Whelsy Boungou
email: w.boungou@psbedu.paris.

Charles Mawusi
email: charles.mawusi@u-bordeaux.fr.

The Impact of Economic Policy Uncertainty on Banks' Non-Interest Income Activities

Keywords: G21, G28, D81, E65 Economic Policy Uncertainty, Bank Diversification, Negative Interest Rate Environment

In this paper, we investigate how banks adjust their business model amid rising global economic uncertainty. Specifically, we analyze the effect of economic policy uncertainty (EPU) on banks' non-interest income activities, using a large panel dataset of 3,913 banks operating in 9 countries over the period 2009-2018. We find no statistically significant effect of EPU on banks' net non-interest income (NNII). Further analysis shows that the lack of effect on NNII is due to a reduction in gross non-interest income which was offset by a reduction in gross non-interest expenses. Our results also suggest that the likelihood of banks to diversify income amid high EPU is conditional on the negative interest rate environment.

Introduction

The economic implications of global uncertainty came into prominence after the 2007-2008 Global Financial Crisis (see, Baker and Bloom, 2013;[START_REF] Stock | Disentangling the Channels of the 2007-2009 Recession[END_REF]. Since the financial crisis, global uncertainty has increased markedly, spurred by events such as the European debt crisis, Brexit, and the trade war between the United States and China (see, Figure 1). This growing trend has significant implications for the growth of economies and the stability of financial systems. Besides, the ongoing novel coronavirus pandemic has reinforced research interest in global uncertainty. Indeed, several recent empirical studies have examined the financial and economic implications of economic policy uncertainty (EPU). Heightened EPU has been found to reduce bank income (Boungou and Mawusi, 2021a), increase bank risk (Wu et al., 2020a), hamper credit growth [START_REF] Nguyen | Economic policy uncertainty and credit growth: Evidence from a global sample[END_REF], decrease non-cash asset holdings [START_REF] Huang | Corporate financial asset holdings under economic policy uncertainty: Precautionary saving or speculating?[END_REF], lower firms' investment [START_REF] Liu | Economic policy uncertainty and firms' investment and financing decisions in China[END_REF], decrease financial stability [START_REF] Hsieh | Economic Policy Uncertainty and Illiquidity Return Premium[END_REF], and impede economic growth [START_REF] Ren | The effects of economic policy uncertainty on China's economy: evidence from time-varying parameter FAVAR[END_REF]. Much less explored is the relationship between EPU and banks' non-interest income activities (thereafter diversification) and in particular, the channels driving diversification. By extension, two main questions arise: (i) Do banks diversify in response to high economic policy uncertainty? (ii) Does the adoption of a negative interest rate policy (NIRP) influence the effect of EPU on bank diversification?

Theoretically, the effects of EPU on bank diversification are ambiguous. According to the real option value theory, uncertainty can increase the risk associated with bank investments while increasing the real option value of diversifying financial assets. Diversification thus presents an opportunity for banks to reduce exposure of their financial assets to uncertainty induced risks. This prevents asset value from falling below liabilities and reduces the prospects of bank failure (see, [START_REF] Yang | Bank diversification and systemic risk[END_REF]. Moreover, uncertainty could induce banks to allocate assets toward "high-yield" investments (Wu et al., 2020a). On the other hand, heightened economic policy uncertainty could increase the real option value of waiting, and in response, banks may decide against diversification and adopt a "wait and see" strategy as they are unsure of future economic conditions (see, [START_REF] Aastveit | Economic uncertainty and the influence of monetary policy[END_REF]. Moreover, excessive diversification could lead to severe systemic risk when uncertainty is high [START_REF] Yang | Bank diversification and systemic risk[END_REF]. 1This paper aims to assess empirically the effect of EPU on banks' diversification. To do so, we use data from 3,913 banks located in 9 countries affected by NIRP over the period 2009-2018. We measure EPU using the country level Economic Policy Uncertainty Indices developed by [START_REF] Arbatli | Policy uncertainty in Japan[END_REF], [START_REF] Armelius | The timing of uncertainty shocks in a small open economy[END_REF], [START_REF] Baker | Measuring economic policy uncertainty[END_REF], [START_REF] Ghirelli | A new economic policy uncertainty index for Spain[END_REF], [START_REF] Hardouvelis | Economic policy uncertainty, political uncertainty, and the Greek economic crisis[END_REF], [START_REF] Kroese | Beleidsonzekerheid in Nederland[END_REF] and [START_REF] Zalla | Economic policy uncertainty in Ireland[END_REF]. All the indices follow the novel news based approach developed by Baker et al. (2013). This approach requires an extensive search, count, and normalization of the words "Economic", "Policy", "Uncertainty" and its related variants as reported in major newspapers of each country. Although the EPU index is highly correlated with other general measures of economic uncertainty, including the Volatility Index (VIX), it is quite unique in that it captures economic, policy, and general uncertainties at the country level. Besides, the EPU is robust to political slant (see, [START_REF] Baker | Measuring economic policy uncertainty[END_REF]. Given that banks non-interest income can be affected by several factors, this measure of economic uncertainty allows us to capture a more holistic effect of uncertainty on banks non-interest income and non-interest expense. [START_REF] Baum | The impact of uncertainty on financial institutions: A cross-country study[END_REF] also point out that an increase in EPU leads to a reduction in the availability of credit in the private sector and thus deterioration in the performance of banks. In doing so, EPU could increase the attractiveness of non-interest income generating activities. In this paper we attempt to address this issue.

Bank diversification on the other hand, is measured using non-interest flows, which consists of net non-interest income and its decomposition. These measures allow us to capture how banks adjust their non-interest income and non-interest expense in a high EPU environment. Our paper contributes to the existing literature in three main ways. To the best of our knowledge, we provide the first empirical evidence of the effects of EPU on banks' diversification decisions. Second, we investigate whether the negative interest rate environment influences the effect of EPU on banks' diversification. Finally, our dataset enables us to examine precisely the response channels of banks to EPU by decomposing net non-interest income into gross non-interest income and gross noninterest expense.

Our results show that EPU does not induce an increase in the banks' net non-interest income. This absence of effect on net non-interest income stems from the counter adjustments to banks' non-interest expenses. Indeed, EPU reduces banks' non-interest income (such as net fees and commissions, and other non-interest income). However, this reduction is offset by the reduction in banks' non-interest expenses, notably personnel expenses and other non-interest expenses.

In addition to examining the impact of EPU on bank diversification, this paper also analyzes whether the negative interest rate environment changes the direction or magnitude of this impact. As the literature on NIRP suggests, negative interest rates impact banks' diversification [START_REF] Boungou | The channels of banks' response to negative interest rates[END_REF] and hence risk-taking decisions [START_REF] Boungou | Negative interest rates policy and banks' risk-taking: Empirical evidence[END_REF]. Our findings show that the negative interest rate environment influences the effect of EPU on banks. We find that in the pre-NIRP period, an increase in EPU implied an increase in bank net non-interest income. However, the fall of interest rates below zero helped to moderate the effects of EPU on banks. While the reduction in non-interest income was greater during the NIRP introduction period, banks' responses in terms of non-interest expense differed before and after the implementation of negative interest rates. Indeed, our findings highlight that in a high EPU environment banks first reduce their other non-interest expense (especially during the pre-NIRP period) and then reduced their personnel expense (post-NIRP period). These adjustments thus make it possible to compensate for the effects of EPU on non-interest income.

The remainder of the paper is organized as follows. Section 2 describes our data and empirical methodology. Section 3 presents our empirical findings. Section 4 concludes.

Data and empirical methodology

Sample and data description

In this paper, we analyze the effect of EPU on banks' income diversification decisions. We collect banking data from the Fitch Connect database and macroeconomic data from the OECD and Datastream. Our database consists of 3,913 banks located in 9 countries (i.e. France, Germany, Greece, Ireland, Italy, Japan, the Netherlands, Spain, and Sweden) over the period from 2009 to 2018. The overall data comprises more than 27,700 observations. We winsorized the data at the 1st and 99th percentile levels to ensure that outliers do not bias our results. Table A1 (in the Appendix) presents detailed descriptive statistics for our measures of bank diversification. In addition, Table A2 provides an aggregated time series of net non-interest income, gross noninterest income and expense to visualize the evolution of these variables over the sample period.

Our country-level measure of EPU comes from the novel indices developed by [START_REF] Arbatli | Policy uncertainty in Japan[END_REF], [START_REF] Armelius | The timing of uncertainty shocks in a small open economy[END_REF], [START_REF] Baker | Measuring economic policy uncertainty[END_REF], [START_REF] Ghirelli | A new economic policy uncertainty index for Spain[END_REF][START_REF] Hardouvelis | Economic policy uncertainty, political uncertainty, and the Greek economic crisis[END_REF], [START_REF] Kroese | Beleidsonzekerheid in Nederland[END_REF] and [START_REF] Zalla | Economic policy uncertainty in Ireland[END_REF].2 Since our banking data are annual frequency data, we calculate the annual average of the monthly EPU. 3 We measure banks' non-interest income activities using net non-interest income that we decompose into gross non-interest income and gross non-interest expense. 4 To explore the different channels through which banks adjust their balance sheets in response to increased economic policy uncertainty, we decompose gross non-interest incomes into net fees and commissions, and other non-interest income (such as net gains (losses) on trading and derivatives, net gains (losses) on other securities, and net insurance income). Secondly, we break down gross non-interest expenses into personnel expenses (such as salaries and employee benefits) and other non-interest expenses (such as information technology costs, marketing, and telecommunication services). All non-interest flows are scaled by total assets.

In our estimates, we include both bank-specific and country-specific controls. Regarding bank-specific controls, we include the natural logarithm of banks' total assets, liquid asset to total asset ratio, equity to asset ratio, and customer deposits to total asset. We include also Herfindahl-Hirschman index, inflation rate, real GDP growth rate, and the slope of the yield curve (the difference between long-term and short-term interest rates) as country-specific controls.

Empirical methodology

To analyze the effect of EPU on bank diversification, we employ the following model:

, , = + , + , , + + + Ɛ , , (1)
Where , , denotes net non-interest income (and its decomposition) of the bank i located in country k in year t.

, is the economic policy uncertainty index for country k in year t.5 , , refers to both bank-specific and country-specific controls.

, , Ɛ , , , denote the time fixed-effects, bank fixed-effects, and idiosyncratic error, respectively. To take into account heteroscedasticity and autocorrelation in our data, standard errors are robust and clustered at the bank-level. 6 To mitigate endogeneity problems, we lag the EPU variable and bank-specific controls by one year (see, Chi and Li, 2017, Wu et al., 2020b, Boungou and Mawusi, 2021a).7

Empirical findings

EPU and bank non-interest flows

Table 1 presents results of the relationship between EPU and banks' net non-interest income. Columns I and II of Table 1 report the results considering only bank-specific controls and country-specific controls respectively. Column III presents the results of our baseline model including both bank-and country-specific controls. The results in column III show a negative and non-significant relationship between EPU and banks' net non-interest income 8 . In other words, it seems that amid high economic policy uncertainty, banks prefer to adopt a "wait and see" strategy. 9 Indeed, we would be tempted to say that if there has been no increase or decrease in diversification decisions. But, is this the case?

Table 1. The relationship between EPU and bank diversification Based on our previous result, it then seems interesting to investigate the response channels of banks, justifying the absence of a significant effect of EPU on net noninterest income. In other words, we examine how banks adjust their business model to circumvent the potential undesirable effects of a high EPU environment. To do so, we go further by decomposing net non-interest income into gross non-interest income (see Table 2) and gross non-interest expense (Table 3). This decomposition allows us to better understand the effects of increased economic policy uncertainty on banks' ability to diversify their income.

Table 2 presents estimates between EPU and banks' non-interest income. We find a negative and statistically significant relationship between EPU and banks' gross noninterest income. Our results indicate that increasing EPU by 1 percentage point (pp) 8 To check that the lack of a significant effect does not stem from our choice of specification, we consider other specifications. The results of this additional analysis are reported in Table A3. The results qualitatively remained unchanged. 9 To check that the lack of effect of EPU on banks' net non-interest income is not related to an artifact of the timing choices in the model, we include two additional lags of the EPU index as robustness checks. The results of this robustness analysis support our baseline (not reported but available on request). leads to a reduction in gross non-interest income by 0.003pp. This reduction in gross non-interest income comes from a reduction in net fees and commissions by 0.001pp and other non-interest income by 0.002pp. This result highlights that in the presence of high EPU, banks struggle to diversify their non-income related activities, justifying a reduction of both net fees and commissions and other non-interest income. Our findings are consistent with the assertion that in the presence of increased uncertainty, banks prefer to wait and see how the situation progresses, instead of taking more risk [START_REF] Aastveit | Economic uncertainty and the influence of monetary policy[END_REF]. These results are also consistent with findings in [START_REF] Tran | Economic policy uncertainty and bank dividend policy[END_REF] which suggest that banks adopt more precautionary behaviors during periods of increased uncertainty.

(I) (II) (III) EPU_t-1 -2.
Table 2. The influence of EPU on banks' gross non-interest income (and its decomposition)

Now, we analyze how banks adjust their non-interest expense following the reduction of non-interest income amidst high EPU. Table 3 reports the results of the effects of EPU on banks' non-interest expense. We highlight a negative relationship between EPU and bank non-interest expense. We find that a 1pp increase in EPU induces a 0.004pp reduction in gross non-interest expense. This reduction in gross non-interest expense comes from the reduction in both personnel expense and other non-interest expenses. We note that the reduction in gross non-interest income (-0.003) is offset by a reduction in gross non-interest expense (-0.004), thus justifying the non-significant effect of EPU on banks' net non-interest income. Indeed, in the face of rising uncertainties, banks reduce their costs, especially salary charges and operating costs to limit the reduction in their non-interest income.

Overall, we find that the EPU has no significant impact on banks' net non-interest income. Indeed, it seems that in the context of high EPU, banks reduce their noninterest expenses to compensate for the reduction in non-interest income. Table 3. The influence of EPU on banks' gross non-interest expense (and its decomposition)

EPU and bank non-interest flows in time of negative interest rates

In this section, we compare the effects of EPU on bank diversification before and during the period of negative interest rates. In other words, we examine whether the negative interest rate environment influence the effects of EPU on banks. Indeed, negative interest rate policy (NIRP) could plunge banks into a period of uncertainty since this is the first time that central banks have introduced negative interest rates. This unprecedented monetary policy tool could thus influence banks' decisions. Indeed, several studies agree that NIRP has squeezed banks' margins (e.g. Boungou and Mawusi, 2021b). In response to this reduction in margins, banks had to adjust their business models, for instance by reducing their costs [START_REF] Boungou | The channels of banks' response to negative interest rates[END_REF]. Since our country sample is composed only of countries that have adopted NIRP, it is easier to distinguish the effects of EPU during the pre-NIRP (2009)(2010)(2011)(2012)(2013) and post-NIRP (2014-2018) periods.

Net non-interest income

Table 4 shows the results of the effects of EPU on bank net non-interest income before and during the period of negative interest rates. We find that the effects of EPU on net non-interest income are of opposite magnitude and direction. During the period before the implementation of negative interest rates, we find a positive and significant relationship between EPU and net non-interest income. An increase in EPU of 1pp increases the net non-interest income by 0.004pp. This result highlights that, during the period preceding the introduction of NIRP, and in the face of increased uncertainty, banks diversify their sources of income. This result is consistent with previous work (among others Wu et al., 2020a, b) which points out that the low-interest rate and high EPU environments were accompanied by an increase in risk-taking by banks. 10Conversely, when we focus on the NIRP implementation period, we find that the increase in EPU has no significant effect on banks' net non-interest income. This result undoubtedly underlines the moderating role of negative interest rate policy in banks' diversification and especially risk-taking decisions [START_REF] Boungou | Negative interest rates policy and banks' risk-taking: Empirical evidence[END_REF]. For instance, [START_REF] Boungou | Negative interest rates policy and banks' risk-taking: Empirical evidence[END_REF] shows that banks located in NIRP-adopter countries took less risk compared to those not affected by this policy. Now we complement our previous result by analyzing whether banks' response channels are the same before and during the period of negative interest rates. In other words, we investigate how NIRP influences the effects of EPU on banks' non-interest income (results reported in Table 5) and non-interest expense (Table 6).

In Table 5, we find a negative relationship between EPU and bank non-interest income before and during the period of negative interest rates. However, this relationship is only statistically significant in the period following the implementation of NIRP. The results in Table 5 suggest that the effects of EPU on banks' non-interest income (reported in Table 1 and discussed in section 3.1) stem mainly from the period of implementation of negative interest rates. An increase in EPU of 1pp during the implementation of NIRP reduced the gross non-interest income of banks by 0.004pp. This reduction is proportionally distributed between net fees and commissions (0.002pp) and other non-interest income (0.002pp). This result also supports the idea of the moderating role played by NIRP in the banks' search for yield.

Table 5. EPU and bank non-interest income in NIRP environment

The results presented in Table 6 shows how banks adjust their non-interest expense in a high EPU environment. We find that banks reduced their gross non-interest expense in response to the high EPU. We show that this reduction was smaller during the NIRP period (-0.004) compared to the pre-NIRP period (-0.006). However, our findings indicate that the reduction in gross non-interest expense (-0.004) was sufficient to offset the reduction in gross non-interest income (-0.004) during the same period.

Table 6. EPU and bank non-interest expense in NIRP environment

The results of the breakdown of gross non-interest expenses show that the reduction in personnel expenses related to high EPU was greater in the context of negative interest rates and, conversely, the reduction in other non-interest expenses was greater before NIRP. This seems to suggest that banks compensate for the reduction in non-interest income by reducing other non-interest expenses (in the period pre-NIRP) and by reducing staff costs (in the period post-NIRP). One explanation could be that faced with the combination of a high EPU and negative interest rates, banks reduce their staff costs with ambitions more oriented towards digital transformation (see, [START_REF] Boungou | The channels of banks' response to negative interest rates[END_REF].

Conclusion

In this paper, we investigate whether economic policy uncertainty (EPU) leads banks to adjust their non-interest income activities (i.e. diversification decisions), and if the adoption of a negative interest rate policy (NIRP) influences the effect of EPU on banks.

Using bank-level data comprising 3,913 banks in 9 countries over the period 2009-2018, we find that there exists a negative relationship between EPU and banks' non-interest income activities (proxy with gross non-interest income and expense).

Using an aggregate measure of bank diversification (proxied as net non-interest income), we find no significant effect of EPU. To document the absence of effect at the aggregate level, we decompose net non-interest income into gross non-interest income and gross non-interest expenses. In doing so, two results emerge: (i) a high EPU reduces the gross non-interest income of banks (such as net fees and commissions, and others non-interest income); (ii) banks reduce their gross non-interest expenses (i.e. personnel expenses and others non-interest expenses) sufficiently to compensate for the decrease in gross non-interest expenses. These results highlight the cautionary behavior of banks in times of high uncertainty.

Finally, we assess the role of NIRP on the dynamics between EPU and the banks' business model. We find that in a negative interest rate environment, the impact of EPU on banks' diversification decisions is lower, compared to the period before the introduction of NIRP. In addition, given the current health crisis affecting the global economy, a possible extension to our analysis could be to extend the coverage of our data beyond 2018 in order to analyze the behavior of banks in the face of rising uncertainties related to the Covid-19 crisis.

Overall, our results are important on at least two levels: (i) while the existing literature focuses on the overall impact of uncertainty on the real economy, the empirical results of this paper provide additional insights to better understand the role of financial intermediaries, in particular banks, in transmitting uncertainty shocks to the real economy; (ii) the cross-country analysis conducted in this paper contributes to the discussion on the increasing uncertainties and negative shocks more commonly observed in the global economy, especially after the Global Financial Crisis.

Compliance with Ethical Standards

No institutional or other funding has been provided for this article. As a result, there is no conflict of interest. Moreover, this article does not contain any studies with human participants or animals performed by any of the authors.

Figure 1 :

 1 Figure 1: The evolution of global uncertainty

Table 4 .

 4 EPU and bank net non-interest income in NIRP environment All dependent variables are scaled by total assets. EPU_t1 refers to Economic Policy Uncertainty index lagged by one year. All estimates include bank fixed-effects and year fixedeffects.The controls include both bank-specific controls (natural logarithm of total assets, liquid assets to total assets ratio, equity to assets ratio, customer deposits to total assets) and countryspecific controls (Herfindahl-Hirschman index, inflation rate, real GDP growth rate, yield curve). Robust standard errors are in brackets. ***, ** and * indicate statistical significance a the 1%, 5%, and 10% levels respectively.

		Gross non-interest	Personnel	Other non-interest
		expense	expense	expense
		(I)	(II)	(III)
	EPU_t-1	-0.004***	-9.42e-04***	-0.003***
		[1.22e-05]	[3.59e-06]	[9.35e-06]
	R2(within)	0.009	0.008	0.007
	Controls	Yes	Yes	Yes
	Observations	27009	27133	27648
	Number of banks	3789	3830	3874
	Note:

include both bank-specific controls (natural logarithm of total assets, liquid assets to total assets ratio, equity to assets ratio, customer deposits to total assets) and country-specific controls (Herfindahl-Hirschman index, inflation rate, real GDP growth rate, yield curve). Robust standard errors are in brackets. ***, ** and * indicate statistical significance a the 1%, 5%, and 10% levels respectively.

  : All dependent variables are scaled by total assets. EPU_t1 refers to Economic Policy Uncertainty index lagged by one year. All estimates include bank fixed-effects and year fixed-effects. The controls include both bank-specific controls (natural logarithm of total assets, liquid assets to total assets ratio, equity to assets ratio, customer deposits to total assets) and country-specific controls (Herfindahl-Hirschman index, inflation rate, real GDP growth rate, yield curve). While Pre-NIRP refers to the period before the implementation of negative interest rates, Post-NIRP refers to the periods after the implementation of negative interest rates. Robust standard errors are in brackets. ***, ** and * indicate statistical significance a the 1%, 5%, and 10% levels respectively. All dependent variables are scaled by total assets. EPU_t1 refers to Economic Policy Uncertainty index lagged by one year. All estimates include bank fixed-effects and year fixed-effects. The controls include both bank-specific controls (natural logarithm of total assets, liquid assets to total assets ratio, equity to assets ratio, customer deposits to total assets) and country-specific controls (Herfindahl-Hirschman index, inflation rate, real GDP growth rate, yield curve). While Pre-NIRP refers to the period before the implementation of negative interest rates, Post-NIRP refers to the periods after the implementation of negative interest rates. Robust standard errors are in brackets. ***, ** and * indicate statistical significance a the 1%, 5%, and 10% levels respectively.

		Gross non-interest income	Net fees & commissions	Other non-interest income
		Pre-NIRP	Post-NIRP	Pre-NIRP	Post-NIRP	Pre-NIRP	Post-NIRP
	EPU_t-1	-0.002	-0.004*	-2.53e-04	-0.002**	-0.002	-0.003*
		[1.81e-05]	[1.94e-05]	[6.58e-06]	[7.16e-06]	[1.45e-05]	[1.48e-05]
	R2(within)	0.051	0.008	0.054	0.006	0.032	0.009
	Controls	Yes	Yes	Yes	Yes	Yes	Yes
	Observations	13266	14159	13283	14170	13406	14352
	Number of banks	3665	3455	3665	3455	3713	3497
		Gross non-interest expense	Personnel expense	Other non-interest expense
		Pre-NIRP	Post-NIRP	Pre-NIRP	Post-NIRP	Pre-NIRP	Post-NIRP
	EPU_t-1	-0.006**	-0.004**	7.27e-04	-0.002***	-0.007***	-0.002
		[2.22e-05]	[2.20e-05]	[5.90e-06]	[5.51e-06]	[1.67e-05]	[1.79e-05]
	R2(within)	0.052	0.010	0.042	0.009	0.049	0.007
	Controls	Yes	Yes	Yes	Yes	Yes	Yes
	Observations	13033	13976	13117	14016	13328	14320
	Number of banks	3620	3415	3621	3418	3708	3490

Note

Note:

Table A2 .

 A2 The evolution of bank diversification over the sample period

				Gross non-
	2009	-0.201	0.031	0.052
	2010	-0.168	0.035	0.052
	2011	-0.01	0.055	0.064
	2012	-0.011	0.058	0.069
	2013	-0.008	0.062	0.07
	2014	-0.006	0.065	0.071
	2015	-0.006	0.064	0.07
	2016	-0.007	0.061	0.068
	2017	-0.005	0.059	0.064
	2018	-0.011	0.029	0.04

interest expense Gross non-interest income Net non-interest income

  Note: This table reports the mean values of net non-interest income, gross non-interest income and gross non-interest expense for each year between 2009 and 2018. All these variables are scaled by total assets.

Wu et al. (2020b) also argue that higher financial uncertainty is associated with lower bank risk-taking.

For more details on how the EPU is calculated, please visit this website: thttp://www.policyuncertainty.com

To test the robustness of this specification, we use the natural logarithm of the EPU index. Even with this additional specification, our baseline result remains unchanged (not reported but available on request).

Net non-interest income is the difference between gross non-interest income and gross non-interest expense.

Using the "World Economic Uncertainty Index" introduced by[START_REF] Ahir | The World Uncertainty Index[END_REF], we obtain similar results (available on request).

Since our sample of countries is relatively short and EPU is measured at the country level, we perform an alternative specification, namely by clustering standard errors at the country level, to demonstrate the robustness of our analysis. Our baseline result remains unchanged even with the clustering of standard errors at the country level. The results of this additional specification have not been reported, but are available on request.

We also use panel data techniques with the two-step system GMM estimators and find similar results (not reported but available on request).

This is consistent with the risk-taking channel of monetary policy in a low interest rate environment (see[START_REF] Borio | Capital regulation, risk-taking, and monetary policy: A missing link in the transmission mechanism?[END_REF].

Appendix


Mathieu Bilhere-Dieuzeide

Julien Chaves-Jacob
email: julien.chaves-jacob@univ-amu.fr

Emmanuel Buhon

Guillaume Biguet-Mermet

Jean-Marc Linares

Mathieu Bilhère-Dieuzeide

Stress-driven method bio-inspired by long bone structure for mechanical part mass reduction by removing geometry at macro and cell-unit scales

Keywords: Bio-inspiration, Bio-mimicry, Bone structure, Mass reduction, Stress-driven, Topology optimization

Introduction

Mechanical part mass reduction is a major issue in design and numerous methods have been developed to deal with this subject. In the last years, a rise of these methods was allowed by the spread of metallic additive manufacturing [START_REF] Thompson | Design for additive manufacturing: trends, opportunities, considerations, and constraints[END_REF]. Indeed, with additive manufacturing, the production costs and the part complexity become uncorrelated. This property allows for parts with complex internal structures to be manufactured. These geometric possibilities can be integrated in design methods that have a mass reduction objective.

The most known and used mass reduction method is the topology optimization (TO). Indeed, it is an industrially proven method developed during the 80s [START_REF] Ding | Shape optimization of structures : a literature survey[END_REF]. It has been implemented and upgraded by CAD software editors. In general, this method is used to satisfy a mass reduction goal while maximizing the mechanical part stiffness under a mechanical loading. In this method, the obtained geometry is largely influenced by the initial part design, especially in 2D simulations [START_REF] Allaire | Structural optimization using sensitivity analysis and a level-set method[END_REF], and by the mechanical loadings [START_REF] Xia | A level set solution to the stress based structural shape and topology optimization[END_REF]. In TO, if the loading localization is different from the nominal one used to initiate the optimization process, a drastic decrease of mechanical performances of the part is observed [START_REF] Audibert | Bio-inspired method based on bone architecture to optimize the structure of mechanical workpieces[END_REF]. Furthermore, the macro-scale topology proposed in this method is based on the design of large hollow or solid material areas organized as slender beams on the studied mechanical part [START_REF] Xia | Topology optimization of thermoelastic structures using level set method[END_REF] [START_REF] Dunning | Simultaneous optimization of structural topology and material grading using level set method[END_REF].

To address the TO behavior, a new type of structure emerged: the uniform lattice structures [START_REF] Dong | A survey of modelling of lattice structures fabricated by additive manufacturing[END_REF]. These structures are composed of cell unit repetitions. At the cell unit scale, the method also proposes hollow or solid material areas organized as slender beams. But at the macroscale, the volume fraction concept is provided in these uniform lattice structures. At this scale, instead of being located as in TO, the void is homogeneously spread over the studied part. This property can influence wide domains such as osteointegration in graft engineering [START_REF] Scerrato | Bio-inspired design of a porous resorbable scaffold for boné reconstruction: a preliminary study[END_REF] and mechanical engineering. In uniform lattice structures, two types of model structures were developed: the first ones are the strut-based structures and the second ones are the Triply Periodic Minimal Surface structures (TPMS). Both structures allow for a better void distribution at the macroscale. Moreover, the unit cell parameters such as their size or their thickness have a great influence on the part's mechanical performances [START_REF] Xiao | Evaluation of topology-optimized lattice structures manufactured via selective laser melting[END_REF] [START_REF] Al-Ketan | Topology-mechanical property relationship of 3D printed strut, skeletal, and sheet based periodic metallic cellular materials[END_REF]: smaller porosities provide higher mechanical performances. Strut-based structures are composed of a beam network arrangement [START_REF] Yu | Structure-property relationship in high strength and lightweight AlSi10Mg microlattices fabricated by selective laser melting[END_REF]. Many studies have shown the effectiveness of these structures that allow numerous options of design [START_REF] Panesar | Strategies for functionally graded lattice structures derived using topology optimization for additive manufacturing[END_REF] [START_REF] Seharing | A review on integration of lightweight gradient lattice structures in additive manufacturing parts[END_REF]. In particular, medical engineering works have demonstrated the influence of the beam orientations [START_REF] Maietta | A further analysis on Ti6Al4V lattice structures manufactured by selective laser melting[END_REF]. On the other hand, the strut-based structures are only composed of sharply jointed beams. These joints generate local tangential discontinuities that give rise to stress concentrations [START_REF] Wang | Optimization of graded filleted lattice structures subject to yield and buckling constraints[END_REF], which can contribute to local cracks. These tangential discontinuities have been reduced by the use of TPMS that present smoother shapes to limit stress concentrations [START_REF] Pan | Design and optimization of lattice structures: a review[END_REF] [18] [START_REF] Shi | A TPMS-based method for modeling porous scaffolds for bionic bone tissue engineering[END_REF].

However, uniform lattice structures are homogeneous structures that can involve a limit in mass reduction optimization. In these methods, the highest mechanical stress value is used to define a unique oversized volume fraction. In consequence, it limits the mass reduction. To go further in mass reduction optimization, gradient lattice structures are proposed. Gradients structures are mechanical stress-driven methods. The local volume fraction is adapted by varying the unit cell thickness according to mechanical stress [START_REF] Seharing | A review on integration of lightweight gradient lattice structures in additive manufacturing parts[END_REF]. High stress areas have a high volume fraction and low stress areas have low volume fraction [START_REF] Kang | Multi-lattice inner structures for high-strength and light-weight in metal selective laser melting process[END_REF]. With this variation, it is possible to reach an optimized mechanical strength-to-weight ratio. This gradient approach can be used with the strut -based structures [START_REF] Jin | Optimal design of three-dimensional non-uniform nylon lattice structures for selective laser sintering manufacturing[END_REF] [22] [START_REF] Kang | Multi-lattice inner structures for high-strength and light-weight in metal selective laser melting process[END_REF] or with the TPMS geometry [START_REF] Panesar | Strategies for functionally graded lattice structures derived using topology optimization for additive manufacturing[END_REF] [START_REF] Li | Design and optimization of graded cellular structures with triply periodic level surface-based topological shapes[END_REF].

To satisfy mass reduction objectives, bio-inspired methods were developed. Nature becomes a great source of inspiration in terms of optimization and structure adaptation. Bio-inspiration can take many forms in a mass reduction process. Internal bone structure was observed by Ghazlan et al. [START_REF] Ghazlan | Performance of a 3D printed cellular structure inspired by bone[END_REF]. In this work, the internal trabecular bone structure was mimicked, in order to design a hybrid concave/convex cell organized in thin-wall structure. The authors compared this hybrid structure with traditional thin-wall structures such as honeycomb or hexagonal re-entrant structures. They showed the effectiveness in compressive stiffness of their structure. In another work, a structure built out of unit cells made of beams was mimicked from the hexactinellid sponge [START_REF] Fernandes | Mechanically robust lattices inspired by deep-sea glass sponges[END_REF]. In this study, the proportion between the cell size and the thickness of the beams has been inspired by the one observed in the sea sponge. Bio-inspiration can also be combined with optimization process [START_REF] Du Plessis | Beautiful and Functional: A Review of Biomimetic Design in Additive Manufacturing[END_REF]. The method bio-inspired by the fish bone presented by Wang et al. [START_REF] Wang | Novel Bionic Design Method for Skeleton Structures Based on Load Path Analysis[END_REF] is an example of these mixed processes. This method mixed observations of fish bone, use of load path optimization process and use of size optimization process. Fillet radii were added to reduce tangential discontinuities and thus stress concentrations. They showed the effectiveness in structural stiffness of their structure. Nevertheless, as in uniform lattice methods, the cell unit size (or the pattern size) was selected manually by the user [START_REF] Li | Anisotropic design and optimization of conformal gradient lattice structures[END_REF]. In most bio-inspired methods, iterations were needed to determine the appropriate pattern size for a given application. A fully bio-inspired mass reduction process was developed by Audibert et al. [START_REF] Audibert | Bio-inspired method based on bone architecture to optimize the structure of mechanical workpieces[END_REF]. A mass reduction method was defined in this work. This method was applied on a plane and thereafter the resulting geometry was extruded along the orthogonal direction. In the rest of this paper, this type of geometries is referred as "2.5D" geometries. In the method, the parameters of patterns (shape and size) and porosities (size and orientation) were bio-inspired by trabecular bone structure. The authors showed the effectiveness of their method compare to TO method in terms of mass reduction and structural stiffness. This method shows a lower limit in terms of material volume fraction that is about 55% (thus the maximum mass reduction is about 45%).

In this paper, a new iterative bio-inspired method based on the bone ontogenesis and remodeling is proposed to improve the mass reduction in 2.5D. This method will be called "Bio-Inspired Mass Reduction Iterative Method" (BIMRIM). A first section will present the biological principles used in the method, more specifically the long bone structure and the medullar cavity development mechanism (Section 2). A second section will present the iterative mass reduction process (Section 3). In a third section, the evaluation of the effectiveness of this method will be shown (Section 4). The TO method was set as the standard for comparing methods. Two parts with the same mass were 3D-printed. An experimental test was carried out on a torsion disc. The torsional stiffness of both parts was compared.

Nature database for mass reduction: osteogenesis and bone remodeling principles mimicry

The design of light and stiff mechanical part is an issue that Nature has been facing since millions of years. Numerous stiff and light solutions are present in the database of Nature [START_REF] Ha | A review of recent research on bio-inspired structures and materials for energy absorption applications[END_REF] [START_REF] Naleway | Structural design elements in biological materials: application to bioinspiration[END_REF]. Among these solutions, endoskeletons are widely spread in Nature, especially in animals. Endoskeletons are composed of bones with complex internal structure. Whether it is the internal structure of bones or designed structures seen in Section 1, they are both too complex to be manufactured through conventional manufacturing. Additive manufacturing offers new possibilities for producing this kind of structures. These manufacturing processes allow to consider the design of complex structures mimicking the bone internal structure.

The method developed thereafter uses additive manufacturing possibilities while being bio-inspired by endoskeleton bones, and more precisely by long bone structure. Indeed, the objectives of the method are to reduce the mass of mechanical part while maintaining high mechanical performances, objectives fulfilled by bones. Moreover, bone is the outcome of millions of years of evolution that led to this structure in several species. In addition, as for most structures of Nature, bones use the minimum amount of energy and resources (food, water, …) to get the best properties possible. For these reasons, this structure is considered as optimized for the objectives it satisfies: being light and stiff.

The skeleton is composed of three types of bones: long bones, short bones and plate-like bones. Long bones are made to resist bending and buckling as defined in [START_REF] Frantzl | Nature's hierarchical materials[END_REF]. Exposure to these mechanical loadings is common in mechanical part under loading. For that reason, BIMRIM is bio-inspired from long bones.

For a human or animal adult, a long bone is composed of three main regions. The diaphysis region is the central part of the bone, followed by the metaphysis region, which makes the transition to the epiphysis region at the bone's ends (Fig. 1). The diaphysis region is composed of cortical bone that consists in a fully dense compact bone material that surrounds the medullar cavity [START_REF] Giorgio | In-depth gaze at the astonishing mechanical behavior of bone: a review for designing bio-inspired hierarchical materials[END_REF]. The epiphysis is made of trabecular bone, which can be described as a combination of thin wall bone material called trabecula, and void called porosities [START_REF] Giorgio | In-depth gaze at the astonishing mechanical behavior of bone: a review for designing bio-inspired hierarchical materials[END_REF].

Bone structure underwent a natural optimization process since bone material is used only in mechanically loaded areas [START_REF] Meyers | Biological materials: structure and mechanical properties[END_REF].

Wolff's law

The main concept used in bone bio-inspired method is the Wolff's law. It describes the ability of the bone structure to adapt its internal structure to the loadings it undergoes during its life [START_REF] Ruff | Who's afraid of the big bad Wolff ? : ''Wolff's law'' and bone functional adaptation[END_REF] [START_REF] Gosman | Pattern in ontogeny of human trabecular bone from SunWatch Village in the prehistoric Ohio Valley: general features of microarchituctural change (dissertation)[END_REF]. In trabecular bone, the local loadings generate specific local internal bone structure with specific bone material parameters [START_REF] Boyle | Three-dimensional micro-level computational study of Wolff's law via trabecular bone remodeling in the human proximal femur using design space topology optimization[END_REF]. The thickness of the local bone material (hence of the trabecula) is determined by the local stress field. Commonly, the trabecular orientation is defined by the orientation of local mechanical principal stresses. High mechanical stresses lead to thick trabecula and dense material [36] [37]. Similarly, low mechanical stresses lead to thin trabecula and porous material.

Medullar cavity appearance

The bone structure evolves during its whole life, mainly due to the subject's movements and locomotion habits. Locomotion is the major factor in the osteogenesis as explained in [START_REF] Ryan | Trabecular bone ontogeny in the human proximal femur[END_REF]. Local mechanical loadings on the long bones are generated by this motion. In turn, these loadings generate mechanical stresses in bone material. Bone-turnover mechanism adapts the bone structure to the loadings. Indeed, bone has a biological feedback system, involving specific cells, that regulates its local microarchitecture, thus its structure [START_REF] Giorgio | In-depth gaze at the astonishing mechanical behavior of bone: a review for designing bio-inspired hierarchical materials[END_REF]. This bone-turnover, that is an iterative and multi-physical process [START_REF] Lekszycki | Functional adaptation of bone as an optimal control problem[END_REF], is composed of growth, modeling, and remodeling of the bone. For human beings, this turnover takes place at least every four months [START_REF] Vuori | Chapter 19 : Sport and Bone[END_REF] (every four months for cortical bone and six months and a half for the trabecular bone [START_REF] Scerrato | Bio-inspired design of a porous resorbable scaffold for boné reconstruction: a preliminary study[END_REF]). In the remodeling step [START_REF] Giorgio | In-depth gaze at the astonishing mechanical behavior of bone: a review for designing bio-inspired hierarchical materials[END_REF], the osteocytes iteratively receive the signal generated from mechanical loading changes and trigger the osteoblasts and the osteocytes to form and resorb bone material [START_REF] Lekszycki | Functional adaptation of bone as an optimal control problem[END_REF]. Bone material is created to face higher mechanical stresses (formation mechanism). Concurrently, the resorption mechanism removes bone material if the mechanical stresses decrease or if the bone material is too aged [START_REF] Huiskes | Effects of mechanical forces on maintenance and adaptation of form in trabecular bone[END_REF].

The medullar cavity appearance process is widely spread on long bones of many species. Fig. 2 illustrates this mechanism for human beings. Bone development process starts in the early embryo life of the fetus. During the first year (Fig. 2a), the baby locomotion is limited and generates random mechanical loading on the long bones involved in locomotion. The bone diaphysis region can be described as a homogeneous porous non-oriented bone. At this age, the diaphysis is holed in its center. It is the beginning of the medullar cavity growth. The epiphysis regions are made of cartilage material. After the first year of life, the bipedal posture becomes the main way of locomotion for humans (Fig. 2b). This new locomotion mode orients the mechanical loadings in the bones. As a result, it orients the bone material structure [START_REF] Ryan | Trabecular bone ontogeny in the human proximal femur[END_REF]. With this motion change, the diaphysis region undergoes two main modifications: the medullar cavity grows and the remaining bone becomes cortical bone. The epiphysis region composed of cartilage is also transformed into trabecular bone. The proposed mass reduction method is bio-inspired by long bones from endoskeletons of terrestrial mammals. In the first stage, the mass reduction method, bio-inspired by trabecular bone structure, is investigated. In the second stage, the medullar cavity is studied and integrated to BIMRIM in order to improve mass reduction.

Determination of material cohesion domain

This subsection investigates the material cohesion threshold of Audibert's method [START_REF] Audibert | Bio-inspired method based on bone architecture to optimize the structure of mechanical workpieces[END_REF].

Method bio-inspired from the trabecular bone structure

Through bone turnover, the trabecular bone is able to form and resorb bone material depending on the mechanical stresses. The following method will focus on the resorption mechanism (material removing).

In the bone, the 3D geometry has an anisotropic mechanical behavior. It can be represented in mechanics by an ellipsoidal fabric tensor [START_REF] Harrigan | Characterization of Microstructural Anisotropy in Orthotropic Materials using a Second Rank Tensor[END_REF] [43] oriented along the principal mechanical stresses. Trabecular bone contains thin bone material walls separated by porosities. The thickness of these walls is named the trabecular thickness, Tbth [START_REF] Parfitt | Bone Histomorphometry: Standardization of Nomenclature, Symbols and Units[END_REF]. The distance between two trabecula is named the trabecular separation, Tbsp [START_REF] Parfitt | Bone Histomorphometry: Standardization of Nomenclature, Symbols and Units[END_REF] (Fig. 3a). A mathematical link has been found between the femoral head diameter and the values of Tbth and Tbsp in multiple species [START_REF] Doube | Trabecular bone scales allometrically in mammals and birds[END_REF]. Mimicking this mathematical link, Audibert et al. [START_REF] Audibert | Bio-inspired method based on bone architecture to optimize the structure of mechanical workpieces[END_REF] proposed to determine a parameter "pattern size p" using the characteristic length of the studied part. This pattern size p is set to discretize the part in local areas (Fig. 3b). In order to preserve the anisotropic mechanical behavior of the bone, it is decided that in the mass reduction method, the mass reduction holes will be extruded ellipses also aligned along the principal stresses. Finally, the remaining material of the part is oriented along the principal stresses. For this aim, the pattern used in the method consists in independent elliptic holes regularly arranged in a Face Centered Cubic (FCC) element of side "pattern size" p. As shown in Fig. 4, one ellipse is located at the center node of the element and one ellipse is located on each corner node of the element. These ellipses are characterized by their orientation and two main parameters: the semi-major and the semi-minor axes of the ellipse respectively noted a, and b (Fig. 4). For the next steps, the mechanical local stress distribution, in the initial part under the nominal loading, has to be known. Usually, Finite Element Analysis (FEA) is used to determine this stress distribution. This distribution will be employed to calculate the orientation and the size of the elliptic holes within the pattern. At each point of the pattern, each independent ellipse will be oriented along the major local principal stress axis. To determine the size of each ellipse (a, and b), two steps must be carried out:

• Audibert et al. highlight a link between the local bone volume fraction vf and local Von Mises stress σVM divided by its yield strength Re, resulting in nondimensionalization. This relation is given by Eq. 1 and illustrated in Fig. 5a. In this relation, it can be noticed that as soon as the ratio σVM/Re is higher than 0.15, a volume fraction vf equal to 1 is observed. At this value, the bone material is cortical bone.

• The ratio of the semi-major and semi-minor axes of the ellipses, FA, is determined using a bioinspired link. Indeed, [START_REF] Harrigan | Characterization of Microstructural Anisotropy in Orthotropic Materials using a Second Rank Tensor[END_REF] [START_REF] Musy | Not only stiffness, but also yield strength of the trabecular structure determined by non-linear μFE is best predicted by bone volume fraction and fabric tensor[END_REF] have determined that in trabecular bone, a relation exists between the anisotropy degree of the fabric tensor and the ratio of principal stresses. Thus, in the method, FA is a function of the ratio between the major principal stress σ1 and the minor principal stress σ3. This law is expressed in Eq. 2 [START_REF] Audibert | Bio-inspired method based on bone architecture to optimize the structure of mechanical workpieces[END_REF] and based on [START_REF] Tsubota | Computer simulation of trabecular remodeling in human proximal femur using large-scale voxel FE models: approach to understanding Wolff's law[END_REF] (Fig. 5b). (1)

= = 1.6 -0.6 exp -1 6.45 ! " # !$ (2)
Finally, knowing the pattern size p, the volume fraction vf and the ratio FA, the dimensions of the semi axes of the ellipse a and b can be determined.

Material cohesion-loss identification

However, this method entails a mass reduction limit due to the 2.5D approach and to the pattern shape. Indeed, if the ellipses are too large, two nearby ellipses will intersect. The resulting hole might alter dramatically the local mechanical performances of the structure. If this phenomenon occurs in a large area of the mechanical part, material cohesion is compromised. Fig. 5c illustrates the relation between this matter cohesion/cohesion-loss inside a 2.5D part and the couple {vf ; FA}. The calculations have been done in the "worst case", where the ellipses are oriented in the diagonal axis of the pattern. Indeed, the ellipses intersect the fastest in this configuration for a given couple {vf ; FA}. It ensures that for a given couple, the pattern remains conservative: material cohesion is maintained if the ellipses of the pattern have different orientations. In Fig. 5c, the abscissa axis represents the local volume fraction, vf, that goes from 0 (no material, only void) to 1 (no void, only material) in the local area. The figure ordinate axis represents the ratio FA of ellipses that goes from 1 (a=b) to the limit of 1.6 (a=1.6*b). In Fig. 5c, the red domain represents all the couples {vf ; FA} that cause material cohesion-loss and the green domain represents all the couples {vf ; FA} that maintain material cohesion. The equation of the frontier between the two domains will be presented in Section 3.2. On Fig. 5c, to ensure the material cohesion (green domain) at any point of the part (for any {vf ; FA} couple), the local volume fraction vf has to be higher than 0.55. For this reason, Audibert et al. define a minimal threshold for the local volume fraction set to vf=0.55. Whereas the Audibert's method has to avoid material cohesion-loss, the BIMRIM will use thresholds in order to identify the part areas where material cohesion-loss may appear. These areas correspond to barely loaded zones that Nature would probably have removed. Section 3.2 will present the iterative process of mass reduction using a variation of this threshold.

To summarize, these two last sub-sections presented a mass reduction method based on trabecular bone structure. In this method, a material cohesion-loss phenomenon appeared. The frontier between the material cohesion domain and the domain where this property is lost will be studied in the next sub-section.

Material cohesion domain definition

To define the volume fraction thresholds, the frontier seen in Fig. 5c from material cohesion domain (green domain) to the other (red domain) will be first studied. In the method presented in Section 3.1.1, the ellipses of the pattern are independent and their sizes and orientations depend on the local mechanical stress field in the considered pattern point. This independency brings an infinite number of possibilities in terms of mass reduction pattern. To be conservative, the equation of the frontier (Fig. 5c) will be calculated with equal ellipses aligned in the diagonal axis of the FCC element (Fig. 3). In this case, the material volume fraction is defined in Eq. 3 with e the distance between two ellipses (Eq. 4).

% & = % & -2π (3) = √2 2 % -2 (4)
The frontier between each material cohesion domain is defined when two ellipses are on the verge of intersecting, creating material cohesion-loss. Therefore, the space between two ellipses is set as e=0 (Eq. 5).

= √2 4 % (5)
Mixing Eq. 2, Eq. 3 and Eq. 5, the direct link between the volume fraction vf and the shape factor FA is obtained (Eq. 6). This equation is plotted in Fig. 5c (blue line on the plot).

= 1 -π 4 (6)

Mass reduction method mimicking medullar cavity controlling material cohesion-loss

In BIMRIM, using the previous material cohesion domain, the material cohesion-loss is avoided and mass reduction is improved.

3.2.1. Iterative process BIMRIM is bio-inspired by the iterations that generate the medullar cavity in long bones. In this method, unlike Nature that is able to form and resorb bone material, material will be only removed to reduce the mass of the mechanical part. The general algorithm of BIMRIM is presented in Fig 6 . In the first part of the BIMRIM algorithm named cavity design process, removable material is successively identified using Mass Removing Thresholds (MRT) (Section 3.2.2). At each iteration, material areas are removed from the part, so the mechanical stresses are redistributed in the remaining material given that the boundary conditions remain constant. In order to keep removing material during the whole process, the MRT evolves as presented in Section 3.2.2. In the second part of BIMRIM algorithm, the method presented in section 3.1.1 is used with a minimal volume fraction set at 0.55 in order to avoid material cohesion-loss. This last iteration generates independent separated elliptic holes.

Definition of Mass Removing Thresholds

In the cavity design process (Fig. 6) of BIMRIM, a MRT represents the limit where the material is considered removable or not. This threshold is based on the mechanical stresses that an area undergoes. If the local volume fraction is below the MRT (corresponding to low mechanical stresses), the material can be removed. At each iteration, the definition of this threshold is based on the material cohesion/cohesion-loss domains.

Two extrema MRTs are defined in terms of volume fraction thresholds that are directly derived from Von Mises stresses (Eq. 1). These two extrema MRT are set as vf_start and vf_end . For each iteration, the MRT values are linearly interpolated between these two values using n iterations of material removing.

The two extrema MRT values used in the cavity design process are defined as follow:

• The starting MRT is set as vf_start=0.215 for the first iteration. With the pattern defined in Fig. 3, if the local volume fraction is lower than this value, material cohesion-loss will necessarily be reached for any ratio FA (Fig. 5c). This MRT value corresponds to circular shape of ellipses (ratio FA=1) as shown in Fig. 7 (green schema).

• The ending MRT is set as vf_end=0.424 for the last iteration. With the pattern defined in Fig. 3, if the local volume fraction is lower than this value, material cohesion-loss will appear on average. The cohesion-loss will depend on the ratio FA and the ellipse orientation (Fig. 5c). This MRT value corresponds to ellipses with average ellipsoid shape (ratio FA=1.36) as shown in Fig. 7 (pink schema).

Between these two MRTs, n iterations will be done with a linear variation of vf_i from vf_start to vf_end with a constant step. At each iteration i with i ϵ⟦1, n⟧, the material areas that have a volume fraction lower than vf_i (Eq. 7) are removed.

_-= 0.215 + / 0.424 -0.215 0 + 1

The complete process is illustrated in Fig. 7. For each iteration, a mechanical part pi has its volume reduced during the mass reduction iteration using the threshold vf_i and becomes the new part pi+1. 8 presents the sample geometry. This sample consists in a torsion disc with four lugs distributed at 90° from one another. Fig. 8a illustrates the full loading case applied on the initial part. This sample undergoes a torque that is simulated by two forces applied on two opposed lugs.

This study case presents a double symmetry (black dotted line on Fig. 8a) in terms of geometry and loading. Thanks to these properties, the computation focuses only on a quarter of the whole disc using sliding contacts on the symmetry planes (Fig. 8b). A node centered in the hole has a rigid connection with all the elements of the inner edge of the hole. A FEA is carried out on this quarter disc using a 2D-shell study.

Saint-Venant principle states that the stress field is incorrectly estimated close to the application points of the forces. To respect this principle, a non-removable area is defined for the cavity design process. This area is represented by a dark red dotted line in Fig. 8b. This restriction is inhibited for the iteration after the cavity design process of BIMRIM (Fig. 6 bottom section). This iteration corresponds to the use of the Audibert's method that generates small elliptic holes in the part.

BIMRIM applied on study case

BIMRIM is applied on this study case. In this experimental validation, the iteration number n (Section 3.2.2) is set to five, which provides a total of eight iterations (i=1: starting iteration, i=2 to 6: 5 iterations, i=7: ending iteration, i=8: Audibert's method [START_REF] Audibert | Bio-inspired method based on bone architecture to optimize the structure of mechanical workpieces[END_REF]).

This whole process has been computed and the result of each iteration is shown in Fig. 9. For each step of BIMRIM, the MRT values are calculated using Eq. 7. The initial part pstart is modified by the cavity design process to obtain the predesign. To finish BIMRIM, the Audibert's method is applied on predesign to generate small elliptic holes so as to obtain the final part pfinal. To take into account the manufacturability of the part pfinal a minimal value of the semi-major axis is set to 0.5.

For each iteration, BIMRIM generates a set of elliptic holes on the whole part pi. In Fig. 9, to simplify the viewing, only ellipses that have a volume fraction lower than the MRT value are displayed (ellipses that may intersect at this threshold). The envelop line of these ellipses is depicted by a red line. This line represents the material that is removed at each iteration. Fig. 9 highlights that at each iteration of the cavity design process, material is removed from the inner part of the disc. In addition, from iteration 5 to the end of the cavity design process (iteration 7), material is also removed from the external part of the disc. These two phenomena generate the square frame shape of predesign.

Fig. 9. BIMRIM process applied on the torsion disc Fig. 10 presents the two main stages of BIMRIM: results of the cavity design process predesign and the final bio-inspired part pfinal. In this study case, BIMRIM reduced the mass of the part by 60.3%.

In this study, TO is used as a standard to evaluate the performances of BIMRIM. The mass of the bioinspired final part is taken as the target mass for the TO method. No geometrical constraints are imposed on this method. The redesign step of the process induces a small mass deviation between the two final geometries (less than 4%). Fig. 10 highlights the fact that the square shape is common for both final geometries (BIMRIM and TO). The main difference between the two designs is the width of branches that is slightly bigger for the part obtained through BIMRIM. This width increase in BIMRIM part is permitted because of the additional mass reduction induced by the small elliptic holes.

Experiment device

The bench test device (Fig. 11) is designed to evaluate the torsional stiffness of both parts: pfinal and pTO. On this device, two lugs of the part are fixed on a multicomponent dynamometer (fixed on the device frame), and the two others are attached to a torsion lever. A displacement is applied on the lever to generate a rotation on the two lugs attached to the lever that will deform the workpiece. This deformation generates a torque which is measured by the dynamometer.

The two parts are 3D-printed with polymer material. The masses of the two printed workpieces are nearly equal (Fig. 12 top section).

Experimental test results

Fig. 12 presents a picture of each printed part with their mass value and the chart of their torsional stiffness. This chart plots the measured torque as a function of the angular displacement applied on the torsion lever, for both sample. A third degree polynomial was best fitted, using least squares method, to the measured data (f1 for BIMRIM and f2 for TO).

It can be observed that both curves present a progressive increase of their slope during the test. This chart shows that the stiffness of the bio-inspired part, pfinal, is substantially higher than that of the TO, pTO. The comparison of the two polynomials (f1 and f2) highlights the fact that regarding their torsional stiffness, the BIMRIM part is 27% stiffer than the TO part.

Discussion

In Section 4.4, the results of experiments highlighted the fact that the BIMRIM part is 27% stiffer than the TO part. The current sub-section investigates on the origin of this gain. The parts can be considered as four perpendicular beams linked by lugs off-centered from the neutral fiber of the beam. It can be assumed that each beam undergoes multiples loadings:

• Traction and compression with a risk of having buckling added to the compression • Bending due to the off-set between the lugs' centers and the beams' axes In both instances, the corresponding quantities are linked to the beam quadratic moment of area. Indeed, both beams differ only in their width. These quadratic moments of area are calculated as follows:

• The TO part's quadratic moment of area, ITO, is calculated from the general formula (Eq. 8). This equation depends on the thickness of the TO beam (thicknessTO) and the width of the TO beam (widthTO). The numerical application leads to ITO=1.08x10 -10 m 4 .

• The BIMRIM part's quadratic moment of area, IBIMRIM, is calculated along the section with parallel planes in the worst case: all the ellipses are located along the same x-abscissa (Fig. 13b). These planes are orthogonal to the general direction of the beam (purple dotted line in Fig. 13a). IBIMRIM is composed of the quadratic moment of area of: first, the filled beam (which depends on the beam's thickness (thicknessBIMRIM) and on the beam's width (widthBIMRIM)) and second, the porosities transferred on the neutral axis of the beam (Eq. 9) calculated using the parallel axis theorem of Huygens (Eq. 10). In this calculation, the porosities are considered uniform. In addition, since the ellipses are two by two equidistant from the beam neutral axis with distances d1 and d2, simplifications of the equations can be done. Finally, with these considerations, the final formula is calculated (Eq. 11). The numerical application leads to IBIMRIM=1. 405x10 -10 m 4 .

The ratio between these two quadratic moments of area shows that IBIMRIM is 30% higher than ITO. This theoretical comparison reinforces the view that the section of the beams is the origin of the part stiffness. 1 ?@A@B-CD E = 1 ?@A@B-CD + F ?@A@B-CD * : &

Conclusions

This paper presents a 2.5D method, bio-inspired from long bone internal structure, to significantly reduce mass of mechanical parts. In the literature, a method bio-inspired from trabecular bone structure was defined to reduce the mass of parts. This stress-driven method adds porosities bio-inspired in three ways: bio-inspired in shape (elliptic holes), bio-inspired in orientation (along the principal stress axes) and bioinspired in size. But this method presents material cohesion-loss limitation. Improving on this method and including cavity design process, a new approach was defined in this paper to reach more significant mass reduction on mechanical parts. This BIMRIM (Bio-Inspired Mass Reduction Iterative Method) is based on long bone medullar cavity structure. The benefits of BIMRIM are as follows:

• It uses iteratively the method bio-inspired from trabecular bone structure in order to identify barely loaded areas of the mechanical part that are removed. This successive removal of material allows for a redistribution of the predicted mechanical stresses in the part. This process determines more accurately the stress field in the remaining material. • The cavity design process identifies and suppresses large unloaded areas. At the end of the algorithm, the method bio-inspired from trabecular bone is applied to finalize the mass reduction process. The generated geometry is similar to long bone, combining trabecular bone areas and medullar cavity areas.

• A physical experimental test is carried out on a torsion disc. In this test, the method leads to a 60% mass reduction from the initial disc. TO method was computed with the same mass objective and was used as a standard to compare the torsional stiffness of BIMRIM part. As a result of the experiments, the BIMRIM part presented a 27% higher torsional stiffness, at equal mass, than the TO part.

The proposed method provides convincing results in 2.5D that opens the way to promising results in 3D.

Fig. 1 .

 1 Fig. 1. Mid-section tomography of a horse humerus on which three parts can be identified: (1) the diaphysis region, (2) the metaphysis region, and (3) the epiphysis region. This tomography has been obtained through a partnership between FUNEVOLMECADEV MNHN team and CBI ISM AMU team

Fig. 2 .

 2 Fig. 2. Evolution of the bone structure depending on the locomotion of the individual

Fig. 3 .

 3 Fig. 3. Schema of the bio-inspired discretization: (a) schema of the natural structure of the femoral head, (b) example of the discretization that could be used on a ring-spanner

Fig. 4 .

 4 Fig. 4. Face Centered Cubic pattern element used in the method

Fig. 5 .

 5 Fig. 5. Relations between the parameters used in the method: (a) Volume fraction vf as a function of the ratio of Von Mises stresses σVM over the yield strength Re, (b) Ratio FA as a function of the ratio of principal stress σ1 over principal stress σ3, (c) FA as a function of vf representing the domains where material cohesion is lost (red) or is maintained (green)

Fig. 6 .

 6 Fig. 6. Bio-inspired algorithm

Fig. 7 .Experimental validation 4 . 1

 741 Fig. 7. BIMRIM process using the different MRT values leading to material removal

Fig. 8 .

 8 Fig. 8. Boundary conditions applied on: (a) the full disc and (b) the simplified quarter disc

Fig. 10 .

 10 Fig. 10. (a) Main stages in the BIMRIM process on the torsion disc with the initial part, the mid-process part obtained with the cavity design process, the final part obtained with the BIMRIM, compared to the TO process (b)

Fig. 11 .

 11 Fig. 11. Schema of the bench test designed to carry out the experimental validation of both BIMRIM and TO parts (left section) and the used test bench (right section)

Fig. 12 .

 12 Fig. 12. Printed parts and their respective mass (BIMRIM in the top left section and TO in top right section), and the obtained curves relating the torque as a function of the torsion angle for both BIMRIM and TO parts (in the bottom section)

Fig. 13 .

 13 Fig. 13. Schema of the branches of the BIMRIM part (a) and the section view A-A which is used to determine the BIMRIM part's quadratic moment of area (b)

  1 ?@A@B-CD -2 * F ?@A@B-CD * G: " & + : & & H([START_REF] Al-Ketan | Topology-mechanical property relationship of 3D printed strut, skeletal, and sheet based periodic metallic cellular materials[END_REF]

Aknowledgment

The experimental equipment was funded by the European Community, French Ministry of Research and Education and Aix-Marseille Conurbation Community. Authors aknowledge the Association Nationale Recherche Technologie (France) for financial support through a Convention Industrielle de Formation par la Recherche fund (Cifre thesis n°2020/0881).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.