Jaume Garcia
email: jaume.garcia@upf.edu

Josep Maria Raya
email: josep.raya@upf.edu

C30 -General < C3 -Econometric Methods: Multiple/Simultaneous Equation Models < C -Mathematical and Quantitative Methods, R21 -Housing Demand < R2 -Household Analysis < R -Urban, Rural

Keywords: housing demand, price and income elasticities, market segmentation, hedonic prices R21, C30 hedonic models, elasticities, housing demand

In this paper we estimate price and income elasticities of demand for housing characteristics using the information of properties appraised in the city of Barcelona (1998Barcelona ( -2001)). We employ a two-stage approach, estimating hedonic price equations for different districts in order to subsequently estimate the corresponding demand equations. The results allow us to analyse the complementarity or substitutability relationships among the characteristics analysed, and also to catalogue these characteristics as goods. By knowing the price, cross and income elasticities of housing characteristics we can make policy recommendations about the type of housing units which are the most desirable to be subsidized.

Introduction

Public-sector budget measures for housing can be divided into indirect aid and direct aid. Direct aid appears in the successive housing plans and takes the form of interest rate subsidies, access to qualified loans and personal grants. Indirect aid is that derived from the tax treatment given to the usual home in the Spanish tax system, and which affects the principle of tax neutrality i .

In Spain 87% of budget aid for housing is allocated to indirect aid to subsidise home purchases, whereas in the European Union this percentage ranges from 10% to 25%, and only in Sweden does indirect aid exceed 50%. This type of public-sector intervention, aimed more at boosting the construction industry than at meeting housing needs, combined with the soaring inflation that has taken place in the housing market in Spain in recent years, has rendered access to housing difficult for certain social groups and has cast doubt on its social function.

A commitment has now been made to shift this imbalance in the publicsector budget towards a greater relative presence of direct aid policies, including, for example, policies for subsidised housing and social renting.

Bearing this in mind, if we understand housing as a basket of different attributes that contribute to the provision of housing services, The possibility of distinguishing between which attributes are seen as necessities and which as luxuries, and knowing the substitutability and complementarity relationships among the various attributes, will allow a more thorough understanding of the housing needs of the population, thus facilitating specific direct aid policies.

In this way, using data on properties appraised in the city of Barcelona, we set out to estimate the price and income elasticities of demand for a series of basic housing characteristics (quantity, quality and location).

For this study we had access to a very large sample, not only in terms of the number of properties included but also as regards the time span, as the appraisals correspond to the period 1998-2001. The main goal of the paper is, therefore, to make policy recommendations about the type of housing units which are the most desirable to be subsidized, based on the information of price, cross and income elasticities of housing characteristics.

The paper is structured as follows. Section 2 sets forth the theoretical framework for the hedonic price model and for obtaining the demand equations. Section 3 describes the empirical model. Section 4 accounts for the data and the variables used in the estimates, while we discuss the

Theoretical framework

The economic literature on hedonic prices arose in the context of the car market. This was the framework for the classical work by [START_REF] Griliches | Price indexes and quality change: studies in new methods of measurement[END_REF], who popularised these models. In this work, he estimated car prices after controlling for the characteristics that affected their price, such as horsepower or fuel consumption. It is worth noting, however, that he did not "invent" this type of model, or even coin the term "hedonic". This must be attributed to Andrew Court, with his work in the early 1940s (Court, 1941). In fact, these works are rather topical at present, as the functional form of these models is under review. ii Once the technique had been popularised in the 1950s [START_REF] Tinbergen | Some remarks on the distribution of labour incomes[END_REF], it took over a decade to provide it with a theoretical foundation. In this case the classical work was that of [START_REF] Rosen | Hedonic prices and implicit markets: product differentiation in pure competition[END_REF] iii . In it he showed how heterogeneous products are composed of various characteristics and how the marginal price implicit in these characteristics can be known by estimating a model (hedonic price model) that accounts for the price of a product in terms of its characteristics. Clearly, housing is a good that fits perfectly into the framework of hedonic price models.

One of the applications of the hedonic price models is the estimation of constant quality price indexes. Desirable information to calculate these indexes would be the price for a sample of repeat sales belonging to a households samples with identical characteristics, allow to the variation in prices of homogenous houses to be obtained. However, given the practical difficulty to get this data, hedonic price models appear as an alternative to the calculation of house price indexes. Mills and Simenauer iv (1996) estimate hedonic price models that allow the comparison of the housing price indexes, over the seven years of their sample of regions. The results show that more than half of the growth in housing prices during this period comes from improvements in the quality. Anyway, Bover and Velilla (2001) addressed the estimation of price indexes with a sample of multiunits of housing (a set of houses that share most of the features) that makes it unnecessary to include in the estimation most of the housing characteristics.

Housing is a unique good, i.e., each housing unit is different in some way from all others. Among its characteristics we find: need, durability, spatial immovability, indivisibility, the practical absence of significant futures and insurance markets, information asymmetries, the importance of the rental market as a result of the high cost of purchasing, and above all multidimensional heterogeneity, which fully justifies the use of the hedonic technique.

Following [START_REF] Rosen | Hedonic prices and implicit markets: product differentiation in pure competition[END_REF], and undertaking the characterisation that he makes in his work of the market for heterogeneous goods, the problem is reduced on the demand side to consumer utility maximisation subject to an exogenous budget constraint. The supply of attributes is important to The consumer of a heterogeneous good such as housing, with a vector of socioeconomic characteristics α that characterises him or her, obtains utility from a vector of n objectively measurable characteristics v ) ,...

( 1 n z z z =
and from a non-housing numerary good, x. Therefore, the utility function:

) , , ,... ( 1 α x z z U U n = (1)
is maximised subject to the budget constraint:

) z ,... z ( p x y n 1 + = (2)
where y is income and

) ,... ( ) ( 1 n z z p z p =
is the function that relates prices to the attributes revealed through the implicit markets for the products.

The consumers are price takers, taking the functional form ) (z p as given and having full information on the parameters of this function. It is important to note at this point that the hedonic price function is not linear, i.e., the consumer can affect the price by varying the quantity consumed of this and/or other characteristics.

Utility maximisation subject to the budget constraint gives rise to the vector of n demand functions for the characteristics:

) ; y ; p ..., ,......... p ( f z n 1 z z i α = ( 3
)
where pz is the vector of the first derivatives of the hedonic price function (4)

If we focus on the demand functions, [START_REF] Rosen | Hedonic prices and implicit markets: product differentiation in pure competition[END_REF] proposes a two-stage estimation method:

• The first stage comprises the hedonic estimation, i.e., that which relates the prices of housing to its characteristics, in order to subsequently compute by derivation a set of implicit marginal prices, the hedonic prices, i z p .

• The second stage serves to estimate the demand equations for each of the housing characteristics using the hedonic price estimates reached in the previous stage. Additionally, a vector of the household's sociodemographic variables α (observed heterogeneity) is usually included, together with its income level y.

The ultimate objective of this second stage is to obtain the price and income elasticities of the various housing characteristics. One problem with this model that is addressed in the literature economic is the identification between the price function and the demand function. In the absence of additional constraints, the second-stage estimate may merely reproduce the information already provided by the first stage. The problem, analysed by Brown and Rosen (1982) and [START_REF] Mcconnell | Identification of preference parameters in hedonic models: consumer demands with nonlinear budgets[END_REF] (1982) propose specifying demand equations of order m for the characteristics, and at the same time a hedonic price equation of order m+2 or higher for the characteristics, such that the implicit marginal price equations will be of order m+1 or higher, solving the identification problem mentioned earlier. However, Ekeland et al (2002 and2004) consider that the demand parameters are always identified in single market data if the marginal price function is nonlinear, which they call a "generic property of equilibrium in the hedonic model". Semiparametric estimation (as in Bin, 2005) or instrumental variables estimation is proposed. Finally, [START_REF] Rouwendal | The hedonic price function as an envelope of bid functions; an exercise in Applied Economic Theory[END_REF] introduces an alternative to the two stage procedures proposed by [START_REF] Rosen | Hedonic prices and implicit markets: product differentiation in pure competition[END_REF]. The author models the hedonic price function explicitly as the envelope of a family of bid-rent functions.

Empirical model

On the basis of the considerations discussed in the section above, we will divide the methodological approach into two parts: the modelling of the hedonic price regression and the modelling of the demand equation.

Following the proposal of Brown and Rosen (1984) As regards the modelling of the effect of the characteristics and the area on price, the hedonic price function specified is linear, and hence so is the budget constraint; consequently the demand equations will be well defined, and additional nonlinearities as a result of the nonlinearity of the budget constraint will be avoided vi . Furthermore, the linear specification does not suffer from endogeneity problems in the implicit marginal prices that would result from using another specification, given that these prices depend on the quantity consumed of each characteristic vii .

The econometric model to estimate corresponding to the hedonic price equation would have the following form: In turn, the second stage consists in estimating a system of demand equations for housing characteristics. Following [START_REF] Parsons | An almost ideal demand system for housing attributes[END_REF] and Bilbao (2001), we chose a system of AIDM-type Marshallian demand equations expressed as a proportion of the budget:

i j , i j j , L k , i k k , H i u L p H p p + + = ∑ ∑ (5)
) P / x log( p log w k j j kj k k β γ α + + = ∑ (6)
where k w is the proportion of the total expenditure that is allocated to ), that are homogeneous of degree 0 in prices, and that satisfy the symmetry conditions, these equations being linear.

characteristic k ( x H p w k k k / = , k H being the quantity demanded of characteristic k), kj k γ α ,

Data and variables

The sample used in this paper comprises 9,297 observations of housing in the city of Barcelona corresponding to the period 1998-2001. As such, the sample has a certain temporal dimension but on no account can be taken as panel data because the properties included in each yearly subsample are different.

The data, duly refined, come from an annual representative sample taken by the appraisal firm TINSA. They contain a great variety of physical characteristics of the housing, together with information about the area of the city to which each property belongs. The prices are therefore appraisal prices viii . Specifically, they are appraisals conducted with the same criterion as official appraisals according to Order ECO/805/2003 (market value) to be used for certain financial purposes (basically, the valuation of the house as important information in order to grant a mortgage loan).

Results

The first stage consists in estimating the function that relates prices of housing to its characteristics. As mentioned earlier, the functional form will be linear and the variable to be explained will be the total value of the housing. Furthermore, in order to avoid arriving at negative prices and also to simplify the results, the dummy variables that refer to location 13 have been replaced by the variable "educational level". This is justified because in [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF], by estimating a hedonic price model (nonlinear at levels) with the same database, it is shown how the variability of the coefficients of the dummy variables referring to location (statistical area) is explained almost solely by the variable "educational level" (correlation close to 0.9). This variable is measured as the average number of years spent in formal education by the inhabitants of any given area. According to the theories of human capital and the available empirical evidence, this type of variable approximates the average income of the inhabitants of the area. As indicated in [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF], this high correlation showing greater willingness to pay in areas with a higher educational level (or higher income) can be explained by the fact that these variables (income or educational level) may be correlated with other types of factors that individuals value when choosing the area in which they wish to live.

The coefficients accompanying each characteristic are marginal prices (which coincide with the average prices of each characteristic). As was made clear earlier, although we only have data for the city of Barcelona, we have sufficient observations for almost all the districts (geographical areas with sufficient internal heterogeneity) to encounter no problem in estimating hedonic price functions by districts, and thus in obtaining the necessary variability in marginal prices to be able to identify their effect on the estimation of the system of demand equations. This segmentation by districts is the first one to be used, each of the equations being The economic literature has focused, by means of capitalization, on the willingness to pay of some neighbourhood amenities such as parks, schools or undergrounds. In this framework, we can emphasize the contributions of Cheshire and Sheppard (1995), [START_REF] Bell | The impact of local sales tax on the value of owner-occupied housing[END_REF], Bilbao-Terol (2000), Bogart and Cromwell (2000), [START_REF] Haider | Effects of transportation Infrastructure and location on residential real estate values[END_REF], Gibbons and Machin (2001), Downes and Zabel (2002) [START_REF] Tse | Estimating neighbourhood effects in house prices: towards a new hedonic model approach[END_REF], [START_REF] Anderson | Open space, residential property values, and spatial context[END_REF], Cheshire and Sheppard (2004). However, the main aim of this paper is not to explain variability in housing prices due to the location but to estimate the price and income elasticities of demand for a series of basic housing characteristics (quantity, quality and location), so we decide to replace the amenities variables for the locational dummies. In fact, the elasticities parameters estimated from a model with these amenities as a explanatory varaibles remain identical and can be obtained by request to the authors. To see the effect of many amenities to the house price with the same dataset you can see [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF] The explanatory power of the models estimated is similar to, and even seemingly greater than, that of the models presented in [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF]. In fact, the adjusted R 2 s stand at around 85%. This is due, firstly, to the fact that the dependent variable is the total price ix of the property rather than the price per square metre, therefore the variability of the dependent variable changes (is greater) and this influences the R 2 value, 15 and secondly to the fact that in [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF] the dependent variable is the log of the price per square metre, a transformation that reduces the variability of the dependent variable, which also influences the R 2 value.

As far as the effects of the different explanatory variables are concerned, the patterns fit those presented in [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF], albeit with differences of magnitude and significance among segments; the only exception is the effect corresponding to the fact of having or not having heating, which has an unexpected sign. This is a consequence of the different specification as regards the functional form (linear in this case and pseudo-logarithmic in [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF] x , and also the fact of approximating the effect of location through the average years of formal education in each statistical area, whereas in [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF] dummy variables are used for 89 groupings of the 248 statistical areas existing in the city of Barcelona. Nevertheless, this linear specification is preferred, as it avoids the above-mentioned endogeneity problems of marginal prices in the system of characteristic demand equations.

Once the prices implicit in each housing characteristic have been obtained, the demands for the characteristics are estimated. As reflected in equation ( 7), the dependent variables in the AIDM are defined as the proportion of expenditure that each household sets aside for each characteristic and are grouped into the following three categories. Therefore, a demand system is estimated with three equations:

• Proportion of expenditure on quantity of housing (w1) is defined as the number of square metres of a property, by the price per square metre • Proportion of expenditure on quality of housing (w2) is defined using the method applied in [START_REF] King | The demand for housing: a Lancastrian approach[END_REF] and Erekson et al (1979) As explanatory variables of the demand equations we should include price indices for the various housing characteristics (pj), and real income or expenditure (x/P), with the aim of calculating the corresponding price and income elasticities. To this effect, the explanatory variables included in the equations, all of them expressed in logs, are:

• The hedonic prices of floor area (quantity price index) and educational level (location price index) calculated in the previous stage.

• The price index for housing quality (quality price index) expressed in natural logarithms. In order to define this index we take a standard property, which in our case is the average housing unit for the city of Barcelona in the period 1998-2001. The characteristics of this average housing unit are calculated by taking an average of the housing characteristics weighted by its weight in the sample. Lastly, the quality price index for each district or year is the sum of the products of the hedonic prices of each of the characteristics that comprise the quality, by the level fixed as standard xiii .

• The total real expenditure, expressed as the quotient between the total budget and Stone's price index as defined above.

The model is estimated using a Seemingly Unrelated Equations (SURE) procedure, which holds two advantages over OLS estimation: we gain efficiency by considering the contemporary correlation between the errors of several equations and we can contrast the symmetry hypotheses of the parameters. Given that the sum of the proportions of expenditure on the Table 1 presents the estimates of the demand equations, imposing homogeneity and symmetry conditions for the two models considered. As is standard practice in studies estimating systems of demand equations, null hypotheses associated with the homogeneity and symmetry conditions are rejected at the usual significance levels. The goodness of fit is high, and all the variables relating to prices and expenditure are individually significant at 1% for all three specifications. The cross [TABLE 1] Table 2 reports the income elasticities,

1 +         = ∧ ∧ i i r w β ε
, and price elasticities,

compensated xiv , 1 * -         = ∧ ∧ i ii p w γ ε
, and uncompensated xv ,

i i p p w - - = ∧ ∧ ∧ β ε ε * , of
housing demand for each of the three characteristics considered and for each of the two models.

The estimated xvi income elasticities classify quantity as a luxury good (elasticity greater than 1), whereas quality and location are considered as necessary goods (although the latter shows an income elasticity that is closer to unity). This may be reflecting a usual result, both in the descriptive information and in the more detailed estimates of hedonic price models, insofar as the price per additional square metre is higher for small properties than for larger ones xvii . It should also be stressed that the magnitude of the estimated income elasticities is very similar for the two specifications used (with and without adding demographic variables). Thus, location presents a more inelastic demand than the other two characteristics considered, while quality presents a compensated price elasticity of around -1, and in the case of floor area (quantity) demand is clearly inelastic, although less so than in the case of location.

Compensated price elasticities are in all cases greater in absolute value than the corresponding uncompensated ones, as a consequence of the income effect they reflect, and the demands relating to quality and quantity can be considered perfectly elastic.

[TABLE 2]

As indicated earlier, we estimated a version of the model with segmentation by districts that includes as explanatory factors information on the sociodemographic characteristics of the districts of the city of Barcelona. The results of the model are also presented in Table 1, and the corresponding elasticities in Table 2.

It should be noted that the proportion of expenditure on quantity (floor area) increases with the proportion of young people in the district, whereas it decreases with the proportions corresponding to the other characteristics considered. Furthermore, the proportion of expenditure on location increases with the proportion of permanent employees and To sum up, the classification of the housing characteristics based on its income elasticity shows us quantity as a luxury good, whereas quality is considered as a necessary good and location a good with an elasticity close to 1. That is, an increase in the income of the citizen results in a less than proportional increase in the quantity demanded of the characteristic "quality", more than proportional increase in the quantity demanded of the characteristic "quantity" and almost proportional in the quantity demanded of the characteristic "location". In terms of policy making, given the former demand patterns obtained, the housing needs of the population are small subsidised housing units keeping certain minimum standards of quality and location. However, if the key segment of this policy is young people the result should be matised, because young people show greater preference in quantity.

Moreover, the compensated price elasticities indicate inelastic demands for all the characteristics, ie, an increase in the price of one of the characteristics results in a less than proportional decrease in the quantity demanded of that characteristic, especially in the case of location. In this work we have obtained demand equations for housing characteristics grouped into three categories: quantity (floor area), quality (recorded through variables relating to the state of repair of the property, renovation work done and other factors) and location, with the main aim of getting to know society's housing needs in order to make policy recommendations about the type of housing units which are the most desirable to be subsidized. In this context, we were extremely interested in knowing the sensitivity of the quantity demanded of a characteristic to variation in its price (price elasticity), variation in the price of another characteristic (cross elasticity) or variations in income (income elasticity).

To this end we have estimated hedonic equations for each of the districts of the city of Barcelona and for each of the four years corresponding to the transactions considered. The hedonic equations have yielded the evaluations (implicit marginal prices) of each of the housing characteristics in each district, together with the differences in the appraisal of a property. Thus, two households with the same income living in the same town and but different districts will have to make a different economic effort to purchase a home with similar In the second stage, we have estimated demand equations for the three characteristics mentioned for the district, which enables us to complete the specification with the inclusion of sociodemographic variables.

Furthermore, it has been observed that the compensated price elasticities indicate inelastic demands. Thus, keeping expenditure and the rest of the prices constant, an increase in the price of one of the characteristics results in a less than proportional decrease in the quantity demanded of that characteristic. In other words, in economic policy terms, in the face of increases in the price of all the housing characteristics, citizens will opt to distribute their budget in such a way as to cause minimum losses in terms of location with respect to their initial desire, at the expense of greater losses regarding the size and quality of their home. Obviously, if citizens can keep their utility constant, an increase in the price of floor area will also lead to a less than proportional decrease in the quantity demanded of floor area. This, given the substitutability and complementarity relationships obtained, will also lead to a decrease in the quantity demanded of location and will be offset by an increase in the quantity demanded of housing quality.

Both the patterns of substitutability and complementarity and the income and price elasticities (compensated and uncompensated alike) yield similar results to those obtained in other similar studies.

Demand models of differentiated goods, such as the one presented here for housing, are frequently used as instruments for the analysis of public policies. In this case, in view of the comments made above, a policy of small subsidised housing units is justified, although with certain minimum standards of quality and in no event in marginal areas, given the demand patterns obtained. In particular, with regard to differences between population groups, young people and women seem to focus their demands on quantity and quality, probably because of the impossibility of obtaining a good location. In contrast, permanent employees and business owners focus their demands on the desired location.

Notwithstanding, demand models of differentiated goods are a first step towards a specific evaluation of public goods and public policies in the sector through simulations.

i Action by the Spanish public sector in terms of housing policy is described in depth in Rodríguez López (1990[START_REF] López García | Algunos aspectos de la economía y la política de la vivienda[END_REF], Pareja and San Martín (1999) and García [START_REF] García Montalvo | La vivienda y el sector de la construcción en España[END_REF]. In particular, Pareja and San Martín (2002) describes action by the Spanish public sector affecting the decision as to which tenure system individuals should adopt.

ii For a detailed analysis of Andrew Court's work and the reasons why his technique was neglected for more than two decades, see [START_REF] Goodman | Andrew Court and the invention of hedonic price analysis[END_REF].

iii The hedonic technique rests on modern consumer choice theory, according to which the consumer derives utility not directly from the good but from its characteristics. See [START_REF] Lancaster | A new approach to consumer theory[END_REF].

iv See, Englund, et al (1999) andChun, et al (1998), among others, as examples of this literature.

v As we will see presently, sometimes socioeconomic variables are also included in order to control for consumer heterogeneity or fiscal variables, with the aim of testing some hypothesis of public intervention.

vi The literature on labour supply models is a good example of the implications of nonlinearity of the budget set in terms of estimation of leisure demand equations (labour supply), and also of the consequences of linearising nonlinear budget sets. See García and Suárez (2003) for a discussion of the empirical importance of these aspects. vii The most common solution to this endogeneity problem has been to use sociodemographic variables as instruments [START_REF] Palmquist | Estimating the demand for the characteristics of housing[END_REF]. viii It should be stressed that the sample used is a selected sample, i.e., a sample of potentially salable properties; indeed, this is why they were valued. In this study this has not been taken into account when estimating the hedonic price model. The evidence provided by Jud and Seaks (1998) shows that in that particular case the problem of sample selection had little effect on the results.

ix The price per square meter is a more correct specification of the dependent variable when the aim is to capture the variability in one market [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF]. However, as [START_REF] Parsons | An almost ideal demand system for housing attributes[END_REF] pointed out, the correct specification of the dependent variable and the functional form is not important when the aim is variability between markets, as it is in this paper. Moreover, the total value and a lineal specification avoid simultaneity and identification problems as well as make interpretation of the results in terms of implicit prices easier, since all the characteristics have a positive coefficient and, therefore, are goods.

x Note that in [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF] the dependent variable is the log of the price per square metre and the floor area appears in an explanatory variable with the definition

      + θ S 1

S ln

, where S is the floor area and θ a parameter that, if equal to 0, corresponds to a model with constant area elasticity. xi In the case of Erekson et al (1979) the expenditure on quality is obtained by the difference between the total value predicted by the hedonic equation, expenditure on floor area and expenditure on location. xii In [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF] the contribution of location to explaining the variability of price per square metre is quantified as representing 53.58% of the explained variation. xiii In this second stage the observations corresponding to District 4 (Les Corts) were removed because it was the least representative sample (347 observations) and because the price index calculated for some observations was negative. xiv Keeping the utility and the rest of the prices constant. xv Keeping the expenditure and the rest of the prices constant.

xvi The evidence provided is robust to alternative market segmentation (segmentation by year), which can be estimated because of the certain temporal dimension of the database. The estimation results and the elasticites are very similar between segmentations and can be obtained from the authors upon request. xvii See [START_REF] Garcia | Precios de la vivienda y localización: evidencia empírica para la ciudad de Barcelona[END_REF] for the econometric evidence. xviii Garcia et al (2006) provides a detailed analysis with some conclusions on urban economics, explaining price differences between areas in terms of their characteristics.

  patterns of each of these attributes of housing will give us greater insight into the aspects that influence how housing is valued.

  Section 5 and end with some brief conclusions in Section 6.

  Such samples, as well as promotions and

  price, but it is exogenous for any given consumer.

  price of housing unit i, k i H , is the k th physical characteristic of housing unit i, and j i L , is the j th characteristic of the area to which housing unit i belongs, while of the k characteristics and the area respectively (parameters to estimate) corresponding to a segment (district or year) y, and i u is the error term. If the segmentation is done by districts in equation (5), dummy variables are included for the years of the period considered in order to capture the time trend in housing prices.

  price of characteristic j, x is the total expenditure allocated to housing characteristics, given that a weak separability assumption is made between housing characteristics and other goods, and P is a price index approximated by a price index previously fixed as that proposed by Stone: p are the prices of the characteristics.

  are imposed. It is therefore a system of demand equations in which the total sum of the proportions of expenditure is equal to unity ( 1 w k = ∑

  the definition of the variables used, the dependent variable is as follows: Total value: total value of the property in euros Whereas the explanatory variables are: Floor area: floor area in square metres, including a proportional part of the communal areas Year: dummy variables defined according to the year to which each observation belongs Age: seven dummy variables in ascending order according to age, for the following categories: new housing; aged 1 to 5 years; 6 to 10; 11 to 20; 21 to 30; 31 to 50; and over 50 Lift: dummy variable with the value 1 or 0 depending whether or not the property has a lift Floor: four categories have been defined with their corresponding dummy variables depending whether the property is ground floor or basement; first floor; second floor; or third floor or higher Attic: dummy variable that takes the value 1 if the property is an attic and 0 otherwise Street: dummy variable that takes the value 1 if the property faces the street and 0 if it faces a courtyard variable that takes the value 1 if the property has heating and 0 otherwise State: state of repair of the property; five dummy variables corresponding to each of the five evaluations of the state of the property (very bad, bad, average, good and very good) Renovation: time elapsed since last renovation work, with four dummy variables corresponding to the four intervals for which this variable is defined: if the property was renovated in the year of the sample or the five previous years; 6 to 10 years previously; 11 to 20 years previously; or over 20 years previously Areas: statistical area (248) in which the property is located Table A.1 in the Annex shows the mean value of the characteristics (except for the area) for each of the years of the sample. It is easy to see that most of the sample is used housing, recently renovated, and with an average floor area of around 85 sq m.

  each district (10) by OLS; the results are presented in Table A.2 in the Annex.

  to which the property belongs, divided by the total expenditure on housing characteristics, which is simply the sum of the expenditures on the three characteristics considered (quantity, quality and location).

  , which consists in grouping together characteristics related to the quality of housing. Thus, expenditure on quality is the sum of the products of the price of each of the characteristics related to quality (age, heating, lift, floor number, whether it faces the street, state of repair, and time elapsed since last renovation work) in each market by the level of each of them chosen by each household xi , dividing the sum of these products by the total expenditure.• Proportion of expenditure on location (w3) is calculated as the product of the price of this characteristic in each market by the years spent in formal education by the average individual belonging to a particular area or year, divided by the total expenditure.The mean values of these proportions are very similar, regardless of whether we use marginal prices estimated by districts. Thus, in the segmentation by districts the proportions of expenditure corresponding to quantity, quality and location are 49.86%, 14.48% and 35.66%. The weight of location is reflecting the importance of this variable in accounting for housing prices xii .

  equal to 1 for all the observations, the estimate is made by removing one of the equations, in this case the one corresponding to expenditure on location, in order to avoid the problem of the singularity of the error variance-covariance matrix as a consequence of the aggregation condition mentioned earlier. Two alternative models are estimated. In the first model the property market is segmented by districts, while the second model takes into account the possible effect that the sociodemographic characteristics of the individuals may have on the demands analysed. To this end, in view of the unavailability of individual data on the characteristics of buyers, we use a set of sociodemographic variables of the various districts, obtained using Barcelona City Council population statistics (model with segmentation by districts and sociodemographic characteristics). The variables included in the specification and recorded at district level are: proportion of women, proportion of young people (aged under 24), proportion of individuals with university studies, proportion of permanent employees and proportion of business owners.

  estimate indicate patterns of substitutability between quantity and quality (positive sign) and of complementarity between quality and location, and also between quantity and location (negative sign).

  with the expected negative sign, in accordance with the decreasing relationship between price and quantity demanded of Although in this case the differences between the estimated elasticities depending on the specification considered, are greater than in the case of the income elasticities, some patterns are also repeated regardless of the model analysed, and are worth mentioning.

  showing greater interest (preferences) in this type of characteristic among these occupational groups.

  . (2001), El otro exceso de gravamen. Un análisis empírico aplicado a las políticas de vivienda directas, Revista de Economía Aplicada 27, 35-61. BIN, O. (2005), A semiparametric hedonic model for valuing wetlands, Applied Economic Letters 12, 597-601. BOGART, W.T. and CROMWELL, B. (2000), How much is a neighbourhood school worth?, Journal of Urban Economics 47, 280-305. BOVER, O. and VELILLA, P. (2001): Hedonic house prices without characteristics: the case of new multiunit housing, Banco de España, Estudios Económicos, 73. BROWN, J.N. and ROSEN, H.S. (1982), On the estimation of structural hedonic price models, Econometrica 50, 765-768. CHESHIRE, P. and SHEPPARD, S. (1995), On the price of land and the value of amenities, Economica 62, 247-267. CHESHIRE, P. and S. SHEPPARD (2004) 'Capitalising the Value of Free Schools: The Impact of Supply Characteristics and Uncertainty,'Economic Journal, 114,

  .; HAURIN, D. and MALPEZZI, D. (1998), New place to place housing price indexes for US metropolitan areas, and their determinants, Real State Economics 26, 235-274. COURT, L.M. (1941), Entrepreneurial and consumer demand theories for commodity spectra, Econometrica 9, 165-192, 241-297. DOWNES, T.A. and ZABEL, J.E. (2002), The impact of school characteristics on house prices: Chicago 1987-1991, Journal of Urban Economics 52, 1-25. EKELAND, I., HECKMAN, J.J. and NESHEIM, L.P. (2002), Identifying hedonic models, The American Economic Review 92, 304-309. EKELAND, I., HECKMAN, J.J. and NESHEIM, L.P. (2002), Identification and estimation of hedonic models, Journal of Political Economy 1, 60-109. ENGLUND, P.; QUIGLEY, J. and REDFEARN, C. (1999): The choice of methodology for computing price indexes: comparisons of temporal aggregation and sample definition, Journal of Real State Finance and Economics 19, 171-196. EREKSON, H., SUMKA, H.J. and WITTE, A.D. (1979), An estimate of a structural hedonic price model of the housing market: an application of Rosen's theory of implicit markets, Econometrica 47, 1151-1173.

Table A .

 A 1: Mean of the characteristics in the period 1998-2001 Table A.2: Estimated hedonic price equations by districts Dependent variable: total value (euros)

		1998 District 1	1999	2000 District 2	2001 District 3
		Coefficient "t"	Coefficient "t"	Coefficient "t"
	Floor area Floor area	86.02 1033.056 (44.84)	83.97 1518.175 (64.35) 84.42	85.86 1227.032 (45.70)
	Heating Heating Street Street Lift	0.326 -7005.169 (2.53) 0.829 1508.631 (0.65) 26655.854 (9.84)	0.353 -8063.990 (4.24) 0.351 0.855 0.846 1169.263 (0.61) 5357.320 )2.39)	-4624.932 (3.41) 0.625 -354.387 (0.28) 0.827 4547.853 (3.10)
	Lift Educational level	0.642 3138.900 (3.24)	0.605 9573.065 (8.59) 0.590	0.590 9713.888 (10.14)
	Year (ref.: 1998) Age New 1999 2000	0.032 24590.530 (9.94) 39836.477 (18.01)	0.030 30067.151 (12.55) 0.043 59571.929 (26.85)	21715.246 (14.60) 0.034 39775.828 (27.61)
	1-6 years 2001	0.026 61303.235 (20.41)	0.016 93757.888 (40.76) 0.020	0.026 61904.247 (40.01)
	6-10 years 11-20 years 21-30 years 31-50 years Over 50 years 31-50 years F o r -1658.018 (0.55) 0.020 0.171 0.327 0.208 0.215 Age (ref.: >50 years) New 13455.273 (1.80) 1-6 years -33690.725 (2.64) 6-10 years 17009.171 (1.55) 11-20 years -441.029 (0.06) 21-30 years -3158.012 (0.70) F o r	0.016 0.126 0.316 0.246 0.249 2489.116 (1.11) 0.023 0.087 0.292 0.279 0.256 34089.407 (6.53) 21244.732 (3.49) 25004.688 (3.26) 14560.564 (4.86) 8834.878 (3.85)	0.013 0.073 0.260 0.294 806.348 (0.53) 0.298 21727.784 (5.23) 7767.997 (2.10) 15744.275 (4.00) 7119.313 (3.49) 5867.687 (3.42)
	State Very bad Bad Average Good Very good State (ref.: very bad) Bad Average Good Very good Floor (ref.: ground) First Second	0.002 0.040 0.613 0.315 0.030 e P e r 13453.121 (4.55) 16467.912 (5.41) P 28784.116 (7.44) 36644.525 (5.32) e 3879.179 (0.77) 4903.844 (1.01) e r	0.170 0.297 0.367 0.119 0.047 13408.640 (3.49) 0.003 0.052 0.639 0.276 0.031 17753.669 (4.63) 27906.759 (6.53) 47441.792 (8.40) 8638.635 (1.24) 7907.102 (1.15)	0.023 0.068 0.614 0.231 0.061 7590.386 (2.56) 16464.967 (5.60) 25396.596 (7.90) 37496.304 (8.56) 1720.541 (0.63) 2175.580 (0.82)
	Third or > Renovation 0-5 years 6-10 years 11-20 years Over 20 years Floor Ground First Second Third or higher Attic Lift*attic Renovation (Ref.: >20) 0-5 years 6-10 years 11-20 years Constant Adjusted R 2 σ Sample size	5700.149 (1.22) 0.455 0.167 0.165 0.180 0.046 0.133 0.157 0.607 R e 13325.130 (2.00) 0.433 0.415 0.176 0.173 0.167 0.173 0.194 0.196 0.051 0.061 0.167 0.136 0.174 0.177 0.497 0.508 v i e w 916.725 (1.62) 20018.175 (1.93) -14117.636 (1.82) 2589.118 (0.29) R 8117.726 (3.74) 10795.896 (4.69) 2051.591 (0.77) 7026.937 (2.57) e 2866.822 (1.16) 1887.911 (0.74) -75912.577 (8.03) -186623.292 (13.65) 0.85 0.85 20.599 2.969 772 1513 v i e w	4882.761 (1.94) 0.448 0.162 0.159 0.188 0.070 0.147 0.180 0.545 1971.686 (0.44) 4679.084 (1.06) 5033.500 (3.20) 1871.377 (1.04) 1260.809 (0.70) -138093.019 (14.40) 0.84 17.024 1225
	Attic	0.055	0.109	0.115	0.055
				O n l O n l
				y y
					34

Table 1 :

 1 Characteristic demand equations

	Model with segmentation by districts
		Quantity	Quality	Location
		Coefficient "t"	Coefficient "t"	Coefficient
	Price index
	Quantity	0.157	131.27	0.048	51.09	-0.205
	Quality	0.048	51.09	-0.007	5.15	-0.041
	Location	-0.205	184.95	-0.041	32.79	0.246
	Real expenditure	0.166	150.31	-0.112	76.63	-0.054
	Constant	0.187	33.35	0.591	87.22	-0.778
	R 2	0.880	0.574
	σ	0.038	0.048
	Model with segmentation by districts and sociodemographic variables
		Quantity	Quality	Location
		Coefficient "t"	Coefficient "t"	Coefficient
	Price index
	Quantity	0.101	52.83	0.080	60.80	-0.181
	Quality	0.080	60.30	0.017	7.76	-0.097
	Location	-0.181	80.88	-0.096	40.47	0.278
	Real expenditure	0.210	199.13	-0.108	73.60	-0.102
	Sociodemographic variables
	% young people	0.533	4.68	3.917	28.79	-4.450
	% women	-0.508	3.01	4.384	18.99	-3.876
	% with higher education	-0.064	1.27	0.986	14.40	-0.932
	% permanent employees	-0.691	10.68	-1.717	20.58	2.408
	% business owners	-0.391	3.14	-3.683	22.95	4.074
	Constant	0.583	7.74	-1.130	10.37	0.547
	R 2	0.922	0.649
	σ	0.031	0.043

Table 2 :

 2 Elasticities of demand

	Model with segmentation by districts
		Quantity	Quality	Location
	Income elasticity	1.36	0.24	1.27
	Compensated elasticity	-0.63	-0.58	-0.22
	Uncompensated elasticity	-1.34	-0.61	-0.68
	Model with segmentation by districts and sociodemographic variables
		Quantity	Quality	Location
	Income elasticity	1.421	0.254	0.714
	Compensated elasticity	-0.797	-0.883	-0.220
	Uncompensated elasticity	-1.506	-0.918	-0.475

* The authors acknowledge funding through the project ECO2008-06395-C05-01, financed by the Ministry of Technology and Science. ://mc.manuscriptcentral.com/


Frank Ralph Jones

Joel P Foreman

David Porter

Shabnam Behzadi

Multi-scale Modelling of the Effect a Viscoelastic Matrix has on the Strength of a Carbon Fibre Composite

Keywords: composite materials, modelling, strength

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The use of thermosetting epoxy resins as the matrix phase in high performance composite systems is commonplace. Such resins offer a number of benefits over alternative systems, such as increased strength, toughness, environmental resistance, good electrical and thermal properties and minimal shrinkage [START_REF] Pham | Epoxy Resins' in Encyclopedia of Polymer Science and Technology[END_REF]. Commercially available resin systems used by the transport industry for mission critical applications often involve blending two or more epoxy resins together with a multifunctional amine curing agent. The composite part is then created by impregnating an array of carbon fibres with the blended resin and curing to a particular schedule. In the past the effect the resin matrix has on the composite properties has been relatively poorly understood. It is only recently that the importance of the viscoelastic nature of the matrix has been shown to directly influence the strength of composite systems [START_REF] Behzadi | [END_REF]. The complex polymeric structure of the resin allows energy to be dissipated as heat via strain rate dependent molecular relaxation processes. This ductile behaviour can be tailored to suit the particular application and this paper shows the results of an attempt to predict this effect using a simple modelling approach.

A number of multifunctional epoxy resins are used in commercial formulations, including those based on glycidyl ethers and amines that typically have functionalities varying between 2 and 4. In particular, diglycidyl ether of bisphenol-A (DGEBA) has two epoxy groups attached to a central phenyl-CMe 2 -phenyl segment via two ether linkages. Tetraglycidyl 4,4' diaminodiphenylmethane (TGDDM) has four epoxy groups attached via amine linkages and triglycidyl p-aminophenol (TGAP) has three epoxy groups attached via one ether and one amine. A typical curing agent for these epoxies is the multifunctional amine diaminodiphenylsulphone (DDS). The chemical structures of the DGEBA, TGDDM, TGAP and DDS monomers are shown in Figure 1.

The mechanism for curing epoxy resins with amines is less than straightforward [3] and is shown in Figure 2. All three reactions involve opening the relatively unstable oxirane ring in the epoxy group to form a CH(OH) group. The initial step in reaction 1 involves a primary amine reacting to form a secondary amine which can then react further to form a tertiary amine in reaction 2. The CH(OH) product of reactions 1 and 2, can also react to form an ether bridge as shown in reaction 3. The likelihood of each reaction occurring is 1>2>3 in the ratio 100:40:10 [3]. The number of epoxy groups in the monomer (the functionality) determines how crosslinked the cured resin will be. Hence, the number of crosslinks between polymer chains increases with increasing functionality (DGEBA<TGAP<TGDDM). The amine curing agent DDS also has a functionality of 4 and hence contributes to the degree of crosslinking in the system.

Fully cured, a typical epoxy resin system is a complex 3-D network which is amorphous and isotropic in nature. From a traditional modelling perspective such as molecular modelling this is quite a disadvantage as such techniques rely on knowing the Cartesian coordinates of the all atoms in the molecule. In a typical epoxy resin system, the coordinates of many thousands of atoms would be required to create a representative sample for modelling. This in turn increases the computing power necessary and hence the cost. Where a representative morphology exists, a repeat unit can reduce the computation required significantly but this is generally not the case in crosslinked polymers. To solve this problem modelling techniques have been developed which use a mean field approach to predicting the properties of polymers, such as Group Interaction Modelling (GIM) [START_REF] Porter | Group Interaction Modelling of Polymer Properties[END_REF] and similar techniques [START_REF] Bicerano | Prediction of Polymer Properties[END_REF]. In GIM, the need for exact atomic coordinates is removed by defining the equations of state and hence predictive constitutive equations in terms of the polymers' representative mer unit(s). In this way, the chemical structure of the polymer is defined by the constituent functional groups of the mer unit (e.g. CH 2 , Phenyl, Epoxy), the resin component ratio (e.g. epoxy to amine ratio) and the degree of cure. This relatively simple information allows the computational time to be drastically reduced while retaining the sound physical basis for accurate predictions [START_REF] Foreman | [END_REF]7,8].

One of the strengths of GIM is the relatively simple way in which the chemistry is defined (i.e. number and type of functional groups). The work presented here will show how the GIM predicted properties of cured epoxy resins change when the chemistry is altered in a systematic way. Three aspects of epoxy resin chemistry are investigated, the epoxy-amine ratio, the isomer of DDS and specific changes in the chemistry of DGEBA. While it is unlikely that a technique like GIM could ever replace experiments, it is worth noting that the calculations presented here were performed in a tiny fraction of the time (and cost) required to measure these quantities experimentally. It is envisaged that a technique such as GIM could be used in the future as a complementary technique, enabling the resin formulator to screen possible candidates for suitability at a fraction of the current cost.

As part of a larger multi-scale modelling effort, the GIM predicted properties such as modulus and stress-strain curves are used in a finite element model that determines the effect a fibre failure has on neighbouring fibres. A fibre failure event triggers an increase in strain concentration in surrounding fibres via the viscoelastic matrix phase, the properties of which are predicted by GIM. This information is then used in a statistical model to predict ultimate composite strength. A Monte Carlo type simulation assesses the propagation of fibre failure events through the composite as a function of the GIM and FEA predicted properties of the matrix and fibres [9,10].

Methods

Group Interaction Modelling

Group Interaction Modelling uses a mean field potential function approach that predicts the bulk properties of polymers from molecular input. The technique uses a modified Lennard Jones potential to define the total interaction energy between neighbouring polymer chains, E total . The potential is defined in terms of thermodynamic energy terms to describe the attractive and repulsive contributions to the total energy function. The equation of state for the system linking energy and volume, V, is the thermodynamic potential function shown in equation [START_REF] Pham | Epoxy Resins' in Encyclopedia of Polymer Science and Technology[END_REF]. The minimum of the potential energy well, E coh , represents the cohesive energy of the system which is acted against by the temperature dependent volumetric term. The polymer chains are arranged in a hexagonal array with an interchain separation distance, r. The polymer chains are assumed to be many orders of magnitude longer than they are wide and therefore, for the purposes of the model, invariant in length. This means that V ∝ r 2 and hence the unusual powers in the Lennard Jones potential in equation [START_REF] Pham | Epoxy Resins' in Encyclopedia of Polymer Science and Technology[END_REF]. The system functions as a strong 1-D oscillator in a weak 3-D field.

M C T coh coh total H H H E V V V V E E + + + - =               -       = 3 0 6 0 2 [1]
The other thermodynamic terms in equation [START_REF] Pham | Epoxy Resins' in Encyclopedia of Polymer Science and Technology[END_REF] are all positive, bringing the system out of the potential well minimum. The thermal energy, H T , is defined using a simplification of the 1-D Debye function for skeletal mode vibrations. The cohesive and thermal energy terms are the two main contributions to the total interaction energy, with subtle modifications from the following terms. H C is a configurational energy term which allows for the presence of morphology such that an amorphous polymer has a higher H C than a semi-crystalline polymer. For amorphous systems, H C takes a value of 0.11E coh . H M is a mechanical energy term which is zero when the system is in equilibrium and positive when the system is perturbed.

A series of linked constitutive equations for the prediction of properties is then derived from the fundamental equation of state given above. Each of the equations defines a particular polymer property in terms of a series of fundamental parameters and previously derived equations. The parameters required are defined on a functional group or mer unit basis and generally obtained using group contributions tables [START_REF] Porter | Group Interaction Modelling of Polymer Properties[END_REF][START_REF] Van Krevelen | Properties of Polymers[END_REF], but other techniques such as molecular modelling are useful too. The parameters are the degrees of freedom, N, the cohesive energy, E coh , the van der Waal's volume, V w , the molecular weight, M, the length, L, and the 1-D Debye temperature, θ 1 . Below the glass transition temperature, N is given a value of 2 for a typical small functional group (such as -O-or -CH 2 -) to account for the degrees of freedom normal to the polymer chain. Correspondingly, above the glass transition temperature N is 3 to account for the extra degree of freedom in the direction of the polymer chain that gets activated. The value of E coh is effectively determined using group contribution tables and represents the minimum energy of the system against which all the other thermodynamic terms act. In practice, the GIM equations balance the attractive E coh with the repulsive thermal contributions from N. Crosslinking is incorporated into the model by decreasing the value of N by 3 for each crosslink site on the mer unit. Similarly, hydrogen bonding is accounted for by increasing the value of E coh by 10,000 J/mol for each hydrogen bond site. The value of θ 1 is obtained from the datasets of Wunderlich [START_REF] Wunderlich | [END_REF] and takes a value of 550 K for polymers containing phenyl rings in the backbone.

The epoxy resin systems investigated in this work are all viscoelastic in nature, and this is incorporated into GIM by treating the elastic and viscous contributions separately using a simple spring-dashpot type model. The viscous contribution is assumed to be due to the two significant transition events which occur in the dynamic mechanical profile. These events can be associated with strain rate dependent molecular relaxation processes which are activated at specific temperatures where energy is dissipated as heat. The glass transition is associated with a large scale increase in the intermolecular motion between polymer chains as the system moves from the glassy to rubbery phase. In this class of epoxy resins, the beta transition is associated with an intramolecular cooperative rotation of the backbone phenyl rings In GIM, these loss events are modelled using a Gaussian distribution function to simulate the temperature distribution of the event. Also, strain rate dependence in GIM is incorporated through the two loss events. The equations control how the speed of a perturbation event affects the temperature at which the event occurs. On a timetemperature equivalent scale, the faster the perturbation occurs, the less chance the relaxation event has to happen. This manifests itself as a relationship between strain rate and transition temperature which is modelled in GIM.

A simple molecular modelling routine is used to estimate the activation energy for each peak then an Arrhenius equation is used to determine the temperature of each transition. The loss associated with both transitions is determined using the loss tangent relationship between lost and stored energy. The cumulative (tan∆) and local (tanδ) loss tangents are determined and expressed as a function of temperature. A distribution term (i.e. the shape of the peak) is determined and the variables are applied to a Gaussian distribution function for each loss tangent profile.

At the heart of GIM is a series of linked constitutive equations designed to predict the terms in equation [START_REF] Pham | Epoxy Resins' in Encyclopedia of Polymer Science and Technology[END_REF]. The heat capacity at constant pressure, C p , is expressed in terms of the temperature, T, and N in equation [START_REF] Behzadi | [END_REF]. It is derived from an empirical approximation to the 1-D Debye model [START_REF] Wunderlich | [END_REF].

        +         = θ T . θ T . R N C p [2]
The heat capacity is the ability to store energy as a function of temperature, so integrating over temperature gives the thermal energy, H T . The zero-point cohesive energy, E coh , is the binding energy at the minimum of the potential energy well at 0K. A temperature dependent value of the cohesive energy is approximated by assuming a 50% loss through the glass transition, applied using the Gaussian distribution function described earlier. The sum of H T and E coh , including the approximations for H C and H M , give a value of E total as a function of temperature.

The linear thermal expansion coefficient, α l , is defined in terms of the previously predicted temperature dependent heat capacity and cohesive energy and is shown in equation [3].

coh p l E R C 3 38 . 1 = α [3]
The thermal expansion coefficient is a change in dimension as a function of temperature, so integrating over temperature, using the van der Waal's volume, V w , as a scaling factor, gives the volume, V. The temperature dependent energy and volume terms of equation [START_REF] Pham | Epoxy Resins' in Encyclopedia of Polymer Science and Technology[END_REF] have now been determined and in principle any property of interest can now be determined. The elastic bulk modulus, B e , can be defined as an energy density term and this relationship is shown for amorphous polymers in equation [START_REF] Porter | Group Interaction Modelling of Polymer Properties[END_REF].

V E B total e 18 = [4]
The term defined in equation [START_REF] Porter | Group Interaction Modelling of Polymer Properties[END_REF] refers to the elastic contribution to the full bulk modulus (i.e. the spring in the spring-dashpot model). The full viscoelastic term is obtained by applying the loss tangent profiles for the glass and beta transitions. An expression for the tensile modulus, E, is obtained using the known relationship between the loss tangent and the temperature gradient of the previously defined elastic bulk modulus [START_REF] Bondi | Physical Properties of Molecular Crystals, Liquids, and Glasses[END_REF]. This leads to an expression for E as shown in equation [START_REF] Bicerano | Prediction of Polymer Properties[END_REF].

( )

M θ L x A where B A ∆ ∆ B E e β g e 1 5 10 5 . 1 tan tan exp - =         + - = [5]
The proportionality constant, A, is expressed in terms of standard GIM parameters defined earlier. The temperature dependent tensile modulus predicted by equation [START_REF] Bicerano | Prediction of Polymer Properties[END_REF] can be compared to experimental values. However, the resin samples undergo brittle failure in tension so compressive moduli are used instead. The comparison is valid as tensile and compressive moduli are very similar at the low strain values used in this work. Once the bulk and tensile moduli are known, Poisson's ratio, ν, can be determined along with the shear modulus.

Finally, the stress-strain curves are obtained by predicting the stress, σ, and strain, ε, separately as a function of a dummy variable. The dummy variable set is chosen as the temperature range from the observation temperature, T 0 , (often room temperature) to a value T 1 which is arbitrarily chosen to be well above T g . The elastic contribution to the strain, ε e , is determined by integrating the thermal expansion coefficient over temperature (which is analogous to the prediction of volume). This is shown in equation [START_REF] Foreman | [END_REF] along with the expression for the full strain which also includes the viscous part of the strain.

        + = = ∫ ∫ 1 0 1 0 tan 1 & T T e T T l e dT dT δ ε ε α ε [6]
Now that the tensile modulus and strain as a function of temperature are known, the tensile stress can be determined. The compressive stress, σ c , is then obtained by correcting by a factor of twice Poisson's ratio to account for expansion in the two axes normal to the compression as shown in equation [7].

Experimental

For validation, the GIM predicted properties are compared to equivalent experimental values where possible. A series of experimental data has been gathered on the compressive modulus and compressive stress-strain curves for TGDDM/DDS, TGAP/DDS and a 50-50 blend of the two at three different strain rates [START_REF] Behzadi | [END_REF]. The epoxy and amine monomers are mixed in the appropriate ratio and heated to 120 o C then de-gassed for at least 30 minutes. The samples are cured in cylindrical moulds according to the following schedule. Ramp to 130 o C at 3 K/min, dwell for 1 hour, ramp to 180 o C at 2 K/min, cure for 2 hours and leave to cool overnight. Uniaxial compression testing was performed using a Hounsfield Universal instrument (H100KS/05) on enough samples to provide an acceptable statistical average.

Results and Discussion

Group Interaction Modelling of the Mechanical Properties of an Epoxy Blend

The GIM method as described above has been applied to TGDDM and TGAP cured with DDS and a 50-50 blend of the two epoxies. Table 1 shows the important GIM input parameters for the constituent functional groups present in either the epoxy or amine monomers. The GIM parameters for the epoxy and amine mer units are summed from their constituent functional groups by reference to the monomer structures in Figure 1. Finally, the mer units for cured TGDDM/DDS, TGAP/DDS and a 50-50 blend of the two are reported, taking into account epoxy-amine ratio, degree of cure and the presence of crosslinking and hydrogen bonding. These values are used in the equations presented in the previous section to predict properties.

The epoxy amine ratio used for TGDDM/DDS was 1:1 due to both monomers having a functionality of 4. The epoxy amine ratio for TGAP/DDS is 4:3 as TGAP has one less epoxy group. However, this is changed slightly due to the increased likelihood of the epoxy-hydroxyl (E-OH) reaction (reaction 3 in Figure 2) in TGAP/DDS. The exact reason for this is not clear but it seems likely due to increased mobility in TGAP over TGDDM. An illustration of this is to consider the molecular structure as the E-OH reaction becomes a possibility. At this stage, the network is forming and reducing the mobility of reactants but the impact this has on the two epoxies is different. In TGDDM the mobility is lower because 3 of 4 epoxy groups are reacted and it is unable to rotate to meet an incoming reactant. In TGAP the mobility is higher because only 2 of 3 epoxy groups are reacted and the molecule is therefore more able to rotate and hence is more able to participate in the E-OH reaction.

In experimental work, the amine is normally present in excess to ensure as high a degree of cure as possible. The experimental samples tested in this work used the standard recipe of 74 and 64 % by weight DDS for TGDDM and TGAP respectively. However, the epoxy amine ratios used in GIM are near to the theoretical values because it is assumed that the parts of the network structure that impart properties are fully cured. A number of potential diluents can be present in the cured resin (such as unreacted hardener, impurities and water) and are known to affect properties [START_REF] Caballero-Martinez | [END_REF]. However, the GIM predictions do not currently require this input in order to be accurate. Work is progressing on developing GIM to predict the effect water (and other diluents) has on the properties of cured epoxy resins [START_REF] Foreman | [END_REF].

The tensile modulus, calculated from equation [START_REF] Bicerano | Prediction of Polymer Properties[END_REF] is presented in Figure 3 as a function of temperature and blend type at a strain rate of 0.00167 /s. The step-wise drops in modulus as the temperature increases are a direct result of the beta and glass transitions. The broad beta transition is centred around ~ -50 o C while the glass transition occurs around 275 o C and has a more abrupt and catastrophic effect on the modulus. At low temperatures (< -100 o C) TGAP has a higher modulus than TGDDM, presumably a manifestation of the former being smaller and able to pack more efficiently at lower temperatures. As the temperature increases the modulus of TGDDM rises above that of TGAP as observed experimentally.

Compressive stress-strain curves, calculated from equations [START_REF] Foreman | [END_REF] and [7] are presented in Figure 4 as a function of blend type at a strain rate of 0.00167 /s. The curves are typical for this class of polymers with an initial elastic section at extremely low strains followed by increasing viscous contributions. Eventually the polymer yields as large scale translation between adjacent polymers causes ductile failure. The yield point is defined here as the point at which the gradient of the curve reaches zero. The TGDDM curve is consistently above that of TGAP which is a reflection of the fact that the simulations were performed at room temperature.

A comparison between the GIM predicted and experimental stress-strain curve for the 50-50 blend at a strain rate of 0.00167 /s is presented in Figure 5. The curves agree very well and in particular the yield condition is predicted with good accuracy. There is some post yield strain softening and hardening present in the experimental curve which is not replicated in the GIM predicted curve. This is not currently part of the model but will be in future versions as predicting such effects requires a simple change in the value of N beyond the yield condition to account for the activating of extra degrees of freedom.

One of the key aspects of GIM is its ability to predict properties as a function of strain rate. Figures 6,7, and 8 compare GIM predicted and experimental moduli and yield stress values as a function of strain rate and blend type. Each pair of lines represents a GIM to experiment comparison and in general the agreement between the two is excellent. In particular the yield stresses are predicted very accurately across all blends and strain rates. The predicted modulus values at the lower rates are slightly inaccurate but this may be in part due to creep effects. The trends observed in both modulus and yield stress is for the values to increase with increasing strain rate. This is consistent with the viscoelastic model presented earlier where loss events are caused by molecular relaxation processes. As the strain rate increases, the molecular processes have less chance to occur and hence less loss is observed which in turn leads to a higher modulus and yield stress.

Modelling Structural Changes of the Epoxy Resin

A significant advantage of GIM over similar techniques is the relatively simple method used to define the polymer system where the representative mer unit is described using its constituent functional groups. This means that the chemistry can be easily altered to investigate any changes to the predicted properties. Figure 9 shows the modulus and yield stress of TGDDM/DDS as a function of the epoxy-amine ratio for a given strain rate and temperature. The values at 50% epoxy (i.e. 50% epoxy mer units, 50% amine mer units) are the same as those reported earlier while increasing or decreasing the amount of epoxy has a consistent effect on the properties. As expected, adding more hardener increases the modulus and yield stress while more epoxy has the opposite effect. It is worth noting that practical issues are an issue here as there would be little benefit in attempting to make a resin where 80% of the monomer molecules were either epoxy or amine. However, an appropriate amount of epoxy could be between 40 and 60% (mer units) and within this range the properties don't vary as much as might be expected. The plateau value of both properties at high epoxy values indicates that there is a limiting low value for each property where ductile effects dominate. As the amount of hardener is increased the brittle nature of the resin increases until a balance between ductile and brittle effects is obtained defining the working range of the resin.

There are two common isomers of the amine hardener used in this work, 4,4-DDS and 3,3-DDS, differing only in relative positions of the amine and sulphonyl groups on the phenyl rings. In 4,4-DDS (as shown in Figure 1) the groups are arranged in the para position which allows the phenyl ring to rotate more easily than in 3,3-DDS where the groups are in the meta position. In GIM parameter terms, the phenyl ring in 3,3-DDS has a higher number of degrees of freedom as it's rotation is not symmetrical. Figure 10 shows the modulus and yield stress of TGDDM/DDS as a function of the percentage of 4,4-DDS for a given strain rate and temperature. The results show that a higher percentage of 4,4-DDS leads to higher modulus and yield stress, a consequence of the fact that the 4,4 isomer dissipates less energy through molecular relaxation than the 3,3 isomer.

Another way to alter the chemistry of the resin is to change the functional groups present. A series of epoxies exist based on whichever variant of bisphenol was used in their manufacture. The most common is the diglycidyl ether of bisphenol-A (or DGEBA, see Figure 1) where two epoxy-ether segments surround a central phenyl-CMe 2 -phenyl group. The central CMe 2 group can be changed to vary the chemistry of the epoxy and is denoted X for convenience. The GIM parameters for six different X groups are given in Table 2. The various functional groups contribute the same number of degrees of freedom to the mer unit as they are all just a single link in the polymer chain. The differences are seen in the cohesive energy and the volume and this leads to different GIM parameters for the five mer units as shown in Table 3. The modulus and yield stress are plotted as a function of the central X group in Figure 11 for a given strain rate and temperature when cured with DDS. The decrease in both modulus and yield stress when moving from DGEBF to DGEBB is a consequence of having more alkyl groups present. These are able to dissipate more energy and the properties are correspondingly lower. The value of DGEBS is high as the SO 2 group is present which has a ratio of cohesive energy to thermal energy which results in high modulus and yield stress. Indeed, the SO 2 group is one of the main reasons DDS is an effective, popular hardener. DGEBC is used as a fire retardant in polycarbonate and its properties are included here to show that GIM is also capable of predicting the properties of more exotic structures. In the case of DGEBA/DDS, the comparison between GIM predicted modulus (2.35 GPa) and experiment (~ 2.3 GPa [18]) is very good.

Modelling of Composite Strength

The work presented here exists as part of a larger multi-scale approach to the prediction of composite properties. GIM has proven effective at predicting the nonlinear properties of an epoxy resin matrix as a function of temperature, strain rate and chemistry. These properties are then used to parameterise a finite element model which predicts the effect a fibre failure has on the surrounding fibres. The impact a fibre failure has on its neighbours is quite different depending on whether the matrix is elastic or viscoelastic [START_REF] Behzadi | [END_REF]. A 1/12 th slice of a 100 µm long unit cell of UD composite is created with hexagonal symmetry in Ansys 11. The unit cell is constructed using the SOLID45 element so that the fibre fraction is 50% as shown in Figure 12. Orthotropic properties were applied to the fibre elements (acquired from experimental measurements on HTA3151 carbon fibres [19]). Isotropic properties were applied to the matrix elements (obtained from the previously described GIM calculations on TGAP/DDS resin). In order to simulate the fibre break, the top layer of elements is removed from the central fibre only. A 1% axial strain is then applied to the system so that the front face displaces outwards and the central fibre remains in place. The strain concentrations associated with the broken fibre are measured in the surrounding fibres. The ineffective length of the broken fibre is defined as twice the length over which 90% strain recovery occurs.

A statistical method is then used to predict the relationship between the strength of the composite and its size using the finite element results as input. A single layer of composite of thickness equal to the ineffective length consisting of a 20x20 array of fibres is assigned random failure strains. The applied strain is increased until the fibre with the minimum failure strain is broken and the strain concentrations are applied to its neighbours. Should any of the neighbouring fibres break as a result of this strain concentration, the process is repeated. Then the applied strain is increased to the next lowest failure strain causing another break and potentially further breaks in neighbouring fibres. The process is repeated until unstable crack growth criteria are achieved (e.g. 3 broken fibres in a row or 2% broken fibres). The applied strain is then equal to the layer failure strain. The whole process is repeated for 50 to 100 iterations until self-consistency is achieved and the average layer failure strain is recorded. The length is scaled up using standard probability tables to give a relationship between composite strength and composite length. A typical set of results is shown in Figure 13 where a plateau value of strength is observed at high lengths. The original predictions by Curtis [20] have been developed here to allow strength predictions using an elastic (no energy dissipation) or viscoelastic (dissipation via beta and glass transitions) matrix. The results indicate that the yielding matrix predictions are nearest to typical experimental values. Ultimately, the multi-scale approach presented here shows how composite strength varies with the nature of the matrix. Subtle changes to the chemistry of the matrix have a corresponding effect on the properties of the composite, and the use of GIM has allowed this to be quantified for perhaps the first time.

Concluding Remarks

A multi-scale modelling approach has provided insights into the importance of the viscoelastic nature of the matrix phase in composite systems. The properties of the matrix phase were predicted using the mean-field based Group Interaction Modelling technique. Group Interaction Modelling allows the pragmatic calculation of a full set of non-linear volumetric, thermomechanical and engineering properties of a thermosetting multifunctional epoxy resin system. The comparison of predicted values with experimental values is very good across a range of temperatures and strain rates. The matrix phase was also altered at a fundamental molecular level to provide predicted properties as a function of chemistry. This provided predictions of properties as a function of the epoxy-amine ratio, the amine isomer and also the specific chemical structure of the epoxy. The matrix phase properties were used in a finite element model to predict the effect a broken fibre has on its neighbours. This was then used in a statistical model to predict the propagation of fibre failure events through a composite system.

[ 13 ]

 13 . It is likened to a 'crankshaft' style motion of two semi-adjacent rings where the handles of the crankshaft are the C-C bonds in the rings parallel to the backbone.

Figure 1 .

 1 Figure Captions

Figure 2 .

 2 Figure 2. General reaction scheme for amine curing an epoxy resin.

Figure 3 .

 3 Figure 3. Plot of GIM predicted tensile modulus against temperature for TGDDM/DDS, TGAP/DDS and a 50-50 blend of the two at a strain rate of 0.00167 /s.

Figure 4 .

 4 Figure 4. Plot of GIM predicted stress-strain curves for TGDDM/DDS, TGAP/DDS and a 50-50 blend of the two at a strain rate of 0.00167 /s.

Figure 5 .

 5 Figure 5. Plot of GIM predicted and experimental compressive stress-strain curves for a 50-50 blend of TGDDM/DDS and TGAP/DDS at a strain rate of 0.00167 /s and room temperature.

Figure 6 .

 6 Figure 6. Plot of GIM predicted compressive modulus and yield stress for TGDDM/DDS as a function of strain rate at room temperature. Experimental points are shown for comparison purposes.

Figure 7 .

 7 Figure 7. Plot of GIM predicted compressive modulus and yield stress for a 50-50 blend of TGDDM/DDS and TGAP/DDS as a function of strain rate at room temperature. Experimental points are shown for comparison purposes.

Figure 8 .

 8 Figure 8. Plot of GIM predicted compressive modulus and yield stress for TGAP/DDS as a function of strain rate at room temperature. Experimental points are shown for comparison purposes.

Figure 9 .

 9 Figure 9. Plot of the GIM predicted compressive modulus and yield stress as a function of epoxy-amine ratio at a strain rate of 0.00167 /s and room temperature. The resin is TGDDM/DDS.

Figure 10 .

 10 Figure 10. Plot of the GIM predicted compressive modulus and yield stress as a function of the percentage of 4,4-DDS over 3,3-DDS at a strain rate of 0.00167 /s and room temperature. The epoxy is TGDDM.

Figure 11 .

 11 Figure 11. Plot of the GIM predicted compressive modulus and yield stress as a function of the type of Bisphenol used to create the epoxy monomer. The central X group is noted for each epoxy and the strain rate is 0.00167 /s at room temperature.

Figure 12 .

 12 Figure 12. The hexagonal unit cell used in the finite element model. A fibre-break is simulated by removing the top layer of cells from the central fibre only and applying a 1% longitudinal strain to the rest of the system.

Figure 13 .

 13 Figure 13. Plot of the relationship between the predicted UD carbon fibre composite failure strain and composite length. The data is presented for a viscoelastic TGAP/DDS matrix and a purely elastic matrix. The dotted line represents a typical experimental failure strain for a UD carbon fibre composite.

Figure 1 .

 1 Figure 1. Chemical structures of DGEBA, TGDDM, TGAP and DDS monomers.

Figure 2 .

 2 Figure 2. General reaction scheme for amine curing an epoxy resin.

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Plot of GIM predicted tensile modulus against temperature for TGDDM/DDS, TGAP/DDS and a 50-50 blend of the two at a strain rate of 0.00167 /s.

Figure 6 .

 6 Figure 6. Plot of GIM predicted compressive modulus and yield stress for TGDDM/DDS as a function of strain rate at room temperature. Experimental points are shown for comparison purposes.

Figure 7 .

 7 Figure 7. Plot of GIM predicted compressive modulus and yield stress for a 50-50 blend of TGDDM/DDS and TGAP/DDS as a function of strain rate at room temperature. Experimental points are shown for comparison purposes.

Figure 8 .

 8 Figure 8. Plot of GIM predicted compressive modulus and yield stress for TGAP/DDS as a function of strain rate at room temperature. Experimental points are shown for comparison purposes.

Figure 9 .Figure 10 .

 910 Figure 9. Plot of the GIM predicted compressive modulus and yield stress as a function of epoxy-amine ratio at a strain rate of 0.00167 /s and room temperature. The resin is TGDDM/DDS.

Figure 11 .

 11 Figure 11. Plot of the GIM predicted compressive modulus and yield stress as a function of the type of Bisphenol used to create the epoxy monomer. The central X group is noted for each epoxy and the strain rate is 0.00167 /s at room temperature.

Figure 12 .Figure 13 .

 1213 Figure 12. The hexagonal unit cell used in the finite element model. A fibre-break is simulated by removing the top layer of cells from the central fibre only and applying a 1% longitudinal strain to the rest of the system.

Table 1 .

 1 GIM parameters for functional groups, mer units and cured mer units.

	Group	N	E coh (J/mol) V w (cm 3 /mol)
	CH 2	2	4,500	10.25
	Phenyl	3	25,000	43.3
	N	2	9,000	4
	Epoxy	4	15,300	22
	CH(OH)	2	20,800	11.5
	SO 2	2	45,000	20.3
	O	2	6,300	5
	TGAP Mer Unit	25	129,700	148.3
	TGDDM Mer Unit	36	191,700	232.9
	DDS Mer Unit	12	113,000	114.9
	E-OH Mer Unit	22	119,700	148.3
	Cured TGAP/DDS	19	121,116	134.0
	Cured 50-50 Blend	21.5 136,732	153.9
	Cured TGDDM/DDS 24	152,350	173.9

Table 2 .

 2 GIM parameters for the central X group for various diglycidyl bisphenol epoxies.

	Variant X	N E coh (J/mol) V w (cm 3 /mol)
	F	CH 2	2 4,500	10.2
	E	CHMe 2 9,000	20.4
	A	CMe 2	2 13,500	30.6
	B	CMeEt 2 18,000	40.8
	S	SO 2	2 45,000	20.3
	C	C=CCl 2 2 32,000	37.0

Table 3 .

 3 GIM parameters for various diglycidyl bisphenol epoxy monomers.

	X	N E coh (J/mol) V w (cm 3 /mol)
	DGEBF CH 2	24 126,700	171.3
	DGEBE CHMe 24 131,200	181.5
	DGEBA CMe 2	24 135,700	191.7
	DGEBB CMeEt 24 140,200	201.9
	DGEBS SO 2	24 167,200	181.4
	DGEBC C=CCl 2 24 154,200	198.1

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgments

This work was carried out as a part of the Weapons and Platform Effectors Domain of the UK MoD Research Program.