International Applied Mechanics, Vol. 43, No. 1, 2007
STRESS STATE OF FLEXIBLE PLATES WITH A HOLE THAT ARE SUBJECT TO BENDING
UDC 539.3
Ya. F. Kayuk
This paper proposes and mathematically proves a method to regularize singular iterations used to solve geometrically nonlinear problems of bending of thin plates with a hole. It is established that the new approximations for deflections and stress functions satisfy the boundary conditions exactly and Karman’s equations asymptotically exactly. The regularization method and the approximations for solutions are based on new methods of summing functional series and sequences Keywords: Karman’s equations, concentration of forces and moments, singular iterations, permanence condition, summation method, approximations Introduction. A linear solution to stress problems for thin plates with holes bent by moments at infinity was obtained by Savin in the monograph [7]. An attempt to solve these problems in a geometrically nonlinear formulation was made in [2–4]. It was proposed to expand the external moment load in powers of a dimensionless parameter. It appeared that the values of iterations at infinite points of the plate increase, especially with increasing number of approximation, etc. Therefore, the need arose to develop special methods for the regularization (summation) of series in a parameter. It should be noted that if problems are solved in a physically nonlinear formulation for not only plates, but also shells with a hole, then, as shown in [10–13], successive approximations (solutions) decay with distance from the hole. Issues of theoretical importance to be resolved are associated with more complete proof of the new regularization (summation) method. These are the permanence (continuity) of the summation methods and the satisfaction of boundary conditions and equilibrium equations, etc. by special approximations for desired solutions. We will consider these issues for the case where the problem posed is solved by the simple iteration method rather than the parameter expansion method. 1. Problem Formulation. Mechanical Subject of Study. Let us consider an undeformed (configuration C 0 ) homogeneous isotropic rectangular plate of thickness h with an arbitrary curvilinear hole with boundary G1 and no corner points. Assume that the external boundary G2 is rather far from G1 . Let distributed bending moments of constant intensity M be specified on a section of the boundary G2 . Figure 1 shows a rectangular plate with a hole whose parallel faces experience the action of bending moments. In C 0 , the median surface p is described by curvilinear orthogonal coordinates a 1 and a 2 . We also assume that the boundary of the hole coincides with one of the coordinate lines, say a 2 (Fig. 1). The bending moments acting on G2 cause the plate to deform, i.e., to go over to a configuration C t . If the deflections are small, i.e. wmax / h < 1, then the stress state of the bent plate is described as a linear boundary-value problem for the deflection function. The perturbation of the state caused by the hole is described by a homogeneous boundary-value problem for the biharmonic equation. The characteristics of this state are then used to determine the concentration of moments near the boundary G1 . It is of interest to examine the influence of large deflections (wmax / h ³ 1) on the perturbed state of the plate.
S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kyiv. Translated from Prikladnaya Mekhanika, Vol. 43, No. 1, pp. 100–116, January 2007. Original article submitted May 19, 2006. 1063-7095/07/4301-0085 ©2007 Springer Science+Business Media, Inc.
85
f
f a1 a2
f
f
Fig. 1
It was established in [7] that deflections in bent plates with a hole increase without limit, while the moments tend to some constant magnitude, which is consistent with the ideas established in mechanics. The situation is absolutely different with a nonlinear formulation of the problem. It was established in [2–4], based on the parameter expansion method, that not only deflections, but also moments in plates with a hole increase without limit, especially with increasing number of approximation. The reason for this phenomenon may be that the deflections of the perturbed state in the linear case already contain singularities at infinity, which is due to the special choice of coordinate system in the pure-bending problem. It is, therefore, no wonder that the order of these singularities increases with the iteration number. Further, the order of differential operators in the equilibrium equations is different. Thus, we introduce the following differential operators: H1 =
1 ¶ A 2 ¶a 2
æ 1 ¶ ç ç A ¶a 2 è 2
1 é ¶ êA 2 ë ¶a 2 1 ì ¶ Q1 = í A1 A 2 î ¶a 1 H12 =
-
ö 1 ¶A 2 ¶ ÷+ ÷ A 2 A ¶a ¶a , 1 1 ø 1 2
æ 1 ¶ ö 1 ¶A 2 ¶ ù ç ÷ ç A ¶a ÷ + A A ¶a ¶a ú , 1ø 1 2 1 2û è 1 ¶ ¶ [ A 2 (H1 + nH 2 )] + (1- n) ¶a ( A1 H12 ) + (1- n) ¶a H12 2 2
ü ¶A 2 1 ¶ H12 . (H1 + nH 2 )ý + (1- n) ¶a 1 A 2 ¶a 2 þ
ü ï ï ï ï ïï ý (curl ), ï -ï ï ï ï ïþ
(1.1)
where A1 and A 2 are the Lamé parameters in the plane p, n is Poisson’s ratio, and curl denotes cyclic permutation on the ordered set (1.2). Then the forces T11 , T12 , T12 = T21 , Q1* , Q2* , the moments M 11 , M 12 = M 21 , M 22 , and the curvatures c jd can be expressed in terms of these operators as follows: ü ï T11 = H1 F, T12 = H12 F, c 11 = -H 2 w, c 12 = H12 w,ï ï M 11 = -D(H1 + n H 2 )w, M 12 = -D(1- n)H12 w, ý (curl ), ï 3 Eh * ï Q1 = -DQ1 w, D = . ïþ 12 (1- n 2 )
(1.2)
where w = w( a 1 , a 2 ) is the deflection of the plate and F = F( a 1 , a 2 ) is the stress function. Then the equilibrium equations for a plate (Karman’s equation) have the following operator form: Lw =
86
1 B ( w, F ), D
1 LF = - EhB ( w, w), 2
(1.3)
where B( w, F ) is a bilinear symmetric differential form for the variables w and F : B ( w, F ) = H 2 wH1 F - 2H12 wH12 F + H1 wH 2 F,
(1.4)
L = DD is a biharmonic operator in the coordinates a 1 and a 2 . Equations (1.3) should be supplemented with the following boundary conditions on G1 : H1 F = 0,
H12 F = 0,
(H1 + nH 2 )w = 0,
Q1 w = 0.
(1.5)
Which boundary conditions are prescribed on G2 depends on the types of fixation and loading. Let the boundary conditions be the same as those in Fig. 1. It is important that the operators of boundary conditions on G1 and G2 be linear, which is the case here. The boundary-value problem (1.3)–(1.5) may not have a unique solution. It is of interest to construct solutions that would decay approaching the outer boundary of the plate G2 . When deflections are small (linear formulation), such solutions can be found using boundary conditions obtained by superposition [7]. These solutions describe the so-called perturbed state of the plate due to the presence of a hole. The deflections associated with this perturbed state are denoted by w* ( a 1 , a 2 ); in this case F * ( a 1 , a 2 ) = 0. If the deflections increase, no pure bending will occur; the tangential stress state will be superimposed on the bending stress state. Hence, the perturbed state w* ( a 1 , a 2 ), F * ( a 1 , a 2 ) will change. Let w = w* + wD ,
F = F* + FD º F
(1.6)
in Eqs. (1.3) and associated boundary conditions. Consider the formula B ( u * + u D , v * + v D ) = B ( u * , v * ) + B ( u * , v D ) + B ( u D , v * ) + B ( u D , v D ).
(1.7)
Substituting (1.6) into the equilibrium equations (1.3) and considering formula (1.7), where u * = w* , u D = wD , v * = F * = 0, v D = F, we obtain LwD =
[(
)] ,
) (
1 B w* , F + B wD , F D
[(
)
(
(1.8)
) (
1 LF = Eh B w* , w* + 2B w* , wD + B wD , wD 2
)] ,
(1.9)
where Lw* = 0, F * = 0, and LF * = 0. The boundary conditions on G1 for Eqs. (1.8) and (1.9) become H1 F = 0,
(H1 + nH 2 )wD
H12 F = 0,
= 0,
Q1 wD = 0.
(1.10)
The boundary conditions on G2 will be homogeneous too. Thus, the perturbed stress state problem for a plate with a hole in a geometrically nonlinear formulation has been reduced to Eqs. (1.8) and (1.9) with homogeneous boundary conditions. Despite the fact that the nonlinear boundary-value problem is “homogeneous,” the external load is described by the function w* appearing in the equilibrium equations. 2. Solving the Nonlinear Perturbed Stress State Problem by the Simple Iteration Method. To solve the problem posed, we will apply a simple method (method of successive approximations):
[(
)
(
) (
1 LF n = - Eh B w* , w* + 2B w* , wDn -1 + B wDn -1 , wDn -1 2 LwDn =
[(
) (
1 B w* , F n + B wDn -1 , F n D
)] ,
)] ,
(2.1) (2.2)
87
where n = 1, 2, 3, ..., wD0 º 0. It is obvious that each iteration yields inhomogeneous boundary-value problems for biharmonic equations whose right-hand sides can be found explicitly. The boundary conditions on G1 and G2 for each of Eqs. (2.1) and (2.2) will be homogeneous. Denote by W the domain occupied by a plate with a hole in Ñ 0 . Let H w and H F be sets of functions defined in the domain W. The set H w includes those functions that satisfy homogeneous boundary conditions on G1 and G2 whose operators coincide with the operators of the boundary conditions for the function w on G1 and G2 . Similarly, H F is a set of functions defined in W and satisfying homogeneous boundary conditions on G1 and G2 whose operators are identical to the operators of the boundary conditions for the function F. Denote by ||K ||w and ||K ||F norms, which can be either differential or integral, on these sets of functions. Supplementing the sets of functions with limiting elements with respect to the norms, we obtain complete normalized spaces of * . We can always derive specific expressions for the norms | |K | | and | |K | | using the results obtained by functions H w* and H F w F Vorovich [1] or Mikhlin [6]. In what follows, we will not specify these norms in order that the final results be in general form, which would be indicative of their importance and reliability. The use of algorithm (2.1) and (2.2) shows that beginning with the iteration n = 1, computed deflections, forces, and moments increase without limit as P º ( a 1 , a 2 ) Þ ¥ (i.e., as the boundary G2 is approached): wDn ( P ) ® ¥ ,
F n (P ) ® ¥,
M 11,n = -D(H1 + nH 2 )wn ® ¥ ,
T11,n = H1 F n ® ¥ ,
T12 ,n = H12 F n ® ¥,
M 12 ,n = -D(1- n)H12 wn ® ¥ ,
Q1*,n = -DQ1 wn ® ¥.
(2.3)
Hence, the approximations {wDn ( P )},
{T11,n ( P )},
{M 11,n ( P )},
{M 12 ,n ( P )},
{Q1*,n ( P )}, K
(n = 1, 2,K)
(2.4)
will diverge on some set of points from W. Then sequences (2.4) should be considered formal solutions of the problem posed. Therefore, to sum and regularize the singular approximations, we need new methods that would be capable of constructing an asymptotically exact solution based on these approximations. The need for such methods is also dictated by practical considerations because formal solutions produced by algorithm (2.1), (2.2) through unwieldy calculations now satisfy the corresponding equations and boundary conditions; hence the natural desire not to reject these solutions, but somehow process or adapt the information they carry to make them physically consistent. 3. Method to Sum Diverging Iterations. Let us consider a sequence {u n ( P )}of the class (2.4); the convergence of this sequence is equivalent to the convergence of the series ¥
J = å Du k ( P ), k =1
Du k ( P ) = u k ( P ) - u k -1 ( P ),
u 0 ( P ) º 0,
k Î N.
(3.1)
The partial sums of this series are denoted by n
S n( u ) ( P ) = å Du k ( P ).
(3.2)
k =1
We introduce the following function of complex variable z: f [z , s( P ), l] =
2lz , 2l + s( P ) - zs( P )
(3.3)
where l > 0 is an arbitrary parameter, and w( P ) is an arbitrary strictly positive (" P Î W) function of variables ( a 1 , a 2 ) º P. Remark. The functions s( P ) can depend on not only the point P, but also arbitrary real parameters a1 , a 2 , a 3 ,K .
88
For | z | =
s( P ) < 1, the function f ( z ,K ) can be expanded into a power series: 2l + s( P ) f ( z , s, l) =
¥
å d m ( l, s ) z m ,
(3.4)
m =1
where d1 =
b , P
d2 =
bs P2
,
d3 =
bs 2 P3
b = 2l,
, …,
P = 2l + s.
(3.5)
To save space, we will not explicitly indicate that the function s depends on coordinates and parameters. From (3.4) we get ¥
å d m ( l, s ) = 1,
d m ( l, s ) > 0
m =1
" m Î N.
(3.6)
Suppose k
[ f ( z , s, l )]
k
¥ é ¥ ù (k ) = ê å d m ( l, s ) z m ú = å d m ( l, s ) z k , êëm =1 úû m =1
(3.7)
where (k ) (k ) ( l, s ) > 0 and d m ( l, s ) = 0 for m < k. dm
(3.8)
ü ï ï ï ï ý ï ï k (k ) ï d ( l , s ) = 1 , " P Î W , m Î N , l > 0 . å m ï m =1 þ
(3.9)
It can be verified that b k -1 m - k k -1 s b , = C km m -1 æ nö n! C nj º çç ÷÷ , è j ø j ! (n - j) ! (k ) dm
Consider the expression ¥
å Du k ( P ) [ f ( z , l, s )]
k
.
k =1
If z = 1, then we obtain the original series J (3.1). If, however, we substitute (3.4) into this expression and order it in increasing powers of z before setting z = 1, then we will obtain a new series J * instead of J: ¥
(
)
J * = å DU k Du s , d k( m ) ,
(3.10)
) = å Du
(3.11)
k =1
where
(
DU k Du s , d k( m )
k
m =1
m
( P ) d k( m ) ( l, s ).
89
Define a subdiagonal triangular matrix: d1(1) d 2(1)
G = d (1) 3 d 4(1) L
d 2( 2 ) d 3( 2 )
d 3( 3 )
d 4( 2 )
d 4( 3 )
L
.
(3.12)
d1(4)
L
L
Considering (3.5) and (3.9), we can establish by calculation that b p bs
b2
p2 bs 2
p2 2b 2 s
b3
b4
G = p3 bs 3
p3 3b 2 s 2
p3 3b 3 s
p4
p4 bs 4
p4 4b 2 s 3
p4 6b 3 s 2
4b 4 s
b3
p5 L
p3 L
p3 L
p3 L
p5 L
,
(3.13)
4b 4 s
where, s = s( P ,K ), p = 2l + s( P ), as before. Matrix (3.13) possesses the following properties: (i) the coefficients of an arbitrary row add up to m = b / p < 1, " l > 0, s( P,K ) > 0 in W; (ii) the coefficients of an arbitrary column d s( r ) , where r is the row number and s is the column number, which is fixed, tend to zero as r ® ¥. These properties can be proved by direct calculation using formulas (3.8) and (3.9). Matrices of general form (3.12) and of specific form (3.13) fall into the class of so-called Toeplitz matrices [9], which is very important. Let us form partial sums of the series (3.10): *
n
(
S n( u ) ( P ,K ) = å DU k Du s , d s( p ) k =1
)
n
º å DU k .
(3.14)
k =1
Considering (3.8) and (3.11), we transform this expression as follows: *
n
n
S n( u ) ( P ,K ) = å DU k = å k =1
n
=å
n
k =1 m =1
å Du m ( P )d k(m ) (l, s) =
k =1 m =1
k
å Du m ( P )d k(m ) (l, s)
n
n
å å Du m ( P )d k(m ) (l, s).
m =1 k =1
Since the dummy indices m and k are equivalent, we change m « k. Then *
n
S n( u ) ( P ,K ) = å
n
k =1 m =1
and, hence,
90
n
n
k =1
m =1
å Du k ( P )d m(k ) (l, s) = å Du k ( P ) å d m(k ) (l, s),
(3.15)
*
n
S n( u ) ( P ,K ) = å Du k ( P )d k (n, l, s),
(3.16)
k =1
where d k (n, l, s) =
n
å d m(k ) (l, s).
(3.17)
m =1
It is expedient to call d k ( n, l, s ) summing coefficients and s( P ,K ) summing functions. It can be verified [5] that the functional sequence {d k ( n, l, s )} with positive coefficients is bounded above and is decreasing in k, " l > 0, s( P,K ) > 0, for arbitrary finite n. Considering (3.13), we can explicitly find the functions d k ( n, l, s ) for arbitrary k, n and l > 0, s( P,K ) > 0. Let, for example, n = 5; then d1 ( 5, l, s ) =
d 3 ( 5, l, s ) =
b æç s s 2 s 3 s 4 1+ + + + p çè p p2 p3 p4
b 3 æç s s2 1+ 3 + 6 ç 3 p p è p2
ö ÷, ÷ ø
ö ÷, ÷ ø
d 2 ( 5, l, s ) =
d 4 ( 5, l, s ) =
b 2 æç s s2 s3 1+ 2 + 3 +4 p p 2 çè p2 p3
b4 æ sö çç 1+ 4 ÷÷ , 4 pø p è
d 5 ( 5, l, s ) =
b5 p5
ö ÷, ÷ ø
, etc.
(3.18)
We will now consider the series (3.1) with the partial sums (3.2). Let the sequence of functions {Du k ( P )} belong to some set of functions H u , where H u may be one of the sets specified above, H w or H F . Suppose that the series (3.1) converges to some function S 0(u ) ( P ) Î H u if lim | | S n( u ) ( P ) - S 0( u ) ( P )| |H u Þ 0.
n ®¥
(3.19)
This means that given e1 > 0, there exists an N such that || S n( u ) ( P ) - S 0( u ) ( P )| |H u < e1
(3.20)
when n > N . Let us now consider the series *
J = lim S n( u ) ( P ,K ). n ®¥
(3.21)
Its partial sums are defined by (3.15) or (3.16). Let us show that if the series J converges in the sense of (3.19), then the *
series J will converge in the same sense. This can be proved with the help of the Bolzano–Cauchy criterion: define two arbitrary parameters r and q ( r > q > 0) and set up the following difference: *
*
B ( r ,q ) = S r( u ) ( P ,K ) - S q( u ) ( P ,K ).
(3.22)
Considering formulas (3.16) and (3.17), we obtain *
*
B ( r ,q ) = S r( u ) ( P ,K ) - S q( u ) ( P ,K ) r
q
k =1
k =1
+ å Du k ( P ) [d k (r, l, s) - 1] + å Du k ( P ) [d k (q, l, s) - 1].
(3.23)
91
n
According to Abel’s formula [9], the sum of pairs of factors H n = å a k b k can be represented as k =1
n -1
H n = å (a k - a k + 1 )B k + a n B n ,
(3.24)
k =1
where B1 = b1 , B 2 = b1 + b 2 , …, B k = b1 + b 2 +K+ B k . Putting a k = d k ( r, l, s ) - 1, b k = D k ( P ), etc., we transform the second and third terms in formulas (3.23) to r
r -1
k =1
k =1
q
q -1
k =1
k =1
å D k ( P ) [dk (r, l, s) - 1] = å [dk (r, l, s) - dk + 1 (r, l, s)] S k(u ) ( P ) + [dr (r, l, s) - 1] S r(u ) ( P ),
å D k ( P ) [dk (q, l, s) - 1] = å [dk (q, l, s) - dk + 1 (q, l, s)] S k(u ) ( P ) + [dq (r, l, s) - 1] S q(u ) ( P ).
(3.25)
(3.26)
Substituting (3.25) and (3.26) into relation (3.23) and estimating it with respect to the norm H u , we obtain || B ( rq ) ||H u £ | | S r( u ) ( P ) - S q( u ) ( P )| |H u + || d r ( r, l, s ) - 1||H u || S r( u ) ( P )| |H u + | | d q ( q, l, s ) - 1| |H u | | S q( u ) ( P )| |H u r -1
+ å || d k ( r, l, s ) - d k + 1 ( r, l, s )| |H u | | S k( u ) ( P )| |H u k =1
q -1
+ å || d k ( q, l, s ) - d k + 1 ( q, l, s )| |H u | | S k( u ) ( P )| |H u .
(3.27)
k =1
Since the series J converges, there exists a k0 such that || S r( u ) ( P ) - S q( u ) ( P )| |H u <
e1 5
if r, q > k 0
(3.28)
for arbitrary e1 > 0. From the convergence of J, it follows that its partial sums are bounded: (u ) || S m ( P )||H u £ | | S 0( u ) ( P )| |H u
(m = 1, 2,K);
(3.29)
then || d r ( r, l, s ) - 1||H u || S r( u ) ( P )| |H u £ | | d r ( r, l, s ) - 1| |H u | | S 0( u ) ( P )| |H u ,
(3.30)
|| d q ( q, l, s ) - 1||H u || S q( u ) ( P )| |H u £ | | d q ( q, l, s ) - 1| |H u | | S 0( u ) ( P )| |H u .
(3.31)
If r and q are sufficiently large, then d r ( r, l, s ) - 1, d q ( q, l, s ) - 1can be made indefinitely small " l > 0, s( P ) > 0. Since the norm is continuous, the expressions || d r ( r, l, s ) -1| |H u and | | d q ( q, l, s ) -1| |H u will be small at the same large r and q. Hence, there exist r, q £ k 0 such that || d r ( r, l, s ) - 1||H u £
e2 -1 , || S 0( u ) ( P )| |H u 5
where e 2 > 0 and e 3 > 0 are arbitrary and rather small. Then inequalities (3.30) and (3.31) become 92
| | d q ( q, l, s ) - 1| |H u £
e3 | | S 0( u ) ( P )| |-H1 , u 5
(3.32)
|| d r ( r, l, s ) - 1| |H u | | S r( u ) ( P )| |H u £
e2 , 5
(3.33)
|| d q ( q, l, s ) - 1| |H u | | S q( u ) ( P )| |H u £
e3 . 5
(3.34)
Let us estimate the fourth and fifth terms on the right-hand side of (3.27): r -1
å || dk ( r, l, s ) - dk + 1 ( r, l, s )| |H u | | S k(u ) ( P )| |H u
k =1
£ || S 0( u ) ( P )||H u
r -1
å | | dk ( r, l, s ) - dk + 1 ( r, l, s )| |H u .
(3.35)
k =1
Considering formulas (3.17), we see that r
d k ( r, l, s ) - d k + 1 ( r, l, s ) =
å d m(k ) ( l, s ) -
m =1
r
å d m(k + 1) ( l, s ).
(3.36)
m =1 ¥
The right-hand side is the difference of the partial sums of the series
å d m(k ) ( l, s ) and
m =1
¥
å d m(k + 1) ( l, s ) with positive
m =1
coefficients; the sums of these series are equal to unity when " l > 0, s( P ) > 0, k Î N . Then expression (3.36) can also be made indefinitely small by appropriately choosing r. Since the norm is continuous, the values | | d k ( r, l, s ) - d k + 1 ( r, l, s )| |H u
(3.37)
will also be rather small " k Î N, l > 0 for the same r. Denote differences (3.37) by x (kr ) = x k( r ) ( l ). It can be seen that x (kr ) Þ 0 as r ® ¥ " k Î N, l > 0. Hence, the sums r -1
å x(kr ) ( l ) can be made rather small when r is large enough.
k =1
Then, given r > N , there exists ( x 4 / 5)/|| S 0( u ) | |H u , where x 4 > 0 is rather small, such that the inequality r -1
å || dk ( r, l, s ) - dk + 1 ( r, l, s )| |H u | | S k(u ) ( P )| |H u
£
k =1
x4 5
(3.38)
x5 . 5
(3.39)
follows from (3.35). Following a similar approach, we establish that q -1
å || dk ( q, l, s ) - dk + 1 ( q, l, s )| |H u | | S k(u ) ( P )| |H u £
k =1
Considering inequalities (3.28), (3.33), (3.34), (3.38), and (3.39), from (3.27) we obtain 5
|| B ( rq ) | |H u £
1 å e i £ e, 5 i =1
where e i £ max e i = e. *
This proves the convergence of the series J " l > 0, s( P ) > 0.
93
*
Let us show that the sum of the series J coincides with the sum of the series J, i.e., with the function S 0(u ) ( P ) " l > 0, s( P ) > 0. *
To this end, we will estimate the difference | | S n( u ) ( P ,K ) - S 0( u ) ( P ,K )| |H u : *
*
S n( u ) ( P ,K ) - S 0( u ) ( P ,K )
= S n( u ) ( P ,K ) - S n( u ) ( P ) + S n( u ) ( P ) - S 0( u ) ( P ) Hu
Hu *
£
S n( u ) ( P ) - S 0( u ) ( P )
Hu
+ S n( u ) ( P ,K ) - S n( u ) ( P )
.
(3.40)
Hu
Since the series J converges, for given x1 > 0 there exists an N such that S n( u ) ( P ,K ) - S 0( u ) ( P )
Hu
<
x1 3
(3.41)
when n > N . Using formulas (3.15), (3.2) and Abel’s formula (3.24), we transform the second term on the right-hand side of inequality (3.40): *
n
S n( u ) ( P ,K ) - S n( u ) ( P ) = å Du k [d k (n, l, s) - 1] k =1
n -1
= å [d k (n, l, s) - d k + 1 (n, l, s)] S k( u ) ( P ) + [d n (n, M , l) - 1] S n( u ) ( P ). k =1
Considering (3.29), we get n -1
*
S n( u ) ( P ,K ) - S 0( u ) ( P )
£ S 0( u ) ( P )
Hu
Hu
+
å [dk (n, l, s) - dk + 1 (n, l, s)] H u
k =1
[dn (n, l, M ) - 1] H u
S 0( u ) ( P )
Hu
.
(3.42)
Following a similar approach, we can state that for given rather small x 2 > 0 and x 3 > 0, there exists an N 1 such that the following inequalities hold when n > N 1 : n -1
å [dk (n, l, s) - dk + 1 (n, l, s)]
k =1
[dn (n, l, M ) - 1]
Hu
£
Hu
x3 3
£
x2 3
. S 0( u ) ( P )
. S 0( u ) ( P )
Hu
.
Hu
,
(3.43)
As a result, we obtain *
S n( u ) ( P ,K ) - S 0( u ) ( P )
£ Hu
94
1 (x1 + x2 + x3 ) £ x , 3
(3.44)
where x = max x i . i This proves the permanence of the summation method. Algorithm (2.1), (2.2) allows explicit calculation of three, or at most four approximations S k( w ) ( P ) and S k( F) ( P ) (k = 1, 4). Therefore, it is not known beforehand whether the sequences {S k( w ) } and {S k( F) } converge. However, the property of permanence allows us to state the following. 1. If we do generalized summation of sequences (2.4), then nothing will be lost—the series are again converging and are summed to the same sum. By appropriate choice of the arbitrary summing functions s( P,K ) present in the coefficients of this series, we can influence the rate of its convergence and, thereby, reduce the number of approximations. 2. If sequences (2.4) are diverging, then these approximations can be summed by varying the functions s( P,K ) [5]. However, it is necessary to check whether the limiting functions satisfy the equilibrium equations and boundary conditions with a prescribed degree of accuracy. These summation methods are a new type of linear summation methods. Note that a linear summation method was also used in [8] to sum weakly converging Fourier series. Interestingly, the modified series J * are similar in analytic structure to the series in [8]. However, the summing functions in [8] were obtained after a sophisticated analytic treatment of a specific problem. A new conceptual idea is to develop linear permanent summation methods such that the coefficients of their matrices (3.12) and (3.13) would contain arbitrary functions and parameters. By varying these parameters and functions, we can influence the convergence and the rate of convergence of the series, to smooth out the singularities of the coefficients, etc. It is shown here and in [5] that summing functions can be constructed relatively easily. This is indicative of the simplicity and efficiency of the summation method proposed. 4. Using the Generalized Summation Method to Solve the Nonlinear Boundary-Value Problem. For the class of problems under consideration, the approximations and associated series (3.1) produced by algorithm (2.1), (2.2) generally diverge as the boundary G2 of the domain W is approached. To find the true solution, it is proposed to use the series J * (3.10). A solution is considered true if for certain n, the *
*
partial sums S n( w ) and S n( F) of this series, etc. calculated using the approximations wDk , F k ( k = 1, n ) and formulas (3.1) satisfy the boundary conditions and equilibrium equations (1.3) with high (asymptotic) accuracy. Suppose that n approximations have been calculated by algorithm (2.1), (2.2). They are further used to set up expressions of the form (3.15): * n S n( w ) ( P ,K ) = DwkD d k k =1
[n, l, s( P )],
DwkD = wkD ( P ) - wkD-1 ( P ),
wD0 ( P ) º 0,
(4.1)
* n S n( F) ( P ,K ) = DF k d k k =1
[n, l, s( P )],
DF kD = F k ( P ) - F k -1 ( P ),
F 0 ( P ) º 0.
(4.2)
å
å
The functions wDk and F k ( k = 1, 2,K ) appearing in these approximations have singularities of certain type. They cause the stress state of the plate not to decrease with distance from the hole. This contradicts the established idea of perturbed states’ decaying with distance from the hole. Hence, the series J * with partial sums (4.1), (4.2), etc. will be slowly converging or even diverging for a certain set of points from W. In stress concentration problems, the deflections and stress function for which approximations (4.1) and (4.2) are proposed are usually of no interest. It is forces and moments that are important. To determine them, it is necessary to evaluate partial derivatives of (4.1) and (4.2). Denote by Dr( w ) and D (pF) linear partial differential operators over (4.1) and (4.2), r and p being the orders of these operators. Estimating the expressions
95
Dr( w )
* S n( w ) ( P ,K )
D (pF)
and
* S n( F) ( P ,K )
(4.3)
is generally a cumbersome procedure. *
Let us discuss an algorithm for regularizing the series J , accelerating their convergence, and deriving expressions (instead of (4.1) and (4.2)) that could be differentiated much easier. The idea of the algorithm is as follows: 1. Generate an iterative process (2.1), (2.2) to evaluate the functions wDk , F k ( k = 1, n ) and the partial derivatives Dr( w ) DwkD ,
D (pF) F k ,
k = 1, 2, ...,
r, p = 1, 2.
(4.4)
2. Consider the functional series ¥
[
]
(4.5)
[
]
(4.6)
J r( w ) = å Dr( w ) DwDk d k n, l, s (rw ) (P; a1 , a 2 ,K) , k =1 ¥
J (pF) = å D (pF) DF k d k n, l, s (pF) (P; c1 , c 2 ,K) , k =1
where the summing functions s( P ,K ) depend on the arbitrary parameters a i and c i (i = 1, 2, ...) and the orders of differentiation r and p. 3. Select the summing functions s r( w ) ( P;K ) and s (pF) ( P;K ) so as to smooth or reduce the orders of singularities of all or some of the coefficients of series (4.5), (4.6), so that the new coefficients would increase with distance from the outer boundary [3]. These summing and simultaneously regularizing functions are denoted by s r( w ) (P; a1 , a 2 ,K),
s (pF) (P; c1 , c 2 ,K).
(4.7)
4. Put P = P0 in (4.7), where P0 is a fixed point from W. This point can be chosen differently (for example, so that the order of regularity would be maximum in its neighborhood). Then, the partial sums of series (4.5) and (4.6) with P = P0 are denoted by * é ù n S n( w,r) = å Dr( w ) DwkD d k ên, l, s (rw ) (P0 ; a1 , a 2 ,K)ú , ê ú k =1 ë û
(4.8)
* é ù n ) ( F) ( F) ê ú. S n( F = D DF d n , l , s P ; c , c , K ( ) å p ,p k k p 0 1 2 ê ú k =1 ë û
(4.9)
Remark. In regularizing and summing series (4.5) and (4.6), it is necessary first to substitute P = P0 into (4.7) and then to set up series (4.5) and (4.6) and expressions (4.8) and (4.9); the direct and inverse operations being not commutative. If functions (4.7) have been selected and used to regularize and sum series (4.5) and (4.6), then expressions (4.8) and (4.9) are regular and finite at infinity [2, 3]. Then, by analogy with (4.8) and (4.9), the deflections and stress function of the perturbed state can be approximated as follows: wnD
96
» S n( w,0)
n
=å
k =1
DwkD d k
* é ù ên, l, s r( w ) (P0 ; a1 , a 2 ,K)ú , ê ú ë û
(4.10)
) F n » S n( F = ,0
* é ù ( F) ê ú. DF d n , l , s P ; c , c , K ( ) å k kê p 0 1 2 ú k=0 ë û n
(4.11)
By identifying the linear operators Dr( w ) and D (pF) in (4.8) and (4.9) with the operators H1 , H 2 , H12 = H 21 , and H1 + nH 2 in (1.2), it is easy to see that the approximations for the true solutions (4.10) and (4.11) satisfy all the boundary conditions (1.2) " l > 0, a i , c i , and P0 ( i = 1, 2, 3,K ). It remains to ascertain to what degree approximations (4.10) and (4.11) satisfy Eqs. (1.8) and (1.9). This is generally a challenging task. It is necessary either to consider specific problems, or to conduct appropriate computation, or to follow certain approaches. However, there is still a chance to assess how solutions (4.10) and (4.11) satisfy the equilibrium equations (1.8) and (1.9). Let us first consider Eq. (1.8) and use (4.10) and (4.11) to estimate the residual
(
)
A1 w* , wD , F = LwD -
[ (
)
)] .
(
1 B w* , F + B wD , F D
(4.12)
Substituting (4.10) and (4.11) into the right-hand side of (4.12) and performing transformations in view of (2.1) and (2.2), we obtain
(
)
A1 w* , wD , F =
n ì 1 ï å í dk D k =1 ï î
* * é ù é ùü ên, l, s r( w ) (P0 ; a1 , a 2 ,K)ú - d k ên, l, s (rF) (P0 ; ñ1 , ñ 2 ,K)ú ïý ê ú ê úï ë û ë ûþ
(
)
´ B w* , DF ê -
(
)
1 ) , B S n( w,0) , S n( F ,0 D
(4.13)
whence
(
A1 w* , wDn , F n
)H
w
£
* * é é ù ùü n ì 1 ï ê (w ) ( F) ê ú d l s d n , , P ; a , a , n , l , s P ; ñ , ñ , K K ) k )ú ïý åí k r ( 0 1 2 r ( 0 1 2 ê ú úï D k =1 ï ê ë û ûþ î ë
(
´ B w* , DF ê
)H
w
+
(
1 ) B S n( w,0) , S n( F , 0 D
)
.
(4.14)
Hw
The first term on the right-hand side can be made indefinitely small by increasing n. This is because the coefficients d k [ n , l;K ] tend to unity as n ® ¥ for each k, given summing functions. This term, however, would be identically equal to zero if p = r = 2 and the summing functions for deflections and the stress functions were identical ( a i = c i ; i = 1, 2,K ). Hence, it can generally be assumed that inequality (4.14) reduces to
(
A1 w* , wDn , F n
)H
w
£
(
1 ) B S n( w,0) , S n( F ,0 D
)
.
(4.15)
Hw
To estimate qualitatively how small the right-hand side of this inequality is, we will nondimensionalize it. It is necessary that the scales used to nondimensionalize the original quantities and parameters be commensurable. Let a be a length scale of the plate (the maximum diameter of the hole, the minimum length of the plate side, etc.). Dimensionless quantities are denoted by “~”. Then we have ~, w = hw
~ F = a 2 hEF,
Lw =
h a4
~, L1 w
LF =
Eh a2
~ L1 F,
(4.16)
where the operator L1 is derived from the biharmonic operator by replacing dS k ( dS k = A k da k ) by dimensionless length ~ elements according to the formulas dS k = adS k ( k = 1, 2).
97
Then inequality (4.15) takes the following dimensionless form:
(
~ ~* ,w ~D ,F A1 w n n
)H
w
(
2
)
(
~ æaö ~D ,F £ 12 1- n 2 ç ÷ B w n n èhø
)H
,
(4.17)
w
where
(
)
(
)
~ ~* ,w ~D ,F ~ D - 12 1- n 2 æ a ö A1 w = = Lw ç ÷ èhø
2
[B (w~ * , F~) + B (w~ D , F~)].
(4.18)
~ ~D ,F Since the bilinear form B ( w n n ) is a bounded function " P Î W, it follows from (4.17) that the error of the true solution to Eq. (1.8) is of O ( a 2 / h 2 ). This error can in many cases be reduced by varying the arbitrary constants in the summing functions. 1
Indeed, considering (4.10), (4.11) and matrices (3.18), it is possible to put b = 2l =
12(1- n 2
)
(h / a) 3 / 2 b1 (b1 > 0) in
the coefficients d k [ n, l, s( P )]. Then the error will be of O ( h / a ). This error can be estimated more accurately with specific solutions. Let the residual for the second equation in (1.9) be given by
(
[ (
)
)
(
) (
1 A 2 w* , wD , F = LF + Eh B w* , w* + 2B w* , wD + B wD , wD 2
)] .
(4.19)
Substituting (4.16) into (4.19), we obtain
(
)
A 2 w* , wD , F =
ìï ~ 1 æ h ö 2 ~D ,w ~D ~* ,w ~ * + 2B w ~* ,w ~D + B w E í L1 F + ç ÷ B w 2èaø a 2 îï
[ (
h
)
(
) (
ü
)]ïýï = ah2 EA 2 (w~ * , w~ D , F~). (4.20) þ
Substituting (4.10) into the right-hand side of (4.19), we get
(
[ (
)
)
(
) (
1 A 2 w* , wDn , F n = LF n + Eh B w* , w* + 2B w* , wnD + B wnD , wnD 2 n * é ù 1 = å d k ên, l; s 2 (P0 ; a1 , a 2 ,K)ú LF k + EhB w* , w* 2 úû k =1 êë
(
)]
)
n * é ù 1 + Eh å d k ên, l; s 2 (P0 ; a1 , a 2 ,K)ú B w* , Dwk + EhB wnD , wDn . 2 úû k =1 êë
(
)
(
)
(4.21)
Expression (4.21) can be rearranged a little using the recurrent formulas that follow from Eqs. (2.1) and (2.2):
(
1 LDF1 = - EhB w* , w* 2
[ (
) (
)
( LF 0
) (
º 0),
(4.22)
1 LDF k = - Eh 2B w* , DwkD-1 + B wkD-1 , wkD-1 - B wkD- 2 , wkD- 2 2
)]
(k = 2, 3,K) .
With formulas (4.22) and (4.23), relation (4.21) becomes * é æ öù 1 A 2 w* , wDn , F n = ê1- d1 ç n, l; s 2 (P0 ; a1 , a 2 ,K) ÷ú EhB w* , w* ç ÷ú 2 êë è øû
(
98
)
(
)
(4.23)
n * é ù - Eh å d k ên, l; s 2 (P0 ; a1 , a 2 ,K)ú B w* , DwkD-1 úû k = 2 êë
(
n * é ù + Eh å d k ên, l; s 2 (P0 ; a1 , a 2 ,K)ú B w* , Dwk úû k =1 êë
(
)
)
n * é ù 1 - Eh å d k ên, l; s 2 (P0 ; a1 , a 2 ,K)ú B wDk -1 , wDk -1 - B wDk - 2 , wDk - 2 2 k = 2 êë úû
[ (
) (
)]+ 12 EhB (wn , wn ).
(4.24)
It can be seen that with a sufficiently large n (or with a1 , a 2 ,Kappropriately chosen), the first three terms will always be small and can be neglected. Using formula (4.20), we nondimensionalize (4.24):
(
)
~ 1æhö ~* ,w ~D ,F A2 w n n » ç ÷ 2èaø
[(
2
* ìï n é ù * ; s P ; a , a ,K d n l , )ú ê íå k 2( 0 1 2 úû îï k = 2 êë
) (
~D ,w ~D ~D ,w ~D ´ B w -B w k -1 k -1 k-2 k-2
ü
)]+ B (w~ n , w~ n )ïýï .
(4.25)
þ
1
Since estimate (4.17) has been derived using lfrom the relation b = 2l =
12(1- n 2
)
(h / a) 3 / 2 b1 (b1 > 0 is an arbitrary
parameter), it should naturally be incorporated into (4.25). Finally, we obtain a second estimate:
(
A 2 w* , wDn , F n
[(
)H
F
£
æhö ç ÷ 2 12(1- n 2 ) è a ø
) (
1
5
)
n
é
k=2
ëê
ù
*
å dk ên, l* ; s 2 (P0 ;K)ú úû
]
~D ,w ~D ~D ,w ~D ~ ,w ~ ) ´ B w -B w + B(w n n k -1 k -1 k-2 k-2
.
(4.26)
HF
It can be seen that the true solution for the stress function satisfies its equation with an error of higher order of smallness. In summary, it should be pointed out that the efficiency and reliability of the regularization method proposed can be judged in solving specific problems, which was partially done in [3, 4].
REFERENCES 1. I. I. Vorovich, Mathematical Problems in the Theory of Thin Shells [in Russian], Fizmatgiz, Moscow (1989). 2. Ya. F. Kayuk and L. M. Krivoblotskaya, “Method for the regularization of singular iterations in nonlinear bending problems for a plate with a hole,” Visn. Donetskogo Univ., Ser. A: Prirodn. Nauky, 1, 83–90 (2002). 3. Ya. F. Kayuk and L. M. Krivoblotskaya, “Concentration of moments around a circular hole in a plate subject to large deflections,” Visn. Donetskogo Univ., Ser. A: Prirodn. Nauky, 2, 87–91 (2002). 4. Ya. F. Kayuk and L. M. Krivoblotskaya, “Singular iterations in nonlinear stress-concentration problems,” Teor. Prikl. Mekh., 36, 98–108 (2002). 5. Ya. F. Kayuk, Some Issues of Parameter Expansion Methods [in Russian], Naukova Dumka, Kyiv (1980). 6. S. G. Mikhlin, Numerical Implementation of Variational Methods [in Russian], Fizmatgiz, Moscow (1966). 7. G. N. Savin, Stress Concentration around Holes, Pergamon Press, New York (1961). 8. A. N. Tikhonov, “On stable methods of summing Fourier series,” Dokl. AN SSSR, 156, No. 2, 268–271 (1964). 99
9. T. M. Fikhtengol’ts, A Course of Differential and Integral Calculus [in Russian], Vol. 2, Fizmatgiz, Moscow (1966). 10. T. A. Storozhuk and I. S. Chernyshenko, “Physically and geometrically nonlinear deformation of spectral shells with an elliptic hole,” Int. Appl. Mech., 41, No. 6, 666–674 (2005). 11. T. A. Storozhuk and I. S. Chernyshenko, “Elastoplastic deformation of flexible cylindrical shells with two circular holes under axial tension,” Int. Appl. Mech., 41, No. 5, 506–511 (2005). 12. M. P. Malezhik, O. P. Malezhik, A. T. Zirka, and I. S. Chernyshenko, “Stress wave fields in plates weakened by curvilinear holes with edge cracks,” Int. Appl. Mech., 42, No. 2, 192–195 (2006). 13. K. I. Shnerenko and V. F. Godzula, “Stress state of a cylindrical composite panel weakened by a circular hole,” Int. Appl. Mech., 42, No. 5, 555–559 (2006).
100
