevate dimensioni non permettono l'utilizzo di guarnizioni deformabili, per cui le due meta della flangia di connessione sono forzate mediante bullonatura e la tenuta e affidata al contatto completo delle due superfici. La previsione della pressione di perdita e un aspetto di progetto di notevole interesse per questa tecnologia. L'azione della pressione interna sollecita la separazione delle superfici della flangia, che invece e contrastata dall'azione di serraggio dei bulloni. Il presente lavoro propone un modello per prevedere la condizione di perdita, basato sulla meccanica della frattura. Dato che le due superfici della flangia sono semplicemente a contatto, esse costituiscono una vera e propria fessura parzialmente aperta. Come ben noto il fattore di intensificazione di una fessura parzialmente aperta e nullo. Imponendo che le due superfici siano parzialmente separate ad una distanza fino al bordo del foro del bullone (che offre un canale di fuoriuscita per il fluido in pressione), e imponendo la condizione di fattore di intensificazione nullo, e possibile determinare la pressione di perdita, analiticamente, mediante la tecnica delle "weight functions" (o "funzioni peso"). Il presente lavoro riporta una positiva validazione del modello proposto mediante sia simulazione numerica sia risultati sperimentali in piena scala e in scala ridotta. Il modello analitico proposto offre uno strumento di progetto di immediata implementazione per comparare diverse geometrie di flangia bullonata

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

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