We consider a quasistatic system involving a Volterra kernel modelling an hereditarily-elastic aging body. We are concerned with the behavior of displacement and stress fields in the neighborhood of cracks. In this paper, we investigate the case of a straight crack in a two-dimensional domain with a possibly anisotropic material law. We study the asymptotics of the time dependent solution near the crack tips. We prove that, depending on the regularity of the material law and the Volterra kernel, these asymptotics contain singular functions which are simple homogeneous functions of degree or have a more complicated dependence on the distance variable to the crack tips. In the latter situation, we observe a novel behavior of the singular functions, incompatible with the usual fracture criteria, involving super polynomial functions of growing in time.
@article{M2AN_2006__40_3_553_0,
author = {Costabel, Martin and Dauge, Monique and Nazarov, Serge\"\i\ A. and Sokolowski, Jan},
title = {Analysis of crack singularities in an aging elastic material},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {40},
year = {2006},
pages = {553-595},
doi = {10.1051/m2an:2006022},
mrnumber = {2245321},
zbl = {1106.74028},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2006__40_3_553_0}
}
Costabel, Martin; Dauge, Monique; Nazarov, Sergeï A.; Sokolowski, Jan. Analysis of crack singularities in an aging elastic material. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) pp. 553-595. doi : 10.1051/m2an:2006022. http://gdmltest.u-ga.fr/item/M2AN_2006__40_3_553_0/
[1] and, Elliptic problems with the parameter and parabolic problems of general type. Uspekhi Mat. Nauk 19 (1963) 53-161 (English transl.: Russ. Math. Surv. 19 (1964) 53-157). | Zbl 0137.29602
[2] and, The theory of creeping heterogeneous bodies. Nauka, Moscow (1983) 336.
[3] , and, Bounds and the asymptote of the stress-strain state of a threedimensional body with a crack in elasticity theory and creep theory. Dokl. Akad. Nauk SSSR 266 (1982) 1361-1366 (English transl.: Sov. Phys. Dokl. 27 (1982) 817-819). | Zbl 0515.73092
[4] , and, Mechanics of growing visco-elasto-plastic bodies. Nauka, Moscow (1987) 472.
[5] and, Stress singularities in viscoelastic media. Q. J. Mech. Appl. Math. 42 (1989) 385-412. | Zbl 0693.73068
[6] and, Stress singularities in angular sectors of viscoelastic media. Int. J. Eng. Sci. 28 (1990) 615-650. | Zbl 0715.73030
[7] and, Stress singularities in viscoelastic media. II. Plane-strain stress singularities at corners. IMA J. Appl. Math. 44 (1990) 163-180. | Zbl 0702.73030
[8] and, Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems. Math. Nachr. 162 (1993) 209-237. | Zbl 0802.35032
[9] and, Crack singularities for general elliptic systems. Math. Nachr. 235 (2002) 29-49. | Zbl 1094.35038
[10] , and, Asymptotics without logarithmic terms for crack problems. Comm. Partial Differential Equations 28 (2003) 869-926. | Zbl 1103.35321
[11] and, The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems. Integr. Equat. Oper. Th. 23 (1995) 294-335. | Zbl 1126.35368
[12] , and, Interface cracks in anisotropic composites. Math. Method. Appl. Sci. 22 (1999) 1413-1446. | Zbl 0933.35052
[13] , Effect of elastic constants on stress in composite under plane deformations. J. Compos. Mater. 1 (1967) 310.
[14] , and, Volterra Integral and Functional equations. Cambridge Univ. Press, Cambridge (1990). | MR 1050319 | Zbl 0695.45002
[15] , Boundary problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obshch. 16 (1967) 209-292 (English transl.: Trans. Moscow Math. Soc. 16 (1967) 227-313). | Zbl 0194.13405
[16] and, Boundary-value problems for the system of elasticity theory in unbounded domains. Korn's inequalities. Uspehi Mat. Nauk 43 (1988) 55-98 (English transl.: Russ. Math. Surv. 43 (1988) 65-119). | Zbl 0669.73005
[17] , and, Elliptic boundary value problems in domains with point singularities. Amer. Math. Soc., Providence (1997). | Zbl 0947.35004
[18] , and, Approximate solutions to integral equations. Nauka, Moscow (1969) 455.
[19] and, Weighted spaces with nonhomogeneous norms and boundary value problems in domains with conical points. In: Elliptische Differentialgleichungen (Meeting in Rostock, 1977), Wilhelm-Pieck-Univ., Rostock (1978) 161-189 (English transl.: Amer. Math. Soc. Transl. Ser. 2 123 (1984) 89-107). | Zbl 0429.35031
[20] and, The coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points. Math. Nachr. 76 (1997) 29-60 (English transl.: Amer. Math. Soc. Transl. Ser. 2 123 (1984) 57-88). | Zbl 0554.35036
[21] , Singularities of stresses in a plane hereditarily-elastic aging solid with corner points. Mech. Solids (Izv. AN SSSR. MTT) 19 (1984) 126-139.
[22] , Singular stress behavior in a bonded hereditarily-elastic aging wedge. Part. I: Problem statement and degenerate case. Math. Method. Appl. Sci. 20 (1997) 13-30. | Zbl 0882.73026
[23] , Singular stress behavior in a bonded hereditarily-elastic aging wedge. Part. II: General heredity. Math. Method. Appl. Sci. 20 (1997) 31-45. | Zbl 0882.73027
[24] , Vishik-Lyusternik method for elliptic boundary-value problems in regions with conical points. 1. The problem in a cone. Sibirsk. Mat. Zh. 22 (1981) 142-163 (English transl.: Siberian Math. J. 22 (1982) 594-611). | Zbl 0502.35039
[25] , Weight functions and invariant integrals. Vychisl. Mekh. Deform. Tverd. Tela. 1 (1990) 17-31. (Russian)
[26] , Self-adjoint boundary value problems. The polynomial property and formal positive operators. St.-Petersburg Univ., Probl. Mat. Anal. 16 (1997) 167-192. (Russian) | Zbl 0945.35029
[27] , The interface crack in anisotropic bodies. Stress singularities and invariant integrals. Prikl. Mat. Mekh. 62 (1998) 489-502 (English transl.: J. Appl. Math. Mech. 62 (1998) 453-464).
[28] , The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes. Uspekhi mat. nauk 54 (1999) 77-142 (English transl.: Russ. Math. Surv. 54 (1999) 947-1014). | Zbl 0970.35026
[29] and, Asymptotic behavior of the solution of a certain integro-differential equation near an angular point of the boundary. Mat. Zametki. 33 (1983) 583-594 (English transl.: Math. Notes 33 (1983) 300-306). | Zbl 0529.45008
[30] and, Neumann problem for selfadjoint elliptic systems in a domain with piecewise smooth boundary. Trudy Leningrad. Mat. Obshch. 1 (1990) 174-211 (English transl.: Amer. Math. Soc. Transl. Ser. 2 155 (1993) 169-206). | Zbl 0778.35033
[31] and, Elliptic problems in domains with piecewise smooth boundaries. Walter de Gruyter, Berlin, New York (1994) 525. | MR 1283387 | Zbl 0806.35001
[32] , and, On the correspondence principle in the plane creep problem of aging homogeneous media with developing slits and cavities. Prikl. Mat. Mekh. 51 (1987) 504-512 (English transl.: J. Appl. Math. Mech. 51 (1987) 392-399). | Zbl 0661.73022
[33] , Les méthodes directes en théorie des équations elliptiques. Masson-Academia, Paris-Prague (1967). | MR 227584
[34] , Lectures on viscoelasticity theory. Springer, NY (1972) 180. | Zbl 0237.73022
[35] and, On certain theorems in the plane problem of the creep theory. Izvestia AN Arm. SSR. Mechanics 4 (1973) 60-76.
[36] , and, Asymptotics of the stress-strain state near the tip of a crack in an inhomogeneously aging bodies. Dokl. Akad. Nauk Armenian SSR 74 (1982) 26-29. (Russian) | Zbl 0564.73043
[37] , and, Asymptotics near the tip of a crack of the state of stress and strain of inhomogeneously aging bodies. Prikl. Mat. Mekh. 47 (1983) 200-208 (English transl.: J. Appl. Math. Mech. 47 (1984) 162-170). | Zbl 0542.73124