Complex Potential Functions and Integro-Differential Equation in Elastic Media Problem in Presence of Heat

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1 American Journal of Fluid Dynamics 0, (4: 3-4 DOI: 0593/afd00040 Complex Potential Functions and Integro-Differential Equation in Elastic Media Problem in Presence of Heat M A Se ddeek, M A Abdou,*, W G El-Sayed, F M El-Saedy 3 Department of Mathematics, Deanship of Educational Services, Al-Qassim University, Saudi Arabia Department of Mathematics, Faculty of Science, Alexandria University, Egypt 3 Department of Mathematics, Faculty of Science, Umm Al-Qura University, Makkah, Saudi Arabia Abstract This paper covered the study of the boundary value problem for isotropic homogeneous perforated infinite elastic media in presence of uniform flow of heat For this, we considered the problem of a thin infinite plate of specific thickness with a curvilinear hole where the origin lie inside the hole is conformally mapped outside a unit circle by means of a specific rational mapping The complex variable method has been applied and it transforms the problem to the integro-differential equation with Cauchy kernel that can be solved to find two complex potential functions which called Gaursat functions Moreover, the three stress components,, xy for the boundary value problem in the thermoelasticity plane are obtained Many special cases of the conformal mapping and three applications for different cases are discussed and many main results derived from the work Keywords Boundary Value Problem of Infinite Plate Weakened by a Hole, Conformal Mapping, Integro-Differential Equation with Discontinuous Kernel, Complex Potential Functions Introduction Several authors wrote about the boundary value problems and their applications in many different sciences, see Gakhov [], Ciarlet et al[], Zebib[3] and Saito and Yamamto[4] From these problems, we established contact problems and mixed problems in the theory of elasticity, see Colton and Kress[5], and Abdou[6] In Abdou[6,7], Abdou and Kha mis[8] and Abdou and Khar-Eldin [9],Co mp lex variab le method is used to express the solutions of these problems in the form of power series by applied Laurant's theorem On the other h some of authors applied the complex variable method to obtain the solutions in the form two complex potential functions which called Gaursat functions by using the properties of Cauchy integrals on the circle, or on any region mapped to outside a unit circle γ by means of a general rational mapping z = cω (ζ ω (ζ doesn't vanish or become infinite outside the unit circle In thermoelastic problems for elastic media, the first and second boundary value problems are equivalent to finding two analytic functions φ (z and ψ (z of one complex argument z = x + iy These functions must satisfy the boundary conditions, K φ( t tφ ( t ψ ( t = f ( t ( * Corresponding author: abdella_777@ yahoocom (M A Abdou Published online at Copyright 0 Scientific & Academic Publishing All Rights Reserved t where denotes the affix of a point on the boundary In terms of z = cω (ζ, c > 0, ω (ζ does not vanish or become infinite for ζ >, the infin ite region mapped to outside a unit circle γ For the first fundamental boundary value problem or it called the stress boundary value problem we have K = and f(t is a given function of stress While for K = k > and f(t is a given function of displacement which called the thermal conductivity then eq( called the second fundamental boundary value problems or the displacement boundary value problems The books written by Noda et al[0], Hetnarski and Ignaczak[], Parkus[] and Popov[3] contain many different methods to solve the problems in the theory of elasticity in one, two and three dimensions The complex potential functions ( t and ( take the following form, see Parkus [] S x + is y φ ( ζ = ln ζ + cγζ + φ( ζ ( + k ( ( S x is y ψ ( ζ = k ( + k ln ζ + cγ ζ + ψ ( ζ (3 S, S are the components of the resultant vector x y of all external forces acting on the boundary and Γ, Γ are complex constants Generally the two complex functions ϕζ ( and ψζ ( are single value analytic functions within the region outside the unit circle γ and ϕ( = 0, ψ( = 0

2 3 M A Seddeek et al: Complex Potential Functions and Integro-Differential Equation in Elastic Media Problem in Presence of Heat In [4], Muskhelishvili used the rational mapping, z = c ( ζ + mζ, c > 0, m is real nu mber, (4 to solving the problem of infinite plate weakened by an elliptic hole El-Sirafy and Abdou in [5], used the rational mapping, ζ + mζ z = c, c > 0, n < (5 nζ to solve the first and second fundamental problems for the infinite plate with general curvilinear hole C confrmally mapped on the domain outside a unit circle The rational mapping, z = c( ζ + mζ + m ζ, 0 m, =, (6 is used by Abdou and Khamis [8] to obtain the solution of the problem of an infinite plate with a curvilinear hole having three poles and they take the circle shape, ellipse shape and square shape as a special cases of the holes Also, the conformal mapping function, ζ + mζ z = c, c > 0, (7 ( nζ ( nζ is studied completely by England [6] In this paper, we consider the boundary value problem for isotropic homogeneous perforated infinite elastic media in presence of uniform flow of heat Then, we use the more general shape of conformal mapping to obtain the complex potential functions for the problem in the form integro- differential equation with singular kennel Many special cases are obtained and several applications are discussed from the work This study is useful for researchers who work on the studies of petroleum tubes industry or water or gas It also benefits the physics scientists who work on the study of the ozone hole Formulation of the Problem Consider a thin infinite plate of thickness h with a curvilinear hole C, where the origin lie inside the hole is conformally mapped on the domain outside a unit circle γ by means of a rational mapping, ζ + mζ + mζ z = c, c > 0, n n ( ( nζ ( nζ where z (ζ does not vanish or become infinite outside the unit circle γ If a temperature distribution Θ = qy is following uniformly in the direction of the negative y a xis, where the increasing a temperature distribution Θ is assumed to be constant a cross the thickness of the plate, ie Θ = Θ( x, y, and q is the constant temperature gradient The uniform flow of heat is distributed by the presence of an insulated curvilinear hole C The heat equation satisfies the relation, Θ = 0, = + ( x y Θ = 0, r = r0 n (3 Figure ( where n is the unit vector perpendicular to the surface Neglecting the variation of the strain and the stress with respect to the thickness of the plate, the thermoelastic potential Φ satisfies the formula, see Parkus [] Φ = (+ ν α Θ (4 where α is a scalar which present the coefficient of the thermal expansion and ν is Poisson s ratio Assume the force of the plate is free of applied loads In this case, the formu la ( fo r the first and second boundary value problems respectively take the following form, s Φ Φ ϕ( t + tϕ ( t + ψ( t = + i + [ ( (] + (5 ix s Y s ds c x y G 0 Φ Φ kφ ( t tφ ( t ψ ( t = u + iv i x y (6 where the applied stresses X (s and Y (s are prescribed on the boundary of the plane s is the length measured from an arbitrary point, u and v are the displacement components, G is the shear modulus Also, here the applied stresses X (s and Y (s must satisfy the following, see Parkus [] dy dx X ( s = xy ds ds dy dx Y ( s = yx ds ds (7 (8 where, xy and are the components of stresses which are given by the following relations, + ixy = Φ Φ Φ (9 G[ + i ] + 4 G[ zϕ ( z + ψ ( z] y x xy + = 4G [ Reφ ( z λθ ] (0 where λ = α ( +ν is the coefficient of heat transfer The rational mapping z = cω (ζ maps the boundary C of the given region occupied by the middle plane of the plate in the z plane onto the unit circle γ in the ζ plane Curvilinear coordinates ( ρ, θ are thus introduce into the z plane which are the maps of the polar conditions in the iθ ζ plane as given by ζ = ρe, 0 θ By using the transformation, Eq( reduce to, z = cω (ζ

3 American Journal of Fluid Dynamics 0, (4: ωζ ( Kϕ( cωζ ( ϕ( cωζ ( ψ( cωζ ( = f( cωζ ( ( ω ( ζ The last formula represent the first and second boundary value problems in ζ plane Figure (5 3 The Rational Mapping The rational mapping ( maps the curvilinear hole C in z plane onto the domain of outside unit circle in ζ plane under the condition ω (ζ does not vanish or become infinite outside the unit circle γ The following graphs give the different shapes of the rational mapping ( n = n = 0, m = im, = 0 + 0i Fi gure (6 n = n = m = m = 0 Fi gure ( n = 00, n = 0, m = im, = 03 03i Fi gure (7 n = n = 0, m = 0987, m = 0 Fi gure (3 n = 03, n = 05, m = + 00 im, = 00 i Fi gure (8 n = 05, n = m = m = 0 Fi gure (4 n = 000, n = 000, m = im, = i Fi gure (9 n = 04, n = 0, = m = im, = 0 n = 09, n = 04, m = 099, m = 005

4 34 M A Seddeek et al: Complex Potential Functions and Integro-Differential Equation in Elastic Media Problem in Presence of Heat Fi gure (0 Fi gure ( 4 Method of Solution In this section, we use the complex variable method to obtain the two complex functions, Goursat functions, ϕζ ( and ψζ ( Moreover, the three stress components xy, and will be complete determined 4 The Stress Components The solution of Eq ( is given by, r0 Θ = q [ R + ], R R = x + y (4 By substitute Eq(4 in Eq(4 and using the definition of Φ in polar coordinates the thermoelastic potential function take the form, ( +ν α q r0 Φ = ln z 4 R (4 Also, the stresses components can be adapted in the form, Φ Φ = G[ ( + λθ y x, (43 Φ Φ = G [ ( λθ + Re[ ϕ ( + ϕ ( + ψ ( ]],(44 y x z z z z Φ xy = G [ + Im( z φ ( z + ψ ( z ] (45 x y Eqs(43, (44 and (45 after some derivatives and algebraic relations adapted as, = G[ η( z + 4 zz+ z Imz + Re[ ϕ ( z M( z, z]],(46 = G[ η( z + 4 zz + z Imz + Re[ ϕ ( z + M( z, z], (47 xy = G [ η ( zz (Im z Re z + Im M ( z, z ], (48 n = 00, n = 04 i m = 07 + im, = 005 i + Re[ ϕ ( z zϕ ( z ψ ( z]] r0 ( + ν η = ( zz, (49 M ( z, z = zφ ( z + ψ ( z (40 After determine the Goursat functions the components of stress are completely determined 4 Goursat Functions To obtain the two complex potential functions, Goursat functions by using the conformal mapping ( in the boundary condition(, we write the expression ω( ζ ω ( ζ in the form, ω ( ζ = α( ζ + β ( ζ ω ( ζ (4 h h α( ζ = +, (4 ζ n ζ n 3 ( n + mn + m( n ( nn ± + h = ( n n [ ( n n n ( m 3 nn n 3 4 m n + ( nn m + nm + n m n ] βζ ( is a regular function for ζ > ω( ζ has a singularity at ζ = n and ω ( ζ ζ = n Using Eq( and Eq(3 in Eq(, we get Using Eq(4 in Eq(44, we have, = 0, ωζ ( Sx is y KcΓ ζ+ Kϕζ ( + [ ζ ( ω ζ π ( + k cγ ϕ ( ζ] + cγ ζ ψζ ( = F( ζ Kϕ( ζ α( ζ ϕ ( ζ ψ( ζ β( ζ ϕ ( ζ N( ζ[ αζ ( + βζ ( ] = F( ζ ckγζ cγ ζ (43 (44 (45 Taking ζ =, we get Kφ( α( φ ( ψ ( = F (, (46 ψ ( ζ = ψ ( ζ + β ( ζ φ ( ζ, (47 F ( ζ ( ζ ζ ζ ( ζ[ αζ ( βζ ( ], (48 = F ck Γ cγ + N + S x is y N( ζ = cγ ζ, (49 ( + k F ( ζ = f ( cω ( ζ = f ( t (40 Assume the function F( with its derivatives must satisfy Holder condition Our aim is to determine the functions φ (ζ and ψ (ζ for the various boundary value problems For this multiply both side of Eq(46 by d where ζ is any point the interior of γ and i( ζ integral over the circle, we obtain

5 American Journal of Fluid Dynamics 0, (4: K ϕ ( αϕ ( ( ψ( π d ( ζ π d ( ζ π d i i i ( ζ γ γ γ (4 F( = π d i ( ζ γ Using Eqs(47,(48 and (49 in Eq(4 then applying the properties of Cauchy integral to have α ( ϕ ( Kϕζ ( π d i ( ζ γ, (4 N( α ( ( ζ ζ = Γ π d A c i ( ζ γ F( A( ζ = d (43 i γ ( ζ The formula (4 represents the integro-differential equation of second kind with Cauchy kernel The references, Fedotov[7], Hanyga and Seredyńska[8], Bavula[9] and AL-Jawary and Wrobel[0], contain many different methods to solve this kind of the equations analytically and numerically in one, two and three dimensions To obtain the integral terms of Eq (4 we use Eq (4 and then apply the residue theorem to have, α( φ ( h b d = c, (44 i γ ( ζ = n ζ where b, =, are complex constants Also, N( α( N( n h d = (45 i γ ( ζ = n ζ So, Eq(4 reduce to, ( ( h Kϕζ = A ζ cγ ζ + [ N ( n + cb ] (46 = n ζ To determine the constants b, differentiating Eq(46 with respect to ζ and substituting in Eq(44, to get α ( [ ( π A cγ i ( ζ γ (47 h hb [ N( n + cb ]] d = ck = ( n = n ζ Substituting Eq(4 in Eq(47, then using the properties of Cauchy integral and applying the residue theorem at the singular points, we obtain ck b + cγ n + h d ( N( n + cb = A( n, (48 n d =, =, (49 ( n The last equation can be written in the form, ckb + c h d b = E, (430 E A = ( n cγ n h d N( n (43 Taking the complex conugate of Eq(430 we get, ck b + c h d b = E (43 From Eq(430 and Eq(43 we have, KE h d E b =, =, (433 c ( K h d To obtain the complex functions ψ (ζ we have form Eq(47 after substituting the expression of ψ ( and F (, and taking the complex conugate of the resulting equation after using the expression of to yields, (434 φ ( = φ ( + N(, (435 S x + is y N( = cγ (436 ( + k Multiplying both sides of Eq(434 by where i( ζ ζ is any point in the interior of γ and integrating over the unit circle, then using the properties of Cauchy integral and calculate sum residue, we obtain ωζ ( ψζ ( = ckγζ ϕ( ζ ω ( ζ, (437 h ζ ϕ + ( n + B( ζ B n ζ = F( B = d (439 i γ 5 Some Applications β ( In this section, we assume different values of the given functions in the first or second fundamental boundary value problems Then, we obtain the expression of Goursat functions After that, the components of stresses can be calculated directly 5 Application : Curvilinear Hole for an Infinite Plate Subected to Uniform Tensile Stress and Flowing He at For ψ ( = F( + ck Γ cγ ω ( + K ϕ( α( ϕ( ϕ( ω ( h + ϕ( n = B( ζ = i F( d, (438 γ ( ζ

6 36 M A Seddeek et al: Complex Potential Functions and Integro-Differential Equation in Elastic Media Problem in Presence of Heat P P iθ K =, Γ =, Γ = e, 0 θ π, Sx = Sy = f = 0, 4 we have, A( ζ = 0, N( ζ =, 4 iθ n e E = h d, 4 iθ (5 n e E =, h d 4 P h d n cos θ n sin θ b = { + } i 4 h d + h d Then Eq(46 becomes iθ L φ( ζ = ζ e +, (5 i= n ζ Also, L = h 4 n cos θ [ + i ] h d n sin θ + h d (53 B( ζ = 0, B = 0, (54 then, h ζ ψζ ( = ζ ϕ( ζ + 4 ( ϕ( ωζ ( n ω ζ = n ζ (55 φ( ζ = φ ( ζ + (56 4 This application discusses the first fundamental boundary value problem of an infinite plate stretched at infinity by the application of a uniform tensile stress of intensity heat in the negative direction of y axis This plate is weakened by a curvilinear hole C which is free from stress For n = 03, n = 05, m = + 00i, m = 00 i, p = 05, the stress components, and xy are obtained in large forms calculated by computer and illustrated in the following two cases : (i When the study is in the normal plate, we have the following shapes for the stress components, see Fig( and Fig(3 (ii In the thermoelasticity plate, we have the following shapes for the stress components by using the substitutions G =, q = 0,,,, see Fig(4 r 0 = 0 75 α = 0 7 ν = and Fig(5 xy ++++ Fi gure ( has positive values at 07π θ 07 π, 7π θ 4π and 6π θ 75π has positive values at 0 θ 07π and 37π θ xy has positive values at π θ 5π and 3 656π θ 94 π max at θ 78π Fi gure (3 max at θ 073

7 American Journal of Fluid Dynamics 0, (4: xy ++++ Fi gure (4 has positive values at 038π θ π, 4π θ 53π and 656π θ has positive values at 0 θ 038π and 53π θ 7 xy has positive values at 0 θ 06 π, 088π θ 74 π, 4π θ 656π and 75π θ max at θ 7π max at θ 407π Figure (5 5 Application : Curvilinear Hole Having two Poles and the Edge is Subect to a Uniform Pressure P If K =, Sx = Sy =Γ=Γ = 0, f ( t = Pt, Θ=qy Then, N( ζ = 0, 3 + m + m f ( = Pc, ( n( n 3 + m + m (57 f ( = Pc, ( n ( n n + mn + m A ( n = = ( n n± ( nζ E (, = A n E = E, (58 E b = c( hd By using these values Eq(46 becomes 3 n + mn + m he ϕζ ( = (59 ( n n ( n ζ ( n ζ( hd = ± = Also, B( ζ =, (50 ζ B = ( n + n (5 Hence Eq(437 becomes ωζ ( ψζ ( = ϕζ ( ζ ( n+ n ω ( ζ (5 h ζ ϕ + ( n n ζ = The previous discussing give the solutions of first fundamental boundary value problem when the edge of hole is subect to a uniform pressure P For n = 03, n = 05, m = + 00i, m = 00 i,

8 38 M A Seddeek et al: Complex Potential Functions and Integro-Differential Equation in Elastic Media Problem in Presence of Heat p = 05, the stress components, and xy are obtained in large forms calculated by computer and illustrated in the following two cases : xy ++++ Fi gure (6 (i When the study is in the normal plate, we have the following shapes for the stress components, see Fig(6 and Fig(7 (ii In the thermoelasticity plate, we have the following shapes for the stress components by using the substitutions G =, q = 0,,,, see Fig(8 r 0 = 0 75 α = 0 7 ν = and Fig(9 has positive values at 038π θ π, 4π θ 53π and 656π θ has positive values at 0 θ 038π and 53π θ 7 xy has positive values at 4π θ 656π and 75π θ 0 θ 06 π, 088π θ 74 π, max at θ 5866π max Fi gure (7 at θ 0987π xy xy ++++ Fi gure (8 has positive values at 0 θ 797π has positive values at 088π θ 09554π and 797π θ 9π has positive values at 0 θ 0 74π and θ 9π

9 American Journal of Fluid Dynamics 0, (4: max at θ 9745π Fi gure (9 53 Application 3: The External Force Acts on the Center of the Curvilinear Hole When, then, K = k N( n N( n max at θ 09876π, Γ = Γ = f = 0, S x is y = n ( + k S x + is y = n ( + k, (53 S x + is y E = h d n, ( + k (54 S x + is y E = h d n ( + k Sx + is y Sx is y b = (55 khdn h d n c( k hd π( + k π( + k Hence Eq(46 reduce to, hn kϕζ ( = π( + k = ( n ζ, (56 kh ( d Sx + is y hd ( + ( S x is y ( k hd ( k hd And the second function given by, ω( ζ h ζ ψ ( ζ = φ( ζ + φ( n, (57 ω ( ζ = n ζ S x + is y φ( ζ = φ ( ζ ζ (58 ( + k The last application gives the solution of the second fundamental boundary value problem when the heat is flowing in the negative direction of y axis and the force acts on the centre of the curvilinear hole For n = 03, n = 05, m = + 00i, m = 0 0 i, p = 05, the stress components, and xy are obtained in large forms calculated by computer and illustrated in the following two cases : (i When the study is in the normal plate, we have the following shapes for the stress components, see Fig(0 and Fig( (ii In the thermoelasticity plate, we have the following shapes for the stress components by using the substitutions G =, q = 0,,,, see Fig( r 0 = 0 75 α = 0 7 ν = and Fig(3 xy ++++ Fi gure (0

10 40 M A Seddeek et al: Complex Potential Functions and Integro-Differential Equation in Elastic Media Problem in Presence of Heat has positive values at 0 θ 797π has positive values at 088π θ 09554π and 797π θ 9π xy has positive values at 0 θ 0 74π and θ 9π max at θ π max at θ 9586π Fi gure ( xy ++++ Fi gure ( has positive values at 0 θ 797π has positive values at 088π θ π and 797π θ 9π xy has positive values at 0 θ 0 74π and θ 9π max at θ π max at θ 9586π Figure (3

11 American Journal of Fluid Dynamics 0, (4: Conclusions From the previous discussion s we have the following result (i The solution of the boundary value problems for isotropic homogeneous infinite elastic media in z plane reduce to obtain the two complex potential functions, Gaursat functions, in ζ plane by using conformal mapping (ii The conformal mapping z = cω( ζ, c > 0, where ω (ζ 0,, for ζ > mapped infinite region to out side a unit circle γ (iii When we applied the conformal mapping z = cω( ζ, c > 0 the boundary value problems reduce to the integro-differentail equation with discontinuous kernel (iv Cauchy method is the best method to solving the integro-differentail equation with Cauchy kernel and obtaining the two complex functions φ (z and ψ (z directly (v The components of stress, and xy is completely determine and plotting after obtaining the two complex functions (vi As a main result of inserting the effect of uniform flow of heat in the negative of y direction we have, see figures (-3 N max xy H > max xy (6 N min xy H < min xy (6 N xy represent the shear components of stress at H normal state, while xy represent the shear components of stress after inserting the effect of heating (vii If we denote to the angle that appears the maximum value of in normal state by θ, and the angle that appears the maximum value of after inserting the effect of heating by θ, we note θ < θ (63 (viii If we denote to the angle that appears the maximum value of in normal state by θˆ, and the angle that appears the maximum value of after inserting the effect of heating by θˆ, we note ˆ θ < ˆ θ (64 (ix All of the previous works in this paper is considered as special cases from this study REFERENCES [] FDGakhov, Boundary value problems, Pergamon 966 [] P G Ciarlet, M H Schultz, and R S Varga, Numerical methods of high-order accuracy for nonlinear boundary value problems I One dimensional problem, Numer Math, Vol 9, 967 [3] A Zebib, A Chebyshev method for the solution of boundary value problems, J Comput PhysVol 53, 984 [4] SSaito, MYamamto, Boundary value problems of quasilinear ordinary differential systems on a finite interval Math Japon Vol 34,989 [5] D Colton, R Kress, Integral Equation Methods in Scattering Theory, John Wiley, New York, 983 [6] M A Abdou, First and second fundamental problems for an elastic infinite plate with a curbilinear hole Alex Eng JVol 33, 994 [7] M A Abdou, Fundamental problems for infinite plate with a curvilinear hole having infinite poles Appl Math Comput,Vol93, 00 [8] M A Abdou and AK Khamis, On a problem of an infinite plate with a curvilinear hole having three poles and arbitrary shape Bull CallMathSoc, Vol9, 000 [9] MA Abdou and EA Khar-Eldin, An infinite plate weakened by a hole having arbitrary shape J Comp Appl Math,Vol 56, 994 [0] N Noda, RB Hetnarski, Y Tanigawa, Thermal Stress, Taylor and Francis, 003 [] RB Hetnarski, J Ignaczak, Mathematical Theory of Elasticity, Taylor and Francis, 004 [] H Parkus, Thermoelasticity, Spring-Verlag, 976 [3] G Ya Popov, Contact Problems for a Linearly Deformable Foundation, Kiev, Odessa, 988 [4] NI Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Holl 953 [5] IH El-Sirafy, MA Abdou, First and second fundamental problems of infinite plate with a curvilinear hole, J Math Phys SciVol 8,984 [6] H Engl A Complex Variables Method in Elasticity, Dover publications, Inc, New-Youk, 97 [7] AI Fedotov, Quadrature-difference methods for solving linear and nonlinear singular integro-differential equations, J Nonlinear Analysis, Vol 7, 009 [8] A Hanyga, M Seredyńska, Positivity of Green's functions for a class of partial integro-differential equations including viscoelasticity, J Wave Motion, Vol 47, 00 [9] VV Bavula, The group of automorphisms of the algebra of polynomial integro-differential operators, J Algebra, Vol348, 0 [0] MA AL-Jawary, LCWrobel, Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary-domain integral and integro-differential equation methods, Engineering Analysis with Boundary Elements, Vol35, 0