Miriam Beblo
email: beblo@zew.de

Denis Beninger
email: beninger@zew.de

François Laisney
email: laisney@zew.de

Franziska Mientus

Alexander Spermann

Family Tax Splitting: A Microsimulation of its Potential Labour Supply and Intra-household Welfare Effects in Germany

Keywords: Collective model, household labour supply, intra-household allocation, tax reform, family splitting JEL Classification: D11, D12, D13, D31, J12, J22

This paper assesses the effects that an introduction of the French family splitting mechanism would have on German families' labour supply and intra-household consumption behaviour. We use simulated real world microdata created by means of a 'deterministic' collective labour supply model. The data are generated by a compound procedure of estimation and calibration based on GSOEP data. In a microsimulation the present tax-benefit system with child benefit/allowance is replaced by a tax scheme with family splitting. The resulting changes in labour supply are surprisingly small, even for women. Welfare effects are also modest, but differ for husbands and wives.

Non-technical summary

In this paper we investigate the effects of an apparently family-friendly tax reform in Germany. We introduce a reform of the present tax-benefit system that replaces marital splitting, complemented with child benefits or allowances, by a tax scheme with family splitting, according to the French example. We are interested in the potential labour supply effects of such a reform, particularly for married mothers. We would like to know whether the lower marginal tax rate resulting from the adoption of the family splitting would provide a significant incentive to work for mothers and fathers. We simultaneously investigate the welfare effects within the family and the possible redistribution of resources between the spouses. For this purpose we use simulated real world microdata created by means of a collective labour supply model. This model accounts for the presence of multiple decision centres with individual preferences within the household. It therefore provides a 'collective' representation of the decision processes of the household, in contrast to the traditional, or unitary, representation. The data are generated by a compound procedure of estimation and calibration based on 1998 data from the German Socio-Economic Panel. The data are used to simulate the introduction of the family splitting and analyze the changes in positive aspects (impact on behaviour) and normative aspects (impact on individual welfare). In a microsimulation, the 1998 German tax-benefit system with child benefit/allowance is replaced by the French family splitting mechanism. Due to acceptability reasons, the minimum tax gain from family splitting is set equal to the present child benefit received.

The resulting changes in labour supply are surprisingly small, even for women. Welfare effects are also modest, but differ for husbands and wives. The main result of our simulation study is that the introduction of family splitting within the German tax-benefit system cannot be expected to have a marked effect on households' behaviour, except for very well endowed families. Indeed the existing system of child benefits or allowances in Germany seems relatively generous for the majority of parents. Its effects are not so different from those of our family splitting reform with a guaranteed bottom transfer, at least for the average German household. A switch from marital to family splitting would have noticeable welfare effects only for high-income households, particularly for the women in these households. Individual working hours would shift marginally but aggregate labour supply effects would be close to zero. One reason for the small behavioural adjustments lies in the small number of children in German families. In particular, better-earning couples in Germany have only few or no children. Since noticeable welfare impacts are simulated for multi-children and high-income households only, the scope for changes seems particularly small. This finding is particularly striking in view of a cut in the fiscal budget due to a decreasing tax revenue. Nevertheless, our study underlines that the issue of intra-household reallocation and redistribution should not be neglected when analyzing reforms of the tax-benefit system, since a higher subsidy for children through family splitting would benefit, if any, primarily the poorer parent in the household.

Introduction

In comparison with France, the employment rate of women with children is still low in Germany. The difference in the proportion of mothers holding a full-time job is particularly striking: for 21 percent of German couples with children under age 15, both parents were employed full-time in 1996 whereas the corresponding figure for France was 36 percent [START_REF] Dingeldey | Das deutsche System der Ehegattenbesteuerung im europäischen Vergleich[END_REF]. This pattern seems to persists even when looking at more recent data. Although the OECD reports a gap in the participation rate of mothers with two and more children of merely 2.5 percentage points in 2000, the proportions of those working part-time differ starkly with 60 percent for Germany and 32 percent for France (OECD 2002, p. 77f). At the same time, fertility in Germany is as low as 1.36 children per woman (OECD 2003, Table GE4.1). France, on the contrary, combines employment of mothers with a relatively high birth rate of 1.89, compared to the average in the European countries.

Reasons for these observed differences may lie, among others, in the different tax-benefit systems of the two countries. Among the usual suspects are particularly those regulations that are related to the family status and the presence of children in the household. While the tax-benefit systems of Germany and France are both set up to follow the principle of ability to pay (Leistungsfähigkeitsprinzip), with the implication that the fraction of household income needed to support a child should not be subject to taxation, the application of this principle differs between the countries. Germany combines joint taxation of married couples with the payment of a lump-sum child benefit, or alternatively a child-related tax exemption. Due to the progression in the tax scheme, low-income households generally opt for the benefit, whereas the tax exemption is chosen by high-income earners. The French system is characterized by a relatively small means-tested child benefit, and joint taxation of all family members. The "quotient familial" (family splitting factor) takes account not only of married adults but also of the number of children in the household. This apparently generous taxation of families is meant to produce horizontal equity between households with and without children (see [START_REF] Sterdyniak | Pour défendre le quotient familial[END_REF]. The fact that high-income households benefit most from the lower tax-liability (in absolute terms) due to progressive taxation is seen as providing an incentive for women to work and contribute to the household income.

Reforms of family taxation are brought up time and again in the public debate in Germany. Experts from various disciplines and political parties have been thinking of introducing variants of family splitting in the actual German tax system. Putting more weight on children's needs when determining a household's tax liability is seen as a promising measure for a family-friendly policy. Starting off from this discussion on the appropriate taxation of families, the aim of our paper is to assess the effects of an application of a French-type family splitting to the German tax-benefit system.

We are interested in the potential labour supply effects of such a reform, particularly for married mothers. We would first like to know whether the lower marginal tax rate resulting from the adoption of the family splitting would provide a significant incentive to work for mothers and fathers. Second, we simultaneously investigate the welfare effects within the family and the possible redistribution of resources between the spouses. Reforms of the tax-benefit system are likely not only to have an impact on households' living conditions but also on individuals' and families' labour supply and consumption behaviour as well as on the intra-household distribution of resources. For the analysis of such intra-family effects, a model is needed that accounts for the presence of multiple decision makers within the household. In this respect, an appealing representation of the decision process of the household is the collective framework, introduced by [START_REF] Chiappori | Rational Household Labour Supply[END_REF][START_REF] Chiappori | Collective Labour Supply and Welfare[END_REF], and [START_REF] Apps | Taxation and the Household[END_REF]. In contrast to the traditional, or unitary, approach this type of model considers several decision makers in the household with individual (and possibly conflicting) preferences. Therefore, drawing on the collective framework opens up the possibility to infer aspects of the within-household welfare implications of policy changes that can be crucial in determining household choices. In an extension of Chiappori's basic model [START_REF] Beninger | Collective models: introduction of the taxation[END_REF] and [START_REF] Donni | Collective Household Labor Supply: Nonparticipation and Income Taxation[END_REF] allow for convex budget constraints. [START_REF] Beninger | Comparison Between Unitary and Collective Models of Household Labour Supply with Taxation[END_REF] provide an application with simulated data. In this paper we draw on [START_REF] Beninger | Welfare Analysis of Fiscal Reforms: Does the Representation of the Family Process Matter? Evidence for Germany[END_REF] who propose a further extension with non-convex budget sets.

Among the existing studies on the effects of family-related reforms of the tax-benefit system in Germany, [START_REF] Althammer | Familienbesteuerung -Reformen ohne Ende?[END_REF] and Wagenhals and Kraus (1998) have looked at the introduction of a family splitting. Their analyses are based on household utility maximization and thereby neglect any redistribution activities within the family that may be caused by the reform. In our study we would like to draw a comprehensive picture including also the intra-family labour supply and welfare effects of a tax reform that introduces family splitting. For this purpose we use simulated real world microdata that have been created by means of a 'deterministic' collective labour supply model [START_REF] Beninger | Welfare Analysis of Fiscal Reforms: Does the Representation of the Family Process Matter? Evidence for Germany[END_REF]. Our real world collective data have been estimated and calibrated so as to reflect the characteristics (with respect to labour supply and disposable household income) of couples of the 1998 wave of the German Socio-Economic Panel (GSOEP).

The outline of the paper is as follows. In Section 2 we describe the simplified version of the 1998 German tax-benefit system with joint taxation of couples that will serve as the baseline of our analysis. Section 3 introduces the tax reform under consideration, that is, our adaptation of the French family splitting to the baseline tax-benefit system. In Section 4 the characteristics of our collective data set are described. The simulation results under family splitting are presented in Section 5. We compare labour supply and consumption of all decision makers in the household in the baseline situation with the simulated effects of the tax reform. Section 6 concludes.

The baseline German tax-benefit system

Germany has a personal income tax system administered at the federal level and regulated by the Personal Income Law (Einkommensteuergesetz). The German tax system is characterized by a comprehensive tax that covers labour earnings as well as income from other sources such as capital investment, housing rents etc. and by joint taxation for married couples. For our exercise we use a simplified form of the 1998 German tax and benefit system that is described in detail in Table 1. 1The function applied to the tax base is smoothly progressive. In Germany, as in France, the tax schedule used is the same for singles and for couples. However for couples, the "Ehegattensplitting" method (marital splitting) is used: the tax rate is applied to half of the joint taxable income, and the outcome is doubled in order to obtain the total income tax liability of the spouses. Tax rate progression and marital splitting lead to a relative advantage for married couples if spouses have unequal incomes.

The most important issue for this study is how children are considered in household taxation. 2 Parents can opt for either a child benefit (DM 220 for the first and the second child, DM 300 for the third and DM 350 from the fourth child on) or a child allowance, that is a lump-sum deduction of DM 6,912 for each child up to age 27, if still in education or doing military or civil service. Due to the progressive tax scheme the child benefit is less, and the tax deduction is more, favourable for high-income households.

Social benefits are means-tested and depend on the number of people in the household. 3 As a simplification, we assume that the maximum social benefit (including housing benefit and special payments) a person can receive is DM 1,000 a month and DM 700 for the partner. 4 In addition there are age-dependent supplementary payments for children. The amount of the transfer depends on the level of earned income ("anrechnungsfreies Erwerbseinkommen") and is degressive, depending on the relevant income measure. In addition, social benefits are related to the geographical location, since they are paid by the local governments and housing benefits depend on the average rent of the locality. We distinguish only between East-and West Germany, and approximate that social benefits are 10% lower in the East. The difference between East-and West Germany stems from the lower costs of living in the East: a substantial fraction of the social benefit is the housing benefit. Finally, social benefit payments depend on the wealth situation of the household, and child benefits are deducted from social benefit payments.

As a graphical illustration of the tax-benefit system described above, Figure 1 depicts a typical situation for a couple with two children. The husband has an hourly wage rate of 25 euro, the wife earns 18 euro per hour. The household does not dispose of any capital inflows or income from rental or leasing. It is therefore eligible for means-tested social benefits at low labour income. The parents receive child benefit for both children. From a yearly gross income of just above 80,000 euro they will opt for child allowance instead, as the tax relief exceeds the lump-sum benefit payment. Figure 1 also reveals the non-convexity of the resulting budget constraint when labour earnings are high enough for social benefit payments to cease. 0.53•X-22,843 X=rounded taxable income, Y=(X-12,312)/10,000, Z=(X-58,590/10,000) solidarity supplement: tax scaled up by a factor of 1.055 (b) Net Income: gross income -twice the tax liability Benefits child benefit: DM 220 for 1st and 2nd child, DM 300 for 3rd, DM 350 from 4th child on (or child allowance alternatively) means tested social benefits (incl. housing benefit and special payments): (c) DM 1,000 in the West, DM 900 in the East. (d) means tested social benefits for partner and children, depending on age Maintenance maintenance payments to children, ex-partner or parents outside the household Disposable income: net income + benefits -maintenance (e) Notes: (a) The tax scheme is given in DM because of the non-linearity of the tax function. Since 2000, the tax scheme is given in DM and euro by the Federal Government. 1 euro = 1.95583 DM. Time unit is the year. (b) The solidarity supplement for the reconstruction of East Germany ("Solidaritätszuschlag") is based on a measure of taxable income that includes the child allowance whether or not parents opt for it. (c) For lack of information on the stock of savings etc., we assume that couples having more than DM 600 capital income or more than DM 4,800 rental income per year are not eligible for social benefit payments. (d) These numbers are based on the average effective maximum social benefit paid in 1998. (e) Social security contributions, although largely compulsory, are taken as consumption expenditures and are not deducted from disposable income. Admittedly, the different types of social security contributions paid in Germany certainly have different consumptive aspects, and our assumption is probably more appropriate for payments to the pension system than for health insurance contributions.

Introducing family splitting

Based on the 1998 German tax scheme with joint taxation of couples, we now introduce a family splitting factor similar to the "quotient familial" of the French tax system while abolishing the existing child benefit/allowance.

Family splitting in France

France introduced a progressive income tax system in 1911. Family splitting has been applied since 1948, with the aim to adapt the fiscal contribution of the household to its financial ability, taking into account the total number of household members, including children. Thus, going beyond joint taxation of married couples, each child in the household adds to the tax quotient of the family. 5 Whereas the parents are each counted as one, the first and the second child both add one half to the quotient. From the third child on, the tax factor is increased by a full point to take account of the over-proportional increase in the costs of living of a large family (a point which is open for discussion) and the decreasing benefit of the splitting (see Table 2). Compared to international equivalence scales used in the literature on inequality measurement, the equivalent scales used by the French family splitting method is more favourable for families with a large number of children.

A main critique addressed to the French family splitting, e.g. by [START_REF] Atkinson | Fiscalité et transferts : une comparaison franco-britannique[END_REF], is that it mainly benefits the richest, thereby neglecting vertical equity. 6 In order to face these critiques, several additional measures have been applied. First there is a maximum tax saving of 1,677 euro per additional halfpoint to the family quotient in 1998. Second, poorer households may receive several complementary benefits (young child allowance -"allocation pour jeune enfant", -family supplement -"complément familial") which can potentially be added to the basic child benefit, depending on family income. [START_REF] Sterdyniak | Pour défendre le quotient familial[END_REF] argues that the French family splitting provides a certain equity among families with comparable endowments. However the costs of children are not fully compensated by the lower tax liability resulting from the family quotient, so that among high-income households, horizontal equity is not fully respected.

The French tax system combines horizontal and vertical equity, up to the critiques mentioned above. With respect to vertical equity it is argued that the current system may encourage highly educated women with a high earnings potential to have children, as the cost of a child is lowered by the progressive benefit of the family splitting. Moreover, mothers are encouraged to continue working because the over-proportional tax loss due to the convexity of the tax function is smoothed by the application of the family quotient. Recent debates on tax policy in France revolve around the linearisation of the tax function, which would abolish the relative advantage of the family splitting factor.

Application to Germany

In our application of the family splitting factor to Germany, we consider that the minimum tax gain from family splitting must not be lower than the current child benefit received, because of acceptability reasons. The latter corresponds to the amount of child benefit the parents would receive only for those children living in the household, according to the 1998 benefit system. Hypothetically, parents have the choice either to receive child benefits -marital splitting is then applied when computing the family's tax liability -or to benefit from family splitting. The ceiling advantage of the family splitting is -somewhat arbitrarily -fixed at twice the potential child benefit. 7 Hence, under the reform, the total transfer a household can receive consists of social benefits only (plus maintenance payments), since the child benefit is already integrated in the tax liability. Note also that the reform is not designed to satisfy revenue neutrality. In a static comparison, Figure 2 illustrates the increase in disposable household income, given gross income and number of children, if a family splitting system as described replaced the baseline 1998 system with marital splitting and complementary child benefits or allowances. The family splitting tax gain increases with household income and number of children. In the middle-income range from 40 to 75 thousand euro annual income, few-children households would benefit most from the reform since the child benefit in the baseline is increasing (higher child benefit for the third, fourth... child than for the first and second) while the marginal tax gain through family splitting is decreasing with the number of children. The pictures reverses for high-income households. The graph underlines substantial gains for multi-children and high-income households (from 80,000 euro annual income) who would be eligible for child benefits otherwise. The difference in net household income reaches a maximum of up to 6,500 euro per year for an annual income of 90,000 to 120,000 depending on the number of children. This maximum is as high as the annual sum of child benefits and refers to the ceiling advantage of family splitting applied. For those families who would opt for child allowances in the base system, the tax gain decreases with rising income and levels out at the difference between the ceiling advantage from family splitting and the maximum tax advantage through child allowances.

Collective microdata for Germany

There exists no data set for Germany that contains, at the same time, information on family members' individual labour supplies, wage incomes and consumption levels. In order to analyze intra-household time allocation and resource distribution we therefore have to "adjust" existing data to our needs. To obtain a data set that represents collectively behaving households in Germany, we proceed according to Beninger, Laisney and Beblo (2003). We start from the 1998 wave of the German Socio-Economic Panel (GSOEP), a representative panel data set of households and individuals living in Germany. The panel gives a wealth of information on the labour market status of individuals and on the various income sources of families. We select German nationals aged 25 to 55. All are employees with contractual working hours of at least 10 hours per week or voluntarily out of employment. The restriction on hours is introduced to avoid extraordinary high wage rates obtained as the ratio of earnings over hours for people with less than 10 hours. We exclude those self-employed, recipients of pension payments, as well as individuals in parental leave, or in education, or registered as unemployed. We thus select 1332 families living in a one-or twogeneration household composed of a married couple and (possibly) dependent children who may also live outside the household.

Beninger et al. estimate preference parameters for single men and women and then predict their labour supply and the corresponding consumption level. This involves a calibration method to determine the partners' relative weights in the household, also called the sharing rule or the power index, as well as a leisure interaction coefficient describing the effect of the leisure of the spouse on each individual's utility. The calibration is done by optimising the fit of predicted to observed hours of work. The introduction of the cross leisure term relaxes the strong assumption of separability of individual preferences in the pairs (c f , l f ) , (c m , l m ) which is usually made in the empirical literature on collective models. Apart from this cross leisure term, married individuals are assumed to have the same preferences as singles with the same characteristics. Beyond the calibration of the power index, an estimation step allows to obtain the expected value of the power index given a set of variables which includes characteristics of the tax-benefit system: this step allows predictions of the changes induced by a reform for the power index in each household. For completeness, we summarize the procedure in the Appendix.

This compound procedure of calibration and estimation yields a microdata set very similar to the GSOEP sample (in terms of individual labour supply choices), but generated according to collective rationality in household behaviour. That is, with our knowledge about individual preferences, and the partners' weights in the household as a function of the tax system, we are now able to predict reactions induced by a reform not only in household behaviour, but also of individual family members regarding their labour supply and consumption decisions.

Simulation results

Simulation results are presented for changes of the power index, that is the spouses' weights in the household, changes in labour supply and intra-household Notes: Means of the pre-and post reform power index by number of children for West Germany. For East Germany, the power index is the same after the reform since the variable yd is not significant in the Eastern part of the country.

welfare.

Changes in the power index

The predicted power index depends on a variable which reflects the wife's relative net earning power. The wife's net earning potential y f measures the expected increase in household disposable income if the wife switches from 0 to 40 hours, the expectation being taken over the male hours distribution (see the Appendix for a formal definition). The variable y m , giving the male's net earning potential, is computed similarly. We then define the 'relative net earning power of the wife's, y d , as:

y d = y f /y m . ( 1 )
Naturally, this ratio depends on the tax-benefit system. As documented in the Appendix, the man's power index is significantly negatively affected by this variable, for West German households, but no effect was found in the East. Thus, for West German households the power index will be affected by changes in the tax system that alter the spouses' earning capacities, such as the introduction of a family splitting factor. The relevance of the power index in intra-household decision making is illustrated in Figure 3. If the reform expands the utility set of the couple, due to lower tax liabilities for instance, the Pareto frontier will move outward. This improvement at the household level does not tell us much about the effect on the individual utilities of the spouses, though. In the collective world, the effect of a change in the budget constraint can be decomposed in two components: the first one is a standard income effect due to the increase of household wealth (shift of the Pareto frontier, dark arrow). The second one is the redistribution effect, accounted for only in the collective framework and reflecting the change of the spouses' relative weights that are affected by the variable y d (move on the Pareto frontier, light arrow). If a tax reform increases the wife's relative earning potential, she will gain weight in the intra-household distribution of resources (move to the left on the Pareto frontier), as depicted in Figure 3.

From Table 3 we see that the husband's power index increases with the number of children in the household. This is true for East and West Germany. The introduction of family splitting, however, affects only the West German households in our sample. Nonetheless, the relative weights of wife and husband hardly vary, meaning that the intra-family allocation and distribution of For East Germany we do not expect any pure collective effect, since the only variable sensitive to the tax system, the wife's relative net earning power, is not significantly related to the power index (see Table 9 in the Appendix).

Changes in participation and working hours

Table 4 compares the working hours of women with children before and after the reform, aggregated in 10-hours categories. The introduction of the family splitting does not seem to have a large impact on the labour supply of married women (and affects men even less). The majority of women is still concentrated at zero, 20 or 40 work hours, with not much variation among these categories after the reform. 8Small changes in hours, however, do occur as becomes evident when looking at more disaggregate numbers. Table 5 documents joint changes in the labour supply of wives and husbands, distinguishing a decrease of working time by more than 5 hours, 1 to 5 hours and the symmetric increases. We see that the reform leads to an hours adjustment of only one of the spouses in most cases. Relatively more adjustments are made by wives than by husbands and most changes are fewer than 6 hours. Nevertheless, a salient feature also of this table is that the most frequent cell is (0,0), with about 93 percent of the households where neither husband nor wive change labour supply due to the adoption of the family tax splitting system. Table 6 summarises the consumption and labour supply effects of the family splitting reform for mothers and fathers of one child, two children or three and more children. Under family taxation the participation rate of married women with one child decreases by 0.6 percent from 75 percent in the baseline situation. About the same percentage of mothers with two children is employed before and after the reform (60 percent). They hardly adjust to the new system, whereas those with three and more children show an increase in participation of 2.6 percent under family splitting. While the participation of husbands does not change at all, their relative change in consumption is slightly larger (for 2 and more children) than that of the wives. Work hours are also adjusted under family taxation. The largest change is observed in multi-children families where mothers increase their working time by 2.3 percent on average, while fathers add less. This change is due original non-participants. When we consider only those women working in the baseline, working hours even decrease with family splitting. Our overall conclusion therefore is that, while changes in consumption levels and hours of work do differ across household types, they are rather small on the whole.

It is important to note that our version of this tax reform is not revenue neutral. In fact, tax revenues are diminished in the family splitting system, since all childless singles and couples still pay the same tax amount, while households with children experience a tax gain on average. Hence, all effects also have to be evaluated in view of a shrinking fiscal budget. Total tax revenues amount to 734.6 million euro before the reform and 591 after the reform (projected for our population of German couples aged 25 to 55). Our results therefore represent an upper bound on behavioural adjustments rather.

Normative aspects of the reform: changes in welfare

We now look at the individual welfare effects of the reform measured for husbands and wives separately. These effects are illustrated by the distribution of percentage changes in individual utility for every decile of the pre-reform distribution of the household equivalent disposable income (in Figures 4 and5). 9The graphs show the mean, the median and the 10th and 90th percentiles of the change-in-utility distribution. The inter-decile and median curves reveal that the spread of the distribution within income levels is higher for females. The expected result is that, the higher the equivalent household income, the larger the welfare impact of family splitting, if any. Well endowed women benefit most from the reform. Among the 10 percent richest households, for instance, the wives' average (mean) gain amounts to 9 percent. Their husbands' welfare, on the contrary, is hardly affected, with an average gain of close to zero percent. The inter-quantile range deviates from zero only for the 20 percent highest income households. Since some wives experience a relatively high gain, respectively loss, the mean effect runs off the median, for the lowest and highest deciles of the female income distribution.

A direct comparison of the welfare effects for both spouses is made in Table 7 where the positions of husbands (winner, indifferent, loser) are cross-tabulated with the positions of their wives. A cut-off of ± 1 percent change defines indifference. While the winners clearly outweigh the losers, the most pronounced result is that of no welfare effect: in 78 percent of the households, neither husband nor wife experiences any change in individual utility through family tax splitting. Almost 19 percent experience a mutual gain. The remaining families face conflicting outcomes: though the household as a whole is better off with the reform, in 2.4 percent of all cases the wife only benefits, while her husband is negatively affected or indifferent. The reverse happens for less than 1 percent of the couples.

Conclusion

The main result of our simulation study is that the introduction of a family splitting component within the German tax-benefit system cannot be expected to have a marked effect on households' behaviour, except for very well endowed families. Indeed the existing system of child benefits or allowances in Germany seems relatively generous for the majority of parents. Its effects are not so different from those of our family splitting reform with a guaranteed bottom transfer, at least for the average German household. A switch from marital to family splitting would have noticeable welfare effects only for high-income households, particularly for the women in these households. Aggregate labour supply effects would be close to zero. Only individual working hours would shift marginally. One reason for these small changes lies in the small number of children in German families. In particular, better-earning couples in Germany have only few or no children. Since noticeable welfare impacts were simulated for multi-children and high-income households only, the scope for changes seems particularly small, unless the demographic impact of such a reform turns out to be large.

First, these findings have to be interpreted in view of a shrinking fiscal budget due to a cut in tax revenues. Second, the treatment of families in the French tax-benefit system, which includes supplementary benefits for children, is much more comprehensive than the family splitting reform we implemented for German households. But as our adoption of the family splitting grants a minimum transfer as high as the current German child benefit, these income-dependent complementary payments would not apply, anyway. Also, the maximum tax saving in the French family splitting mechanism is much lower than in the German application. As a result, the behavioural adjustments and welfare changes of our microsimulation are in every sense an upper bound of what can be expected from the introduction of a family tax splitting system in Germany.

One drawback of our approach is that we do not distinguish between leisure and housework when modelling the labour supply decision. Especially housewives may be misleadingly assigned more leisure time, and thus more power in the household. In this case and if this applies to women of high-income households, our result of a welfare gain for these women will again represent an upper bound, meaning that the real effects may be even smaller.

Even if the overall effects can hardly be seen from the surface, it has become evident that the issue of intra-household reallocation and redistribution should not be neglected when analyzing reforms of the tax-benefit system. Our study underlines (for rich households) that a higher subsidy for children through family splitting would benefit, if any, primarily the poorer parent, that is, the wife in most families.

Figure 1 :

 1 Figure 1: The 1998 German tax-benefit system. This figure iIllustrates the situation of a couple with two children. The wife and the husband earn 18 respectively 25 euro per hour. They potentially receive means tested social benefit.

Figure 2 :

 2 Figure 2: Change in disposable household income, family splitting versus 1998 German tax-benefit system (marital splitting + child benefit/allowance).

Figure 3 :

 3 Figure 3: Shift of, and move on the Pareto frontier due to a reform

Figure 4 :

 4 Figure 4: Relative welfare gain/loss for wives by decile of equivalized household income

Figure 5 :

 5 Figure 5: Relative welfare gain/loss for husbands by decile of equivalized household income

Table 1 :

 1 Simplified tax-benefit system for household taxation, Germany 1998 (married couples) Taxable income -Tax reliefs Tax schedule: Tax rate applied to half the tax base:

	Taxable income
	earnings
	capital income
	income from rental and leasing
	maintenance payments from ex-partner
	Tax reliefs
	DM 4,000 standard deduction for earnings (a)
	DM 12,000 standard deduction for capital income
	DM 216 standard special expense deduction
	child allowance (or child benefit alternatively)
	exemption for social security contribution ("Vorsorge")
	maintenance payments to ex-partner
	Tax base: Income (X) bracket Income tax liability
	0-12,365	0
	12,366-58,643	(91.19•Y+2,590)•Y
	58,644-120,041	(151.96•Z+3,343)•Z+13,938
	>120,041

Table 2 :

 2 Equivalence scale for the "quotient familial" in 1998 : Until 1982 the household gained a full point only for the third child.

	Number of children	Number of shares
	0	2
	1	2 . 5
	2	3
	For each child from the 3rd child on +1

Note

Table 3 :

 3 Power index pre-and post-reform

		West		East
	Number of children 1998 tax family splitting 1998 tax
	1	.50768	.50755	.4454
	2	52292	.52289	.4790
	≥ 3	.53199	.53215	.5090

Table 4 :

 4 : Rows: pre-reform, Columns: post-reform. Working hours are aggregated in 10hours categories. Entries in the body of the table and in the margins give percentages, the last cell contains the number of observations.

			Wives' working hours pre-and post-reform
		0	10	20	30	40	50	60	total
	0	33.98	0.48	0.10					31.95
	10	0.29 13.80	0.10					9.94
	20		0.19 18.15	0.58				19.02
	30			0.10 13.03	0.19			12.55
	40			0.10	0.10 16.99 0.10		26.06
	50					0.19 1.35		0.39
	60							0.19	0.10
	total	31.76	10.33	19.31	12.55	25.39 0.58 0.10 n=1036

Notes

Table 5

 5

	: Change in working hours of couples: wife vs. husband
		-5 to -1	0 1 to 5 > 5	total
	< -5	0.10	0.19	0.10	0	0.39
	-5 to -1	0.10	1.16	0.10	0	1.35
	0	0 . 1 0 93.05	0.19 0.10	93.44
	1 to 5	0.10	3.38	0.48	0	3.96
	> 5	0.10	0.39	0.39	0	0.87
	total	0.48	98.17	1.25 0.10 n=1036
	Notes: Rows (columns) give the percent change in the working hours of wives (hus-
	bands).

Table 6 :

 6 Aggregate effects by number of children

	Average change in	1 child wife husband wife husband wife husband 2 children ≥ 3 children
	Consumption (%)	0.42	0.10 0.23	0.35	0.17	0.50
	Participation (%)	0.61	0.00 0.31	0.00	2.63	0.00
	Working time, all (%)	0.47	0.04 0.32	0.05	2.29	0.17
	Work. time, participants (%) 0.34	0.04 0.18	0.05 -0.71	0.17
	Notes: 1036 observations: 410 households with 1 child, 478 households with two chil-
	dren and 148 households with at least three children.

Table 7 :

 7 Intra-household welfare effects (wife versus husband)

		loss	0	gain	total
	loss	0.29	0	0.29	0.58
	0	0 78.19	0	78.19
	gain 2.41	0 18.82	21.24
	total 2.70 78.19 19.11 n=1036
	Notes: Rows (columns) give the loss or gain in welfare due to the tax reform of wives
	(husbands).

This simplification is an adaptation, to our particular sample and emphasis, of the microsimulation program developed at ZEW (for a description see

[START_REF] Jacobebbinghaus | Dokumentation des ZEW Steuer-Transfer-Mikrosimulationsmodells[END_REF].

For convenience, the tax rules, and also our tax program, are written in DM rather than in euro as the non-linear 1998 German tax scheme is only available in DM (1 euro = 1.95583 DM). All other nominal magnitudes in the paper will be given in euro.

In our static setting we ignore unemployment insurance and unemployment benefits which are both related to former earnings. Both transfers actually require the search of, and the willingness to take up, a job.

The maximum social allowances we apply for both parts of Germany are based on the average effective maximum social benefits paid in 1998 (see Statistisches Bundesamt, 2001).

The system applies to cohabiting couples as well.

Therefore Glaude (1991) e.g. proposed an alternative tax-benefit system, closer to those applied in other European countries. Family splitting would be replaced by a generous meanstested child benefit. Poor families would be further supported by revenue and household size dependent benefits.

[START_REF] Chiappori | Rational Household Labour Supply[END_REF] Variations of this ceiling advantage do not change the results, though.

We allow for a change in working time by ± 0,1,2,5,10 or 20 hours.

The equivalence scale for the household disposable income is a modified OECD scale: 1 for the first adult in the household (wife or husband), 0.7 for the second parent, 0.6 for children aged 16 or above, 0.5 for each child between 7 and 15, and 0.4 for each child under age 7.

This research has been supported by the Fritz-Thyssen-Stiftung. It builds on calibrated data produced in the EU project "Welfare analysis of fiscal and social security reforms in Europe: does the representation of family decision processes matter?" which was partly financed by the EU, General Directorate Employment and Social Affairs, under grant VS/2000/0778. We thank Olivier Bargain for information on the French tax system and and Susanne Steffes for comments.

Appendix

Definition of the net earning potential The wife's net earning potential is defined as follows: let p k f and p k m denote the observed sample frequencies of the discretised weekly labour supplies h k of wives and husbands, respectively. Denote R fk 0 mk the household disposable income when the husband works h k hours and the wife works h k 0 hours, where k ∈ {0, 10, 20, 30, 40, 50, 60}. The variable y f is defined as:

Estimation of the power index

We first define an index k m which parameterises the household's position on the Pareto frontier, and which is defined as:

where Ūm , U min m and U max m are the effective, minimum and maximum utility levels the husband can reach as illustrated in Figure 3. Equivalently, we can define an alternative parameterisation k f with a similar formula for the wife's utility levels. In order to take account of the local curvature of the Pareto frontier, and to obtain power indices that add up to 1, we compute for each household a scalar α solving the equation k α f + k α m = 1, and we call ω f = k α f and ω m = k α m the wife's and the husband's power index, respectively. These numbers are both between 0 and 1. We then set up a logistic equation relating the calibrated power index of the husband, ω m , to a set of explanatory variables. This equation of the type ln [ω m / (1 -ω m )] = xγ + ² will allow us to obtain predicted values ωm for the male weight ωm between 0 and 1 given x. Important variables to include in x are variables capturing the way in which the tax benefit system influences the relative earning power of the spouses. If these turn out to contribute significantly to the prediction of ω m , they will allow us to describe changes in the power index induced by tax reforms.

The list of explanatory variables x used to explain the male weight in the household includes, apart from variable y d described in the text, the husband's age, the age difference between the spouses, the husband's schooling level and job status, as well as child-related variables. The estimation results, obtained separately for East and West Germany, are given in Table 8. The husband's power index is significantly negatively related to the wife's relative net earning potential in the West, but no significant impact was found in the East.


Rainer Glüge

Rainer Gluge
email: gluege@ovgu.de

Effective plastic properties of laminates made of isotropic elastic plastic materials

The effective elastic behavior of laminates is well understood. In this contribution, we make use of the fact that the elastic homogenization can be extended relatively easy to the plastic case (He and Z.-Q. Feng, 2012), which is interesting for bilayer metallic structures. With the aid of stress concentration tensors, the effective yield limit is calculated, and its properties are examined. It turns out to be anisotropic, pressuredependent, non-smooth, and evolves with the plastic deformation. Using the constitutive equations of von Mises elastic plastic materials, the effective plastic flow behavior is obtained as a set of coupled nonlinear ordinary differential equations with algebraic conditions. A general solution can hardly be given, but it can be integrated numerically. Further, it is possible to examine some properties of its solution without explicitly stating it, such as its poles. The latter allow to extract closed form-expressions for the ultimate loading stress in monotonic stress-driven tests. All analytical results are compared to representative volume element simulations, and found to match very well.

Introduction

Composite materials are of constant interest to engineers and researchers, as they often display desirable properties that can moreover be adjusted by the microstructure design. A nice example is the combination of stiff brittle glas layers with a small fraction of soft ductile glue, which results in a tremendous increase of fracture resistance, while the high effective stiffness of the glas is retained [START_REF] Fratzl | Hindered Crack Propagation in Materials with Periodically Varying Young's Modulus-Lessons from Biological Materials[END_REF].

The simplest conceivable composites are laminates [START_REF] Milton | The Theory of Composites[END_REF]. As such, a large amount of literature is dedicated to them, which is compiled in textbooks such as [START_REF] Reddy | Mechanics of Laminated Composite Plates and Shells: Theory and Analysis[END_REF]. Many contributions focus on the full 3D effective elastic properties (Backus, 1962;[START_REF] Bogetti | Three-dimensional effective property and strength prediction of thick laminated composite media[END_REF][START_REF] Chou | Elastic constants of layered media[END_REF][START_REF] Dumontet | Homogénéisation d'un mateériau à structure périodique stratifié, de comportement élastique lineáire et non linéaire et viscoélastique[END_REF][START_REF] Francfort | Homogenization and optimal bounds in linear elasticity[END_REF][START_REF] He | Homogenization of layered elastoplastic composites: Theoretical results[END_REF][START_REF] Liu | The effective Young's modulus of composites beyond the Voigt estimation due to the Poisson effect[END_REF][START_REF] Pagano | Exact moduli of anisotropic laminates[END_REF][START_REF] Postma | Wave propagation in a stratified medium[END_REF][START_REF] Salamon | Elastic moduli of a stratified rock mass[END_REF][START_REF] Tartar | Estimations fines des coefficients homogénéisés[END_REF], in some cases restricted to isotropic constituents. A compact summary of the effective stiffness of laminates made up of arbitrary many anisotropic layers is given in [START_REF] Glüge | The effective stiffness and stress concentrations of a multi-layer laminate[END_REF], and shown to be a generalization of classical laminate theory [START_REF] Berthelot | Classical Laminate Theory[END_REF].

The stiffness of laminates often serves as a reference solution for iterated microstructures, either through interface orientation averages [START_REF] Kalisch | Analytical homogenization of linear elasticity based on the interface orientation distribution -a complement to the self-consistent approach[END_REF] or nested (rank n) laminates [START_REF] Debotton | Transversely isotropic sequentially laminated composites in finite elasticity[END_REF][START_REF] Hariton | High-rank nonlinear sequentially laminated composites and their possible tendency towards isotropic behavior[END_REF]. Likewise, many contributions deal with material failure and fatigue under cyclic loading, like delamination and fiber fracture in carbon fiber laminates [START_REF] Bogetti | Predicting the nonlinear response and progressive failure of composite laminates[END_REF][START_REF] Hu | Failure analysis of fiber-reinforced composite laminates subjected to biaxial loads[END_REF][START_REF] Jalalvand | Homogenization of diffuse delamination in composite laminates[END_REF]. For such composites, these properties are more important than the plastic properties.

However, there are also laminates that contain metallic layers, which considerably deform plastically before material failure occurs. Such laminates arise naturally as lamellar eutectics [START_REF] Trepczyńska-Lent | Rod and lamellar growth of eutectic[END_REF], in forming processes [START_REF] Sakai | Development of a high strength, high conductivity copper-silver alloy for pulsed magnets[END_REF] or by martensitic transformations or by twinning [START_REF] Bhattacharya | Microstructure of Martensite: Why it Forms and how it Gives Rise to the Shapememory Effect[END_REF]. They may be also manufactured by accumulative rolling or explosive welding of different metallic or intermetallic layers [START_REF] Bataev | Structural and mechanical properties of metallic-intermetallic laminate composites produced by explosive welding and annealing[END_REF][START_REF] Hausöl | Microstructure and mechanical properties of accumulative roll bonded AA6014/AA5754 aluminium laminates[END_REF] or by gluing thin metallic layers and fiber reinforced layers [START_REF] Sinmazçelik | A review: Fibre metal laminates, background, bonding types and applied test methods[END_REF]. They often display an increased strength and ductility compared to the pure phases.

There is much less literature on the plastic properties of such laminates. The probably most advanced works in this regard are deBotton and Ponte-Castañeda (1992), El Omri et al. (2000), [START_REF] Ponte-Castañeda | On the homogenized yield strength of two-phase composites[END_REF], Pruchnicki (1998a,b,c), and [START_REF] Sawicki | Yield condition for layered composites[END_REF] and [START_REF] He | Homogenization of layered elastoplastic composites: Theoretical results[END_REF]. [START_REF] Sawicki | Yield condition for layered composites[END_REF] considered the effective yield limit by using the stress concentration tensors, as will be shown later. Ponte-Castañeda and deBotton (1992) (Sec. 5) consider as well the yield limit, while deBotton and Ponte-Castañeda (1992) approach the post-yield behavior, using the deformation theory of plasticity (i.e. a nonlinear elastic law mimicing plasticity) and a variational homogenization approach. Consequently, their approximations may serve in case of monotonic processes. Pruchnicki (1998c) used the same nonlinear elastic approch with an asymptotic expansion for the homogenization. It was apparently Pruchnicki (1998a,b) who firstly considered the evolution of internal state variables in this context. He derived closed form expressions for the effective elasticity, the effective yield function and the effective hardening rule. Later, El Omri et al. (2000) and [START_REF] He | Homogenization of layered elastoplastic composites: Theoretical results[END_REF] recognized that the analytical homogenization of the elasticity of laminates can be transferred relatively easy to the elastoplastic rate form. A detailed comparison to the works of Pruchnicki, El Omri et al. (2000) and [START_REF] He | Homogenization of layered elastoplastic composites: Theoretical results[END_REF] is postponed to Sec. 4.4.

Outline. After desribing the notation and the setting (Sects 2 and 3), we build on the closed-form expressions of the effective stiffness and stress concentration tensors for purely elastic laminates (given in Glüge and Kalisch (2014), Sec. 3.4). Assigning yield limits to the phases, this allows already to examine the properties of the effective yield limit (Sec. 4.1). Further, with the constitutive equations of isotropic von Mises plasticity and some very mild assumptions, it is then possible to derive the ordinary differential equations that govern the Effective plastic properties of laminates made of isotropic elastic plastic materials April 21, 2016 plastic response (Sec. 4.2). Being a coupled nonlinear ODE system, its general solution is not accessible, but its poles can be extracted easily. These allow to examine properties beyond the onset of plastic flow, like the ultimate loading in monotonic tensile tests (Sec. 4.2.1), for which we give the explicit solution. We finally put our results into context with the findings of other researchers (Sec. 4.4).

Notation and setting

Notation. A direct notation is preferred. Vectors are denoted as bold minuscules (like a), second-order tensors as bold majuscules or bold greek letters (like A, ε, σ), and fourth-order tensors as blackboard bold majuscules (like A). The dyadic product and single scalar contractions are denoted like (a

⊗ b ⊗ c) ••(d ⊗ e)=(b • d)(c • e) a,
with • being the usual scalar product between vectors. The upper index T denotes the transpose of a secondorder tensor, (a⊗b) T := b⊗a. The upper index S is used to define symmetric parts for second-and fourth-order tensors,

A S := 1 2 (A + A T ), (1)
A S := 1 4 (A ijkl + A jikl + A ijlk + A jilk ) e i ⊗ e j ⊗ e k ⊗ e l . (2)
Components are given w.r.t. orthonormalized bases {e i } or {E I }, where

E 1 := e 1 ⊗ e 1 , E 4 := √ 2(e 2 ⊗ e 3 ) S , (3)
E 2 := e 2 ⊗ e 2 , E 5 := √ 2(e 3 ⊗ e 1 ) S , (4)
E 3 := e 3 ⊗ e 3 , E 6 := √ 2(e 1 ⊗ e 2 ) S . (5)
E 1...6 form a basis is for symmetric second order tensors. These can be further decomposed into their deviatoric and dilatoric parts A ′ and A • ,

A = A - tr(A) 3 I A ′ + tr(A) 3 I A • . ( 6
)
We make use of Einstein's summation convention with implicit summation from 1 to 3 over minuscule and 1 to 6 over majuscule indices that appear pairwise in a product. For example, the second order identity tensor and the identity on symmetric order tensors are given by

I = e i ⊗ e i = E 1 + E 2 + E 3 , I = 1 2 (e i ⊗ e j ⊗ e i ⊗ e j + e i ⊗ e j ⊗ e j ⊗ e i )=E I ⊗ E I . (7)
Setting. We consider laminates that consist of two isotropic elastic, isotropic von Mises-plastic layers without hardening in the small strain setting. The two materials are distinguished by the upper indices + and -.

Further, without loss of generality, we take the laminate normal n parallel to the e 3 -direction, see Fig. 1.

Basic equations

In the following subsections, we collect the basic equations that we need. While jump conditions (Sec. 3.1), mixture rule (Sec. 3.2) and von Mises plasticity (Sec. 3.3) are well known, the analyis of the stress concentration tensors in case of isotropic phases (Sec. 3.4) appears to be not documented elsewhere1 . The latter play an important role by relating the effective stresses to the partial stresses, which enter the individual constitutive laws. It turns out that the stress concentration tensors affect largely the effective plastic behavior.

Jump conditions at the interface

The well known kinematic and dynamic compatibility conditions at the interface are

E + -E -=(a ⊗ n) S , (8)
(T + -T -) • n = o, (9)
respectively, with the strains E ± and stresses T ± on either side of the laminate. One can decompose both E and T into its continuous and its discontinuous parts. With our choice e 3 = n, this decomposition is

E ± = E 1 E 1 + E 2 E 2 + E 6 E 6 Econt + E ± 3 E 3 + E ± 4 E 4 + E ± 5 E 5 E ± discont , (10)
T ± = T ± 1 E 1 + T ± 2 E 2 + T ± 6 E 6 T ± discont + T 3 E 3 + T 4 E 4 + T 5 E 5 Tcont . (11)
In terms of block matrices we have

E ± = ⎡ ⎢ ⎣ E ± 11 E ± 12 E ± 13 E ± 12 E ± 22 E ± 23 E ± 13 E ± 23 E ± 33 ⎤ ⎥ ⎦ e i ⊗ e j , T ± = ⎡ ⎢ ⎣ T ± 11 T ± 12 T ± 13 T ± 12 T ± 22 T ± 23 T ± 13 T ± 23 T ± 33 ⎤ ⎥ ⎦ e i ⊗ e j , (12)
where the shaded, continuous parts of the component matrices are common to E ± and T ± . Hence, we need not to destinguish the continuos parts on either side of the phases. The continuous and discontinuous parts of E ± and T ± are perpendicular to each other [START_REF] Laws | On interfacial discontinuities in elastic composites[END_REF], i.e.,

T ± discont ••E ± discont =0, (13)
T cont ••E cont =0. ( 14
)
One can define fourth order interface projection tensors that filter these parts from E ± and T ± (He and Z.-Q. [START_REF] He | Homogenization of layered elastoplastic composites: Theoretical results[END_REF][START_REF] Hill | Interfacial operators in the mechanics of composite media[END_REF].

Rules of mixture

Presuming homogeneity in each layer, the effective stresses and strains are related to the partial stresses and strains by

T = v -T -+ v + T + , (15)
E = v -E -+ v + E + , (16)
where v + +v -= 1 are the volume fractions. This holds actually only in absence of localization, which we presume throughout the text. The mathematical requirements for this to hold are certain convexity requirements on the April 21, 2016

(incremental) material potentials [START_REF] Schröder | Anisotropic polyconvex energies[END_REF]. For linear elasticity in the small strain setting this is not an issue, but for plasticity, some minimum hardening is needed to suppress localization [START_REF] Bartels | Quasi-Static Small-Strain Plasticity in the Limit of Vanishing Hardening and Its Numerical Approximation[END_REF].

Material law

The equations of isotropic, von Mises elastic plastic material behavior are well known, and therefore reproduced only in due brevity:

strain decomposition:

E = E e + E p (17)
elastic law:

T =3KE • e +2GE ′ e (18)
yield criterion:

φ(T)= 3 2 T ′ -σ F (19) loading condition: φ(T)=0, φ(T)| Ep=const. > 0 (20) associative flow rule: Ėp = λN, N = φ ′ (T)= 1 T ′ T ′ (21) consistency condition: φ(T) = 0 during plastic flow: λ = N •• Ė (22)
They can be found in textbooks on continuum mechanics, like [START_REF] Bertram | Solid Mechanics-Theory, Modeling, and Problems[END_REF]. The simple equation for the consistency parameter λ is due to the elastic isotropy and the lack of hardening. The flow stress σ F is the stress at which plastic deformations occur in a tensile test, K and G are the elastic compression and shear moduli, respectively. Here, we abbreviate the elastic law with the more general linear mapping

T = C ••E, (23)
C =3KP • +2GP ′ , (24)
with the isotropic projectors P • = 1 3 I ⊗ I, P ′ = I -P • , which extract the dilatoric and deviatoric parts of a second order tensor,

E • = P • ••E, (25)
E ′ = P ′ ••E. ( 26
)
The key ingredient that allows to extend the analytical homogenization from elasticity to plasticity is the similarity of the elastic law (Eq. 23) to the stress rate in case of plastic yielding. This has been recognised by El Omri et al. (2000) He and Z.-Q. Feng (2012)(Sec. 3.1 in [START_REF] He | Homogenization of layered elastoplastic composites: Theoretical results[END_REF]). Upon inserting the strain decomposition (Eq. 17) and then the flow rule (Eq. 21) into the rate form of the elastic law we obtain

Ṫ = C ••( Ė -Ėp ) (27) = C ••( Ė - 3 2σ F T ′ ⊗ T ′ •• Ė) (28) = C ••(I - 3 2σ F T ′ ⊗ T ′ ) •• Ė (29) = C(T ′ ) •• Ė. ( 30
)
The tangential modulus C depends on T, which eventually gives a system of ordinary differential equations, but the algebraic structure is the same as in case of total form elasticity and rate-form plasticity (see Sec. 4.2).

Stress concentration tensors

In linear elasticity, the jump conditions, mixture rules and elastic laws (Eqs 8,9,15,16,23) constitute a linear system for E when T is given, or vice versa, the coefficients of which being the effective stiffness tetrad C,

T = C ••E, (31)
C = v + C + + v -C --v + v -∆C ••Z ••∆C, (32)
∆C = C + -C -, (33)
Z = n ⊗ Z ⊗ n, (34)
Z =(v + A -+ v -A + ) -1 , (35)
A ± = n • C ± • n. (36)
Further, one can derive stress and strain concentration tensors2 L ± that relate the effective stresses and strains to the partial stresses and strains,

E ± =(I ∓ v ∓ Z S ••∆C) ••E, (37)
T ± = C ± ••(I ∓ v ∓ Z S ••∆C) ••C -1 L ± ••T. (38)
A derivation of these tensors is given in [START_REF] Glüge | The effective stiffness and stress concentrations of a multi-layer laminate[END_REF], but the concept is rather old. They have been derived and used by [START_REF] Sawicki | On application of effective moduli theory to layered soil[END_REF][START_REF] Sawicki | Yield condition for layered composites[END_REF]. Here, we want to discuss their properties in more detail. For isotropic materials and the laminate normal e 3 , the partial stresses in terms of the effective stresses and the stress concentration tensors take the form

T ± = L ± ••T, (39)
L ± = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ L ± 11 L ± 12 L ± 13 L ± 12 L ± 11 L ± 13 1 1 1 L ± 11 -L ± 12 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ E I ⊗ E J , (40)
L ± 11 = G ± G V 1+X ± , (41)
L ± 12 = G ± G V X ± , (42)
L ± 13 = -v ∓ (A ± ) 2 B ∓ -(A ∓ ) 2 B ± + B ± B ∓ (∆A -2∆B) 6N ± , (43)
L + 66 = L ± 11 -L ± 12 = G ± G V , (44)
X ± = v ∓ G ∓ (A ∓ B ± -A ± B ∓ ) 3N ± , (45)
N ± = v ∓ G ∓ K ∓ A ± + v ± G ± K ± A ∓ , (46)
where the abbreviations are

∆(•)=(•) + -(•) -, ( 47
) (•) V = v + (•) + + v -(•) -, (48)
A ± = K ± + 4 3 G ± , (49)
B ± = K ± - 2 3 G ± . ( 50
)
April 21, 2016

The L ± -tensors have been derived for purely elastic isotropic phases, i.e. they are valid as long as [START_REF] Glüge | The effective stiffness and stress concentrations of a multi-layer laminate[END_REF][START_REF] Sawicki | On application of effective moduli theory to layered soil[END_REF] . They take different values otherwise. Thus, the effective yield surface that is discussed later holds only for the plastically undeformed material.

E + p = E - p = O holds

Properties of the concentration tensors.

A key ingredient to the proper analysis of the effective flow criterion are the stress concentration tensors L ± . They satisfy by

T = v + T + + v -T -naturally I = v + L + + v -L -. (51)
Further, they are because of G ± > 0a n dK ± > 0 strictly positive definite. Four out of the six eigenvalues are found directly on the principle diagonal in Eq. ( 40). The remaining two eigenvalues satisfy

λ 1 λ 2 = G ± G V 2 (1 + 2X ± ), (52)
λ 1 + λ 2 = G ± G V (2 + 2X ± ). ( 53
)
Positivity of λ 1,2 is ensured by X ± > -1/2. Due to the positivity of K ± and G ± , also A ± > 0a n dN ± > 0 hold. Taking the limit K ± → 0 from above leads to the smallest possible value for X ± , and yields X ± →-1/2 from above. The eigenvalues of L ± are

λ ± 1,6 = G ± G V , (54)
λ ± 2 = G ± G V (1 + 2X ± ), (55)
λ 3,4,5 =1. (56)
Thus, there are only three different eigenvalues and eigenspaces. The numbering corresponds (up to λ 1,2 )t o the position on the principle diagonal in L ± in Eq. ( 40). The 2nd-order right eigentensors of

L ± are V 1 = 1 √ 2 (E 1 -E 2 ), ( 57
) V 2 = 1 √ 2 (E 1 + E 2 ), ( 58
) V ± 3 = κ ± -y ± κ ± +2y ± (E 1 + E 2 )+E 3 , (59)
V 4 = E 4 , (60)
V 5 = E 5 , (61)
V 6 = E 6 , (62)
κ ± =3K ± K ∓ (G ∓ -G ± ), ( 63
)
y ± =2G ± G ∓ (K ∓ -K ± ). ( 64
)
From the eigentensors, only V ± 3 depends on the elasticity constants. Further, the eigentensors do not depend on the volume fractions, unlike the eigenvalues. Note that V ± 3 is not normalized for the sake of simplicity. Also, all eigentensors are orthogonal to each other except V 2 and V ± 3 . We can therefore give easily the left eigentensors V Li which are the dual vectors to the right eigentensors, i.e. V Li ••V j = δ ij . Of these, only V L2,3 differ from the right eigentensors,

V ± L2 = 1 √ 2 (E 1 + E 2 ) - √ 2 κ ± -y ± κ ± +2y ± E 3 , (65)
V L3 = E 3 . ( 66
)
The right eigentensors have the following interpretations in terms of stresses:

Effective plastic properties of laminates made of isotropic elastic plastic materials April 21, 2016

V 2 to the eigenvalue λ ± 2 : isotropic stress inside the laminate plane (2D hydrostatic stress)

V 1 , V 6 with the eigenvalue G ± /G V : plane stress inside the laminate plane in absence of in-plane isotropic stress (2D deviatoric stress)

V ± 3 , V 4 , V 5 with the eigenvalue 1: shear stress parallel or perpendicular to the laminate plane (V 4 , V 5 ), superimposed with a specific ratio (that depends on κ ± and y ± ) of tension/compression along the laminate normal and isotropic stress inside the laminate plane (V ±

3 )

It is due to the latter mixing of modes (in-plane hydrostatic stress with deviatoric stress states) that the effective yield surface is pressure dependent, as will be seen in Sec. 4.1.

3.4.2 Special choices for K ± and G ±

In case of T being parallel to one of V 1,2,4,5,6 , all three stress tensors T, T + and T -are coaxial. For T parallel to V ± 3 , coaxiality between the three (which implies V + 3 = V - 3 ) occurs for special material combinations:

κ ± = y ± : This constraint can be written as 3∆G G + G -= 2∆K K + K -, (67)
or in terms of Young's modulus and Poisson's ratio

ν + E + = ν - E -. (68)
For such material combinations, V 3 becomes E 3 , i.e. an effective tension/compression along the laminate normal is coaxial to both stress concentrations. There are still three different eigenvalues, but the eigenspaces are orthogonal (hence L ± are symmetric): We observe a decoupling of in-plane hydrostatic tension/compression and tension/compression along the laminate normal.

y ± =0↔ K + = K -: For such material combinations, V 3 becomes I. There are still three different eigenvalues. An effective hydrostatic pressure is coaxial to both stress concentrations, i.e. it does not enter the local yield criteria, and therefore the effective yield criterion is independent of the effective hydrostatic pressure in this case. However, an effective deviatoric stress can affect the hydrostatic stresses in T ± . This is due to the non-orthogonality of the eigenbasis.

κ ± =0↔ G + = G -: For such material combinations, all eigenvalues except λ 2 become 1, and

V 3 becomes -1 2 E 1 -1 2 E 2 + E 3 .T h u s ,V 1,3,4,5
,6 form a deviator basis, i.e. all deviatoric stress states are conducted directly to the stress concentrations. Again, due to the non-orthogonality of the eigenbasis, a hydrostatic stress state in the effective stresses T affects the deviatoric parts of the stress concentration tensors.

4 The effective plastic properties 4.1 The effective yield limit φ

We proceed similar to [START_REF] Sawicki | Yield condition for layered composites[END_REF]. We have the elastic range limits

φ ± (T ± ) ≤ 0, ( 69
)
in which we can express T ± by the concentration tensors and the effective stresses (see Eq. 39). Plastic deformations occur if one of the two phases yields, thus we join φ ± to an effective elastic range limit by The minus is introduced to keep the usual orientation of φ such that φ ≤ 0 indicates the elastic range. A similar approach has been used, e.g., by [START_REF] Goda | 3D plastic collapse and brittle fracture surface models of trabecular bone from asymptotic homogenization method[END_REF], who approached the effective brittle fracture criterion of bone as the intersection of local fracture criteria of oriented trusses.

φ(T):=-φ + (T + )φ -(T -) (70) = -φ + (L + ••T)φ -(L -••T). (71
Properties of the effective yield limit φ.

Given elastic ranges φ ± that are convex in the stress space, the effective elastic range φ is as well convex in the stress space. This follows from the fact that the transformations L ± are linear and therefore affine mappings on the stress space, which inherit the convexity from φ ± . Considering φ ± (T) < 0 as two convex sets, the intersection of both gives a new convex set φ(T).

The effective yield surface φ(T) has an edge where the two smooth yield surfaces φ ± (T) coincide, i.e. we have a loss of smoothness compared to the partial yield surfaces.

The effective yield surface depends on the effective hydrostatic pressure unless K + = K -, see the second bullet point in Sec. 3.4.2.

If K + = K -, an effective hydrostatic pressure may cause yielding in both phases, but in opposite flow directions (see the analysis of the stress concentration tensors in Sec. 3.4). Therefore, the plastic deformations remain very small, and should be considered as eigenstrains that cancel out on average. This has also been noted by [START_REF] He | Homogenization of layered elastoplastic composites: Theoretical results[END_REF].

Illustrating example. Consider a laminate made of steel (+) and aluminum (-), with the material parameters

E + = 200GPa, ν + =0 .3, E -= 70GPa, ν -=0 .34, σ + F = 200Mpa, σ + F = 100MPa
, and v + = v -=0 .5. Due to the transverse isotropy around the laminate normal e 3 , we need only one angle 0 ≤ θ ≤ π/2 to represent all relevant uniaxial stress states,

T = σv ⊗ v, v =cosθe 1 +sinθe 3 , (72)
see Fig. 2. The elastic ranges are depicted in Fig. 3. Although the effective flow criterion appears not convex in the e 1 -e 3 -plane, it is convex in the stress space. If we superimpose this tensile test with a hydrostatic pressure -pI, the partial yield surfaces and hence the effective yield limit change considerably, as depicted in Fig. 4.

The effective flow behavior

As outlined in Sec. 3.3, we can go from the purely elastic to the elastic-plastic case by considering the stress and strain rates and the tangential stiffnesses

C ± , Ṫ± = C ± ••( Ė± -Ė± p ) (73) = C ± ••( Ė± - 3 2σ ± F T ± ′ ⊗ T ± ′ •• Ė± ) (74) = C ± ••(I - 3 2(σ ± F ) 2 T ± ′ ⊗ T ± ′ ) C ± •• Ė± . ( 75
)
Effective plastic properties of laminates made of isotropic elastic plastic materials Figure 3: Elastic range for uniaxial stress states for an example material combination. The vertical axis corresponds to the laminate normal, the horizontal axis is parallel to the laminate. For each point in the plane, the direction of tension is given by its direction from the origin, the distance from the origin is the stress magnitude σ. The two left plots indicate the change of the elastic range of the steel (σ F = 200MPa) and aluminum (σ F = 100MPa). The circular sections are the original isotropic von Mises yield limits φ ± (T ± ), the shaded regions are the yield limits φ ± (T). The right graph contains the range in which both phases deform elastically, which is the intersection of the shaded regions from the first two plots. The circular sections in the right plot are the arithmetic and the harmonic averages of the von Mises yield limits σ ± F . Unlike in elasticity, they do not necessarily correspond to energetic bounds. The approximate equivalence of the effective yield limit and the harmonic mean for θ =0a n dθ = π/2 is a coincidence, due to the specific elasticity parameters. The dots in the right plot represent the elastic range limit as found in representative volume element simulations, see Sec. 4.3.

With the yield conditions T ± ′ = 3/2σ ± F , the tensors I -N ± ⊗ N ± , N ± = T ± ′ / T ± ′ project all stresses into their parts perpendicular to the current yield surface normals φ ± ′ (T ± ). Moreover, the time derivatives of the jump conditions and mixture rules are

Ė+ -Ė-=(ȧ ⊗ n) S , ( 76
) ( Ṫ+ -Ṫ-) • n = o, (77)
Ṫ = v -Ṫ-+ v + Ṫ+ , (78)
Ė = v -Ė-+ v + Ė+ , (79)
with the auxillary quantity ȧ that does not appear in L ± or C ± . The latter equations imply the following assumptions:

The orientation of n is constant, as already implied by using the small strain setting.

The volume fractions are constant.

Homogeneity in each layer is preserved during plastic flow.

The last point is probably most critical, as it is known that a substantial amount of hardening is needed to suppress the formation of microstructures in plasticity, see, e.g., [START_REF] Carstensen | Non-convex potentials and microstructures in finite-strain plasticity[END_REF]. However, this is mostly an issue in finite plasticity and when boundary conditions allow for localization [START_REF] Glüge | Generalized boundary conditions on representative volume elements and their use in determining the effective material properties[END_REF].

Presuming the persistance of the plane interface together with isochoric plastic deformations already reduces possible localization modes to a sub-lamination by shearing parallel to the interface. This quasi-1D-localization can be excluded by assuming weak hardening. There is a large amount of literature that deals with complex microstructures due to strain localization, phase transitions and the like. A good entry point is [START_REF] Bhattacharya | Microstructure of Martensite: Why it Forms and how it Gives Rise to the Shapememory Effect[END_REF]. Accepting these fairly reasonable presumptions, we observe that the elastic solution is conducted to the plastic case, but all involved fourth order tensors depend now on T ± ′ through the tangent moduli C ± (T ± ).

Effective plastic properties of laminates made of isotropic elastic plastic materials Figure 4: Yield surface for the example material in tension superimposed with a hydrostatic pressure. The bold lines mark the effective yield limit that results from the intersection of the individual elastic ranges.

Effective plastic properties of laminates made of isotropic elastic plastic materials [START_REF] Taylor | [END_REF] We denote this distinction by a tilde. The former constant stress concentrations become a system of coupled ordinary differential equations with the partial yield limits as two algebraic conditions,

Ṫ± = L ± (T ± ) •• Ṫ, (80)
L ± = C ± ••(I ∓ v ∓ Z S ••∆ C) •• C -1 , (81)
C = v + C + + v -C --v + v -∆ C •• Z ••∆ C, (82)
∆ C = C + -C -, (83)
Z = n ⊗ Z ⊗ n, (84)
Z =(v + A -+ v -A + ) -1 , (85)
A ± = n • C ± • n, (86)
C ± = ⎧ ⎨ ⎩ C ± if φ ± (T ± ) < 0 C ± ••(I -T ± ′ ⊗ T ± ′ / T ± ′ 2 )i fφ ± (T ± ′ )=0and φ± (T ± ′ )| C ± =C ± > 0 . ( 87
)
Eqs ( 80) to (87) give 12 evolution equations for the partial stresses T ± , which are either trivial when both phases deform elastically, or somewhat complicated in case of plastic yielding. In the latter case, we are confronted with nonlinear coupled ODEs that are homogeneous of degree 0 in T ± , as one expects from rate-independent material behavior.

Ultimate loading

A closed form solution to the latter system of nonlinear ODEs is rather hard to find, even when Ṫ is constant. Nevertheless, we can examine equilibrium points without giving explicit solutions by examining the effective tangent modulus C. A characteristic point is the ultimate loading, which is attained when the strain increment does not result in a stress increment, i.e.

C(T

± ) •• Ė = O. ( 88
)
The latter equation implies that C(T ± ) has a zero eigenvalue, which is most easily identified by examining the determinant of C. Two cases can be distinguished:

Yielding takes place in both phases, i.e. both C ± depend on T ± , and in both phases φ ± (T ± ) = 0 holds. It turns out that in this case the determinant of C is always zero.

Yielding takes place in one phase, where we take the + phase, see Fig. 5. The denominator of the determinant of C is a quadratic form in the entire T + ′ . With T + ′ on the yield surface, the denominater cannot become zero. However, the numerator of the determinant of

C is num(det( C)=-96G -3 G + 2 v -G V (3G V K -K + (T + ′ 2 11 + T + ′ 2 12 + T + ′ 11 T + ′ 22 + T + ′ 2 22 )+ (89) K V G -G + (4T + ′ 2 12 +(T + ′ 11 -T + ′ 22 ) 2 )). ( 90
)
This is a negative semidefinite quadratic form in T + ′ 11 ,T + ′ 22 ,T + ′ 12 .T h u s , C is singular whenever

T + ′ 11 =0,T + ′ 22 =0,T + ′ 12 =0. ( 91
)
We know from the jump conditions for T ± that T + and T -have these components in common (Eq. 9).

Since we look at the deviator, only the shear stress components T + 13 and T + 23 are non-zero. This means just that C becomes singular when the deviatoric part of the partial stresses of the plastically deforming phase is a pure shear parallel to the laminate.

Effective plastic properties of laminates made of isotropic elastic plastic materials

σ + F T + T - σ - F < T - ∞ : simultaneous yielding σ - F > T - ∞ : asymptotic yielding T - ∞ E T ±
Figure 5: Sketch of the partial stress norms over effective strain norms in case of plastic yielding.

Conclusion.

Yielding will in general occur firstly in one phase, say the + phase. Then, two cases need to be distinguished:

1. T + evolves into a pure shear parallel to the laminate, while T -remains inside the elastic range φ -.

2. After some plastic flow in the + phase, the -phase also reaches its yield limit.

We refer to these cases of ultimate loading as asymptotic yielding in one phase and simultaneous yielding in both phases.

Ultimate loading in uniaxial tensile tests

We will use the results of the latter section to determine the ultimate loading in tensile tests, which are as before parametrized by the angle θ. The general problem is that we search points of the solution of the DAE system without considering the complete evolution of the system. Therefore, we are restricted to simple cases of monotonic loading with a constant loading direction. Further, we cannot use the kinematic compatibility condition, as it gives only three additional equations, but the jump of the plastic strains has 5 degrees of freedom. Therefore, the elastic strains and hence the elasticity parameters do not appear in the ultimate loading limit. In the remainder, examples use the same parameters as in Sec. 4.1.

Asymptotic yielding in one phase. The necessary condition is that one partial stress is on the partial yield limit (say φ + (T + ) = 0). Further, it was found that T + must be a pure shear parallel to the laminate (Eq. 91), such that only T + ′ 13 and T + ′ 23 can be nonzero. Additional symmetry arguments render this sufficient to determine the ultimate loading: Due to the effective tension direction being constant inside the e 1 -e 3 plane (see Fig. 2) and the transversal isotropy, we can presume T + ′ 23 = 0 throughout the test. We can determine T + 13 directly:

σ + F = 3/2 T + ′ (92) = 3/2 T + 13 (e 1 ⊗ e 3 + e 3 ⊗ e 1 ) (93) = √ 3T + 13 . ( 94
)
We obtain T + 13 = σ + F / √ 3 MPa, while all other components of T + are zero. With T = σv ⊗ v (see Eq. 72), we have T 13 = σ cos θ sin θ. Further, the mixture rule with v + = v -=1 /2 together with the compatibility of the Effective plastic properties of laminates made of isotropic elastic plastic materials stresses

T + 13 = T - 13 implies σ cos θ sin θ = σ + F / √ 3, (95)
which allows to write σ in terms of θ. It can be seen that this solution contributes considerably to the quality of the ultimate loading estimate (Fig. 6). It is easy to see that in case of compression the same magnitude for the ultimate loading is attained.

Figure 6: Elastic range and ultimate loading in tensile tests with the yield limits σ + F = 100MPa and σ - F = 200MPa. The vertical axis corresponds to the laminate normal, the horizontal axis is parallel to the laminate. For each point in the plane, the direction of tension is given by its direction from the origin, the distance from the origin is the stress magnitude σ. The dots are obtained through representative volume element simulations (see Sec. 4.3), the shaded region is the effective elastic range. The circular section is the arithmetic mean (1) of both yield limits σ ± F , the hyperbolic curve (3) is the ultimate loading in case of asymptotic yielding in one phase (Eq. 95) and the dotted curve (4) is ultimate loading in case of simultaneous yielding in both phases (Eq. 106). The arrow (2) indicates the intersection of the partial yield limits φ + (L + ••T)=φ -(L -••T), which lies on the ultimate loading limit, since it is equivalent to simultaneous yielding in both phases.

Simultaneous yielding in both phases. We now search for partial stresses T ± that are on the yield surface in both phases, are dynamically compatible, result in a uniaxial effective stress state, do not break the test symmetry.

In equations we have respectively

3 2 T + ′ ••T + ′ = σ + F 2 (96) 3 2 T -′ ••T -′ = σ - F 2 (97) T + 13 = T - 13 (98)
T + 23 = T -

23

(99)

T + 33 = T -

33

(100)

v + T + + v -T + = σv ⊗ v, v =cosθe 1 +sinθe 3 (101)
T - 12 = T + 12 =0. ( 102
)
April 21, 2016

By the latter equations, all partial stress components are determined as functions of σ and θ. Since in essence quadratic, there is more than one solution. For a given tension direction θ but varying σ we find infinitely many points T ± that satisfy the above constraints. For example, let θ =0,σ + F = 100MPa, σ - F = 200MPa and v + = v -=1/2. Then we find solutions for 50MPa ≤ σ ≤ 150MPa, namely

σ = 50MPa → T + = -100MPa e 1 ⊗ e 1 , T -= 200MPa e 1 ⊗ e 1 (103) σ = 150MPa → T + = 100MPa e 1 ⊗ e 1 , T -= 200MPa e 1 ⊗ e 1 . (104)
The first case corresponds to phases under eigenstresses that partly cancel out, the second case corresponds to partial stresses that sum up to the largest possible external loading. We are interested in the latter case, hence we search to maximize σ for each angle θ over the range of admissible partial stresses. The general solution for T ± of Eqs ( 96) to ( 102) is lengthy. It contains a square root term

(-3(σ - F 2 -σ + F 2 ) 2 + 15(σ - F 2 + σ + F 2 )σ 2 -30σ 4 +9σ 2 (σ - F 2 + σ + F 2 -2σ 2 )cos4θ. ( 105
)
The extremal cases are separating the real from the complex solutions, hence we can determine it by setting the argument of the latter root to zero and solve for σ. The resulting biquadratic equation has four solutions. These correspond to maximal and minimal external loading (maximal eigenstresses cancellation) in tension and compression. Being interested in the ultimate loading in tension we consider the largest value for σ, i.e. we take plus signs in front of the two involved square roots

σ = 1 2 σ - F 2 + σ + F 2 + -3σ - F 4 +26σ - F 2 σ + F 2 -3σ + F 4 +3(σ - F 2 + σ + F 2 ) 2 cos 4θ 5+3cos4θ . ( 106
)
As before, in a compression test (a minus in front of the outer square root and a plus in front of the inner square root), the ultimate loading has the same magnitude as in the tensile test, i.e. there is no strength differential effect.F o rθ =0a n dθ = π/2 we find the ultimate loading to be the arithmetic mean of the yield limits σ ± F . For angles around θ = π/4, the solution becomes complex. This is exactly when the case of asymptotic yielding needs to be considered, see Figs. 6 and8. Consequently, we divide the stress-strain plots into three families, given in Fig. 7.

To summarize, we can estimate the ultimate loading stress by considering

(1) the arithmetic mean of the partial yield stresses in tests that lead to coaxial partial stresses, like tension parallel or perpendicular to the laminate,

(2) stress states that lead to partial stresses such that both partial yield limits are met,

(3) the equilibrium points in case of yielding in only one phase (asymptotic yielding), (4) the maximal effective loading in case of yielding in both phases, see Fig. 6. ( 3) and ( 4) completely define the ultimate loading curve in tensile and compression tests, where the case (4) contains ( 1) and ( 2) as special cases. Naturally, the range of ( 3) is larger for phases with very different yield limits, while the converse is true for the range of (4).

To conclude, we determine the angle θ at which the curves (3) and (4) meet. From the preceding analysis we have for the ultimate loading in case of asymptotic yielding in one phase

T -= 1 v -(T -v + T + ) (107)
T + = σ + F / √ 3M P a( e 1 ⊗ e 3 + e 3 ⊗ e 1 ) (108)
T = σv ⊗ v, v =cosθe 1 +sinθe 3 (109) σ = σ + F / √ 3M P a/ sin θ/ cos θ. (110)
Effective plastic properties of laminates made of isotropic elastic plastic materials We just have to put T -on the yield surface to determine the angles θ that separate the range of asymptotic yielding in one phase from the ranges of eventual yielding in both phases,

3 2 T -′ ••T -′ =(σ - F ) 2 → θ = π/4 ± 0.2549. ( 111
)
This corresponds basically to considering the intersection of the two different yield limits. The corresponding section has been highlighted in Fig. 8, where the elongation to ultimate loading is depicted. In case of yielding in one phase, it is infinite, since the ultimate loading stress is approached asymptotically. In case of one phase yielding until the second phase yields, one has to track the evolution of the internal variables E ± .

Representative volume element simulations

To check the analytical results, representative volume element simulations have been devised. A cubical section of the laminate with one interface dividing the + and the -phase has been subjected to periodic boundary conditions and monotonic tensile tests, see Fig. 9. We used a regular hexahedral mesh with quadratic shape functions. To capture potential localizations or pattern formation, we used fine meshes, but did not observe such events. Thus, the presumptions made in Sec. 4.2 appear feasible. The stress-strain curves (Fig. 7) have been extracted, and on each the effective stresses and strains at the yield point and the ultimate loading have been determined (Figs. 3,6,8).

Comparison to other works

Pruchnicki (1998a,b). The main differences to the works of Pruchnicki are that he carried out the scale transition by the asymptotic expansion method, where laminates are treated as a special case, and he considers tangential slip (Pruchnicki, 1998a) and boundary layers [START_REF] Pruchnicki | Homogenized nonlinear constitutive law using Fourier series expansion[END_REF].

Conversely, we (and El Omri et al. (2000) and He and Z.-Q. Feng (2012)) consider perfectly bonded interfaces. Consequently, Pruchnickis results depend on additional material parameters that do not appear here. Nevertheless, the following common features are observed: Symmetry of the ultimate loading limit w. r. t. the 45 tensile direction, while this symmetry is not displayed in the yield limit, compare Fig. 6 here and Fig. 11 in [START_REF] Pruchnicki | Homogenized nonlinear constitutive law using Fourier series expansion[END_REF].

Asymptotic yielding in tension close to the 45 tensile tests and finite ultimate elongation due to yielding in both phases when tension is applied approximately parallel or perpendicular to the laminate, compare

Effective plastic properties of laminates made of isotropic elastic plastic materials April 21, 2016 Effective plastic properties of laminates made of isotropic elastic plastic materials Fig. 7 here and Fig. 10 in [START_REF] Pruchnicki | Homogenized nonlinear constitutive law using Fourier series expansion[END_REF], which illustrates also the anisotropic hardening that has been diagnosed here and in Pruchnicki's works.

El Omri et al. (2000) and [START_REF] He | Homogenization of layered elastoplastic composites: Theoretical results[END_REF]. He and Z.-Q. Feng (2012) derived the general ordinary differential equations that govern the evolution of the partial plastic variables, and thus the effective plastic response, as we did (Eqs (80) to ( 87)). They show that the effective material law can be cast in the framework of generalized standard materials with a finite number of internal variables, which they exploit in their derivation. However, it appears that they miss to point out that the requirement of homogeneity of the phases during plastic flow is only given in absence of softening. Further differences are that we use a different method to derive the governing differential equations and that He and Z.-Q. Feng (2012) derive the effective flow rule for the general anisotropic case, while we focus here on examining the system of evolution equations, which finally leads to analytical expressions for the effective ultimate loading in uniaxial tests in maninates of isotropic, von Mises plastic phases.

The latter result is in line with deBotton and Ponte-Castañeda (1992) and Ponte-Castañeda and deBotton (1992) and He and Z.-Q. Feng (2012): deBotton and Ponte-Castañeda (1992) and Ponte-Castañeda and deBotton (1992) found that the ultimate loading surface cannot be given as a single quadratic form. He and Z.-Q. Feng (2012) state that the ultimate loading surface is a piecewise quadratic form. Our examination of laminates of isotropic von Mises plastic phases in monotonic tensile tests, for which we gave the ultimate loading explicitely (Eqs 95,106), confirms these findings. Additionally, we have discussed the source of this behavior, which is the case distinction into asymptotic yielding in one phase and simultaneous yielding in both phases. This arises naturally when looking for states for which the effective tangent modulus has a zero eigenvalue. Further, we could show that in these uniaxial monotonic tests, there is no strength differential effect on the macroscale, neither for the yield nor the ultimate loading limit. The work of El Omri et al. ( 2000) is somewhat similar to [START_REF] He | Homogenization of layered elastoplastic composites: Theoretical results[END_REF] and the present work. They consider tensile and shear tests, where the tensile tests show similar stress-strain curves as determined by us, namely anisotropic hardening and kinks or asymptotic behavior (Fig. 5 in El Omri et al. (2000)). However, they restrict their examples to phases with equal elasticities.

Summary

We have examined the plastic properties of laminates of isotropic, elastic plastic layers. It has been found that most of the desirable properties of the von Mises yield limit are lost in the homogenization. The effective yield limit is given as the intersection of the partial yield limits, which is therefore non-smooth. Due to the properties of the stress concentration tensors, which map the effective stress state to the partial stresses, the effective yield limit depends on the effective hydrostatic pressure and is anisotropic. However, it remains convex in the effective stresses. The yield surface has been analyzed for the virgin material. The results have been confirmed numerically by representative volume element simulations.

Further, with some mild restrictions on the plastic deformations (basically the persistance of the laminate structure and small strains), the homogenization of the rate-form plastic equations is similar to the homogenization of the total-form elastic equations. Instead of constant stress concentration tensors one has then stress rate concentration tensors that depend on the partial stresses, i.e. one obtains a set of nonlinear ordinary differential equations for the partial stresses, which characterize the internal state of the microstructure. For example, one observes effective hardening behavior while only one phase yields plastically, unlike in pure von Mises material without hardening. The ODE system is homogeneous of degree zero in the partial stresses. A closed form solution can hardly be attained. Nevertheless, it is possible to extract the ultimate loading (i.e. the plastic reserve) in tensile tests as a function of the tension direction. Two fundamental cases have to be distinguished, namely Effective plastic properties of laminates made of isotropic elastic plastic materials April 21, 2016 yielding in one phase until a shear stress parallel to the laminate is approached asymptotically in the yielding phase, while the other phase remains elastic, and yielding in one phase until the second phase starts yielding.

For both cases, one can give closed form expressions for the effective ultimate loading. It contains special cases, namely the arithmetic mean of the yield stresses in tension parallel and perpendicular to the laminate and the intersection of the yield surface, which confirms the plausibility of the result. The analytical findings match exactly to results from representative volume element simulations. Further, the effective ultimate loading does not depend on the elasticity parameters, unlike the effective yield limit. For the monotonic uniaxial tests, we can exclude a strength differential effect for the yield and the ultimate loading stress.

In case of a complex loading path or when hardening is included, no closed form expressions can be given. Still, the time integration of the partial stresses and possibly some hardening variables is much more convenient than considering representative volume elements [START_REF] Nezamabadi | Compressive failure of composites: A computational homogenization approach[END_REF].

Figure 1 :

 1 Figure 1: We consider laminates made of two isotropic von Mises elastic plastic materials.

Figure 2 :

 2 Figure 2: Sketch of the parametrization of a uniaxial stress state in a transversely isotropic laminate by one angle θ and the absolute stress σ in direction of v.

Figure 7 :

 7 Figure 7: Stress-strain curves in the plastic range for uniaxial tensile tests in direction of θ, with the discretization θ = nπ/40, with an integer n. The left plot covers n =0...6, the center plot covers n =7...13, the right one covers n =14...20. The arrows connecting the points of ultimate loading indicate the parametrization with n.

Figure 8 :Figure 9 :

 89 Figure 8: Elongation to yielding (inner curve) and elongation to ultimate loading (outer curve) for uniaxial tensile tests in direction of θ, with the discretization θ = nπ/40, with an integer n. The shaded region indicates the range of θ in which the ultimate loading is approached asymptotically. It is larger when both phases have markedly different yield limits. The gray circle sections serve as markings. The dashed lines indicate the transition to asymptotic behavior.

Similar but more involved analysis exist for cracks and bounded inclusions, see[START_REF] Barthélémy | Compliance and Hill polarization tensor of a crack in an anisotropic matrix[END_REF] and[START_REF] Nazarov | Elasticity polarization tensor, surface enthalpy, and Eshelby theorem[END_REF]. In the present laminate case the analysis is particularly simple, as the phases remain homogeneous, see Sec. 4.2.Effective plastic properties of laminates made of isotropic elastic plastic materials

Effective plastic properties of laminates made of isotropic elastic plastic materials

The inverse concentration tensors are also referred to as polarization tensors[START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF].Effective plastic properties of laminates made of isotropic elastic plastic materials


Isabelle Andriot
email: isabelle.andriot@inrae.fr

Karine Gourrat

Rémy Reynaud

Sylvie Cordelle

Caroline Peltier

Olivier Berdeaux

Géraldine Lucchi

Characterization of key aroma compounds in Burgundy truffle

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Truffles are a fungus of the genus Tuber. Some of them have an important economic value due to their gastronomic qualities appreciated in "grande cuisine". Périgord truffle (Tuber melanosporum or Black truffle) and White Alba truffle (Tuber magnatum pico) are actually the noblest ones. While these two species are wellvalued, Burgundy truffle (Tuber uncinatum) is less well-characterized in its production area and truffle producers encounter some difficulties to sell their harvest at its fair value. The aroma compounds in truffles are investigated with sensory evaluation and two physico-chemical methods. The figures 1 and 2 showed that:

French

Sensory analysis

-The Burgundy truffles (summer) are weakly odorant, -The Lot truffles are characterised by forest odours and sweet odours, -The Burgundy truffles (autumn) are characterised by negative odours and spicy odours. The results presented in figure 3 and4 show that it is possible to discriminate the truffles according to places and seasons (1,2,3) . However, the analyses of frozen mix truffles as a control have shown a sample dispersion according to time from June to November (data not shown).

PTR-MS analysis

Dissimilarity

We can suspect truffle conservation problems and / or an instrumental bias. The results obtained by GC-MS confirm the results obtained by PTR-MS and sensory analyses.

GC-MS analysis

The truffles are separated according to harvest places and seasons (Figure 5). A volcano plot with a foldchange higher than 3 and p-value<10 -5 allows compounds to be discriminated depending on their peak area in the TIC of the chromatogram. The results are presented in Table 1.

Pentane-2,3-dione and butan-2-one are characteristic of the truffle freshness (4) .

( The same results are found with the PTR-MS for the truffles of october and november.

Sensory analyses, PTR-MS, GC-MS are, three of them, good methods to differentiate truffles according to harvest places and the seasons.

The Burgundy truffles harvested in summer are less odorant than Burgundy truffles harvested in autumn. These results will have to be combined to microbiota and genetic analyses. This work will bring a scientific contribution to the creation of an IGP (Indication Géographique Protégée) request.



  16 panellists were trained with 8 sessions  Truffles were evaluated only by olfaction Peridium Gleba Classical descriptive profile (QDA)

Figure 1 :Figure 2 :

 12 Figure 1: Canonic variables analysis of truffles (QDA) Truffles of Lot (south of France , summer) Burgundy Truffles (summer) / Burgundy Truffles (autumn)

Figure 3 :

 3 Figure 3: PCA of truffles from different harvests and places Figure 4: Dendogram of truffles (the same truffles as PCA)

Figure 5 :

 5 Figure 5: PCA of truffles from different places in Burgundy, the same truffles analysed by sensory analyses and PTR-MS analyses. Unfortunately, the Lot truffles are not enough for the GC-MS

  ). Vita, F., et al. (2015). ." Sci Rep 5(12629). (2). Gioacchini Anna, M., et al. (2005). ." Rapid Communications in Mass Spectrometry 19(17): 2365-2370. (3). March, R. E., et al. (2006). International Journal of Mass Spectrometry 249-250: 60-67. (4). Vahdatzadeh M, et al. (2019). Food Microbiology 84:103251

Table 1 : Compounds over-expressed, and the corresponding truffles

 1

This study is part of the research project BIJOU, supported by FEADER and Bourgogne Franche-Comté Region.