ISSN 10283358, Doklady Physics, 2014, Vol. 59, No. 7, pp. 335–340. © Pleiades Publishing, Ltd., 2014. Original Russian Text © T.O. Korepanova, V.P. Matveenko, I.N. Shardakov, 2014, published in Doklady Akademii Nauk, 2014, Vol. 457, No. 3, pp. 286–291.
MECHANICS
Analytical Constructions of Eigensolutions for Isotropic Conical Bodies and Their Applications for Estimating Stress Singularity T. O. Korepanova, Academician V. P. Matveenko, and I. N. Shardakov Received March 5, 2014
DOI: 10.1134/S1028335814070118
One of the important results of the classical theory of elasticity is the possibility of existence of singular solutions, substantiation of the occurrence of which is given in [1]. A reasonably complete review of the works related to the construction and analysis of singular solutions in twodimensional problems of the theory of elasticity is given in [2, 3]. It is necessary to note that, in these reviews, there are none of the numerous works of Russian mechanics researchers. For three dimensional problems, it is possible to single out two classes of regions: the edge of a spatial wedge and the apex of a polyhedral wedge or cone. Interest in the former problems was exhausted by the results of a number of works including [4]. Works devoted to investigation of singularities of stresses in apices of a polyhedral wedge and a cone have begun to appear in recent years. Similarly to other areas of the theory of elasticity, both obtaining particular numerical results and testing numerical methods have importance in constructing singular solutions. In the threedimensional problems, appli cations of analytical methods are related mainly to cir cular cones. The most complete investigation in this field is [5], in which an analytical solution is obtained for a solid circular cone under the boundary condi tions on the lateral surfaces in stresses and displace ments. In this study, we present the total spectrum of ana lytical eigensolutions for various versions of isotropic cones and the examples of calculation of singularity indices of stresses for solid, hollow, and compound cones under various versions of the boundary condi tions on lateral surfaces.
Institute of Mechanics of Continua, Ural Branch, Russian Academy of Sciences, Perm, 614061 Russia email:
[email protected]
On the resource (www.icmm.ru/compcoeff), we listed all formulas and the possibility of calculating with them, which make it possible for a reader to obtain independently the numerical results for the considered problems. We consider a homogeneous circular cone, the ver tex of which coincides with the origin of spherical coordinates r, θ, and ϕ, and the base of which is per pendicular to the axis θ = 0 . The cone occupies the volume 0 ≤ r ≤ ∞, θ1 ≤ θ ≤ θ0 , and 0 ≤ ϕ ≤ 2π , while its boundary is determined by the coordinate surfaces θ = θ1, θ = θ 0 . The version θ1 = 0 = 0 corresponds to a solid cone. We formulate the problem for constructing the eigensolutions satisfying the homogeneous equations of equilibrium: (1 + S )grad div U − curl curl U = 0
(1)
1 , ν is the Poisson ratio, and U is the 1 − 2ν displacement vector) and to one of the homogeneous boundary conditions on the surfaces θ = θ1 and θ = θ 0 in the displacements (here, S =
ur = 0,
uθ = 0,
uϕ = 0,
(2)
the stresses σ rθ = 0,
σθθ = 0,
σθϕ = 0
(3)
or the mixed boundary conditions, the mechanical content of which corresponds to an ideal sliding on the lateral surface,
uθ = 0,
σ r θ = 0,
σθϕ = 0.
(4)
For the body of rotation under consideration and boundary conditions (2)–(4), the eigensolutions with taking into account the dependence on the radius pre sented in [1] can be presented in the form of the Fou rier series on the circular coordinate ϕ 335
336
KOREPANOVA et al. ∞
ur ( r, θ, ϕ) = u0(θ)r α + ∑ [uk (θ)r αsin(k ϕ)] , α
k =1 ∞
uθ ( r, θ, ϕ) = v 0(θ)r + ∑ [v k (θ)r αsin(k ϕ)],
(5)
k =1 ∞
uϕ ( r, θ, ϕ) = w0(θ)r α + ∑ [w k (θ)r αcos(kϕ)]. k =1
If θ1 = 0, the region under consideration is restricted by only one coordinate surface θ = θ 0 , and the following condition of regularity should be fulfilled at θ = 0 :
∂ur = 0, ∂θ
uθ = 0,
uϕ = 0.
(6)
Within the framework of the formulation of the problem under consideration, we can consider the compound cone occupying the region V = V (1) + V (2) , where the subregion V(1) {subregion V(2)} is fabricated from a material with the shear modulus μ(1) (μ(2)) and the Poisson ratio ν(1) (ν(2)), and its configuration is determined by the relations 0 ≤ r ≤ ∞ , 0 ≤ ϕ ≤ 2π, and θ2 ≤ θ ≤ θ0 (θ1 ≤ θ ≤ θ2 ). In the special cases, θ 1 and θ 0 can be equal to 0 and π , respectively. For a compound cone, the eigensolutions of Eq. (5) are constructed for each of the subregions, and the conditions of an ideal fastening or the condition of ideal sliding can be set on the contact boundary θ = θ 2 . After substituting Eqs. (5) into equilibrium Eqs. (1) and passing to the new independent variable x = 1 − cos θ, we obtain the following equations for 2 each of the harmonics of the Fourier series:
d 2uk (x) du (x) + (1 − 2x ) k 2 dx dx 2 4 xR1(x − 1) + k x(1 − x)R2 dv k (x) uk (x) + + 4 x(x − 1) x(1 − x) dx R2 ⎡ 1 − x v (x) − kwk (x)⎤ = 0, + k 2 ⎦⎥ x (1 − x ) ⎣⎢ 2
d 2wk (x) dw (x) + (1 − 2x ) k dx dx 2 2 4 xG 2 ( x − 1) + G1k + 1 kG3 + wk (x) + uk (x) 4 x ( x − 1) 2 x (1 − x ) x (1 − x )
x(1 − x)
(7a)
( )
d 2v k (x) dv (x) + G1 (1 − 2x ) k G1x (1 − x ) dx dx 2 2 4 xG2 ( x − 1) + k + G1 + v k (x) + G3 x (1 − x ) d uk (x) 4 x ( x − 1) dx (7b) k (1 − G1 ) dwk (x) ( G1 + 1) k ( 2x − 1) + + wk (x) = 0, dx 2 4 x ( x − 1)
⎡( G − 1) k dv k (x) ( G1 + 1) k ( 2x − 1) ⎤ v k (x)⎥ = 0. +⎢ 1 + dx 2 4 x ( x − 1) ⎣ ⎦ Here we introduce the designations
(7c)
2 (1 − ν)(1 − α)( α + 2) , R2 = 3 − α − 4ν , 2ν − 1 −1 + 2ν 2 (1 − ν) 2 ( α + 4 − 4ν) , G2 = α (1 + α) , G3 = . G1 = 1 − 2ν 1 − 2ν Boundary conditions (2)–(4) and regularity condi tion (6) are also transformed with taking into account Eqs. (5). The version for the zero harmonic of the Fourier series is considered separately because it does not explicitly follow from the algorithm of construction of partial solutions of the set of differential Eqs. (7) for an arbitrary value of k ≠ 0 . At k = 0 , two problems take place: the axisymmetrical rotation and the axisym metrical deformation. In the first of them, the compo nent of the displacement vector w0 is determined by Eq. (7c). At the axisymmetrical deformation, the dis placementvector components u0 and v0 are deter mined by Eqs. (7a) and (7b). The solutions for the function w0 are constructed in the form of the generalized power series R1 =
w0 ( x ) =
∞
∑A
mx
m +β
,
(8)
m =0
where Am are the coefficients of series and β is another characteristic index. For finding the coefficients Аm of the series and the index β, we substitute Eq. (8) in Eq. (7c). Equating to zero the expressions with the identical exponents х, we obtain the recurrent relation for Аm:
(2β + 2m + 1)(2β + 2m − 1) Am + 4 [α ( α + 1) − ( 2β + 2m − 1) (β + m − 1)] Am−1 − 4 ( α + 2 − m − β)( α − 1 + m + β) Am−2 = 0,
(9)
m = 0, 1, 2, ... . From the condition of existence of the nonzero solution with respect to А0, it follows the characteristic equation
(2β + 1)(2β − 1) = 0 ,
(10)
the roots of which are β1 = 1 and β 2 = − 1 . 2 2 DOKLADY PHYSICS
Vol. 59
No. 7
2014
ANALYTICAL CONSTRUCTIONS OF EIGENSOLUTIONS
The fulfilled transforms make it possible to obtain the first partial solution, which has the form of the generalized power series for Eq. (7c): ∞
∑A
w0(1) ( x ) =
(1) m+1/2 . m x
v 0(3) ( x ) =
∑
(11)
The second partial solution has the form ∞
∑{[A
(2) m
+ Bm(2)lnx]x m
−1/2
},
⎧⎪ ∞ (4) ⎫ m−1 ⎪ (4) × ⎨ (Pm + Dm lnx)x ( ) ⎬ , ⎪⎩m=0 ⎪⎭
(12)
∑
m =0
where
where the coefficients Am(1) , Am(2) , Am(3) , Pm(1) , Pm(2) , Pm(3) , Pm(4) , Dm(3), and Dm(4) are determined from the recurrent relations presented in the resource (www.icmm.ru/ compcoeff) and can be calculated in there. For the calculation of the coefficients, it is necessary to intro duce the number m of terms in the series, the Poisson ratio ν , the number k of the harmonic, and α of inter est, which can be both complex and real.
( m − 1)(2m − 3) − α − α (2) (2) Bm = Bm−1 m ( m − 1) (2m − 3 + 2α)(2m − 5 − 2α) (2) − Bm−2, 4m ( m − 1) 2
(2) (2) Am = 1 − 2m Bm m ( m − 1)
( m − 1)(2m − 3) − α 2 − α (2) (2) + Am−1 + 4m − 5 Bm−1 m ( m − 1) m ( m − 1) (2m − 3 + 2α)( 2m − 5 − 2α) (2) 2 ( m − 2) (2) − Am−2 − Bm−2, 4m ( m − 1) m ( m − 1) m > 1,
B0(2) = 1,
(13)
where S 0(1) and S 0(2) are the constants determined from the set combination of boundary conditions (2)–(4). We write the partial solutions of Eqs. (7a) and (7b) corresponding to the version of the axisymmetrical deformation as ∞
∑
Am(1) x (
m+1)
∞
u0(2)(x) =
,
m =0
u0(3)(x) =
∑A
(2) m m x ,
m =0
∞
∑ {[A
(3) m
+ Bm(3)ln(x)]x (
m+1)
},
(14)
m=0 ∞
u0 (x) = (4)
∑ {[A
(4) m
+ Bm ln(x)]x }, m
(4)
m=0
v0 ( x) = (1)
∞
x (1 − x ) (1) m Pm x , 2 [(α − 1) S − 2] (α + α ) m=0
∑ ∞
(2) v0
x (1 − x ) (2) m Pm x , ( x) = 2 [(α − 1) S − 2] (α + α ) m=0
DOKLADY PHYSICS
u0 ( x ) = C0 u0 ( x ) + C0 u0 ( x ) (1) (1)
∑
Vol. 59
No. 7
2014
(15)
(2) (2)
+ C0 u0 ( x ) + C0 u0 ( x ) , (3) (3)
(4) (4)
v 0 ( x ) = C0(1)v 0(1) ( x ) + C0(2)v 0(2) ( x )
(16)
+ C0 v 0 ( x ) + C0 v 0 ( x ) , (3) (3)
A0(2) = 1, A1(2) = 1. The general solution of differential Eq. (7c) has the form
u0(1)(x) =
The general solution for u0 and v 0 takes the form
B1(2) = −α ( α + 1) ,
w0 ( x ) = S 0(1)w0(1) ( x ) + S0(2)w0(2) ( x ),
x (1 − x ) [(α − 1) S − 2] (α + α 2)
∞ ⎧⎪ ⎫ (3) (3) m⎪ × ⎨1 + S + (Pm + Dm lnx)x ⎬ , ⎪⎩ x ⎪⎭ m =0 x (1 − x ) v 0(4) ( x ) = [(α − 1) S − 2] (α + α 2)
m =0
w0(2) ( x ) =
337
(4) (4)
where C0(1) , C0(2) , C0(3), and C0(4) are the constants deter mined from the set combination of boundary condi tions (2)–(4). For constructing the partial solutions of the set of Eqs. (7) fulfilling a number of transforms [6], we obtain a set of two differential equations with respect to w k and v k :
d 4wk (x) d 3wk (x) d 2wk (x) + + f ( x ) f ( x ) 3 2 dx 4 dx 3 dx 2 (17) dwk (x) + f1(x) + f 0(x)wk (x) = 0, dx d 2v k (x) d 3 w k ( x) v ψ 2 ( x) + ψ = φ ( x ) ( x ) ( x ) 0 3 k 2 3 dx dx (18) 2 d w k ( x) dw k (x) + φ 2 ( x) + φ1(x) + φ 0(x)w k (x), dx dx 2 f 0 ( x ) = 1 xα ( α + 1)( x − 1) 2 × [2x ( α + 3)( α − 2)( x − 1) + k 2 − 1] 2 2 + 1 ( k − 1) ( k + 1) , f1 ( x ) 16 2 2 = x (1 − x )( 2x − 1) ⎡4x(α + α − 3) ( x − 1) + 1 k − 1⎤ , ⎢⎣ 2 2⎥⎦ f 4(x)
338
KOREPANOVA et al. ∞
f 2 ( x ) = − 1 x 2 ( x − 1) 2 ×[4x(α 2 + α − 18) ( x − 1) + k 2 − 13], 2
f3 ( x ) = 6x ( x − 1) ( 2x − 1) ,
(3) v k (x)
(4) v k (x)
4
x (1 − x ) uk (x) = 2x ( x − 1) k ( S α + 2S + 2)
∞
∑ m =0 ∞
w k(2)(x)
∑
=
∞
w k(3)(x) =
∑[( A
(4) [( Am
(4) m −(k +1)/2 Bm ln(x))x ].
=
∑
+
(6) v k (x)
=
∑P
m =0 ∞ (2)
∑[P
∑
(6) (Pm
(1) m+(k +1)/2 , m x
∑P m =0
(2) m +(k −1)/2 , m x
(5) m+(k +1)/2 ], m x
(23) +
(6) m +(k −1)/2 Dm ln x)x .
Then, using the partial solutions w k(1), wk(2), wk(3), w k(4), v k(1), v k(2), v k(3) , v k(4), v k(5) , v k(6) , and obtained relation (20), we determine six partial solutions: ∞
2 x (1 − x ) (1) m +(k +1)/2 Em x , kx ( x − 1) ( S ( α + 2) + 2) m=0
∑ ∞
(2) uk
2 x (1 − x ) (2) m (k 1)/2 = Em x + − , kx ( x − 1) ( S ( α + 2) + 2) m=0
∑
uk(3) =
2 x (1 − x ) kx ( x − 1) ( S ( α + 2) + 2)
∞
×
∑ (E
(3) m
+ Gm(3)lnx)x m−(k −1)/2,
m =0
2 x (1 − x ) kx ( x − 1) ( S ( α + 2) + 2)
×
∑ (E
(4) m
(24)
+ Gm(4)lnx)x m−(k +1)/2,
m =0
∞
uk(5)
x (1 − x ) = E m(5) x m+(k +1)/2, kx ( x − 1) ( S ( α + 2) + 2) m=0
∑
uk(6) =
x (1 − x ) kx ( x − 1) ( S ( α + 2) + 2)
∞
∞
v k (x) =
=
m =0
uk(4) =
Consecutively substituting the obtained partial solutions w k(1), w k(2), wk(3), and w k(4) in the righthand side of Eq. (18) and solving it as inhomogeneous, we find four partial solutions v k(1), v k(2), v k(3) , and v k(4): (1)
,
m =0
m =0
v k (x) =
+
∞
(1)
(21)
+ Bm(3)ln(x))x m−(k −1)/2],
m −(k −1)/2
(4) m −(k +1)/2 Dm ln(x)]x .
∞
+ − [ Am(2) x m (k 1)/2],
(3) m
(3)
(22)
(4) [Pm
∞
m =0 ∞
(4) w k ( x)
∑
=
(5) v k (x)
[ Am(1) x m+(k +1)/2],
m =0
+ Dm ln x]x
Further, solving Eq. (18) as homogeneous, we find two more partial solutions v k(5) and v k(6) :
}
w k(1)(x) =
(3) m
m =0
uk =
⎧ d 2wk (x) dw (x) × ⎨4 x 2 ( x − 1) + 4 x ( 2x − 1) k 2 (20) dx dx ⎩ − [4αx ( x − 1)(1 + α) + k 2 ( S + 1) + 1]wk (x) dv (x) − ⎡2kSx ( x − 1) k + k ( 2x − 1)( S + 2) v k (x)⎤ . ⎢⎣ ⎥⎦ dx At the first stage, we determine four partial solu tions from the solution of Eq. (17):
∑[P
m =0 ∞
(19)
4 f 4 ( x ) = x ( x − 1) , ψ 0(x) = xα ( α + 1)( x − 1) + 1 (1 − k 2 ), 4 2 2 ψ 2(x) = x ( x − 1) , φ0(x) = − 1 x [4 xα ( α + 1t)( x − 1) − k 2 + 1] 2x − 1, k 2 2 2 φ1(x) = 1 x [4 x(α + α − 4) ( x − 1) − k + 1] x − 1, k 2 2 3 ( x − 1) 2 3 ( x − 1) φ2(x) = −5x ( 2x − 1) , φ3(x) = 2x . k k The relation establishing the dependence of the function uk on the functions w k and v k and their deriv atives is also the result of the implemented transforms: 3
3
=
×
∑ (E
(6) m
+ G m(6)ln x)x m−(k −1)/2,
m =0
where the coefficients Am(1) , Am(2) , Am(3) , Am(4) , Bm(3) , B m(4) , Pm(1) , Pm(2) , Pm(3) , Pm(4) , Dm(3), Dm(4) , Pm(5) , Pm(6) , Dm(6), E m(1), E m(2), E m(3) , E m(4), E m(5) , E m(6), G m(3), G m(4) , and Gm(6) are presented and can be calculated on the resource (http://195.69.156.95:81/coeff/index2.html). The general solutions for uk , v k , and w k have the form DOKLADY PHYSICS
Vol. 59
No. 7
2014
ANALYTICAL CONSTRUCTIONS OF EIGENSOLUTIONS Reαn 1.0
339
Reαn 1.0
(a)
(b)
0.9
0.9
1 2 3
0.8
0.7
0.7
0.6 θ1 = 20°
0.6 60°
θ1 = 80°
90°
120°
150°
180°
0.4 80°
105°
130°
155°
180° θ0
Fig. 1. Dependence of Re αn on the angle θ0 at the fixed angles θ1 of a hollow cone and the zero stresses on the lateral surfaces; k = (1) 0, (2) 1, and (3) 2.
uk ( x ) = Ck uk ( x ) + Ck uk ( x ) + Ck uk ( x ) (1) (1)
(2) (2)
(3) (3)
+ Ck uk ( x ) + Ck uk ( x ) + Ck uk ( x ) , (4) (4)
(5) (5)
(6) (6)
v k ( x ) = Ck(1)v k(1) ( x ) + Ck(2)v k(2) ( x ) + Ck(3)v k(3) ( x ) + Ck v k ( x ) + Ck v k ( x ) + C k v k ( x ) , (4) (4)
(5) (5)
(6) (6)
(25)
wk ( x ) = Ck(1)wk(1) ( x ) + Ck(2)wk(2) ( x ) + Ck wk ( x ) + Ck wk ( x ) , (3) (3)
(4) (4)
where Ck(1) , Ck(2) , Ck(3), Ck(4) , Ck(5), and C k(6) are the con stants determined from the set combination of bound ary conditions (2)–(4). For the considered version of the conical body on the basis of the obtained general solutions for k = 0 , k ≥ 1, and the set combination of boundary condi tions, we obtain homogeneous sets of the linear alge braic equations with respect to the constants S 0(1), S 0(2), C0(1) , C0(2) , C0(3), C0(4) , and Ck(i) for k > 0 , i ∈ (1,2,…,6) . The coefficients of these sets of equations depend on the angles at the vertex of conical bodies, the elastic char acteristics of materials, and the characteristic index α. From the condition of existence of the nonzero solu tion of the set of linear algebraic equations, we find the indices α determining the character of the singularity of stresses in the vertex of conical bodies. On the basis of the method considered, we con structed the solutions for a solid cone under the boundary conditions in displacements and stresses. The results for the values of Re α n < 1 determining the singular solutions are identical to the results of [5]. DOKLADY PHYSICS
Vol. 59
No. 7
2014
It should be noted that, for a solid cone, the singu lar solutions under the boundary conditions in stresses take place for the zero, first, and second harmonics of the Fourier series, and under the boundary conditions in the displacements, they occur at the zero and first harmonics of the Fourier series. The new results about the character of the singular ity of stresses in the vertex of a solid cone are obtained under the boundary conditions on the lateral surface corresponding to ideal sliding. Here the singular solu tions take place at the zero, first, and second harmon ics of the Fourier series and at the angle θ0 smaller than π . In Fig. 1, we show the dependences of eigenvalues of Re α n < 1 on the angle of external conical cavity θ0 for various internal opening angles θ1 of the cone. On the conical surfaces, we set the zero boundary condi tions in stresses. The solid line corresponds to real eigenvalues, and the dashed line corresponds to the complex ones. On the basis of the solutions obtained, the results can be obtained also for other versions of the boundary conditions on the lateral surfaces. For a compound cone, the application of an algo rithm for determining the characteristic indices is pos sible in several versions. The first version is the com pound cone with one boundary conical surface θ = θ 0 and the contact boundary θ = θ2. For the solution of this problem, it is necessary to use the regular partial solutions for the internal subregion (0 ≤ r ≤ ∞ , 0 ≤ ϕ ≤ 2π, θ1 ≤ θ ≤ θ2 ) and the irregular partial solu tions presented by relations (12), (14), (15), (21)–(24)
340
KOREPANOVA et al. Reαn 1.0
Reαn 1.0
(a)
(b)
0.9 0.9 0.8 0.8 0.7
1 2
0.5 −3
0.7
3
0.6
−1
1
3 log(G1/G2)
0.6
0
0.1
0.2
0.3
0.4
0.5 ν1, ν2
Fig. 2. (a) Dependences of Re αn on log(G1/G2) for θ2 = 60 , θ0 = 120 , ν1 = ν2 = 0.3 ; (b) dependences of Re αn for the various
Poisson ratios ν1, ν2 , for θ2 = 60 , θ0 = 120 , k = (1) 0, (2) 1, and (3) 2.
for the external subregion (0 ≤ r ≤ ∞ , 0 ≤ ϕ ≤ 2π, θ2 ≤ θ ≤ θ0). The second version is the compound cone with two boundary conical surfaces θ = θ0, θ = θ1, and the contact boundary θ = θ2. In this case, we use the whole set of partial solutions w 0(1), w0(2), u0(1) , u0(2) , u0(3), u0(4), v 0(1), v 0(2), v0(3) , v 0(4), wk(1),..., wk(4) , uk(1),v k(1),..., uk(6),v k(6), pre sented by relations (12), (14), (15), and (21)–(24) both for the internal and the external subregion. As an example, we considered a compound cone with the zero boundary conditions in stresses on the external lateral face and the ideal conditions of fasten ing of parts from various materials. In Fig. 2a, we show the dependence of Re α n < 1 on the ratio of shear G modula 1 of materials. The dependence of the eigen G2 G values on the Poisson ratio for 1 = 1 is shown in Fig. 2b. G2 The solid line designates the eigenvalues for ν1 = 0.3 in dependence on ν2 and by the dashdotted line for ν 2 = 0.3 in dependence on ν1.
ACKNOWLEDGMENTS This work was supported by the Program of Funda mental Research of UrO25P, project no. 12P1 1018, and by the Council of the President of the Rus sian Federation for Support of Leading Scientific Schools, grant no. NSh2590.2014.1. REFERENCES 1. V. A. Kondrat’ev, Tr. MMO 16, 209 (1967). 2. G. B. Sinclair, Appl. Mech. Rev. 57 (4), 251 (2004). 3. G. B. Sinclair, Appl. Mech. Rev. 57 (4), 385 (2004). 4. S. E. Mikhailov, Izv. AN SSSR, MTT, No. 5, 103 (1979). 5. V. A. Kozlov, V. G. Maz’ya, and J. Rossmann, Math. Surv. and Monographs 85 (2001). 6. E. Kamke, Differentialgleichungen: Losungsmethoden und Losungen. Gewohnliche Differentialgleichungen (Lipzig, 1959).
Translated by V. Bukhanov
DOKLADY PHYSICS
Vol. 59
No. 7
2014
