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Journal of Elasticity
Published in: Journal of Elasticity 1/2015

01-01-2015

Power-Law Solutions of Anisotropic Multi-material Elasticity Problems

Author: M. Steigemann

Published in: Journal of Elasticity | Issue 1/2015

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Abstract

Singular solutions play a major role in many areas of linear elasticity theory. For example, the displacement field near corners of a structure composed of anisotropic linear elastic materials can be approximated asymptotically by a series of power-law functions. Especially for problems in fracture mechanics the knowledge of the singular exponents and also the functions themselves is fundamental. The focus of this work is to derive explicit formulas for such solutions. The main tool is a complex approach developed by Costabel and Dauge in there works. We show, how solutions can be derived for multi-material structures also covering multiple roots in the characteristic material equations and logarithmic solutions in the case of multiple singular exponents.
previous article Numerical Implementation of the Equivalent Inclusion Method for 2D Arbitrarily Shaped Inhomogeneities
next article Singular Shear-Force States in Elementary Plate Theory

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Literature
1.
Bogy, D.: On the problem of edge-bonded elastic quarter-planes loaded at the boundary. Int. J. Solids Struct. 6, 1287–1313 (1970) CrossRefMATH
2.
3.
Carpinteri, A., Paggi, M.: On the asymptotic stress field in angularly nonhomogeneous materials. Int. J. Fract. 135, 267–283 (2005) CrossRefMATH
4.
Cook, T., Erdogan, F.: Stresses in bonded materials with a crack perpendicular to the interface. Int. J. Eng. Sci. 10, 677–697 (1972) CrossRefMATH
5.
Costabel, M., Dauge, M.: Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems. Math. Nachr. 162, 209–237 (1992) CrossRefMathSciNet
6.
Costabel, M., Dauge, M.: General edge asymptotics of solutions of second order elliptic boundary value problems i. Proc. R. Soc. Edinb. A 123, 109–155 (1993) CrossRefMathSciNetMATH
7.
Costabel, M., Dauge, M.: Computation of corner singularities in linear elasticity. In: Costabel, M., Dauge, M., Nicaise, S. (eds.) Boundary Value Problems and Integral Equations in Nonsmooth Domains. Lect. Notes in Pure and Appl. Math., vol. 167, pp. 59–68. Dekker, New York (1994)
8.
Costabel, M., Dauge, M., Duduchava, R.: Asymptotics without logarithmic terms for crack problems. Commun. Partial Differ. Equ. 28(5–6), 869–926 (2003) CrossRefMathSciNetMATH
9.
Costabel, M., Dauge, M., Lafranche, Y.: Fast semi-analytic computation of elastic edge singularities. Comput. Methods Appl. Mech. Eng. 190, 2111–2134 (2001) ADSCrossRefMATH
10.
Dempsey, J., Sinclair, G.: On the stress singularities in the plane elasticity of the composite wedge. J. Elast. 9(4), 373–391 (1979) CrossRefMathSciNetMATH
11.
Dundurs, J.: Effect of elastic constants on stress in a composite under plane deformation. J. Comput. Math. 1, 310–323 (1967)
12.
Dundurs, J.: Discussion: Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading. ASME J. Appl. Mech. 35(3), 650–652 (1969) CrossRef
13.
Gohberg, I., Kaashoek, M., Schagen, F.: On the local theory of regular analytic matrix functions. Linear Algebra Appl. 182, 9–25 (1993) CrossRefMathSciNetMATH
14.
Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Classics in Applied Mathematics. SIAM, Philadelphia (2009) CrossRefMATH
15.
Golub, G., Van Loan, C.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996) MATH
16.
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman Press, London (1985) MATH
17.
Gu, M., Eisenstat, S.: An efficient algorithm for computing a strong rank-revealing QR factorization. SIAM J. Sci. Comput. 17(4), 848–869 (1996) CrossRefMathSciNetMATH
18.
He, N., Hutchinson, J.: Crack deflection at an interface between dissimilar elastic materials. Int. J. Solids Struct. 25, 1053–1067 (1989) CrossRef
19.
Kondratiev, V.: Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc. 10, 227–313 (1967)
20.
Kozlov, V., Maz’ya, V., Rossmann, J.: Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Problems. Mathematical Surveys and Monographs, vol. 85. American Mathematical Society, Providence (2001) MATH
21.
Leguillon, D., Sanchez-Palencia, E.: Computation of Singular Solutions in Elliptic Problems and Elasticity. Wiley/Masson, New York/Paris (1987) MATH
22.
Lekhnitskii, S.: Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco (1963) MATH
23.
Maz’ya, V., Plamenevsky, B.: The coefficients in the asymptotic forms of the solutions of elliptic boundary-value problems in domains with conical points. Math. Nachr. 76, 29–60 (1977) CrossRefMathSciNetMATH
24.
Maz’ya, V., Rossmann, J.: Über die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten. Math. Nachr. 138, 27–53 (1988) CrossRefMathSciNetMATH
25.
Nazarov, S., Plamenevsky, B.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. de Gruyter Expositions in Mathematics, vol. 13. de Gruyter, Berlin (1994) CrossRefMATH
26.
Paggi, M., Carpinteri, A.: On the stress singularities at multimaterial interfaces and related analogies with fluid dynamics and diffusion. Appl. Mech. Rev. 61(2), 1–22 (2008) CrossRefMathSciNet
27.
Sih, G., Paris, P., Irwin, G.: On cracks in rectilinearly anisotropic bodies. Int. J. Fract. 1, 189–203 (1965) CrossRef
28.
Specovius-Neugebauer, M., Steigemann, M.: Eigenfunctions of the 2-dimensional anisotropic elasticity operator and algebraic equivalent materials. ZAMM Z. Angew. Math. Mech. 88(2), 100–115 (2008) CrossRefMathSciNetMATH
29.
Ting, T.: Anisotropic Elasticity: Theory and Applications. Oxford University Press, Oxford (1996) MATH
30.
Vu-Quoc, L., Tran, V.: Singularity analysis and fracture energy release rate for composites: Piecewise homogeneous-anisotropic materials. Comput. Methods Appl. Mech. Eng. 195, 5162–5197 (2006) ADSCrossRefMathSciNetMATH
31.
Westergaard, H.: Bearing pressures and cracks. J. Appl. Mech. 6, A49–A53 (1939)
32.
Williams, M.: Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. Appl. Mech. 19, 526–528 (1952)
33.
Yosibash, Z.: Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection with Failure Initiation. Interdisciplinary Applied Mathematics, vol. 37. Springer, Berlin (2011)
34.
Zak, A., Williams, M.: Crack point singularities at a bi-material interface. J. Appl. Mech. 30, 142–143 (1963) ADSCrossRef
Metadata
Title
Power-Law Solutions of Anisotropic Multi-material Elasticity Problems
Author
M. Steigemann
Publication date
01-01-2015
Publisher
Springer Netherlands
Published in
Journal of Elasticity / Issue 1/2015
Print ISSN: 0374-3535
Electronic ISSN: 1573-2681
DOI
https://doi.org/10.1007/s10659-014-9479-0

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