Jerneja Penca

Mechthild Donner
email: mechthild.donner@inrae.fr

Hugo De Vries

Business models for sustainable food systems: a typology based on a literature review

Keywords: business models, food systems, sustainability, circular economy, transition, literature review . Introduction

The current state of the agri-food systems calls for more attention to sustainability. This article explores how diverse business models can contribute to sustainability in food systems, and develops a new business model typology in this domain in which the wider business ecosystem is taken into account. A systematic literature review is done and articles are analyzed according to the business model types, their various sustainability dimensions and roles within the food system. Nine di erent business model types for sustainable food systems are identified: circular business models, place-based social food networks, new logistics or online food distribution business models, disruptive, su ciency, inclusive, and family business models, the focal company, and regional food hubs. Collaborative approaches, clear sustainability visions, companies' values and continuous innovation are important factors for agri-food business models striving for sustainability. More in-depth single-country but also cross-country comparative and cross-disciplinary research is needed for understanding business models and their transitions toward sustainable outcomes within diverse and complex food systems.

bases to generate food security and nutrition for future generation is not compromised". Here, strong references are made to the United Nations Sustainability Development Goals (SDG), especially Zero Hunger (SDG 2), Reduced Inequalities (SDG 10), and Sustainable Production and Consumption (SDG 12). This definition is largely based on the initial definition provided by the Brundtland Committee in 1987. Since then, the literature on 'sustainability' has shown exponential growth curves as already signaled in 2011 [START_REF] Bettencourt | Evolution and structure of sustainability science[END_REF].

The [START_REF] Brundtland | Our common future: Report of the 1987 World Commission on Environment and Development[END_REF] definition states that "Sustainable development is development that meets the needs of the present without compromising the ability of future generations to meet their own needs". For food systems, this implies that the outcomes of food systems, namely sufficient and healthy food, should always and everywhere be available for all next generations, without exceeding the environmental carrying capacity of the planet [START_REF] Rockström | A safe operating space for humanity[END_REF]; the social dimension has been added by [START_REF] Raworth | A doughnut for the Anthropocene: humanity's compass in the 21st century[END_REF] resulting in the doughnut model with a social foundation and ecological ceiling. In music, one might explain sustainability by saying 'music should never die out'. In mathematics, one could describe this as curves-sinusoidal patterns in 2 dimensionsendlessly balancing between upper and lower limits [START_REF] De Vries | A new conceptual 'cylinder' framework for sustainable bioeconomy systems and their actors[END_REF]; hence for SFS, food patterns that evolve between all sustainability indicators, each with a lower and upper limit; consequently, social, ecological and economic foundations and ceilings. The latter may provide a suggestion to translate the SDG as objectives into sustainability indicators, however all with a lower and upper limit. Since each food system can be described in terms of a context (playing field), resources (pieces), resources handling (moves), constraints and incentives (rules), time (its evolution characteristics), (un)sustainable outcomes (win or lose), and its food actors (players), the interactions between actors play a fundamental role in the final outcomes of food systems [START_REF] De Vries | Sustainable food systems science based on physics principles[END_REF]. In this context, single business models evolve toward more integrated and collaborative business models depending on various actors and embedded within larger business ecosystems (Donner and de Vries, 2021). This review addresses the current state of the art in the literature on business models related to sustainable food systems, to provide options for firm managers to develop more sustainable business models in the future. Several literature reviews on sustainable business models in general have been published until now [START_REF] Bocken | A literature and practice review to develop sustainable business model archetypes[END_REF][START_REF] Geissdoerfer | Sustainable business model innovation: a review[END_REF][START_REF] Nosratabadi | Sustainable business models: a review[END_REF][START_REF] Preghenella | Exploring business models for sustainability: a bibliographic investigation of the literature and future research directions[END_REF]; however, only one review on sustainable business models in the agri-food sector was found [START_REF] Barth | Towards a conceptual framework of sustainable business model innovation in the agri-food sector: a systematic literature review[END_REF], which focuses on internal business model innovation and does not link business models to the larger food system.

The term "business model" has already been used largely in the 1990s by entrepreneurs proposing their internet business ideas to investors [START_REF] Magretta | Why business models matter[END_REF][START_REF] Zott | The business model: recent developments and future research[END_REF]. Later, it developed into a more analytical and conceptual tool for organizing, implementing and innovating enterprises [START_REF] Geissdoerfer | Sustainable business model innovation: a review[END_REF]. A business model explains how an enterprise operates [START_REF] Casadesus-Masanell | From strategy to business models and onto tactics[END_REF]. It can be used as a framework for describing all the activities of a firm from its value creation, value proposition, and value delivery to value capture. In their largely recognized Canvas Model, [START_REF] Osterwalder | Business model generation: a handbook for visionaries, game changers, and challengers[END_REF] differentiate between nine business model building blocks: (i) the key activities, partners and resources as the strategic elements, (ii) the customer segments, relationships and channels as the market elements, and (iii) the costs and revenue streams as the financial components. The business model concept can be applied in two ways, either in a static analytical approach, or in a transformational approach for innovation in the organization [START_REF] Demil | Business model evolution: in search of dynamic consistency[END_REF].

Sustainable business models aim at capturing "economic value while maintaining or regenerating natural, social, and economic capital beyond its organizational boundaries" [START_REF] Schaltegger | Business models for sustainability: Origins, present research, and future avenues[END_REF]. Therefore, in their triple-layered business model, [START_REF] Joyce | The triple layered business model canvas: a tool to design more sustainable business models[END_REF] extend the [START_REF] Osterwalder | Business model generation: a handbook for visionaries, game changers, and challengers[END_REF] Canvas model (based on economic value creation with its financial components costs and revenues) by two additional layers: an environmental layer and a social layer. This allows exploring the sustainability outcomes of a firm from a triple bottom line perspective, including economic, environmental and social dimensions. Herein, the benefits created should outweigh the negative impacts. The assessment is proposed to be done via multiple indicators. The environmental layer can be evaluated by means of lifecycle analysis, considering the full life cycle of a product or service from resource extraction, processing, distribution, and usage, to end of life [START_REF] Svoboda | Note on Life Cycle Analysis in Pollution Prevention[END_REF][START_REF] Sonesson | Environmental Assessment and Management in the Food Industry: Life Cycle Assessment and Related Approaches[END_REF]. It provides an evaluation of the environmental impacts via different types of indicators such as CO 2 emissions, energy consumption, resource depletion, water use, etc. The social layer refers to the benefits a company can create for its stakeholders and the larger society. Its assessment is thus based on a larger stakeholder perspective, seeking to balance the interests of different stakeholder groups including employees, shareholders, communities, customers, suppliers, governmental bodies, interest or other societal groups. With regard to a food company, this may imply offering healthy products, inclusion of farmers, transparency in governance and decision-making, well-being for employees, or community engagement [START_REF] Joyce | The triple layered business model canvas: a tool to design more sustainable business models[END_REF]. Several classifications of sustainable business models exist in the literature, but they often are more general in nature and not focusing on the agri-food system. For example, [START_REF] Bocken | A literature and practice review to develop sustainable business model archetypes[END_REF], reviewing different sectoral approaches in literature and practice, propose eight different archetypes of business models for sustainability, grouped into technological, social and organizational innovations. [START_REF] Ulvenblad | An overview of sustainable business models for innovation in Swedish agri-food production[END_REF] build on these archetypes to map sustainable business model innovations in the Swedish agri-food sector. [START_REF] Lüdeke-Freund | The sustainable business model pattern taxonomy-45 patterns to support sustainability-oriented business model innovation[END_REF], based on a Delphi expert survey, identified 45 different sustainable business model patterns belonging to eleven different groups along economic, social and ecological sustainability dimensions. Within the specific German wine sector, Dressler and Paunovi (2020) differentiate between the advanced sustainability model encompassing all three sustainability dimensions, a proactive sustainability model focusing on environmental but also including some social aspects, and a basic sustainability model with a rather narrow ecological focus.

The fact that current food systems are unsustainable requires among others rethinking the functioning of current business models. In addition, the vulnerability of current food systems to shocks like COVID-19, to crises like climate change and loss of biodiversity, imposes further reflections on private sector operations. In particular, attention is asked to (i) business models in which the wider business ecosystem is considered and (ii) an enlarged but reasonable diversity of business models in order to increase the overall resilience of food system actors to shocks and crises. Currently, business models have primarily been evolving in the context of linear value chains, hence relatively narrow ecosystems and individual business model types with standardized suppliers and customers. In case of a disruption of a single resource (e.g., wheat), a full value chain may stop functioning. Consequently, broader food systems approaches are advocated today [START_REF] Halberg | SCAR SWG Food systems Policy Brief: The added value of a Food Systems Approach in Research and Innovation[END_REF].

The aim of this article is to reveal which and understand how diverse business models and their business ecosystems, are linked to and can contribute to environmental, social, and economic sustainability in food systems from a practical point of view, and to develop a new business model typology, based on a systematic literature review. The guiding research questions are: What types of business models currently exist for the transition toward sustainable food systems? How do these businesses contribute to sustainability in a food system? How are the business models embedded in and influenced by the larger food system? What are the governance principles and enablers of these business models? What does this imply for extending the frontiers for future research in SFS at the edge of organizational, technological and social innovations?

.

Methods and materials

To explore how business models are linked to and can contribute to sustainable food systems, a systematic literature review was performed, considered as a rigorous method for analyzing the state of the art of a scientific topic [START_REF] Petticrew | Systematic Reviews in the Social Sciences: A Practical Guide[END_REF], and for identifying and critically evaluating research being relevant for a specific topic and research questions [START_REF] Snyder | Literature review as a research methodology: an overview and guidelines[END_REF]. Two leading academic databases were used for the literature search, Scopus and Web of Science [START_REF] Zhu | A tale of two databases: the use of Web of Science and Scopus in academic papers[END_REF]. Keywords were chosen for the data collection, including business model, food system, sustainable, and case. The last keyword was added as we were looking for empirical cases and evidence reported in the literature. No limitation on the date of publication was defined. The search was conducted in September 2022.

. . Selection process of documents

To describe and illustrate the stepwise process of the selection of articles for our study, the PRISMA flowchart was used as a reference (Figure 1).

In total, 123 records were found on Scopus and 110 records on Web of Science. After applying a filter and removing records other than journal articles (reports, book chapters, conference proceedings, etc.) and in languages other than English, 78 records remained on Scopus and 107 on Web of Science. As a next step, the documents of the two databases were copied and listed in an excel file, including authors, article title, abstract, source title, publication year, and DOI link. Duplicate documents were removed. The remaining number of articles was 115.

Then, all titles and abstracts of each article were read by both authors. The following exclusion criteria were agreed upon and applied: references that did not fit the topic and purpose of our study, articles that were mere literature review articles or theoretical in nature, articles that were not reporting empirical cases, articles without a clear reference to the food system nor to one or more of the sustainability dimensions. After this screening of titles and abstracts, 40 articles were retained.

The following step consisted of trying to retrieve the full articles of the 40 selected documents. The existing institutional access to articles was used, and authors were directly contacted when access was not possible. The three articles that were finally not available were ruled out.

After reading the full texts, the number of articles included in the study was 37. These final articles retained for analysis are listed in the Appendix.

. . Overview of the publications

To display the overall nature of the articles studied, a descriptive quantitative analysis of the sample was initially carried out and several graphs were created with Excel. First, the evolution in time of the published papers on business models for sustainable food systems was analyzed and illustrated in Figure 2. It shows the number of studies published from 2011 until September 2022, with an increase since 2018. Most of the articles appeared in 2021 (12 articles).

Another peak was in 2019, with seven published articles. The general development of the number of publications corresponds to a tendency toward increased research on the topic of sustainable business models in the past years [START_REF] Nosratabadi | Sustainable business models: a review[END_REF].

As shown in Figure 3, the articles were published in journals dedicated to various disciplines such as business, economics, environmental, food or nutrition science, but also in interdisciplinary journals. Most of the articles appeared in Sustainability and the British Food Journal (each four articles), followed by Business Strategy and the Environment and the Journal of Cleaner Production (each three articles).

. . Criteria for the qualitative content analysis

After the descriptive analysis, a qualitative content analysis of the 37 articles was carried out to answer the research questions. Qualitative content analysis is useful for conducting a literature review in a transparent and systematic way [START_REF] Seuring | Conducting content-analysis based literature reviews in supply chain management[END_REF]. The following criteria for the analysis of the articles were defined, according to the main themes to be explored and questions to be answered:

• Objectives and major research topics of the articles.

• Methodology used/unit(s) of analysis/country.

• Results:

-Business model type and elements studied.

-Sustainability dimensions and indicators.

-Role within the food system.

• Main findings.

• Implications for future research or practice.

A matrix table was used to display and analyze all articles, as shown with one example in Table 1. . Results

. . Objectives and major research topics

In general, the objectives of the studied documents are to explore, analyze and/or measure various individual or collective business strategies, models, governance structures, and innovations that contribute to either one or several dimensions of sustainability.

The topics of the articles can be grouped into four main categories.

The first category refers to sustainable business strategies and innovation (article reference numbers 4,5,28,29,31,33,34,37 of the Appendix). For example, [START_REF] Boccia | Innovation in the food distribution system[END_REF] presented two new and alternative distribution business models from Italy that stimulate more responsible, conscious and ethical consumption: the Ethical Purchasing Group, linking farmers directly to consumers, and EATALY, combining sales, culture and quality in one marketplace. Another example of business model innovation for sustainable consumption is given by [START_REF] Bocken | Sufficiency business strategies in the food industry-The case of Oatly[END_REF], presenting the Swedish company Oatly which developed a sufficiency strategy combing healthy food with environmental benefits by offering plant-based alternatives to dairy. The second category of topics deals with the measurement and evaluation of either one or several dimensions of sustainability (references 2, 3, 16, 18, 19, 29, 36). Here, [START_REF] Hamilton | Sustainable food lab learning systems for inclusive business models worldwide[END_REF] for example assessed some of the outcomes of the Sustainable Food Lab in the USA, a consortium of large-scale food businesses, and public and associative actors. These outcomes were tangible (e.g., positive environmental outcomes) as well as intangible (leadership and cooperation) in nature. Next, [START_REF] Kazancoglu | Performance evaluation of reverse logistics in food supply chains in a circular economy using system dynamics[END_REF] developed and applied a system dynamics model to a Turkish food supply chain, for evaluating the environmental performance of reverse logistics to minimize food loss and waste.

The third major category of topics is about local, alternative food supply and marketing, contributing to the inclusion of smallholders and to local sustainable development, especially in rural areas (7,8,9,13,17,25). For example, [START_REF] Mair | Critical thinking for sustainable development at the Creemore 100 mile store[END_REF] studied the 100 Mile Store in Creemore, Canada, as an alternative business model run by women only and selling organic, locally-sourced and fair-trade products, thus contributing to various sustainable development goals.

The fourth category covers numerous articles that are dealing with circular economy business strategies for food waste and agricultural by-product valorization (6,10,11,12,14,20,27,28,34). Among those articles, Del Vecchio et al. ( 2022) analyzed the case of Fiusis, an Italian company valorizing olive by-products for local energy supply and contributing to rural development, enabled by collaborations along the supply chain and multistakeholder implication.

. . Methodologies, units of analysis, and countries

As the aim was to get insights into empirical cases of business models for sustainable food systems, one of the keywords for the literature search was "case". Consequently, the methodologies principally used in the sample are different types of business case studies (e.g., single, multiple, comparative, longitudinal case studies), for a better understanding of the business model working mechanisms, or to test or apply analytical or assessment frameworks. The case study methodology in general allows generating relevant knowledge for managers (De Massis and Kotlar, 2014), and has been defined as "an empirical inquiry that investigates a contemporary phenomenon in depth and within its real-life context" [START_REF] Yin | Case Study Research: Design and Methods (applied social research methods)[END_REF]. In the literature studied, multiple data are collected for analyzing a case, including interviews with farmers, food company managers or various stakeholders, focus group discussions, surveys, secondary data collection, and observations in the field, often combined with literature reviews and reviews of practices.

The units of analysis concern either (groups of) farmers, cooperatives, food processing companies (start-ups, SMEs, large companies), retailers, online food delivery services, food chains, food networks, consortia, food markets, food-tourism linkages, or food-energy nexus. Hence, the main stages of the agri-food chain including production, processing and delivery models are covered: farming (5 cases), processing industry (14 cases), retail/logistics (9 cases), network/hub (4 cases), and diverse cases or food system stages (5 cases).

Next, the number of business cases reported in the articles per continent and country is presented (Figure 4). Most cases originated from Europe (23 cases in total). Here, Italy was the leading country (7 cases), followed by Spain (3 cases), and Sweden, the Netherlands, Finland, Poland and Belgium (each 2 cases). One case was respectively reported from Germany, Denmark, and Portugal. The second continent was Asia-Pacific (5 cases), including Australia, New Zealand, Vietnam, Philippines, and Turkey, followed by Canada (3 cases), and the USA, South America and North Africa / Maghreb (with 2 cases per each continent). Only 5 studies were dealing with cross-country cases.

. . Business model types and elements

Nine different types of business models (BM) for sustainable food systems could be identified via the analysis of the 37 articles, visualized in Figure 5.

-Circular BM that reduce and/or valorize food waste or agricultural by-products (references 6, 10, 11, 12, 14, 20, 27, 28, 34 of the Appendix), based on circular economy (recovering, recycling, upcycling, cascading) and bioeconomy principles (waste conversion from higher to lower value) [START_REF] Donner | Implementation of circular business models for olive oil waste and by-product valorization[END_REF]. These BM mainly deliver economic and environmental benefits. -Alternative, place-based and social food networks (references 4, 7, 13, 15, 17, 25, 36). This type of BM links (smallscale) producers to consumers via direct sales, promotes and valorizes local agri-food resources and know-how, and aims at farmers' inclusion and local sustainable (economic and social) development. -Disruptive BM (22,33,37) that are characterized as fundamentally new or different from existing BM by changing their way of acting toward increased sustainability. Some examples are community-supported food production ( 22), virtual food spaces (33), or food companies that adopt a stewardship way of leadership engaging with many stakeholders for health and wellbeing (37). -Alternative logistics or online food distribution BM (29,31,33). This type includes novel logistic concepts for food retail in cities (31), online food delivery services (29) or online food markets that emerged during the COVID pandemic (33), and evaluates their various sustainability impacts. -Regional food consortia, food labs, or food hubs (21,16,26) are BM that are characterized by cooperation between different private and public actors, and therefore can create synergies and joint strategies for increased sustainability in the food system. -Sufficiency BM (5) are also a new type that is meant to encourage sustainable consumption, for example via plantbased diets. -An inclusive BM (8) has been defined as a "commercially viable model that benefits low-income communities by including them in a company's value chain on the demand side as Regarding single business model elements or business strategies toward sustainability, the following observations are worth mentioning. Among the value propositions, origin-based, organic or plant-based food products, sometimes in combination with new ways of distribution or sustainable consumption were put forward. Alternative or reverse logistics were mentioned as an element for reducing the negative environmental impact and for waste prevention. Next, different types of partnerships, collaboration and shared governance seemed to be important for achieving sustainable BM including more equally distributed value. The business strategies concerned the motives, practices, success and risk factors of sustainable or circular BM, business ethics such as corporate social responsibility, the implementation of environmental management systems, or assessments of the sustainability impact of BM.

. . Sustainability dimensions and indicators

Another research question concerned the sustainability dimensions and indicators in the studied literature. Here, results show that some articles explore more in detail one dimension of sustainability (2,8,17,21). Other articles focus on two sustainability dimensions, and then often combine economic with environmental issues, e.g. for circular economy approaches (6,10,11,12,19,27,28,30); however, the circular economy sometimes is also considered as contributing to all sustainability dimensions (14, 20). A combination of economic and social perspectives is explicitly represented by local food approaches (3, 18). Next, the topic of food and nutritional security tends to be dealt with economic issues (1,4,5,26,29,33). Finally, most often, economic, environmental and social dimensions are integrally considered when talking about sustainability.

Although all articles are dealing with sustainable food systems, only some of them define it. For [START_REF] Hingley | Local and sustainable food supply: The role of European retail consumer co-operatives[END_REF], "the idea of sustainable food system is normatively oriented albeit methodically loose: there is hardly a prescription for various multi-stakeholder and multi-level measures appropriate for progress toward increased sustainability in terms of food system." [START_REF] Hubeau | Sustainability experiments in the agri-food system: Uncovering the factors of new governance and collaboration success[END_REF] state that "sustainability is a contested and evolving concept with uncertainty about values, interests and methodological approaches" and define it as "a highly innovative initiative to improve the sustainability state of the whole chain through new arrangements of collaborations".

With regard to the sustainability indicators, the following results were obtained.

Regarding the economic dimension, it is referred to in terms of the economic valorization of by-products, the adding value to waste, a profit and cost optimization based on green principles, or a win-win stakeholders' cooperation.

The environmental dimension shows multiple characteristics. From the production side, it includes e.g., agroecology practices, organic products, the quantities of inputs used in agriculture, the quantities of waste, spillage, or emissions, renewable energy use, and the quantity and quality of recirculated products. From a consumption side, it refers to sustainable consumption, i.e., the use of eco-friendly products, 'without exceeding the carrying capacity of the planet' , anti-consumption practices such as reduce, reuse, recycle, or a reduction of food travel. In more general terms, the environmental dimension also refers to landscape protection and valorization.

The social dimension is indicated as an equal distribution of value, providing healthy nutrition for all, inclusion of smallholders, local cooperation and producer-consumer relations, local direct sales, social responsibility, addressing societal needs and democratic ways. It also refers to the protection of local know-how and of cultural heritage.

The nutrition and health dimension encompasses food security, quality and healthy food products. Surprisingly, the economic dimension of health-related problems is not considered, although the costs for diseases due to obesity are substantially increasing.

. . The roles of business models within the food system

To understand how business models are linked and can contribute to sustainable food systems, their roles within the food system were analyzed.

In general, the need for a system-based approach for achieving food systems sustainability was highlighted (e.g., 19, 24). More in detail, five key aspects of the system were identified in the studied documents: (i) the role of the institutional context, (ii) the role of actors and stakeholders, (iii) the role of partnerships and collaborations, (iv) the importance of circular economy principles, (v) the importance of alternative localized food system approaches, including a comparison between local small-scale vs. global largescale food systems.

(i) The institutional context plays a role in the compliance with but also protection of PDO (Protected Designation of Origin) and organic labels (1); it is important for circular and disruptive sustainable BM via policies, subsidies, investments and research funding (12,20,22), as it enables, for example, green energy production via legislation and funding (10).

Another example are urban freight policies that influence food retail and logistics in cities ( 31). (ii) The role of actors and stakeholders: here, it was mentioned that production and consumption are the twin pillars for sustainable food systems (25), that the engagement of all supply chain actors and stakeholders is needed to promote sustainability at a system level (18,6) or to get bioeconomy firms accepted ( 12), and that different actors from the same ecosystem can achieve a collective outcome ( 27). Particularly, it was highlighted that farmers must act as researchers by acquiring knowledge to maintain a dynamic and resilient system (15), that food companies must adopt a responsible attitude toward their environment to achieve sustainability (2), that an increased engagement of consumers is needed for the transition to a sustainable and circular food system (5). (iii) Partnerships and collaborations of food and bioeconomy enterprises are diverse (11), partnerships between governments, universities and the private sector play a key role in promoting SDG (32), collaboration with research is important for developing biotechnologies ( 12), and relationships between suppliers and a company facilitate circular economy and waste management ( 30). (iv) The circular economy is presented as an alternative approach to the current linear agri-food system and as a harmonious process of material circulation ( 14), creating symbiotic networks ( 21). Food and agricultural waste management are herein very important (28), as it can combine agricultural with bioeconomy systems (34). (v) Alternative localized food systems are defined by the low number of intermediaries and the reduction of the spatial distance between producers and consumers (3). They can revitalize primary production in communities ( 7), include smallholders ( 8), and support a healthier world (9). Cooperation and relations are enabled through geographical proximity ( 13). The local food system is also considered as a social force for open food supply chain coordination ( 17), and traditional farmers' markets can contribute to a sustainable local food system (33). Small vs. large-scale food systems: local, alternative and ethical consumption models are often considered more sustainable than large-scale distribution, especially from a social and food quality point of view (4,7,8,9,13,15); however, for environmental outcomes, large scale food companies can enable greater positive impacts ( 16). Moreover, "globalization can foster sustainable food systems and promote collective ecological action through knowledge transfer and shared concern for local environments and communities" [START_REF] Häger | Transformation toward sustainability on a Costa Rican coffee farm: environmental, socioeconomic, and psychological perspectives[END_REF].

Table 2 presents in detail the new business model typology for sustainable food systems. It is based on the main results in terms of characteristics, sustainability dimensions and role in food systems of the nine business model types.

. Discussion

In this section, first, the main findings and implications are summarized and discussed, concerning new business models and governance principles (Section 4.1), the enabling factors for a sustainability transition from a business management and policy perspective (Section 4.2), the connection between business models and food system building blocks (Section 4.3), and future research options (Section 4.4).

. . New business models and governance principles

The most dominant types of business models that aim to contribute to sustainability in food systems, are first of all local food networks, as they can boost overall sustainable regional development (7, 13), bring producers and consumers geographically close together for finding new ways of producing, marketing and consuming food (9). Secondly, circular business models for food waste and by-product valorization have become increasingly important. They should be dynamic and integrate actors from different sectors and key stakeholders (12), for example, symbiotic networks or industrial ecology parks sharing knowledge, facilities and closing energy, water and material loops (21,27). Thirdly, disruptive business models such as focusing on alternative protein-source products, alternative distribution chains, sustainable production and community-oriented food can offer opportunities for combing economic value with social and environmental benefits as well as healthy food (5,22). While research for local food networks has been ongoing for several decades, although not necessarily under the term business models, the topics of circular economy and disruptive business models for food production, processing, distribution, and recycling are more recent. All these research streams are influenced by institutional agendas and food legislation, such as food origin protection, organic food, novel food, food certifications and labels, or circular economy.

Concerning governance principles, for local food networks, shared ownership for producers with processing and distribution facilities has been proposed as having a positive impact on local economies (3). Other propositions included decreasing power asymmetries between value chain partners via more inclusive business models (8), or retail cooperatives governed by the principle of social responsibility (17). However, [START_REF] Hamilton | Sustainable food lab learning systems for inclusive business models worldwide[END_REF] cautioned against mere bottom-up governance of food supply chains, as one should ask "to what degree this is practical, under what conditions, and to what degree is this notion utopian beyond the scale and scope of local markets?" Also, some other authors highlighted the importance of leadership ( 16) and of the focal company (32) for achieving sustainable food systems.

. . Enabling factors from a business management and policy perspective

Here, the first question is what farmers and food business managers can do to enable more sustainable business models. In general, collaborative approaches, clear sustainability visions, companies' values such as corporate social responsibility principles and continued innovation are critical factors for the transition of business models to sustainability, as highlighted by several authors (14,24,27,31,37). Paying attention to the circular economy, such as reverse logistics that can minimize food wastes and losses (19), combined with the engagement of stakeholders and cooperation within the value chain (6), and marketing strategies to increase consumer awareness for biobased or local products (11,23), are enabling factors. Especially start-ups that disrupt existing and create new institutions (norms, beliefs) are important for a circular economy transition (28). Furthermore, implementing environmental management systems within food companies helps reduce costs and the use of inputs (2). Food origin and quality labeling can correspond to new ways of responsible or ethical consumption, or contribute to ensuring food security, preventing illegal practices (4, 7). Digitalization strategies and new online tools (e.g., virtual food spaces) can support circular economy and sustainability processes (10, 33). For farmers, agroecological practices, bioenergy production and diversification of products, markets and activities can help increase resilience (15,34).

Business model type Characteristics Main sustainability dimensions

Role in the food system

Circular BM reducing or valorizing food waste or agricultural by-products (6,10,11,12,14,20,27,28,34) Value creation based on circular economy and bioeconomy principles.

Economic, environmental

Strong dependence on the (political-legal) business ecosystem and stakeholder engagement.

Alternative, place-based and social food networks (4,7,13,15,17,25,36) Link (small-scale) producers to consumers via direct sales. Promote and valorize agrifood resources and local know-how. Aim at farmers' inclusion and local sustainable development.

Economic, social Local food system with geographical and social proximity between different actors.

Disruptive BM (22,33,37) Fundamentally new or different from existing BM by changing their way of operating toward sustainability.

Can contribute to various dimensions of sustainability Aim to induce institutional changes of the system.

New logistics or online food distribution BM (29,31,33) Novel logistic concepts for food retail in cities, online food delivery services or online farmers markets emerged during the COVID pandemic.

Food security (social dimension)

Local food system with geographical proximity between different actors.

Regional food consortia, food labs, food hubs (21,16,26) Cooperation between different private and public actors, synergies and joint strategies for increased sustainability in the food system.

Can contribute to various dimensions of sustainability Regional clustering and networks as driving forces and support for the food system toward sustainability.

Sufficiency BM (5) Encourage sustainable consumption.

Healthy diets (social dimension), environmental dimension

Aim to enhance the transition to a sustainable food system by addressing consumers.

Inclusive BM (8) Benefit low-income communities by including them in a company's value chain.

Social, economic

Local agrifood system approach, based on territorial collective action.

Family BM (30) Values and emotions that normally influence family firms' business decisions might influence transformations toward sustainable production.

Environmental, social

Strong ties between the business and its stakeholders, especially suppliers and the community.

The focal company BM (32) Provide leadership in the value chain, can play a crucial role toward the different actors and therefore contribute to sustainable supply chain management.

Social, environmental Public-private-research partnerships with a clear strategy -thanks to a leadership role -are crucial for the firms' and industries' transition.

Another question is how policymakers can support farmers and food businesses toward sustainability. From their study among Spanish farmers and their value creation strategies, [START_REF] Alvarez | Valuecreating strategies in dairy farm entrepreneurship: a case study in northern Spain[END_REF] concluded that policymakers should promote education and training in entrepreneurial skills to foster value-creating strategies including environmental value. [START_REF] Croft | Does 'the local' provide a pathway to revitalizing primary production in regional communities? A case study of professional fishing on the NSW South coast[END_REF] and [START_REF] Mejía | Strategic supply chain planning for food hubs in central colombia: an approach for sustainable food supply and distribution[END_REF] proposed that regional development agencies should support the transition to new sustainable business models by providing (tax, legal, and financial) incentives. Similarly, several other authors (10,11,20,21) concluded that public incentives, policies and financial support were critical for the success of circular economy business models.

. . The connection between business models and food system building blocks

In Table 2, the nine different types of business models, with their characteristics, main targeted sustainability dimensions and their role in food systems are presented. As underlined in the introduction, each food system can be described in terms of a context (playing field), resources (pieces), resources handling (moves), constraints and incentives (rules), time (its evolution characteristics), (un)sustainable outcomes (win or lose), and its food actors (players), i.e., the seven building blocks of a game [START_REF] De Vries | Sustainable food systems science based on physics principles[END_REF]. Therefore, we can pose the question: Which of the seven building blocks is most directly related to a business model type and its main role? This is depicted in Figure 6. The alternative logistics business model type is directly targeting changes in distribution, hence in "moves". The sufficiency business model is primarily outcome-driven. The disruptive business model strives to overcome existing constraints. The circular business model aims at zero loss in resources. The family-owned, focal company and inclusive business models are strongly actor strategy focused. The regional hubs and place-based social networks strive for lasting embeddedness in their territories.

Since the seven building blocks together allow food systems to evolve "in time"-like playing games, all business models may have a first entry point as shown in Figure 6; however, all types are related to all seven blocks. For example, the circular business model has the ambition to recycle all resources, without any losses; its first entry point is "resources/pieces". But, for being able to reach zero loss, it may change its processes (moves), its sourcing locations (playing fields), face new regulations and incentives (rules), include other actors (players), change time schemes, and reach multiple outcomes.

FIGURE

The first entry point of business models for sustainable food systems, according to the seven building blocks of a game (own design).

.

. Future research options

The literature review provides a series of options for future research. Major ones are here listed:

-Further explore viable business sufficiency strategies, and joint strategies involving various stakeholders (5, 36); this allows understanding of how diverse internal and external interactions and strategies between actors impact the evolution of food systems. -Organizational and cultural factors for circular economy adoption require more investigation (6). As Figure 6 reveals, the main focus of the circular economy is currently on the most efficient resource usage for food and bio-based products, with the most appropriate advanced technologies like bioreactors, ultrasound, microwave, pulsed electrical fields or enzyme immobilization-assisted extractions [START_REF] Sharma | Sustainable processing of food waste for production of bio-based products for circular bioeconomy[END_REF][START_REF] Castro-Muñoz | A comprehensive review on current and emerging technologies toward the valorization of bio-based wastes and by products from foods[END_REF] -Cross-disciplinary research for understanding complex issues (36). One should note that the evolution of food systems is the sum of all activities by actors, with different business models, in different environments, handling diverse resources, and confronted with boundary conditions in time. Insights into the interaction between actors in food systems thus require knowledge of disciplines and the capacity of systems to learn, adapt and strive for sustainable outcomes. -Since our findings primarily reveal insights in agri-foodoriented sustainable business models, a future cross-greenblue economy review study would be interesting; this is foreseen in the future Partnership on Sustainable Food Systems (EC, 2022).

.

Conclusion

This article aimed to understand how diverse business models can contribute to sustainability in food systems and to develop a new business model typology in this domain taking into account the wider business ecosystem, based on a systematic literature review.

Insights from the review highlight that from a business management point of view, collaborative approaches, clear sustainability visions, companies' values such as corporate social responsibility principles and continued innovation are important factors for the transition of business models to sustainability. Furthermore, implementing circular economy principles, such as reverse logistics that can minimize food wastes and losses, together with stakeholders' engagement and cooperation within the value chain, and marketing or labeling strategies for increasing consumer awareness of biobased or local products are possible pathways to sustainability.

Results have also led to a development of a new business model typology (Figure 5, Table 2). Nine different business model types for sustainable food systems currently emerged from the literature: circular BM, alternative, place-based and social food networks, new logistics or online food distribution, disruptive BM, sufficiency BM, inclusive BM, family BM, the focal company BM, and regional food hubs. Actually, the three most representative types are alternative place-based and social food networks, striving for lasting embeddedness in their territories, circular business models targeting zero agri-food waste, as well as disruptive business models aiming to overcome existing constraints.

The nine BM can be positioned in a concept for food systems based on the structure of a game with seven building blocks (players, playing field, pieces, moves, rules, time, and outcomes). All business models have a unique entry point, i.e., the building block of a food system game to which it is directly related e.g. for circular BM the "pieces" (resources efficiency). However, all BM also target the other six building blocks (Figure 6); this becomes important when a business model evolves from one type to another as response to external changes. Even more, in a cluster of stakeholders, it may be the combination of BM that accelerate the transition toward sustainable food systems. This makes the food system game concept well suited for presenting the sustainable business model typology, even more since all are familiar with the concept of (playing a) game.

Several questions remain, passing frontiers for future research. Primarily, general questions targeting the diversity of business models are relevant; in case of shocks (e.g., COVID-19 or the war in Ukraine) or crisis (e.g., climate change), which diversity of business models in which business ecosystem context guarantee actors that they are jointly resilient to these? Next, more specific questions principally concern the organizational and cultural factors of business models in the agri-food and bioeconomy domain, and the joint strategies of different private and public actors toward sustainable food systems. Also then, a further exploration is needed of the linkages between different concepts such as the circular economy and corporate social responsibility.

Overall, more in-depth single country [including literature in national languages, as e.g., underlined in the SAPEA report (2020)] but also cross-country comparative business case studies and research passing disciplinary frontiers are to be reviewed and performed for a better understanding of the needed diversity in business models for confronting shocks and crises. New insights contribute to transitions toward sustainability within diverse, complex and inter-connected food systems.

FIGURE

  FIGURESelection process of documents (Source: adapted from Page et al.,).

FIGURE

  FIGURENumber of publications on business models for sustainable food systems from to September (own illustration).

FIGURE

  FIGURENumber of publications per journal (own illustration).

FIGURE

  FIGURENumber of cases studied per country (own illustration).

FIGURE

  FIGURENine types of business models for sustainable food systems (own illustration).

  clients and consumers, and/or on the supply side as producers, entrepreneurs or employees" (UNDP 2008). In the article by[START_REF] Daburon | Toward territorialised dairy inclusive businesses: insights from an Egyptian case study: making dairy businesses inclusive[END_REF], the Danone Egypt Ecosystem Project is aiming at dairy smallholders' inclusion.-Family BM (30). As not only economic but also non-economic factors such as values and emotions normally influence family firms' business decisions, they might be transformed toward sustainable production models. -The focal company BM (32), which provides leadership in the value chain, can play an exemplary role toward the different actors and therefore contribute to sustainable supply chain management.

  TABLE Extract of the matrix table including criteria used to display and analyse the articles.

	Implications for	future research	or practice			Policymakers should	consider the	heterogeneity of agrarian	diversification initiatives;	political measures aimed	at education and training	of entrepreneurial skills	essential for farmers.
	Main findings					Farmers' value-creating	strategies are diverse and	associated with	contextual factors.	Farmers "tend to align	with their entrepreneurial	capabilities and context,	enabling them to succeed	with any of the strategies	pursued."
			Role within the	food system		Strategies are influenced	by contextual factors	such as the region,	compliance with the	conditions for producing	with a quality label	(PDO, organic), and the	intensification level of	the farms' productive	system.	Labels benefit from	institutional protection	that allows them to	reduce the competitive	pressure.
	Results		Sustainability	dimensions and	indicators	4 different value-creation	strategies:	'Ecological'	(social-ecological	motivation),	'Single-product'	(mass-intensive	production, economic	motivation),	'Innovative' (innovation,	diversification),	'Traditional'	(established, quality	schemes as PDO).
			Business model	type and	elements	Farmers business	models.	Value proposition: offer	different products and	quality schemes to	consumers	Customers: direct/	indirect sales	Key resources/	capabilities: technical-	entrepreneurial	skills.
	Methodology/	unit of analysis/	country			49 farmers in Northern	Spain, interviews,	qualitative-quantitative	analysis.
	Research	objective and	topic			Analyse the various	value-creating and	diversification strategies	of dairy farms in the	North of Spain.
	Author/ Date					Alvarez et al. (2021)

  TABLE A detailed presentation of the new business model typology for sustainable food systems.

Frontiers in Sustainable Food Systems frontiersin.org

Funding

The project has partially received funding from the Horizon Europe research and innovation program under grant agreement no. 101059497 -FOODPathS.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

Conception of the study and methodology: MD. Data analysis, original draft preparation, and review and editing: MD and HV. Funding acquisition: HV. Both authors have read and agreed to the submitted version of the manuscript.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Appendix


Equivalent pin-forces or equivalent moments for the modelling of piezoelectric patches: a parametric study Taha Ajnada1,2 , Romain Corcolle 3 , Yves Bernard 1,2 and Laurent Daniel 1,2 Abstract-Many piezoelectric (PZ) actuation or sensing systems consist of PZ patches bonded on elastic beams or blades.

In order to optimise the design of such systems, Finite Element Analysis (FEA) can be used. However, this option is relatively time consuming and not necessarily appropriate to the first steps of the design process.

The replacement of FEA by simple analytical tools is desirable in the early design stages in order to explore the optimal configurations for the device (beam dimensions, patch position and properties).

Two main modelling approaches can be found in the literature, based on the Solid Mechanics beam theory. The first approach consists in replacing the PZ patch by two opposite forces positioned at the surface of the beam, each at one extremity of the PZ patch. The second approach consists in replacing the PZ patch by two opposite moments positioned at the neutral axis of the beam, each at one extremity of the PZ patch.

The object of this paper is to detail these options, and to evaluate their range of validity. For this purpose, a parametric study is conducted on a cantilever beam structure to compare the different approaches for standard dimensions and material properties.

The results of corresponding FEA simulations are taken as reference. It is shown that the validity of analytical models is restricted to a narrow range of material properties and dimensions. This range is chosen when the error between the normalised displacements obtained analytically and those obtained by numerical calculation does not exceed 6%. Within this range, the two-moment model is revealed a more precise choice than two pin-forces. As a consequence, its validity range is larger compared to other analytical approaches. This is due to the introduction of the flexural stiffness of the PZ patch and of a realistic strain profile across the section of the structure. These results can be used to obtain analytical expressions of stress and strains in PZ actuation and sensing devices. PZ materials are used in many actuation, sensing or transduction systems such as bistable MEMS ( [START_REF] Saif | On a tunable bistable MEMS -theory and experiment[END_REF]), MEMS micro-valves ( [START_REF] Chakraborty | MEMS microvalve for space applications[END_REF]), micropumps for drug delivery ( [START_REF] Qifeng | Study on a piezoelectric micropump for the controlled drug delivery system[END_REF], vibration energy harvesters ( [START_REF] Wang | Comparison of electromagnetic and piezoelectric vibration energy harvesters with different interface circuits[END_REF]), fuel injectors ( [START_REF] Yu | Macroscopic spray characteristics of kerosene and diesel based on two different piezoelectric and solenoid injectors[END_REF]), haptic devices ( [START_REF] Ghenna | Vector control of piezoelectric transducers and ultrasonic actuators[END_REF]) or textile sensors ( [START_REF] Nilsson | Poling and characterization of piezoelectric polymer fibers for use in textile sensors[END_REF]), and their advantages are many. First of all, response times are short. The materials can therefore be exploited at high frequencies (up to MHz ( [START_REF] Zhang | Piezoelectric 1-3 Composites for High Frequency Ultrasonic Transducer Applications[END_REF])). It is the property that is exploited, for example, in the case of fuel injection in automotive industry. It also means that a wide range of speeds can be exploited. The stress levels that the materials can withstand are high ( [START_REF] Bell | Expanding the application space for piezoelectric materials[END_REF]). This makes it possible, for example, to generate high forces or torques in complete and compact systems that incorporate PZ materials. Electromechanical conversion is intrinsic and it is simple to generate the electric field necessary for the conversion. This leads to compact and lightweight devices with simple structures ( [START_REF] Wiwattananon | Understanding piezoelectric composite-based actuators with nonlinear and 90 • domain walls effects[END_REF]). The above statement, combined with the exploitable speeds and stress levels, creates space for devices with high power-to-weight ratio. The use of PZ materials, e.g. in travelling wave motors, leads to actuators with low rotational speeds ( [START_REF] Zhang | Piezoelectric materials for high power, high temperature applications[END_REF]). This property eliminates the need to use gearboxes for high torque and low speed applications. The small displacements generated (under high voltage levels) lead to good controllability of the actuators and high resolution, reaching the nanoscale ( [START_REF] Shi | Investigation on a linear piezoelectric actuator based on stick-slip/scan excitation[END_REF]), with very fast response times, below 1ms. The electromechanical conversion is increasingly improved ( [START_REF] Thompson | Performance of tonpilz transducers with segmented piezoelectric stacks using materials with high electromechanical coupling coefficient[END_REF]). It takes place silently and with a negligible magnetic field (compared to conventional solutions). Operation at cryogenic temperatures is possible even if the performance of the actuators is lower at low temperatures. PZ materials can operate at high temperatures without failure. Their usage temperature range is though limited by their Curie point T C , about 160 • C -350 • C for ferroelectric ceramics ( [START_REF] Zhang | Piezoelectric materials for high temperature sensors[END_REF]). The lifetime of these materials is relatively high, with very low consumption, especially in static use. Finally, the use of PZ materials under vacuum or in clean rooms is permitted due to their composition ( [START_REF] Shchagin | Ceramic piezoelectric transformer in vacuum for acceleration of electrons and production of x-rays[END_REF]).

In many applications, the PZ material is a patch bonded on a structure. The working principle of the device then relies on the interaction between the PZ patch and the structure (actuation, sensing or both). In order to model the action of a PZ patch bonded on an elastic structure, several approaches are available. A variational formulation, combined to the coupled constitutive relations of PZ materials, can be used ( [START_REF] Corcolle | Modeling of a beam structure with piezoelectric materials: introduction to ssd techniques[END_REF], [START_REF] Wang | Thickness-variable composite beams for vibration energy harvesting[END_REF]). The problem cannot usually be solved analytically, and the implementation of a numerical model is required. With the objective of establishing analytical relationships, [START_REF] Crawley | Use of piezoelectric actuators as elements of intelligent structures[END_REF] modelled the action of symmetrical bonded patches on an elastic structure, by using two equivalent pin-force, tangent to the surface of the structure. Alternately, [START_REF] Piefort | Finite element modelling of piezoelectric structures[END_REF] or [20] modelled the action of a PZ patch by two opposite moments, applied at the location of the patch extremities on the elastic structure. There are also other models for predicting the system (substrate + actuator) response for thin and thick structures. We mention three of them in the following. First, the consistent plate model ( [START_REF] Crawley | Induced strain actuation of isotropic and anisotropic plates[END_REF]) that formulates the strain energy relations for a laminated plate, under the assumptions of thin and classical laminated plate theory. Second, the strain energy model ( [START_REF] Wang | Modelling of finite length spatially distributed induced strain actuators for laminate beams and plates[END_REF]) that models laminate beams and plates with attached or embedded finite length spatially-distributed induced strain actuators. Third, modes selective excitation ( [START_REF] Dimitriadis | Piezoelectric actuators for distributed vibration excitation of thin plates[END_REF]) that predicts the behaviour of two dimensional patches of PZ material bonded to the surface of an elastic distributed structure and used as vibration actuators. Other works give an explicit analytical solution of deformation, vibration and optimal shape control of laminated cantilever PZ composite plates. The explicit formulation concerns the shape as a consequence of the applied loads, among others the PZ action. The resulting shape is then emphasised, rather than the causes (loads and more precisely the PZ action) that gave rise to it ( [START_REF] Soheil Gohari | New explicit solution for static shape control of smart laminated cantilever piezo-compositehybrid plates/beams under thermo-electro-mechanical loads using piezoelectric actuators[END_REF], [START_REF] Soheil Gohari | Analytical solution of the electro-mechanical flexural coupling between piezoelectric actuators and flexible-spring boundary structure in smart composite plates[END_REF]).

The objective of this paper is to assess the range of validity of two main analytical approaches to model and give an explicit formulation of the effect of a PZ patch on an elastic beam. The first approach replaces the PZ patch by two equivalent opposite forces and the second by two equivalent opposite moments. A parametric evaluation is performed on a case study consisting of a PZ patch perfectly bonded to a cantilever beam structure. In an experiment, the actuator is usually bonded to the structure via an adhesive. The Physics of the electromechanical interaction between the piezoelectric element, the adhesive and the host structure is not yet been fully understood. Many fundamental problems related to its practical implementation have not been resolved. The proposed models need to be experimentally verified and improved ( [START_REF] Xu | A modified electro-mechanical impedance model of piezoelectric actuator-sensors for debonding detection of composite patches[END_REF], [START_REF] Bhalla | Electromechanical impedance modeling for adhesively bonded piezo-transducers[END_REF]). The experimental results show that increasing the adhesive thickness changes the electromechanical impedance and resonance frequency of the piezoelectric element, as well as the amplitude of the sensor signal ( [START_REF] Xinlin | Effect of adhesive on the performance of piezoelectric elements used to monitor structural health[END_REF]). For an actuator under the assumption of perfect bonding to the host structure, which is our study case, [START_REF] Crawley | Use of piezoelectric actuators as elements of intelligent structures[END_REF] showed that the shear stress between the actuator and the host beam was transferred mainly over an infinitesimal region at the actuator ends. This is consistent with the pin-forces and pin-moments models at the actuator ends that we have developed. The corresponding Finite Element Analysis (FEA) is performed using COMSOL Multiphysics to serve as a reference solution for the problem. Consistently with the studied analytical models, we consider in the FEA simulation that the system is without adhesive layer, i.e. the PZ actuator is perfectly bonded to the beam. In our study, the direct solver implemented in Comsol Multiphysics MUMPS (MUltifrontal Massively Parallel sparse direct Solver) was used. MUMPS has the particularity to support cluster computing, allowing to use more memory than is typically available on a single machine.

In a first part, the study case is presented. The two analytical approaches are then presented and followed by the FEA implementation. A parametric study is then performed for different material and geometrical parameters. This allows defining the range of validity of the analytical models by comparison to FEA results.

II STUDY CASE

The considered problem (figure 1) consists in an elastic beam (e.g. steel), with length l b and thickness t b , clamped on its left side. The longitudinal linear elastic behaviour of the beam (along direction x) is characterised by the Young modulus E b . A PZ patch (e.g. PZT), with length l p and thickness t p is bonded on the upper side of this cantilever beam. The PZ patch and the cantilever beam have the same width b. x 1 is the x-coordinate such that the patch is bonded to the beam on the zone [x 1 x 1 + l p ]. The Young modulus of the PZ patch along direction x is noted E p . The PZ patch is equipped with electrodes on its upper and lower sides. V 0 is the voltage applied between these electrodes, the polarisation of the PZ patch being upward as shown in figure 1. The transverse PZ coefficient of the PZ patch is denoted by d 31 .

Fig. 1: Case study of a PZ patch bonded to an elastic cantilever beam: geometric and material parameters.

The PZ patch is excited by the application of a voltage along its polarisation direction z (Figure 1). Under free boundary conditions, the patch strains by the PZ deformation Λ, proportional to the applied voltage V 0 . For a patch bounded on a blade, under plane stress assumption, the longitudinal strain ε p in the patch is given by the behavioural law:

ε p = σ p E p + Λ (1)
where Λ the stress-free PZ strain of the PZ patch along the x-direction:

Λ = d 31 V 0 t p (2)
The different models (analytical and FEA) for this study case problem will be compared using the vertical displacement δ at the free (right) end of the cantilever beam. The longitudinal strain in the beam is noted ε b . The assumption of plane stress is considered for the actuator and for the structure as well.

III EQUIVALENT PIN-FORCE MODEL( [29], [30])

The pin-force model ( [START_REF] Bailey | Distributed piezoelectric-polymer active vibration control of a cantilever beam[END_REF], [START_REF] Crawley | Detailed Models of Piezoceramic Actuation of Beams[END_REF]) describes the action of the PZ patch on the beam by two opposite forces with module F, placed at the extremities of the PZ patch (figure 2). Fig. 2: Simplification of the action of a PZ patch on an elastic structure using the pin-force model, case where the PZ stiffness is not considered.

Several variants of the pin-force model can be obtained by following different assumptions. A first pin-force model can be developed assuming that the strain remains uniform throughout the section of the structure (figure 3). This assumption can be used when the structure is made of symmetrically bonded patches on each side of the beam. Although this assumption is obviously not relevant to describe the study case of figure 1 subjected to bending, the corresponding result is given below because it serves as a basis for more general configurations. In the absence of any external longitudinal force, the pin-force F 0 under uniform strain assumption, is given by (see ( [START_REF] Bailey | Distributed piezoelectric-polymer active vibration control of a cantilever beam[END_REF], [START_REF] Crawley | Detailed Models of Piezoceramic Actuation of Beams[END_REF])):

F 0 = - bE b t b 1 + ψ Λ (3)
with

ψ = E b t b E p t p (4)
Such model cannot describe bending, so that the obtained deflection δ is zero whatever the applied voltage.

A more refined pin-force model can be developed by maintaining the strain constant across the section of the PZ patch but assuming a linear variation of the strain across the section of the elastic beam (figure 4). In that case, the application of a voltage V 0 will result in a deflection δ . Fig. 4: A pin-force representation assuming uniform strain along the cross section of the PZ patch and a linear variation of the strain along the cross section of the elastic beam.

The pin-force F 1 in that case is given by (see ( [START_REF] Bailey | Distributed piezoelectric-polymer active vibration control of a cantilever beam[END_REF], [START_REF] Crawley | Detailed Models of Piezoceramic Actuation of Beams[END_REF])):

F 1 = - bE b t b 3 + ψ Λ (5)
In the construction of this model F 1 , it should be mentioned that the stiffness added by the presence of the PZT patch is not taken into consideration.

A more precise pin-force model ( [START_REF] Chaudhry | The pin-force model revisited[END_REF]) can be obtained by adding the stiffness of the PZ patch to the stiffness of the structure, the resulting model is F 2 (figure 5). The pin-force model F 2 in that case is given by (see Appendix B):

F 2 = - bE b t b 3 + ψ + t 2 p 3t 2 b Λ (6)
As mentioned in ( [START_REF] Chaudhry | The pin-force model revisited[END_REF]), the F 2 pin-force model contains some contradiction in it from a theoretical standpoint. By assuming a constant strain distribution in the actuator, we are in fact specifying that the actuator does not bend. However, the assumption of constant strain in the actuator seems acceptable, but neglecting the actuator flexural stiffness is what really leads to the discrepancy in the pin-force model for thin structures.

IV EQUIVALENT TWO-MOMENT MODEL ([32])

The two-moment model in ( [START_REF] Gibbs | Excitation of thin beams using asymmetric piezoelectric actuators[END_REF]) describes the action of the PZ patch on the beam by two opposite moments with module M, placed at the extremities of the PZ patch (figure 6). By comparison to the previous two-force model, this model introduces a linear variation of the strain across the sections of both the elastic beam and the actuator. The extension and bending of the elastic beam is described by the strain field ε b . This strain field can be written as the sum of a uniform strain (extension part) and a linearly varying strain (bending part). This strain can be seen as the result of the application of the two opposite moments. The moment M in that case is given by (see ( [START_REF] Gibbs | Excitation of thin beams using asymmetric piezoelectric actuators[END_REF])): , where z n defines the neutral axis of the beam (z position for which the strain ε b is zero):

M bE p = -K f I p + K e -1 -z n K f T -z n (K e -
z n = E p t 2 p + t p t b 2 (E b t b + E p t p ) (8)
and where

K f = 2 t b + t p ( 1
- E b E p t 3 b t p + E 2 b t 4 b + E 2 p t 4 p + E b E p t b t 3 p E 2 p t 4 p + E 2 b t 4 b + E b E p 4t 3 b t p + 6t 2 b t 2 p + 4t b t 3 p ) K e = t 4 p + E b E p t 3 b t p t 4 p + E 2 b E 2 p t 4 b + E b E p 4t 3 b t p + 6t 2 b t 2 p + 4t b t 3
p are introduced to simplify the moment formula.

V EXPRESSION OF THE DEFLECTION IN THE STUDY

CASE

Equations ( 5) and ( 6) give the expressions for a pin-force model without considering the stiffness of the PZ patch (F = F 1 ) and with considering it (F = F 2 ), respectively. Equation ( 7) defines a two-moment model with the moments applied at the patch extremities. The modelling options for the modelling of a cantilever beam, taken as the reference study case, are illustrated in figure 7. Using the standard Euler-Bernoulli theory, the expression of the transverse displacement w(x) of the beam neutral axis (along z-direction) can be directly derived form the expressions of F 1 , F 2 or M. The deflection δ at the free (right) end of the cantilever beam is then simply given by:

δ = w(x = l b ) (9)
This value δ will be used in the following as the comparison indicator between the different modelling approaches.

According to Euler-Bernoulli theory, the transverse displacement w(x) must obey the following equation:

w ′′ = d 2 w dx 2 = - M f z (EI) (10)
M f z is the bending moment of the beam and (EI) is the bending stiffness of the beam. It is defined as:

(EI) = E b I b (11)
for the F 1 model, and as:

(EI) = E b I b + E p I p ( 12
)
for the F 2 and M models. The difference between these definitions comes from the fact that the bending stiffness of the PZ patch is not neglected in the two latter models. The definition of the bending moment M f z naturally leads to consider three distinct areas on which the transverse displacement is defined. The expression of the transverse displacement w(x) is given below according to the three considered approaches. It is noted w F1 (x), w F2 (x) and w M (x) for the F 1 , F 2 and M models, respectively. For 0 ≤ x < x 1 :

w F1 (x) = w F2 (x) = w M (x) = 0 (13)
For

x 1 ≤ x < x 2 :                w F1 (x) = - F 1 t b 4E b I b . (x -x 1 ) 2 w F2 (x) = - F 2 t b 4 (E b I b + E p I p ) . (x -x 1 ) 2 w M (x) = - M 2 (E b I b + E p I p ) . (x -x 1 ) 2 (14)
For

x 2 ≤ x ≤ l b :                                          w F1 (x) = - F 1 t b 2E b I b .(x 2 -x 1 ).x + F 1 t b 4E b I b .(x 2 2 -x 2 1 ) w F2 (x) = - F 2 t b 2 (E b I b + E p I p ) .(x 2 -x 1 ).x + F 2 t b 4 (E b I b + E p I p ) .(x 2 2 -x 2 1 ) w M (x) = - M E b I b + E p I p .(x 2 -x 1 ).x + M 2 (E b I b + E p I p ) .(x 2 2 -x 2 1 ) (15)
The deflection δ F1 , δ F2 and δ M for the F 1 , F 2 and M models, respectively, can then be easily obtained at the position x = l b :

                                         δ F1 = w F1 (l b ) = - F 1 t b 2E b I b .(x 2 -x 1 ).l b + F 1 t b 4E b I b .(x 2 2 -x 2 1 ) δ F2 = w F2 (l b ) = - F 2 t b 2 (E b I b + E p I p ) .(x 2 -x 1 ).l b + F 2 t b 4 (E b I b + E p I p ) .(x 2 2 -x 2 1 ) δ M = w M (l b ) = - M E b I b + E p I p .(x 2 -x 1 ).l b + M 2 (E b I b + E p I p ) .(x 2 2 -x 2 1 ) (16)
It can be noticed that all models give a similar form for the expression of the deflection δ so that equation ( 16) can be re-written as follows:

                 δ F1 =K. 1 3 + E. T δ F2 =K. 1 3 + E. T + 1 3 T 2 δ M =K. T (1 + T ) E. T 3 + 4 T 2 + 6 T + 4 + E (17)
where

K = 6d 31 V 0 t b t p l b l p 1 - l p 2l b - x 1 l b (18)
and T = t b t p and E = E b E p are dimensionless parameters characteristic of the structure geometry and material choice, respectively.

Comparing the three analytical models then reduces to comparing the dimensionless functions f F1 , f F2 , f M , defined by equation [START_REF] Piefort | Finite element modelling of piezoelectric structures[END_REF]:

                 f F1 ( T , E) = 1 3 + E. T f F2 ( T , E) = 1 3 + E. T + 1 3 T 2 f M ( T , E) = T (1 + T ) E. T 3 + 4 T 2 + 6 T + 4 + E (19)
It is then evident that, if the deflection δ in the cantilever beam problem is taken as the comparison indicator, the difference between the three approaches only depends on the dimensionless parameters T and E, namely the ratio of the thicknesses and the ratio of the Young modulus of the elastic beam and PZ patch. The functions f ( T , E) will be named "normalised deflections" in the following.

It can also be noticed that all analytical models provide a very similar equation for w(x), namely a horizontal line between 0 and x 1 , a parabolic function between x 1 and x 2 and again a straight line between x 2 and l b . Since the continuity of the derivative of w(x) is imposed, the equation w(x) is actually fully defined by δ = w(l b ) and the geometric parameters: For 0 ≤ x < x 1 :

w F1 (x) = w F2 (x) = w M (x) = w δ (x) = 0 (20) For x 1 ≤ x < x 2 : w F1 (x) = w F2 (x) = w M (x) = w δ (x) = - δ x 2 -x 1 . 1 x 2 + x 1 -2l b . (x -x 1 ) 2 (21) For x 2 ≤ x ≤ l b : w F1 (x) = w F2 (x) = w M (x) = w δ (x) = δ x 2 + x 1 -2l b . (x 2 + x 1 -2x) ( 22
)
For a given deflection, the deformation of the beam is therefore the same for all models.

The problem being purely elastic, the deformation of the beam provides the internal stress. Therefore, if the tip deflection is the same for two models, the deformed shape is also the same, as well as the internal stresses. This is the reason why the tip deflection of the beam has been chosen as a convenient comparison indicator.

VI PARAMETRIC STUDY

The normalised deflections f have been calculated for the three analytical models of interest for different thickness ratios T varying from 0.1 to 10, and for different Young modulus ratios E varying from 0.1 to 5. The results obtained with the three models for the normalised deflections f are presented in figure 8. For thickness ratios T ≤ 2, the responses are significantly different between the three models, with relative errors of more than 30% on average, regardless of Young modulus ratio E. For T ≥ 2, F 1 and F 2 models are close but remain different from the results of the two-moment model. Finally, on the area defined by T ≥ 4 and E ≥ 2, the three analytical approaches provide very similar results, with relative errors of less than 6%, so that they could be indifferently used in that range of parameters.

VII VALIDATION USING FEA

The analytical approaches state that the normalized displacement only depends on T and E, the thickness and Young modulus ratios of the beam and the PZ patch. This result is tested using a FEA simulation that will provide a reference solution. In agreement with the analytical models, the FEA study is implemented under the assumption of plane stress. Regarding the FEA simulations, the dimensions and material parameters have been set as follows. The PZ material under consideration is PZT so that the transverse PZ coefficient d 31 is 62.10 -12 m/V and the Young modulus E p is 81 GPa, which are typical values for PZT 1 . The range of exploration for the dimensionless parameter E has been set to {0.1 to 5}. The beam Young modulus is defined by E p . E and ranges from 8 GPa to 405 GPa. This wide range includes typical values for steel or aluminium. For the thickness of the PZ patch, it has been considered in the range {0.1 mm to 3.1 mm}, which corresponds to typical commercially available thicknesses for PZT patches. The range of exploration for the dimensionless parameter T has been set to {0.1 to 10}. This choice covers thicknesses for the beam from 0.01 mm to 30 mm. All other parameters have been given fixed values. The set of parameters used for the FEA simulations are summarised in table I.

For each FEA simulation, the normalised deflection f FEA is obtained by dividing the deflection δ by the parameter K defined by equation [START_REF] Crawley | Use of piezoelectric actuators as elements of intelligent structures[END_REF]. Then, for each value of t p , the normalised deflection is plotted against T and E. From figure 9, it is clear that the normalised deflection is not only a function of the dimensionless ratios T and E, as stated by the analytical approaches, but is also affected by the thickness of the patch. Figure 10 shows the relative error (defined as | f (FEA,0.1)f (FEA,3.1) |/ f (FEA,3.1) between the two extreme curves (t p = 0.1 mm and t p = 3.1 mm). This leads to define an error threshold, below which this assumption is considered valid. We propose, according to the error map and isovalues, that for a relative error below 20%, the normalised deflection will be considered as correctly defined by the parameter ratios. The comparison between the analytical models and the FEA model will only be investigated in this range of parameter variation. Outside this range, even in the case where the error between the analytical models and the FEA model is relatively small, nothing can be concluded. The statement of depending only on parameter ratios is not restricted to a specific area in the analytical case, unlike the FEA model. The reason behind this lies in the simplifying assumptions of the analytical model: bringing the action of the patch to a point of application, simplifying the behaviour law of the PZ material to only isotropic and x-axis behaviour, while the FEA model takes into account all components of the coupling and compliance matrices of the PZT.

The 2D FEA simulations are now compared to analytical results. The reference result f FEA is chosen to be the extreme two curves from figure 9. Relative error is then defined by the maximum of the error between the value of the analytical normalised displacement f anal : ( f F1 , f F2 or f M ) and the value of the FEA normalised displacement on one of the extreme curves f (FEA,3.1) or f (FEA,0.1) :

max | f anal -f (FEA,3.1) | f (FEA,3.1) , | f anal -f (FEA,0.1) | f (FEA,0.1)
. Error maps of the three analytical models ( f F1 , f F2 , f M ) with respect to that reference are presented in the following figures. It is reminded that the analytical model validity is investigated only in the region where the definition of the normalised deflection f as a function of only ratios T and E is considered valid. This range has been reported from figure 10 on all figures with dashed red isovalues. In order to allow a clearer comparison between the three analytical models, all colour maps have been saturated above error values of 10%. A value of 10% (yellow) designates an error equal or higher than 10%.

The inaccuracy of the F 1 model is very high as shown in figure 11. In the investigation zone, the error between the analytical F 1 model and FEA model is always higher than 10%. The F 2 model is slightly better as shown in figure 12. For T ≥ 1 and E ≥ 2 in the investigation zone, the relative error generally remains less than 6%. The M model, associated to less drastic assumptions, logically shows the best results (see figure 13). In the full range tested for the two dimensionless parameters T and E, in the investigation zone, the relative error does not exceed 3% in a large part of the investigation zone, and stays below 6% almost everywhere.

To summarise the analysis performed this paper, figure 14 shows the areas of validity of the analytical models F 1 , F 2 and M, based on the FEA results. Taking into account classical beam materials: steel and aluminium, we can highlight the following findings. F 1 model is not applicable for either the Steel-PZT or Aluminium-PZT combination. It cannot be used either for other types of materials based on the FEA result. F 2 model can be used for a Steel-PZT combination in a very narrow range of thickness ratios (around T = 2). By contrast, M model can be applied to both steel and aluminium cases. The only requirement is that the condition on the investigation zone, validated by FEA, is respected. This concerns ratios 0.1 ≤ T ≤ 2 for steel beam and 0.5 ≤ T ≤ 3 for aluminium beam.

VIII CONCLUSIONS

After the analysis carried out in this work, the following conclusions can be drawn: 1) Several approaches can be built depending on the assumptions made on the variation of the strain across the cross-section of the beam and the actuator. Three options were retained and named F 1 , F 2 and M-model.

2) The comparison between these models can be summarised to a function of dimensionless parameters: thicknesses and Young modulus ratios.

However, FEA results showed that the action of a PZT can be accurately modelled by such a function only in a given range of dimensionless parameters. 3) Three dimension plots and error maps of analytical and numeric models show shortcomings with respect to the relevance of the F 1 model, which is by construction very simple and is based on many simplifying assumptions. 4) Pin-force model with uniform strain distribution on only the active element can be used in the case where the flexural stiffness of the actuator is added. This concerns the case where the thickness of the beam is 1 to 2 times larger than the PZ, and for many types of beam materials. 5) Moment model, which models the action of a PZ patch by two moments applied at its ends, has been defined as the best model. Its are validated by comparing its prediction on deflection with the results obtained from the FEA study.

The key factors explaining the superiority of the M model is the introduction of the flexural stiffness of the PZ patch on one hand, and the introduction of a realistic variation of the strain across the cross-section of the structure, on the other hand. It is believed that the conclusions obtained in this work on a simple structure can be extrapolated to more complex structures on which PZ patches are glued. However, the study dealt with the case of static excitation of the actuator. The dynamic case where the actuator is supplied with frequency voltage is a prospect for future work.

  keywords: PZT, beam theory, PZ patch, cantilever, finite element modelling, equivalent models NOMENCLATURE E b Young modulus of beam material s E 11 Elastic compliance under constant electric field of PZ material along x-direction E p Young modulus of PZ material along x-direction E p = 1/s E 11 l b Lenght of the beam l p Lenght of the patch b Width of beam and patch (identical values) t b Thickness of the beam t p Thickness of the PZ patch δ Beam's free end displacement ε b Strain across the beam section ε p Strain across the PZ section I b Quadratic moment of the beam I b = bt 3 b /12 d 31 Transverse PZ constant of PZ material V 0 Applied voltage Λ Stress-free PZ strain of the PZ patch along the x-direction (Λ = d 31 V 0 /t p ) I INTRODUCTION

Fig. 3 :

 3 Fig. 3: A first pin-force representation assuming uniform strain along the cross section of the beam and PZ patch (ε b = ε p ).

Fig. 5 :

 5 Fig.5: A pin-force representation assuming uniform strain along the cross section of the PZ patch and a linear variation of the strain along the cross section of the elastic beam. Actuator flexural stiffness is added to the model.

Fig. 6 :

 6 Fig.6: A two-moment representation assuming a linear variation of the strain along both the cross section of the elastic beam and PZ patch. Actuator flexural stiffness is included in the model.

Fig. 7 :

 7 Fig. 7: Two modelling options for a cantilever beam actuated by a PZ patch: pin-force (left) and two-moment (right) models.

Fig. 8 :

 8 Fig. 8: Comparison of the pin-force (F 1 and F 2 ) and twomoment (M) models for the normalised deflection f in the cantilever beam reference problem.

Figure 9

 9 Figure 9 shows the ( T , E, f FEA ) plot for the two extreme values of t p . All intermediate calculations are positioned evenly between the two extreme curves. A third curve of

Fig. 9 :

 9 Fig. 9: 3D FEA results of the normalised deflection f FEA for different PZT thicknesses t p .

Fig. 10 :

 10 Fig. 10: Relative error between ( t b t p , E b E p , f FEA ) two extreme curves, isobase map. Light yellow denotes values greater than or equal to 50%. The two horizontal lines are corresponding to E of the steel and aluminium association with PZT.

Fig. 11 :

 11 Fig. 11: Relative error between the F 1 pin-force model and the FEA solution taken as reference. Range delimited by dotted red lines: all models assume the same statement (dependence only on t b t p and E b E p ). A mask is added outside the domain validated by FEA. The two horizontal lines are corresponding to E of the steel and aluminium association with PZT.

Fig. 12 :

 12 Fig. 12: Relative error between the F 2 pin-force model and the FEA solution taken as reference. Range delimited by dotted red lines: all models assume the same statement (dependence only on t b t p and E b E p ). A mask is added outside the domain validated by FEA. The two horizontal lines are corresponding to E of the steel and aluminium association with PZT.

Fig. 13 :

 13 Fig. 13: Relative error between the M pin-moment model and the FEA solution taken as reference. Range delimited by dotted red lines: all models assume the same statement (dependence only on t b t p and E b E p ). A mask is added outside the domain validated by FEA. The two horizontal lines are corresponding to E of the steel and aluminium association with PZT.

Fig. 14 :

 14 Fig. 14: Conclusion on the validity domains of the three analytical models, with respect to the FEA calculation.

TABLE I :

 I Dimensions and material parameters used for the FEA simulation of the cantilever beam case study.

Université Paris-Saclay, CentraleSupélec, CNRS, Laboratoire de Génie Electrique et Electronique de Paris, 91192, Gif-sur-Yvette, France

Sorbonne Université, CNRS, Laboratoire de Génie Electrique et Electronique de Paris, 75252, Paris, France

3 Division of Engineering and Computer Science, NYU Shanghai, 1555 Century Avenue, Shanghai, 200122, People's Republic of China

http://www.noliac.com/

ACKNOWLEDGEMENT

This work is supported by the Automotive Mechatronics Chair, a cooperation between Faurecia, CentraleSupélec and Esigelec.