We study the structure of extremals of a class of second order variational problems without convexity, on intervals in R + . The problems are related to a model in thermodynamics introduced in [7]. We are interested in properties of the extremals which are independent of the length of the interval, for all sufficiently large intervals. As in [12, 13] the study of these properties is based on the relation between the variational problem on bounded, large intervals and a limiting problem on R + . Our investigation employs techniques developed in [10, 12, 13] along with turnpike techniques developed in [16, 17].On etudie la structure des extremales d'une classe de problemes variationnels non convexes du deuxieme ordre, sur des intervalles de R + . Ces problemes sont relies a un modele thermodynamique introduit dans [7]. Nous nous interessons aux proprietes des extremales qui ne dependent pas de la longeur des intervalles, pourvu que ceux-ci soient assez grands. Comme dans [12,13] l'etude de ces proprietes s'appuie sur la relation entre le probleme variationnel sur de grands intervalles bornes et un probleme limite sur R + . Notre travail emploie des techniques developpees dans [10,12,13] ainsi que dans [16,17].