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A strip cut in a composite elastic wedge
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Journal of Applied Mathematics and Mechanics xxx (2016) xxx–xxx
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A strip cut in a composite elastic wedge夽 D.A. Pozharskii Don State Technical University, Rostov-on-Don, Russia
a r t i c l e
i n f o
Article history: Received 11 January 2016 Available online xxx
a b s t r a c t Integral equations of new three-dimensional problems concerning a cut in the plane of symmetry of a composite elastic wedge are obtained. The wedge consists of three wedge-shaped layers with a common apex, connected by a sliding clamp: in the middle there is a compressible layer with a cut, to the sides of which two identical incompressible layers are pressed. For the case of a strip cut emerging on the edge of the composite wedge, using the method of paired integral equations, the problem is reduced to Fredholm integral equations of the second kind. An expression is obtained for the stress intensity factor. © 2016 Elsevier Ltd. All rights reserved.
An earlier examination has been made of similar problems for a power-law-deforming1 and transversally isotropic space.2 For a homogeneous three-dimensional wedge, the kernels of integral equations of problems of cuts3 have been obtained in quadratures (sliding or rigid clamping of the wedge faces) or depend on the solution of an auxiliary Fredholm integral equation of the second kind (free faces). For a composite three-dimensional wedge, when the wedge-shaped layer with a cut is incompressible,4 the kernels of the integral equations depend on the solution of an auxiliary integral equation (sliding or rigid clamping) or a system of two integral equations (free faces). Here, the operators in the kernels of the integral equations of problems of cuts are mutually inverse to the operators in the kernels of the integral equations of the corresponding contact problems.5,6 Unlike the problem studied earlier,4 in the case considered below, the material of the layer around the cut has an arbitrary Poisson’s ratio. This case does not boil down to the well-known way4 of rearranging the elastic constants in the solutions,4 but is fundamentally new, as mixed three-dimensional problems for an elastic wedge of compressible material (in contrast to incompressible material) generally reduce to functional difference equations.5 Here, for all three types of boundary condition, the kernel of integral equations of the problem of cuts depends on the solution of a single auxiliary Fredholm integral equation of the second kind, and the operator in the kernel is inverse to the operator in the integral equation of the corresponding contact problem.7
1. Formulation and integral equations of problems In cylindrical coordinates r, , z, we will consider a three-dimensional wedge consisting of wedge-shaped layers (the left-hand part of Fig. 1)
夽 Prikl. Mat. Mekh. Vol. 80, No. 4, pp. 489–495, 2016. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010 0021-8928/© 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010
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Fig. 1.
hinged along their faces = −␣ and = 3␣. The outer faces = −␣ −  and = 3␣ +  are under conditions of sliding or rigid clamping or are stress-free (problems A, B, and C respectively). The layers 1 and 3 have elasticity parameters G1 (shear modulus) and 1 = 0.5 (Poisson’s ratio, incompressible material), and the layer 2 has the parameters G and . In the middle half-plane of the composite wedge = ␣ there is a cut at the region (r, z) ∈ , which is in the open state under the action of the normal load = −q(r, z), (r, z) ∈ , = ␣ ± 0. It is required to determine the magnitude of the cut opening u = ±f(r, z), (r, z) ∈ , = ␣ ± 0. The normal stress intensity factor can then be found. With account taken of the symmetry relative to the plane of the cut, below we will consider the region −␣ −  ≤ ≤ ␣, the boundary conditions in which will be written in the form (for displacements and stresses in region 1 we use the superscript 1)
B
C)
(1.1)
Applying the technique of reducing the problem of the theory of elasticity to the Hilbert boundary-value problem as generalized by I.N. Vekua5 and complex integral Fourier and Kontorovich–Lebedev transformations, we will reduce the boundary-value problems (1.1) to the following integral equation relative to the function f(r, z):
(1.2) where Ki (r) is the Macdonald function, and the operator A in the kernel is defined as
(1.3)
Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010
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3
where the function () satisfies the Fredholm integral equation of the second kind (0 ≤ < ∞)
(task A)
(task B)
(task B) (1.4) Here, to the upper (lower) sign there corresponds the upper (lower) function. With fairly low values of the parameter (1–2), it is possible, using the method of successive approximations, to present the solution of integral equation (1.4) in the form of a power series of (1–2) uniformly converging in the Banach space CM (0, ∞) of continuous functions bounded on the half-axis. After such presentation and its substitution into formula (1.3), it becomes clear that the kernel of integral equation (1.2) is symmetrical:
The main term of the kernel of integral equation (1.2) is related to the first term on the right-hand side of formula (1.3) and is singled out after the expansion
(the term in square brackets decreases exponentially as → ∞, as in this case, by virtue of definitions (1.4), we have W() ∼ th2␣) in the form8,9
(1.5) Then integral equation (1.2) becomes integro-differential, the main term of which corresponds to the case of a crack in an unbounded elastic solid.10 Problem A with 2 = 2 − 4␣ corresponds to the case of a cut in the composite space of two wedge-shaped layers, one of which is incompressible. In particular, when ␣ = /4,  = /2 (Problem A), we have two half-spaces connected by a sliding clamp, one of which is Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010
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4
incompressible, while the other contains a cut that is perpendicular to the boundary (the right-hand side of Fig. 1). In this case, integral equation (1.4) has an exact solution, and the kernel of integral equation (1.2), (1.5) can be represented without quadratures:8
(1.6) When = 0 (G G1 ), formula (1.6) reduces to the formula obtained earlier for the case of a plane cut in a half-space with a free boundary that is perpendicular to this boundary (Ref. 9, formula (1.15)). Let us consider a further case where all wedges are incompressible, = 0.5. Integral equation (1.4) likewise has an exact solution in this case
Then, instead of (1.3), we will obtain
(1.7) It is simple to ascertain that integral equation (1.2), (1.7) for problems A, B, and C coincides with well-known integral equations (Ref. 4, formulae (1.2) to (1.7), where it must be assumed that 1 = 0.5 and where the differences in notation must be taken into account). 2. Strip cut We will examine a strip cut
emerging on the edge of a composite wedge. Let the function q(r, z) be periodic, even, and expandable into a Fourier series with respect to z. Then it is sufficient to study the case
and then to draw up the superimposition of solutions for different target values n ≥ 1, with allowance for the solution for the limiting case of plane strain (n = 0). Assuming in integral equation (1.2) that f(r, z) = f(r) cos(pz), we will reduce it to the following one-dimensional integral equation relative to the function f(r):
(2.1) To solve integral equation (2.1), we will apply the method of paired integral equations associated with integral Kontorovich–Lebedev transformation.11 In deriving the dual integral equations, it is necessary to know the inverse operator to operator A of form (1.3) for problems A, B, and C. In finding these operators, use is made of the relation between problems A, B, and C of cuts and the corresponding contact problems for a two-layer composite wedge of twice as small opening angle (sliding clamp between wedge-shaped layers; the face of the compressible layer is in contact with a rigid punch, while the face of the incompressible layer is subject to the conditions of a sliding or rigid clamp or is stress-free).7 The symbols of the kernels of the integral equations of these problems are mutually inverse. As a result, the following theorem is proved. Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010
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Theorem.
5
The operator A {d(s)} : CM (0, ∞) → CM (0, ∞) of the form (1.3) has an inverse that is equal to B = (A )−1 , where
(2.2) while the functions ± () are determined from the system of Fredholm integral equations of the second kind (0 ≤ ≤ ∞)
0 Example.
(2.3)
When = 0.5, system (2.3) has the exact solution
Then, from formulae (2.2), we will find
(2.4) It is simple to ascertain that the expressions in square brackets in formulae (2.4) and (1.7) are mutually inverse, i.e., the theorem holds. Using the theorem, we will reduce Eq. (2.1) to paired integral equations
(2.5) relative to the new function F(u) linked to the sought function by the relation
We will seek the solution of paired integral equations (2.5) in the form
Then, the first equation of system (2.5) is satisfied identically, while the second equation, using well-known results,11 is reduced to a Fredholm integral equation of the second kind with a symmetrical kernel relative to (t)
(2.6) Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010
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6 Table 1 Problem
A
B
C
Case
1
2
3
1
2
3
1
2
3
=1 =2 =3
0.533 0.570 0.653
0.940 1.04 0.998
1.45 1.26 1.03
0.494 0.523 0.594
0.821 0.944 0.951
1.27 1.20 1.01
4.89 3.78 2.65
2.27 1.81 1.29
1.82 1.39 1.07
For numerical solution of the Fredholm integral equations of the second kind (1.4), (2.3), and (2.6), it is possible to use the mechanical (x), the tables given in Ref. 12 are useful. To improve the quadrature method. In calculating the modified Bessel functions ReK1 ⁄2 + i agreement of the integral on the right-hand side of integral equation (2.6), it is possible to use the formula11
The normal stress intensity factor (SIF)
is expressed via solution of Eq. (2.6) by the formula (2.7) The Table 1 gives values of the dimensionless SIF
calculated by means of formulae (2.4), (2.6), and (2.7) with = 0.5, = 1, q(r) = q = const and different values of the dimensionless parameter = pa for a composite quarter-space (4␣ = 2 = /4, case 1), a composite half-space (4␣ = 2 = /2, case 2), and a composite three-quarterspace (4␣ = 2 = 3/4, case 3). The greatest SIF values are observed in Problem C (free faces), and the lowest SIF values in Problem B (rigid clamping), which is in agreement with well-known results.13 For a more pronounced wave-like load on the flanks of the cut (with increase in ) in Problem C for the three examined cases of angles ␣ and , the SIF decreases, which was noted earlier for a strip cut in a homogeneous wedge with free faces.3 Acknowledgements This work was supported by the Russian Ministry of Education and Science (project 2941, Problem A) and the Russian Foundation for Basic Research (15-01-00331, Problems B and C). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Mkhitaryan SM. The stress state of a power-law-deformed infinite space with a cut in the form of a strip or half-plane. Dokl Akad Nauk Arm SSSR 1982;74(1):30–6. Artamonova YeA, Pozharskii DA. A strip cut in a transverselly isotropic elastic solid. J Appl Math Mech 2013;77(5):551–8. Pozharskii DA. A spatial problem for an elastic wedge with a strip cut. J Appl Math Mech 1994;58(5):913–8. Aleksandrov VM, Pozharskii DA. Problems of cuts in a composite elastic wedge. J Appl Math Mech 2009;73(1):103–8. Aleksandrov VM, Pozharskii DA. Non-Classical Three-Dimensional Problems of the Mechanics of Contact Interactions of Elastic Solids. Moscow: Faktorial; 1998. Aleksandrov VM, Pozharskii DA. Three-dimensional contact problems with friction for a composite elastic wedge. J Appl Math Mech 2010;74(6):692–8. Pozharskii DA. Three-dimensional contact problems for a composite elastic wedge. J Appl Math Mech 2016;80(1):99–103. Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. Amsterdam: Elsevier; 2007. Pozharskii DA. An elliptical crack in an elastic three-dimensional wedge. Izv Ross Akad Nauk MTT 1993;(6):105–12. Gol’dshtein RV, Klein IS, Eskin GI. Variational difference method for solving certain integral and integro-differential equations of three-dimensional problems of elasticity theory. Preprint No. 33. Moscow: Institute for Problems in Mechanics, USSR Acad. Sci.; 1973. 11. Lebedev NN, Skal’skaya IP. Dual integral equations related to the Kontorovich–Lebedev transformation. J Appl Math Mech 1974;38(6):1033–44. 12. Rappoport YuM. Tables of Modified Bessel Functions K1/2+i (x). Moscow: Nauka; 1979. 13. Aleksandrov VM, Smetanin BI, Sobol’ BV. Fine Stress Concentrators in Elastic Bodies. Moscow: Nauka; 1993.
Translated by P.S.C.
Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010
ARTICLE IN PRESS
JAMM-2369; No. of Pages 6
Journal of Applied Mathematics and Mechanics xxx (2016) xxx–xxx
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
A strip cut in a composite elastic wedge夽 D.A. Pozharskii Don State Technical University, Rostov-on-Don, Russia
a r t i c l e
i n f o
Article history: Received 11 January 2016 Available online xxx
a b s t r a c t Integral equations of new three-dimensional problems concerning a cut in the plane of symmetry of a composite elastic wedge are obtained. The wedge consists of three wedge-shaped layers with a common apex, connected by a sliding clamp: in the middle there is a compressible layer with a cut, to the sides of which two identical incompressible layers are pressed. For the case of a strip cut emerging on the edge of the composite wedge, using the method of paired integral equations, the problem is reduced to Fredholm integral equations of the second kind. An expression is obtained for the stress intensity factor. © 2016 Elsevier Ltd. All rights reserved.
An earlier examination has been made of similar problems for a power-law-deforming1 and transversally isotropic space.2 For a homogeneous three-dimensional wedge, the kernels of integral equations of problems of cuts3 have been obtained in quadratures (sliding or rigid clamping of the wedge faces) or depend on the solution of an auxiliary Fredholm integral equation of the second kind (free faces). For a composite three-dimensional wedge, when the wedge-shaped layer with a cut is incompressible,4 the kernels of the integral equations depend on the solution of an auxiliary integral equation (sliding or rigid clamping) or a system of two integral equations (free faces). Here, the operators in the kernels of the integral equations of problems of cuts are mutually inverse to the operators in the kernels of the integral equations of the corresponding contact problems.5,6 Unlike the problem studied earlier,4 in the case considered below, the material of the layer around the cut has an arbitrary Poisson’s ratio. This case does not boil down to the well-known way4 of rearranging the elastic constants in the solutions,4 but is fundamentally new, as mixed three-dimensional problems for an elastic wedge of compressible material (in contrast to incompressible material) generally reduce to functional difference equations.5 Here, for all three types of boundary condition, the kernel of integral equations of the problem of cuts depends on the solution of a single auxiliary Fredholm integral equation of the second kind, and the operator in the kernel is inverse to the operator in the integral equation of the corresponding contact problem.7
1. Formulation and integral equations of problems In cylindrical coordinates r, , z, we will consider a three-dimensional wedge consisting of wedge-shaped layers (the left-hand part of Fig. 1)
夽 Prikl. Mat. Mekh. Vol. 80, No. 4, pp. 489–495, 2016. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010 0021-8928/© 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010
G Model JAMM-2369; No. of Pages 6 2
ARTICLE IN PRESS D.A. Pozharskii / Journal of Applied Mathematics and Mechanics xxx (2016) xxx–xxx
Fig. 1.
hinged along their faces = −␣ and = 3␣. The outer faces = −␣ −  and = 3␣ +  are under conditions of sliding or rigid clamping or are stress-free (problems A, B, and C respectively). The layers 1 and 3 have elasticity parameters G1 (shear modulus) and 1 = 0.5 (Poisson’s ratio, incompressible material), and the layer 2 has the parameters G and . In the middle half-plane of the composite wedge = ␣ there is a cut at the region (r, z) ∈ , which is in the open state under the action of the normal load = −q(r, z), (r, z) ∈ , = ␣ ± 0. It is required to determine the magnitude of the cut opening u = ±f(r, z), (r, z) ∈ , = ␣ ± 0. The normal stress intensity factor can then be found. With account taken of the symmetry relative to the plane of the cut, below we will consider the region −␣ −  ≤ ≤ ␣, the boundary conditions in which will be written in the form (for displacements and stresses in region 1 we use the superscript 1)
B
C)
(1.1)
Applying the technique of reducing the problem of the theory of elasticity to the Hilbert boundary-value problem as generalized by I.N. Vekua5 and complex integral Fourier and Kontorovich–Lebedev transformations, we will reduce the boundary-value problems (1.1) to the following integral equation relative to the function f(r, z):
(1.2) where Ki (r) is the Macdonald function, and the operator A in the kernel is defined as
(1.3)
Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010
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3
where the function () satisfies the Fredholm integral equation of the second kind (0 ≤ < ∞)
(task A)
(task B)
(task B) (1.4) Here, to the upper (lower) sign there corresponds the upper (lower) function. With fairly low values of the parameter (1–2), it is possible, using the method of successive approximations, to present the solution of integral equation (1.4) in the form of a power series of (1–2) uniformly converging in the Banach space CM (0, ∞) of continuous functions bounded on the half-axis. After such presentation and its substitution into formula (1.3), it becomes clear that the kernel of integral equation (1.2) is symmetrical:
The main term of the kernel of integral equation (1.2) is related to the first term on the right-hand side of formula (1.3) and is singled out after the expansion
(the term in square brackets decreases exponentially as → ∞, as in this case, by virtue of definitions (1.4), we have W() ∼ th2␣) in the form8,9
(1.5) Then integral equation (1.2) becomes integro-differential, the main term of which corresponds to the case of a crack in an unbounded elastic solid.10 Problem A with 2 = 2 − 4␣ corresponds to the case of a cut in the composite space of two wedge-shaped layers, one of which is incompressible. In particular, when ␣ = /4,  = /2 (Problem A), we have two half-spaces connected by a sliding clamp, one of which is Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010
G Model JAMM-2369; No. of Pages 6
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4
incompressible, while the other contains a cut that is perpendicular to the boundary (the right-hand side of Fig. 1). In this case, integral equation (1.4) has an exact solution, and the kernel of integral equation (1.2), (1.5) can be represented without quadratures:8
(1.6) When = 0 (G G1 ), formula (1.6) reduces to the formula obtained earlier for the case of a plane cut in a half-space with a free boundary that is perpendicular to this boundary (Ref. 9, formula (1.15)). Let us consider a further case where all wedges are incompressible, = 0.5. Integral equation (1.4) likewise has an exact solution in this case
Then, instead of (1.3), we will obtain
(1.7) It is simple to ascertain that integral equation (1.2), (1.7) for problems A, B, and C coincides with well-known integral equations (Ref. 4, formulae (1.2) to (1.7), where it must be assumed that 1 = 0.5 and where the differences in notation must be taken into account). 2. Strip cut We will examine a strip cut
emerging on the edge of a composite wedge. Let the function q(r, z) be periodic, even, and expandable into a Fourier series with respect to z. Then it is sufficient to study the case
and then to draw up the superimposition of solutions for different target values n ≥ 1, with allowance for the solution for the limiting case of plane strain (n = 0). Assuming in integral equation (1.2) that f(r, z) = f(r) cos(pz), we will reduce it to the following one-dimensional integral equation relative to the function f(r):
(2.1) To solve integral equation (2.1), we will apply the method of paired integral equations associated with integral Kontorovich–Lebedev transformation.11 In deriving the dual integral equations, it is necessary to know the inverse operator to operator A of form (1.3) for problems A, B, and C. In finding these operators, use is made of the relation between problems A, B, and C of cuts and the corresponding contact problems for a two-layer composite wedge of twice as small opening angle (sliding clamp between wedge-shaped layers; the face of the compressible layer is in contact with a rigid punch, while the face of the incompressible layer is subject to the conditions of a sliding or rigid clamp or is stress-free).7 The symbols of the kernels of the integral equations of these problems are mutually inverse. As a result, the following theorem is proved. Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010
G Model
ARTICLE IN PRESS
JAMM-2369; No. of Pages 6
D.A. Pozharskii / Journal of Applied Mathematics and Mechanics xxx (2016) xxx–xxx
Theorem.
5
The operator A {d(s)} : CM (0, ∞) → CM (0, ∞) of the form (1.3) has an inverse that is equal to B = (A )−1 , where
(2.2) while the functions ± () are determined from the system of Fredholm integral equations of the second kind (0 ≤ ≤ ∞)
0 Example.
(2.3)
When = 0.5, system (2.3) has the exact solution
Then, from formulae (2.2), we will find
(2.4) It is simple to ascertain that the expressions in square brackets in formulae (2.4) and (1.7) are mutually inverse, i.e., the theorem holds. Using the theorem, we will reduce Eq. (2.1) to paired integral equations
(2.5) relative to the new function F(u) linked to the sought function by the relation
We will seek the solution of paired integral equations (2.5) in the form
Then, the first equation of system (2.5) is satisfied identically, while the second equation, using well-known results,11 is reduced to a Fredholm integral equation of the second kind with a symmetrical kernel relative to (t)
(2.6) Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010
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JAMM-2369; No. of Pages 6
D.A. Pozharskii / Journal of Applied Mathematics and Mechanics xxx (2016) xxx–xxx
6 Table 1 Problem
A
B
C
Case
1
2
3
1
2
3
1
2
3
=1 =2 =3
0.533 0.570 0.653
0.940 1.04 0.998
1.45 1.26 1.03
0.494 0.523 0.594
0.821 0.944 0.951
1.27 1.20 1.01
4.89 3.78 2.65
2.27 1.81 1.29
1.82 1.39 1.07
For numerical solution of the Fredholm integral equations of the second kind (1.4), (2.3), and (2.6), it is possible to use the mechanical (x), the tables given in Ref. 12 are useful. To improve the quadrature method. In calculating the modified Bessel functions ReK1 ⁄2 + i agreement of the integral on the right-hand side of integral equation (2.6), it is possible to use the formula11
The normal stress intensity factor (SIF)
is expressed via solution of Eq. (2.6) by the formula (2.7) The Table 1 gives values of the dimensionless SIF
calculated by means of formulae (2.4), (2.6), and (2.7) with = 0.5, = 1, q(r) = q = const and different values of the dimensionless parameter = pa for a composite quarter-space (4␣ = 2 = /4, case 1), a composite half-space (4␣ = 2 = /2, case 2), and a composite three-quarterspace (4␣ = 2 = 3/4, case 3). The greatest SIF values are observed in Problem C (free faces), and the lowest SIF values in Problem B (rigid clamping), which is in agreement with well-known results.13 For a more pronounced wave-like load on the flanks of the cut (with increase in ) in Problem C for the three examined cases of angles ␣ and , the SIF decreases, which was noted earlier for a strip cut in a homogeneous wedge with free faces.3 Acknowledgements This work was supported by the Russian Ministry of Education and Science (project 2941, Problem A) and the Russian Foundation for Basic Research (15-01-00331, Problems B and C). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Mkhitaryan SM. The stress state of a power-law-deformed infinite space with a cut in the form of a strip or half-plane. Dokl Akad Nauk Arm SSSR 1982;74(1):30–6. Artamonova YeA, Pozharskii DA. A strip cut in a transverselly isotropic elastic solid. J Appl Math Mech 2013;77(5):551–8. Pozharskii DA. A spatial problem for an elastic wedge with a strip cut. J Appl Math Mech 1994;58(5):913–8. Aleksandrov VM, Pozharskii DA. Problems of cuts in a composite elastic wedge. J Appl Math Mech 2009;73(1):103–8. Aleksandrov VM, Pozharskii DA. Non-Classical Three-Dimensional Problems of the Mechanics of Contact Interactions of Elastic Solids. Moscow: Faktorial; 1998. Aleksandrov VM, Pozharskii DA. Three-dimensional contact problems with friction for a composite elastic wedge. J Appl Math Mech 2010;74(6):692–8. Pozharskii DA. Three-dimensional contact problems for a composite elastic wedge. J Appl Math Mech 2016;80(1):99–103. Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. Amsterdam: Elsevier; 2007. Pozharskii DA. An elliptical crack in an elastic three-dimensional wedge. Izv Ross Akad Nauk MTT 1993;(6):105–12. Gol’dshtein RV, Klein IS, Eskin GI. Variational difference method for solving certain integral and integro-differential equations of three-dimensional problems of elasticity theory. Preprint No. 33. Moscow: Institute for Problems in Mechanics, USSR Acad. Sci.; 1973. 11. Lebedev NN, Skal’skaya IP. Dual integral equations related to the Kontorovich–Lebedev transformation. J Appl Math Mech 1974;38(6):1033–44. 12. Rappoport YuM. Tables of Modified Bessel Functions K1/2+i (x). Moscow: Nauka; 1979. 13. Aleksandrov VM, Smetanin BI, Sobol’ BV. Fine Stress Concentrators in Elastic Bodies. Moscow: Nauka; 1993.
Translated by P.S.C.
Please cite this article in press as: Pozharskii DA. A strip cut in a composite elastic wedge. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.09.010