Numerical Implementation of an Interfacial Edge Dislocation Solution in a Multi-Layered Medium
A novel method is presented for obtaining the stress
field induced by an edge dislocation in a multilayered composite. To
demonstrate the applications of the obtained solution, we consider the
problem of an interfacial crack in a periodically layered bimaterial
medium. The crack is modelled as a continuous distribution of edge
dislocations and the Distributed Dislocation Technique (DDT) is
utilized to obtain numerical results for the energy release rate (ERR).
The numerical implementation of the dislocation solution in
MATLAB is also provided.
[1] J. L. Sackman, J. M. Kelly and A. E. Javid “A layered notch filter for
high-frequency dynamic isolation,” J. Pressure Vessel Technol., vol.
111, no.1, pp. 17-24, 1989.
[2] H. Holleck, M. Lahres and P. Woll, “Multilayer coatings – influence of
fabrication parameters on constitution and properties,” Surf. Coat.
Technol., vol. 41, pp. 179-190, 1990.
[3] B.A. Roeder C.T. Sun, “Dynamic penetration of Alumina/aluminium
laminates: experiments and modelling,” Int. J. Impact Eng., vol. 25, no.
2, pp. 169-185, 2001.
[4] L. Chen and M. J. Pindera, “Plane analysis of finite multilayered media
with multiple aligned cracks - Part I: theory,” J. Appl. Mech., vol. 74, no.
1, pp. 128-143, 2006.
[5] A. Khanna and A. Kotousov, “Stress analysis of a crack in a fiberreinforced
layered composite,” Compos. Struct., vol. 118, pp. 139-148,
2014.
[6] I. Sellinger, P. M. Weiss, A. Nguyen, Y. Lu, R. A. Assink, W. Gong and
C.J. Brinker, “Continuous self-assembly of organic–inorganic
nanocomposite coatings that mimic nacre,” Nature, vol. 394 , pp. 256-
260, 1998.
[7] H. Gao, B. Ji, I. L. Jäger, E. Arzt and P. Fratzl, “Materials become
insensitive to flaws at nanoscale: Lessons from nature,” Proc. Natl.
Acad. Sci. USA, vol. 100, no. 10, pp. 5597-5600, 2003.
[8] A. A. Daneshy, “Hydraulic Fracture Propagation in Layered
Formations,” SPE J., vol. 18, no.1, pp. 33-41, 1978.
[9] A. Gudmundsson, T. H. Simmenes, B. Larsen and S. L. Philipp, “Effects
of internal structure and local stresses on fracture propagation,
deflection, and arrest in fault zones,” J. Struct. Geol., vol. 32, pp. 1643-
1655, 2010.
[10] A. Khanna and A. Kotousov, “Controlling the Height of Multiple
Hydraulic Fractures in Layered Media,” SPE J. SPE-176017-PA, 2015.
[11] A. Khanna, A. Kotousov, J. Sobey and P. Weller, “Conductivity of
narrow fractures filled with a proppant monolayer,” J. Petrol. Sci. Eng.,
vol. 100, pp 9-13, 2012.
[12] A. Khanna, A. Keshavarz, K. Mobbs, M. Davis and P. Bedrikovetsky,
“Stimulation of the natural fracture system by graded proppant
injection,” J. Petrol. Sci. Eng., vol. 111, pp. 71-77, 2013.
[13] V.V. Bolotin, “Delaminations in composite structures: its origin,
buckling, growth and stability,” Compos. Part B – Eng., vol. 27, no.2,
pp. 129-145, 1996.
[14] A.C. Garg, “Delamination - a damage mode in composite structures,”
Eng. Fract. Mech., vol. 29, no. 5, pp. 557-584, 1988.
[15] J. Cook, J.E. Gordon, C.C. Evans and D.M. Marsh, “A mechanism for
the control of crack propagation in all-brittle systems,” P. Roy. Soc.
Lond. A – Mat., vol. 282, no. 1391, pp. 508-520, 1964.
[16] M-Y. He and J.W. Hutchinson, “Crack deflection at an interface
between dissimilar elastic materials,” Int. J. Solids. Struct., vol. 25, no.
9, pp. 1053-1067, 1989.
[17] V. Gupta, A.S. Argon and Z. Suo, “Crack deflection at an interface
between two orthotopic media,” J. Appl. Mech. vol. 59, no. 2S, pp. S79-
S87, 1992.
[18] J. Li, “Debonding of the interface as ‘crack arrestor’,” Int. J. Fract., vol.
105, no. 1, pp. 57-79, 2000.
[19] K. Okumura and P-G. de Gennes, “Why is nacre strong? Elastic theory
and fracture mechanics for biocomposites with stratified structures,”
Eur. Phys. J. – E, vol. 4, no. 1, pp. 121-127, 2001.
[20] M.P. Cleary, “Primary factors governing hydraulic fractures in
heterogeneous stratified porous formations,” No. UCRL-13884; CONF-
781112-10 (Massachusetts Inst. of Tech., Cambridge, USA 1978).
[21] B. Bilby and J. Eshelby, “Dislocations and the theory of fracture,” in:
Fracture: an advanced treatise, vol. 1., H. Liebowitz, Ed. New York,
USA: Academic Press, 1968.
[22] A. Kotousov, L. Bortolan Neto and A. Khanna, “On a rigid inclusion
pressed between two elastic half spaces,” Mech. Mater. vol. 68, pp. 38-
44, 2014.
[23] L. Bortolan Neto and A. Khanna, “The performance of hydraulic
fractures partially filled with compressible proppant,” Aust. J.
Multidisciplinary Eng., vol. 10, no. 2, pp. 185, 2013.
[24] A. Khanna, L. Bortolan Neto and A. Kotousov, “Effect of residual
opening on the inflow performance of a hydraulic fracture,” Int. J. Eng.
Sci. vol. 74, pp. 80-90, 2014.
[25] L. Bortolan Neto, A. Khanna and A. Kotousov, “Conductivity and
performance of hydraulic fractures partially filled with compressible
proppant packs,” Int. J. Rock Mech. Min. Sci., vol. 74, pp. 1-9, 2015.
[26] F. Erdogan G. Gupta, “The stress analysis of multi-layered composites
with a flaw,” Int. J. Solids Struct., vol. 7, no. 1, pp. 39-61, 1971.
[27] F. Erdogan G.D. Gupta, “Layered composites with an interface flaw,”
Int. J. Solids Struct., vol. 7, no. 8, pp. 1089-1107, 1971.
[28] F. Erdogan, “Fracture problems in composite materials,” Eng. Fract.
Mech. vol. 4, no. 4, pp. 811-840, 1972.
[29] L. Kucherov, Delamination in periodically layered bi-material
composites. Delamination in periodically layered bi-material composites
(Ph.D. Thesis, Tel Aviv University, Israel 2003).
[30] L. Kucherov and M. Ryvkin, “Interface crack in periodically layered
bimaterial composite,” Int. J. Fract., vol. 117, no. 2, pp. 175-194, 2002.
[31] P.A. Kelly, J.J. O’Connor and D.A. Hills, “Stress field due to a
dislocation in layered media,” J. Phys. D – Appl. Phys. vol. 28, no. 3, pp.
530-534, 1995.
[32] N.A. Fleck, J.W. Hutchinson and Z. Suo, “Crack path selection in a
brittle adhesive layer,” Int. J. Solids Struct., vol. 27, no. 13, pp. 1683-
1703, 1991.
[33] C-H. Kuo, “Elastic field due to an edge dislocation in a multi-layered
composite,” Int. J. Solids Struct., vol. 51, pp. 1421-1433, 2014.
[34] N.L. Muskhelishvili, Some Basic Problems of Mathematical Theory of
Elasticity. Groningen, The Netherlands: P. Noordhoff, 1958.
[35] T-Y. Zhang and J.C.M. Li, “Interaction of an edge dislocation with an
interfacial crack,” J. Appl. Phys., vol. 72, no. 6, pp. 2215-2226, 1992. [36] C-Y. Hui and D.C. Lagoudas, “Stress fields of interface dislocations,” J.
Appl. Mech. vol. 57, no. 1, pp. 247-248, 1990.
[37] A.P.S. Selvadurai, Partial Differential Equations in Mechanics 2: The
Biharmonic Equation, Poisson's Equation. Springer Science & Business
Media, 2000.
[38] D.A. Hills, P.A. Kelly, D.N. Dai and A.M. Korsunsky, Solution of crack
problems: the distributed dislocation technique. Dordrecht, The
Netherlands: Kluwer Academic Publishers, 1996.
[39] J.R. Rice, “Plane problems of cracks in dissimilar media,” J. Appl.
Mech., vol. 55, no. 1, pp. 98-103, 1988.
[1] J. L. Sackman, J. M. Kelly and A. E. Javid “A layered notch filter for
high-frequency dynamic isolation,” J. Pressure Vessel Technol., vol.
111, no.1, pp. 17-24, 1989.
[2] H. Holleck, M. Lahres and P. Woll, “Multilayer coatings – influence of
fabrication parameters on constitution and properties,” Surf. Coat.
Technol., vol. 41, pp. 179-190, 1990.
[3] B.A. Roeder C.T. Sun, “Dynamic penetration of Alumina/aluminium
laminates: experiments and modelling,” Int. J. Impact Eng., vol. 25, no.
2, pp. 169-185, 2001.
[4] L. Chen and M. J. Pindera, “Plane analysis of finite multilayered media
with multiple aligned cracks - Part I: theory,” J. Appl. Mech., vol. 74, no.
1, pp. 128-143, 2006.
[5] A. Khanna and A. Kotousov, “Stress analysis of a crack in a fiberreinforced
layered composite,” Compos. Struct., vol. 118, pp. 139-148,
2014.
[6] I. Sellinger, P. M. Weiss, A. Nguyen, Y. Lu, R. A. Assink, W. Gong and
C.J. Brinker, “Continuous self-assembly of organic–inorganic
nanocomposite coatings that mimic nacre,” Nature, vol. 394 , pp. 256-
260, 1998.
[7] H. Gao, B. Ji, I. L. Jäger, E. Arzt and P. Fratzl, “Materials become
insensitive to flaws at nanoscale: Lessons from nature,” Proc. Natl.
Acad. Sci. USA, vol. 100, no. 10, pp. 5597-5600, 2003.
[8] A. A. Daneshy, “Hydraulic Fracture Propagation in Layered
Formations,” SPE J., vol. 18, no.1, pp. 33-41, 1978.
[9] A. Gudmundsson, T. H. Simmenes, B. Larsen and S. L. Philipp, “Effects
of internal structure and local stresses on fracture propagation,
deflection, and arrest in fault zones,” J. Struct. Geol., vol. 32, pp. 1643-
1655, 2010.
[10] A. Khanna and A. Kotousov, “Controlling the Height of Multiple
Hydraulic Fractures in Layered Media,” SPE J. SPE-176017-PA, 2015.
[11] A. Khanna, A. Kotousov, J. Sobey and P. Weller, “Conductivity of
narrow fractures filled with a proppant monolayer,” J. Petrol. Sci. Eng.,
vol. 100, pp 9-13, 2012.
[12] A. Khanna, A. Keshavarz, K. Mobbs, M. Davis and P. Bedrikovetsky,
“Stimulation of the natural fracture system by graded proppant
injection,” J. Petrol. Sci. Eng., vol. 111, pp. 71-77, 2013.
[13] V.V. Bolotin, “Delaminations in composite structures: its origin,
buckling, growth and stability,” Compos. Part B – Eng., vol. 27, no.2,
pp. 129-145, 1996.
[14] A.C. Garg, “Delamination - a damage mode in composite structures,”
Eng. Fract. Mech., vol. 29, no. 5, pp. 557-584, 1988.
[15] J. Cook, J.E. Gordon, C.C. Evans and D.M. Marsh, “A mechanism for
the control of crack propagation in all-brittle systems,” P. Roy. Soc.
Lond. A – Mat., vol. 282, no. 1391, pp. 508-520, 1964.
[16] M-Y. He and J.W. Hutchinson, “Crack deflection at an interface
between dissimilar elastic materials,” Int. J. Solids. Struct., vol. 25, no.
9, pp. 1053-1067, 1989.
[17] V. Gupta, A.S. Argon and Z. Suo, “Crack deflection at an interface
between two orthotopic media,” J. Appl. Mech. vol. 59, no. 2S, pp. S79-
S87, 1992.
[18] J. Li, “Debonding of the interface as ‘crack arrestor’,” Int. J. Fract., vol.
105, no. 1, pp. 57-79, 2000.
[19] K. Okumura and P-G. de Gennes, “Why is nacre strong? Elastic theory
and fracture mechanics for biocomposites with stratified structures,”
Eur. Phys. J. – E, vol. 4, no. 1, pp. 121-127, 2001.
[20] M.P. Cleary, “Primary factors governing hydraulic fractures in
heterogeneous stratified porous formations,” No. UCRL-13884; CONF-
781112-10 (Massachusetts Inst. of Tech., Cambridge, USA 1978).
[21] B. Bilby and J. Eshelby, “Dislocations and the theory of fracture,” in:
Fracture: an advanced treatise, vol. 1., H. Liebowitz, Ed. New York,
USA: Academic Press, 1968.
[22] A. Kotousov, L. Bortolan Neto and A. Khanna, “On a rigid inclusion
pressed between two elastic half spaces,” Mech. Mater. vol. 68, pp. 38-
44, 2014.
[23] L. Bortolan Neto and A. Khanna, “The performance of hydraulic
fractures partially filled with compressible proppant,” Aust. J.
Multidisciplinary Eng., vol. 10, no. 2, pp. 185, 2013.
[24] A. Khanna, L. Bortolan Neto and A. Kotousov, “Effect of residual
opening on the inflow performance of a hydraulic fracture,” Int. J. Eng.
Sci. vol. 74, pp. 80-90, 2014.
[25] L. Bortolan Neto, A. Khanna and A. Kotousov, “Conductivity and
performance of hydraulic fractures partially filled with compressible
proppant packs,” Int. J. Rock Mech. Min. Sci., vol. 74, pp. 1-9, 2015.
[26] F. Erdogan G. Gupta, “The stress analysis of multi-layered composites
with a flaw,” Int. J. Solids Struct., vol. 7, no. 1, pp. 39-61, 1971.
[27] F. Erdogan G.D. Gupta, “Layered composites with an interface flaw,”
Int. J. Solids Struct., vol. 7, no. 8, pp. 1089-1107, 1971.
[28] F. Erdogan, “Fracture problems in composite materials,” Eng. Fract.
Mech. vol. 4, no. 4, pp. 811-840, 1972.
[29] L. Kucherov, Delamination in periodically layered bi-material
composites. Delamination in periodically layered bi-material composites
(Ph.D. Thesis, Tel Aviv University, Israel 2003).
[30] L. Kucherov and M. Ryvkin, “Interface crack in periodically layered
bimaterial composite,” Int. J. Fract., vol. 117, no. 2, pp. 175-194, 2002.
[31] P.A. Kelly, J.J. O’Connor and D.A. Hills, “Stress field due to a
dislocation in layered media,” J. Phys. D – Appl. Phys. vol. 28, no. 3, pp.
530-534, 1995.
[32] N.A. Fleck, J.W. Hutchinson and Z. Suo, “Crack path selection in a
brittle adhesive layer,” Int. J. Solids Struct., vol. 27, no. 13, pp. 1683-
1703, 1991.
[33] C-H. Kuo, “Elastic field due to an edge dislocation in a multi-layered
composite,” Int. J. Solids Struct., vol. 51, pp. 1421-1433, 2014.
[34] N.L. Muskhelishvili, Some Basic Problems of Mathematical Theory of
Elasticity. Groningen, The Netherlands: P. Noordhoff, 1958.
[35] T-Y. Zhang and J.C.M. Li, “Interaction of an edge dislocation with an
interfacial crack,” J. Appl. Phys., vol. 72, no. 6, pp. 2215-2226, 1992. [36] C-Y. Hui and D.C. Lagoudas, “Stress fields of interface dislocations,” J.
Appl. Mech. vol. 57, no. 1, pp. 247-248, 1990.
[37] A.P.S. Selvadurai, Partial Differential Equations in Mechanics 2: The
Biharmonic Equation, Poisson's Equation. Springer Science & Business
Media, 2000.
[38] D.A. Hills, P.A. Kelly, D.N. Dai and A.M. Korsunsky, Solution of crack
problems: the distributed dislocation technique. Dordrecht, The
Netherlands: Kluwer Academic Publishers, 1996.
[39] J.R. Rice, “Plane problems of cracks in dissimilar media,” J. Appl.
Mech., vol. 55, no. 1, pp. 98-103, 1988.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:70373", author = "Aditya Khanna and Andrei Kotousov", title = "Numerical Implementation of an Interfacial Edge Dislocation Solution in a Multi-Layered Medium", abstract = "A novel method is presented for obtaining the stress
field induced by an edge dislocation in a multilayered composite. To
demonstrate the applications of the obtained solution, we consider the
problem of an interfacial crack in a periodically layered bimaterial
medium. The crack is modelled as a continuous distribution of edge
dislocations and the Distributed Dislocation Technique (DDT) is
utilized to obtain numerical results for the energy release rate (ERR).
The numerical implementation of the dislocation solution in
MATLAB is also provided.", keywords = "Distributed dislocation technique, Edge dislocation,
Elastic field, Interfacial crack, Multi-layered composite.", volume = "9", number = "7", pages = "1247-8", }
