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The need to classify uncertainty into epistemic and aleatory has been the subject of much debate and discussion (Hoffman and Hammonds 1994; Apostolakis 1999; Der Kiureghian and Ditlevsen 2009; Lemaire 2014). It therefore seemed useful to create a section dedicated to the challenges and the usefulness of this distinction. This debate arises from the controversy between Niels Bohr and Albert Einstein concerning the nature of randomness observed at the quantum scale. For the former, it was a fundamental randomness, while for the latter, it was linked to an incomplete knowledge of phenomena at these scales. To illustrate his point of view, Einstein stated that “God does not play dice”. Subsequent theoretical and experimental work by physicists seems to have proved Niels Bohr right: randomness observed at these scales is intrinsic to quantum nature. Now that the debate at the microscopic scale has been closed, one may raise the same question at the macroscopic level of mechanics, which is the subject of this book. Are there uncertainties at the macroscopic level of an intrinsically aleatory nature that can affect mechanical systems? Or are all uncertainties at the macroscopic level related only to a lack of knowledge?

It is the author’s view that macroscopic mechanical systems can be affected by uncertainty of an inherently aleatory nature. This is notably related to two aspects: free will and the chaotic nature of some systems. Free will refers to the ability of human beings to freely self-determine and thus to be responsible for their actions. An opposite school of thought, articulated by Pierre Simon Laplace through “Laplace’s demon”, would establish that perfect knowledge of the present and past would make it possible to perfectly predict the future, including with regard to human actions. In such a vision, all uncertainties would obviously be of an epistemic nature, and an improvement in knowledge would make it possible to reduce them. Since all our societies are based, particularly in terms of justice systems, on the responsibility of each individual for their actions, we shall admit here the existence of the faculty of free will. Free will thus leads to a degree of aleatory uncertainty, which cannot be reduced by improvement in knowledge.

An example from the mechanical field is now provided to illustrate the uncertainty associated with free will. Anyone who has seen a construction excavator at work has quickly realized that it is an illusion to try to predict the load that its articulated arms will be subjected to during its lifetime, as the conditions under which the machine is used can be extremely unexpected. The ingenuity (if we look on the positive side) of the operators of the machine means that they will always find new uses for the machine, which the engineer designing the machine would never have thought of. For example, using the bucket to push a truck that the excavator has just loaded and that skids up the slope to get out of the hole, and also using it for different kinds of demolitions. Consequently, it is not possible to consider the uncertainty about the loads being applied on the arms of the machine as an uncertainty of a purely epistemic nature, solely linked to lack of knowledge and therefore reducible by improving knowledge. As a matter of fact, this would mean that an improvement in knowledge would ultimately be able to predict the ways in which operators will decide to use the machine, which is obviously contrary to free will.

Another uncertainty of an intrinsically aleatory nature at the macroscopic level is that related to chaotic systems. Chaos theory has shown that such systems cannot be predicted beyond a more or less long-term horizon. A classic example of a chaotic system is the atmospheric system, which is reflected by the fact that the state of the system (namely what is commonly known as weather) cannot be predicted beyond a certain period of time; this is the famous butterfly effect. Note that it is the system itself that is chaotic and not just our modeling of the system. An improvement of our models will therefore not change anything in the system itself, which will remain chaotic. Because of their chaotic nature, and therefore unpredictability in the more or less long term, chaotic systems are a source of aleatory uncertainty, which cannot be reduced by improvement in knowledge.

In order to illustrate this with a mechanical case, let us again consider the case of the gusts that an aircraft will experience. When designing a new model of aircraft, it is useful for the engineer to know the amplitude of the gusts that the aircraft will experience during its lifetime in order to be able to size the aircraft structure as efficiently as possible. The amplitude of these gusts can then be considered as aleatory, irreducible uncertainty. Indeed, due to the chaotic system that is the atmosphere, it is not possible to predict air movements on a scale of a few meters over a time scale of several decades (that is, the lifetime of the aircraft). Moreover, it should be noted that the chaotic nature of the atmospheric system is further amplified by interaction with human free will. Uncertainty about human response to climate change makes it difficult to predict the state of the atmospheric system on the time and space scales under consideration. The engineer therefore has no means of reducing this uncertainty about the amplitude of gusts by improving knowledge.

After having seen two sources of uncertainty that are inherently aleatory in nature at the macroscopic scale, one may ask whether other uncertainties can be classified as aleatory. While the previous two sources are intrinsically aleatory, the classification of other sources of uncertainty may depend on the situation. It should not be forgotten that the classification between epistemic and aleatory uncertainty introduced in the previous section is specific to the problem under consideration. When we considered the favorable outcome of the “coin toss” as aleatory uncertainty, the rules of the game are fixed (in particular, considering that the toss cannot be biased). If it is considered that biasing the toss is part of the rules of the game, then the uncertainty about the outcome is obviously reducible.

In mechanical problems, the engineer is responsible for articulating the problem, and, in particular, for setting “the rules of the game”, namely what can and cannot be changed. The classification is then specific to the problem under consideration and an uncertainty on the same physical quantity can then be classified as being epistemic for certain problems and aleatory for others.

To illustrate this, let us take the example of the uncertainty on the limit stress at failure of a spar of an aircraft wing. Without further details on the problem under consideration, it is not possible to say whether this uncertainty should be classified as epistemic or aleatory. Indeed, in the first case, assume that the problem considered is that of designing a new wing, keeping the same raw material supplier and the existing quality control process, and that we are focusing on the uncertainty in the limit stress at failure of the spar of one of the future aircraft that will be manufactured. Since the raw material batches obtained from the supplier have some variability, which cannot be reduced by staying within the scope of the problem under consideration (that is, without changing supplier and the quality control process), the uncertainty about the limit stress at failure in question can be classified as aleatory, irreducible uncertainty. In the second case, assume that we are studying the limit stress at failure of the spar of a particular aircraft that has already been manufactured. The uncertainty about this limit stress at failure is of interest to ensure that this aircraft will be able to withstand a higher load associated with a particular mission being considered for this aircraft. In this case, the failure limit stress has a single “true” value, but which is not known. The uncertainty associated with the failure limit stress is this time classified as epistemic uncertainty. It can be reduced by improving knowledge, for example, here by testing a sample kept by the manufacturer from the batch of material used for the spar.

In light of this ambivalence, the interest of classifying uncertainties into epistemic and aleatory for a given problem can be questioned. From the author’s perspective, the usefulness of this classification is most apparent from an engineering point of view in terms of distinguishing between reducible and irreducible uncertainties. This distinction is important because the solutions to be provided by the engineer to these two types of uncertainties are very different. For the former, the solutions essentially consist of seeking to reduce uncertainties, while for the latter, the solutions essentially consist of designing the systems in their presence (for example, by integrating partial safety factors, redundancies or by explicitly calculating the reliability of the system).

The interest of alternative approaches to modeling epistemic uncertainties is also questionable compared to the probabilistic approach typically used for aleatory uncertainties. Clearly, since the distinction between epistemic and aleatory uncertainty is based on the engineer’s decisions regarding the problem to be considered, this, in itself, cannot justify different modeling methods for these two types of uncertainties. Taking the example of the limit stress at failure of a spar, illustrating the dependence of the classification on the problem under consideration, we can elaborate by noting that in the first case probabilistic modeling is typically adopted to model the variability of the stress at failure. It is obvious that when we shift to case 2, there is no reason not to maintain the probabilistic approach to model the epistemic uncertainty before trying to reduce it by way of testing (using Bayesian approaches, for example). There are thus many cases where probabilistic modeling is well suited to modeling epistemic uncertainties.

On the other hand, there are also situations where probabilistic modeling may not be the most appropriate way to model uncertainties. These are typically cases where tests or simulations on the uncertain quantity of interest are difficult, if not impossible, to obtain. In such cases, the available data are not sufficient to construct a suitable probabilistic model. Alternative approaches that are less data-intensive and based on expert opinions have been proposed and will be reviewed in the following sections. Finally, it should be noted that, in practice, for the same quantity, there is often an epistemic (reducible) and an aleatory (irreducible) part of uncertainty, making the analysis even more complex.