Analysis of the stress field in a wedge using the fast expansions with pointwise determined coefficients

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1 Journal of Physics: Conference Series PAPE OPEN ACCESS Analysis of the stress field in a wedge using the fast expansions with pointwise determined coefficients To cite this article: A D Chernyshov et al J. Phys.: Conf. Ser. 93 elated content - Peculiarities of stress field formation during cutting isotropic material by mining machine cutters V V Gabov and D A Zadkov - The Casimir effect of eissner Nordström blackhole Xiao Kui and Liu Wen-Biao - Method of otational Turning With Multifaceted Cutters N Indakov, Y Gordeev and A Binchurov View the article online for updates and enhancements. This content was downloaded from IP address on 5// at 9:4

2 IOP Conf. Series: Journal of Physics: Conf. Series () doi :./4-59/93// Analysis of the stress field in a wedge using the fast expansions with pointwise determined coefficients А D Chernyshov, V V Goryainov,3 and А А Danshin 4 Voronezh State University of Engineering Technologies, evolution av., 9, Voronezh, ussia Voronezh State Technical University, let Oktyabrya st., 4, Voronezh, ussia 3 Voronezh State University, University sq.,, Voronezh, ussia 4 PJSC ostelecom, evolution av., 35, Voronezh, ussia chernyshovad@mail.ru, gorvit@mail.ru, hb9@pisem.net Abstract. The stress problem for the elastic wedge-shaped cutter of finite dimensions with mixed boundary conditions is considered. The differential problem is reduced to the system of linear algebraic equations by applying twice the fast expansions with respect to the angular and radial coordinate. In order to determine the unknown coefficients of fast expansions, the pointwise method is utilized. The problem solution derived has explicit analytical form and it s valid for the entire domain including its boundary. The computed profiles of the displacements and stresses in a cross-section of the cutter are provided. The stress field is investigated for various values of opening angle and cusp s radius.. Introduction A lot of research is devoted to consideration of elastic infinite wedge-shaped domains. The stress field for the antiplane deformation of elastic wedge is examined in [-3]. The plane deformation of the wedge is discussed in [4-]. Some particular results are obtained by applying Mellin transforms [,5,] and using functions of a complex variable [,3,]. The exact solutions to the problems of plane deformation of the elastic wedge under zero loading on its lateral edges can be derived using Wiener- Hopf method [4]. The wedge has straight cracks on its axis of symmetry. An intrusion of the wedge into the plastic half-space has been investigated in the monograph []. The loaded wedge with smooth edges was considered in [9]. The particular solutions for the truncated circular sector are given in []. Some studies are devoted to three-dimensional problems for an elastic wedge [,]. In [] the explicit matrix algorithm for three-dimensional wedge problem solution is proposed. In [] one of the wedge s surfaces is reinforced with a coating of Winkler type. On the other surface some arbitrary boundary conditions are set. In this case, the methods of nonlinear boundary integral equations and of successive approximations are used. In this work the new analytical method of fast expansions [3] will be applied. It allows to obtain the high accuracy solution to the problem under study in an explicit analytical form. The method of fast expansions is applicable for solution to the problems associated with partial differential [4], integro-differential [3] and ordinary differential [5] equations.. Materials and methods The equilibrium equations in terms of displacements in cylindrical coordinates for the wedge-shaped Content from this work may be used under the terms of the Creative Commons Attribution 3. licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd

3 IOP Conf. Series: Journal of Physics: Conf. Series () doi :./4-59/93// domain Ω ( < r r, θ θ ) have form U U + ν V 3 ν V U =, ν r r r ν r r θ ν r θ r θ + ν U 3 ν U V V V = ν r r θ ν r θ r r r ν r θ The displacements U and V are defined at the rear edge r The rest edges are loaded by external forces = : ( θ ), ( θ ) U = U V = V () () U V V U V U + = Φ, + + ν = Φ3 ( r ), r θ r r r r θ r θ = θ = U V V U V U + = Φ, + + ν = Φ4, r θ r r r r θ r θ θ = θ = θ U V U U V V + ν + = F ( θ ), + = F ( θ ). r r θ r r θ r r = r r r (3) (4) elying on a physical meaning of the problem, the stresses and strains should be bounded throughout the wedge-shaped domain Ω (otherwise the cutter would be broken). For this reason, the additional conditions must be imposed ( ) ( { ( θ θ ) ) } U, V C, C r r, U, V <. (5) The formulation of the elastic problem ()-(5) is given in a classical form, but it is not complete. The requirement to the functions U and V of being smooth and bounded is essential, since it imposes specific restrictions on the boundary conditions formulation. For the case of mixed boundary conditions ()-(4), the relations (5) lead to the necessity of the "consistency" conditions formulation. If r =, θ =, these conditions aren't satisfied, a discontinuity will occur at the angular points ( r =, θ = θ ), ( r = r θ = ) ( r r, θ θ ),, = =, that is, the conditions (5) will be broken. Hence, the accuracy of the solution will be essentially affected. The boundary conditions () and (4) at the arcs r = r, r = are to be conformed with the conditions (3) at the rays θ, θ θ at the arc r are rewritten as: = =. For this purpose the predetermined functions U ( θ ), V ( θ ) = are replaced by some new functions U ( θ ), V ( θ ) [, ], U U, V V and the boundary conditions () θ θ = θ = θ () The functions U ( θ ), V ( θ ) are supposed to coincide with the original functions U ( θ ), V ( θ ) outside the ε neighborhoods of the angular points. And inside the ε neighborhoods these functions are such that they take on some unknown values at points θ = and θ = θ :

4 IOP Conf. Series: Journal of Physics: Conf. Series () doi :./4-59/93// θ [ ε θ ε ] U ( θ ) U ( θ ) V ( θ ) V ( θ ) ( θ ) ( θ ),, =, =, U = U, V = V, U = U, V = V, () where the constants U, V, U, V are assumed to be unknown in advance and they are to be determined throughout the solution process. Further, the boundary functions of sixth order [3] will be used to determine the displacements U U θ, V θ U θ, V θ from () at and V, so the functions will be smoothly conjugated with the points (, ε ) and (, θ ε ) up to sixth order derivatives as it is required by the condition (5). Assuming fixed condition at the cutter s rear edge r =, the relations () reduce [ ] θ ε, θ ε, U θ =, V θ =, U = U, V = V, U θ = U, V θ = V. () Thus, under the "fixed boundary" we will imply the equality of the displacement components to zero only for ε θ θ ε, and inside the ε neighborhoods of the angular points the displacements U θ, V θ satisfying the conditions (5) and (). are determined by some additional relations Assuming the smallness of ε neighborhoods the concrete relations for U ( θ ), V ( θ ) are not given. This is sufficient to derive the solution to the problem in a finite form. r,, r, θ some "consistency" conditions are also to be At the cusp of the cutter at points satisfied. In order to derive these conditions we will make use of the following considerations. For the boundary conditions defined in ()-(4) the unique solution to the boundary problem exists. However, discontinuities of the stresses might occur at the angular points if this circumstance isn t envisaged in advance when the boundary conditions are being specified. That is, the stresses will depend on the direction of approach to the angular points for arbitrarily chosen functions F ( θ ) and drawback can be avoided by specifying two "consistency" conditions at every point (,), (, ) F θ. This r r θ. The first two conditions can be obtained using the symmetry property of the stress tensor σ rθ = σθr. This leads to the following relations: ( r ) F ( ), ( r ) F ( θ ) Φ = Φ = (9) These conditions impose the restrictions on a choice of the function F ( θ ). The remaining two conditions are derived by formally substituting the radial loading F ( θ ) at the cusp from relations (4) by some new function F ( θ ). This function is supposed to coincide with the original function F ( θ ) inside the interval δ θ θ δ, and it takes on some unknown values F = F and F ( θ ) = F at the angular points [, ],, ( ), θ δ θ δ F θ = F θ F = F F θ = F, () where the constants F, F are assumed to be determined throughout the solution process. F θ Following the condition (5), the function points ( r, δ ) and ( r ), θ δ δ neighborhoods the concrete relation for F ( θ ) solution. It is only sufficient to suppose that the function F ( θ ) Further, we need to define ( r ) should be smoothly conjugated with F ( θ ) at the up to sixth order derivatives. Assuming the smallness of i is not given to simplify the construction of the satisfies the conditions (5) and (). Φ in the boundary conditions (3) to solve the problem. As an example, the simplest rapidly decreasing functions were chosen. The main loading throughout the 3

5 IOP Conf. Series: Journal of Physics: Conf. Series () doi :./4-59/93// sample processing is applied to the upper edge ( θ = θ ) of the cutter The lower edge ( r r Φ = Φ, Φ 4 ( r ) = Φ4 θ = ) is assumed to be zero loaded, that is ( r ) Under the conditions (9)-(), the loadings F ( θ ), F simplest linear form Φ = Φ =. 3 () θ on the cutter's cusp are expressed in the θ [, ] F F F, F ( ) F, F F, F r θ θ δ θ δ θ = θ = = θ = θ = Φ. () θ θ The constant coefficients F, Φ, Φ 4 in relations (), () depend on pressing force applied to the sample being processed and they are determined experimentally. From the previous considerations, it follows that instead of relations (4), the consistent boundary conditions should be used where the function F ( θ ) U V U U V V F, F r ν r θ r θ + + = + = r θ r r θ, (3) r must satisfy the conditions (). The form of function F ( θ ) is not changed as it satisfies the consistency conditions (9) due to relations (). Thus, the original boundary value problem ()-(4) reduces to its altered formulation involving consistent boundary conditions (3), (), (3) with additional unknown constants to be determined throughout the solution process. U, U, V, V, F, F According with the method of fast expansions [3] the displacements U ( r, θ ) and (, ) expressed as the superposition of boundary function and truncated Fourier sine series where functions of sixth order r θ U = M r + u r m V = M r + r m N N U m= θ m= θ (4) V r θ can be V (, θ ) m sin π, (, θ ) υm sin π, (5) U V N is the number of Fourier series terms retained, M ( r θ ) and M,, θ r θ are the boundary 3 3 U θ θ θ θ θθ θ θθ M ( r, θ ) = A ( r ) + A + A3 ( r ) + A4 θ θ θ 3 θ θ θ θθ θθ θ θθ θ θ + A5 + + A + 4 θ 45 θ θ θ θθ θθ θ θ + A + 54θ A θ θθ θ θ 3θ θ +, 54θ 5 () 4

6 IOP Conf. Series: Journal of Physics: Conf. Series () doi :./4-59/93// 3 3 V θ θ θ θ θθ θ θθ M ( r, θ ) = B ( r ) + B ( r ) + B3 + B4 θ θ θ 3 θ θ θ θθ θθ θ θθ θ θ + B5 + + B + 4 θ 45 θ θ θ θθ θθ θ θ + B + 54θ () θ θθ θ θ 3θ θ + B +. 54θ 5 This representation of the displacement components involves + N unknown functions υ A r A r, B r B r, u r, r, m = N, () m m depending only on the radial coordinate r. Following the same approach the unknown functions () can also be represented in a fast expansion form: r r A r = M r + a n z z = j = M r = a z + a z j N A ( j ) ( j) A( j [ ] ) ( j ) j n+ sin π,,,, ( ) n= r ( j) z z z ( j) z z ( j) z z z z + a3 + a4 + a ( j) z z z ( j) z z z z z + a + + a a j z z z z +, 54 5 N 3 B ( j ) ( j) B( j ) ( j ) ( j) ( j) z z z B j r = M r + bn + sin nπ z, j =, M r = b ( z) + b z + b3 n= ( j) z z ( j) z z z z ( j) z z z + b4 + b5 + + b ( j) z z z z z ( j) z z z 3z + b + + b +, N 3 u ( m ) ( m) u ( m ) ( m ) ( m) ( m) z z z um r = M r + un+ sin nπ z, m = N, M r = u ( z) + u z + u3 n= ( m) z z ( m) z z z z ( m) z z z + u4 + u5 + + u ( m) z z z z z ( m) z z z 3z + u + + u +, (9) () () 5

7 IOP Conf. Series: Journal of Physics: Conf. Series () doi :./4-59/93// N 3 υ ( m ) ( m) υ ( m ) ( m ) ( m) ( m) z z z υm r = M r + υn+ sin nπ z, m = N, M r = υ ( z) + υ z + υ3 n= ( m) z z ( m) z z z z ( m) z z z + υ4 + υ5 + + υ ( m) z z z z z ( m) z z z 3z + υ + + υ Substituting the relations (), () and (9)-() into the differential equations () and into the boundary conditions (3), (), (3) and differentiating with respect to r and θ gives the representation N + N + unknown constant coefficients of the original boundary value problem containing ( j) ( j) ( m) ( m) () a, b, u, υ, j =, n = N +, m = N. (3) n n n n As the resultant expressions are quite cumbersome, they are not given here and further referred to as double fast expansion form of the original problem. In order to determine the unknown constant coefficients (3), the pointwise method, which was developed and tested in [5] will be employed. Following this technique it is necessary to discretize the wedge-shaped domain Ω in a uniform grid with ( N + )( N + ) nodes ( s, k ) interval [,θ ] is divided by N + inner points θk kθ ( N 5 ), k,,.., N 5 [ r, ] - by N + inner points rs r s( r ) ( N 5 ), s,,.., N 5 of each node ( s, k ) ( N + )( N + ) linear algebraic equations in terms of unknown coefficients (3). r θ so that the = + = + and the interval = + + = +. Substituting the values r θ into the double fast expansion of governing differential equations () gives Further, it is required to discretize the boundary of the domain Ω in the same way. Dividing the,θ by N + inner computational points θk = kθ ( N + ), k =,,.., N + and interval [ ] substituting these values into the relations () and (3) gives 4 N + linear equations. Similarly, dividing the interval [ r, ] by s s =,,.., N + and substituting these values into the relations (3) gives 4 N + inner computational points r r s ( r ) ( N ) = + +, N + more linear equations. Thus, from the boundary conditions (3), () and (3) we obtain 4( N + ) + 4( N + ) linear equations in terms of unknown coefficients (3). N + N + linear algebraic equations in the All in all, we derive the consistent system of unknowns (3) and additional equations for the constants U, U, V, V, F, F following from the consistency conditions. This system then has been solved numerically in Maple software. 3. esults and discussion As an example the numerical results for solution to the boundary value problem (), (3), (), (3) for the cutter made of instrumental quick-cutting steel of grade are provided. This steel has the following values of elastic moduli [, ] σ, 5. = Pa,, 33 ν =, E =. Pa, λ =., µ =.5, where σ, is the.% offset yield stress. Since the solution determined by the expressions (5)-(), (9)-() approximately satisfies the differential equations () and the boundary conditions (3), (), (3), then we will use a relative residual { i } = ( r, ) max f ( r, ) to estimate an accuracy of the solution derived, where f (, ) δ δ θ θ i r θ denotes

8 AMSCM IOP Conf. Series: Journal of Physics: Conf. Series 93 () 3459 doi:./4-59/93// the i-th term of the equations () or boundary conditions (3), (), (3). The distributions of the relative residual δ D of the approximate solution (5) to the equations () retaining only three terms ( N = 3) in first expansion, and thirty terms ( N = 3 ) in the second one are shown in figure. These results were computed for the following values of loadings and wedge dimensions Φ =, Φ 4 =, F =, r = 5 m, = m, θ = 5 π (4) It can be seen (refer to figure ) that the maximal values of relative residual δ D are concentrated nearby wedge's cusp and they don't exceed. and.5 for the first and second differential equations, respectively. The distribution of relative residual δ G along the region's boundary as well as residual s magnitudes for all boundary conditions (3), (), (3) are essentially the same. The typical behavior of relative residual δ G at the edges θ = θ and r = r of the domain Ω is shown in figure. Figure. The relative residual of the differential equations: (a) (); (b) (). Figure. The relative residual of the boundary conditions: (a) δ G θ =θ ; (b) δ G r = r.

9 AMSCM IOP Conf. Series: Journal of Physics: Conf. Series 93 () 3459 doi:./4-59/93// It is apparent that the accuracy achieved (refer to figures, ) is acceptable for the most technical purposes. It should be noted that the similar high accuracy approximate solutions were accomplished in [3-5] where the method of fast expansions has also been involved. The computed profiles of the displacements U ( r,θ ), V ( r, θ ) and stress components σ r, σ θ, σ rθ in a cross-section Ω of the cutter are shown in figure 3 and figure 4, respectively. Figure 3. The displacement components: (a) U ( r,θ ) ; (b) V ( r, θ ). Figure 4. The stress components: (a) σ r ; (b) σ θ ; (c) σ rθ.

10 IOP Conf. Series: Journal of Physics: Conf. Series () doi :./4-59/93// The numerical results corresponding to the parameters' values (4) satisfy the yield condition [] * * * ( σ r ) + ( σθ ) + ( σ z ) + ( σ rθ ) σ, = 5. Pa ɶ σ, ɶ σ = = 3.5 Pa, where σ = σ ( σ + σ ), σ = σ ( σ + σ ), θ θ r θ σ z = σ z σ r + σθ, σ z = The distribution of σɶ in the cutter's cross-section is shown in figure 5. It is obvious that the maximal stress σɶ in a cutter occurs at its edge θ = θ in the vicinity of point r =.35 m. * r r r θ * * Figure 5. The distribution of σɶ. The computed values of σɶ for various opening angles θ and radii r are presented in table. Examining these data it can be concluded that the displacements and stresses decline with an increase of opening angle θ and they raise with a decrease of the cusp s radius r. These results are consistent with the general elasticity and strength of materials theories. θ 4 π.959 π 3 5. π 4.34 π.5 Table. The values of σɶ, Pа. r, m Conclusion Thus, the application of the method of fast expansions allowed to obtain the solution to the problem in an explicit analytical form which is valid for the entire domain Ω including its boundary. The displacement and stress profiles derived can be utilized in the analysis of the stress state arising in wedge-shaped cutters. eferences [] Shahani A and Adibnazari S Analysis of perfectly bonded wedges and bonded wedges 9

11 IOP Conf. Series: Journal of Physics: Conf. Series () doi :./4-59/93// with an interfacial crack under antiplane shear loading Int. J. of Solids and Structures vol 3 is 9 pp 39-5 [] Shahani A 999 Analysis of an anisotropic finite wedge under antiplane deformation J. of Elasticity vol 5 is pp -3 [3] Shahani A and Ghadiri M Analysis of anisotropic sector with a radial crack under antiplane shear loading Int. J. of Solids and Structures vol 4 is - pp 3-39 [4] Nekislykh E M and Ostrik V I Problems on elastic equilibrium of a wedge with cracks on the axis of symmetry Mechanics of Solids vol 45 is 5 pp 43-5 [5] Stampouloglou I H and Theotokoglou E E 3 The plane elasticity problem of an isotropic wedge under normal and shear distributed loading - application in the case of a multimaterial problem Int. J. of Solids and Structures vol 4 is pp 539- [] Linkov A and ybarska-usinek L Plane elasticity problem for a multi-wedge system with a thin wedge Int. J. of Solids and Structures vol 4 is 4 pp [] Chue C H and Liu Ch I A general solution on stress singularities in an anisotropic wedge Int. J. of Solids and Structures vol 3 is 3 39 pp 9-9 [] Bykovtsev G I and Ivlev D D 99 Theory of plasticity (Vladivostok: Dal'nauka) p 5 [9] Chernyshov A D 95 On deformation of continuous media in a wedge-like region with smooth faces J. of Applied Mathematics and Mechanics vol 39 is pp 4-53 [] Timoshenko S P and Goodier J N 9 Theory of elasticity (McGraw-Hill, Inc., 3rd edn) p 5 [] Guo L, Zhang Z M, Wang W and Wong P L An explicit matrix algorithm for solving three-dimensional elastic wedge under surface loads Int. J. of Solids and Structures vol pp 3-4 [] Aleksandrov V M and Pozharskii D A Three-dimensional contact problems for an elastic wedge with a coating J. of Applied Mathematics and Mechanics vol is pp -5 [3] Chernyshov A D 4 Method of fast expansions for solving nonlinear differential equations Computational Mathematics and Mathematical Physics vol 54 is pp - [4] Chernyshov A D, Goryainov V V and Marchenko A N Study of temperature fields in a rectangular plate with a temperature-dependent internal source with the aid of fast expansions Thermophysics and Aeromechanics vol 3 is pp 3-45 [5] Chernyshov A D, Goryainov V V and Chernyshov O A 5 Application of the fast expansion method for spacecraft trajectory calculation ussian Aeronautics vol 5 is pp - [] [] [] Ishlinskiy A Yu and Ivlev D D The mathematical theory of plasticity (Moscow: Fizmatlit) p 4