Gentes Annie

Renon Anne-Lyse

How to frame the un-known? The odd alliance of design and "fundamental physics" in a design school

Keywords: Design Education, Interdisciplinarity, Expansive Learning, Design Theory, Knowledge

How to frame the un-known? The odd alliance of design and "fundamental physics" in a design school

Annie Gentes, Anne-Lyse Renon

Interdisciplinarity in design education

Design education is organized so as to teach students how to be creative (Cross, Christiaans, and Dorst, 1994;Folkmann, 2010;van Dooren, Boshuizen, van Merriënboer, Asselbergs, and van Dorst, 2013;Lu, 2015;[START_REF] Tovey | Designerly Thinking and Creativity[END_REF], build a theoretical and visual culture (Brookes,1992;Dutton, 1991;Gall, 2008;Chin, 2011;Hadjiyanni, 2014), solve problems with methodological and analytical techniques (Schon & Wiggins,1992;Goldschmidt & Smolkov 2006;Adams, turns, and Atman, 2003;Ozkan & Dogan, 2013;Daalhuizen, 2014), to create industrial and social value, community of practice (Lave andWenger, 1991, Wenger, 1998), and multidimensional treatment (Engeström, 1987). One of the issues of the design curricula is to help creative skills intersect with theoretical knowledge (Gentes et alii, 2015[START_REF] Tovey | Designerly Thinking and Creativity[END_REF]. Some programs actually engage the students into scientific literacy with the usual argumentation as reported by Fenstein (2010): sciences are helpful even for students who do not intend to engage into a scientific career because they are part of a general education (Donnelly, 2006). Science literacy is also supposed to help people make better reasoning and therefore helps them better manage their lives. In fact, multiple interdisciplinarities exist as Huutoniemi et alii (2009), who analyze their typology and indicators, stress. In education, the definitions also vary from the lowest degree of integration to reinventing and refiguring the fields of knowledge [START_REF] Klein | A Platform for a Shared Discourse of Interdisciplinary Education[END_REF]: Lenoir, Geof-froy, and Hasni identified 8 distinct forms of interdisciplinarity (Lenoir and alii, 2001). As far as design education is concerned, [START_REF] Findeli | Rethinking Design Education for the 21st Century: Theoretical, Methodological, and Ethical Discussion[END_REF] points out that schools of design (such as Ulms or Chicago) relied on a balance between art, sciences, and technologies and taught various disciplines that were judged necessary, in particular because design was considered as an applied science. However, he remarks the impossibility of listing the infinite number of sciences that could be applied by design. Ezio [START_REF] Manzini | New design knowledge[END_REF] also observed that contemporary design schools could be considered as "agents of (sustainable) change". In his analysis, design educational programs play a fundamental role in "new scenarios for the future", and the new challenge regarding design education is not so much to accumulate research and knowledge about everything but to know how to manage the "values of design research". One of the consequences of what he calls "new design knowledge" is that all current disciplines can come into play to support a designing activity. This issue of managing interdisciplinarity is tackled in specific ways in professional settings [START_REF] Jacobs | Interdisciplinary curriculum: Design and implementation[END_REF]Luecht and alii, 1989;Austin and alii, 2001). For Manzini, co-design and the collaboration of large teams makes it possible to achieve complex projects. But in this article, we are interested in understanding how a design school can train its students towards this interdisciplinary co-design, or how to engage in a dialog with disciplines. As pointed out by Wooyoung Sung et al. (2015), most of industrial design education is based on the "studiobased design pedagogy". The format is perfectly adapted to situations where the problem is relatively well identified. But when faced with complexity, "industrial design educators may need to consider an approach that is more interdisciplinary and that seeks solutions beyond those found in one design domain or other". Interdisciplinarity in this educational context goes hand in hand with the increasing scope of design projects and the wider range of design productions.

In our opinion, the question is whether the interdisciplinarity is conceptual (search for meaning) or instrumental (functional aim) or, as we make the hypothesis, "expansive". By expansive, we mean "constructing and implementing a radically new, wider and more complex object and concept for their activity" [START_REF] Engeström | Studies of expansive learning: Foundations, findings and future challenges[END_REF]. From this point of view, the goal would not be so much about teaching designerly way of knowing [START_REF] Cross | Designerly Ways of Knowing: Design Discipline Versus Design Science[END_REF], but to provide with different modalities supporting "expansive learning" [START_REF] Engeström | Expansive learning at work: Toward an activity theoretical reconceptualization[END_REF].

Our hypothesis is that interdisciplinarity in design can be better understood if we look at the characteristics and properties of these interdisciplinary situations to understand how they can actively support invention. How do students learn how to handle interdisciplinarity in action, through workshops, documents, and artifacts? To answer these questions, we describe 3 design workshops called "Form and Material", organized and supervised by a professor in fundamental physics (Z), and two designers (X) and (Y). In the analysis, we focus on how the different actors frame the workshops: explain it to students, organize the interactions, pace the design work. How do they manage not only the knowledge but also the non-knowledge (Mathieu and Schmidt, 2014) that goes with collaborating with other disciplines? How does this interdisciplinary framework support an expansive learning rather than a cumulative one?

Based on the observations and interviews, we present five properties of this framework: affective, cognitive, reflexive learning, economics, and political. These properties presuppose a model of design and design education. Finally, we will propose a conclusion about this a fortiori strategy in design education and how it relies on expansive learning.

Field work: interdisciplinarity in practice Context: an unlikely encounter between design and fundamental physics

The analysis of the literature on design shows numerous collaborative programs between Science and Design [START_REF] Cross | Science and design methodology: a review[END_REF][START_REF] Cross | Designerly Ways of Knowing: Design Discipline Versus Design Science[END_REF][START_REF] Bruffee | Collaborative learning: Higher education, interdependence, and the authority of knowledge[END_REF]Stahl, 2006;[START_REF] Renon | Graphic design and Objectivity: Looking at Meta-Atlases, in: Seeing Architecture. Design contribution to Knowledge Building[END_REF]. While interdisciplinarity is advocated by educational institutions and sustained by the analysis of professional design practices, many students who are introduced to design interdisciplinarity are afraid of the vast array of disciplines that they should know and use. The question is how to train for an almost infinite set of knowledge? Is it even possible to do so or is it a myth? And how are students prepared to raise up to the challenge of not understanding the depth of other disciplines that they must work with?

To address these questions, we chose to study a series of collaborations, engaged since 2011, between a professor in fundamental physics who is also one of the authors of this paper and designers in a design school. Investigation of fundamental physics problems by design is not entirely new [START_REF] Kelly | Some Educational Activities in Physics[END_REF][START_REF] Chi | Encoding process characteristics of experts and novices in physics[END_REF]Chi and alii, 1981;Chii and alii, 1989). However, it is still an institutional exploratory space for design education. We chose this experiment first because it is a test bed for pedagogical explorations of the relations between science and design. We also chose it because one of the authors (JB) could participate in the different stages of the project, from its definition to its implementation allowing for a longitudinal approach.

The opening of workshops "Form and Material" to fundamental physics

The "Form and Material" workshops gather each time about fifteen to twenty design students of mixed backgrounds and levels but with no specific qualification in science. They are supervised by two professional designers (X) and (Y) and a physicist (Z). For each workshop, a physics theme is chosen by the designers and the physicist together among the areas of expertise of the physicist: superconductivity in 2011, quantum physics in 2013 and light and optics in 2014. Focus is put on fundamental topics and not so much on technologies or applications. For example, the quantum physics project focuses on basic quantum phenomena such as wave-particle duality or tunneling effect. The light project focuses on the electromagnetic and quantum nature of light, not on technologies of lightings. During the workshops, students are first given outreach seminars by the physicist together with visits to the physics lab and open discussions about the physics at play. The students are then asked to conceive a design project inspired by the scientific material during a four-month period with two days per week devoted to the workshop.

There is a certain latitude in the definition of what the students' projects are going to be useful for. Their project can address a pedagogical goal and serve for outreach purpose, for example videos displaying physics phenomena, or devices demonstrating physics experiments. But the projects can also end up in artefacts inspired by science but with no educational purpose, for example lightings, clothes, or jewels. Students are encouraged to experiment various formats and domains. These workshops explore a wide variety of subjects (physics education, but also security, games, food, household use, sound, art, sport…), thanks to the help of the teachers-designers who make sure that every student explores a different path. The resulting projects are then shown to the rest of the school in a collective presentation and exhibition. They are also displayed in videos gathered in a website (ref : www.supraconductivite.fr www.designquantique.fr www.lightsciencedesign.fr) and further used in various outreach activities: exhibits in science museums, outreach talks, science fairs… A detailed description of the artefacts produced by the students can be found in [START_REF] Bobroff | Design and Superconducting Levitation[END_REF] for superconductivity and in [START_REF] Jutant | Objets de médiation de la science, objets de design : le cas du projet Design Quantique[END_REF] for quantum physics.

Research methods

One of the workshops has already been described and analyzed through a participative observation and a semio-pragmatic analysis of the documents and artifacts produced by the students by [START_REF] Jutant | Objets de médiation de la science, objets de design : le cas du projet Design Quantique[END_REF]. Elaborating on Jutant and Bobroff, who pointed out the diversity of popularization strategies deployed by the students, we wanted to analyze the framework of these activities that give the ideological background, the legitimization of the production, the specific "episteme", that is the presuppositions that found the practice and a priori knowledge of this experience ("that defines the conditions of possibility of all knowledge, whether expressed in a theory or silently invested in a practice", Foucault, 1966, 168). We therefore did qualitative interviews with the actors post workshops with a focus on how the proposition of the workshops had been framed [START_REF] Becker | Participant observation and interviewing: A comparison[END_REF][START_REF] Tedlock | From participant observation to the observation of participation: The emergence of narrative ethnography[END_REF].

Analysis of the "odd alliance"

The experiment that we describe was not planned as such, as the director of the school at that time points out. The "natural partnerships" of the design school are more with engineering sciences. The interviews with the different actors of the projects confirms this exploratory dimension of the project. An encounter between fundamental physics and design can be surprising. As it was a first occurrence in the school, a number of methods were used to make sure that the students would be able to tackle the challenge. A first 5-day collaboration with the physics professor was undertaken to "test" the feasibility of this collaboration and to reassure the students and the different actors of the project. As this first step was successful and the students were enthusiastic, the direction of the school and the faculty decided to do another, longer, 4 month workshop the following semester of the same year. All the subsequent workshops followed the same format which include elements of speech (such as the goals of the students productions), and pedagogical organization.

1.

The students have the freedom to explore the subject with any medium they choose, as they are supposed to take a "posture" of designer. It is not a question of truth or error but how to acquire a "position" towards a body of knowledge.

2.

The physicist is present all along the workshop. It starts during the presentation of the different workshops to the students, since the physicist and the designers present the "physics and design" workshop together. Then the physicist attends the workshop about two to three times per month. According, to him and the supervising designers, it allows a more trustful and open dialog with the students. After a few weeks, they don't hesitate to ask questions:

"It seems that my presence has a comforting effect: the fact that I'm enthusiastic about their productions and accessible on the science side seems to reassure and motivate the students."

In any case, the presence of the physics teacher is very important. When he is present at the workshop, the students want to do their best to show him their productions. So, each time he comes, we observed moments of acceleration of production, and new exploration and consolidation periods.

3.

The physical presence of all the actors during the workshop emphasizes the collaboration.

Even before producing anything, the students can anticipate a certain form of complementarity. There is a dramaturgy of the collaboration as well as an effective contribution of all the participants. In addition, there is the staging of an equality of disciplines. Science is not above design (the physicists: "I guess this perhaps reassures the students that I'm reachable and enthusiast about this collaboration".) The claim of the experiment is that each body of knowledge (design and science), looks at the others' competence with "ignorant eyes" [START_REF] Rancière | The ignorant schoolmaster: Five lessons in intellectual emancipation[END_REF]. As the physicist says:

"I am not a designer, I will not teach the students how to do design. In the same way, they are not physicists, and I won't expect them to become so."

In other words, the actors insist that the identity of the participants is not changed by the experiment. Still a collaboration is presented as possible. The workshop is the way to materialize this collaboration in practice.

4.

To encourage students in exploration, the physicist qualifies different levels of integration between their activity and scientific knowledge. As the physicist says:

" -I also insist that I don't expect them to understand every aspects of the physics at play: they can be "superficial" in their understanding. Also I make it clear that I will be there often and available to discuss and provide explanations as much as needed."

As he mentions, sometimes the students want to make "pedagogical" projects which explain physics:

" -In this case only (not the most common), I'm more demanding on the science exposed in their projects, and I ask for a validation process where I'm allowed to correct the scientific part if needed."

The productions can and will be used in scientific communication contexts such as exhibitions, websites, science museums … The work done is therefore validated outside of the workshop. This gives an additional value to the students' productions. This validation is a guarantee that their work is meaningful in a scientific context.

5.

According to the actors, the framework also manages a passage of the abstract to the concrete. One of the supervising designers said that she was disconcerted by the choice of quantum physics in particular, because, as she says:

" -It was very abstract and made it difficult for the students to project themselves in objects".

To counterbalance the abstract dimension of the project, the students were invited to visit the physics lab "to anchor the workshop in tangible places of scientific practice." Another method was to resort to usual and well-known design methods. As one of the designers pointed out:

" -We asked the students to use a method they know well, the scenarios of use, so that the project appeared "same as usual". We wanted to reassure the students on the objects they would have to produce, and by this way remove inhibitions they may have with the scientific knowledge they are not supposed to have."

Discussion: the five properties of the "design and physics" experiment

From the interviews and the observations, the framework appears to have several properties that build a specific dispositive made of language, organization, places, interactions, that structure the distribution of power between the actors and the disciplines [START_REF] Foucault | Discipline and Punish: the Birth of the Prison[END_REF].

1.

It is an affective dispositive. The new workshop is considered as a destabilizing environment. Indeed, the design students with no scientific background are faced with fundamental modern physics involving abstract concepts which may involve sophisticated mathematics or high-tech tools. Destabilization also occurs about the image of science itself, not embedded in applications or technologies, but from the point of view of fundamental research. However, the director shows his confidence that designers can elaborate within such a difficult environment. For him, trust in the design students' capacity to grasp elements that are beyond their usual skills and knowledge is at the core of this operation. As we have seen in the previous section, the charismatic ("enthusiastic") presence of the physics professor is a part of the affective dispositive as well as the reassuring collective or individual discussions. The figures of power and knowledge also frame the affective challenge with legitimate authority. The underlying model of design is that it can be a psychological challenge that has to be managed with care and attention.

2.

It is a reflexive and an "expansive learning" [START_REF] Engeström | Expansive learning at work: Toward an activity theoretical reconceptualization[END_REF] dispositive: the emphasis cannot be on "contents" since there is little chance that the design students will be able to catch notions that require years of training in physics. They get some elements of contents through the course given by the physicist, but they are mostly encouraged to gather their creative and making skills. There is therefore an abrupt shift from relying on learning something or learning how to make something, to using skills learnt in different classes and to put these skills into the project. The director is acutely aware that it is a particular challenge because he observes that students have difficulties to put into practice something they have learnt in one class to another class or workshop. The dispositive is therefore not only centered on the capacity to reuse some competence learnt elsewhere, but also, because of its extreme qualities, it is a reflexive space on this particular practice since physics is not a class "proper". The class is a test bed of designing through experience which is one of the design activity profiles analyzed by [START_REF] Cross | Designerly Ways of Knowing: Design Discipline Versus Design Science[END_REF] and [START_REF] Kruger | Solution Driven versus Problem Driven Design: Strategies and Outcomes[END_REF]. Designers explore their own past and tap into their previous realizations so as to find similarities with the new design projects. During the workshop, the students have to do the same and actually think about this way of doing design. It is also congruent with the model of the "reflective practitioner" described by [START_REF] Schön | The reflective practitioner: How professionals think in action[END_REF]Schön ( , 1987) ) that is a rationale that is based on doing and stepping back.

3.

It is a cognitive dispositive [START_REF] Rusbult | A Model of Integrated Scientific Method and its Application for the Analysis of Instruction[END_REF]. Though the professor in physics tries to give as many vivid metaphors as he can -for example, he presents the quantum tunneling effect as if, when an object is projected onto a wall, a small tunnel opens up and lets the object go through; or he presents superconducting levitation as a giant invisible wave embedded in the material which swirls when a magnet approaches and repels the magnet -the students have to work past their non-knowledge to be able to produce an artifact or a representation. Some students even acknowledged the fact that not understanding the topic in-depth was a liberating factor in terms of creativity, as designers and physicist looks at the others' competence with a priori "ignorant" eyes (to use Rancière's expression in the "Ignorant Schoolmaster"). This is congruent with a theory of "projection" and transfer in design [START_REF] Chow | Piercean Abduction, Signes & Transfer. 8th European Academy Of Design Conference -1st[END_REF][START_REF] Chow | Piercean Abduction, Signes & Transfer. 8th European Academy Of Design Conference -1st[END_REF]. Designers bring together elements (whether facts, aesthetics features, concepts, methods) that apparently have nothing in common in a surprising way and create a new concept/ artefact. This unexpected encounter of seemingly unrelated elements is not only as in Pierce's logics [START_REF] Peirce | Prolegomena to an Apology for Pragmaticism[END_REF] the way to find new hypotheses for facts. More importantly from a design perspective, it allows to create an unknown object. As the physicist says: "I'm here to discover new types of innovative and often unexpected points of view on my own scientific field, in terms of formats, representations, and understandings, which I can then reuse in various outreach contexts". Based on Peirce's definition of creative abduction [START_REF] Roozenburg | On the pattern of reasoning in innovative design[END_REF], we can consider that conception happens dynamically with concepts that are neither true nor false. These concepts or projects of artefacts, force the participants to look for solutions or knowledge that could bridge the gap between the fields that are brought together.

4.

It is an economics dispositive. Even if the body of knowledge is not expected to change with the experiment, students work with the uncertainty of the possible applications of scientific knowledge, and more broadly speaking with the uncertainties of the identity of objects. For example, a student conceived a wooden artefact to mimic some mathematical representations of wave functions. This artefact originally designed for a specific use in an outreach context in science museums happened to be used later in education as an introductory tool to help physics students think about the concept of the wave function and, on the other side, in a design exhibit (Biennale de Saint-Etienne). This seems to be an adaptation to the general mode of uncertainty that affects contemporary economics under the rule of radical innovation and that was pointed out by researchers in design [START_REF] Morello | Design predicts the future when it anticipates experience[END_REF] as well as in management and organization sciences [START_REF] Hatchuel | 13 Building Innovation Capabilities: The Development of Design-Oriented Organizations[END_REF]. Contemporary objects have no stabilized identity and designers cannot count on traditions of use for their objects. The director is quite clear about this: "nothing is going to be the same in twenty or thirty years from now. I want to make sure that designers will have the skills to adjust to an ever-changing environment". The shift from knowing something and knowing how to make something to knowing a posture of continuous adjustment to a changing set of environmental data is at the core of the dispositive. This kind of collaboration is the way to materialize this "changing environment" in practice.

5.

It is a political scientific dispositive. The workshop organizes a form of emancipation [START_REF] Rancière | The ignorant schoolmaster: Five lessons in intellectual emancipation[END_REF][START_REF] Rancière | The Emancipated Spectator[END_REF] from academic disciplines. First, the disciplines are represented by the professors participating in the workshops. The professors reinforce a sense of disciplinary identity by repeating that they will not change or become a hybrid between design and science. But at the same time, they offer a representation of the relations between disciplines that frees the participant of a strict and closed definition of disciplines. First, contrary to what happens most of the times between sciences, there is no hierarchy between disciplines. As suggested in the previous section of this article, the workshops stage and put into practice an equal collaboration. Second, since it is assumed that they will not become physicists, students are allowed to disregard the usual path to learning physics. This is made possible by a clear initial agreement with the scientific partner that the produced artefacts do not need to be necessarily scientifically accurate. There is still a relation to science. The workshop is like a shortcut that privileges borrowing facts, theories, images, from a discipline, rather than using a structured disciplinary body of knowledge. This seems to be the case for all the actors that agree to play out of their leagues since the physicist is no designer, the designer is no physicist and there is a general agreement that there are other ways of building knowledge than accumulating it.

Conclusion: design and science in education: a framework for expansive learning

Studying these experiments, we had multiple goals:

• Beyond the particularism of these examples, what are the properties of these experiences and can they be replicated in different institutions?

• What kind of learning is targeted by educational frameworks that bring together design and sciences?

• How do these experiments teach us something about design as conception?

Contrary to what could be expected, the physicist is not there to fill up the gaps of knowledge in physics.

While the interviews show that some students are more literate in physics at the end than at the beginning, the purpose of the curriculum is not to turn them away from design in the direction of physics, in a movement from "incompetence" to "competence". The introduction of physics in design education is not primarily for the sake of "contents", nor is it entirely for the sake of physics. In these workshops, the interdisciplinarity of design does not rely on an illusion of universal knowledge either within one person or even a group. The interdisciplinarity is not thought in terms solely of the addition of knowledge bases, or people representing these different knowledge bases as observed in innovative companies.

In the framework that we analyzed, interdisciplinarity does not appear to be conceptual in the sense of articulating two disciplinary fields together that would finally fit thanks to the emergence of new mutual concepts. The field of physics is not presented as being challenged by the field of design nor the field of design is impacted directly by the discipline of physics.

Finally, interdisciplinarity is not "instrumental" in the sense that physics as a science would need design to accomplish some of its goals, or design would use physics to pursue its tasks.

To come back to our initial question whether the interdisciplinarity displayed in these workshops is conceptual (search for meaning) or instrumental (functional aim), we can therefore say that it is neither. But something is nonetheless accomplished through the introduction of fundamental physics in a design curriculum. By bringing a discipline without a priori overlap with design knowledge, the workshop is an exploration of what is fundamental about design practice and knowledge. The whole framework makes an a fortiori demonstration of what design and design learning is about.

As we have seen in the discussion, it makes a demonstration of the capacity of designers to cope with five major properties of design situation: design can be a psychological challenge because it shows the limits of design knowledge not only on a personal level but because of the actual disparities between disciplines; it is a reflexive process where designers tap into their personal history and experience to create new representations at the crossroads of disciplines; it is a cognitive challenge since it deals with non-knowledge in the projection towards an X (unknown object); it is an economic challenge since there is no stability of objects in a society of continuous innovation; finally it is a political claim about the relationships between disciplines that neglects their boundaries and hierarchy.

The framework is designed so as to rehearse and cope with these difficulties. It points to a model of design and learning in design that involve expansive learning as defined by [START_REF] Engeström | Studies of expansive learning: Foundations, findings and future challenges[END_REF]. By expansive, we mean "constructing and implementing a radically new, wider and more complex object and concept for their activity" [START_REF] Engeström | Studies of expansive learning: Foundations, findings and future challenges[END_REF].

First, the framework relies on the reasoning that who can do more can do less. Namely, if a student follows this type of workshops, he will "a fortiori" be able to participate in any other interdisciplinary project, especially those that involve science. The five different properties of the workshop are probably more or less present in the other workshops but the latter pushes their logic beyond the ordinary. If one can learn how to design in such conditions one will be able to design in all circumstances.

The response to the challenges of design situations as they are staged through the workshops is to promote expansive learning because it is learning about expansion: the tools, the frameworks, the personal and group dynamics, the way to learn… The design students are not supposed to learn something that they wouldn't know yet, but to construct their own knowledge and imagine objects and practices, by their "nonknowledge".

The introduction of physics in the workshops therefore played a reflexive role on design practice, not because design knowledge must include more and more disciplines but because it can deal with all the principle challenges of any design situation.

In our research, understanding how the situation of learning was framed was therefore fundamental but is not enough to see how design in practice solves the tensions that such a strange encounter brings. As students are not asked to adopt reproductive gestures, but productive postures, our future research (similar workshops are programmed in the course of 2016 with the same protagonists) will evaluate how the students actually use their capacity of projection, transfer and hybridization, build artifacts, scenarios, and other students' productions, as well as the nature of the displays (in their "plastic artwork" properties), and the evaluation of the objects (in their diversities) to solve the interdisciplinary tensions.


Vinicius G Lopes
email: vinicius.g.lopes@uerj.br-joao.peterson@uerj.br-americo@ime.uerj.br

João Victor

L L Peterson

Americo Cunha

J V L L Peterson

Nonlinear characterization of a bistable energy harvester dynamical system

This chapter explores the nonlinear dynamics of a bistable piezomagneto-elastic energy harvester with the objective of determining the influence of external force parameters on the system response. Time series, phase space trajectories, Poincaré maps and bifurcation diagrams are employed in order to reveal system dynamics complexity and nonlinear effects, such as chaos incidence and hysteresis.

Introduction

Technological advances of the last decades have brought a wide variety of new portable electronic devices (smart phones, remote and micro sensors, smart medical implants, etc) and, at the same time bring great facilities to everyday life, created a huge demand for autonomous sources of energy. In this context, the energy harvesting technologies have been seeing as a potential solution in scenarios where conventional power supplying may not be practical or even available.

Energy harvesting technologies are based on physical-chemical properties of special (energy harvesting) materials, which provides electrical potential in response to an external stimulus, such as light (photoelectric), heat (pyroelectric) or mechanical vibration (piezoelectric), among others. A popular and comprehensible example of application may be found in solar energy generation, where the photovoltaic panels, gathering the minor units called cells which contains the silicon-based photoelectric material, when exposed to the sunlight incidence, performs its conversion into the electrical output power.

The piezoelectric energy harvesting systems have been widely discussed on recent works. An interesting aspect concerns into the shape versatility of such materials making them suitable for many design applications, as shown by [START_REF] Puspitarini | Designing a sustainable energyharvesting stairway: Determining product specifications using triz method[END_REF], also contributing to diversify its application possibilities -a review on it is addressed in [START_REF] Wei | A comprehensive review on vibration energy harvesting: Modelling and realization[END_REF]. Among the most promising ones, some examples can be found in different knowledge areas, as in medicine for powering pacemakers [START_REF] Kumar | Piezoelectric energy harvester for pacemaker application: a comparative study[END_REF], the electrical engineering and telecommunications, for industrial facilities [START_REF] Hadas | Virtual prototypes of energy harvesting systems for industrial applications[END_REF] and Internet of Things (IoF) wireless devices power supplying [START_REF] Kamalinejad | Wireless energy harvesting for the internet of things[END_REF]; in mechanics, as way for power recovering from friction losses on vehicle suspensions [START_REF] Abdelkareem | Vibration energy harvesting in automotive suspension system: A detailed review[END_REF] or from skyscrapers oscillations [START_REF] Xie | Energy harvesting from high-rise buildings by a piezoelectric harvester device[END_REF] or even for damping structural vibrations [START_REF] Gripp | An adaptive piezoelectric vibration absorber enhanced by a negative capacitance applied to a shell structure[END_REF][START_REF] Trindade | Semi-modal active vibration control of plates using discrete piezoelectric modal filters[END_REF].

The first vibration energy harvesting systems proposed were based on linear configurations, as the cantilever beam layout shown on Fig. 1. The relative simplicity of this system motivates new analysis and layouts, for example, as seen in [START_REF] Lu | A novel composite multi-layer piezoelectric energy harvester[END_REF]. Although, as shown by Cottone et al. [START_REF] Cottone | Nonlinear energy harvesting[END_REF] and, in more recent papers [START_REF] Kato | A broadband energy harvester using leaf springs and stoppers with response stabilization control[END_REF][START_REF] Kluger | Robust energy harvesting from walking vibrations by means of nonlinear cantilever beams[END_REF][START_REF] Vijayan | Non-linear energy harvesting from coupled impacting beams[END_REF][START_REF] Wang | A low-frequency, wideband quad-stable energy harvester using combined nonlinearity and frequency upconversion by cantilever-surface contact[END_REF], nonlinear configurations, such as the bistable inverse pendulum depicted in Fig. 2, may present better power recovering performances, when compared to the linear counterparts. The richness of such devices dynamics, combined with the new available improvements in numerical simulation tools and computing resources, have been propelling a wide variety of analysis considering different devices layouts, as seen in [START_REF] Foong | On mechanical damping of cantilever beam-based electromagnetic resonators[END_REF][START_REF] Ju | Impact-based piezoelectric vibration energy harvester[END_REF][START_REF] Pan | Design and analysis of a broadband vibratory energy harvester using bi-stable piezoelectric composite laminate[END_REF][START_REF] Stoykov | Vibration energy harvesting by a timoshenko beam model and piezoelectric transducer[END_REF] for composite and cantilever vibrating beams, or in [START_REF] Friswell | Non-linear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass[END_REF][START_REF] Halvorsen | Statistics of a noise-driven elastic inverted pendulum[END_REF][START_REF] Litak | Regular and chaotic vibration in a piezoelectric energy harvester[END_REF][START_REF] Litak | Vertical beam modal response in a broadband energy harvester[END_REF] for inverse pendulum layouts and new designs as those discussed in [START_REF] Bradai | Nonlinear analysis of an electrodynamic broadband energy harvester[END_REF][START_REF] Dai | Nonlinear analysis and characteristics of inductive galloping energy harvesters[END_REF][START_REF] Harris | Recurrence plot and recurrence quantification of the dynamic properties of cross-shaped laminated energy harvester[END_REF][START_REF] Naseer | Piezomagnetoelastic energy harvesting from vortex-induced vibrations using monostable characteristics[END_REF][START_REF] Zhou | Improving energy harvesting from random excitation by nonlinear flexible bi-stable energy harvester with a variable potential energy function[END_REF].

The external forcing influence over harvesters energetic performance and dynamic characteristics have also been widely investigated in [START_REF] Borowiec | Energy harvesting of cantilever beam system with linear and nonlinear piezoelectric model[END_REF][START_REF] Lopes | Analysis of the nonlinear dynamics of a bistable energy harvesting system with colored noise disturbances[END_REF][START_REF] Martens | Stationary response of nonlinear magneto-piezoelectric energy harvester systems under stochastic excitation[END_REF][START_REF] Pereira | Random effects in a nonlinear vibration-based piezoelectric energy harvesting system[END_REF], which consider mixed harmonic and random noise forcing, and also in [START_REF] Belhaq | Energy harvesting from quasi-periodic vibrations[END_REF][START_REF] Ghouli | Quasiperiodic energy harvesting in a forced and delayed Duffing harvester device[END_REF], that explore the possibility of energy scavenging in a quasi-periodic regime from a delayed dynamics. The multidisciplinary character of the theme also motivates the extension of the discussion to the coupled electrical circuits, as seen on [START_REF] Basqueroto | Analysis of bistable and chaotic piezoelectric energy harvesting device coupled with diode bridge rectifier[END_REF][START_REF] Clementino | Modeling of piezoeletric energy harvesting considering the dependence of the rectifier circuit[END_REF][START_REF] Liao | Unified modeling, analysis and comparison of piezoelectric vibration energy harvesters[END_REF][START_REF] Silva | Equivalent electrical circuit framework for nonlinear and high quality factor piezoelectric structures[END_REF].

The same nonlinearities responsible to improve the devices energetic performance, on the other hand, contributes to raise the systems dynamics com- plexity, even leading to chaotic and unstable responses for some operational configurations. In [START_REF] Lopes | Numerical study of parameters influence over the dynamics of a piezo-magneto-elastic energy harvesting device[END_REF][START_REF] Lopes | On the nonlinear dynamics of a bi-stable piezoelectric energy harvesting device[END_REF][START_REF] Peterson | Dynamic analysis and characterization of a nonlinear bi-stable piezo-magneto-elastic energy harvester[END_REF] the authors demonstrates the presence of hysteresis effects on the nonlinear dynamical response of a piezo-magneto-elastic energy harvesting system, that is associated to the co-existence of multiple solutions, for the same operational conditions depending on the external forcing parameters values. Such phenomena highlight the importance of a detailed investigation of the nonlinear dynamics of devices in order to improve the comprehension of its behavior and limitations.

In this sense, this chapter presents an analysis of the dynamics of a nonlinear bistable piezo-magneto-elastic energy harvesting system, aiming to investigate the influence of different harmonic forcing parameters on system output voltage. Numerical simulations reveals the effects of the amplitude and frequency excitation conditions on the chaos and regularity incidence on system electrical response. The bifurcation diagrams analysis allows to clearly identify the hysteresis phenomena and map those dynamics regions susceptible to chaos incidence. The voltage time series results sampled from such diagrams deepens the analysis of chaotic dynamics effects on the electrical output.

The rest of this chapter is organized as follows. The second section presents the energy harvesting system and its mathematical modeling. Next, in sections 3 and 4, the dynamical system is analyzed by time series and bifurcation diagrams, respectively. Final remarks close the text in the fifth section.

2 Bistable energy harvester

Physical system

The physical system of interest in this work is the piezo-magneto-elastic energy harvesting device depicted in Fig. 3. This electromechanical system consists of an elastic beam in vertical configuration, made of ferromagnetic material, with the upper extreme clamped into a rigid base, and the bottom end moves freely. A pair of magnets is placed at the rigid base lower part. Two piezoelectric laminae are placed on beam's highest part, being also connected to a resistive circuit. The rigid base is periodically excited by an external source, which, together with the magnetic field generated by magnets, induces large amplitude vibrations in the beam. Once this movement is perceived by the piezoelectric laminae, the mechanical energy is converted into electrical power, which is dissipated in the resistor. This energy harvesting dispositive, proposed by Erturk et al. [START_REF] Erturk | A piezomagnetoelastic structure for broadband vibration energy harvesting[END_REF], is based on the classical magneto-elastic beam of Moon and Holmes [START_REF] Holmes | A nonlinear oscillator with a strange attractor[END_REF][START_REF] Moon | A magnetoelastic strange attractor[END_REF]. x Fig. 3 Illustration of the piezo-magneto-elastic energy harvester.

Mathematical model

If the piezoelectric laminae and the electric circuit are not coupled to the system, the dynamic behavior of the beam is well approximated (for a limited frequency band) by its first mode shape, whose amplitude evolves according to a single degree-of-freedom Duffing oscillator [START_REF] Kovacic | The Duffing equation: nonlinear oscillators and their behavior[END_REF]. However, once the piezoelectric transducer is attached to the beam, it starts to influence the system dynamic behavior so that piezoelectric interaction effect must be taken into account [START_REF] Erturk | A piezomagnetoelastic structure for broadband vibration energy harvesting[END_REF]. In this sense, [START_REF] Erturk | A piezomagnetoelastic structure for broadband vibration energy harvesting[END_REF] shows that the harvesting device dynamics can described by the following initial value problem

ẍ(t) + 2 ξ ẋ(t) - 1 2 x(t) 1 -x 2 (t) -χ v(t) = f cos (Ω t), (1)
v(t) + λ v(t) + κ ẋ(t) = 0, (2)
x(0) = x 0 , ẋ(0) = ẋ0 , v(0) = v 0 , (3)
where t denotes time; x is the beam tip amplitude of oscillation; v is the voltage in the resistor; ξ is the damping ratio; f is the amplitude of the external force induced by the rigid base oscillation; Ω is the external excitation frequency; λ is a reciprocal time constant; the piezoelectric coupling terms are represented by χ, in the mechanical equation, and by κ in the electrical one; x 0 represents the beam edge initial position; ẋ0 is the beam edge initial velocity; and v 0 denotes the initial voltage over the resistor. The upper dot is an abbreviation for time derivative. All parameters described above are dimensionless. For all the simulations reported below, unless something is said on the contrary, the following dimensionless parameters are used: ξ = 0.01, κ = 0.5, χ = 0.05, λ = 0.05, Ω = 0.8. The standard initial conditions is (x 0 , ẋ0 , v 0 ) = (1, 0, 0). Different values of f are used, being indicated below. The dynamic is investigated for a temporal window defined by the interval 0 ≤ t ≤ 5000.

3 Time series analysis

Effects of excitation amplitude

The voltage time series, analyzed for 0.019 ≤ f ≤ 0.275, with amplitude increments of 0.032, are shown in Fig. 4. The steady state behavior is shown in the red window and, when pertinent, the transient behavior is depicted in a yellow box.

Figure 4(a) show the response of the system for excitation amplitude f = 0.019. The energy generated with such small excitation amplitude is minimal, and the response present regular behavior throughout the analyzed range. In Fig. 4(b), as the forcing amplitude increases, the response assumes a larger amplitude. With f = 0.083 the response assumes a chaotic behavior, as can be seen in the yellow and red boxes. The voltage generated by the system increases. In 4(d) the dynamic presents regular steady state response starting from approximately 1000 time units, whereas it transient response is still chaotic. The amplitude of voltage output rises, to approximately 1 nondimensional volt, now that the energy input is considerably higher. The subsequent images (4(e), 4(f), 4(g), 4(h) and 4(i)) present more or less the same response. All of them have wide amplitude range, between -1 and 1, and regular response at steady state, with little transient response. In Fig. 5 it is possible to see a comparison between the chaotic attractor for f = 0.083, and the regular one, for f = 0.115. This figure also show the respective projections in the planes of displacement vs velocity, displacement vs voltage and velocity vs voltage, and corresponding Poincaré sections depicted by red dots. One can notice that a minimum change of the order of 0.032, that corresponds to 3.2% of the characteristic displacement used in the nondimensionalization, in the value of excitation amplitude can resolve in a totally different response. For f = 0.083, the dynamic presents chaotic behavior, trapped in the chaotic strange attractor, whereas for f = 0.115 the system resolve in a regular attractor. Once for the chaotic configuration the system response presents non periodic pattern, its Poincaré section is represented by a large number of red dots not organized according to an invariant topological structure. The regular behavior has only one frequency, and thus, its Poincaré section has only one dot. The projections show that the regular attractor has higher energy output, characterized by its large amplitude in displacement. A wide overview of the dynamics is best carried out by mapping the various attractors of the dynamical system. For this purpose the dynamics is integrated, for an excitation frequency Ω = 0.8 and a total of 1200 different initial conditions, uniformly spaced in the region -3 ≤ x 0 ≤ 3, -3 ≤ ẋ0 ≤ 3 and v 0 = 0, and the attractors corresponding to the steady-state of each case are identified. Projections of the identified attractors are shown in Fig. 6.

The attractors projections for forcing amplitude f = 0.019 are depicted on Fig. 6(a). They are four in number, all with single period, being those depicted on magenta and cyan colors the more energetic orbits. For f = 0.051, which results are depicted on Fig. 6(b), similar magenta and cyan basins are observed, as an expressive enlargement of blue and red basins shapes. Three regular orbits, depicted on brown, golden and green colors appears around those ones, reveling more energetic orbits which may be associated to the forcing amplitude increase. For f = 0.083, on Fig. 6(c), instead, the green orbit remains while the others disappears. A chaotic dynamics orbit, depicted on gray color, emerges corroborating the results observed on the phase attractor in Fig. 5, for the same model parameters. Similar results can be observed for f = 0.211, on Fig. 6(g

Effects of excitation frequency

Now the effects of the excitation frequency over the system dynamics is explored trough attractors projections, calculated in a similar way as in the previous section, for f = 0.083, f = 0.115 and 0.1 ≤ Ω ≤ 0.9, and shown in Figs. 7 and8, respectively. The Fig. 7(a), for Ω = 0.1, reveals two low energy regular attractors with low amplitude orbits for beam displacement and velocity. Similar results can be observed for f = 0.115, on Fig. 8(a), but with a chaotic attractor, on gray color. For Ω = 0.2, on Fig. 7(b), a clear narrowing of the basins shape is observed, as the chaos incidence, depicted on gray color, corresponding to larger amplitudes. The analysis of the attractors reveals a larger velocity range in regular response if compared with the previous case. Despite of the presence of a green regular attractor, no expressive changes are observed for f = 0.115 results, on Fig. 8(b), in comparison with those for f = 0.083. For Ω = 0.3, on Fig. 7(c), the red basin prevails over the blue one; furthermore, the tendency of narrowing as the system moves away from the unforced system equilibrium points (presents in (-1, 0) and (1, 0)) continues. The chaos still remains, as the attractors keep to increase in amplitude. A similar behavior is observed by considering f = 0.115, as show on Fig. 8(c). For Ω = 0.4, on Fig. 7(d), one can note the predominance of the blue basin, although the red and blue basins are even more mixed. An expressive incidence of chaos is noted, despite of the presence of a new basin, depicted on green color, is noteworthy. The attractors show that such basin is more energetic due to a greater amplitude both in displacement and velocity. In addition, it is observed the emergency of two other basins, depicted on cyan and magenta colors, both with more energy than others, presenting two periods. For f = 0.115, on Fig. 8(d), similar results can be observed except for the double period basins, which have disappeared. For Ω = 0.5, in Fig. 7(e), an expressive growth of the green basin occurs, along with the greater mix of peripheral regions, as the system gets closer to the resonance frequency. The attractors show the expansion of the green, blue and red basins and the inexistence of cyan and magenta ones. Also noteworthy is the transformation of the blue and red basins, now with two periods. No differences can be noted for f = 0.115 results, on Fig. 8(e). The Fig. 7(f), for Ω = 0.6, shows the predominance of the green basin with blue and red islands. The attractors present a continuous growth of the green orbit, alongside with small growth of the blue and red orbits, which returned to present a single period and, in counterpart with the previous case, there is absence of chaotic responses. Similar results are observed for f = 0.115, on Fig. 8(f). For Ω = 0.7, in Fig. 7(g), a growth of the blue and red islands, as an increasing relevance of the green basin can be observed. The attractors continue to have more energetic orbits. Similar results can be observed for f = 0.115, depicted on Fig. 8(g). In the system resonance frequency of Ω = 0.8, depicted on Fig. 7(h), the chaotic responses predominates along with the most energetic orbit, on green color. The attractors show that such orbit is the only regular one. Similar results are observed for f = 0.115, on Fig. 8(h). Finally, for Ω = 0.9, which results are depicted on Fig. 7(i), the blue and red basins return subtly, still with presence of chaos. The attractors show the most energetic orbits. For f = 0.115, on Fig. 8(i), despite of the similarities, the red and blue basins present double period orbits.

Bifurcation analysis

The system bifurcation diagrams are built by sampling 1200 values for external excitation parameters from regularly spaced observation interval. Results are referred as forward and backward for an ascending and descending values sampling ordering, respectively. For each sampled amplitude, the dynamical system is integrated for a constant frequency. The first 90% of system response time series is neglected, as a way to avoid the transient response. From displacement, velocity and voltage results, the amplitude forward and backward bifurcation diagrams are obtained, according the chosen sampling ordering. A similar process can be carried out for frequency values, with constant amplitudes, leading to frequency forward and backward bifurcation diagrams. For ongoing results, typically, cool colors are reserved for forward results, while the warm colors scale depict the backward ones.

Influence of excitation amplitude

Regarding amplitude analysis, both forward and backward bifurcation diagrams are built for nine Ω values regularly sampled from 0.1 ≤ Ω ≤ 0.9, considering the amplitude sampling interval as 0.1 ≤ f ≤ 1.4. The output voltage results are shown on Fig. 9.

The blurred regions depict chaotic dynamic on system response, for those on that output voltage may assume a wide range of different values. An overview of both, forward and backward bifurcation diagrams, points that the system is strongly sensitive to non-regular dynamics for the highest amplitude and lowest frequency values analysed. Some similar phenomena can be seen on the first portion of amplitude control interval on highest frequencies. From Ω = 0.5 and higher frequency values, some expressive discrepancies are observed on voltage output by comparison of forward and backward sam- pling orderings. This concerns to the nonlinear effects introduced by the pair of magnets on harvester, which are also responsible for the bistable configuration. For Ω = 0.6 and Ω = 0.7, the system voltage response keeps regular for all tested amplitudes on the interval. A multiple period region appears for the last forcing amplitude values, after some slim band of chaotic dynamics. Similar formations are observed between the chaotic regions on the two highest frequencies analysed. The Fig. 10 presents both forward and backward bifurcation diagrams for Ω = 0.2 and Ω = 0.8 extracted from frequencies observation window overview in Fig. 9. In both cases, for Ω = 0.2, the system presents chaotic dynamics about 0.260 ≤ f ≤ 0.270. A pitchfork bifurcation revealed on forward diagram near f = 0.23 is hidden by system dynamics on backward, and reappears about f = 0.27, which highlights the system response dependence from the problem initial conditions. Discontinuous regions are also observed on the last portion of amplitudes interval, suggesting a voltage behavior more independent of initial conditions, thus, reliable for the early amplitudes on this frequency. It is important to take account that such conclusions are true only for the model parameters defined. A similar analysis can be carried out for the amplitude bifurcation diagram for Ω = 0.8. A pair of chaotic voltage bands can be seen on forward diagram for 0.07 ≤ f ≤ 0.08 and 0.09 ≤ f ≤ 1.00 filled by a richer multiple region. Now, instead, a strong regular dynamics region fills the middle and the last portions of the amplitude intervals both for forward and backward results. Extending the previous discussion, Fig. 11 shows the voltage waveforms, obtained by sampling the system dynamics for specific forcing amplitude values taken from Fig. 10. For example, for Ω = 0.2 and f = 0.2146, the harvester output voltage is regular, both on forward and backward diagrams, as seen before, besides the signal waveforms are quite distorted. For f = 0.2695 and the same excitation frequency, although both forward and backward waveforms have no period defined due to the chaotic motion.

Influence of excitation frequency

For the frequency analysis, the forward and backward diagrams are built for nine amplitude regularly sampled values from 0.019 ≤ f ≤ 0.275, for 0.01 ≤ Ω ≤ 1.4 set as the frequency sampling interval. The Fig. 12 presents the harvester output voltage results. For both forward and backward cases, the system reveals itself more susceptible to the chaos occurrence, when compared with those for amplitude analysis, depicted in Fig. 9. Despite of it, all diagrams present a common regular behavior in the middle portion of frequencies interval. The blurred regions became more evident for the last amplitude values in the observation windows, for which both low and high frequencies leads the system to chaos condition. Particularly, for f ≥ 0.179, voltage results present multiple chaotic regions on backward diagrams, what emphasizes the influence of sampling ordering on explicit IPV solution. For higher amplitudes, multiple period regions and pitchfork bifurcations appear close to the chaotic regions, although it still means regular dynamics. A similar phenomena occurs on the early portion of frequencies interval 0.083 ≤ f ≤ 0.147, where some discontinuities are also observed, both on forward and backward cases. By comparison with the same regions on higher amplitudes diagrams, the discontinuities, thus, may be inferred as a way to predict the chaos occurrence. For the lowest amplitude, system voltage results remains regular for all investigated frequencies. A small pitchfork bifurcation can still be noted on backward results about f = 0.6. This indicates that for different operation frequencies, a small forcing amplitude value may be interesting once it prevents chaotic dynamics.

The Fig. 13 presents the forward and backward diagrams for f = 0.083 and f = 0.179, sampled from amplitude observation window overview in Fig. 12. For f = 0.083, forward and backward methods reveals a similar dynamics profile regarding the chaos and regularity. The exception is confined to the region between 0.4 ≤ f ≤ 0.6, where a pitchfork bifurcation on backward case gives rise to a discontinuity on forward one. The chaotic dynamics region about Ω = 0.8 is small if compared with those observed for higher amplitude values, both on forward and backward results. For f = 0.179, as an overview of Fig. 12 suggests that the system presents a rich dynamics related to chaos occurrence. A strongly nonlinear region can be observed on the first middle of the frequencies interval, filled with large discontinuities region on its early portion. The system susceptibility to initial conditions for higher amplitudes, inducted by the sampling ordering, becomes more evident when forward and backward results are compared. A massive chaotic region rises around of Ω = 1.0 on backward case, in contrast with regular dynamics on forward diagram on same region. The system results investigation may be improved by sampling amplitude and frequency values from diagrams in Fig. 13, as shown by the voltage time series presented in Fig. 14. The voltage time series for f = 0.179 and Ω = 0.3786, for example, highlights the importance of a deep investigation. System voltage waveforms presents a regular pattern, both on forward and backward diagrams. Although, the multiple period region distortion is more evident on backward case, once forward voltage waveform seems to preserve some characteristics of the harmonic forcing. In this sense, best results are observed for f = 0.179 with Ω = 1.0585, where forward amplitudes overcomes backward one. The chaotic dynamics consequences is observed better for f = 0.083 with Ω = 0.8037, in Fig. 14. A periodic behavior is not clearly seen on voltage waveforms, neither on forward or backward cases.

Concluding remarks

This chapter presents a nonlinear dynamic analysis of a piezo-magneto-elastic energy harvesting system. Its nonlinear behavior is explored by time series, Poincaré maps, phase space trajectories and bifurcation diagrams. The voltage output time series and the phase attractors analysis allow to relate the length of the transient dynamics response to the external forcing frequency and amplitude parameters values. The same graphs also provide a way to link the magnitude of the harmonic forcing and the amount of output power.

The bifurcation diagrams analysis reveals a hysteresis effects on the system dynamics which may be associated to the model nonlinearities, introduced by the pair of magnets placed on the rigid structure lower part, also responsible to the bistable configuration. The voltage time series sampled from such diagrams provides a detailed overview of the chaos incidence distortions on system output voltage waveforms.

Fig. 1

 1 Fig. 1 Illustration of a linear vibration energy harvester.

Fig. 2

 2 Fig. 2 Illustration of a nonlinear vibration energy harvester.

Fig. 4

 4 Fig. 4 Time series of voltage for 0.019 ≤ f ≤ 0.275.

Fig. 5

 5 Fig. 5 Attractors in phase space for f = 0.083 (top) and f = 0.115 (bottom), phase space trajectories projections and corresponding Poincaré sections.

  ). For f = 0.147, f = 0.179, f = 0.243 and f = 0.275, which results are depicted, respectively, on Figs. 6(e), 6(f), 6(h) and 6(i), a single period orbit is observed, what reveals that, for such configurations, the system dynamics is regular.

Fig. 6

 6 Fig. 6 Phase space trajectories projections for 0.019 ≤ f ≤ 0.275 and Ω = 0.8.

9 Fig. 7

 97 Fig. 7 Phase space trajectories projections for 0.1 ≤ Ω ≤ 0.9 and f = 0.083.

9 Fig. 8

 98 Fig.8Phase space trajectories projections for 0.1 ≤ Ω ≤ 0.9 and f = 0.115.

Fig. 9

 9 Fig. 9 Bifurcation diagram of voltage as function of excitation amplitude f , for several values of excitation frequency Ω .

Fig. 10

 10 Fig. 10 Bifurcation diagrams for Ω = 0.2 (left) and Ω = 0.8 (right). Forward diagrams on blue colors and backward, on red and yellow.

Fig. 11

 11 Fig. 11 Voltage time series for Ω = 0.2/Ω = 0.8 and several values of f . Forward diagrams on blue colors and backward, on red.

Fig. 12

 12 Fig. 12 Bifurcation diagram of voltage as function of excitation frequency Ω , for several values of excitation amplitude f .

Fig. 13

 13 Fig. 13 Bifurcation diagrams for f = 0.083 (left) and f = 0.179 (right). Forward diagrams on blue colors and backward, on red.

Fig. 14

 14 Fig. 14 Voltage time series for f = 0.083/ f = 0.179 and several values of Ω . Forward diagrams on blue colors and backward, on red.

Acknowledgments

The authors acknowledge the support given to this research by the funding agencies Carlos Chagas Filho Research Foundation of Rio de Janeiro State (FAPERJ) under grants E-26/010.002.178/2015 and E-26/010.000.805/2018, and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001.


J Champagne

C Ovalle

S Cantournet

F Lequeux

H Montes

K Le-Gorju

Hydrostatic pressure effect on the non-linear mechanics of filled rubbers: experiments and physico-mechanical approach

It has been known for several decades that rubbers exhibit a hydrostatic pressure-temperature superposition, interpreted by the free-volume theory. Moreover, our experimental results, on a filled rubber, demonstrate a hydrostatic pressure-temperature superposition which is two orders of magnitude larger than the one observed in pure rubbers. A new physico-mechanical model is proposed to account for the hydrostatic pressure effect responsible for a local glass transition temperature shift. It is suggested that it comes from the change of distances between aggregates. The predicted shift, given by the model, is found to be consistent with experimental data.

INTRODUCTION

The addition of "active" fillers -carbon black or silicain pure rubbers leads to a strong enhancement in mechanical properties [START_REF] Bills | The tensile properties of highly filled polymers. effect of filler concentrations[END_REF]. Reinforcement goes beyond the mere increase in stiffness, showing improvements in the elongation at break, hysteresis and abrasion resistance. Hence, filled rubbers are of prior importance for many industrial applications. In some specific applications, the material is confined between stiffer parts resulting in hydrostatic pressure, generally excluded in modelling.

The hydrostatic pressure effect on mechanical properties has been widely investigated in pure rubbers for the last 70 years. Measurements of stress relaxation modulus [START_REF] Fillers | The effect of pressure on the mechanical properties of polymers[END_REF]), Young's modulus [START_REF] Paterson | Effect of pressure on young's modulus and the glass transition in rubbers[END_REF] or even dielectric properties [START_REF] Cheng | Pressure-temperature study of dielectric relaxation of a polyurethane elastomer[END_REF]) under superposed hydrostatic pressure and temperature were performed using high pressure chambers. More recently, hydrostatic pressure effect was investigated through shearcompression tests [START_REF] Alkhader | A new shear-compression test for determining the pressure influence on the shear response of elastomers[END_REF]). Unanimously, experimental results show that glass transition temperature rises with hydrostatic pressure by amounts varying from 0.15 to 0.3 K.MPa -1 depending on the material. These values are consistent with theoretical predictions based on the free-volume theory [START_REF] Ferry | The free volume interpretation of the dependence of viscosities and viscoelastic relaxation times on concentration, pressure, and tensile strain[END_REF]. This effect will be referred as "matrix pressure effect" in the following. To our knowledge, hydrostatic pressure effect has never been studied on filled rubbers.

In this paper, hydrostatic pressure effect on the thermo-mechanical behaviour of filled rubbers is shown to be two orders of magnitude greater than the one observed in pure rubbers. Observations are described under a physical framework allowing to develop a new physico-mechanical model accounting for local glass transition temperature.

MATERIAL AND TEST SAMPLE

The studied material is a poly(dimethylsiloxaneco-diphenylsiloxane) rubber highly filled with silica nanoparticles. Based on Nuclear Magnetic Resonance (NMR) measurements, the molar proportion of diphenylsiloxane groups is estimated at 5%. Classical Thermo-Gravimetric Analysis (TGA) does not allow to scale the reinforcement due to a ceramic production from polymer during thermal degradation. However, the mass proportion of fillers is estimated at 30%. From Differential Scanning Calorimetry (DSC) and Dynamic Thermo-Mechanical Analysis (DTMA), the glass transition temperature T g is found to be -115 • C.

Test samples, also known as poker-chip [START_REF] Gent | Internal rupture of bonded rubber cylinders in tension[END_REF], consist of a thin rubber disc bonded between two alloy (Alumec79 ASTM 7022) cylinders. A macroscopic confinement c is defined by means of the disc height h d and the disc radius r d : ( 1967). This critical value was investigated through a parametric study conducted in an in-house Finite Element Analysis code (Z-set).

c = r d h d (
The numerical problem was modelled in an axisymmetric way [Fig. 1]: uniaxial compression simulations were performed by controlling the vertical displacement u z . The material was modelled by a quasi-incompressible neo-Hookean formalism where the strain energy potential W (C 10 , K v ) depends respectively on an elastic and a bulk modulus:

ρ 0 W (C 10 , K v ) = C 10 (I 1 -3) + K v U(J) (2)
where C 10 ∼ 1 MPa, K v ∼ 1 GPa, I 1 is the first strain invariant and U(J) = 1 2 1 2 (J 2 -1)log(J ) takes into account the incompressibility effect. Besides, to avoid volumetric locking and pressure oscillations, a three-field mixed finite element formulation is used [START_REF] Al Akhrass | Integrating a logarithmic-strain based hyperelastic formulation into a three-field mixed finite element formulation to deal with incompressibility in finitestrain elastoplasticity[END_REF].

Simulations were performed on ten numerical samples for a wide range of macroscopic confinements, from 1 to 10. To analyse the state of stress, the triaxiality ratio [START_REF] Lemaitre | Mécanique des matériaux solides[END_REF], defined by the hydrostatic pressure p H and the Von Mises equivalent stress σ V M eq , was computed:

χ = p H σ V M eq (3)
with p H = -1 3 Tr(σ) and σ V M eq = 3 2 σ dev : σ dev 1 2 . The following approximation is made. If the specified criterion χ > 1 is fulfilled, then the stress tensor is assumed to be hydrostatic.

Numerical results

To compare all results, the volumetric fraction of hydrostatic state V χ>1 , i.e. the region of the numerical sample verifying the criterion, is reported [Fig. 2]. The volumetric fraction increases towards 1. A critical macroscopic confinement c * = 4 seems to be a relevant threshold, implying at least 80% of hydrostatic state in the material. Furthermore, a parabolic pressure field along the radial direction ρ was seen. The stress state is not hydrostatic on the free edges because the material is not constrained in the radial direction [START_REF] Dorfmann | Shear, compressive and dilatational response of rubberlike solids subject to cavitation damage[END_REF]. Therefore, on a sufficiently confined sample, the total stress has a contribution of more than 80% originating from a hydrostatic state, i.e. a compression load is equivalent to a hydrostatic pressure test. The experimental setup [Fig. 3] was designed to be adapted in an Instron 8801 testing machine (±50 kN, ±75 mm). For symmetry reasons, two samples are loaded simultaneously. The hydraulic cylinder displacement, in the perpendicular direction, provides a sinusoidal shear strain amplitude ∆γ. To limit heat build-up due to the thermo-mechanical coupling [START_REF] Ovalle Rodas | A finite strain thermo-viscoelastic constitutive model to describe the self-heating in elastomeric materials during low-cycle fatigue[END_REF], frequency was kept to a minimal value of 1 Hz. The axial spring ensures the static compression load, leading to an average hydrostatic pressure:

p H = F c πr 2 d (4)
Thus, the pressure effect on the thermo-mechanical behaviour can be studied.

Experimental results

Shear hysteresis loops obtained for different applied hydrostatic pressures are highlighted [Fig. 4a.]. Test were performed at the same temperature and shear amplitude (T = -15 • C, ∆γ = 16%). It can be seen that hydrostatic pressure has an increasing effect, even at low values ∼ 1 MPa, on the thermo-mechanical behaviour of the material: stiffness and dissipation.

The temperature effect on the shear mechanical properties, at a constant hydrostatic pressure, is also investigated. Temperature is seen as an influential parameter with the ability to soften stiffness and to decrease dissipation [START_REF] Lion | On the large deformation behaviour of reinforced rubber at different temperatures[END_REF].

From these results, both physical parameters, i.e. hydrostatic pressure and temperature, are found to have the same effect on the mechanical behaviour, i.e. two different sets of parameters (p 1 H , T 1 ), (p 2 H , T 2 ) could be found to exhibit an identical mechanical behaviour [Fig. 4b.]. The existence of a hydrostatic pressure-temperature equivalence is highlighted. According to the complete set of results, the order of magnitude of the hydrostatic pressure effect is about 10 K.MPa -1 . Besides, in an analogous study on an pure polyurea [START_REF] Alkhader | A new shear-compression test for determining the pressure influence on the shear response of elastomers[END_REF], the hydrostatic pressure effect was found to be much smaller, around 0.24 K.MPa -1 , which is consistent with the aforementioned matrix pressure effect existing in pure rubbers. Thus, the present study highlights the fillers major role in the hydrostatic pressure effect.

MICRO-MECHANICAL APPROACH

Physico-mechanical model for filled rubbers

The next step is to understand, under a physical framework, the hydrostatic pressure effect which was experimentally brought to light, in order to integrate its contribution into a micro-mechanical model, described below.

The addition of fillers into a rubber matrix brings outstanding enhancement in the mechanical properties. This phenomenon is explained by the filler network in which some polymer regions are confined between neighbouring aggregates [Fig. 5]: far away from the filler, the local behaviour is rubbery, similar to pure matrix; by contrast, in confined polymer regions, a strong filler-matrix interaction constricting the polymer to a glassy-like state, i.e. polymer chain mobility reduction, has been demonstrated [START_REF] Berriot | Filler-elastomer interaction in model filled rubbers, a 1h nmr study[END_REF][START_REF] Papon | Glass-transition temperature gradient in nanocomposites: evidence from nuclear magnetic resonance and differential scanning calorimetry[END_REF]. As the glass transition is related with an increase of mobility, confined polymer regions, also called "glassy bridges", are characterized by a local glass transition temperature T local g . The glassy bridges play a major role in the local behaviour and lead to high dynamics heterogeneities in the filled rubber [START_REF] Montes | Influence of the glass transition temperature gradient on the nonlinear viscoelastic behavior in reinforced elastomers[END_REF], represented by a glass transition temperature distribution.

A local continuum constitutive equation was formulated. Two main features characterize this model. Firstly, the polymer chain mobility reduction due to the filler-matrix interaction is modelled by a glass transition temperature gradient [Fig. 5]. T local g (h) re-

T local g (h) = T bulk g 1 + 4δ h (5)
where T bulk g is the pure matrix glass transition temperature and δ is a characteristic length of fillermatrix interaction. Locally, the glass transition takes place at temperatures much higher than in the pure matrix, which reflects the confinement effect.

Secondly, deformation induces an enhancement of mobility of the amorphous macromolecules [START_REF] Loo | Chain mobility in the amorphous region of nylon 6 observed under active uniaxial deformation[END_REF]. Thus, the stress affects the chain conformations, modifying the polymer structure from glassy to rubbery [START_REF] Robertson | Theory for the plasticity of glassy polymers[END_REF]. The pressure dependence commonly follows the aforementioned matrix pressure effect deduced from the free-volume theory:

∂ T g ∂ p = α K (6)
Besides, shear stress is proven to affect the glassy polymer dynamic (Lee & Ediger 2010) but no universal formulation for the deviatoric component has been proposed yet. In polymers, the yield stress is known to obey the pressure modified Von Mises expression [START_REF] Rottler | Yield conditions for deformation of amorphous polymer glasses[END_REF] normalized by the material parameter K ∼ 1 MPa.K -1 [START_REF] Ward | Mechanical properties of solid polymers[END_REF]. Yielding is related to an increase of the molecular motion, as well as a decrease of pressure. Thus, in order to take into account volumetric and deviatoric stress, ( 6) is extended to a general stress expression described with respect to an unstretched state:

T local g (σ) = T 0 g - σ V M eq -α p K (7)
where T 0 g is the glass transition temperature limit without solicitation. T local g (σ) allows to reproduce a glassy-to-rubbery flow-like transition, i.e. a stress softening.

These two contributions on the local glass transition temperature, i.e. the bridge height ( 5) and the local stress (7), are considered as additive (Colombo & 𝑝 " al. 2015). Thus, a unique glass transition temperature function reflecting the local behaviour is written:

T local g (h, σ) = T bulk g 1 + 4δ h confinement - σ V M eq (h) -α p(h) K stress softening (8)

Pressure effect on local behaviour

For the following, it will be assumed that filled rubber reached the percolation threshold. In this case, the hydrostatic pressure is carried out by the filler network [Fig. 6]. Thus, the bridge height, called h(p H ) in the following, decreases with the hydrostatic pressure.

In order to take into account the hydrostatic pressure dependency, ( 8) is rewritten as:

T local g (h(p H ), σ) = T bulk g 1 + 4δ h(p H ) T a g (9) - σ V M eq (h(p H )) -α p(h(p H )) K T b g + α K p H T c g
Total hydrostatic pressure effect is thus given by three contributions. Their derivative versus hydrostatic pressure gives the amplitude of the hydrostatic pressure effect on the local glass transition temperature for each mechanism: The first two contributions are described in the following subsections. The major contribution will be identified.

∂ T local g ∂ p H = ∂ T a g ∂ p H - ∂ T b g ∂ p H + ∂ T c g ∂ p H (

Contribution on local confinement

A theoretical treatment allows to predict the pressure effect on local confinement. First, the bridge height is assumed to depend on the initial bridge height h 0 and the local strain:

h(p H ) = h 0 (1 -ε local zz ) = h 0 (1 - σ local zz K v ) (11)
where ε local zz and σ local zz are respectively the local strain and stress in the bridge direction.

When the hydrostatic pressure acts on the filler network, the force transmitted through the aggregate F z can be expressed at the aggregate scale according to the percolation hypothesis:

F z πR 2 ag p H (12)
and at the bridge scale:

F z πR 2 c σ local zz with R 2 c R p h 0 ( 13
)
where R ag , R p , R c are respectively the aggregate radius, the filler radius and the size of the aforementioned T g gradient.

Combining (9-T a g , 11, 12) and ( 13) yields:

T a g = T bulk g (1 + 4δ h 0 . 1 1 - R 2 ag K v R p h 0 p H ) (14)
Assuming that

R 2 ag K v R p h 0 p H 1, one obtains: ∂ T a g ∂ p H 4T bulk g δ R 2 ag K v R p h 2 0 ( 15
)
which is analytically expressed as a decreasing function of the square of the initial bridge height [Fig. 9].

Contribution on local stress

On the other side, the pressure effect on local stress is quantified by means of a Z-set analysis. This contribution is approximated to the rate of change of (9-T b g ) with respect to the non-pressurized state:

∂ T b g ∂ p H σ V M eq (h(p H )) -α p(h(p H )) K p H ( 16
)
It is computed in the confined region through a parametric study on the initial bridge height. It consists in simulating the closer approach of two aggregates due to hydrostatic pressure [Fig. 6] to quantify the stress disruption effect in the confined region. The numerical model is axi-symmetric: uniaxial compression simulations were performed by controlling the hydrostatic pressure acting on the aggregates. In a similar way as the initial numerical study, the matrix is modelled by a neo-Hookean formalism and the elastic modulus C 10 dependence is studied. Besides, the filler is assumed to be a stiffer elastic material (E = 70 GPa, ν = 0.17).

For an infinite bridge height h 0 → ∞, i.e. no confinement, a homogeneous uniaxial compression state is expected everywhere in the matrix:

σ = 0 0 0 0 0 0 0 0 -σ zz ⇒ p = σ zz 3 σ V M eq = σ zz (17)
In this limit case, the asymptotic behaviour is thus found to be: 9]. For bridges of interest, i.e. 0.75 to 8 nm, the triaxiality ratio [Fig. 7] never exceeds 1 α in confined regions, i.e. ρ < R p , further:

∂ T b g ∂ p H → 2.7 K.MPa -1 [Fig.
χ < 1 α ⇒ σ V M eq -α p > 0 (18)
which induces a negative contribution of the local stress on the total hydrostatic pressure effect (10). Furthermore, this quantity is found to be strictly independent of the elastic modulus, i.e. rubbery-like or glassy-like. The uniaxial stress field [Fig. 8] obtained for an initial glassy bridge h 0 = 2nm allows to clearly emphasize the local stress disruption in the confined region which is responsible for this pressure effect. By contrast, a homogeneous uniaxial compression state is observed if sufficiently far away from the bridge area, i.e. ρ > R p . Numerical results [Fig. 9] show a sharp increase of the pressure effect on the local stress for very tiny bridges (h 0 < 1.5 nm). For higher bridges, a slight decrease, consistent with the aforementioned asymptotic behaviour (17), is observed.

Discussion

From a micro-mechanics framework, hydrostatic pressure brings a contribution on local confinement, on local stress and on the matrix. All these contributions were quantified [Fig. 9]. It clearly appears that the local confinement effect, due to the closer approach of aggregates, dominates.

The total hydrostatic pressure effect (10) accounts for the local glass transition temperature shift expected for filled rubbers [Fig. 9]. A local shift of about 10 K.MPa -1 is predicted for intermediate bridge heights 3 nm ≤ h ≤ 4 nm, which is consistent with our experimental observations. This hydrostatic pressure effect is about two orders of magnitude larger than the one measured in pure rubbers.

From a physical framework, the hydrostatic pressure influences the initial filler network by decreasing the bridge heights (closer approach of aggregates). It leads to a significant shift of the local be- haviour T local g for shorter bridges. This mechanism drives the hydrostatic pressure-temperature superposition observed on the real sample. Finally, in order to describe quantitatively the macroscopic behaviour of the material, a scale-up has to be done, which is in progress.

CONCLUSIONS

In order to study the hydrostatic pressure effect on the thermo-mechanical behaviour of filled rubbers, an experimental setup was designed. Shear mechanical properties have been measured under superposed hydrostatic pressure and temperature. A hydrostatic pressure-temperature superposition is highlighted. The amplitude of this effect is about 10 K.MPa -1 , which is two orders of magnitude larger than the one observed in pure rubbers.

From a micro-mechanics framework, the local glass transition temperature shift involved by the hydrostatic pressure -through local confinement, local stress and the matrix-has been discussed. Our results proved that the local confinement effect, due to the closer approach of aggregates, is essentially responsible for this local shift. More importantly, this predicted shift is consistent with our experimental results, which explains the observed hydrostatic pressure-temperature superposition.

Figure 1 :

 1 Figure 1: Symmetry and boundary conditions.

Figure 2 :

 2 Figure 2: Macroscopic confinement effect on the stress state.

Figure 3 :

 3 Figure 3: Experimental setup used for shear-compression tests.

Figure 5 :

 5 Figure 5: Schematic view of dynamics heterogeneities in filled rubbers.

Figure 4 :

 4 Figure 4: Shear thermo-mechanical response: a) Pressure influence and b) mechanical superposition obtained for two sets (p H , T ).

∂

  p H is the contribution on local confinement caused by the bridge height decreasing due to the neighbouring aggregates closer approach, ∂ T b g ∂ p H is the contribution on the local stress caused by a stress disruption in the glassy bridge due to high confinement and ∂ T c g ∂ p H is the aforementioned matrix pressure effect.

Figure 6 :

 6 Figure 6: Geometrical parameters and simulated part.

Figure 7 :

 7 Figure 7: Triaxiality ratio in the confined regions (bridges).

Figure 8 :

 8 Figure 8: Unixial stress field σ zz for h 0 = 2 nm, R p = 5nm.

Figure 9 :

 9 Figure 9: Total hydrostatic pressure effect on the local glass transition temperature with respect to the initial bridge height.