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.

中文翻译：

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使用松弛方法实现残余应力测量中可靠的不确定性量化：寻找平均残余应力是一个适定问题

背景

在之前的工作中，通过松弛方法识别残余应力的问题被证明在数学上是不适定的。在实践中，这意味着求解过程受到偏差-方差权衡的影响，必须引入一些理论上无法计算的偏差才能获得具有可管理信噪比的解。

客观的

因此，出现了一个重要的问题：如果解的不确定性的一部分无法访问，如何量化解的不确定性？理论上，额外的物理知识可以提供偏差的表征，但利用目前可用的技术，这一过程实际上是不可能的。

方法

对现有方法中的偏差进行了简要回顾，表明排除这些偏差需要一些在实践中永远无法获得的知识。然后，引入了一段距离内平均应力的概念，结果表明，找到它们会产生适定问题。数值例子说明了理论讨论

结果

由于求平均应力是一个适定问题，因此偏差与方差的权衡消失了。用通常的方法可以估计结果的不确定性，并可以获得准确的置信区间。

结论

在更广泛的范围内，我们认为残余应力和松弛方法暴露了点应力值概念的局限性，相反，当可以假设自然无应力状态时，点应力值几乎可以完美地工作，就像在经典连续介质力学中一样（例如，在弹性理论）。因此，我们被迫关注压力的影响，而不是对其进行逐点评估。