Fictional mechanism explanations: clarifying explanatory holes in engineering science

This paper discusses a class of mechanistic explanations employed in engineering science where the activities and organization of nonstandard entities (voids, cracks, pits…) are cited as core factors responsible for failures. Given the use of mechanistic language by engineers and the manifestly mechanistic structure of these explanations, I consider several interpretations of these explanations within the new mechanical framework (among others: voids should be considered as shorthand expressions for other entities, voids should be reduced to lower-level mechanisms, or the explanations are simply abstract mechanistic explanations). I argue that these interpretations fail to solve several philosophical problems and propose an account of fictional mechanism explanations instead. According to this account, fictional mechanism explanations provide descriptions of fictional mechanisms that enable the tracking of counterfactual dependencies of the physical system they model by capturing system constraints. Engineers use these models to learn about and understand properties of materials, to build computational simulations of their behaviour, and to design new materials.


Introduction
The engineering sciences make extensive use of mechanistic explanations ( van Eck, 2015van Eck, , 2018. However, there is a class of explanations that, while referring to 'mechanisms', do not fit neatly into the new mechanical framework. More precisely, within the field of reliability engineering, there is a class of explanations that utilize what are referred to as 'failure mechanisms'. Failure mechanism explanations attempt to elucidate the physical, chemical, or other processes which have led to the failure of a component or an artefact (Liao et al., 2000). Sometimes these failure mechanisms are studied in terms of micromechanical modelling, and, in many cases, the use of these micromechanical models leads to explanations that feature nonstandard entities as the core factors responsible for the failure process. I use the term 'nonstandard entities' to capture the sort of things that, when treated as 'entities', generate tensions with certain ontological commitments of the new mechanical philosophy, such as entities being systems composed of parts, or mechanisms being particulars and part of the causal structure of the world. As an example, take Kahn and Liu's characterization of the micro-mechanics of ductile fracture: The fracture of metals is the macroscopic manifestation of the evolution and accumulation of microscopic defects, such as micro-voids and shear bands et al. […] It is assumed that micro-void initiation, growth and coalescence are responsible for the fracture of ductile materials. (Khan & Liu, 2012, p. 1).
These micromechanical explanations employ the properties and activities of nonstandard entities and suggest that they play a causally productive role. Some use the volume, shape, and evolution of voids and how they merge to lead to ductile failure. Others use the depth and geometry of pits. Intriguingly, engineering scientists treat these nonstandard entities as components of mechanisms, and characterizations of these nonstandard entities figure in the explanations they develop, which have a clear mechanistic structure: explanations describe how the growth (activity) and coalescence (interaction) of micro-voids (entities) are organized so as to be responsible for the fracture (phenomenon). Yet, the entities that engage in these activities are voids.
Voids are usually taken to be causally relevant but not causally operative (Martin, 1996, p.62). Yet, the explanations described above appear to take voids to be causally operative. How to square this observation with the common view that mechanisms are fully determinate particulars, in which entities that carry out activities have active (productive) causal powers (c.f. Glennan, 2017, p.2 and p.14)? Nonstandard entities such as voids are certainly not the objects most new mechanists have in mind: '[w]hat the new mechanists refer to as "entities" are concrete material objects that are located in space and time.' (Kaiser, 2018, p. 117).
The main aim of this paper is to clarify and elucidate the workings of these (micromechanical) failure mechanism explanations, focussing specifically on void explanations of ductile fracture. I first argue that they are bona fide explanations and that they are mechanistic (in the sense that they are model-based explanations where the model describes a mechanism). This result, I argue, admits of two interpretations, viz. the models either depict extant mechanisms which are part of the causal structure of the world, or they depict fictional mechanisms. In the first case, two broad strategies present themselves: to opt for a non-literal interpretation, whereby the explanation is 'not really' talking about voids, or to bite the bullet and take these explanations at face value, embracing the idea that voids have causally productive powers. The first strategy would take these explanations to be providing indirect descriptions of the mechanism. This could be realised in different ways, such as changing the level of abstraction (e.g., voids might be referring to some underlying embodied mechanism which can fulfil the role played by voids in the explanation) or by showing how voids are simply short-hand expressions, placeholders, that refer to standard entities (such as 'metal lining that encloses the void'). The second strategy tries to solve the problem by embracing the idea that voids engage in activities. In other words, it takes these explanations to be providing a direct description of an extant mechanism; but this requires buying into a 'non-standard ontology'. After due consideration, I argue that both strategies fail to capture the structure of these explanations and do not provide insight into how they work. I then defend an account of fictional mechanism explanations instead. On this account, the growth and coalescence of micro-voids are understood as fictional mechanisms which enable tracking relevant patterns of counterfactual dependence of extant physical systems. This tracking is afforded and informed by the representation of relevant structural constraints that make a difference to how the phenomenon of ductile failure manifests itself.
The paper is structured as follows. Section 2 gives a brief characterization of microvoid explanations of ductile failure. Section 3 argues that these are bona fide explanations that, given their mechanistic structure, seem to be in tension with several ontological commitments of standard new mechanical accounts. Section 4 describes the two possible strategies that take these explanations to be providing a depiction (whether direct or indirect) of an extant mechanism: to either show that voids are standard entities in some sense (section 4.1) or to somehow argue the explanations are providing a literal description of the mechanism responsible for the phenomenon by adopting a non-standard ontology where voids can interact (section 4.2). Section 5 defends an account of fictional mechanism explanations.
2 Explaining ductile failure using micro-voids When a material is subjected to tensile stress resulting in permanent deformation it is said to have suffered ductile failure. If one conducts a tensile test by stretching a rod made up of a ductile material, one will observe the mid-section of the material become thinner as it elongates. This is sometimes called 'necking'. If the tension continues, the material is pulled apart, leading to ductile fracture. Some materials are more ductile in nature (e.g., aluminium alloys or gold), which makes them prone to suffer these failures under sufficient tension, as opposed to brittle materials which would 'snap'. Understanding this phenomenon is of great relevance to engineering design, as many materials require ductility (e.g. the wings of an airplane). In fact, impurities such as magnesium are sometimes carefully added to metals to intervene on ductility.
The first explanations for the phenomenon of ductile failure developed in the 50s, with the works of Tipper (1949) and Puttick (1959), who documented the microscopic process that leads to ductile failure. When observed under a microscope (see Fig. 1), ductile fractures exhibit rough and dimpled surfaces (as opposed to brittle materials, which are smoother). Their main insight was that the formation of small cavities was responsible for the failure. Nowadays it is possible to observe nucleation (the creation of voids) in situ by using tomography.
One of the first models of the behaviour of these voids was proposed by McClintock et al. in McClintock et al., 1966, who considered the growth of homogeneously distributed cylindrical holes in a continuum material. The characterization of each hole was made as a single hole in an infinite material, without representing hole interaction. However, despite the modelling being quite idealized, the 'mechanism' they try to model is quite straightforward: "The growth and eventual merging of holes from microscopic inclusions within materials results in fracture." (McClintock et al., 1966, p.1) Building on this work, Rice and Tracey (1969) found out that the volume change of the voids made a greater difference than their shape change. Gurson (1977) additionally suggested a damage variable that was linked to the void volume fraction f v (the division between void and effective volume, which would represent porosity). Tvergaard (1981) added the voids interaction effect to Gurson's model (at the same  (Clemintime, 2019). The small cavities that appear like dimples or darker regions in the image correspond to so-called micro-voids time Le Roy et al. (1981) also modelled the interaction of the voids). One of the novelties they introduced is that the presence of neighbouring voids decreases the maximal loading as the stress distribution changes. Needleman and Tvergaard (1984) further improved the understanding of how voids interact. When two voids are close enough, the material loses the capacity of sustaining the loading. This can be detailed by a variable which accounts for threshold porosity (f c ). When the void volume (f v ) approximates this threshold (f v ∼ f c ), voids start to merge. From there, if f v increases, the surrounding material fails.
The 'mechanism' that explains ductile failure can be summarized into three stages (see Fig. 2): 1. 'Nucleation' refers to the creation of new voids. Voids might already be present in some materials when load is applied, but in others, they will form as the load is being applied. Most models assume that all second-phase particles (impurities) lead to nucleation. It is possible to further describe the mechanisms of how nucleation 1 comes about in terms of condensation of lattice vacancies due to high stress concentrations, pile-up of slip lines, inclusions, or second-phase particles, etc.
2. Growth is the 'activity' by which voids augment their volume. 'Coalescence' refers to the interaction of the voids (their merging) and its implications. The specifics will depend on the material at hand. The more ductile the metal, the faster voids increase their radius before coalescence occurs. This explains why these materials  (Bbanerje, 2008) 'stretch' more. In less ductile metals coalescence starts earlier and there is less void growth. Coalescence also involves the void's effects on the surrounding areas. For example, when two voids are big enough and close enough to each other, the material separating them loses its ability to stand certain stresses (e.g., it can be 'stretched'). The way in which voids can coalesce ('organization') is also quite varied, and some authors take these differences in organization to comprise different mechanisms. 2 One mode of coalescence is called 'internal necking', where 'the ligament between the two voids shrinks with a shape typical of a necking process (…) [and] the voids evolve towards a diamond shape' (Scheyvaerts et al., 2010, p.98). Another is called 'void sheeting', where there are secondary small voids forming bands with bigger voids. Yet another mode of coalescence 3 is referred to as 'necklace coalescence', which occurs 'in rows of closely spaced voids gathering within elongated clusters.' (ibidem).
3. Failure then occurs when either the voids grow too large (and the material deforms, or fractures) or coalescence forms a big enough crack (Tvergaard & Hutchinson, 2002).
The practical use of these models depends heavily on stochastic methods or idealizations by assuming that all voids have the size of the average void and are spaced homogeneously. In many cases isotropy is also assumed. Recent approaches involve the use of computer simulations to track the evolution of voids. These developments are leading to interesting discoveries such as the design of improved materials (e.g., Osovski et al., 2019, provide a methodology for designing material microstructures with improved fracture resistance). The next section shows why these explanations pose a challenge to the new mechanical framework and considers ways to resolve the issue from within the framework.

Understanding explanatory holes mechanistically
In this section I first show that these micromechanical explanations constitute cases of bona fide explanation (section 3.1), and then proceed to show that they are a species of mechanistic explanation (section 3.2). I close by elaborating how these explanations, given their reliance on non-standard entities, pose a challenge for new mechanical accounts (section 3.3).

Void explanations are explanations
One might object that these 'explanations' simply are not cases of explanation. Two observations resist this interpretation: engineering practice itself and philosophy of explanation. These explanations, by the lights of the engineering scientists using them, are considered appropriate to account for and to understand ductile failure. Similarly, they are used to design (improved) materials and artefacts.
In addition, they possess several hallmarks generally associated with explanation as posited by several familiar philosophical accounts of explanation; namely: explanations make their associated explanandum unsurprising (Hempel, 1965), they show how the explanandum depends on the explanans (Woodward, 2003;Glennan, 2017), and they unify descriptions of phenomena that were previously not connected (Glennan, 2017;Kitcher, 1981;Mäki, 2001).
Explanations employing voids make the failure unsurprising, since introducing certain initial conditions into their models leads to expectable failure (in Section 5 I outline how this works in greater detail). Void explanations show how the phenomenon of ductile failure depends on certain material configurations (e.g., the initial organization of cavities). Void explanations unify; they show how the failure of different objects made up of different (ductile) materials via diverse types of loading are explainable via the same 'mechanism' (void nucleation, growth, and coalescence).
Given that void explanations are bona fide explanations, the natural follow-up question is: what type of explanations are they?

Void explanations describe mechanisms
There are good reasons to consider these explanations mechanistic (i.e., model-based explanations in which the models represent or depict mechanisms). The mechanisms depicted in these model-based explanations seem to fit the main characterizations of mechanisms. Consider Stuart Glennan's definition of minimal mechanism which has its roots in the Illari-Williamson definition (Illari & Williamson, 2012), and is considered by many the ecumenical view of mechanisms 4 : A mechanism for a phenomenon consists of entities (or parts) whose activities and interactions are organized so as to be responsible for the phenomenon. (Glennan, 2017, p.17).
The entities are captured by nouns and the activities expressed by verbs. The distinction between activities and interactions is introduced to account for the distinction between unary activities (the ones an entity carries on its own, such as when a person daydreams) and n-ary activities (which involve the interaction of two or more entities, such as the colliding of billiard balls). The activity and organization of the entities is responsible for the phenomenon, in the sense of determining, producing, underlying, constituting, or maintaining it.
In our void explanations, the role of the entities is played by voids, which will be analysed in the next subsection, as it is the source of most problems. 5 These entities engage in unary activities: they grow. Growth is a unary activity in the sense that voids do not engage with other entities in order to grow. The voids also interact (in the model) by coalescing. Considering coalescence an n-ary activity might seem problematic, as the voids stop being singular voids after the merging, thus a 'binary activity' of two voids merging would resolve into a singular void, which cannot engage in a binary activity on its own. The problem disappears when realizing that the interaction happens before (and during) the coalescence, and not after (i.e., interactions can occur and cease to occur at various stages of the mechanism, and the same entities do not need to be fully present throughout the entire process). These voids and their interactions need to be organized in a certain way so as to be responsible for the failure. Section 2 detailed several modes of coalescence which specify the required organizations. For example, voids can be organized forming a 'necklace', i.e., they gather closely in clusters and these clusters are aligned. This organization also plays an important role in failure, since a different configuration would not have the same result (e.g., if voids were spaced at a sufficient distance).
In sum, the ecumenical definition of mechanisms can be applied as follows: the mechanism for ductile failure consists of voids (entities) whose growth (unary activity) and coalescence (interaction) is organized so as to be responsible for the failure (phenomenon).
Furthermore, and perhaps more importantly, these explanations are treated (unproblematically) as mechanistic by engineering scientists. They take these explanations to describe mechanisms, and the role these mechanisms are taken to play is regarded similar to the operation of any other mechanism (e.g., when talking about interacting or masking mechanisms). One only need survey some relevant engineering journals to find an abundance of examples of this 'mechanism talk', for instance: An important failure mechanism in ductile metals and their alloys is by growth and coalescence of microscopic voids. (Benzerga & Leblond, 2010, p. 170). The effects of irradiation on ductile fracture mechanisms -void growth to coalescence -are assessed in this study based on model experiments. (Barrioz et al., 2017, abstract).
In sum, we have at least three reasons to take these explanations to be mechanistic: engineers' use of the word 'mechanism' paired with the use of mechanistic lingo ('responsibility', 'interaction', etc.); the fact that these 'mechanisms' are sometimes taken to interact with other mechanisms and processes (see second quote above); and the fact that their structure matches to a substantial extent the ecumenical definition of mechanism as offered by philosophers of science.

The problem: Voids cannot productively interact
Voids seem far removed from the notion of entity most mechanists use. Perhaps one could emphasize the 'activity' or the 'property' side of voids in the mechanism and disregard the relevance of entities. Unfortunately, most accounts of mechanisms rely heavily on the role played by entities. For instance, Machamer et al. assert that "no activities without entities, and entities do not do anything without activities" (Machamer et al., 2000, p. 8); and Illari & Williamson affirm that 'entities and activities are always equally important in that they must both be present to produce the phenomenon' (Illari & Williamson, 2012, p.126). This idea, sometimes called the interdependency thesis, is quite prevalent throughout the literature. In fact, entities often constitute one of the criteria for classifying types of mechanism. In any case, in engineering explanations, voids (qua 'entities') are indeed mentioned explicitly, so this observation needs to be addressed. This constitutes the first part of the problem: voids cannot be 'ignored'; not just because they are mentioned explicitly, but because otherwise the model collapses: activities (in this case growth and coalescence) need actors (in this case voids).
If entities cannot be ignored, we could maybe argue that voids can nonetheless play the role of entities. After all, the entities one finds in mechanisms can be quite diverse (from institutions to DNA), so perhaps nonstandard entities such as voids can also fit the mechanists' requirements for being an entity.
To assess whether this is the case, let us first distinguish between defining what an entity is and giving necessary conditions for being considered an entity. Glennan has a definition of entities (in a mechanism) on the one hand, and some necessary conditions on the other. He defines entities as follows: '[Entities are] objects-things that have reasonably stable properties and boundaries (…) Entities and activities are not abstract; they are fully determinate particulars located somewhere in space and time; they are part of the causal structure of the world' (Glennan, 2017, p.20).
In line with other authors, he also subscribes to the interdependency thesis, since activities must have actors. But aside from defining entities, Glennan provides four characteristics entities must fulfil to be considered apt for 'entity-hood' as characterized by the new mechanists (Glennan, 2017, p.34). First, entities are what engage in activities and interactions. Second, they have locations in space and are stable bearers of causal powers over time. Third, these causal powers are what allow them to engage in activities. Fourth, most or all entities are systems composed of parts and most or all the powers of entities are mechanism dependent (he introduces the phrasing 'most or all' to deal with the fundamental entities of certain branches of Physics).
Carl Craver does something similar (Craver, 2007, p. 131-132). To distinguish mechanisms that refer to real (as opposed to fictional) components, he gives four necessary conditions for being an entity (part/component) in a mechanism. First, they should have stable properties; second, they should be robust, in the sense of being detectable through many different methods; third, one should be able to manipulate them to intervene on other entities; and fourth, they should be plausible in normal conditions (more specifically they should be physiologically plausible, since Craver is studying mechanisms in the brain).
An examination of these definitions and criteria reveals several discrepancies when it comes to voids and other nonstandard entities. Voids are not systems composed of parts. They do not seem to be fully determinate particulars (they are ontologically parasitic). But even if we take voids to be determinate particulars, how are we to understand their interaction as being a 'productive process'? It would be difficult to defend that voids have causal powers (in the sense of being causally operative). Furthermore, it is hard to see how one could manipulate voids to intervene on other entities: how would one interact with these voids and how would these voids interact with something else? This problem extends to most mechanistic accounts, since they ultimately define entities in 'standard' terms: new mechanists think of concrete material objects when talking about entities. If the first part of the problem was that entities cannot be discarded when explaining through mechanisms, the second part of the problem is that voids do not satisfy key requirements for being an entity (as defined by the new mechanists). In other words, voids seem to be indispensable for these explanations, yet they seem to clash with the notion of entities new mechanists have.
I consider three possible solutions. First, one could argue that voids fit current definitions of entity and they satisfy entity-hood criteria. This would involve showing how voids actually refer to something else (some standard entity) that does meet such desiderata or showing that the mechanism of void coalescence is actually referring to some other mechanism. Second, one could modify the mechanists' definitions of entities and problematic criteria. On this view, the mechanism of void growth and coalescence is part of the causal structure of the world. This would entail accepting a non-standard ontology where voids interact. These strategies are explored in the following section in turn. Section 5, on the other hand, considers the possibility that the models in these explanations do not refer to extant mechanisms but describe fictional mechanisms instead. I argue that this view is the most promising solution for clarifying the structure and value of these model-based explanations.

Two possible new mechanical solutions
In this section I consider two possible solutions that take the models used in the explanations to be depicting extant mechanisms. I first consider the view that these models provide indirect depictions of extant mechanisms. I then consider the interpretation that these models provide direct depictions of extant mechanisms.

Standard ontology
Showing how these explanations are, in some sense, referring to standard entities can be done in at least two ways. The first is to argue that the explanation misidentifies the level of abstraction (e.g., voids should be reduced to lower-level mechanisms), and the second is two show that voids are shorthand expressions for something else.
Changing levels of abstraction One might reduce the 'mechanism' to the lower-level mechanisms responsible for voids in terms of standard entities, neglecting the level at which engineers suggest the relevant explanatory mechanism should be sought. That is, to simply consider that the stresses that enlarge the voids or that 'make them interact', together with other processes of material objects, are the relevant explanatory mechanisms. Engineers would probably not deny the operation of such lower-level mechanisms since they consider the mechanism in which voids 'figure' as part of a multilevel mechanism. However, engineers pitch these explanations at the level of abstraction of voids when offering explanations for ductile failure. A reconceptualization of these engineering explanations along these lines downplays the importance engineering scientist assign to the epistemic and (re)design benefits offered by explanations that rely on voids (e.g., allowing the creation of fine-grained computational simulations that can lead to the design of new materials, or to being able to assess ductility based on impurities that will lead to nucleation). I therefore consider this an unattractive option.
Another route can be found in Glennan (2017, pp. 196-200), which suggests a viable way of dealing with non-productive causation within the mechanistic framework. Such an account might be a useful way to describe void explanations by bracketing the productive role played by voids. On Glennan's view, disconnection can be a genuine case of causal production, unlike omission or prevention. He builds on Shaffer's (Schaffer, 2000) example of a gun's firing to show how, despite many steps of the firing process containing disconnections (e.g., springs getting released), there is a plausible mechanistic story to be told. His suggestion is that guns have the capacity to fire bullets, a capacity which is mechanism dependent. Pulling the trigger of a gun precipitates such a capacity in virtue of the activities of each component. And while there is disconnection in some stages, the system as a whole acts mechanistically (disconnections are part of how the mechanism is wired). The core idea is that disconnection is only problematic if one only focuses on the interaction between individual components instead of the whole mechanism. This interpretation could be useful towards understanding void explanations in the following way: what produces ductile failure is not voids (qua voids), but rather, the whole system composed of voids, their surroundings, the metallic 'channels' that connect voids, the exerted stresses, etc. Such a system has the capacity of producing ductile failure thanks to there being certain disconnections (voids) at some stages. Ductile failure occurs because of how the mechanism is wired (with voids playing a dysconnectivity role).
I believe that the 'actual' mechanism might be adequately described with the above characterization. However, this characterisation fails to capture what makes void explanations explanatory. The micromechanical models engineers put forward not only seem to work without the inclusion of these extra entities and assumptions; their explanatory power seems to hinge on descriptions of void coalescence. In other words, even if the characterization above is an accurate story of what is happening at the ontic level, there is still a story to be told at the epistemic level about how models of void coalescence are explanatory and offer epistemically valuable information. Characterizations of 'actual mechanisms' do not offer this explanatory story.
Generally, it seems as though changing the level of abstraction is at odds with the explanatory practice itself. If we want to account for these explanations as used in the practice, and remain committed to the standard new mechanical framework, another route must be taken. A prima facie reasonable way forward seems to argue that voids are actually shorthand expressions referring to standard entities.
Voids are shorthand expressions for standard entities It could be the case that when engineers use the term 'void' they are really referring to something other than a void. Under this interpretation, the heavy lifting in the explanations is then not done by appeals to voids, but actually by what these voids are stand-ins for. Some possible standard entities that the voids could stand in for include the air that fills a void or the metal-lining that encloses a void.
There are however several problems with the idea that 'void' is a shorthand expression for a standard entity. The first is the way engineers use the term 'void', since for the most part they mean exactly that, the 'empty' space within a material that does not contain the material in question. In this sense, voids are not used as stand-ins for standard entities in explanations. The second problem is to find suitable entities that voids could be shorthand expressions for. No entity seems to have the right properties. For example, one could consider that 'void' is really referring to the surface of material that encloses the void (the metal-lining), but surfaces have no volume, and the void is not made of metal (as is the surface). Not to mention that if you chip away at the metal lining you get rid of it (but the void grows!).
Given that the explanation hinges on the volume (growth) of voids, this does not seem a feasible solution. Another plausible candidate would be the air (or some other substance) that fills a gap where there is no metal. But then we run into the issue that the air does not coalesce (it already forms a whole), neither does it form a crack, nor does it produce any effect on the ductility of the material. To see why, consider that its removal would make no difference.
A further possibility could be to say that voids refer to properties of standard entities. Lewis and Lewis (1970) made a clear case of how problematic treating holes as properties can be. They further discuss problems of considering holes as perforations (e.g., saying that a paper is triply perforated if it contains 3 holes). If one decides to go down this route, one should be aware that a good characterization of voids as properties needs at the very least to account for why they are so easy to operationalize (e.g., we can easily count them, measure their volume, speak about how they have grown or been filled, or about creating them by puncturing a material). This means that building a complicated account trying to show how holes or voids really refer to something other than themselves is rife with difficulties, given the great burden of explicating such characteristics, qualities, and properties. 6

Non-standard ontology
Another approach is to bite the bullet and take these explanations at face value, i.e., voids do in fact interact. Given the nature of voids, such an approach could draw on the literature on omission and prevention as a starting point. A first attempt reveals that traditional accounts of omission do not quite do the trick. For example, Lewis (2004) discusses the Void's ability to bring about someone's death. For Lewis, it is not what the Void is doing as much as what it is not doing (e.g., the internal pressure of the air in the lungs is normally counterbalanced by the atmosphere, but this counterbalance disappears in the Void, leading to empty lungs). In contrast, the properties of the voids do seem to play a productive causal role in the engineering void explanations. To some extent one could argue that it is the lack of material (with all its cohesive properties) that causes the failure of the metal, but void explanations, as they are presented, do not rely on the lack of cohesive properties as much as on the fact that the coalescence of voids forms cracks.
In a more recent approach, Barros (2013) incorporates absences into a mechanistic account by considering them to be causes of the mechanism failure. A working mechanism provides a baseline which if disturbed (e.g., by taking away oxygen from the lighting of a match) produces failure. But in void explanations, the materials' failure is what is being explained mechanistically. The baseline in such a case would be the piece of metal just sitting there, from which we cannot articulate a failure mechanism in terms of subtracting causal factors.
More broadly, void explanations do not hinge on absences or omissions, rather, they hinge on void interaction. Absences 7 and omissions thus do not seem to be apt notions to understand voids.
As it stands, no account on offer seems to have the right ingredients from which to build an account where voids can interact in a physical sense. Such an account would therefore have to be built bottom up. The main issue is facing up to the challenge of how something without materiality can engage in activities. How can it exchange something it does not have? A void does not have quantities, marks, or energy. A possibility would be to argue that voids can only interact with other voids (and not with material entities) and, hence, no conservation laws would be violated. But even then, what exactly would be exchanged in the interaction?
Problems do not end there. A suitable account would also have to show how voids can interact, but other discontinuities, such as shadows, corners, or rainbows, cannot (on pain of an ontological explosion). I struggle to see how such a project would succeed. I suggest there might be an easier solution to the problem: the mechanism is fictional.

Fictional mechanism explanations
Void explanations describe mechanisms, but voids cannot productively interact. In this section I argue that we can resolve this tension by considering these explanations to be fictional mechanism explanations. I first outline what a fictional mechanism explanation is (Section 5.1). I then explain how it is possible for a fictional mechanism to capture physical dependencies (Section 5.2).

Fictional mechanism explanations
I argue that in void explanations, the model of void growth and coalescence should be read as fictional. Fictional in the sense of misrepresenting the ontology of the physical system for whom it is a model. 8 The 'mechanism' of void growth and coalescence is not part of the causal structure of the world. Nonetheless, I argue that it is epistemically useful and has explanatory power. 7 Absence is usually defined as the state of not being present. When we think of a void, we are not referring to some entity that is absent but could be present. Voids have primary qualities (being countable, having volume, etc.), whereas absences do not. 8 By fictional I do not simply mean that these models distort or misrepresent features of the system they represent, as is the case with idealizations. I mean that they are introducing entities and activities that are not part of our ontic store. Other examples of models employing fictions have been analysed in the literature, including Maxwells' ether (Morrison, 2009), semiclassical electron trajectories (Bokulich, 2012), and gravitational force (Bokulich, 2016).
Void explanations make use of models that, despite not depicting extant mechanisms, capture counterfactual dependencies of physical systems that are deemed important. Void explanations' models do so in virtue of mirroring the constraints of the physical system. This mirroring is achieved using clear translational keys between the model and the system, which enable knowing what relevant features are being tracked and how they are being tracked. For example, when the volume of 'empty' space in the physical system grows larger, a void grows in the model. When two voids merge in the model, the channel of metal that was separating two 'empty' regions no longer separates these two regions, leading to a larger 'empty' region. These physical features that are tracked by the model shape the structural constraints 9 of extant mechanisms. For example, when subjecting a piece of metal to a given tensile load, the presence of 'empty' pockets without metal means that the areas surrounding these pockets are subjected to a greater tension (there are less atomic bonds holding the structure together). The presence of these 'empty' areas, which are represented by voids in the model, constrains the outcomes of how stresses produce strains locally.
Models of fictional mechanisms are a useful tool to describe the dynamics of the systems' constraints. If we were to simply describe the constraints of the system without using any mechanical notions, we would lose important information about the evolution of the system. The elements of mechanisms (entities that act and when organized interact in a certain way) provide the right sort of ingredients to model the evolution of the physical structure of these constraints. Models of fictional mechanisms support counterfactuals about the evolution of physical system constraints while providing a tool to visualize these constraints. In order to understand how a fictional mechanism can track the constraints of the system over time we need to consider how the process of translating from model to target works. The next section explores this in greater detail. One key takeaway is that a realistic representation (in terms of extant mechanisms) would either not contain or would fail to highlight the right sort of information about these constraints.
The tracking of constraints, which can be fleshed out in greater detail using computer simulations, provides crucial understanding of ductile failure and enables the design of new materials. For example, engineers can estimate the likelihood of failure given a certain distribution of voids, they can understand why certain metal specimens of different materials fail sooner under the same load, and they can use these models to think about how certain modifications will affect the end result, such as the effect of adding impurities such as magnesium to a metal, which will generate a different number or distribution of micro-voids.
Generalizing these ideas, we can provide the following definition for a fictional mechanism explanation: An explanation is a fictional mechanism explanation if and only if (i) it includes a model of a mechanism that enables tracking relevant counterfactual dependencies of a physical system by capturing its relevant constraints through (ii) misrepresenting the system's ontology, yet in such a way that (iii) there is a set of well-defined translational keys that justify inferences from the model to its putative target.
Void explanations fulfil these clauses. The fictional mechanism of void growth and coalescence enables tracking relevant counterfactual dependencies of ductile materials by capturing relevant physical constraints. When the crack in the model is big enough, the piece of metal will pull apart. When enough voids coalesce in the model, necking occurs. If there are more voids, or the distribution of voids is arranged in a certain way in the model, less tension is required to pull the metal piece apart. The model, however, misrepresents the ontology of the system. There are no voids coalescing in ductile materials. Nevertheless, this fictional mechanism, provided with adequate translational keys can explain the phenomenon of ductile failure.
I now turn to the question of how it is possible for a fictional mechanism to capture real features of the physical system it represents and clarify the role played by translational keys.

How can a fictional mechanism capture worldly dependencies?
Earlier I argued that the way fictions provide information about physical systems is through translational keys. Here I consider in more detail how these keys work in the case of void explanations.
When considering ductile failure, strictly speaking it is the stresses to which the material is subjected that cause the observed effect (whether fracture, bending, or stretching). However, if one wishes to explain how this effect comes about, it becomes necessary to consider the relevant microstructures within the material that catalyse these stresses into actual strains. When modelling ductile materials, the mechanism of void nucleation, growth, and coalescence plays a key role in characterizing the behaviour of the material. The fictional mechanism keeps track of the microscopic changes that account for macroscopic phenomena. The physical structures represented by the elements of the fictional mechanism modulate the causal path between cause (stress) and effect (strain). As an analogy consider how the different geomorphic features of the terrain modulate the path a river takes from source to mouth. These features are not producing the descent of water, but they constrain the possible paths it can take. If one wishes to explain certain characteristics rivers have, one needs to consider these features.
At the physical level there is the expansion of material and the breaking of bonds, which results in 'empty' regions. The enlargement of these regions is captured by the voids' growth. These regions affect the properties of their surroundings. They make neighbouring regions less cohesive, increase the general porosity of the material, and interrupt plastic flow. Provided a constant global tension, a decrease in the number of atomic bonds subjects the remaining bonds to a greater local tension. Similarly, the metal surrounding these regions can expand without much resistance into these areas, which in some instances can help explaining properties of deformation under compressive forces. These 'empty' regions present a clear physical constraint on the system's evolution under tensile load. They are causally relevant, even if they do not engage in activities (the region represented by a void is not causally responsible for its own growth, nor is the merging of two 'empty' regions a physical processrather, it is the correlate of the physical process of a metallic channel fracturing).
The connection between the model and the physical system is cashed out in terms of translational keys. At each point in time, there is a void for each 'empty' region and with its same relevant characteristics (volume, location, etc.). Furthermore, the merging of two voids leads to a larger void that can be found in the system as a larger empty region as the result of a rupture of metallic channels. Void coalescence keeps track of what channels of metal become broken. These empty regions are empirically observed to develop 'as if' they coalesced, that is why the model correctly captures their evolution. The fine-grained details of the manifestation of ductility physically depends on these empty regions. By modelling them, we highlight the relevant constraints that make a difference to how ductile failure manifests. This is how the fictional mechanism of void growth and coalescence can explain ductile failure.
Giving a description of the actual mechanisms would provide the wrong sort of counterfactual information. The mechanism of void growth and coalescence should therefore not be understood as an indirect depiction of the actual mechanism. It is not representing a mechanism; it is representing the evolution of physical structures that are shaped by mechanisms and which in turn also provide structural constraints to the physical processes that occur. The model describes a space of constraints on causal processes without depicting the processes themselves. Detailed this way it becomes easier to see why an interpretation of voids constituting short-hand expressions tracking the 'actual mechanism' or of absences that produce the effect will not work: the (fictional) mechanism of void coalescence is not tracking the mechanism responsible for producing the effect; it is tracking the structural constraints of the space within which other mechanisms and processes are occurring.
As a closing remark, I would like to point out that the question as to whether a fiction can be explanatory has been examined in the literature through different angles (Bokulich, 2012(Bokulich, , 2016Nguyen, 2021;Schindler, 2014). The account that informed the one developed here is due to Alisa Bokulich (2012). She defends that a model containing fictions explains 'by showing how the elements of the model correctly capture the pattern of counterfactual dependence of the target system' (Bokulich, 2012, p. 730). To demarcate proper explanatory models from calculation devices, she also argues that models must also satisfy a 'justificatory step'. This step should specify to what extent the model is an adequate representation given certain purposes. Models containing fictions explain if they can capture real features of the system they represent. Whether a model is explanatory or not does not depend on whether it is a realistic representation, but on whether it can capture real features, thereby producing an understanding that might otherwise not be available by giving a realistic representation.
The present account of fictional mechanisms fulfils Bokulich's desiderata for fictions to be explanatory. The model's elements track dependencies in the physical system, and the model captures realistic features given certain purposes, which can be fleshed out in terms of translational keys.

Conclusion and further work
Engineers often explain ductile failure in terms of a failure mechanism characterized by the nucleation, growth, and coalescence of micro-voids. The aim of this paper was to understand the structure of such explanations and to clarify how they work. I offered an account of fictional mechanism explanations. According to this account, models of fictional mechanisms are used to track relevant dependencies in the physical system by virtue of adequately capturing the system's constraints.
The results of this analysis bear significance beyond Material Engineering. Explanations employing nonstandard entities are present in a variety of scientific domains. For instance, some areas of microphysics make use of crack propagations to explain the breaking of chemical bonds; food scientists study the properties of holes-crumb-with regard to its effects on the properties of bread; In the field of semiconductors, the direction of non-natural current in terms of positive charges is thought of as the movement of holes that were previously filled by electrons, as opposed to the movement of natural current which involves the movement of actual electrons; and in astronomy, there is talk of cosmic voids, regions of extremely low density or of dark energy, merging to produce bigger voids or to explain certain distributions (Russell, 2013).
This paper considered one subclass of micromechanical explanations ('void explanations' of ductile failure). Follow up work concerns investigating other subclasses of nonstandard entity explanations in engineering science (creep-voids, fatigue cracks, corrosion pits, etc.) and in domains other than material science. Fictional mechanism explanations may play a bigger role in science than recognized by contemporary philosophy of explanation. If this is the case, their structure, scope, and function require further elucidation.