Analysis of crack singularities in an aging elastic material

Martin Costabel; Monique Dauge; SergeïA. Nazarov; Jan Sokolowski

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

  • Volume: 40, Issue: 3, page 553-595
  • ISSN: 0764-583X

Abstract

top
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 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 ln r growing in time.

How to cite

top

Costabel, Martin, et al. "Analysis of crack singularities in an aging elastic material." ESAIM: Mathematical Modelling and Numerical Analysis 40.3 (2006): 553-595. <http://eudml.org/doc/249733>.

@article{Costabel2006,
abstract = { 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 $\frac12$ 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 ln r growing in time. },
author = {Costabel, Martin, Dauge, Monique, Nazarov, SergeïA., Sokolowski, Jan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Crack singularities; creep theory; Volterra kernel; hereditarily-elastic.; asymptotic solution; anisotropic material law},
language = {eng},
month = {7},
number = {3},
pages = {553-595},
publisher = {EDP Sciences},
title = {Analysis of crack singularities in an aging elastic material},
url = {http://eudml.org/doc/249733},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Costabel, Martin
AU - Dauge, Monique
AU - Nazarov, SergeïA.
AU - Sokolowski, Jan
TI - Analysis of crack singularities in an aging elastic material
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/7//
PB - EDP Sciences
VL - 40
IS - 3
SP - 553
EP - 595
AB - 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 $\frac12$ 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 ln r growing in time.
LA - eng
KW - Crack singularities; creep theory; Volterra kernel; hereditarily-elastic.; asymptotic solution; anisotropic material law
UR - http://eudml.org/doc/249733
ER -

References

top
  1. M.S. Agranovich and M.I. Vishik, Elliptic problems with the parameter and parabolic problems of general type. Uspekhi Mat. Nauk19 (1963) 53–161 (English transl.: Russ. Math. Surv.19 (1964) 53–157).  
  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 SSSR266 (1982) 1361–1366 (English transl.: Sov. Phys. Dokl.27 (1982) 817–819).  
  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.  
  6. C. Atkinson and J.P. Bourne, Stress singularities in angular sectors of viscoelastic media. Int. J. Eng. Sci.28 (1990) 615–650.  
  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.  
  8. M. Costabel and M. Dauge, Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems. Math. Nachr.162 (1993) 209–237.  
  9. M. Costabel and M. Dauge, Crack singularities for general elliptic systems. Math. Nachr.235 (2002) 29–49.  
  10. M. Costabel, M. Dauge and R. Duduchava, Asymptotics without logarithmic terms for crack problems. Comm. Partial Differential Equations28 (2003) 869–926.  
  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.  
  12. R. Duduchava, A.M. Sändig and W.L. Wendland, Interface cracks in anisotropic composites. Math. Method. Appl. Sci.22 (1999) 1413–1446.  
  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).  
  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).  
  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. Nauk43 (1988) 55–98 (English transl.: Russ. Math. Surv.43 (1988) 65–119).  
  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).  
  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. 2123 (1984) 89–107).  
  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. 2123 (1984) 57–88).  
  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.  
  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.  
  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).  
  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)  
  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. nauk54 (1999) 77–142 (English transl.: Russ. Math. Surv.54 (1999) 947–1014).  
  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. Notes33 (1983) 300–306).  
  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. 2155 (1993) 169–206).  
  31. S.A. Nazarov and B.A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries. Walter de Gruyter, Berlin, New York (1994) 525.  
  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).  
  33. J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson-Academia, Paris-Prague (1967).  
  34. A.C. Pipkin, Lectures on viscoelasticity theory. Springer, NY (1972) 180.  
  35. G.S. Vardanyan and V.D. Sheremet, On certain theorems in the plane problem of the creep theory. Izvestia AN Arm. SSR. Mechanics4 (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 SSR74 (1982) 26–29. (Russian)  
  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).  

NotesEmbed ?

top

You must be logged in to post comments.