André De Palma
email: andre.depalma@enscachan.fr.

Nathalie Picard
email: nathalie.picard@u-cergy.fr

Ignacio Inoa
email: ignacio.inoa@u-cergy.fr.

Discrete Choice Decision-Making with Multiple Decision Makers within the Household

There is still a long way to achieve the goal of providing a theoretical and empirical framework to model and apply economics of the family. Decision-making within the family has been neglected too long in transportation. Two special issues by Bhat and Pendyala, 2005 [17] and by Timmermans and Junyi Zhang, 2009 [81] provide the most notable exceptions. The objective of this paper is to set-up a flexible framework to discuss the development of integrated transportation models involving interacting and interdependent actors; updating previous reviews from the point of view of economics of the family . Transportation is very keen to have access to this type of models, since their applications are numerous. Let mention, for example, residential location choice, workplace choice, car ownership, choice of children's school, mode choice, departure time choice activity patterns and the like. The (non unitary) economics of the family models are totally different models, which do not merely extend existing discrete choice models. They introduce new concepts, which are specific to within family interactions: negotiation, altruism, or repeated interaction and Pareto optimality. This review is completed with the study of different types of accessibility measures including recent work on timegeography measures of accessibility.

Introduction

Different research streams concerned with household decision-making have developed independently in different disciplines. The corresponding papers consider topics such as labour supply, transportation decisions, time and task allocation, or residential and employment choices. Literature in these fields has been dominated by models in which the household is treated as a single decision-making unit or unitary models (see Timmermans, 2009 [79] for an extensive review on past research in the transportation literature, and Vermeulen, 2002a [START_REF] Vermeulen | Collective household models: Principles and main results[END_REF],

2002b [START_REF] Vermeulen | Where does the unitary model go wrong? simulating tax reforms by means of unitary and collective labour supply models. the case for belgium[END_REF] for a literature review on unitary and collective household models, and Bianchi and Robinson, 1997 [START_REF] Bianchi | What did you do today? Children's use of time, family composition, and the acquisition of social capital[END_REF] for sociological study of time use within the household). In collective models, the different household members are engaged in a joint decision process involving bargaining. Until recently, interactions within the household were not explicitly modelled and the decision-making process outcome was considered as resulting from a representative individual (as if the household were a black box which needs not to be opened). In such models, household interactions were either introduced through explanatory variables defined at the household level, or simply disregarded in models of activity-travel demand (see Srinivasan and Bhat, 2005 [75]). Examples of household-level explanatory variables like number of household members, of active members, of children, household income, and other household dummy variables (e.g., occupational status, property status, age, etc.) are provided in Townsend, 1987 [START_REF] Townsend | The effects of household characteristics on the multi-day time allocations and travel/activity patterns of households and their members[END_REF] or in [START_REF] Golob | A model of activity participation and travel interactions between household heads[END_REF] [START_REF] Golob | A model of activity participation and travel interactions between household heads[END_REF], among others.

However, many household decision-making processes involve more than one decision maker who cannot be reduced to a single agent. Note that even when the decision problem under scrutiny does not involve multiple decision makers, most of the individual decisions depend, directly or indirectly, on the choices made by other household members (Timmermans and Zhang, 2009 [81]). A growing body of research in different fields, ranging from transportation demand to labour economics, has recently started to explicitly take into account the interactions between household members, and to model the corresponding strategies. The distinction between discrete and continuous household decisions is at the core of these new theoretical and empirical developments. Van Soest, 1995 [83] proposed a discrete choice model of labour supply disregarding the negotiation process within the household, whereas collective models of time allocation and labour supply focus on this negotiation process in discrete (Vermeulen, 2006 [86]) or in continuous-discrete applications (Blundell et al., 2007 [22]). The modelling of within-family interactions took off in the transportation literature with the special issues on modelling intra-household interactions edited by Bhat and Pendyala, 2005 [17] and by Timmermans and Zhang, 2009 [81]. In a nutshell, Bhat and Pendyala, 2005 [17] focus on contributions based on utility-maximizing models, whereas Timmermans and Zhang, 2009 [START_REF] Harry | Modeling household activity travel behavior: Examples of state of the art modeling approaches and research agenda[END_REF] present work that "adopt diverse methodologies" as group decision theory, and microsimulation approaches. There are a few contributions using experimental economics methodology to compare the decisions taken by the husband alone, the wife alone, and then by the spouses together (see [START_REF] Bateman | An experiment on risky choice amongst households[END_REF] [START_REF] Bateman | An experiment on risky choice amongst households[END_REF], Beharry-Borg, Hensher and Scarpa, 2009 [START_REF] Beharry-Borg | An analytical framework for joint vs separate decisions by couples in choice experiments: The case of coastal water quality in tobago[END_REF] or de Palma, Picard and Ziegelmeyer, 2011 [START_REF] André De Palma | Individual and couple decision behavior under risk: evidence on the dynamics of power balance[END_REF]). However, they do not analyse the decision mechanism within the couple.

A careful review of the literature reveals two main streams of research using the discrete choice framework to model household decision-making processes in multi-person households (several decision makers), with an explicit modelling of within-family interactions. In these studies, household decision-making models are developed in a discrete choice framework or with discrete choice (econometric/empirical) applications. The first research stream is related to collective models and their discrete labour supply model applications. The second research stream covers the transportation, activity-demand and location literature. These streams will be described in the following sections.

Section 2 explains the difference between individual and family models, and provides a transition from unitary to collective and negotiation models, with a special focus on labour supply models. The transportation and activity-travel demand literature with intra-household interactions is presented in Section 3. Section 4 describes the location and accessibility contributions considering multiple decision makers within the households, while Section 5 concludes.

Individual versus Family Models

There are two cases when it becomes relevant (and often necessary) to switch from individual decision-making models to family decision-making models. The first case is when the choices themselves are relevant to the family (marriage, divorce, number of children). The second case is when individual choices generate externalities to the other members of the household (competition among family members for using a joint car, joint leisure, chores sharing, spatial mobility, retirement plans…).

Current research on family economics aims at providing the best representation of the household decision-making process, when the household is made of several decision-makers. In the simplest (unitary) models, the household is considered as the unique decision unit, whereas in the most elaborate models, each household member is characterized by specific preferences, and the household decisions result from the confrontation (or aggregation) of these individual preferences.

Individual Choices about Consumption and Work

Traditionally, family economics describes household behaviour, focusing on choices concerning consumption and work. In order to examine these questions, the household is described as a small production unit combining domestic time with intermediary goods bought on the market in order to produce some commodities, which are in turn consumed by household members).

These developments have allowed economists to answer old but important questions related to major socio-demographic changes over the twentieth century, such as the change in domestic working hours (Gronau, 1977 [42]) and female labour supply, the growing divorce rate (Becker, Landes, and Michael, 1977 [11]), or the diminishing fertility rate (Becker and Lewis, 1973 [12]).

The description of household behaviour may rely on contributions from various disciplines such as sociology, demography or ethnology (see Picard, 1999 [66]). The topics under study go far beyond consumption or work choices. However, the way of representing household decisions in family economics until the late 80's is not very far from that of "traditional" models of consumption and labour supply, since these models generally neglect the multiplicity of decision-makers.

The methodology traditionally used in family economics is quite straightforward. A (unique) household utility function describes the household preferences, taking as arguments the quantity of goods consumed (including local amenities when location choices are at stake) and/or leisure time. This function is maximized under a unique budget constraint aggregating the resources and expenses of all household members (and possibly a unique aggregated time constraint). The maximization of household utility determines the relevant demand functions. This procedure can be used for assessing the effects of economic policies on individual behaviour and welfare. For example, Hausman, 1981 [46] estimates the effect of a variation in marginal taxation rates on hours worked, and measures the cost of the US taxation system in terms of welfare.

Family Economics in Unitary models

In the "unitary model", the household is considered as a unique decision-maker, and no attention is devoted to the complexity of the decision-making process and of the numerous transactions between family members. As pointed by Nobel Prize laureate Amartya Sen, 1983 [72], in the unitary models literature, the household members are incorporated into a "glued together family". Following the impulsion originated from seminal work by Nobel Prize laureate Gary Becker (1965 [6]; 1973 [START_REF] Becker | A theory of marriage: Part i[END_REF]; 1974 [START_REF] Becker | A theory of social interactions[END_REF]; 1991 [START_REF] Becker | A Treatise on the Family[END_REF]), recent developments in family economics broadens the classical research field to new questions such as the marriage decision or choices related to the number of children, their education, and the allocation of tasks and time among household members.

In unitary models, any difference in the demographic structure (e.g. differences between a single person and a couple with or without children) is either totally ignored, or simply reflected in ad hoc equivalence scales.

In addition, any conflicting situations that might arise among members (affecting the decisionmaking process) are disregarded. The unitary model indeed neglects the diverging interests that may arise among household members, and implicitly assumes that these members pursue consensual objectives, leading to a poor understanding (and thus modelling) of decision mechanisms (and therefore, of resources allocation) within the household. This generates three kinds of drawbacks. The first drawback is a wrong interpretation of empirical results. For example, Lise and Seitz, 2011 [START_REF] Lise | Consumption inequality and intra-household allocations[END_REF] show that failing to consider changes in the intrafamilial distribution of consumption leads to a major overestimation of inequality growth over the last 40 years in the UK. This bias should be taken into account in any poverty-reducing policy, and its recognition would lead to a fundamental rethinking of the change in intrafamilial distribution of consumption. The second drawback is a wrong assessment of the effects of economic policies on the well-being of each household member, concerning for example a change in the income taxation system. Lundberg, Pollak, and Wales, 1997 [START_REF] Shelly | Do husbands and wives pool their resources? evidence from the united kingdom child benefit[END_REF] illustrate the importance of these questions for family policy, and show that a shift of family benefits from the father to the mother during the 70s in the UK was followed by a rise in the demand for women's and children's clothes. This indisputable empirical result is inconsistent with the "unitary model" of the household in which every member has the same objective function. The third drawback is a poor predictive power of unitary models and a biased evaluation of behavioural effects of economic policies.

Unitary models implicitly or explicitly assume that the household is the basic decision unit, and acts as a unique decision-maker. In multi-person households, this implies that members are supposed to act as if the household preferences could be represented by a unique utility function, maximized under a unique budget (and time) constraint. The consensus model proposed by Samuelson, 1956 [70] could provide some justification of this unitary description, under very restrictive and unrealistic assumptions. Another attempt to legitimate the unitary approach was made by Becker, 1974 [8], with his famous "rotten kid" theorem. It basically states that, if there is a "benevolent dictator" in the family, then all family members, even if they are selfish, will act to maximize the same utility function as the benevolent dictator. The key assumption is that the benevolent dictator transfers money to each family member. All members then want to please the benevolent dictator in order to receive a larger transfer.

However, Bergstrom, 1989 [START_REF] Theodore | A fresh look at the rotten kid theorem-and other household mysteries[END_REF] later stressed that this attempt to justify unitary models relies on too arbitrary and unrealistic assumptions (especially the assumption about transfers).

Moreover, theoretical predictions derived from unitary models, such as income pooling, are often in contradiction with empirical data. Income pooling means that household decisions depend on household total resources, but not on the distribution of these resources among family members. This prediction comes from the assumption of a unique budget constraint defined at the household level, and a unique household utility function to be maximized.

Income pooling has been rejected, for example, by Thomas, 1990 [78], who showed that the relative contributions of men and women to the household income influence household decisions.

Family Economics in collective and other bargaining models

Collective models and other within-households bargaining models aim at answering the theoretical and empirical criticisms addressed to unitary models of family decision-making. They developed in two major directions.

"Strategic" models rely directly on the theory of non-cooperative games (see, e.g. Ashworth and Ulph, 1981 [START_REF] Ashworth | Household models[END_REF], Leuthold, 1968 [55]), while "collective" models proposed by Chiappori 1988 [25],

1992 [START_REF] Chiappori | Collective labor supply and welfare[END_REF] rely on the basic assumption that the household decision process leads to Paretoefficient allocations. The bargaining process may then be either explicit (as in McElroy and Horney, 1981 [START_REF] Mcelroy | Nash-bargained decisions: Toward a gen-eralization of the theory of demand[END_REF], or in Lundberg and Pollak, 1993 [57]), or non-specified (as in Chiappori, 1988 [25], 1992 [START_REF] Chiappori | Collective labor supply and welfare[END_REF]). In the latter case, the bargaining process is very general and not restrictive, only assuming Pareto-optimality. Pareto-optimality hypothesis seems natural for analysing household decisions since family members, who interact over a long period, are probably able to find mechanisms leading to efficient decisions.

Collective models can be used to study the welfare level of each household member, and therefore to analyse and measure in a consistent way the redistributive effects of any economic policy, not only at the household level but also at the individual level. Indeed, recent theoretical results show that, under some rather plausible conditions, individual utility functions can be recovered from household behaviour (and disentangled from bargaining power effects, whereas bargaining effects induce a bias in the measurement of preference parameters in unitary models). This evaluation of economic policies offers promising research avenues, especially in the context of urban development or transportation policies.

Labour supply models within the family

Examples of applications of the two-adult model are provided by Hausman and Ruud, 1984 [45]; Ransom, 1987 [START_REF] Michael | An empirical model of discrete and continuous choice in family labor supply[END_REF]; Bloemen, 1989 [START_REF] Hans | The added worker effect in a microeconomic model of the family with market rationing[END_REF]; Kapteyn, Kooreman, and van Soest, 1990 [START_REF] Kapteyn | Quantity rationing and concavity in a flexible household labor supply model[END_REF]. In these models, hours worked by the two spouses are treated as mixed discrete and continuous random variables, whereas in van Soest, 1995 [START_REF] Van Soest | Structural models of family labor supply: A discrete choice approach[END_REF] they are treated as discrete, which allows easily incorporating nonlinear taxes, joint filing, fixed costs of working, unemployment benefits, hours restrictions, unobserved wage rates of non-workers, and random preferences, etc., without challenging model tractability. The models are estimated using smooth simulated maximum likelihood relying on Gourieroux and Monfort, 1993 [START_REF] Gourieroux | Simulation-based inference: A survey with special reference to panel data models[END_REF].

The method is illustrated by several policy simulations. Unfortunately, it assumes a unitary model, which neglects the effect of the policy on respective bargaining powers, and may therefore lead to severe bias in labour supply reactions.

In the collective models (Chiappori, 1988 [25]; Chiappori, 1992 [START_REF] Chiappori | Collective labor supply and welfare[END_REF]), the only assumption made about the decision-making process is the Pareto efficiency (optimality of decisions). That is, the decision-making process involving more than on household member leads to allocations such that it would not be possible to make one household decision-maker better-off, without making at least another household decision-maker worse off. The collective models are very general in the sense that they do not rely on a restrictive specific bargaining process, and they do not assume any restrictive functional form for each member preferences. Collective models have proved at many occasions to perform better than the usual unitary models to explain observed behaviour (Vermeulen, 2002a [84]).

Collective models of labour supply have been developed for two-earner households (e.g., Fortin and Lacroix, 1997 [START_REF] Fortin | A test of the unitary and collective models of household labour supply[END_REF]; Moreau and Donni, 2002 [62]; Chiappori, Fortin, and Lacroix, 2002 [START_REF] Chiappori | Marriage market, divorce legislation, and household labor supply[END_REF]) in a continuous framework. However, the interest here is limited to the models in which utility functions are directly estimated and household labour supply is considered as a discrete choice problem. The contributions of Van Soest, 1995 [START_REF] Van Soest | Structural models of family labor supply: A discrete choice approach[END_REF], Bingley and Walker, 1997 [START_REF] Bingley | The labour supply, unemployment and participation of lone mothers in in-work transfer programmes[END_REF] and Keane and Moffitt, 1998 [START_REF] Keane | A structural model of multiple welfare program participation and labor supply[END_REF] are the first ones to use a discrete choice framework to study labour supply, but this was done in the context of a unitary model.

Van Soest, 1995 [START_REF] Van Soest | Structural models of family labor supply: A discrete choice approach[END_REF] developed a joint labour supply model for two spouses, in which the budget set is discrete. The discretization of the budget set allowed for incorporation of nonlinear taxation and non-convexities, which gave rise to a series of discrete collective labour supply models, starting with Laisney, 2002 [START_REF] Laisney | Welfare analysis of fiscal and social security reforms in europe: does the representation of family decision processes matter?[END_REF] and subsequently with Vermeulen et al., 2006 [START_REF] Vermeulen | Collective models of labor supply with nonconvex budget sets and nonparticipation: A calibration approach[END_REF], Vermeulen, 2006 [86], and Blundell et al., 2007 [22]. Laisney, 2002 [53] considers household labour supply as a discrete choice problem, and integrates non-participation and nonlinear taxation. Vermeulen et al., 2006 [87], develop a discrete choice collective model and solve it using a procedure mixing calibration and estimations. Blundell et al., 2007 [22], consider a model in which the male labour supply is discrete, whereas the female labour supply is continuous. Vermeulen, 2006 [86], models female labour supply in a discrete choice framework considering male labour supply as given, and including non-participation and nonlinear taxation.

Other discrete collective models of labour supply include Callan, Van Soest, and Walsh, 2009 [START_REF] Callan | Tax structure and female labour supply: Evidence from ireland[END_REF]; Bloemen, 2010 [START_REF] Hans | Income taxation in an empirical collective household labour supply model with discrete hours[END_REF]; Haan, 2010 [START_REF] Haan | A multi-state model of state dependence in labor supply: Intertemporal labor supply effects of a shift from joint to individual taxation[END_REF]; Michaud and Vermeulen, 2011 [START_REF] Michaud | A collective labor supply model with complementarities in leisure: Identification and estimation by means of panel data[END_REF]; Pacifico, 2012 [START_REF] Pacifico | On the role of unobserved preference heterogeneity in discrete choice models of labour supply[END_REF].

Intra-household interaction and group decision-making models

The main research stream applying discrete choice models to household decisions involving multiple decision-makers has been developed in the transportation literature. It includes the socalled intra-household interaction and group decision-making models of transportation, activitydemand, and location choices. Intra-household interaction models study how the household decides in the long run and in the short run. More precisely, these models take into account the interdependencies between residential location and workplace of household members (long term), or between activities and travel patterns (short term). Intra-household interaction models are interested in studying, for example, which activities are conducted in a day or over several days by the household members (sharing maintenance responsibilities, household cars, and pick-up and drop-off); when, where, by whom and with whom the activities are performed (joint or independent engagement on activities); and how job types and job locations of the different household members affect professional and residential mobility and location decisions of the household.

The general research stream concerned with intra-household interaction and group decisionmaking models of transportation, activity-demand, and location choices research stream can be subdivided into two interconnected subfields of study. The first subfield is concerned with choices in a long-term decision context (i.e. residential location and mobility; job location and mobility, car ownership). Abraham and Hunt, 1997 [START_REF] Abraham | Specification and estimation of nested logit model of home, workplaces, and commuter mode choices by multiple-worker households[END_REF]; Freedman and Kern, 1997 [START_REF] Freedman | A model of workplace and residence choice in twoworker households[END_REF]; Sermons and Koppelman, 2001 [START_REF] Sermons | Representing the differences between female and male commute behavior in residential location choice models[END_REF] and Waddell, 1996 [92] have analysed residential and professional location choices as resulting from a multiple-worker household decision-making process.

The second subfield is made of studies concerned with choices in a short-term decision context: the activity-travel demand literature (i.e. mode choice, travel behaviour, car sharing, and task allocation and activity based models). This literature develops models of task allocation, decisions related to joint travel and activity participation, mode choice, car sharing and so on.

Discrete choice modelling on these topics has been studied by Wen and Koppelman, 1999 [START_REF] Wen | Integrated model system of stop generation and tour formation for analysis of activity and travel patterns[END_REF], 2000 [START_REF] Wen | A conceptual and methodological framework for the generation of activity-travel patterns[END_REF]; Gliebe and Koppelman, 2002 [START_REF] Gliebe | A model of joint activity participation between household members[END_REF]; Scott and Kanaroglou, 2002 [START_REF] Scott | An activity-episode generation model that captures interactions between household heads: development and empirical analysis[END_REF]; Vovsha, Petersen, and Donnelly, 2003 [START_REF] Vovsha | Explicit modeling of joint travel by household members: Statistical evidence and applied approach[END_REF]; 2004a [START_REF] Vovsha | Model for allocation of maintenance activities to household members[END_REF]; 2004b [START_REF] Vovsha | Impact of intrahousehold interactions on individual daily activity-travel patterns[END_REF]; Bradley and Vovsha, 2005 [START_REF] Bradley | A model for joint choice of daily activity pattern types of household members[END_REF];

Srinivasan and Athuru, 2005 [START_REF] Karthik | Analysis of within-household effects and between-household differences in maintenance activity allocation[END_REF]; Srinivasan and Bhat, 2006 [76], among others.

The work of Gliebe and Koppelman, 2002 [START_REF] Gliebe | A model of joint activity participation between household members[END_REF]; Scott and Kanaroglou, 2002 [START_REF] Scott | An activity-episode generation model that captures interactions between household heads: development and empirical analysis[END_REF]; Vovsha, Petersen, and Donnelly, 2003 [START_REF] Vovsha | Explicit modeling of joint travel by household members: Statistical evidence and applied approach[END_REF]; Srinivasan and Bhat, 2006 [76] concerns the decision to participate in an activity jointly or independently from other household members. Gliebe and Koppelman, 2002 [START_REF] Gliebe | A model of joint activity participation between household members[END_REF] model independent activity participation, allocation of time to joint activities, and the interplay between individual and joint activities using a proportional share model. Scott and Kanaroglou, 2002 [71] develop a trivariate (by household type) ordered probit to model the number of non-work, out-of-home activity episodes for household heads.

The work concerned with task allocation (of maintenance activities) is better represented by discrete choice model systems that are embedded in tour-based travel demand modelling systems. On the one hand there is the discrete choice system of Vovsha, Petersen, and Donnelly, 2003 [START_REF] Vovsha | Explicit modeling of joint travel by household members: Statistical evidence and applied approach[END_REF]; 2004a [START_REF] Vovsha | Model for allocation of maintenance activities to household members[END_REF]; 2004b [START_REF] Vovsha | Impact of intrahousehold interactions on individual daily activity-travel patterns[END_REF] that is the joint travel model component that makes part of the Mid-Ohio Regional Planning Commission. On the other hand there is the discrete choice system of Bradley and Vovsha, 2005 [START_REF] Bradley | A model for joint choice of daily activity pattern types of household members[END_REF] that is part of the activity-based model of the Atlanta region. Bradley and Vovsha, 2005 [23] survey the contributions on activity-travel demand literature, in which either intra-household decision-making is not considered explicitly, or discrete choice model techniques are not used.

The attention is restricted here to theoretical or empirical developments that consider a discrete choice modelling strategy in activity-travel demand models accounting for interpersonal dependencies in multiple decision-makers households. However, the activitytravel demand literature has also used seemingly unrelated regressions (SUR) and structural equation modelling (SEM) to account for household interactions (see Srinivasan and Bhat, 2005 [75]). These approaches usually develop a SUR or SEM system of two or more equations corresponding to the time invested in activities by the household head and the other members in consideration (household head spouse and/or children or other active household members).

Other classifications of activity-travel demand models that account for interpersonal dependencies in households with multiple decision makers have been proposed. For instance, Timmermans, 2009 [START_REF] Timmermans | Household decision making in travel behaviour analysis[END_REF] classifies activity-based travel demand models that explicitly consider interactions within households with multiple decision makers into three categories: microsimulation, rule-based and utility-maximizing models. Micro-simulation models simulate a household member daily activity-travel pattern using algorithms that replicate the observed patterns from data (including time constraints and actual decision-making outcome) giving timing and sequence of activities schedules that account for household's and personal's characteristics (see, e.g., Pribyl and Goulias, 2005 [START_REF] Pribyl | Simulation of daily activity patterns incorporating interactions within households: Algorithm overview and performance[END_REF]). The second category of models is referred to as the rule-based models. They build multi-agent computational processes in which the individual activity-travel decisions reflect "if-then" decision tree structures, regarding which activities, with whom, and for how long the activities are conducted (see, e.g., Arentze and Timmermans, 2004 [3]).

Timmermans' last category of models corresponds to the utility-maximizing models. He further subdivides utility-maximizing models into those using the discrete choice approach (based on the random utility models) and in those using the time allocation approach. Time allocation models are based on a group utility function. This function is a linear function of individualspecific terms and of interaction terms that reflect the interactions between different individuals in a multiplicative form. The household then allocates its time to activities such that its utility would be maximized given individual time constraints (see, e.g. Zhang and Fujiwara, 2006 [98]).

Location and Accessibility

Lee et al, 2010 [START_REF] Brian | Reexamining the influence of work and nonwork accessibility on residential location choices with a microanalytic framework[END_REF] categorized accessibility measurement approaches into four groups: the proximity based (measured in term of travel time, distance, etc.), the gravity-based (derived from the denominator in the gravity model), the cumulative opportunities approach (as a special case of the gravity-based measure), and the utility-based approach (denominator of the MNL model). See Lee et al, 2010 [START_REF] Brian | Reexamining the influence of work and nonwork accessibility on residential location choices with a microanalytic framework[END_REF] for more details on this classification and for further references.

The utility-based approach allows for the development of disaggregated or individual-specific accessibility measures captured in the log-sum variable when the nested logit formulation is used. See Ben-Akiva and Lerman, 1979 [START_REF] Ben | Disaggregate travel and mobility choice models and measures of accessibility[END_REF]; Srour et al., 2002 [77]; Waddell and Nourzad, 2002 [93]; and Zondag and Pieters, 2005 [START_REF] Zondag | Influence of accessibility on residential location choice[END_REF] among others.

A major determinant of household location is accessibility. Accessibility to jobs measures the spatial proximity of the residential location to the job location. In the absence of income effects, and with a Logit demand function, the accessibility is measured by a log-sum term, which is a measure of consumer surplus. In the Logit case, accessibility corresponds to the expected maximum utility, which is equal to the Logarithm of the denominator of the Logit demand function. It is easy to see that when the utility is additive in income, the derivative of the accessibility is the demand function. This is a direct application of Roy's identity (Anderson, de Palma, and Thisse, 1992 [START_REF] Anderson | Discrete Choice Theory of Product Differentiation[END_REF]). The same property is true for the Generalized Extreme Value case, provided that the utility is additive in income (and, as a consequence, there are no income effects). The reader is referred to Anderson, de Palma, and Thisse, 1992 [START_REF] Anderson | Discrete Choice Theory of Product Differentiation[END_REF]; and de Palma and Kilani, 2007 [START_REF] De | Invariance of conditional maximum utility[END_REF] for details.

In the homogenous case, all agents have the same preferences (this means in particular, the same values of time and the same preferences for jobs), and therefore have the same measure of accessibility. In the heterogeneous case, the accessibility depends on household characteristics and in particular on the value of time.

Research on residential location has commonly used accessibility as an aggregated measure of ease of access to jobs or people in choice models where the household is considered as a single decision-making unit (individual or unitary approach). By contrast, Chiappori, de Palma, Picard and Inoa, 2013 [START_REF] Chiappori | Couple residential location and spouses workplaces[END_REF] have studied residential location of households including two active spouses. In the context of a collective model assuming Pareto-optimality, they measure simultaneously spouses' respective values of time and bargaining powers. They show that neglecting bargaining powers can lead to a bias in the estimated values of time which may reach 20%. The order of magnitude of this bias is comparable to the male-female difference in values of time. Similarly, Picard et al., 2013 [67] measure simultaneously spouses' respective values of time and bargaining powers in a joint mode choice model.

In the individual or unitary approach, accessibility has been (1) studied in single and multipleworker location choice models, (2) measured using different approaches, and (3) used as an indicator of non-work activities. We develop in the following subsections these three issues on the use and measurement of accessibility measures.

Accessibility Measures From Multiple Worker Location Choice Models

The study of the interactions between household members within a household in residential location choice models has allowed the differentiation of accessibility measures by sociodemographic characteristics, identifying differences between females and males, and between multiple-worker households and one-worker households. The reference studies of multipleworker residential location choice models are the works of Timmermans et al., 1992 [80],

Abraham and Hunt, 1997 [START_REF] Abraham | Specification and estimation of nested logit model of home, workplaces, and commuter mode choices by multiple-worker households[END_REF], Freedman and Kern, 1997 [START_REF] Freedman | A model of workplace and residence choice in twoworker households[END_REF], and Sermons and Koppelman, 2001 [START_REF] Sermons | Representing the differences between female and male commute behavior in residential location choice models[END_REF]. Note that Timmermans et al., 1992 [80] studied the residential location choice of two-worker households but using a nine-step (decompositional) joint choice model.

Abraham and Hunt, 1997 [START_REF] Abraham | Specification and estimation of nested logit model of home, workplaces, and commuter mode choices by multiple-worker households[END_REF] used a logit model structure with a system for weighting the contributions of different workers to the household utility in a three-level nested logit (residential location, workplace, and mode choice). Freedman and Kern, 1997 [START_REF] Freedman | A model of workplace and residence choice in twoworker households[END_REF], analyses residential location and workplace with a joint logit model where a two-worker household jointly chooses residential location and both spouses workplace to maximize utility, subject to budget and time constraints. Sermons and Koppelman, 2001 [73] develop a multinomial logit model of residential location choice to study differences between males and females in sensitivity to commuting time for two-worker households.

In general, these studies showed that females are more sensitive to commuting time and accessibility measures than males. Demographic characteristics such as presence of children, workplace status, and spouses' occupation and workplace location, are determinants for commuting time and accessibility measures, and therefore residential location choices in a multiple worker household.

Individual-specific Accessibility Measures

Despite the variety of contributions to the study of residential location, little has been said regarding the influence of job type on the individual-specific accessibility to jobs, and therefore on the residential location and workplace choices when individuals are considered forwardlooking. Household members choose a workplace conditional on their current residential location, while also considering the future changes on workplaces when choosing their residential location.

Inoa, Picard, and de Palma, 2014 [START_REF] Inoa | Effect of an accessibility measure in a model for choice of residential location, workplace, and type of employment[END_REF] have elaborated a three-level nested logit model that allows to study the interdependency of residential location and workplace, while accounting for variation of preferences for job types across individuals. Residential location is the upper level choice, and workplace and job type are the middle and lower level choices, respectively. With this nested structure, an individual-specific accessibility measure is constructed, which corresponds to the expected maximum utility across all potential workplaces and job types.

When considering accessibility to jobs, the choice of a particular workplace depends on the distribution of jobs by type, which are valued differently by different workers. Their modelling of the job type choice allows them to compute an individual-specific measure of attractiveness to job types (log-sum variable) and to use it in the workplace location choice model.

Using data from the Paris Region Census, Inoa, Picard, and de Palma, 2014 [START_REF] Inoa | Effect of an accessibility measure in a model for choice of residential location, workplace, and type of employment[END_REF] find that the individual-specific job type attractiveness measure is a more significant predictor of workplace location than the standard total number of jobs measure. Most importantly, the individualspecific accessibility measure is an important determinant of the residential location choice, and its impact on the residential location choice strongly depends on gender, fertility, age, and education. Some resulting individual-specific accessibility maps are displayed in Annex. They show that accessibility is more equally distributed over the region for the lowest education level, whereas it is more concentrated in the Central business district for the most educated workers.

Time Geographic Measures of Accessibility

Activity Pattern Models

The literature on residential location has not restricted accessibility to the concept of proximity to jobs. It has also studied the accessibility to different (non-work) activity opportunities and measured their respective influence on residential location. Activity-travel demand and task allocation models are concerned with the activity patterns of households and individuals all over a day (and even all over a week, in the new activity-based time use data sets). Capturing nonwork accessibility is therefore essential when modelling in-home and out-of-home activity patterns and trip chaining (Neutens et al., 2012 [63]). Accessibility measures adapted for these models can be found in the framework of time geographic measures of accessibility.

Hägerstrand, 1970 [START_REF] Hagerstrand | What about people in regional science[END_REF] introduced the concept of time-space prism (TSP) in order to describe the temporal and spatial constraints in which individuals travel to and participate in activities.

Time-space prisms define the locations that an individual can reach given a time budget. The area shaped by the potential locations that the individual can reach in a given time gap is referred to as the potential path area (PPA). A thorough study of the time-geographic measures can be found in the work of Miller, 1991 [START_REF] Harvey | Modeling accessibility using space-time prism concepts within geographical information systems[END_REF] and Kwan, 1998 [52]. Kim and Kwan, 2003 [50] provide a review on accessibility measures used in empirical settings derived from the timespace prism.

There exist only a few applications where time-geographic measures of accessibility have been used in the literature under consideration here, that is, discrete choice location models and activity-travel demand and time allocation models that consider intra-household interactions. Among these applications are the work of Lee et al., 2010 [START_REF] Brian | Reexamining the influence of work and nonwork accessibility on residential location choices with a microanalytic framework[END_REF]; Yoon and Goulias, 2009 [START_REF] Seo | Impact of individual accessibility on travel behavior and its propagation through intra-household interaction[END_REF], 2010 [START_REF] Seo | Constraint-based assessment of intrahousehold bargaining on time allocation to activities and travel using individual accessibility measures[END_REF]; Kitamura et al., 2001 [51]; Ettema, 2006 [36]. Lee et al., 2010 [54] developed a discrete choice residential location model that includes a disaggregated accessibility measure to non-work activities (derived from the TPS framework), while also accounting for the accessibility to jobs. Yoon and Goulias, 2009 [START_REF] Seo | Impact of individual accessibility on travel behavior and its propagation through intra-household interaction[END_REF], 2010 [START_REF] Seo | Constraint-based assessment of intrahousehold bargaining on time allocation to activities and travel using individual accessibility measures[END_REF] developed a structural equations model of activity and time allocation that consider intrahousehold interactions in households without child only and then in households with and without child where the accessibility measure used is based on time geography. Using timegeographic accessibility measure, Kitamura et al., 2001 [51] studied the influence of travel patterns and residential location on car ownership; and Ettema, 2006 [36] developed a discrete continuous Tobit model of activity participation and duration.

Dynamic Transport Models

The time geographic models described above neglect the interactions between households.

However, such interactions are important in the case of congested cities and they evolve across time and geography. Such externalities are analyzed in dynamic transport models.

METROPOLIS is a dynamic model which describes mode choice, route choice and departure time choice (see de Palma and Marchal, 2002 [START_REF] De | Real cases applications of the fully dynamic metropolis tool-box: an advocacy for large-scale mesoscopic transportation systems[END_REF]; de Palma, Kilani, and Lindsey, 2005 [START_REF] André De Palma | Congestion pricing on a road network: A study using the dynamic equilibrium simulator metropolis[END_REF]; and de Palma and Lindsey, 2006 [START_REF] De | Modelling and evaluation of road pricing in paris[END_REF]). It is dynamic in the sense that congestion depends on the time of the day. It uses a nested logit model, where the mode choice is made at step one, while departure time choice is made at step two.

The departure time choice is given by a continuous logit model, with the log-sum formula as the welfare measure. In this case, the mode choice model, at the upper level, depends on the accessibility at the lower stage. Consider a user going from origin i to destination j. The attractiveness of this car user (the formula is similar for the public transportation users) is:

    0 exp / , T ij ij A t C u du     
where C ij (u) represents the generalized cost of an individual using private transportation, and u (and t) denote the departure time. Following Vickrey, 1969 [88], this generalized cost is given by:

          * ( ) * , ij ij ij ij C u tt u t u tt u u tt u t           
where  is the value of time,  is the unit schedule delay early parameter,  is the unit schedule delay late parameter, t * is the desired arrival time at destination and tt ij (u) is the travel time given the departure time u. Such measure is potentially useful for other applications, such as activity choice, or residential location. In the latter case: the log-sum aggregation over the destinations j provides a potential user benefit of residential location at i.

In the case of couples, the value of time of the man may not be the same in the case his spouse is at home at the time he leaves and in the case she leaves before him, and vice versa. In that case, the generalized cost function of the man depends on the departure time of the woman, and the generalized cost function of the woman depends on the departure time of the man. The resulting within-family externalities have been studied by de Palma, Picard, and Lindsey, 2012 [START_REF] André De Palma | Departure time choice within the family[END_REF]. They showed that, although cooperation is clearly beneficial for couples themselves, it may exacerbate congestion and thus worsen negative externalities between families.

Interactions within families outside the household

Compton and Pollak, 2009 [START_REF] Compton | Proximity and coresidence of adult children and their parents: Description and correlates[END_REF] have analysed interactions within larger families, living in different households. They describe and analyse the patterns of proximity and co-residence involving adult children and their mothers using data from the National Survey of Families and Households (NSFH) and the U.S. Census. Their idea is that the ability of family members to engage in intergenerational transfers of hands-on care requires close proximity or co-residence.

They find that, in spite of the decline in intergenerational co-residence in the United States, most Americans still live within 25 miles of their mothers, and even closer for the lowest educational levels. Individual characteristics such as age, race and ethnicity affect both the probability of co-residence and close proximity, and their effect depends on gender and marital status, indicating the need to model the corresponding categories separately.

Compton and Pollak, 2011 [START_REF] Compton | Family proximity, childcare, and women's labor force attachment[END_REF] further show that close geographical proximity to mothers or mothers-in-law has in turn a substantial positive effect on the labour supply of married women with young children. They argue that proximity increases labour supply through the availability of childcare. Their interpretation of availability is there broad enough to include not only regular scheduled childcare during work hours but also an insurance aspect of proximity (e.g., a mother or mother-in-law can provide irregular or unanticipated childcare). Using large American datasets, they find that the predicted probability of employment and labour force participation is 4-10 percentage points higher for married women with young children living in close proximity to their mother or their mother-in-law compared to those living further away.

Conclusion and extensions

There is still a long way to achieve the goal of providing a theoretical and empirical framework to model and apply economics of the family models. Decision-making within the family has been neglected too long in transportation. Two special issues by Bhat and Pendyala, 2005 [17] and by Timmermans and Junyi Zhang, 2009 [81] provide the most notable exceptions. We also refer the reader to the discussion on group behavior, held at 11 th International Conference on Travel

Behavior Research (Kyoto), and organized by J. Zhang and A. Daly (Timmermans, 2009 [79]).

The objective of this paper was to set-up a flexible framework to discuss the development of integrated transportation models involving interacting and interdependent actors. showed that cross-fertilization between the economics of the family and the transportation field is needed (). We hope that this chapter will provide a first step to fulfil this gap. Some preliminary discussion on how to integrate bargaining and collective models in transportation and urban economics can be found in Ben-Akiva et al., 2012 [START_REF] Ben-Akiva | Process and context in choice models[END_REF].

  Transportation is very keen to have access to this type of models, since their applications are numerous. Let mention, for example, residential location choice, workplace choice, car ownership, choice of children's school, mode choice, departure time choice activity patterns and the like. The (non unitary) economics of the family models are totally different models, which do not merely extend existing discrete choice models. They introduce new concepts, which are specific to within family interactions: negotiation, altruism, or repeated interaction and Pareto optimality.It is our belief that that activity pattern, mode choice, allocation of time, residential and job location choices, as well as departure time choice, cannot be analysed in the family, without the idea that there are almost always conflicting interests, given the budget and the time constraints of the different family members. Therefore, some coordination and cooperation is needed to achieve common goals, even if the cost borne by spouses is often not identical. The transportation field has made large advances in this area, but the connection with the theoretical and the econometric model are still open. The group behavior discussion in the 11 th International Conference on Travel Behavior Research


Farouk Benmeddour
email: farouk_benmeddour@yahoo.fr

Fabien Treyssede

Laurent Laguerre

Fabien Treyssède

Numerical prediction of guided waves interaction with non-axisymmetric cracks in elastic cylinders

Keywords: Guided waves, Cylinders, FEM-SAFE, Ultrasonics, Hybrid method

Numerical prediction of guided waves interaction with non-axisymmetric cracks in elastic cylinders

Introduction

The motivation of this work is highlighted by the non-destructive testing need of cables, which are widely used in civil engineering. Environmental and operational conditions are likely to cause defects such as corrosions and cracks. Guided waves might be an appropriate technique since they can carry-out energy over long distances. Cables are generally constituted of a central cylinder surrounded by six helical wires. Given the complexity of such a geometry, only a cylindrical structure is studied in this work [START_REF] Pochhammer | Über die fortpflanzungsgeschwindigkeiten kleiner schwingungen in einem unbegrenzten isotropen kreiszylinder[END_REF]. The interaction of elastic guided waves with damages is complex since they are multi-modal and dispersive. Models are often required for a better understanding of these interaction phenomena.

A wide range of work has been reported on the interaction of guided waves with defects in plates [START_REF] Benmeddour | Study of the fundamental Lamb modes interaction with asymmetrical discontinuities[END_REF], cylinders [START_REF] Engan | Torsional wave scattering from a diameter step in a rod[END_REF] and tubes [START_REF] Demma | The reflection of guided waves from notches in pipes: a guide for interpreting corrosion measurements[END_REF]. However, researchers generally use two dimensional models based on a given symmetry to avoid a heavy and complex three dimensional analysis. Hence, these methods can not deal with non-axisymmetric damages. The special case of the free-end cylinder has been studied first by Gregory and Gladwell [START_REF] Gregory | Axisymmetric waves in a semi-infinite elastic rod[END_REF] then by Rattanawangcharoen et al. [START_REF] Rattanawangcharoen | Reflection of waves at the free edge of a laminated circular cylinder[END_REF] and Taweel et al. [START_REF] Taweel | Wave reflection from the free end of a cylinder with an arbitrary cross-section[END_REF].

The aim of the paper focusses on the interaction of guided waves with non-axisymmetric cracks in a cylinder, which require a fully three dimensional approach. In section 2, a hybrid method involving the classical finite element method (FEM) and the semi-analytical finite element (SAFE) technique is developed. The average power flow is directly derived from SAFE matrices. Hence, the reflection and transmission power flows of guided modes can be readily determined. Section 3 deals with the interaction of the first fundamental compressional Pochhammer-Chree mode with non-axisymmetric cracks. Results for cracks with different opening angles are obtained and discussed. A comparison with results found in the literature for the free-end cylinder is performed with success.

Analytical development of the hybrid method

2.1.General description of the problem

In this section, a three-dimensional hybrid FEM-SAFE method is proposed. Hybrid methods have successfully been applied to two-dimensional problems (see Ref. [START_REF] Rattanawangcharoen | Axisymmetric guided waves in jointed laminated cylinders[END_REF] for instance). For conciseness, this method is only presented for the special case of a free-end cylinder. It can readily be generalized to an infinite waveguide by adding a right section. Figure 1 depicts a damage located in an arbitrary volume V at the free-end of an arbitrary waveguide section. The arbitrary section have a uniform geometry along the propagation direction. An incident wave is launched in the positive direction (in). The reflected waves (re), from both the damage and the free-end, travel in the negative direction. The arbitrary section is situated in the left while the damage and the free-end are situated in the interior region I of the volume. The two regions are connected each other with the left surface S L where L designates left. The whole volume V is bounded by the traction free surface S T and the left section S L .

Figure 1. Representation of a damage in an arbitrary volume at the free-end of an arbitrary

waveguide section.

2.2.The Global system of the hybrid method FEM-SAFE

The volume V is studied with the standard FEM. It can be shown that the variational formulation for the FEM can be expressed as:

〈U L  U I 〉 [ D L L D LI D IL D II ]{ U L U I } =〈 U L U I 〉 { F L 0 } , (1)
where {U L } and {U I } are the left and interior displacements of the degrees of freedom

(dofs), [ D L L ] , [ D LI ] , [ D IL ] and [ D II ] are the selected dofs of the global dynamic stiffness matrix [ D]=[ K ]- 2 [ M ] (  is the angular frequency).
The far field is investigated by using eigenmode expansions based on the SAFE method. The SAFE method is first used in order to determine the elastic guided modes for the inlet cross-section of the FEM region. This technique reduces three dimensional waveguide analyses to two dimensions by using a spatial Fourier transform along the propagation direction. Therefore, only the cross section of the waveguide is analysed, which allows a fast and accurate computation of eigenmodes (wavenumbers and modeshapes). These modes correspond to either ingoing (+) or outgoing (-) waves. The variational formulation of the left section can be written as (see Ref. [START_REF] Hayashi | Guided wave dispersion for a bar with an arbitrary cross-section, a rod and rail example[END_REF] for instance):

〈U L 〉 [ K 1 ] - 2 [ M S ] -j k [ K 2 ] -[ K 2 ] t *  k 2 [ K 3 ] {U L }=0 , (2)
where the superscript t designates the transpose and  *  is the complex conjugate.

The next step consists in performing an eigenmode expansions of the FEM solutions at the inlet cross-section. These solutions can then be naturally decomposed as incident and reflected normal modes. The application of the eigenmode expansions on the displacements and forces leads to:

{U L }=[B L  ]{ L  }[B L -]{ L -}, {U L }=[ B L -]{  L -}, {F L }=[T L  ]{ L  }[T L -]{ L -} , (3) with [ B L ± ]= [ {U L1 ± } {U L2 ± } ... { U L N L ± ± }] , [T L ± ]= [ { F L1 ± } {F L 2 ± }... { F L N L ± ± }]
and N L ± is the maximum number of modes used in the computation. { L ± } is the vector of the ingoing (+)

and the outgoing (-) coefficients of the mode shapes displacements {U L ± } , respectively.

Based on Ref. [START_REF] Treyssède | Elastic waves in helical waveguides[END_REF], it can be shown that {F L } for each mode can be computed directly from the SAFE matrices:

{F L n ± }=-[ K 2 ] jk n ± [ K 3 ] {U Ln ± } , ( 4
)
where n is the mode number. The substitution of the above quantities (3) in the equation (1) gives the global system of the hybrid method:

[ B L -t 0 0 I d ][ D L L D LI D IL D II ][ B L - 0 0 I d ] -[ T L - 0 0 0 ]  {  L - U I } =-[ B L -t 0 0 I d ][ D L L D LI D IL D II ] [ B L  0 ] -[ T L  0 ]  { L  } , (5)
where I d is the identity matrix. The enforced coefficients vector { L  } are known while { L -} and {U I } are variables to be computed by solving the global system (5).

2.3.The power balance

The power reflection and transmission coefficients are computed by dividing the reflected and the transmitted power by the incident power, respectively ( R nm =P n re  / P m inc  and T nm =P n tr  /P m inc where: m is the incident mode and n is the reflected or the transmitted mode). The power incident, reflected and transmitted coefficients are determined by using the average power flow [START_REF] Demma | The reflection of guided waves from notches in pipes: a guide for interpreting corrosion measurements[END_REF]. This average power flow can be derived directly from SAFE matrices [START_REF] Treyssède | Elastic waves in helical waveguides[END_REF] as, for example, for the m th incident single eigenmode:

P m inc  =  2 ℑ  Lm  * {U Lm  } t * {F Lm  } Lm   , (4)
where ℑ designates the imaginary part. The power balance is satisfied when the sum of the power reflected and transmitted coefficients equals unity.

NDTCE'09, Non-Destructive Testing in Civil Engineering

Nantes, France, June 30th -July 3rd, 2009

Numerical results

3.1.General description

A set of computations is carried out to analyse non-axisymmetric cracks and the free-end of an elastic cylinder. Cracks with different opening angles are studied. The opening angle  takes values: n /2 , where n=1, 2, 3 and 4 . For the special case of n=4 , the damage is a free-end cylinder. For conciseness, the range of dimensionless frequency ( = r /c L , r is the cylinder radius and c L is the longitudinal celerity) is restricted to vary between 1.8 and 2.4. This range is chosen to enable comparisons with results found in the literature (see Section 3.2). The launched wave is the compressional L(0,1) Pochhammer-Chree mode. This mode is applied at the left section of the cylinder with a unit amplitude.

3.2.Non-axisymmetric cracks' analysis

Figures 2a-d illustrate the power reflection coefficients of the reflected compressional modes L(0,1), L(0,2) and L(0,3) with respect to dimensionless frequency. It is shown that the power reflection coefficients are sensitive to the crack opening angle. For all computations, the sum of the power reflected and transmitted coefficients at each frequency equals unity with an error less than 0.1%.

Figure 2d depicts the power reflection coefficients when the L(0,1) mode encounters a free-end cylinder. As expected and due to the axisymmetric geometry, no mode conversions to the flexural modes are observed. These results are compared to those of Gregory and Gladwell [START_REF] Demma | The reflection of guided waves from notches in pipes: a guide for interpreting corrosion measurements[END_REF], which are represented with black points. A very good agreement is found, which validates the hybrid FEM-SAFE method.

Figures 3a andb show the power transmission coefficients of the compressional modes when the L(0,1) interacts with a crack of opening angle =n /2 ( n=1, 2 ) respectively. Unlike power reflection coefficients, the power transmission coefficients of the converted modes (L(0,2) and L(0,3)) are insignificant (less than 0.1, see Fig. 3). Figures 4a andb depict the power reflection and transmission coefficients when the L(0,1) mode encounters a crack with = . It is shown that the power reflection and transmission coefficients of the converted flexural modes (F(1,1), F(1,2), F(1,3), F(2,2) and F(3,2)) have the same order of magnitude.

Conclusions

A three-dimensional hybrid method combining the FEM and the SAFE techniques is developed. The interaction of the fundamental compressional guided wave with nonaxisymmetric cracks and a free-end cylinder is performed. The power balance is satisfied and a comparison with other results is carried out with success. For the considered frequency range, the interaction of the L(0,1) mode with the free-end cylinder and three cracks show significant mode conversions to compressional modes (L(0,2) and L(0,3)). Relatively low mode conversions to flexural modes are observed (F(1,1), F(1,2), F(1,3), F(2,2) and F(3,2)). This work shows that the proposed three-dimensional hybrid FEM-SAFE method is a powerful predicting tool for analysing arbitrary damaged waveguides. This method is relatively easy to implement in an existing FE code and could help optimising inspection strategies for non-destructive evaluation. Experimental measurements are in progress and will be presented in future works.

  NDTCE'09, Non-Destructive Testing in Civil EngineeringNantes, France, June 30th -July 3rd, 2009

Figure 2 .

 2 Figure 2. Power reflection coefficients of the compressional modes when the L(0,1) mode encounters a crack with an opening angle  equal to /2 (a),  (b), 3 / 2 (c) and 2  (d).

Figure 3 .

 3 Figure 3. Power transmission coefficients of the compressional modes when the L(0,1) mode encounters a crack with an opening angle  equal to /2 (a) and  (b).

Figure 4 .

 4 Figure 4. Power reflection (a) and transmission (b) coefficients of the flexural modes when the L(0,1) mode encounters a crack with an opening angle = .

  NDTCE'09, Non-Destructive Testing in Civil Engineering Nantes, France, June 30th -July 3rd, 2009

Acknowledgements

This research work is part of the project ACTENA (Auscultation des Câbles Tendus Non Accessibles) funded by the French ANR (Agence National de Recherche) and EDF (Électricité De France) and piloted by LCPC (Laboratoire Central des Ponts et Chaussées).


Ahmed Frikha
email: ahmed.frikha@lcpc.fr

Fabien Treyssede

Patrice Cartraud

F Treyssède

Modelling of elastic wave propagation in prestressed helical waveguides

The goal of this paper is to numerically investigate the effect of prestress on the propagation of elastic waves in straight and helical waveguides. The simulation technique, based on a semi-analytical finite element (SAFE) method, reduces the problem to the two-dimensional cross-section. The threedimensional equilibrium equations of prestressed structures are written in the covariant and contravariant bases and used for the SAFE method. The eigenproblem can be solved by fixing the wavenumber k and finding the angular frequency ω or inversely. Dispersion curves for single straight and helical wires will be computed to quantify the effect of prestress on the wave propagation.

Introduction

Guided waves are commonly used to control large components such as plates, tubes and strands since they have the advantage to spread over long distances with little loss of energy. Because of the dispersive and multimodal behavior of guided waves, simulation becomes a very helpful tool for a proper analysis of these tests. The study of wave propagation in prestressed helical structures such as strands and springs requires the development of the equation of dynamics in a helical coordinate system. In the framework of modelling elastic guided waves, this paper presents a numerical approach dedicated to helical prestressed structures.

Curvilinear coordinate system

Let us considering a helical waveguide with a constant cross-section along the axis. The helix centerline curve can be described by the following position vector :

r(s) = R cos( 2π l s)e X +R sin( 2π l s)e Y +( L l s)e Z (1)
where l = √ L 2 + 4π 2 R 2 is the curvilinear length of one helix step, (e X , e Y , e Z ) represents the Cartesian basis. R and L are respectively the radius of the centerline in the (e X , e Y ) plane and the helix step along e Z (see Figure1). The variable s, corresponding to the arc length, varies between 0 and l. One considers the Serret-Frenet basis (e n , e b , e t ) associated to the helix centerline. The unit tangent, normal and binormal vectors to the centreline are respectively obtained from e t = ∂r/∂s, ∂e t /∂s = κe n and ∂e n /∂s = τ e b -κe t (e b = e t ∧ e n ). For a helix, both the curvature κ = 4π 2 R/l 2 and the tortuosity τ = 2πL/l 2 are constant. Any cartesian vector x can be expressed in the Serret-Frenet basis as follows :

x(x, y, s) = r(s) + x e n (s) + y e b (s) (2)
This mapping defines the non-orthonormal covariant basis (g 1 , g 2 , g 3 ). g 1 , g 2 and g 3 , are given by: g 1 = ∂x/∂x, g 2 = ∂x/∂y and g 3 = ∂x/∂s. We respectively define the contravariant basis (g 1 , g 2 , g 3 ) , which vectors are given by g i • g j = δ j i . Components of the covariant (resp. contravariant) metric tensor are defined by g mn = g m • g n (resp. g mn = g m • g n ). This covariant metric tensor is given by:

g =   1 0 -τ y 0 1 τ x -τ y τ x τ 2 (x 2 + y 2 ) + (1 -κx) 2   (3)
Note that it does not depend on the third spatial variable s. The Christoffel symbol of the second kind, defined by Γ k ij = g i,j • g k , are given by:

Γ k 11 = Γ k 12 = Γ k 21 = Γ k 22 = 0, Γ 2 23 = Γ 2 32 = Γ 3 23 = Γ 3 32 = 0, Γ 1 33 = κ(τ y) 2 1 -κx + κ(1 -κx) -τ 2 x, Γ 2 33 = - κτ 2 xy 1 -κx -τ 2 y, Γ 3 33 = κτ y 1 -κx , Γ 1 13 = Γ 1 31 = - κτ y 1 -κx , Γ 1 23 = Γ 1 32 = -τ, Γ 2 13 = Γ 2 31 = κτ x 1 -κx + τ, Γ 3 13 = Γ 3 31 = - κ 1 -κx (4)
All covariant (resp. contravariant) quantities are indexed down (resp. top). The notation (• ), i (i = 1, 2, 3) is used for derivatives with respect to x, y and s. For more details on curvilinear coordinate systems, the reader can refer to Ref. [START_REF] Chapelle | The Finite Element Analysis of Shells -Fundamentals[END_REF] for instance.

2 Variational formulation of prestressed structures Assuming a linearly elastic material, static prestressed state with small strains, small superimposed perturbations with a time harmonic e -iωt dependence, the 3D variational formulation governing prestressed dynamics is given by [START_REF] Bathe | Finite Element Procedures[END_REF]:

V 0 δǫ : C : ǫ dV 0 + V 0 tr(∇δu• σ 0 • (∇u) T ) dV 0 -ω 2 V 0 ρδu • u dV 0 = 0 (5)
for any kinematically admissible trial field δu. u and ǫ respectively denote the displacement vector and the Green-Lagrange strain tensor. ρ is the material density and V 0 represents the prestressed structural volume. C and σ 0 denote respectively the matrix of material properties and the Cauchy prestressed tensor (assumed to be known). tr(• ) and ∇(• ) respectively denote the trace and the gradient tensor with respect to predeformed variables. Each quantity of Eq. ( 5) must be written in the covariant and contravariant coordinate systems. In the covariance basis, components of ∇u, denoted γ ij , can be written as follows:

γ ij = (u i,j + u j,i ) -Γ k ij u k .
The strain tensor ǫ is written in terms of the gradient tensor as follows:

ǫ = 1/2(∇u + ∇u T ).

Semi analytical finite element method

In order to still speak of propagation modes, the application of a SAFE method relies on a spatial Fourier transform in the s direction by assuming an e iks separation. Three criteria are implicitly needed. We assumed that the cross-section of the guide does not vary along s nor the material properties. The third criterion is related to the variational formulation and requires that the coefficients of every differential operator do not depend on s, which is true if the metric tensor does not depend on s either. As shown previously, this occurs for the metric tensor defined by Eq. ( 3). A SAFE method can thus be applied (see Ref. [START_REF] Treyssède | Elastic waves in helical waveguides[END_REF] for more details on unprestressed structures). The displacement vector and its trial field are rewritten as: u = u(x, y)e i(ks-ωt) , δu = δu(x, y)e -i(ks-ωt) . e iks are separated from all field components, and ∂/∂s replaced by ik, where k denotes the axial wavenumber.

Then, the FE discretization of Eq. ( 5) finally arrives at the following eigenvalue problem for the column vector U containing displacement degrees of freedom:

{K 1 -ω 2 M + ik(K 2 -K T 2 ) + k 2 K 3 }U = 0 (6)
The eigenproblem obtained by fixing ω and finding k is quadratic. At fixed real k, the eigenproblem given by Eq. ( 6) is linear for finding ω 2 . This approach, only valid in the absence of damping, allows to find propagating (k purely real), evanescent (k purely imaginary) and inhomogeneous waves (k fully complex).

Conclusion

A SAFE method is proposed to analyse the effect of prestress on wave propagation in straight and helical waveguides. This SAFE method is based on the equation of prestressed structural dynamics, which are written in a curvilinear helical coordinate system. Results will be presented during the conference and checked with some reference solutions. The effect of prestress on the propagation of guided waves will be outlined.

Figure 1 :

 1 Figure 1: Helix centreline and its local basis


Ahmed Frikha
email: frikhaahmed@yahoo.fr

Fabien Treyssede

F Treyssède

P Cartraud

NUMERICAL MODELING OF ELASTIC WAVE PROPAGATION IN PRESTRESSED HELICAL WAVEGUIDES WITH THE SAFE METHOD

come

Numerical modeling of elastic wave propagation in prestressed helical waveguides with the safe method

Introduction

Guided wave is recognized as being among the most popular techniques of Non Destructive Testing. Due to the complexity of signals, this technique is often restricted to simple geometries such as plates and tubes. In the area of non-prestressed strands inspection, recent experimental studies have been performed as shown in the work of [START_REF] Rizzo | Wave propagation in multi-wire strands by wavelet-based laser ultrasound[END_REF] and Laguerre et al. (2002 a). In the presence of preload, only a few experimental studies have been conducted [START_REF] Kwun | Effects of tensile loading on the properties of elastic-wave propagation in a strand[END_REF] and Laguerre et al. 2002 b). The complexity of the results is partly related to the helical geometry of peripheral wires and to the axial load. In the case of wave propagation in helical waveguide, models using periodic finite element as shown in the work of [START_REF] Treyssède | Numerical investigation of elastic modes of propagation in helical waveguides[END_REF], or the SAFE method in the work of [START_REF] Treyssède | Elastic waves in helical waveguides[END_REF] have been developed and validated. The SAFE method is considered among the simplest techniques for numerically computation of elastic modes. This technique has been widely used for straight waveguides with arbitrary cross-section [START_REF] Gavric | Computation of propagative waves in free rail using a finite element technique[END_REF], [START_REF] Hayashi | Guided wave dispersion curves for a bar with an arbitrary cross-section, a rod and rail example[END_REF] and [START_REF] Damljanovic | Propagating and evanescent elastic waves in cylindrical waveguides of arbitrary cross-section[END_REF]). For toroidal waveguides, this method has been used by [START_REF] Demma | The effect of bends on the propagation of guided waves in pipes[END_REF]. This method has been extented to twisted waveguides by [START_REF] Onipede | Propagating waves and end modes in pretwisted beams[END_REF]. Recent studies show the development of the SAFE method for strands to seven strands [START_REF] Treyssède | Elastic waves in helical waveguides[END_REF]. In this paper, the SAFE method is extended to study the seven strands subjected to axial loading. The prestressed state is computed using a multi-scale method.

Theory

Variational formulation of dynamic motion in prestressed structures

One consider a linear elastic material with harmonic dependence of e -i t . Based on the assumption of small perturbations, the 3D variational formulation describing the dynamic motion of a prestressed structure is given by [START_REF] Bathe | Finite Element Procedures[END_REF] as follows:

∫ V 0   : C: d V 0  ∫ V 0 tr ∇  u⋅ 0 ⋅∇ u T d V 0 - 2 ∫ V 0   u⋅u d V 0 =0, (1)
for  u kinematically admissible. u ,  , C and  0 denote respectively the displacements vector, the strain tensor, the elastic moduli tensor and the prestress tensor .  is the material density ICAMEM2010 18-20 December, 2010, Hammamet, Tunisia and V 0 represents the volume of the prestressed structure. The operators tr . and ∇ are respectively the trace and the gradient relative to the prestressed state.

Helical coordinate system

One consider a helical waveguide with a constant cross-section along the helical axis. This axis can be described by the vector position as follows (see Figure 1):

r s=R cos 2  l se X Rsin  2  l se Y  L l se Z , ( 2
)
where l= L 2 4  2 R 2  is the curvilinear length of one helix turn. e X , e Y ,e Z  is the Cartesian basis. R and L are respectively the radius of the helix in the Cartesian plane e X ,e Y  and the length of one helix turn along the e Z axis. The parameter s, corresponding to the curvilinear position, varies between 0 and l. For helix, the curvature =4 2 R/l 2 and the tortuosity =2 L/ l 2 are constants. The vector x can be written in the Frenet-Serret basis as follows:

x  x , y , s=r  sx e n s y e b  s.

(3)

Then we define a covariant non-orthogonal basis denoted g 1 , g 2 , g 3  associated to the helix. g 1 , g 2 ,and g 3 are given by g 1 =∂x / ∂ x , g 2 =∂ x /∂ y and g 3 =∂x / ∂ s . We define respectively the contravariant basis g 1 ,g 2 ,g 3  , where its vectors are given by g i .g j = i j . The covariant metric tensor g , whose coefficients are defined by g mn =g m .g n , is given by:

g= [ 1 0 - y 0 1  x - y  x  2 x 2  y 2 1- x  2]
.

(4) ICAMEM2010 18-20 December, 2010, Hammamet, Tunisia Note that the covariant metric tensor does not depend on the curvilinear variable s. In the case where the cross-section and the material properties of the waveguide do not vary along s, we can conclude that the helical waveguide is translationally invariant in the helical system.

For helical waveguide with circular cross-section, the section does not vary along to the curvilinear axis. Consequently, the translation invariance is verified in this system ( =4  2 R/l 2 , =2  L /l 2 ). The rotating coordinate system ( =0, =2 / L ) allows to consider the translation invariance along the axis for a helical and straight wires (see [START_REF] Treyssède | Investigation of elastic modes propagating in multi-wire helical waveguides[END_REF]). Therefore, this coordinate system verify the translationally invariance for the seven wires.

Semi-analytical finite element method

All quantities of the variational formulation presented in equation ( 1) must be written in the helical coordinate system. The covariant components of ∇ u in the contravariant basis, noted  ij , are written as follows:

 ij =u i , j - ij k u k ,
where the Christoffel symbol of second kind is defined by

 ij k =g i, j . g k .
The strain tensor  is given by: =1 /2 ∇ u∇ u T  . In the Frenet-Serret basis, the vector of the gradient tensor components, {}= {  nn  nb  nt  bn  bb  bt  tn  tb  tt } , and the deformation vector, {}= {  nn  bb  tt 2  nb 2  nt 2  bt } , are written as a function of the displacement vector, {u }= { u n u b u t } , as follows:

{}=G xy ∂/∂ s G s {u },{}=L xy ∂/ ∂ s L s  {u} , ( 5
)
where L xy and L s are described in the work of [START_REF] Treyssède | Elastic waves in helical waveguides[END_REF] . G xy and G s are given by the following two matrices: 6) where

G xy = [ ∂ ∂ x 0 0 ∂ ∂ y 0 0  -  1- x  1- x 0 ∂ ∂ x 0 0 ∂ ∂ y 0  1- x  0 0 0 ∂ ∂ x 0 0 ∂ ∂ y -  1- x 0  ] , G s = [ 0 0 0 0 0 0 1 1- x 0 0 0 0 0 0 0 0 0 1 1- x 0 0 0 0 0 0 0 0 0 1 1- x ] , (
=  1- x  y ∂ ∂ x -x ∂ ∂ y  .
Since the system is translationally invariant along the s axis, a Fourier transform in the s direction can be performed. The displacement vector and its field test are given by: u=u x , y e i ks- t  ,  u= u x , y e -i ks- t  . The exponential e iks can be separated from all components, and ∂/∂ s is replaced by ik, where k is the axial wavenumber (along the helical axis). The SAFE method can be applied (see [START_REF] Treyssède | Elastic waves in helical waveguides[END_REF] for non-prestressed structures).

The finite element discretization of the variational formulation, given by equation ( 1), can be written as an eigen-problem as follows:

{K 1 - 2 Mi k K 2 -K 2 T k 2 K 3 }U=0, (7)
where U contains the degrees of freedom in displacement.

Stress and strain vectors are related by { }=C{} , where { }={ nn  bb  tt 2  nb 2  nt 2 bt } .

C represents the matrix that relates the vector of deformations to the stress vector (and not the tensor C ijkl ).

Elementary matrices of equation ( 7) are given by:

K 1 e = ∫
) 8
where  0 is given by the prestress matrix  0 as follows:

 0 = [  0 0 0 0  0 0 0 0  0 ] .
The resolution of equation ( 8) allows to determine the modes of propagation. For fixed  , the eigen-problem is quadratic in k . For a given real wave number k, we obtain a linear eigen-problem in  2 .

Note that the elastic wave propagation is determined in the prestressed geometry. This static state is computed using a helical homogenization. This technique based on asymptotic expansion reduces the problem to a 2D problem in the cross-section. The static model was validated for helical springs by comparing its results with those of the analytical model of [START_REF] Ancker | Pitch and curvature corrections for helical springs[END_REF]. For seven wire strands, this model has been validated by comparing its results with those of a 3D finite element model developed by [START_REF] Ghoreishi | Validity and limitations of linear analytical models for steel wire strands under axial loading, using a 3D FE model[END_REF].

Dispersion analysis of a seven-wire strand

The material is assumed to be isotropic, with no material damping. For a steel wire, a typical value of 0.30 will be chosen for the Poisson coefficient. We consider waveguides with a circular cross-section of radius a. The adimensionalized angular frequency is given by  a/ c s , c s denoting the shear velocity. Six-node triangles meshes will be used. FE computations are held at fixed real wavenumbers k.

For helical springs, the SAFE model is validated by comparing its results with the model based on the Timoshenko beam approximation. This reference model was developed in the work of [START_REF] Frikha | Effect of axial load on the propagation of elastic waves in helical beams[END_REF]. In this paper, only the results of a seven-wire strand are presented.

Figure 2: Mesh of the seven wire strand cross-section ICAMEM2010 18-20 December, 2010, Hammamet, Tunisia A seven wire strand is made of a straight core of radius a and one layer of helical wires. R h is the helix radius. In this section, R h is assumed equal to 1.967a with an angle of 7.9°( =0,=0.0705 ). The material properties are as follows: E=2.17e11 Pa, =0.28 , =7800 kg / m 3 . Stick contact conditions are assumed for simplicity (no slip, no separation and no friction are considered). The mesh of the section of the strand is presented in Figure 2. Figure 3-left shows the dispersion curves of a non-prestressed seven wire strand (in black) and of its central wire (in grey) for the adimensional frequency range [0;2]. Due to a strong inter-wire coupling, one observes a far more complex behaviour than for single wires. The most striking phenomenon is the cut-off of the fastest mode "notch frequency" (compressional-like L(0,1) mode) around  a / c s =0.35 , corresponding to 83kHz as experimentally observed by [START_REF] Kwun | Effects of tensile loading on the properties of elastic-wave propagation in a strand[END_REF] for a nominal diameter of strand equal to 12.7 mm. The same phenomenon is observed experimentally in the work of Laguerre et al. (2002 b) with a frequency around 68kHz for a radius of the central wire equal to 2.7 mm.

As the axial load increases, this notch frequency shifts to higher frequencies (Figure 3-right). Taking into account the prestress, the deformation of the geometry and the contact without interpenetration between wires, this frequency is located in the vicinity of  a / c s =0.443 under 60% of the Ultimate Tensile Strength load. This frequency corresponds to 105kHz for a nominal diameter of strand equal to 12.7 mm and 86kHz for a radius of the central wire equal to 2.7 mm. This shift of the notch frequency is in agreement with the experimental results of [START_REF] Kwun | Effects of tensile loading on the properties of elastic-wave propagation in a strand[END_REF] and Laguerre et al. (2002 b). This phenomenon is due to the contact between the central and the peripheral wires.

Conclusion

In this paper, a SAFE method was proposed to analyze the effect of prestressing on the wave propagation in a helical waveguide. The prestressed static state was modeled by a method using helical homogenization. The modeling of this state has been validated by comparison with a 3D FE model proposed in the literature. Dispersion inside a non-prestressed and prestressed typical seven wire strand has then been investigated by assuming stick contact conditions. These dispersion curves were compared with experimental results. For an increasing load, the missing band shifts to higher frequencies. This trend is consistent with results obtained experimentally.

Figure 1 :

 1 Figure 1: Central axis of helical structure and its local basis

Figure 3 :

 3 Figure 3: Dispersion curves of non-prestressed strand (left) and a prestressed strand with 60% of the Ultimate Tensile Strength (right).

th International Conference on Advances in Mechanical Engineering and Mechanics ICAMEM2010 18-20 December, 2010, Hammamet, Tunisia

th International Conference on Advances in Mechanical Engineering and Mechanics


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