Junction problem for Euler-Bernoulli and Timoshenko elastic inclusions in elastic bodies
Authors:
A. M. Khludnev and T. S. Popova
Journal:
Quart. Appl. Math. 74 (2016), 705-718
MSC (2010):
Primary 35Q74, 49J40
DOI:
https://doi.org/10.1090/qam/1447
Published electronically:
July 20, 2016
MathSciNet review:
3539029
Full-text PDF Free Access
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Abstract: In the paper, we consider an equilibrium problem for a 2D elastic body with thin Euler-Bernoulli and Timoshenko elastic inclusions. It is assumed that inclusions have a joint point, and we analyze a junction problem for these inclusions. Existence of solutions is proved, and different equivalent formulations of the problem are discussed. In particular, junction conditions at the joint point are found. A delamination of the elastic inclusion is also assumed. In this case, inequality type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. We investigate a convergence to infinity and to zero of a rigidity parameter of the elastic inclusions. Limit problems are analyzed.
References
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- Alexander Khludnev, Contact problems for elastic bodies with rigid inclusions, Quart. Appl. Math. 70 (2012), no. 2, 269–284. MR 2953103, DOI 10.1090/S0033-569X-2012-01233-3
- A. M. Khludnev, Singular invariant integrals for elastic body with delaminated thin elastic inclusion, Quart. Appl. Math. 72 (2014), no. 4, 719–730. MR 3291824, DOI 10.1090/S0033-569X-2014-01355-9
- Alexander Khludnev and Atusi Tani, Overlapping domain problems in the crack theory with possible contact between crack faces, Quart. Appl. Math. 66 (2008), no. 3, 423–435. MR 2445521, DOI 10.1090/S0033-569X-08-01118-7
- A.M. Khludnev and V.A. Kovtunenko, Analysis of cracks in solids, Southampton-Boston, WIT Press, 2000.
- A.M. Khludnev, Elasticity problems in non-smooth domains, Moscow, Fizmatlit, 2010.
- A. M. Khludnev, On a crack located at the boundary of a rigid inclusion in elastic plate, Izvestiya RAS, Mech. of Solids (2010), no. 5, 98–110.
- Alexander Khludnev and Günter Leugering, On elastic bodies with thin rigid inclusions and cracks, Math. Methods Appl. Sci. 33 (2010), no. 16, 1955–1967. MR 2744613, DOI 10.1002/mma.1308
- A.M. Khludnev and M. Negri, Crack on the boundary of a thin elastic inclusion inside an elastic body, Z. Angew. Math. Mech. 92 (2012), no. 5, 341–354.
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- A. M. Khludnev, V. A. Kovtunenko, and A. Tani, On the topological derivative due to kink of a crack with non-penetration. Anti-plane model, J. Math. Pures Appl. (9) 94 (2010), no. 6, 571–596. MR 2737389, DOI 10.1016/j.matpur.2010.06.002
- Victor A. Kovtunenko, Shape sensitivity of curvilinear cracks on interface to non-linear perturbations, Z. Angew. Math. Phys. 54 (2003), no. 3, 410–423. MR 2048661, DOI 10.1007/s00033-003-0143-y
- P. A. Krutitskii, The Neumann problem for the equation $\Delta u-k^2u=0$ in the exterior of non-closed Lipschitz surfaces, Quart. Appl. Math. 72 (2014), no. 1, 85–91. MR 3185133, DOI 10.1090/S0033-569X-2013-01319-4
- N. P. Lazarev and E. M. Rudoy, Shape sensitivity analysis of Timoshenko’s plate with a crack under the nonpenetration condition, ZAMM Z. Angew. Math. Mech. 94 (2014), no. 9, 730–739. MR 3259385, DOI 10.1002/zamm.201200229
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- H. Le Dret, Modeling of the junction between two rods, J. Math. Pures Appl. (9) 68 (1989), no. 3, 365–397 (English, with French summary). MR 1025910
- H. Le Dret, Modeling of a folded plate, Computational Mechanics 5 (1990), 401–416.
- P. K. Mallick, Fiber-Reinforced Composites. Materials, Manufacturing, and Design, Marcel Dekker, Inc., 1993.
- M. Nasser and A. Hassen, Embedded beam under equivalent load induced from a surface moving load, Acta Mechanica 67 (1987), 237-247.
- Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387, DOI 10.1515/9783110848915.525
- N. V. Neustroeva, Unilateral contact of elastic plates with a rigid inclusion, Vestnik of Novosibirsk State University (math., mech., informatics) 9 (2009), no. 4, 51–64.
- T. A. Rotanova, On unilateral contact between two plates one of which has a rigid inclusion, Vestnik of Novosibirsk State University (math., mech., informatics) 11 (2011), no. 1, 87–98.
- E. M. Rudoy, Differentiation of energy functionals in the problem for a curvilinear crack with possible contact between crack faces, Izvestiya RAS, Mech. of Solids 6 (2007), 113–127.
- E. M. Rudoy, Asymptotics of energy functional for elastic body with a crack and rigid inclusion. 2D case, Appl. Math. Mechs. 75 (2011), no. 2, 719–729.
- Giuseppe Saccomandi and Millard F. Beatty, Universal relations for fiber-reinforced elastic materials, Math. Mech. Solids 7 (2002), no. 1, 95–110. MR 1900936, DOI 10.1177/1081286502007001226
- Isabelle Titeux and Evariste Sanchez-Palencia, Junction of thin plates, Eur. J. Mech. A Solids 19 (2000), no. 3, 377–400. MR 1763831, DOI 10.1016/S0997-7538(00)00175-3
References
- Blaise Bourdin, Gilles A. Francfort, and Jean-Jacques Marigo, The variational approach to fracture, Springer, New York, 2008. Reprinted from J. Elasticity 91 (2008), no. 1-3 [MR2390547], with a foreword by Roger Fosdick. MR 2473620, DOI 10.1007/978-1-4020-6395-4
- Philippe G. Ciarlet, Hervé Le Dret, and Robert Nzengwa, Junctions between three-dimensional and two-dimensional linearly elastic structures, J. Math. Pures Appl. (9) 68 (1989), no. 3, 261–295 (English, with French summary). MR 1025905
- Gianni Dal Maso and Rodica Toader, A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. Ration. Mech. Anal. 162 (2002), no. 2, 101–135. MR 1897378, DOI 10.1007/s002050100187
- Antonio Gaudiello and Elvira Zappale, Junction in a thin multidomain for a fourth order problem, Math. Models Methods Appl. Sci. 16 (2006), no. 12, 1887–1918. MR 2287334, DOI 10.1142/S0218202506001753
- Antonio Gaudiello and Ali Sili, Asymptotic analysis of the eigenvalues of an elliptic problem in an anisotropic thin multidomain, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 4, 739–754. MR 2819710, DOI 10.1017/S0308210510000521
- Antonio Gaudiello and Elvira Zappale, A model of joined beams as limit of a 2D plate, J. Elasticity 103 (2011), no. 2, 205–233. MR 2772851, DOI 10.1007/s10659-010-9281-6
- H. Itou and A.M. Khludnev, On delaminated thin Timoshenko inclusions inside elastic bodies, Math. Meth. Appl. Sciences, DOI 10.1002/mma.3279.
- Alexander Khludnev, Contact problems for elastic bodies with rigid inclusions, Quart. Appl. Math. 70 (2012), no. 2, 269–284. MR 2953103, DOI 10.1090/S0033-569X-2012-01233-3
- A. M. Khludnev, Singular invariant integrals for elastic body with delaminated thin elastic inclusion, Quart. Appl. Math. 72 (2014), no. 4, 719–730. MR 3291824, DOI 10.1090/S0033-569X-2014-01355-9
- Alexander Khludnev and Atusi Tani, Overlapping domain problems in the crack theory with possible contact between crack faces, Quart. Appl. Math. 66 (2008), no. 3, 423–435. MR 2445521, DOI 10.1090/S0033-569X-08-01118-7
- A.M. Khludnev and V.A. Kovtunenko, Analysis of cracks in solids, Southampton-Boston, WIT Press, 2000.
- A.M. Khludnev, Elasticity problems in non-smooth domains, Moscow, Fizmatlit, 2010.
- A. M. Khludnev, On a crack located at the boundary of a rigid inclusion in elastic plate, Izvestiya RAS, Mech. of Solids (2010), no. 5, 98–110.
- Alexander Khludnev and Günter Leugering, On elastic bodies with thin rigid inclusions and cracks, Math. Methods Appl. Sci. 33 (2010), no. 16, 1955–1967. MR 2744613, DOI 10.1002/mma.1308
- A.M. Khludnev and M. Negri, Crack on the boundary of a thin elastic inclusion inside an elastic body, Z. Angew. Math. Mech. 92 (2012), no. 5, 341–354.
- A. Khludnev and M. Negri, Optimal rigid inclusion shapes in elastic bodies with cracks, Z. Angew. Math. Phys. 64 (2013), no. 1, 179–191. MR 3023082, DOI 10.1007/s00033-012-0220-1
- A. M. Khludnev, V. A. Kovtunenko, and A. Tani, On the topological derivative due to kink of a crack with non-penetration. Anti-plane model, J. Math. Pures Appl. (9) 94 (2010), no. 6, 571–596. MR 2737389, DOI 10.1016/j.matpur.2010.06.002
- Victor A. Kovtunenko, Shape sensitivity of curvilinear cracks on interface to non-linear perturbations, Z. Angew. Math. Phys. 54 (2003), no. 3, 410–423. MR 2048661, DOI 10.1007/s00033-003-0143-y
- P. A. Krutitskii, The Neumann problem for the equation $\Delta u-k^2u=0$ in the exterior of non-closed Lipschitz surfaces, Quart. Appl. Math. 72 (2014), no. 1, 85–91. MR 3185133, DOI 10.1090/S0033-569X-2013-01319-4
- N. P. Lazarev and E. M. Rudoy, Shape sensitivity analysis of Timoshenko’s plate with a crack under the nonpenetration condition, ZAMM Z. Angew. Math. Mech. 94 (2014), no. 9, 730–739. MR 3259385, DOI 10.1002/zamm.201200229
- N. Lazarev, Shape sensitivity analysis of the energy integrals for the Timoshenko-type plate containing a crack on the boundary of a rigid inclusion, Z. Angew. Math. Phys. 66 (2015), no. 4, 2025–2040. MR 3377729, DOI 10.1007/s00033-014-0488-4
- H. Le Dret, Modeling of the junction between two rods, J. Math. Pures Appl. (9) 68 (1989), no. 3, 365–397 (English, with French summary). MR 1025910
- H. Le Dret, Modeling of a folded plate, Computational Mechanics 5 (1990), 401–416.
- P. K. Mallick, Fiber-Reinforced Composites. Materials, Manufacturing, and Design, Marcel Dekker, Inc., 1993.
- M. Nasser and A. Hassen, Embedded beam under equivalent load induced from a surface moving load, Acta Mechanica 67 (1987), 237-247.
- Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, de Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387, DOI 10.1515/9783110848915.525
- N. V. Neustroeva, Unilateral contact of elastic plates with a rigid inclusion, Vestnik of Novosibirsk State University (math., mech., informatics) 9 (2009), no. 4, 51–64.
- T. A. Rotanova, On unilateral contact between two plates one of which has a rigid inclusion, Vestnik of Novosibirsk State University (math., mech., informatics) 11 (2011), no. 1, 87–98.
- E. M. Rudoy, Differentiation of energy functionals in the problem for a curvilinear crack with possible contact between crack faces, Izvestiya RAS, Mech. of Solids 6 (2007), 113–127.
- E. M. Rudoy, Asymptotics of energy functional for elastic body with a crack and rigid inclusion. 2D case, Appl. Math. Mechs. 75 (2011), no. 2, 719–729.
- Giuseppe Saccomandi and Millard F. Beatty, Universal relations for fiber-reinforced elastic materials, Math. Mech. Solids 7 (2002), no. 1, 95–110. MR 1900936, DOI 10.1177/1081286502007001226
- Isabelle Titeux and Evariste Sanchez-Palencia, Junction of thin plates, Eur. J. Mech. A Solids 19 (2000), no. 3, 377–400. MR 1763831, DOI 10.1016/S0997-7538(00)00175-3
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Additional Information
A. M. Khludnev
Affiliation:
Lavrentyev Institute of Hydrodynamics of the Russian Academy of Sciences, and Novosibirsk State University, Novosibirsk 630090, Russia
Email:
khlud@hydro.nsc.ru
T. S. Popova
Affiliation:
North-Eastern Federal University, Yakutsk, 677000, Russia
MR Author ID:
1055465
Email:
ptsokt@mail.ru
Keywords:
Thin inclusion,
rigid inclusion,
non-penetration condition,
crack,
variational inequality,
junction conditions
Received by editor(s):
January 20, 2015
Published electronically:
July 20, 2016
Article copyright:
© Copyright 2016
Brown University
