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A. Annexe, Ce calcul est nécessaire afin de pouvoir projeter le champ de contrainte de l'inclusion ellipsoïdale et déterminer les composantes normales et tangentielles à l'interface, au point considéré. L'équation standard d'un ellipsoïde est donné par l'équation (A-1) en notant a, b et c les demi-axes ellipsoïdaux, Il est possible d'en donner une équation paramétrée (A-2) en fonction de deux angles u et v, comme précisé par la Figure II.10. (A-1) (A-2

. La-déformation-totale-dans, et introduit la définition du tenseur d'interaction T. Ce dernier est alors défini selon l'équation (B-12), en notant I 4 le tenseur d'identité d'ordre 4. Finalement, les champs de contrainte et de déformation dans l'hétérogénéité sont identifiés et exprimés en fonction du tenseur d'Eshelby ainsi que des tenseurs de rigidité de la matrice et de l'inclusion. Dans le cas d'un milieu non infini, il reste cependant à exprimer ? 0 (respectivement ? 0 ) en fonction de la déformation (respectivement la contrainte) appliquée aux limites du VER. Plus précisément, il s

. L. Démontrons-désormais-ce-théorème, équation (D-6) permet d'écrire l'expression (D-8) en considérant la définition de la déformation en fonction du vecteur déplacement u. L'application successive du théorème de flux-divergence permet alors d'écrire l'équation (D-9) puis (D-10)

D. Le-théorème and . Hill, Mandel statue que sur un domaine D de volume V, la moyenne du produit de la contrainte et de la déformation est égale au produit de la contrainte moyenne et de la déformation moyenne. Ceci est représenté par l'équation (D-11), qui est au coeur de la définition d'un comportement effectif et particulièrement important lors de l

. Intéressons-nous-À-la-démonstration-de-ce-théorème, Tout d'abord, l'écart entre les deux grandeurs concernées peut s'exprimer selon l'équation (D-12) L'application du théorème de fluxdivergence aboutit alors à l'expression (D-13), puis (D-14) et finalement (D-15) Cela permet alors d'écrire l'équation (D-16) Il est important de noter que trois conditions aux limites particulières permettent de retrouver l'égalité (D-11), en appliquant les conditions aux limites en traction ou en déplacement vues précédemment