This paper considers the problem of two adherents joined by a soft thin adhesive along their common surface. Using the asymptotic expansion method, the authors obtain a simplified model in which the adhesive is treated as a material surface and is replaced by returning springs. The authors show weak and strong convergence of the exact solution toward the solution of the limit problem. The singularities of the limit problem are analyzed, and it is shown that typically they are logarithmic. Furthermore, the authors investigate the phenomenon of boundary layer by studying the correctors, the extra terms, which must be added to the classical asymptotic expansion to verify the boundary conditions. The correctors show that, contrary to the adherents, in the adhesive there are power-type singularities, which are at the base of the failure of the assemblage.
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References
1.
[1] Goland, M. and Reissner, E.: The stresses in cemented joints. J. Appl. Mech., ASME, 11, AI 7-A27 (1944).
2.
[2] Gilibert, Y. and Rigolot, A.: Analyse asymptotique des assemblage coiks 'a double recouvrement sollicites au cisaillement en traction. J. Mecanique Appliquge, 3, 341-372 (1979).
3.
[3] Adams, R. D., Comyn, J., and Wake, W C.: Structural Adhesive Joints in Engineering, Chapman & Hall, London, 1997.
4.
[4] Klarbring, A.: Derivation of a model of adhesively bonded joints by the asymptotic expansion method. Int. J. Eng. Sci., 29, 493-512 (1991).
5.
[5] Destuynder, P., Michavila, F., Santos, A., and Ousset, Y.: Some theoretical aspects in computational analysis of adhesive lap joints. Int. J. Num. Methods Eng., 35, 1237-1262 (1992).
6.
[6] Edlund, U. and Klarbring, A.: Analysis of elastic and elasto-plastic adhesive joints using a mathematical programming approach. Comput. Meth. Appl. Mech. Eng., 78, 19-47 (1990).
7.
[7] Geymonat, G., Krasucki, F., Marini, D., and Vidrascu, M.: A domain decomposition method for a bonded structure. Math. Models Methods Appl. Sci.8, 1387-1402 (1998).
8.
[8] Suquet, P.: Discontinuities and plasticity, in Non-Smooth Mechanics and Applications, pp. 280-340, ed., Moreau and Panagiotopoulos, Springer-'krlag, Wien, 1988.
9.
[9] Ganghoffer, J. E., Brillard, A., and Schultz, J.: Modelling of the mechanical behaviour ofjoints bonded by a nonlinear incompressible elastic adhesive. Eur J. Mech., A/Solids, 16, 255-276 (1997).
10.
[10] Geymonat, G. and Krasucki, F.: A limit model of a soft thin joint, in Partial Differential Equations and Applications, ed., Marcellini, Talenti, and Wsentini, Marcel Dekker, New York, 1996.
11.
[11] Licht, C. and Michaille, G.: Une mod6lisation du comportement d'un joint coll6 elastique.C. R. Acad. Sci. Paris, I, 322, 295-300 (1996).
12.
[12] Brezis, H., Caffarelli, L. A., and Friedman, A.: Reinforcement problems for elliptic equations and variational inequalities. Ann. Mat. Pur Appl., 4(123), 219-246 (1980).
13.
[13] Acerbi, E. and Buttazzo, G.: Reinforcement problems in the calculus of variations. Ann. Inst. Poincarg, Analyse non lingare, 3, 273-284 (1986).
14.
[14] Acerbi, E. and Buttazzo, G.: Limit problems for plates surrounded by soft material. Arch. Rat. Mech. Anal., 92, 355-370 (1986).
15.
[15] Lene, F. and Leguillon, D.: Homogenized constitutive law for a partially cohesive composite material. Int. J. Solids Struct., 18, 443-458 (1982).
16.
[16] Lions, J. L.: Perturbations singulieres dans les problemes aux limites et en contr6le optimal, in Lecture Notes in Mathematics, no. 323, Springer-'krlag, Paris, 1973.
17.
[17] Ciarlet, P G.: Plates and Junctions in Elastic Multi-Structures. An Asymptotic Analysis, Masson and Springer-Xtrlag, Paris, 1990.
[20] Lebon, F., Licht, C., Michaille, G., and Khaoua, A. O.: Comportament asymptotique d'une bande dissipative mince de faible rigidite, simulation du collage, in Proceedings of the 8th Congres National Mecama4Aussois, 1995.
21.
[21] Caillerie, D.: The effect of a thin inclusion of high rigidity in an elastic body. Math. Meth. Appl. Sci., 2, 251-270 (1980).
22.
[22] Lemrabet, K.: Probleme de ventcel pour le systeme de l'elasticite dans un domaine de R3. C. R. Acad. Sci. Paris, I, 304, 151-154 (1987).
23.
[23] Leguillon, D. and Sanchez-Palencia, E.: Computation of Singular Solutions in Elliptic Problems and Elasticity, RMA no. 5, Masson, Paris, 1987.
24.
[24] Grisvard, PR: Singularities in Boundary Wilue Problems, RMA no. 22, Masson and Springer-verlag, Paris, 1992.
25.
[25] Grisvard, P.: Singularit6s en elasticit6. Arch. Rat. Mech. Anal., 107, 157-180 (1989).
26.
[26] Nazarov, S. A. and Plamenevsky, B. A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin, 1994.
27.
[27] Williams, M. L.: Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. App. Mech., ASME, 19, 526-528 (1952).
28.
[28] Teniou, D.: Divers problemes theoriques et numeriques liUs au systeme de l'6lasticit6 dans des domaines non reguliers, These d'Etat, Universit6 de Rennes, France, 1987.
29.
[29] Grisvard, P: Problemes aux limites dans les polygones, mode d'emploi. Bullettin de la Direction des etudes et recherches de I EDJ, C, 1, 21-59 (1986).
30.
[30] Kondratiev, V A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskovkogo Mat. Obschetsva, 16, 209-292 (1967).
31.
[31] Bui, H. D. and Taheri, S.: La singularit "sepine" dans les bi-materiaux en thermo-61astoplasticit6. C. R. Acad. Sci. Paris, II, 309, 1527-1533 (1989).
32.
[32] Deperrois, A. and Dang Xbn, K.: Inclusions des surface et singularit6 6pine. C. R. Acad. Sci. Paris, II, 311, 1285-1290 (1990).
33.
[33] Destuynder, P., Michavila, F., Ousset, Y., and Santos, A.: Utilisation du taux de restitution de l'energie dans l'analyse 'a deux echelles de l'endommagement d'un joint colle. C. R. Acad. Sci. Paris, I, 310, 161-165 (1990).
34.
[34] Timoshenko, S. and Goodier, J. N.: Theory of Elasticity, McGraw-Hill, New York, 1951.
35.
[35] Leguillon, D.: Une singularite logarithmique pour un probleme non-homogene: la singularit6 6pine, Memoire d'habilitation, Universit6 P et M. Curie (Paris 6), Paris, 1993.
36.
[36] Bogy, D. B.: Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions. J. App. Mech., ASME, 38, 377-386 (1971).
37.
[37] Rice, J. R. and Sih, G. C.: Plane problems of cracks in dissimilar media. J. AppL. Mech., ASME, 33, 418-423 (1965).
38.
[38] Rice, J. R.: Elastic fracture mechanics concepts for interfacial cracks. J. Appl. Mech., ASME, 55, 98-103 (1988).
39.
[39] Lenci, S.: La teoria dell'interfaccia debole con applicazioni ai compositi, Ph.D. thesis, University of Florence, Italy, 1997.
40.
[40] Destuynder, P.: Sur une justification des modeles de plaques et de coques par les methodes asymptotiques, These de doctorat d'Jttat es Sciences Math6matiques, Universit6 Paris 6, 1980.
41.
[41] Sanchez-Hubert, J. and Sanchez-Palencia, E.: Introduction aux Methods Asymptotiques et a I Homoggngisation, Masson, Paris, 1992.
42.
[42] Adams, R. D. and Harris, J. A.: The influence of local geometry on the strength of adhesive joints. Int. J. Adhesion Adhesives, 7, 69-80 (1987).
43.
[43] Hutchinson, J. W., Mear, M. E., and Rice, J. R.: Crack paralleling an interface between dissimilar materials. J. Appl. Mech., ASME, 54, 828-832 (1987).
44.
[44] Yang, M. and Kim, K.-S.: The behaviour of subinterface cracks with crack-face contact. Eng. Fract. Mech., 44, 155-165 (1993).
45.
[45] Leguillon, D.: Un exemple d'interaction singularite-couche limite pour la modelisation de la fracture dans les composites. C. R. Acad. Sci. Paris, II, 319, 161-166 (1994).
