201 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Thermoelastic Problem of an Infinite Plate Weakened by a Curvilinear Hole R. T. Matoog* Department of Mathematics, Faculty of Applied Sciences, University of Umm Al-Qura, KSA Email: rmatoog_777@mail.com Abstract Mathematical model is considered, to discuss the analytic solution of the first and second boundary value problems (BVPs), for an infinite plate weakened by a curvilinear hole C having two poles. The elastic plate carries a steady uniformly distributed axial current of density J and is placed in an ambient medium of steady temperature .eT Using a conformal mapping function, the curvilinear hole is conformally mapped on the domain outside (inside) a unit circle .γ Then, the Gaursat functions (GFs) are determined. Moreover, the three components of stresses, in the presence of temperature T distributed around the curvilinear hole are completely determined. Many special and new cases are derived from the work. In addition, many, applications for the first and second BVPs are discussed. Moreover, the three stresses components, in each application, are computed. Keywords: Boundary value problems; thermoelastic plate; Gaursat functionsl; rational mapping; curvilinear hole ; AMS (2010): 74B10; 30C20. 1. Introduction Several authors have discussed the solutions of the BVPs, in two-dimensional problems, using some different methods; see [1-5]. In the two- dimensional problems, in the theory of elasticity, some authors have used complex variables method to obtain, in a closed form, the solution of the first and second fundamental problems, see [6-10]. Others authors have used Laurent’s theorem to obtain the Gaursat functions for an infinite plate weakened by a curvilinear hole C, see [11-15]. ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 202 In thermoelastic, problems for the first and second BVPs are equivalent to finding the thermoelastic potential ,Φ in addition is finding the two analytic GFs, ( )zϕ and ( )zψ of one complex argument iyxz += . These analytic functions must satisfy the boundary conditions, 1 1 1( ) ( ) ( ) ( )K t t t t f tϕ ϕ ψ′− − = (1.1) where t denotes the affix of a point on the boundary. In terms of )(ζωcz = , 0>c , ( )ω ζ′ does not vanish or become infinite for 1>ζ , the infinite region outside a unit circle γ . For the stress BVPs, 1K = − and ( )f t is a given function of stress. While for 1K k= > , ( )f t is a given function of displacement which called the thermal conductivity, we have the second fundamental BVPs or called the displacement BVPs. The GFs 1( )tϕ and )(1 tψ take the following form see [16] 1( ) ln ( ) 2 (1 ) x yS iS c k ϕ ζ ζ ζ ϕ ζ π + = − + Γ + + (1.2) and, 1 ( ) ( ) ln ( ). 2 (1 ) x yS iS k c k ψ ζ ζ ζ ψ ζ π ∗− = + Γ + + (1.3) Where ,x yS S are the components of the resultant vector of all external forces acting on the boundary and , ∗Γ Γ are complex constants. Generally the two GFs ( )ϕ ζ and ( )ψ ζ are single value analytic functions within the region out side the unit circle γ and ( ) 0 , ( ) 0ϕ ψ∞ = ∞ = . In this paper, we consider the BVP for isotropic homogeneous perforated infinite elastic media in presence of uniform flow of heat. Then, we use a general shape of conformal mapping to obtain the GFs for the problem in the form of integro-differential equation with singular kennel. Many special cases are obtained and several applications are discussed from the work. 2. Formulation of the problem Consider a thin infinite plate of thickness h with a curvilinear hole C , where the origin lies inside the hole is conformally mapped on the domain outside a unit circle γ by means of a rational mapping function, 1 2 1 2 1 21 1 1 2 , 0 , (1 ) (1 ) m mz c c n n n n ζ ζ ζ ζ ζ − − − − + + = > ≠ − − (2.1) where )(ζz′ does not vanish or become infinite outside the unit circle γ . 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 203 The uniform flow of heat is distributed by the presence of an insulated curvilinear hole C. Neglecting the variation of the strain and stress with respect to the thickness of the plate, and assuming the force of the plate are free of applied loads. In this case, the thermoelastic potential Φ satisfies the formula, see Parkus [16] 2 (1 )υ α∇ Φ = + Θ (2.2) where α is a scalar which present the coefficient of the thermal expansion and ν is poisson's ratio. In the steady state, the temperature T as measured from a reference temperature rT , satisfies Poisson’s equation and the heat flow vector Q is given by Fourier law for heat conduction 2 ( , ) ,x yT Q K T K ζ ∇ = − = − ∇ (2.3) Where, ( , )x yζ is the rate of heat supply per unit volume and K is the coefficient of heat conduction. More extensive forms for Q may be found in [7, p. 101]. For our purposes, it is convenient to write H Tζ ζ ζ= + (2.4) Where, 2 ,H Jζ σ = represents the contribution of the magnetic field (Joule heat supply), J represents a steady uniform distributed axial current of density, which carries by the plate and σ the coefficient of electrical conductivity. In addition, Tζ is the rate of heat supply from all other sources. In this work Tζ will be neglected. The general solution (2.3) is h pT T T= + (2.5) Where hT is the harmonic part of T i.e. is the solution of the boundary value problem 2 2 2 2 2 2( ) 0,hi T x y ∂ ∂ ∇ = ∇ = + ∂ ∂ , ( ) 0hTii r ∂ = ∂ On the boundary ,or r r x iy= = + (2,6) 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 204 Moreover, pT is any particular solution of Poisson’s Eq. (2. 3). It easily to have 2 0 0{Im } . Im 4 r JT T z z z z Kσ = + − (2.7) The thermoelastic potential functionΦ, satisfies, see Parkus [16], 2 (1 ) .Tυ α∇ Φ = + (2.8) where z is defined by Eq. (2.1). Substituting from (2.8) into (2.2) and integrating the result, the thermoelastic potential becomes 2 2 2 2 0 0( ) (1 ) { Im [ln( ) }. 4 Jz T r z z z z z K υ α σ Φ = + − − (2.9) In this case, the formulas (1.1) for the first and second BVPs respectively, take the following forms, 1 1 1 0 1( ) ( ) ( ) [ ( ) ( )] 2 s t t t t i iX s Y s ds c x y G ϕ ϕ ψ ∂Φ ∂Φ′+ + = + + − + ∂ ∂ ∫ (2.10) 1 1 1( ) ( ) ( )k t t t t u iv i x y ϕ ϕ ψ ∂Φ ∂Φ′− − = + − − ∂ ∂ (2.11) 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 and Φ represents the thermoelastic potential function. In addition, the applied stresses ( )X s and ( )Y s satisfy the following, see [16] ( ) ; ( ) .xx xy yx yy dy dx dy dxX s Y s ds ds ds ds σ σ σ σ= − = − (2.12) where ,xx xyσ σ and yyσ are the components of stresses given by the following relations, 2 2 2 2 12 [ ( 2 ) Re[ 2 ( ) ( ) ( ) ]] 2xx G T z z z z y x σ λ ϕ ϕ ψ∂ Φ ∂ Φ ′ ′′ ′′= − − + + − − ∂ ∂ , (2.13) 2 2 2 2 12 [ ( 2 ) Re[ 2 ( ) ( ) ( ) ]] 2yy G T z z z z y x σ λ ϕ ϕ ψ∂ Φ ∂ Φ ′ ′′ ′′= − − + + + ∂ ∂ , (2.14) and, 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 205 2 2 [ Im( ( ) ( ) ) ]xy G z z z x y σ ϕ ψ∂ Φ ′′ ′′= + + ∂ ∂ . (2.15) Here, )1(2 νλ α += is the coefficient of heat transfer and ,T Φ are given respectively, by equations (2.7), and (2.9). 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 −ζ plane as given by ie θζ ρ= , 0 2θ π≤ ≤ . By using the transformation ( )z cω ζ= , Eq. (1.1) reduce to, 1 1 1 ( )( ( )) ( ( )) ( ( )) ( ( )) ( ) K c c c f cω ζϕ ω ζ ϕ ω ζ ψ ω ζ ω ζ ω ζ − − = ′ (2.16) The last formulas represent the first and second BVPs in ζ − plane. In this study, we use the rational mapping (2.1) to map the curvilinear hole C of the infinite plate outside a unit circleγ . Then, we use the properties of Cauchy integral to obtain the two GFs. After that, we can determine the components of stresses. Many applications and special cases are derived from the work. 3. The rational mapping The mapping function (2.1) 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 (2.1) for some different values of n's and m's. Figure 1: 1 2 1 2( 0, 0.987, 0)n n m m= = = = Figure 2: 1 2 1 2( 0.5, 0)n n m m= = = = 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 206 Figure 3: ( 1 2 1 20, 0.5, 0, 0.55n n m m= = = = ) Figure 4: ( 1 2 1 20.9, 0.4, 0.99, 0.05n n m m= = = − = − ) 4. Gaursat functions To obtain the two complex potential functions, GFs by using the conformal mapping (2.1) in the boundary conditions (1.1), we write the expression )( )( 1−′ ζω ζω in the form, 1 1 ( ) ( ) ( ) ( ) ω ζ α ζ β ζ ω ζ − − = + ′ (4.1) where, 1 2 1 2 ( ) h h n n α ζ ζ ζ = + − − , (4. 2) and, 3 2 2 2 1 2 1 2 2 3 4 1 1 2 1 1 2 2 1 2 1 1 2 2 2 ( )(1 ) (1 ) ( )[1 2( ) ( 3 ) 2 ( ) ] j j j j j j j j j j n m n m n n n h n n n n n m n n n m n n n m n m n m n± + + − − = − − + − − − + + + ( ( 0,1)j = (4.3) 1( )β ζ − is a regular function for 1ζ > . The term )( )( 1−′ ζω ζω has two singularities at 1n=ζ and 2n=ζ . Using equations (4.1) and (4.2) in equation ( 2.16), and integrating both sides, we get 2 1 1 ( ) ( ) [ ( ) ]j j j jj h K A c N n c b n ϕ ζ ζ ζ ζ ∗ − = − = − Γ + + −∑ (4.4) 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 207 where, 1 ( )( ) ; ( ) ; ( ) ( ( )) ( ) 2 ( ) 2 (1 ) x yS iSFA d N c F f c f t i k γ σζ σ ζ ζ ζ ω ζ π σ ζ π − = = Γ − = = − +∫ (4.5) The function ( )F σ with its derivatives must satisfy Holder condition. The complex constants jb can be determined in the following form 2 2 2 2 2 2 2, ( ) ( ) , , 1, 2 ( ) (1 ) jj j j j j j j j j j j j j j j KE h d E n b E A n c n h d N n d j c K h d n ∗− ′= = − − Γ − = = − − (4.6) Using the results of (4.5) and (4.6), the function ( ),ϕ ζ is completely determined. The value of the function ( )ψ ζ takes the form 21 1 1 1 ( )( ) ( ) ( ) ( ) , ( ) 1 j j jj h cK n B B n ζω ζψ ζ ζ ϕ ζ ϕ ζ ω ζ ζ − − − ∗ ∗ = = Γ − + + − ′ −∑ (4.7) Where 1 ( ) 1 ( )( ) ( ) ( ); ( ) ; 2 ( ) 2 F FN B d B d i i γ γ σ σϕ σ ϕ σ σ ζ σ σ π σ ζ π σ∗ ′= + = = −∫ ∫ . 5. Some applications In this section, we assume different values of the given functions in the first or second BVPs. Then, we obtain the expression of GFs. Hence, the stresses components can be calculated directly. 5.1. Application 1: Curvilinear hole for an infinite plate subjected to uniform tensile stress in the presence of heat For 21, , , 0 2 , 0, 4 2 i x y P PK e S S fθ θ π∗ −− = − Γ = Γ = ≤ ≤ = = = we have 2 1 2 1 ( ) 2 ji i j LcP e n θϕ ζ ζ ζ − = = + −∑ , (5.1) 1 2 1 1 1 ( )( ) ( ) ( ) 4 ( ) 1 j j j j hcP n n ζω ζψ ζ ζ ϕ ζ ϕ ω ζ ζ − − − ∗ ∗ = = − − + ′ −∑ (5.2) where, 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 208 2 21 2 cos 2 2 sin 2 [ ]; ( ) ( ) 4 1 1 4 j j j j j j j j n ncP cPL h i h d h d θ θ ϕ ζ ϕ ζ∗ − ′= + = + − + . (5.3) The application discusses the first BVP of an infinite plate stretched at infinity by the application of a uniform tensile stress of intensityΓ . The plate is weakened by a curvilinear hole C which is free from stress and the intensity temperature T, in the all edge of the curvilinear hole C. Figure(5) describes the following: 1) xxσ has a positive values at 0.127 0.7 , 1.27 1.4π θ π π π   and 1.6 1.75π θ π  . 2) yyσ has a positive values at 0 0.127 1.37 2andθ π π θ π    . 3) xyσ has a positive values at πθπ 5.13  and 1.656 1.94π θ π  . 4) yy xx σ σmax at 1.146θ π≈ . xx yy σ σmax at 1.56θ π≈ . For 3.01 =n , 2 0.25n = , 1 1m = , 2 0.01,m = − and 0.25p = the stress components xxσ , yyσ and xyσ are obtained in large forms calculated by computer and illustrated in Fig.(5) and Fig. (6), respectively. σ + + + xyσ ___ yyσ ___ xxσAppl.1 of components stress Case1: Figure 5 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 209 σ 1 02 , 0.1, 0.75, 0.7, 1G T r α ν= = = = = + + + xyσ ___ yyσ ___ xxσ Appl.1 of Thermoelastic 2: Case Figure 6 From Figure (6), we deduce 1*) xxσ has a positive values at 0.127 0.7 , 1.27 1.4π θ π π π   and 1.6 1.75π θ π  2*) yyσ has a positive values at 0 0.127 1.37 2andθ π π θ π    . 3*) xyσ has a positive values at 3 1.5π θ π  and 1.656 1.94π θ π  . 4*) max xx yy σ σ at 1.178θ π≈ . max yy xx σ σ at 0.732θ π≈ . Application 2: Curvilinear hole having two poles and the edge is subject to a uniform pressure P . If 1, 0, ( ) .x yK S S f t P t∗= − = = Γ = Γ = = Then, we have 32 2 1 2 1 11 ( ) , ( )( ) ( )(1 ) j j j j j jj j j j j j n m n m h E cP n n n n h d ϕ ζ ζ ζ= =± + + = − − − − −∑ ∑ . (5.4) 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 210 1 2 1 1 1 2 1 ( )( ) ( ) ( ) ( ) ( ) 1 j j j j h cP n n cP n n ζω ζψ ζ ϕ ζ ζ ϕ ω ζ ζ − − − = ′ ′= − − − + + ′ −∑ . (5.4) The previous discussing give the solutions of first boundary value problem when the edge of hole is subject to a uniform pressure P . For 3.01 =n , 2 0.25n = , 1 1m = , 2 0.01m = − , 0.25.p = Figure 7 From figure 7, we have 0xxσ > at 0.38 1.1 , 1.4 1.53π θ π π θ π    and 1.656 2π θ π  . 0yyσ > at 0 0.38 1.53 1.72andθ π π θ π    . 0xyσ > at 0 0.16 , 0.828 1.274 ,θ π π θ π    1.4 1.656π θ π  and 1.75 2π θ π  . yy xx σ σmax at 1.17θ π≈ . xx yy σ σmax at 1.40127θ π≈ . In figure (8), we deduce that: xxσ > 0 at 0.38 1.1 ,1.4 1.53π θ π π θ π    and 1.656 2 .π θ π  Also, yyσ .> 0 at 0 0.38 1.53 1.72andθ π π θ π    . Finally, xyσ > 0 at 0.828 1.274 ;π θ π  1.4 1.656 ;π θ π  1.4 1.656π θ π  and 1.75 2π θ π  . yy xx σ σmax at 1.52866θ π≈ . xx yy σ σmax at 1.0987θ π≈ . σ + + xyσ ___ yyσ __ xxσ Appl.2. of components essstr case1: 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 211 Figure 8 Application 3: The external force acts on the center of the curvilinear. When, , 0,K k f∗= Γ = Γ = = we have 2 22 2 2 2 2 2 2 1 ( )1( ) (1 ( )) 2 (1 ) ( ) ( ) ( ) j j j j x y j j x y j j j j j j h n kh d S iS h d k S iS k n k h d k h d ϕ ζ π ζ=  + − = − + − + − − −   ∑ , (5.5) 1 2 1 1 ( )( ) ( ) ( ) ( ) 1 j j j j h n n ζω ζψ ζ ϕ ζ ϕ ω ζ ζ − − ∗ ∗ = = − + ′ −∑ , (5.6) where, 1( ) ( ) 2 (1 ) x yS iS k ϕ ζ ϕ ζ ζ π − ∗ + ′= − + . (5.7) The last application gives the solution of the second fundamental BVP in the presence of temperature around the edge of the hole. The force acts on the center of the curvilinear hole. For 1 0.3n = , 2 0.25n = , 1 1m = , 2 0.01m = − , 0.25p = , the stress components xxσ , yyσ and xyσ are obtained in large forms calculated by computer and illustrated in figures (9) and (10). In figure (9): xxσ positive at 0 1.7197θ π  . yyσ at 0.828 0.9554θ π  and 1.7197π θ 1.91π . xyσ at 0 0.1274 ,θ π  1.91θ π . max xx yy σ σ at σ 1 02 , 0.1, 0.75, 0.7, 1G T r α ν= = = = = + + xyσ ___ yyσ ___ xxσAppl.2 of Thermoelastic 2: Case 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 212 0.98726θ π≈ . max yy xx σ σ at 1.9745θ π≈ .In the thermoelastic plate, we have the shapes of figure(10), for the stress components by using the substitutions 2 1 =G , 1.0=q , 75.00 =r , 7.0=α , 1=ν . In figure (10) xxσ > 0 at 0 1.7197 .θ π  0yyσ > at 0.828 0.9554θ π  and 1.7197 1.91π θ π  . 0xyσ > at 0 0.1274θ π  and 1.91θ π . max xx yy σ σ at 0.9809θ π≈ . max yy xx σ σ at 1.9586θ π≈ . σ + + + xyσ ___ yyσ __ xxσ Appl.3. of components stress Case1: . Figure 9 σ 1 02 , 0.1, 0.75, 0.7, 1G T r α ν= = = = = + + xyσ ___ yyσ ___ xxσAppl.3 of Thermoelastic 2: Case Figure 10 214-201, No 1, pp 82) Volume 2017( )Research Journal for Engineering, Technology, and Sciences (ASRJETS American Scientific 213 5. Conclusion From the previous discussions we have the following results (i) The solution of the BVPs 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. ( ) , 0z c cω ζ= > , where ( ) 0,ω ζ′ ≠ ∞ , for 1>ζ mapped infinite region to out side a unit circle γ . 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