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Interaction of a rigid punch with a base-secured elastic rectangle with stress-free sides
Journal of Applied Mathematics and Mechanics 74 (2010) 475–485
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Interaction of a rigid punch with a base-secured elastic rectangle with stress-free sides夽 N.A. Bazarenko Rostov-on-Don, Russia
a r t i c l e
i n f o
Article history: Received 21 August 2008
a b s t r a c t The plane contact problem of the indentation of a rigid punch into a base-sucured elastic rectangle with stress-free sides is considered. The problem is solved by a method tested earlier and reduces to a system of two integral equations in functions describing the displacement of the surface of the rectangle outside the punch and the normal or shear stress on its base. These functions are sought in the form of the sum of trigonometric series and an exponential function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result of this are regularized by introducing small positive parameters. Because the matrix elements of the systems, and also the contact stresses, are defined by poorly converging numerical and functional series, the previously developed method of summation of these series is used. The contact pressure distribution and the dimensionless indenting force are found. Examples of a plane punch calculation are given. © 2010 Elsevier Ltd. All rights reserved.
Unlike the case examined earlier,1 securing of the rectangle is taken into account; a method developed earlier to solve contact problems for bodies of finite size (for a rectangle1 and for cylinders of finite length2,3 ) is used. By employing the generalized orthogonality of homogeneous solutions (formula (1.10) in Ref. 1), it is possible to satisfy some boundary conditions, and, as previously,2,3 to reduce the problem to a system of two integral equations (rather than to a single integral equation, as in the case of an unsecured rectangle1 ). 1. Formulation of the problem and homogeneous solutions In a Cartesian system of coordinates x, y, we will consider the problem of the indentation into an elastic rectangle, sucured at its base y = 0, of a rigid punch having a width 2a and a base y = b − ␦(x), where ␦(x) is an even function in x (Fig. 1). We will assume that in the contact area of the punch and rectangle there are no friction forces, and outside the contact area there is no additional load. The boundary conditions can then be written in the form (1.1) (1.2) where u and are vector components of the displacement of points of the elastic medium. The plane strain problem is solved in term of stresses by expressing the required quantities in terms of the biharmonic function
Investigating the stress function in the form = (x)Y(y), where
夽 Prikl. Mat. Mekh. Vol. 74, No. 4, pp. 667–680, 2010. E-mail address: [email protected]. 0021-8928/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2010.09.014
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Fig. 1.
we obtain
(1.3) Satisfying the first two equations of boundary condition (1.1), we obtain
(1.4) elsewhere.2,4
The asymptotics of the roots Zn and the iteration scheme for calculating them have been given Taking into accout Eqs (1.3) and (1.4), and integrating the Hooke’s relations, we find the eigenfunction Fn (x) and the stress–strain state corresponding to the non-zero eigenvalue Zn (n = 1, 2, . . .):
(1.5) where G is the shear modulus and is Poisson’s ratio. For root Z0 = 0 we have the corresponding equations
(1.6) From relations (1.5) and (1.6) we obtain the homogeneous solutions
(1.7) Here and below, the prime on the summation symbol denotes the abbreviated notation
The functions Fn (x) (n = 1, 2, . . .) satisfy the equations
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477
and the generalized orthogonality condition1
(1.8) Integrating by parts, condition (1.8) can be given the form
(1.9) 2. Method of solution We will give two versions of the solution of the contact problem that enable us to study the effect of the normal and shear stresses concentration at the corner points of the base of the rectangle.5,6 We will introduce the notation (the version)
(2.1) where (x) and g(x) are the required functions, even in x. As there is divergence among the functional series (1.7) which, when y = ys (y0 = 0, y1 = b), define the left-hand sides of conditions (1.1), (1.2) and (2.1), the above-mentioned boundary conditions are replaced by the following
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7) where
(2.8) Equations (2.2), (2.3) and (2.6) are equivalent to the system of relations
(2.9)
(2.10)
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The constants f˜0,1 and f0,0 are determined directly from the first equation of system (2.9) when x = 1 and from Eq. (2.5) when x = −1:
(2.11) Below, using the generalized orthogonality conditions (1.8) and (1.9), we define the constants f˜n,1 and fn,0 . Multiplying the first equation (x) (Eq. (2.10) by F (x)), and then adding and integrating of system (2.9) by m shzm x (Eq. (2.5) by chzm x/chzm ) and the second equation by Fm m with respect to x, we find
(2.12) Now, using relations (2.8), we express the quantities fn,1 and f˜n,0 (n = 0, 1, . . .) in Eqs (2.4) and (2.7) in terms of the constants f˜n,1 and fn,0 , and we then replace these constants by integrals (2.11) and (2.12). As a result, conditions (2.4) and (2.7) take the form
(2.13) (when s = 1, a ≤ x ≤ 1, and when s = 0, |x| ≤ 1). Here
Suppose the specified function ␦() and the required functions g() and () are defined by the series
(2.14)
(2.15) In Eq. (2.15), br = r and || ≤ 1. From the condition ␦(a) = g(a), we find
(2.16) Substituting expressions (21.4)–(2.16) into integral equations (2.13) and equating the coefficients of ␦k (k = 0, 1, . . .) to zero, we obtain a system of functional equations
(2.17)
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(when s = 1, a ≤ x ≤ 1, and when s = 0, |x| ≤ 1). Here
(2.18)
where Iv (zn ) (v = 1, 2, 3) are modified Bessel functions.7 Formulae for calculating the integrals Qh,n , Q˜ h,n and Ikn are known.1,2,8 It is not difficult to show (see Section 3) that the functional series (2.18) evenly converge in the segment [0,1], and consequently they can be integrated term by term. Multiplying Eq. (2.17) when s = 1 by coslm (x − a), and when s = 0 by cosbm x (m = 0, 1, . . .), and integrating respectively over with respect to the segments [a, 1] and [0,1], we obtain two infinite systems of algebraic equations in the unknown (k) (k) quantities Xh and X˜ h (k, h = 0, 1, . . .) (2.19) Taking into account the integrals
˜ ˜ and B˜ and the vectors b(k) and b we obtain expressions for the elements of the matrices A, B, A
(k)
Another approach to solving the problem is possible (the version), where, instead of (x), the function (x) is required, where xy (x,0) = −2(x), |x| ≤ 1. In this version, boundary conditions (2.2)–(2.4) are retained, while conditions (2.5)–(2.7) are replaced by the
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following:
(2.20)
(2.21)
(2.22) where
From Eqs (2.20) and (2.21), applying generalized orthogonality condition (1.8), we find
(2.23) Taking into account integrals (2.11), (2.12) and (2.23), and also the relations
conditions (2.4) and (2.22) can be written in the form
(2.24) (when s = 1, a ≤ x ≤ 1, and when s = 0, |x| ≤ 1). Here
Suppose the required function () is defined by the series
Satisfying the condition (0) = 0 (() is an odd function in ), we find
The system of functional equations of the form (2.17) for the version has the form
(2.25) (when s = 1, a ≤ x ≤ 1, and when s = 0, |x| ≤ 1). Here
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The functional equations (2.25) lead to two systems of algebraic equations of the form (2.19) and to the relations
Integral equations (2.13) and (2.24) are a consequence of the ill-posed problem, and therefore the two systems (2.19) (and the analogous systems for the version) are ill-posed and must be regularized by introducing the small positive parameters ␣ and ˛. ˜ 9 The regularized systems have the form (2.26) Hence, we define the regularized solutions
Y(k)
˜ and Y
(k)
(k)
˜ (Z(k) and Z
for the version) and the functions
(2.27)
(2.28) Then, by means of formulae (2.13) and (2.17), we find the functions (x,ys ) (s = 0, 1), in terms of which we express the stress y (x,b) = (x,b) and the displacement u(x,0) = − (x,0):
(2.29)
3. Summation of the series s (x) and f˜ s (x), are defined only by poorly converging numerical and As the elements of matrices A, B, . . ., and also the functions fh, h, functional series, a method for calculating the residues of these series was developed based on the use of asymptotic summation formulae
(3.1) where
The expressions J(s,) and J(s,0) are given elsewhere.1,2 The conditions of applicability of the first and second formulae of system (3.1) are essentially conditions of even convergence of series (3.1). Theorem.
Suppose the conditions
are satisfied. Then the first functional series (3.1) converges evenly in D. However, if
then the series converges evenly in D0 for any values of (M).
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N.A. Bazarenko / Journal of Applied Mathematics and Mechanics 74 (2010) 475–485 , and We will prove the legitimacy of term-by-term integration with respect to the variable of the series defining the kernels K0,0 (K0,0 For this, we will consider, for example, the kernel
K0,0 ).
(3.2) We will expand, in a series in powers of the small parameter n , the nth term of the residue Rp – the expression −i0n, (x)Un (). Such an expansion can be obtained using the representations
(3.3) Taking formula (3.3) into account, we find
(3.4) Multiplying the corresponding expansions (3.4) and discarding terms of a higher order of infinitesimals than indicated accuracy a special representation for the nth term of the residue
2n ,
we find with the
where A0 = 22 (1 − x)(1 − ), 0 = x + and s0 = (2 − x − )/2 is the smallest exponent. Let D denote the set of points M(x, ) such that x, ∈ [0,1], |x − | > 0. Because, in this set, the conditions of the theorem are satisfied for all sk (M) and k (M) (k = 0, . . ., N), the functional series (3.2), which specifies the kernel K0,0 (x, ), converges evenly in D. The even convergence , of the series defining the kernels K1,1 was established earlier.1
, Thus, the kernels Kh,s (x, ) (h, s = 0, 1) of integral equations (2.13) and (2.24) can be integrated with respect to , and the functional equations (2.17) and (2.25) with respect to the variable x. It was also established that all the kernels are continuous and bounded in the , ¯ (x, ) in the band D*{|x − | → 0} have a singularity of the type (x − )ln|x − |. The logarithmic region D{x, ∈ [0, 1]}, and here the kernels Kh,h
singularity was found taking into account the asymptotics as |x − | → 0 of the special function (z,s,),10 in terms of which the residues of the series investigated were expressed. The series summation procedure was described in detail earlier3 and illustrated by calculations of the values of the function
It was shown3 that K() = 0, || < 1 and K(1) = 1/2. The necessary monitoring of the accuracy of the calculation of the residue Rp = ap+1 + ap+2 + . . . was carried out with respect to the quantity p = |r − R| (for an ideal calculation, p = 0), where
1 (h, x) and R Thus, in checking the residues Rp, p, (m, h) occurring in the formulae
with h = m = 0 and x = 1/2, the following values are obtained
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483
Table 1 Y¯ s × 105
Ys × 105 Versions s
1
2
3
1
2
3
0 1 2 3 75 76 77 78 79 80
−379277 287746 −42379 350158 −618 84 −597 31 −587 −41
−249889 66252 63045 228079 −584 215 −573 146 −569 62
−196940 29636 43168 157944 −498 256 −491 188 −490 112
−28778 64131 110207 −295979 245 −232 237 −224 234 −221
−65849 255200 −253881 −49056 195 −185 189 −179 186 −176
−113019 467501 −643407 186581 362 −344 355 −336 355 −337
Fig. 2.
4. Determination of the contact pressure and indenting force We will give examples of the calculation of a plane punch (␦(x) = ␦0 , k = 0, a = 1/4) for the following versions: (1) b = 1/4, (2) b = 1/2, (3) (0) (0) (0) b = 3/4. The infinite systems (2.26) in the unknown quantities Yh and Y˜ h (h = 0, 1, . . .) (the zero superscript on Yh and (0) (x) will be omitted below) were truncated and solved for a few values of ␣ and ˛. ˜ For each version we chose the pair (␣, ˛) ˜ of lowest values of the regularization parameters (the values of the pairs were (3 × 10−18 , 3 × 10−15 ), (5 × 10−19 , 6 × 10−15 ) and (2 × 10−19 , 4 × 10−15 ) for 1, 2 and 3 version respectively) for which there was no pronounced amplitudes of oscillations of regularized solutions Yh and Y˜ h (h = 0, . . ., 80), and the error was fairly small:
The search for the optimal pair (␣, ˛) ˜ is made significantly easier by the fact that the amplitude of oscillations of solutions Yh and Y˜ h are defined by one parameter: ␣ and ˛ ˜ respectively. Table 1 gives values of the constants Yh × 105 and Y˜ h × 105 (h = 0, . . ., 3; h = 75, . . ., 80). Figs. 2 and 3 show graphs of the functions (0) (x) ≡ (x) and (0) (x) ≡ (x) that were obtained using formulae (2.27) and (2.28); the number of the curve is the same as the number of the version. On the graphs, the spike of stresses in the vicinity of the point x = 1 is notable. In order to find the contact pressure q(x) = −y (x,b) (|x| ≤ a), we will return to relations (2.29) with k = 0 (the subscript is omitted below):
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Fig. 3.
Hence it follows that the dimensionless function of the contact pressure distribution (x) ˜ and the indenting force N0 are defined by the expressions
(4.1) Taking into account the equations
we obtain N0 = 4a−1 ␣1 . The third derivative (x,b) at the central node x = x0 is found numerically:9
Here
Table 2 gives values of the function (t) ≡ (at) ˜ (t = x/a) with t = tk = k/6, and Table 3 gives values of the parameters ( = 0.3)
and of the functions (x) and (x) at extremum points and at x = 1. In these tables, the right-hand columns for version 2 relate to a rectangle lying without friction on a rigid base, i.e., not secured at y = 0 (see Ref. 1, version 2, p. 351). Comparing the values of (tk ) and r for version Table 2
k
(tk ) Versions
0 1 2 3 4 5
1
2
9.292 9.281 9.266 9.312 9.625 11.080
4.513 4.544 4.648 4.877 5.380 6.784
3 3.853 3.885 3.993 4.223 4.712 6.034
3.174 3.207 3.314 3.537 3.997 5.200
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485
Table 3
Parameters
1 2 3 max min (1) max min (1)
Versions 1
2
5.516 2.323 1.285 4.565 0.018 0.288 1.119 0.048 0.117
3.114 1.128 0.872 1.946 0.065 0.658 0.503 0.104 0.228
3 2.730 0.963 0.794 – – – 0 0 0
2.323 0.794 0.691 1.098 0.157 1.130 0.310 0.163 0.385
2 in the left-hand columns with the corresponding values in the right-hand columns, we see that the difference amounts to 10–17%. Consequently, imposing additional constraints on the rectangle leads to an increase the contact pressure. In the upper right-hand part of Fig. 2, graphs of the function (t) obtained from formula (4.1) are also given. In order to explain these graphs, we will single out the root singularity, for example, of the function y (x,b) (version 2) in the left-hand half-vicinity of the point x = a:
Here L4 (x) = a0 + a1 z+a1 z2 + a3 z3 + a4 z4 is a generalized interpolation polynomial for the function y(x) = z ϕ(x) ˜ (z ≡ the interpolation nodes
√
a − x) specified at
Calculating the values yk = y(xk ) with h = 0.0005 (k = 0) and h = 0.006 (k = 1, . . ., 4)
and then the coefficients a0 , a1 , . . ., a4 , we find
In conclusion, for version 1–3 we give the inequality
indicating that the two solutions of the contact problem are fairly close. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Aleksandrov VM, Bazarenko NA. The contact problem for a rectangle with stress-free sides. Prikl Mat Mekh 2007;71(2):340–51. Bazarenko NA. The contact problem for a hollow and a solid cylinder with stress-free ends. Prikl Mat Mekh 2008;72(2):328–41. Bazarenko NA. Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert. Prikl Mat Mekh 2010;74(3):126–39. Vorovich II, Aleksandrov VM, Babeshko VA. Non-classical Mixed Problems of Elasticity Theory. Moscow: Nauka; 1974. Bazarenko NA. Solution by the operator method of the plane problem of elasticity theory for a strip with periodically repeating cuts. Izv Ross Akad Sci MTT 2007;4:156–67. Neuber H. Kerbspannungslehre. Berlin: Springer; 1958. Abramowitz M, Stegun IA. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Washington: Government Printing Office; 1964. Gradshteyn IS, Ryzhik IM. Tables of Integrals, Sums, Series, and products. San Diego: Academic; 2000. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978. Bateman H, Erdélyi A. Higher Transcendental Function. New York: McGraw-Hill; 1955.
Translated by P.S.C.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Interaction of a rigid punch with a base-secured elastic rectangle with stress-free sides夽 N.A. Bazarenko Rostov-on-Don, Russia
a r t i c l e
i n f o
Article history: Received 21 August 2008
a b s t r a c t The plane contact problem of the indentation of a rigid punch into a base-sucured elastic rectangle with stress-free sides is considered. The problem is solved by a method tested earlier and reduces to a system of two integral equations in functions describing the displacement of the surface of the rectangle outside the punch and the normal or shear stress on its base. These functions are sought in the form of the sum of trigonometric series and an exponential function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result of this are regularized by introducing small positive parameters. Because the matrix elements of the systems, and also the contact stresses, are defined by poorly converging numerical and functional series, the previously developed method of summation of these series is used. The contact pressure distribution and the dimensionless indenting force are found. Examples of a plane punch calculation are given. © 2010 Elsevier Ltd. All rights reserved.
Unlike the case examined earlier,1 securing of the rectangle is taken into account; a method developed earlier to solve contact problems for bodies of finite size (for a rectangle1 and for cylinders of finite length2,3 ) is used. By employing the generalized orthogonality of homogeneous solutions (formula (1.10) in Ref. 1), it is possible to satisfy some boundary conditions, and, as previously,2,3 to reduce the problem to a system of two integral equations (rather than to a single integral equation, as in the case of an unsecured rectangle1 ). 1. Formulation of the problem and homogeneous solutions In a Cartesian system of coordinates x, y, we will consider the problem of the indentation into an elastic rectangle, sucured at its base y = 0, of a rigid punch having a width 2a and a base y = b − ␦(x), where ␦(x) is an even function in x (Fig. 1). We will assume that in the contact area of the punch and rectangle there are no friction forces, and outside the contact area there is no additional load. The boundary conditions can then be written in the form (1.1) (1.2) where u and are vector components of the displacement of points of the elastic medium. The plane strain problem is solved in term of stresses by expressing the required quantities in terms of the biharmonic function
Investigating the stress function in the form = (x)Y(y), where
夽 Prikl. Mat. Mekh. Vol. 74, No. 4, pp. 667–680, 2010. E-mail address: [email protected]. 0021-8928/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2010.09.014
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Fig. 1.
we obtain
(1.3) Satisfying the first two equations of boundary condition (1.1), we obtain
(1.4) elsewhere.2,4
The asymptotics of the roots Zn and the iteration scheme for calculating them have been given Taking into accout Eqs (1.3) and (1.4), and integrating the Hooke’s relations, we find the eigenfunction Fn (x) and the stress–strain state corresponding to the non-zero eigenvalue Zn (n = 1, 2, . . .):
(1.5) where G is the shear modulus and is Poisson’s ratio. For root Z0 = 0 we have the corresponding equations
(1.6) From relations (1.5) and (1.6) we obtain the homogeneous solutions
(1.7) Here and below, the prime on the summation symbol denotes the abbreviated notation
The functions Fn (x) (n = 1, 2, . . .) satisfy the equations
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477
and the generalized orthogonality condition1
(1.8) Integrating by parts, condition (1.8) can be given the form
(1.9) 2. Method of solution We will give two versions of the solution of the contact problem that enable us to study the effect of the normal and shear stresses concentration at the corner points of the base of the rectangle.5,6 We will introduce the notation (the version)
(2.1) where (x) and g(x) are the required functions, even in x. As there is divergence among the functional series (1.7) which, when y = ys (y0 = 0, y1 = b), define the left-hand sides of conditions (1.1), (1.2) and (2.1), the above-mentioned boundary conditions are replaced by the following
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7) where
(2.8) Equations (2.2), (2.3) and (2.6) are equivalent to the system of relations
(2.9)
(2.10)
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The constants f˜0,1 and f0,0 are determined directly from the first equation of system (2.9) when x = 1 and from Eq. (2.5) when x = −1:
(2.11) Below, using the generalized orthogonality conditions (1.8) and (1.9), we define the constants f˜n,1 and fn,0 . Multiplying the first equation (x) (Eq. (2.10) by F (x)), and then adding and integrating of system (2.9) by m shzm x (Eq. (2.5) by chzm x/chzm ) and the second equation by Fm m with respect to x, we find
(2.12) Now, using relations (2.8), we express the quantities fn,1 and f˜n,0 (n = 0, 1, . . .) in Eqs (2.4) and (2.7) in terms of the constants f˜n,1 and fn,0 , and we then replace these constants by integrals (2.11) and (2.12). As a result, conditions (2.4) and (2.7) take the form
(2.13) (when s = 1, a ≤ x ≤ 1, and when s = 0, |x| ≤ 1). Here
Suppose the specified function ␦() and the required functions g() and () are defined by the series
(2.14)
(2.15) In Eq. (2.15), br = r and || ≤ 1. From the condition ␦(a) = g(a), we find
(2.16) Substituting expressions (21.4)–(2.16) into integral equations (2.13) and equating the coefficients of ␦k (k = 0, 1, . . .) to zero, we obtain a system of functional equations
(2.17)
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(when s = 1, a ≤ x ≤ 1, and when s = 0, |x| ≤ 1). Here
(2.18)
where Iv (zn ) (v = 1, 2, 3) are modified Bessel functions.7 Formulae for calculating the integrals Qh,n , Q˜ h,n and Ikn are known.1,2,8 It is not difficult to show (see Section 3) that the functional series (2.18) evenly converge in the segment [0,1], and consequently they can be integrated term by term. Multiplying Eq. (2.17) when s = 1 by coslm (x − a), and when s = 0 by cosbm x (m = 0, 1, . . .), and integrating respectively over with respect to the segments [a, 1] and [0,1], we obtain two infinite systems of algebraic equations in the unknown (k) (k) quantities Xh and X˜ h (k, h = 0, 1, . . .) (2.19) Taking into account the integrals
˜ ˜ and B˜ and the vectors b(k) and b we obtain expressions for the elements of the matrices A, B, A
(k)
Another approach to solving the problem is possible (the version), where, instead of (x), the function (x) is required, where xy (x,0) = −2(x), |x| ≤ 1. In this version, boundary conditions (2.2)–(2.4) are retained, while conditions (2.5)–(2.7) are replaced by the
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N.A. Bazarenko / Journal of Applied Mathematics and Mechanics 74 (2010) 475–485
following:
(2.20)
(2.21)
(2.22) where
From Eqs (2.20) and (2.21), applying generalized orthogonality condition (1.8), we find
(2.23) Taking into account integrals (2.11), (2.12) and (2.23), and also the relations
conditions (2.4) and (2.22) can be written in the form
(2.24) (when s = 1, a ≤ x ≤ 1, and when s = 0, |x| ≤ 1). Here
Suppose the required function () is defined by the series
Satisfying the condition (0) = 0 (() is an odd function in ), we find
The system of functional equations of the form (2.17) for the version has the form
(2.25) (when s = 1, a ≤ x ≤ 1, and when s = 0, |x| ≤ 1). Here
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The functional equations (2.25) lead to two systems of algebraic equations of the form (2.19) and to the relations
Integral equations (2.13) and (2.24) are a consequence of the ill-posed problem, and therefore the two systems (2.19) (and the analogous systems for the version) are ill-posed and must be regularized by introducing the small positive parameters ␣ and ˛. ˜ 9 The regularized systems have the form (2.26) Hence, we define the regularized solutions
Y(k)
˜ and Y
(k)
(k)
˜ (Z(k) and Z
for the version) and the functions
(2.27)
(2.28) Then, by means of formulae (2.13) and (2.17), we find the functions (x,ys ) (s = 0, 1), in terms of which we express the stress y (x,b) = (x,b) and the displacement u(x,0) = − (x,0):
(2.29)
3. Summation of the series s (x) and f˜ s (x), are defined only by poorly converging numerical and As the elements of matrices A, B, . . ., and also the functions fh, h, functional series, a method for calculating the residues of these series was developed based on the use of asymptotic summation formulae
(3.1) where
The expressions J(s,) and J(s,0) are given elsewhere.1,2 The conditions of applicability of the first and second formulae of system (3.1) are essentially conditions of even convergence of series (3.1). Theorem.
Suppose the conditions
are satisfied. Then the first functional series (3.1) converges evenly in D. However, if
then the series converges evenly in D0 for any values of (M).
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N.A. Bazarenko / Journal of Applied Mathematics and Mechanics 74 (2010) 475–485 , and We will prove the legitimacy of term-by-term integration with respect to the variable of the series defining the kernels K0,0 (K0,0 For this, we will consider, for example, the kernel
K0,0 ).
(3.2) We will expand, in a series in powers of the small parameter n , the nth term of the residue Rp – the expression −i0n, (x)Un (). Such an expansion can be obtained using the representations
(3.3) Taking formula (3.3) into account, we find
(3.4) Multiplying the corresponding expansions (3.4) and discarding terms of a higher order of infinitesimals than indicated accuracy a special representation for the nth term of the residue
2n ,
we find with the
where A0 = 22 (1 − x)(1 − ), 0 = x + and s0 = (2 − x − )/2 is the smallest exponent. Let D denote the set of points M(x, ) such that x, ∈ [0,1], |x − | > 0. Because, in this set, the conditions of the theorem are satisfied for all sk (M) and k (M) (k = 0, . . ., N), the functional series (3.2), which specifies the kernel K0,0 (x, ), converges evenly in D. The even convergence , of the series defining the kernels K1,1 was established earlier.1
, Thus, the kernels Kh,s (x, ) (h, s = 0, 1) of integral equations (2.13) and (2.24) can be integrated with respect to , and the functional equations (2.17) and (2.25) with respect to the variable x. It was also established that all the kernels are continuous and bounded in the , ¯ (x, ) in the band D*{|x − | → 0} have a singularity of the type (x − )ln|x − |. The logarithmic region D{x, ∈ [0, 1]}, and here the kernels Kh,h
singularity was found taking into account the asymptotics as |x − | → 0 of the special function (z,s,),10 in terms of which the residues of the series investigated were expressed. The series summation procedure was described in detail earlier3 and illustrated by calculations of the values of the function
It was shown3 that K() = 0, || < 1 and K(1) = 1/2. The necessary monitoring of the accuracy of the calculation of the residue Rp = ap+1 + ap+2 + . . . was carried out with respect to the quantity p = |r − R| (for an ideal calculation, p = 0), where
1 (h, x) and R Thus, in checking the residues Rp, p, (m, h) occurring in the formulae
with h = m = 0 and x = 1/2, the following values are obtained
N.A. Bazarenko / Journal of Applied Mathematics and Mechanics 74 (2010) 475–485
483
Table 1 Y¯ s × 105
Ys × 105 Versions s
1
2
3
1
2
3
0 1 2 3 75 76 77 78 79 80
−379277 287746 −42379 350158 −618 84 −597 31 −587 −41
−249889 66252 63045 228079 −584 215 −573 146 −569 62
−196940 29636 43168 157944 −498 256 −491 188 −490 112
−28778 64131 110207 −295979 245 −232 237 −224 234 −221
−65849 255200 −253881 −49056 195 −185 189 −179 186 −176
−113019 467501 −643407 186581 362 −344 355 −336 355 −337
Fig. 2.
4. Determination of the contact pressure and indenting force We will give examples of the calculation of a plane punch (␦(x) = ␦0 , k = 0, a = 1/4) for the following versions: (1) b = 1/4, (2) b = 1/2, (3) (0) (0) (0) b = 3/4. The infinite systems (2.26) in the unknown quantities Yh and Y˜ h (h = 0, 1, . . .) (the zero superscript on Yh and (0) (x) will be omitted below) were truncated and solved for a few values of ␣ and ˛. ˜ For each version we chose the pair (␣, ˛) ˜ of lowest values of the regularization parameters (the values of the pairs were (3 × 10−18 , 3 × 10−15 ), (5 × 10−19 , 6 × 10−15 ) and (2 × 10−19 , 4 × 10−15 ) for 1, 2 and 3 version respectively) for which there was no pronounced amplitudes of oscillations of regularized solutions Yh and Y˜ h (h = 0, . . ., 80), and the error was fairly small:
The search for the optimal pair (␣, ˛) ˜ is made significantly easier by the fact that the amplitude of oscillations of solutions Yh and Y˜ h are defined by one parameter: ␣ and ˛ ˜ respectively. Table 1 gives values of the constants Yh × 105 and Y˜ h × 105 (h = 0, . . ., 3; h = 75, . . ., 80). Figs. 2 and 3 show graphs of the functions (0) (x) ≡ (x) and (0) (x) ≡ (x) that were obtained using formulae (2.27) and (2.28); the number of the curve is the same as the number of the version. On the graphs, the spike of stresses in the vicinity of the point x = 1 is notable. In order to find the contact pressure q(x) = −y (x,b) (|x| ≤ a), we will return to relations (2.29) with k = 0 (the subscript is omitted below):
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N.A. Bazarenko / Journal of Applied Mathematics and Mechanics 74 (2010) 475–485
Fig. 3.
Hence it follows that the dimensionless function of the contact pressure distribution (x) ˜ and the indenting force N0 are defined by the expressions
(4.1) Taking into account the equations
we obtain N0 = 4a−1 ␣1 . The third derivative (x,b) at the central node x = x0 is found numerically:9
Here
Table 2 gives values of the function (t) ≡ (at) ˜ (t = x/a) with t = tk = k/6, and Table 3 gives values of the parameters ( = 0.3)
and of the functions (x) and (x) at extremum points and at x = 1. In these tables, the right-hand columns for version 2 relate to a rectangle lying without friction on a rigid base, i.e., not secured at y = 0 (see Ref. 1, version 2, p. 351). Comparing the values of (tk ) and r for version Table 2
k
(tk ) Versions
0 1 2 3 4 5
1
2
9.292 9.281 9.266 9.312 9.625 11.080
4.513 4.544 4.648 4.877 5.380 6.784
3 3.853 3.885 3.993 4.223 4.712 6.034
3.174 3.207 3.314 3.537 3.997 5.200
N.A. Bazarenko / Journal of Applied Mathematics and Mechanics 74 (2010) 475–485
485
Table 3
Parameters
1 2 3 max min (1) max min (1)
Versions 1
2
5.516 2.323 1.285 4.565 0.018 0.288 1.119 0.048 0.117
3.114 1.128 0.872 1.946 0.065 0.658 0.503 0.104 0.228
3 2.730 0.963 0.794 – – – 0 0 0
2.323 0.794 0.691 1.098 0.157 1.130 0.310 0.163 0.385
2 in the left-hand columns with the corresponding values in the right-hand columns, we see that the difference amounts to 10–17%. Consequently, imposing additional constraints on the rectangle leads to an increase the contact pressure. In the upper right-hand part of Fig. 2, graphs of the function (t) obtained from formula (4.1) are also given. In order to explain these graphs, we will single out the root singularity, for example, of the function y (x,b) (version 2) in the left-hand half-vicinity of the point x = a:
Here L4 (x) = a0 + a1 z+a1 z2 + a3 z3 + a4 z4 is a generalized interpolation polynomial for the function y(x) = z ϕ(x) ˜ (z ≡ the interpolation nodes
√
a − x) specified at
Calculating the values yk = y(xk ) with h = 0.0005 (k = 0) and h = 0.006 (k = 1, . . ., 4)
and then the coefficients a0 , a1 , . . ., a4 , we find
In conclusion, for version 1–3 we give the inequality
indicating that the two solutions of the contact problem are fairly close. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Aleksandrov VM, Bazarenko NA. The contact problem for a rectangle with stress-free sides. Prikl Mat Mekh 2007;71(2):340–51. Bazarenko NA. The contact problem for a hollow and a solid cylinder with stress-free ends. Prikl Mat Mekh 2008;72(2):328–41. Bazarenko NA. Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert. Prikl Mat Mekh 2010;74(3):126–39. Vorovich II, Aleksandrov VM, Babeshko VA. Non-classical Mixed Problems of Elasticity Theory. Moscow: Nauka; 1974. Bazarenko NA. Solution by the operator method of the plane problem of elasticity theory for a strip with periodically repeating cuts. Izv Ross Akad Sci MTT 2007;4:156–67. Neuber H. Kerbspannungslehre. Berlin: Springer; 1958. Abramowitz M, Stegun IA. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Washington: Government Printing Office; 1964. Gradshteyn IS, Ryzhik IM. Tables of Integrals, Sums, Series, and products. San Diego: Academic; 2000. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978. Bateman H, Erdélyi A. Higher Transcendental Function. New York: McGraw-Hill; 1955.
Translated by P.S.C.