On the properties of integral equations arising in contact problems for porous elastic strip

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European Journal of Mechanics A/Solids 22 (2003) 489–496

On the properties of integral equations arising in contact problfor porous elastic strip

Antonio Scaliaa,∗, Mezhlum A. Sumbatyanb

a Dipartimento di Matematica, Università di Catania, Viale A. Doria n. 6, 95125 Catania, Italyb Research Institute of Mechanics and Applied Mathematics, Stachki Prospect 200/1, Rostov-on-Don 344090, Russia

Received 12 February 2002; accepted 31 January 2003

Abstract

The paper is concerned with a Fredholm integral equation of the first kind arising in contact problems for elastic founwith voids. In the structure of its kernel there are two principal parameters. The first one is of a relative thickness offoundation. The second one is coupled with the porosity of the material. We study uniqueness and solvability of tintegral equation, and then give explicit analytical asymptotics for the leading asymptotic term of the strip complianctwo limiting cases of thick and thin strip. Finally, analytical results are compared with results obtained by direct nutreatment. 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

1. Introduction

Since the classical works of Hertz, many papers were devoted to study contact problems on rigid punch abovfoundations of various kinds. Sadowsky first developed an exact analytical solution for linear isotropic elastic half-plashowed that under frictionless contact the distribution of the contact normal pressure is explicitly given as (see Sneddp(x)= P/π(a2 − x2)1/2 (so-called “parabolic” contact diagram), whereP is the force applied to the punch, and 2a its width.

In early works some authors reduce such contact problems to a Riemann boundary-value problem, but afterequivalent formulation in terms of respective integral equations becomes more standard approach, so far as varioumethods have been recently proposed just to solve (both analytically and numerically) integral equations of vario(Fredholm, with a weak singularity, with the singularity of Cauchy type; for some standard techniques, see Muskhe1953). Particularly, some asymptotic approaches were developed to study contact problems for semi-infinite elastic fo(see, for instance, Aleksandrov, 1968).

At the same time, in many fields of engineering practice, it is well known that experimental results do not justify the padiagram of the contact stress, since the materials of the real foundations (particularly, in civil engineering) cannot be coas ideal elastic solids.

For this reason Cowin and Nunziato (1983) proposed a new model, to adequately describe elastic materials withour previous works (Scalia and Sumbatyan, 2000; Scalia, 2002) we studied contact problems in the case of a half-planfoundations made of such a material. We reduced the problem to respective Fredholm integral equations of the firstapplied a co-location numerical technique to determine distribution of the contact pressure and compliance of the fou

The main purpose of the present work is to study analytically (more pricisely, asymptotically) these integral equationin contact problems for such semi-infinite foundations.

* Corresponding author.E-mail addresses:[email protected] (A. Scalia), [email protected] (M.A. Sumbatyan).

0997-7538/03/$ – see front matter 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
doi:10.1016/S0997-7538(03)00027-5

490 A. Scalia, M.A. Sumbatyan / European Journal of Mechanics A/Solids 22 (2003) 489–496

2. Review of previous results

ds then,

e).

nt,

nt

,

idth 2

If we consider a two-dimensional plane-strain stationary problem for homogeneous and isotropic material with voiaccording to the Cowin–Nunziato theory, the two componentsux, uy of the displacement vectoru= {ux,uy,0} and the relativechange in the volume fractionφ (i.e., porosity) satisfy the 3× 3 system of partial differential equations

∂2ux

∂x2+ c2 ∂

2ux

∂y2+ (

1− c2) ∂2uy

∂x∂y+H ∂φ

∂x= 0,

∂2uy

∂y2+ c2 ∂

2uy

∂x2+ (

1− c2) ∂2ux

∂x∂y+H ∂φ

∂y= 0,

l21

(∂2φ

∂x2+ ∂2φ

∂y2

)− l21

l22

φ −(∂ux

∂x+ ∂uy

∂y

)= 0,

(1)

where

c2 = µ

λ+ 2µ, H = β

λ+ 2µ, l21 = α

β, l22 = α

ξ, (2)

µ andλ are elastic moduli;α, β andξ constants connected with the porosity of the material (all the quantities are positivThe components of the stress tensor are given by the constitutive equations:

σxx

λ+ 2µ= ∂ux

∂x+ (

1− 2c2)∂uy∂y

+Hφ,σyy

λ+ 2µ= (

1− 2c2)∂ux∂x

+ ∂uy

∂y+Hφ, (3)

τxy

µ= ∂ux

∂y+ ∂uy

∂x.

It should also be noted that parametersc andH are dimensionless, and parametersl1 andl2 have dimension of length.If the foundation is the strip of finite thicknessd: 0 � y � d, |x| < ∞, placed upon an absolutely rigid basey = 0

without friction, and a normal loadp(x), |x| <∞ is applied over the upper boundary (y = d) of the strip, then it is shown inScalia (2002) that the settling of the free surfaceuy is given as follows (sh,ch,cth are hyperbolic sine, cosine and cotangerespectively)

y = h: uy(x, d)=w(x)= 1

4πµ(1− c2)

∞∫−∞

p(ξ)dξ

∞∫−∞

L(s)

sexp

[−is(x − ξ)]ds, (4)

where

L(s)= r(1− c2)(1−N)2 sh2(sh)

D(s),

D(s)= r[sh(sh)ch(sh)+ sh](1−N)(1−N − c2) + 2Nc2s2[r ch(sh)− s cth(rh)sh(sh)

]sh(sh),

r = r(s)=√

1−N + s2, N = l22

l21

H(0<N < 1− c2)

, h= d

l2.

(5)

This representation can be obtained by applying Fourier transform, along the variablex, to the system (1) and the evideboundary conditions

y = 0: τxy = 0, uy = 0,∂φ

∂y= 0,

y = d: τxy = 0, σyy = p(x), ∂φ

∂y= 0.

(6)

Note that all geometric quantities in formulas (5), (6) (including displacement:w = uy/l2) are written in a dimensionless formbeing related to the valuel2. Since the parameterN in (5) is 0<N < 1, so this involves thatr = r(s) > 0.

Therefore, solution of the traction boundary-value problem is given in quadratures. If there is a rigid punch of the wawith the plane base penetrating (under applied forceP ) into the porous strip, thenp(x) = 0, |x| > a andw(x) ≡ w, |x| < a,

A. Scalia, M.A. Sumbatyan / European Journal of Mechanics A/Solids 22 (2003) 489–496 491

is a known value, so the considered contact problem results in the following Fredholm integral equation of the first kind with

oted

heg

ier

use

respect to functionp(x):

b∫−bp(ξ)K(x − ξ)dξ = 2πµ

(1− c2)

w, |x|< b, b= a

l2, K(x)=

∞∫0

L(s)

scos(sx)ds, (7)

where the two (positive) non-dimensional geometric parametersb andh are independent upon each other. It should be nalso that in the caseN = 0 the kernel (4), (5) reduces to the well known classical linear elastic kernel with

L(s)= sh2(sh)

sh(sh)ch(sh)+ sh = ch(2sh)− 1

sh(2sh)+ 2sh. (8)

3. General properties of the kernel K(x)

First of all, it can be proved (see Scalia, 2002) that the denominatorD(s) (5) of the kernel Fourier image is positive on treal semi-axis:D(s) > 0 ⇒ L(s) > 0 for s > 0. This guarantees integrability in expression(7)2 at least over arbitrary (alonpositive values ofs) finite-length interval.

It is clear that the estimate

L(s)= L∞(s)+ O(e−2s), s→ +∞, L∞(s)= r(1− c2)(1−N)2

r(1−N)(1−N − c2)+ 2Nc2s2(r − s) (9)

leads to the following uniqueness theorem.

Theorem 1. Eq. (7) cannot have more than one solution in the functional Lebesgue spaceLp(−b, b), 1< p < 2.

The proof is based upon positiveness of the symbolic functionL(s)/s and some well known properties of the Fourtransform. If there are two different solutionsp1(x) andp2(x), then their differenceg(x) = p1(x)− p2(x) satisfies Eq. (9)with a trivial right-hand side and therefore, according to Parseval identity, one obtains

b∫−bg(ξ)K(x − ξ)dξ = 0, |x|< b ⇒

b∫−b

b∫−bg(x)g(ξ)K(x − ξ)dx dξ = 1

2

∞∫−∞

L(s)

sG2(s)ds = 0, (10)

where the functionG(s) is a Fourier image of the functiong(x). Obviously, the last integral over the real axis exists, becathe estimate (9) yields|L(s)| �M ⇒ |L(s)/s| �M/|s|, s → ∞, andG(s) ∈ Lq(−∞,∞), q = p/(p − 1). Of course, wehave also used the evident value of the symbolic function at the origin:

lims→0

L(s)

s=A= (1− c2)(1−N)h

2(1−N − c2) , (11)

that implies regularity of the integrand in (10) ats → 0. So far asL(s)/s > 0, |s| <∞, identity (10) involvesG(s) ≡ 0 ⇒g(x)≡ 0 that completes the proof.

In order to study solvability of Eq. (7), we need to extract the principle part (atx→ 0) of the kernelK(x). Since (9) implies

L(s)→ L∞(∞)= (1− c2)(1−N)2(1−N)(1−N − c2)+N(1 −N)c2 = 1, s→ +∞, (12)

let us rewrite the kernel in the equivalent form as

K(x)=∞∫

0

[L(s)− 1]cos(sx)+ exp(−s)s

ds +∞∫

0

cos(sx)− exp(−s)s

ds = − ln |x| +K1(x),

K1(x)=∞∫0

[L(s)− 1] cos(sx)+ exp(−s)s

ds,

(13)


where we have used the table value from Gradshteyn and Ryzhik (1994) of the integral

orm

y-type

f

of

setting

ied to the

∞∫0


ds = − ln |x|. (14)

Obviously, the kernelK1(x) is at least differentiable:

K ′1(x)= −

∞∫0

[L(s)− 1

]sin(sx)ds ∈ C(0,1), (15)

asL(s)− 1 = O(s−2), s→ +∞, and so the integral in (15) is finite. Therefore, if one rewrites Eq. (7) in the equivalent f

−b∫

−bp(ξ) ln|x − ξ |dξ = 2πµ(1 − c2)

bw−

b∫−bp(ξ)K1(x − ξ)dξ, |x|< b, (16)

then one may apply, after differentiation of (16), an explicit-form inversion formula from Gakhov (1966) for the Cauchsingular integral (|x|< b)

p(x)= 1

π(b2 − x2)1/2

[P −

b∫−b

(b2 − ξ2)1/2

ξ − x dξ

b∫−bp(t)K ′

1(t − ξ)dt], (17)

where

P =b∫

−bp(x)dx (18)

is the impression force.

Theorem 2. Eq. (7) is (uniquely) solvable inLp(−b, b) (1<p < 2) for arbitrary positive values of parametersb andh.

The proof directly follows from representation (16), since(b2 − x2)−1/2 ∈Lp(−b, b), 1< p < 2, and due to regularity othe Cauchy-type integral operator on the segment(−b, b).

Representation (17) allows one to state stronger (than simply belonging toLp(−b, b)) conclusions about the structurethe solution:

Theorem 3. Solution of Eq.(7) has the following form

p(x)= g(x)

(b2 − x2)1/2, (19)

where the functiong is smooth: g(x) ∈ C1(−b, b).

4. Asymptotic properties for large thickness h

It would be quite natural to directly obtain respective integral equation for the case of a half-plane foundation, byh→ ∞. As follows from (5), this leads again toL(s)∼L∞(s), h→ ∞. Unfortunately, for such a symbolic functionL∞(s)/srespective integral representation for the kernel (7) diverges, as behaviour of the integrand is of the order 1/s at s → 0. Thisconforms to the well-known defect of the plane problems for unbounded domains when any non-balanced load, applboundary line, yields logarithmic (i.e., infinite) growth of the displacements with distance. Therefore, the caseh→ ∞ requiresa more refined treatment.

Let us write the kernel in the following form

K(x)=∞∫

0

L(s)− [1− exp(−2sh)]L∞(s)s

cos(sx)ds +∞∫

0

[1− exp(−2sh)]L∞(s)s

cos(sx)ds = I1 + I2 (20)


that is called “detaching” (or “removal”) method in asymptotic expansions for kernels with removable singularities. The first

ee

r atdepend on

integralI1 here is regular and the following lemma gives respective power expansion.

Lemma 1. For the integralI1 in (20) the following asymptotic expansion holds

I1 =∞∑n=0

dn

(x

h

)2n, h→ ∞, (21)

uniformly over|x|< 2b.

Really, by the change of variabless′ = sh (the primes are omitted later on) we obtain

I1 =∞∫

0

L(s/h)− [1− exp(−2s)]L∞(s/h)s

cos

(sx

h

)ds. (22)

So far as expression in the square brackets makes the integrand regular for all 0< s <∞ and since the fraction in (22) for largs does not exceed (in modulus)M exp(−s), with a positive constantM being independent onh, hence one can apply for thcosine function a Taylor expansion in powers.

Obviously, the main term of the expansion (21) is obtained from (22) by puttingh= ∞:

d0 = limh→∞ I1 = −1

2

∞∫0

exp(−s)sh(s)+ 2s

sh(s)ch(s)+ s1− exp(−2s)

sds. (23)

Similar idea can be applied to estimate the integralI2 (cf. Aleksandrov, 1968). Let us rewrite it as follows

I2 =∞∫

0

[1− exp(−2sh)

]L∞(s)cos(sx)−L∞(0)exp(−s)s

ds

+L∞(0)∞∫

0

1− exp(−2sh)

sexp(−s)ds = I21 +L∞(0)I22, (24)

where

L∞(0)= B = (1− c2)(1−N)(1−N − c2) . (25)

The integralI22 is given by the table in Gradshteyn and Ryzhik (1994)

I22 =∞∫

0

1− exp(−2sh)

sexp(−s)ds = ln(2h+ 1)= ln(2h)+ O

(1

h

). (26)

In order to estimate asymptotic behaviour of the integralI21, let us notice that the fraction under this integral is regulas → 0, and so the integral can be represented as a difference of two independent ones. The first of them does notparameterh and, as to the second one, a standard Watson method of integration by parts proves that

∞∫0

exp(−2sh)L∞(s)cos(sx)−L∞(0)exp(−s)

sds = O

(1

h

), h→ ∞. (27)

This leads to the following result.

Lemma 2. The asymptotic estimate of the integralI2 in (20), uniform with respect to|x| < 2b, has the following form

I2 = B ln(2h)+K∗(x)+ O

(1

h

), h→ ∞, K∗(x)=

∞∫0

L∞(s)cos(sx)−B exp(−s)s

ds. (28)

It is obvious now that the two lemmas give an explicit asymptotics of the total kernel:


Theorem 4. At h→ ∞ the kernelK(x) possesses, uniformly over|x|< 2b, the following asymptotic behaviour( )

because

l

rithmic

K(x)=K∞(x)+ O1

h, K∞(x)= B ln(2h)+ d0 +K∗(x). (29)

It is clear from (29) that the kernel does not have any finite-form representation for asymptotically large thickness,(as can be predicted beforehand) there is a logarithmic infinite increase with the parameterh increasing.

It is interesting to notice that, if one applies derivative to the both sides of Eq. (7), then one comes, underh→ ∞, to acorrect integral equation, as the one we developed in Scalia and Sumbatyan (2000):

b∫−bp(ξ)dξ

∞∫0

L∞(s)sin[s(x − ξ)]ds = 0, |x|< b, (30)

and distribution of the contact pressurep(x) can be uniquely defined from this equation if the applied forceP is known.However, Eq. (30) does not allow one to extract any relation between the settlingw and the applied forceP .

5. Compliance of the foundation for large h

Taking into account representation (29), one can rewrite the main integral equation (7) in the following form

b∫−bp(ξ)

[K∗(x − ξ)+B ln(2h)+ d0

]dξ = 2πµ

(1− c2)

w, |x| < b. (31)

Let us consider the auxiliary equation

b∫−bq(ξ)K∗(x − ξ)dξ = 1, |x| < b, (32)

with the kernelK∗(x) (28) free of parameterh. It is obvious that

p(x)= 2πµ(1− c2)

w{1− [

B ln(2h)+ d0]P

}q(x), (33)

where the impression forceP is given by (18). After applying integration over the interval−b < x < b, relation (33) involvesthe following representation for compliance

C = 2πµ(1− c2)wP

= 1+ [B ln(2h)+ d0]QQ

, (34)

where

Q=b∫

−bq(x)dx. (35)

Formula (34) gives the leading asymptotic term of the compliance ath→ ∞ sinceq(x) (and soQ), as a solution of integraequation (32), is free ofh.

The following result establishes that Eq. (32) is a classical Fredholm equation of the first kind with a (weak) logasingularity in the kernel.

Theorem 5. The kernelK∗ (28) is at least differentiable for|x|< 2b, x �= 0, and in a vicinity of the origin

K∗(x)= − ln |x| + d1 + O(x2 ln |x|), x→ 0, d1 =

∞∫0

L∞(s)− 1+ (1−B)exp(−s)s

ds. (36)

The proof directly follows from the representation (cf. (25))

K∗(x)=∞∫

0

[L∞(s)− 1]cos(sx)+ [1−L∞(0)]exp(−s)s

ds +∞∫0


ds, (37)

with the use of (14).


6. Asymptotic analysis in the case of thin strip

dof

opf

).

e

entationslts with

Theorem 6. Solution of equation

b∫−bg(ξ)K(x − ξ)dξ = 1, |x|< b, (38)

can be represented as a superposition of the three functions(seeAleksandrov, 1968)

g(x)= ϕ(b+ x)+ ϕ(b− x)− ϕ0, |x|< b, (39)

where the functionϕ(x) satisfies the following equation

∞∫0

ϕ(ξ)K(x − ξ)dξ = 1+∞∫0

[ϕ(2b+ ξ)− ϕ0

]K(x + ξ)dξ, 0< x <∞, (40a)

and the constantϕ0 is a solution of the convolution equation

∞∫−∞

ϕ0(ξ)K(x − ξ)dξ = 1, |x|<∞ ⇒ ϕ0(x)≡ ϕ0 = 1

πA. (40b)

Correctness of this statement directly follows from the eveness of the solutiong(x) in (38), after carrying the right-hanintegral in (40a) over the left side. It should also be noted that solutionp(x) of the main equation (7) is expressed in termsfunctiong(x) asp(x)= 2πµ(1− c2)wg(x).

Theorem 7. At h→ 0 asymptotic behaviour of the right-hand integral in(40a)is given as follows

∞∫0

[ϕ(2b+ ξ)− ϕ0

]K(x + ξ)dξ = O

(e−2b/h), h→ 0, x > 0, (41)

and so the leading term of the asymptotic expansion of functionϕ(x) ath→ 0 can be obtained as a solution of the Wiener–Hequation

∞∫0

ϕ(ξ)K(x − ξ)dξ = 1, 0< x <∞. (42)

In order to prove this theorem, it is sufficient first to solve Eq. (42) and then to verify correctness of the estimate (41Application of the Wiener–Hopf method to Eq. (42) is rather standard and routine (Noble, 1958). This involves

ϕ(x)= 1

πA+ ih

π21−(0)

∞∑n=1

exp(−πnx/h)n1+′(−πni/h)

, (43)

whereL(s)/s = 1+(s) 1−(s) is a factorization of the symbolic functionL(s)/s. From its explicit form (5) one can concludthat1+′(−πni/h) is an algebraic function of parameterh, that justifies the asymptotic estimate (41) to be valid.

Representation (43), with the use of (39), permits explicit asymptotic estimate of the integral measure

G=b∫

−bg(x)dx =

b∫−bϕ0(x)dx

[1+ O(h)

] = 2b

πA

[1+ O(h)

], h→ 0, (44)

that gives respective asymptotic formula for compliance of the thin strip:

C = πA

2b= π(1− c2)(1−N)h

4(1 −N − c2) b . (45)

When both the asymptotic cases of thick and thin strip are studied, formulas (34) and (45) give respective represof the leading-term approximation for compliance of the strip. It is interesting to compare these explicit analytical resu


ss

Elementrithm). Ity

ojects.

Fig. 1. Compliance versus thickness of the strip:N = 0.4, c2 = 0.35, b = 1.0. 1 – direct numerical treatment by BEM; 2 – small-thickneasymptotics; 3 – large-thickness asymptotics.

direct numerical solution of the main integral equation (7). We have performed this comparison by using BoundaryMethods (Banerjee and Butterfield, 1981). An example of such a test is presented in Fig. 1 (lg there is a common logacan be seen that asymptotic results obtained for the cases of large and smallh give a good approximation not only for limitllarge and small thickness but also for moderate values.

Acknowledgements

The paper has been supported by Italian Ministry of University (M.U.R.S.T.) through its national and local (60%) pr

References

Aleksandrov, V.M., 1968. Asymptotic methods in contact problems of elasticity theory. J. Appl. Math. Mech. 32, 691–703.Banerjee, P.K., Butterfield, R., 1981. Boundary Element Methods in Engineering Science. McGraw-Hill, New York.Cowin, S.C., Nunziato, J.W., 1983. Linear elastic materials with voids. J. Elasticity 13, 125–147.Gakhov, F.D., 1966. Boundary Value Problem. Pergamon Press, Oxford.Gradshteyn, I.S., Ryzhik, I.M., 1994. Table of Integrals, Series, and Products, 5th edition. Academic Press, New York.Muskhelishvili, N.I., 1953. Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen.Noble, B., 1958. Methods Based on Wiener–Hopf Technique. Pergamon Press, New York.Scalia, A., 2002. Contact problem for porous elastic strip. Int. J. Engrg. Sci. 40, 401–410.Scalia, A., Sumbatyan, M.A., 2000. Contact problem for porous elastic half-plane. J. Elasticity 60, 91–102.Sneddon, I.N., 1966. Mixed Boundary Value Problems in Potential Theory. North-Holland, Amsterdam. Wiley, New York.

On the properties of integral equations arising in contact problems for porous elastic strip

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