Literature Review on Cracks Located at the Interface of Dissimilar Materials (Interface Cracks)

  • Sergey KozinovEmail author
  • Volodymyr Loboda
Part of the Springer Tracts in Mechanical Engineering book series (STME)


There is a significant breakthrough in use of composite materials in modern technology, construction, mechanical engineering and electronics in the past decade. Such popularity takes place due to ability to achieve required (electro-)mechanical properties of the composite. Analyzing stresses, strains and electric displacements in the elements of mechanical or electromechanical structures, the most important issue is to determine stress and electric displacement concentrations near the defects or cracks, which are characterized by intensity factors (IFs). Based on the experimental investigations, fracture of composites is found to be primary caused by the cracks located at the interface of the composite components.


  1. 1.
    Muskhelishvili N (1977) Some basic problems of the mathematical theory of elasticity. Springer, DordrechtCrossRefGoogle Scholar
  2. 2.
    Panasyuk V (1968) Limit equilibrium of brittle bodies with cracks. Naukova Dumka, Kyiv (translation in English: Michigan information service, Detroit, 1971) (in Russian)Google Scholar
  3. 3.
    Parton V, Kudryavtsev B (1988) Electromagnetoelasticity. Gordon and Breach Science Publishers, New YorkGoogle Scholar
  4. 4.
    Cherepanov G (1979) Mechanics of brittle fracture. McGraw-Hill International Book Co., New YorkzbMATHGoogle Scholar
  5. 5.
    Cruse T (1988) Boundary element analysis in computational fracture mechanics. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  6. 6.
    Hahn H (1976) Bruchmechanik: Einführung in die theoretischen Grundlagen. Mechanik, Teubner-Studienbüche, StuttgartzbMATHGoogle Scholar
  7. 7.
    Sneddon L, Lowengrub M (1969) Crack problems in the classical theory of elasticity. Wiley, New YorkzbMATHGoogle Scholar
  8. 8.
    Sih G (1973) Methods of analysis and solutions of crack problems. Mechanics of fracture, vol 1. Noordhoff International Publisher, LeydenGoogle Scholar
  9. 9.
    Kassir M, Sih G (1975) Three dimensional crack problems. Mechanics of fracture, vol 2. Noordhoff International Publisher, LeydenGoogle Scholar
  10. 10.
    Altenbach H, Altenbach J, Rikards R (1996) Einführung in die Mechanik der Laminatwerkstoffe. Deutscher Verlag für Grundstoffindustrie, StuttgartGoogle Scholar
  11. 11.
    Qin Q (2001) Fracture mechanics of piezoelectric materials. WIT Press, Southampton and BostonGoogle Scholar
  12. 12.
    Kienzler R (1993) Konzepte der Bruchmechanik. Vieweg, WiesbadenGoogle Scholar
  13. 13.
    Atluri S (1986) Computational methods in the mechanics of fracture. Elsevier Science Publisher, Noorth-HollandzbMATHGoogle Scholar
  14. 14.
    Schwalbe K, Scheider I, Cornec A (2013) Guidelines for applying cohesive models to the damage behaviour of engineering materials. Springer, HeidelbergCrossRefGoogle Scholar
  15. 15.
    Freund L (1990) Dynamic fracture mechanics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  16. 16.
    Gdoutos E (1990) Fracture mechanics criteria and applications. Kluwer, Dordrecht, The NetherlandsCrossRefzbMATHGoogle Scholar
  17. 17.
    Kanninen M, Popelar C (1985) Advanced fracture mechanics. Oxford University Press, New YorkzbMATHGoogle Scholar
  18. 18.
    Tada H, Paris P, Irwin G (1985) The stress analysis of cracks handbook, 2nd edn. Paris Production Inc., St. LouisGoogle Scholar
  19. 19.
    Murakami Y (1987) Stress intensity factors handbook, vols 1–5. Pergamon Press, OxfordGoogle Scholar
  20. 20.
    Williams ML (1959) The stresses around a fault or cracks in dissimilar media. Bull Seism Soc America 49:199–204MathSciNetGoogle Scholar
  21. 21.
    Rice JR, Sih GC (1965) Plane problems of cracks in dissimilar media. J Appl Mech 32:418–423CrossRefGoogle Scholar
  22. 22.
    England A (1965) A crack between dissimilar media. Trans ASME J Appl Mech 32:400–402CrossRefGoogle Scholar
  23. 23.
    Erdogan F (1965) Stress distribution in bonded dissimilar materials with cracks. Trans ASME J Appl Mech 32(15):2027–2040Google Scholar
  24. 24.
    Mossakovsky V, Rybka M (1964) Generalization of the Griffith-Sneddon criterion for the case of a nonhomogeneous body. J Appl Math Mech 28(6):1277–1286Google Scholar
  25. 25.
    Nahta R, Moran B (1993) Domain integrals for axisymmetric interface crack problems. Int J Solids Struct 30:403–410CrossRefzbMATHGoogle Scholar
  26. 26.
    Martin-Moran C, Barber J, Comninou M (1983) The penny-shaped interface crack with heat flow. Part 1: perfect contact. J Appl Mech 50:29–36CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Martin-Moran C, Barber J, Comninou M (1983) The penny-shaped interface crack with heat flow. Part 2: imperfect contact. J Appl Mech 50:770–776CrossRefzbMATHGoogle Scholar
  28. 28.
    Zhao M, Dang H, Fan C, Chen Z (2016) Analysis of an arbitrarily shaped interface cracks in a three-dimensional isotropic thermoelastic bi-material. Part 1: theoretical solution. Int J Solids Struct 97:168–181CrossRefGoogle Scholar
  29. 29.
    Rice J (1988) Elastic fracture mechanics concept for interfacial cracks. J Appl Mech 55:98–103CrossRefGoogle Scholar
  30. 30.
    Clements D (1971) A crack between dissimilar anisotropic media. Int J Engen Sci 9:257–265CrossRefzbMATHGoogle Scholar
  31. 31.
    Hwu C (1993) Fracture parameters for the orthotropic bimaterial interface cracks. Eng Fract Mech 45:89–97CrossRefGoogle Scholar
  32. 32.
    Kattis M (1999) Nonplanar interfacial cracks in anisotropic bimaterials. Int J Fract 98:313–327CrossRefGoogle Scholar
  33. 33.
    Quan W, Sun CT (1998) Methods for calculating stress intensity factors for interfacial cracks between two orthotropic solids. Int J Solids Struct 35:3317–3330CrossRefzbMATHGoogle Scholar
  34. 34.
    Ting TCT (1986) Explicit solution and invariance of the singularities at an interface crack in anisotropic composites. Int J Solids Struct 22:965–983CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Ting TCT (1990) Interface cracks in anisotropic bimaterial. J Mech Phys Solids 38:505–513CrossRefGoogle Scholar
  36. 36.
    Ting TCT (2000) Recent developments in anisotropic elasticity. Int J Solids Struct 37:401–409CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    Comninou M (1977) The interface crack. J Appl Mech 44:631–636CrossRefzbMATHGoogle Scholar
  38. 38.
    Comninou M (1978) The interface crack in a shear field. ASME J Appl Mech 45:287–290CrossRefzbMATHGoogle Scholar
  39. 39.
    Dundurs J, Comninou M (1979) Some consequences of inequality conditions in contact and crack problems. J Elast 9:71–82CrossRefzbMATHGoogle Scholar
  40. 40.
    Comninou M, Dundurs J (1983) Partial closure of cracks at the interface between a layer and a hald-space. Eng Fract Mech 18:315–323CrossRefGoogle Scholar
  41. 41.
    Ni L, Nemat-Nasser S (1991) Interface cracks in anisotropic dissimilar materials: an analytical solution. J Mech Phys Solids 39:113–144CrossRefMathSciNetzbMATHGoogle Scholar
  42. 42.
    Ni L, Nemat-Nasser S (1992) Interface cracks in anisotropic dissimilar materials: general case. Quaterly Appl Math 2:305–322CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Huang Y, Wang W, Liu C, Rosakis A (1998) Intersonic crack growth in bimaterial interfaces: an investigation of crack face contact. J Mech Phys Solids 46:2233–2259CrossRefzbMATHGoogle Scholar
  44. 44.
    Simonov IV (1984) On the steady motion of a crack with slip and separation sections along the interface of two elastic materials. J Appl Math Mech 48(3):347–353.
  45. 45.
    Simonov I (1985) Brittle cleavage of a piecewise-homogeneous elastic medium. J Appl Math Mech 49(2):207–214CrossRefMathSciNetzbMATHGoogle Scholar
  46. 46.
    Simonov IV (1986) Crack at an interface in a uniform stress field. Mech Compos Mater 21:650–657.
  47. 47.
    Beom H, Atluri S (1996) Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media. Int J Fract 75:163–183CrossRefGoogle Scholar
  48. 48.
    Atkinson C (1977) On stress singularities and interfaces in linear elastic fracture mechanics. Int J Fract 13:807–820CrossRefMathSciNetGoogle Scholar
  49. 49.
    Atkinson C (1982) The interface crack with a contact zone (an analytical treatment). Int J Fract 18:161–177MathSciNetGoogle Scholar
  50. 50.
    Gautesen A, Dundurs J (1987) The interface crack in a tension field. J Appl Mech 54:93–98CrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    Gautesen A, Dundurs J (1988) The interface crack under a combined loading. ASME J Appl Mech 55:580–586CrossRefzbMATHGoogle Scholar
  52. 52.
    Dundurs J, Gautesen A (1988) An opportunistic analysis of the interface crack. Int J Fract 36:151–159CrossRefGoogle Scholar
  53. 53.
    Gautesen A (1992) The interface crack in a tension field: an eigenvalue problem for the gap. Int J Fract 55:261–271Google Scholar
  54. 54.
    Gautesen A (1993) The interface crack under a combined loading. Int J Fract 60:349–361Google Scholar
  55. 55.
    Loboda V (1993) The quasi-invariant in the theory of interface cracks. Eng Fract Mech 44:573–580CrossRefGoogle Scholar
  56. 56.
    Kharun I, Loboda V (2003) A set of interface cracks with contact zones in combined tension-shear field. Acta Mechanica 166:43–56CrossRefzbMATHGoogle Scholar
  57. 57.
    Kharun I, Loboda V (2004) A thermoelastic problem for interface cracks with contact zones. Int J Solids Struct 41:159–175CrossRefzbMATHGoogle Scholar
  58. 58.
    Kharun I, Loboda V (2002) Interface cracks with contact zones in the field of concentrated forces and moments. Math Methods Phys-Mech Fields 45(2):103–113 (in Ukrainian)Google Scholar
  59. 59.
    Wang SS, Choi I (1983) The interface crack between dissimilar anisotropic composite materials. J Appl Mech 50:169–178CrossRefMathSciNetzbMATHGoogle Scholar
  60. 60.
    Qu J, Xue Y (1998) Three-dimensional interface cracks in anisotropic bimaterials: the non-oscillatory case. J Appl Mech 65:1048–1055CrossRefGoogle Scholar
  61. 61.
    Nakhmein E, Nuller B (1976) A method for solving of contact periodic problems for the elastic strip and ring. USSR AS, MTT 40(3):53–61 (in Russian)Google Scholar
  62. 62.
    Nakhmein E, Nuller B (1986) Contact between an elastic half-plane and a partly separated stamp. J Appl Math Mech 50(4):507–515CrossRefzbMATHGoogle Scholar
  63. 63.
    Nakhmein E, Nuller B (1988) The pressure of a system of stamps on an elastic half-plane under general conditions of contact adhesion and slip. J Appl Math Mech 52(2):223–230CrossRefMathSciNetzbMATHGoogle Scholar
  64. 64.
    Nakhmein E, Nuller B (1992) Combined periodic boundary-value problems and their applications in the theory of elasticity. J Appl Math Mech 56:82–89CrossRefMathSciNetGoogle Scholar
  65. 65.
    Herrmann K, Loboda V (1999) On interface crack models with contact zones situated in an anisotropic bimaterial. Arch Appl Mech 69:317–335CrossRefzbMATHGoogle Scholar
  66. 66.
    Herrmann K, Loboda V (2001) Contact zones models for an interface crack in a thermomechanically loaded anisotropic bimaterial. J Therm Stress 24:479–506CrossRefGoogle Scholar
  67. 67.
    Kharun I, Loboda V (2004) A problem of thermoelasticity for a set of interface cracks with contact zones between dissimilar anisotropic materials. Mech Mater 7:585–600CrossRefGoogle Scholar
  68. 68.
    Grinchenko V, Ulitko A, Shulga N (1989) Electroelasticity. In Mechanics of coupled fields in the elements of constructions, 5 vol, Naukova Dumka (in Russian)Google Scholar
  69. 69.
    Zhang T, Zhao M, Tong P (2002) Fracture of piezoelectric ceramics. Adv Appl Mech 38:147–289CrossRefGoogle Scholar
  70. 70.
    Chen Y-H, Lu T (2003) Cracks and fracture in piezoelectrics. Adv Appl Mech 39:121–215CrossRefGoogle Scholar
  71. 71.
    Zhang T, Gao C-F (2004) Fracture behaviors of piezoelectric materials. Theor Appl Fract Mech 41:339–379CrossRefGoogle Scholar
  72. 72.
    Schneider G (2007) Influence of electric field and mechanical stresses on the fracture of ferroelectrics. Ann Rev Mater Res 37:491–538CrossRefGoogle Scholar
  73. 73.
    Kudryavtsev B, Parton V, Rakitin V (1975) Fracture mechanics of piezoelectric materials. Rectilinear tunnel crack on the boundary with a conductor. J Appl Math Mech 39(1):136–146CrossRefMathSciNetzbMATHGoogle Scholar
  74. 74.
    Fil’shtinskii L, Fil’shtinskii M (1994) Green’s function for a composite piezoceramic plane with a crack between phases. J Appl Math Mech 58(2):355–362CrossRefMathSciNetGoogle Scholar
  75. 75.
    Wang T, Han X (1999) Fracture mechanics of piezoelectric materials. Int J Fract 98:15–35CrossRefGoogle Scholar
  76. 76.
    Gao C-F, Wang M (2000) Collinear permeable cracks between dissimilar piezoelectric materials. Int J Solids Struct 37:4969–4986CrossRefzbMATHGoogle Scholar
  77. 77.
    Beom H (2003) Permeable cracks between two dissimilar piezoelectric materials. Int J Solids Struct 40:6669–6679CrossRefzbMATHGoogle Scholar
  78. 78.
    Gao C-F, Hausler C, Balke H (2004) Periodic permeable interface cracks in pizoelectric materials. Int J Solids Struct 41:323–335CrossRefzbMATHGoogle Scholar
  79. 79.
    Zhou Z-G, Wang B (2006) Investigation of behavior of Mode-I interface crack in piezoelectric materials by using Schmidt method. Appl Math Mech 27:871–882Google Scholar
  80. 80.
    Deeg W (1980) The analysis of dislocation, crack and inclusion problems in piezoelectric solids. PhD thesis, Stanford UniversityGoogle Scholar
  81. 81.
    Sosa H (1991) Plane problems in piezoelectric media with defects. Int J Solids Struct 28:491–505CrossRefzbMATHGoogle Scholar
  82. 82.
    Suo Z, Kuo CM, Barnett DM, Willis JR (1992) Fracture mechanics for piezoelectric ceramics. J Mech Phys Solids 40:739–765Google Scholar
  83. 83.
    Lekhnitsky S (1963) Theory of elasticity of an anisotropic elastic body. San Francisco: Holden-DayGoogle Scholar
  84. 84.
    Stroh AN (1962) Steady state problems in anisotropic elasticity. J Math Phys 41:77–103CrossRefMathSciNetzbMATHGoogle Scholar
  85. 85.
    Gao C-F, Fan W (1999) Exact solution for the plane problem in piezoelectric materials with an elliptic hole or a crack. Int J Solid Struct 36:2527–2540CrossRefMathSciNetzbMATHGoogle Scholar
  86. 86.
    Pak Y (1992) Linear electro-elastic fracture mechanics of piezoelectric materials. Int J Fract 54:79–100CrossRefGoogle Scholar
  87. 87.
    Ru CQ, Mao X, Epstein M (1998) Electric-field induced interfacial cracking in multilayer electrostrictive actuators. J Mech Phys Solids 46:1301–1318CrossRefMathSciNetzbMATHGoogle Scholar
  88. 88.
    Ru CQ (1999) Effect of electrical polarization saturation on stress intensity factors in a piezoelectric ceramic. Int J Solids Struct 36:869–883CrossRefzbMATHGoogle Scholar
  89. 89.
    Shen S, Nishioka T, Hu SL (2000) Crack propagation along the interface of piezoelectric bimaterial. Theor Appl Fract Mech 34:185–203CrossRefGoogle Scholar
  90. 90.
    Wang XD (2000) Analysis of strip electric saturation model of crack problem in piezoelectric materials. Int J Solids Struct 37:6031–6049CrossRefzbMATHGoogle Scholar
  91. 91.
    Wang XD, Meguid SA (2000) On the electroelastic behaviour of a thin piezoelectric actuator attached to an infinite host structure. Int J Solids Struct 37:3231–3251CrossRefzbMATHGoogle Scholar
  92. 92.
    Hao T, Shen Z (1994) A new electric boundary condition of electric fracture mechanics and its application. Eng Fract Mech 47:793–802CrossRefGoogle Scholar
  93. 93.
    Dunn M (1994) The effect of crack faces boundary conditions on the fracture mechanics of piezoelectric solids. Eng Fract Mech 48:25–39CrossRefGoogle Scholar
  94. 94.
    McMeeking R (1999) Crack tip energy release rate for a piezoelectric compact tension specimen. Eng Fract Mech 64:217–244CrossRefGoogle Scholar
  95. 95.
    Xu X, Rajapakse RKND (2001) On a plane crack in piezoelectric solids. Int J Solids Struct 38:7643–7658CrossRefzbMATHGoogle Scholar
  96. 96.
    Wang BL, May YW (2003) On the electrical boundary conditions on the crack surfaces in piezoelectric ceramics. Int J Eng Sci 41:633–652CrossRefGoogle Scholar
  97. 97.
    Gruebner O, Kamlah M, Munz D (2003) Finite element analysis of cracks in piezoelectric materials taking into account the permittivity of the crack medium. Eng Fract Mech 70:1399–1413CrossRefGoogle Scholar
  98. 98.
    Govorukha V, Loboda V, Kamlah M (2006) On the influence of the electric permeability on an interface crack in a piezoelectric bimaterial compound. Int J Solid Struct 43:1979–1990CrossRefzbMATHGoogle Scholar
  99. 99.
    Li Q, Chen Y (2008) Solution for a semi-permeable interface crack in elastic dielectric/piezoelectric bimaterials. ASME J Appl Mech 75:1–13Google Scholar
  100. 100.
    Landis C (2004) Electrically consistent boundary conditions for electromechanical fracture. Int J Solids Struct 41:6291–6315CrossRefzbMATHGoogle Scholar
  101. 101.
    Li W, McMeeking R, Landis C (2008) On the crack face boundary conditions in electromechanical fracture and an experiment protocol for determining energy release rates. Eur J Mech A/Solids 27:285–301CrossRefzbMATHGoogle Scholar
  102. 102.
    Ricoeur A, Kuna M (2009) Electrostatic traction at dielectric interfaces and their implication for crack boundary conditions. Mech Res Commun 36:330–335CrossRefzbMATHGoogle Scholar
  103. 103.
    Qin Q, Mai Y-W (1999) A closed crack tip model for interface cracks in thermopiezoelectric materials. Int J Solids Struct 36:2463–2479CrossRefzbMATHGoogle Scholar
  104. 104.
    Herrmann K, Loboda V (2000) Fracture mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterials by consideration of various contact zone models. Arch Appl Mech 70:127–143CrossRefzbMATHGoogle Scholar
  105. 105.
    Herrmann K, Loboda V, Govorukha V (2001) On contact zone model for an interface crack with electrically insulated crack surfaces in a piezoelectric bimaterial. Int J Fract 111:203–227Google Scholar
  106. 106.
    Comninou M (1977) Interface crack with friction in the contact zone. J Appl Mech 44(4):780–781CrossRefGoogle Scholar
  107. 107.
    Comninou M, Dundurs J (1980) Effect of friction on the interface crack loaded in shear. J Elast 10(2):203–212CrossRefMathSciNetzbMATHGoogle Scholar
  108. 108.
    Leguillon D (1999) Interface crack tip singularity with contact and friction. Comptes Rendus de l’Académie des Sciences - Series IIB - Mechanics-Physics-Astronomy, vol 327, no 5, pp 437–442Google Scholar
  109. 109.
    Antipov Y (1995) An interface crack between elastic materials when there is dry friction. J Appl Math Mech 59(2):273–287CrossRefMathSciNetzbMATHGoogle Scholar
  110. 110.
    Ostrik V (2003) Friction contact of the edges of an interface crack under the conditions of tension and shear. Mater Sci 39(2):214–224CrossRefMathSciNetGoogle Scholar
  111. 111.
    Sapsathiarn Y, Senjuntichai T, Rajapakse R (2012) Cylindrical interface cracks in 1-3 piezocomposites. Compos: Part B 43:2257–2264Google Scholar
  112. 112.
    Loboda V, Kharun I (2001) Plane problem of a crack on the interface of orthotropic plates with friction of crack lips. Mater Sci 37(5):735–745CrossRefGoogle Scholar
  113. 113.
    Kaminsky A, Kipnis L, Kolmakova V (1995) Slip lines at the end of a cut at the interface of different media. Int Appl Mech 31(6):491–495CrossRefzbMATHGoogle Scholar
  114. 114.
    Kaminsky A, Kipnis L, Kolmakova V (1999) On the Dugdale model for a crack at the interface of different media. Int Appl Mech 35(1):58–63Google Scholar
  115. 115.
    Kaminsky A, Kipnis L, Dudik I (2004) Initial development of the prefracture zone near the tip of a crack reaching the interface between dissimilar media. Int Appl Mech 40(2):176–182CrossRefGoogle Scholar
  116. 116.
    Kaminsky A, Dudik I, Kipnis L (2006) On the direction of development of a thin fracture process zone at the tip of an interfacial crack between dissimilar media. Int Appl Mech 42(2):136–144CrossRefGoogle Scholar
  117. 117.
    Kaminsky A, Dudik I, Kipnis L (2007) Initial kinking of an interface crack between two elastic media. Int Appl Mech 43(10):1090–1099CrossRefMathSciNetzbMATHGoogle Scholar
  118. 118.
    Loboda V, Sheveleva A (2003) Determining prefracture zones at a crack tip between two elastic orthotropic bodies. Int Appl Mech 39(5):566–572CrossRefzbMATHGoogle Scholar
  119. 119.
    Sulim G, Grilitskii D, Belokur I (1977) Periodic problem for composite plane with cracks. Mater Sci 13(1):72–75CrossRefGoogle Scholar
  120. 120.
    Nakhmein E, Nuller B, Ryvkin M (1982) Deformation of a composite elastic plane weakened by a periodic system of the arbitrarily loaded slits. J Appl Math Mech 45(6):821–826CrossRefzbMATHGoogle Scholar
  121. 121.
    Kudryavtsev B, Rakitin V (1976) Periodic set of cracks at the interface of piezoelectric and solid conductor. USSR Acad Sci Mech Solids 2:121–129 (in Russian)Google Scholar
  122. 122.
    Kaloerov S, Boronenko O (2006) Magnetoelastic problem for a body with periodic elastic inclusions. Int Appl Mech 42(9):989–996CrossRefzbMATHGoogle Scholar
  123. 123.
    Häusler C, Gao C-F, Balke H (2004) Collinear and periodic electrode-ceramic interfacial cracks in piezoelectric bimaterials. ASME J Appl Mech 71:486–492CrossRefzbMATHGoogle Scholar
  124. 124.
    Schmueser D, Comninou M (1979) The periodic array of interface cracks and their interaction. Int J Solids Struct 15:927–934CrossRefzbMATHGoogle Scholar
  125. 125.
    Ru C (2000) Electrode-ceramic interfacial cracks in piezoelectric multilayer materials. Trans ASME J Appl Mech 67:255–261CrossRefzbMATHGoogle Scholar
  126. 126.
    Liu M, Hsia K (2003) Interfacial cracks between piezoelectric and elastic materials under in-plane electric loading. J Mech Phys Solids 51:921–944CrossRefzbMATHGoogle Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of Mechanics and Fluid DynamicsTU Bergakademie FreibergFreibergGermany
  2. 2.Department of Theoretical and Computational MechanicsOles Honchar Dnipro National UniversityDniproUkraine

Personalised recommendations