Analysis of crack singularities in an aging elastic material
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 3, p. 553-595

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 $\frac{1}{2}$ or have a more complicated dependence on the distance variable $r$ 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 $lnr$ growing in time.

DOI : https://doi.org/10.1051/m2an:2006022
Classification:  35Q72,  74D05,  74G70
Keywords: crack singularities, creep theory, Volterra kernel, hereditarily-elastic
@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},
publisher = {EDP-Sciences},
volume = {40},
number = {3},
year = {2006},
pages = {553-595},
doi = {10.1051/m2an:2006022},
zbl = {1106.74028},
mrnumber = {2245321},
language = {en},
url = {http://www.numdam.org/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, Volume 40 (2006) no. 3, pp. 553-595. doi : 10.1051/m2an:2006022. http://www.numdam.org/item/M2AN_2006__40_3_553_0/

[1] M.S. Agranovich and M.I. Vishik, 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] N.Kh. Arutyunyan and V.B. Kolmanovskii, The theory of creeping heterogeneous bodies. Nauka, Moscow (1983) 336.

[3] N.Kh. Arutyunyan, S.A. Nazarov and B.A. Shoikhet, 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] N.Kh. Arutyunyan, A.D. Drozdov and V.E. Naumov, Mechanics of growing visco-elasto-plastic bodies. Nauka, Moscow (1987) 472.

[5] C. Atkinson and J.P. Bourne, Stress singularities in viscoelastic media. Q. J. Mech. Appl. Math. 42 (1989) 385-412. | Zbl 0693.73068

[6] C. Atkinson and J.P. Bourne, Stress singularities in angular sectors of viscoelastic media. Int. J. Eng. Sci. 28 (1990) 615-650. | Zbl 0715.73030

[7] J.P. Bourne and C. Atkinson, Stress singularities in viscoelastic media. II. Plane-strain stress singularities at corners. IMA J. Appl. Math. 44 (1990) 163-180. | Zbl 0702.73030

[8] M. Costabel and M. Dauge, Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems. Math. Nachr. 162 (1993) 209-237. | Zbl 0802.35032

[9] M. Costabel and M. Dauge, Crack singularities for general elliptic systems. Math. Nachr. 235 (2002) 29-49. | Zbl 1094.35038

[10] M. Costabel, M. Dauge and R. Duduchava, Asymptotics without logarithmic terms for crack problems. Comm. Partial Differential Equations 28 (2003) 869-926. | Zbl 1103.35321

[11] R. Duduchava and W.L. Wendland, 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] R. Duduchava, A.M. Sändig and W.L. Wendland, Interface cracks in anisotropic composites. Math. Method. Appl. Sci. 22 (1999) 1413-1446. | Zbl 0933.35052

[13] J. Dundurs, Effect of elastic constants on stress in composite under plane deformations. J. Compos. Mater. 1 (1967) 310.

[14] G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional equations. Cambridge Univ. Press, Cambridge (1990). | MR 1050319 | Zbl 0695.45002

[15] V.A. Kondratiev, 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] V.A. Kondratiev and O.A. Oleinik, 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] V.A. Kozlov, V.G. Maz'Ya and J. Rossmann, Elliptic boundary value problems in domains with point singularities. Amer. Math. Soc., Providence (1997). | Zbl 0947.35004

[18] M.A. Krasnosel'Skii, G.M. Vainikko and P.P. Zabreiko, Approximate solutions to integral equations. Nauka, Moscow (1969) 455.

[19] V.G. Maz'Ya and B.A. Plamenevskii, 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] V.G. Maz'Ya and B.A. Plamenevskii, 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] S.E. Mikhailov, Singularities of stresses in a plane hereditarily-elastic aging solid with corner points. Mech. Solids (Izv. AN SSSR. MTT) 19 (1984) 126-139.

[22] S.E. Mikhailov, 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] S.E. Mikhailov, 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] S.A. Nazarov, 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] S.A. Nazarov, Weight functions and invariant integrals. Vychisl. Mekh. Deform. Tverd. Tela. 1 (1990) 17-31. (Russian)

[26] S.A. Nazarov, 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] S.A. Nazarov, 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] S.A. Nazarov, 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] S.A. Nazarov and B.A. Shoikhet, 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] S.A. Nazarov and B.A. Plamenevskii, 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] S.A. Nazarov and B.A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries. Walter de Gruyter, Berlin, New York (1994) 525. | MR 1283387 | Zbl 0806.35001

[32] S.A. Nazarov, L.P. Trapeznikov and B.A. Shoikhet, 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] J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson-Academia, Paris-Prague (1967). | MR 227584

[34] A.C. Pipkin, Lectures on viscoelasticity theory. Springer, NY (1972) 180. | Zbl 0237.73022

[35] G.S. Vardanyan and V.D. Sheremet, On certain theorems in the plane problem of the creep theory. Izvestia AN Arm. SSR. Mechanics 4 (1973) 60-76.

[36] V.P. Zhuravlev, S.A. Nazarov and B.A. Shoikhet, 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] V.P. Zhuravlev, S.A. Nazarov and B.A. Shoikhet, 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