 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
|
| In English (Mech. Solids): | | 4810 |
|
| << Previous article | Volume 50, Issue 1 / 2015 | Next article >> |
| K.B. Ustinov, "On Separation of a Layer from the Half-Plane: Elastic Fixation Conditions for a Plate Equivalent to the Layer," Mech. Solids. 50 (1), 62-80 (2015) |
| Year |
2015 |
Volume |
50 |
Number |
1 |
Pages |
62-80 |
| DOI |
10.3103/S0025654415010070 |
| Title |
On Separation of a Layer from the Half-Plane: Elastic Fixation Conditions for a Plate Equivalent to the Layer |
| Author(s) |
K.B. Ustinov (A. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia, ustinov@ipmnet.ru) |
| Abstract |
The solution of the homogeneous problem on a half-infinite crack passing along the interface between a thin layer and an elastic half-plane made of materials with distinct properties is obtained and analyzed. Following [1-4], the two-sided Laplace transform is used to reduce the problem to a matrix Riemann problem. The class of combinations of elastic constants of the materials for which the matrix coefficient can be factorized by the method proposed in [1-4] is singled out. This factorization is used to generalize the problem studied in [1-4] to the case of distinct elastic constants of the layer and the half-plane (although they satisfy an additional condition). An asymptotic expression for the crack shore displacements far from the tip are obtained. It is shown that the leading terms of the asymptotics of the crack shore displacements correspond to the cantilever (plate) displacements under boundary conditions of the rigid fixation type, i.e., to the conditions that the displacements and the angle of rotation at the fixation point are proportional to the components of the resultant and the bending moment of the load. Some expressions for the entries of the matrix of elastic fixation coefficients are obtained. |
| Keywords |
separation, interface crack, factorization, elastic fixation |
| References |
| 1. | A. N. Zlatin and A. A. Khrapkov,
"A Semi-Infinite Crack Parallel to the Boundary of the Elastic Half-Plane,"
Dokl. Akad. Nauk SSSR
31, 1009-1010 (1986)
[Sov. Phys. Dokl. (Engl. Transl.)
31 (12), 1009-1010 (1986)]. |
| 2. | A. N. Zlatin and A. A. Khrapkov,
"Elastic Half-Plane Weakened by a Crack Parallel to Its Boundary,"
in Studies in Elasticity and Plasticity,
Vol. 16: Problems of Contemporary Fracture Mechanics
(LGU, 1990),
pp. 68-75
[in Russian]. |
| 3. | A. N. Zlatin and A. A. Khrapkov,
"Vector Riemann Problem with Zero Index of Matrix-Coefficient Exponent,"
Izv. VNIIG im. B. E. Vedeneeva
181 12-16 (1985). |
| 4. | A. A. Khrapkov,
Wiener-Hopf Method in Mixed Elasticity Theory Problems
(VNIIG, St. Petersburg, 2001)
[in Russian]. |
| 5. | A. I. Kalandiia,
Mathematical Methods of Two-Dimensional Elasticity
(Nauka, Moscow, 1973)
[in Russian]. |
| 6. | V. M. Alexandrov and S. M. Mkhitaryan,
Contact Problems for Bodies with Thin Coatings and Interlayers
(Nauka, Moscow, 1983)
[in Russian]. |
| 7. | M. A. Grekov,
Singular Plane Problem of Elasticity
(Izd-vo SPb Univ., St. Petersburg, 2001)
[in Russian]. |
| 8. | H.-H. Yu and J. W. Hutchinson,
"Influence of Substrate Compliance on Buckling Delamination of Thin Films,"
Int. J. Fract.
113 (1), 39-55 (2000). |
| 9. | G. N. Chebotarev,
"To Solution of Riemann Boundary-Value Problem in Closed Form
for Systems of n Pairs of Functions,"
Uch. Zap. Kazan Univ.
1166 (4), 31-58 (1956). |
| 10. | A. E. Heins,
"System of Wiener-Hopf Equations,"
in Proc. of Symposia in Applied Mathematics. II
(McGraw Hill, 1950),
pp. 76-81. |
| 11. | D. S. Jones,
"Commutative Wiener-Hopf Factorization of a Matrix,"
Proc. Roy. Soc. A
393 (4), 185-192 (1984). |
| 12. | N. G. Moiseev,
"Factorization of Matrix Functions of Special Form,"
Dokl. Akad. Nauk SSSR
305 (1), 44-47 (1989)
[Sov. Math. Dokl. (Engl. Transl.)
39, 264-267 (1989)]. |
| 13. | Y. A. Antipov and N. G. Moiseev,
"Exact Solution of the Plane Problem for a Composite Plane
with a Cut across the Boundary between Two Media,"
J. Appl. Math. Mech.
55 (4), 531-539 (1991). |
| 14. | I. D. Abrahams,
"On the Noncommutative Factorization of Wiener-Hopf Kernels
of Khrapkov Type,"
Proc. Roy. Soc. London A
454, 1719-1743 (1998). |
| 15. | V. G. Daniele,
"On the Factorization of Wiener-Hopf Matrices in Problem Solvable with Hurd's Method,"
Trans. ANTENNAS Propagate
26, 614-616 (1978). |
| 16. | Y. A. Antipov and V. V. Silvestrov,
"Factorization on a Riemann Surface in Scattering Theory,"
Quart. J. Mech. Appl. Math.
55, 607-654 (2002). |
| 17. | Y. A. Antipov and V. V. Silvestrov,
"Vector Functional Difference Equation in Electromagnetic Scattering,"
IMA J. Appl. Math.
69 (1), 27-69 (2004). |
| 18. | J. W. Hutchinson and Z. Suo,
"Mixed Mode Cracking in Layered Materials,"
in Advances in Applied Mechanics, Vol. 29,
Ed. by J. W. Hutchinson and T. Y. Wu
(1992),
pp. 63-191. |
| 19. | Z. Suo and J. W. Hutchinson,
"Interface Crack between Two Elastic Layers,"
Int. J. Fract.
43, 1-18 (1990). |
| 20. | K. B. Ustinov,
"On Shear Separation of a Thin Strip from the Half-Plane,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 6, 141-152 (2014)
[Mech. Solids (Engl. Transl.)
49 (6), 713-724 (2014)]. |
| 21. | M. D. Thouless, A. G. Evans, M. F. Ashby, and J. W. Hutchinson,
"The Edge Cracking and Spalling of Brittle Plates,"
Acta Metall.
35, 1333-1341 (1997). |
| 22. | R. L. Salganik,
"The Brittle Fracture of Cemented Bodies,"
Prikl. Mat. Mekh.
27 (5), 957-962 (1963)
[J. Appl. Math. Mech. (Engl. Transl.)
27 (5), 1468-1478 (1963)]. |
| 23. | B. M. Malyshev and R. L. Salganik,
"The Strength of Adhesive Joints Using the Theory of Crack,"
Int. J. Fract. Mech.
1 (2), 114-128 (1965). |
| 24. | G. Doetsch,
Handbook der Laplace-Transformation
(Birkhäuser, Basel, 1946; Fizmatlit, Moscow, 1958). |
| 25. | K. B. Ustinov, A. V. Dyskin, and L. N. Germanovich,
"Asymptotic Analysis of Extensive Crack Growth Parallel to Free Boundary,"
in 3rd Int. Conf. Localized Damage 94, Southampton: Comput. Mech.Publ.
(1994),
pp. 623-630. |
| 26. | K. B. Ustinov,
On Shear Separation of a Thin Strip from the Half-Plane,
Preprint No. 1047
(IPMekh RAN, Moscow, 2013)
[in Russian]. |
| 27. | B. Cotterell and Z. Chen,
"Buckling and Cracking of Thin Film on Compliant Substrates under Compression,"
Int. J. Fract.
104 (2), 169-179 (2000). |
| 28. | R. L. Salganik and K. B. Ustinov,
"Deformation Problem for an Elastically
Fixed Plate Modeling a Coating Partially
Delaminated from the Substrate (Plane Strain),"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 4, 50-62 (2012)
[Mech. Solids (Engl. Transl.)
47 (4), 415-425 (2012)]. |
|
| Received |
23 July 2013 |
| Link to Fulltext |
|
| << Previous article | Volume 50, Issue 1 / 2015 | Next article >> |
|
 If you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
|
|
Russian 
English