Abstract
The objective of this contribution is to find effective properties of hyperelastic bodies weakened by periodically distributed microfissures. Problems investigated are internal Signorini's problems with friction. Two such problems have been studied. The first problem is purely static and the friction law is of the deformational plasticity type. To find the overall properties an implicit variational inequality has been homogenized. The second problem concerns homogenization in the quasi-static case and the sliding rule of the flow law type. The variational formulation is obtained in the form of an implicit variational inequality coupled with a variational inequality. In both cases the macroscopic behaviour is elastic-plastic of nonstandard type.
Similar content being viewed by others
References
Aboudi J. (1987): Stiffness reduction of cracked solids. Eng. Fracture Mech. 26, 637–650
Aboudi J.; Benveniste Y. (1987): The effective moduli of cracked bodies in plane deformations. Eng. Fracture Mech. 26, 171–184
Achenbach J. D.; Li Z. L. (1886): Propagation of horizontally polarized transverse waves in a solid with a periodic distribution of cracks. Wave Motion 8, 371–379
Andrieux S.; Bamberger Y.; Marigo J.-J. (1986): Un modèle de matériau microfissuré pour les bétons et les roches. J. Méc. Théor. Appl. 5, 471–513
Attouch H. (1984): Variational properties of epi-convergence. Applications to limit analysis problems in mechanics and duality theory. In: Lecture notes in mathematics, vol. 1091, pp. 80–104. Berlin, Heidelberg, New York: Springer
Attouch, H.; Murat, F. (1985): Homogenization of fissured elastic materials. Publ. AVAMAC, Université de Perpignan, Mathématiques, No 85-03
Azé, D. (1984): Epi-convergence et dualité. Applications à la convergence de de variable primales et duales pour des suites de problèmes d'optimisation convexe. Publ. AVAMAC, Université de Perpignan, Mathématiques, No 84-12
Azé, D. (1985): Convergence des variables duales dans des problèmes de transmission à travers des couches minces par des méthodes d'epiconvergence. Publ. AVAMAC, Université de Perpignan, Mathématique, No 85-06
Barański, W.; Gajl, O. (1985): On asymptotic behaviour of solutions of the equilibrium problems for fissured elastic body. Preprint, Inst. Constr. Engg. Łodz Technical University
Biroli M. (1977): G-convergence for elliptic equations, variational inequalities and quasi-variational inequalities. Rend. del Sem. Mat. e Fisico di Milano 47, 269–328
Biroli M. (1979a): G-convergence for elliptic variational and quasi-variational inequalities. In: Proc. of the Int. Meeting on Recent Methods in Non-Linear Analysis, Rome, May 8–12, 1978 (Eds: De Giorgi E.; Magenes E.; Mosco U.), pp. 361–383, Bologna: Pitagora Editrice
Biroli M. (1979b). An estimate on convergence of approximation by iterations of a solution to a quasi-variational inequality and some consequences on continuous dependence and G-convergence. Annali di Mat. Pura ed Applicata 122, 83–101
Biroli M. (1980). Estimates in homogenization for variational and quasi-variational inequalities. In: Proc. of the Meeting on Free Boundary Problems, Pavia, Settembre-Ottobre 1979, Instituto Nazionale di Alta Matematica, Roma, Vol. II, pp. 45–61
Biroli M.; Marchi S.; Norando T. (1981): Homogenization estimates for quasi-variational inequalities. Boll. U.M.I., 18-A, 267–276
Boccardo L.; Capuzzo Dolcetta I. (1978): Disequazioni quasi-variazionali con funzione d'ostacolo quasi-limitato: esistenza di soluzioni e G-convergenza. Boll. U.M.I., 15-B, 370–385
Boccardo L.; Capuzzo Dolcetta T.; Matzeu M. (1979): Nouveaux résultats d'existence, perturbation et homogénéisation de quelques problèmes non linéaires. In: Proc. of the Int. Meeting on Recent Methods in Non-Linear Analysis, Rome, May 8–12, 1978, (Eds: De Giorgi E.; Magenes E.; Mosco U.), pp. 115–130, Bologna: Pitagora Editrice
Bouchitté, G. (1987): Calcul des variations en cadre non reflexif. Représentation et relaxation de fonctionnelles intégrales sur un espace de measure. Applications en plasticité et homogéneisation. Thèse de Doctorat d'Etat, Université de Perpignan
Bourgeat A. (1984): Homogenized behaviour of two-phase flows in naturally fractured reservoirs with uniform fractures distribution. Comp. Meth. Appl. Mech. Eng. 47, 205–216
Christensen R. M. (1979): Mechanics of composite materials. New York: Wiley
Coussy O. (1986): The influence of interface cracks on wave motion in fiber reinforced elastic solids. J. Méc. Théor. Appl. 5, 803–823
De Borst R. (1987): Smeared cracking, plasticity, creep and thermal loading— a unified approach. Comp. Math. Appl. Mech. Eng. 62, 89–110
Dyskin, A. V. (1985): On the computation of effective deformational characteristics of fissured materials. Mekhanika Tv. Tela No 4, 130–135 (in Russian)
Ekeland I.; Temam R. (1976): Convex analysis and variational problems. Amsterdam: North-Holland
Felder E. (1985): Formulation thermodynamique des interactions superficielles entre deux corps. J. Méc. Théor. Appl. 4, 283–303
Goldstein R. V.; Zhitnikov Yu. V. (1986): Equilibrium of strips and fissures with regions of closing and opening in an elastic medium. Prikl. Mat. Mekh. 50, 826–834 (in Russian)
Goldstein, R. V.; Zhitnikov, Yu. V. (1987): An analysis of a plane crack taking into account slip and stick zones in regions of closing in the case of complex loading. Mekh. Tv. Tela No 2, 141–148 (in Russian)
Goldstein, R. V.; Zhitnikov, Yu. V. (1988): Stress state of an elastic solid weakened by an elliptic crack with interacting faces in the case of complex loading. Mekh. Tv. Tela No 1, 126–132 (in Russian)
Hashin Z. (1983). Analysis of composite materials-survey. Trans. ASME, J. Appl. Mech. 50, 481–505
Hashin Z. (1986): Analysis of stiffness reduction of cracked cross-ply laminates. Eng. Fract. Mech. 25, 771–778
Hill R. (1965): A self consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222
Horii H.; Nemat-Nasser S. (1983): Overall moduli of solids with microcracks: load-induced anisotropy. J. Mech. Phys. Solids 31, 155–171
Kachanov M. L. (1982): A microcrack model of rock inelasticity. Part I: frictional sliding on microcracks. Mech. of Materials 1, 19–27. Part II: Propagation of microcracks, ibid., pp. 29–41
Kachanov M. (1983): Elastic solids with many cracks: a simple method of analysis. Int. J. Solids Struct. 23, 23–43
Kachanov M.; Montagut E. (1986): Interaction of a crack with certain microcrack arrays. Eng. Fract. Mech. 25, 625–636
Kemeny J.; Cook N. G. W. (1986): Effective moduli, non-linear deformation and strength of a cracked elastic solid. Int. J. Rock. Mech. Min. Sci. and Geomech. Abstr. 23, 107–118
Laurent P. J. (1972): Approximation of optimisation. Paris: Hermann
Leguillon D.; Sanchez-Palencia E. (1982): On the behaviour of a cracked elastic body with (or without) friction. J. Méc. Théor. Appl. 1, 195–209
Lewiński T.; Telega J. J. (1988a): Asymptotic method of homogenization of fissured elastic plates. J. Elasticity 19. 37–62
Lewiński T.; Telega J. J. (1988b): Homogenization of fissured Reissner-like plates. Part I: Method of two-scale asymptotic expansions. Arch. Mech. 40, 97–117. Part III: Some particular cases and illustrative example, ibid., 299–307
Lewiński, T.; Telega, J. J. (1989): Overall properties of plates with partially penetrating fissures. C. R. Acad. Sci. Paris, Sér. II
Litewka A. (1986): On stiffness and strength reduction of solids due to crack development. Eng. Fract. Mech. 25, 637–643
Onami M. (1987): Introduction to micromechanics. Moskva: Metalourgya (in Russian)
Rockafellar R. T. (1970): Convex analysis. Princeton: Princeton University Press
Sanchez-Palenciz E. (1980). Non-homogeneous media and vibration theory. Berlin, Heidelberg, New York: Springer
Sumarac D.; Krajcinovic D. (1987): A self-consistent model for microcrack-weakened solids. Mech. Materials 6, 39–52
Suquet, P. M. (1982): Plasticité et homogénéisation. Thèse d'Etat, Université Paris VI
Suquet P. M. (1985): Approach by homogenization of some linear and nonlinear problems in solid mechanics. In: Boehler J.-P. (ed.): Plastic behaviour of anisotropic solids, pp. 77–117, Paris: Editions du CNRS
Telega J. J. (1987): Variational methods in contact problems of mechanics. Uspekhi Mekhaniki (Adv. in Mechanics) 10, 3–95 (in Russian)
Telega J. J. (1988): Topics on unilateral contact problems of elasticity and inelasticity. In: Moreau J. J.; Panagiotopoulos P. D. (eds): Nonsmooth mechanics and applications, pp. 341–462, Wein-New York: Springer
Telega J. J.; Lewiński T. (1988): Homogenization of fissured Reissner-like plates. Part II: Convergence. Arch. Mech. 40, 119–134
Willis J. R. (1982): Elasticity theory of composites. In: Hopkins H. G.; Sewell M. J. (eds): Mechanics of solids, pp. 653–686. Oxford: Pergamon Press
Author information
Authors and Affiliations
Additional information
Communicated by M. Kleiber, March 2, 1989
Rights and permissions
About this article
Cite this article
Telega, J.J. Homogenization of fissured elastic solids in the presence of unilateral conditions and friction. Computational Mechanics 6, 109–127 (1990). https://doi.org/10.1007/BF00350517
Issue Date:
DOI: https://doi.org/10.1007/BF00350517