Tommaso Grossi

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.

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.

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.

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.