Applied Mathematics and Mechanics (English Edition, Vol 21, No 12, Dec 2000)
Published by Shanghai University, Shanghai, China
Article ID: 0253-4827(2000) 12-1401-06
T H E V A R I A T I O N A L P R I N C I P L E S A N D A P P L I C A T I O N OF NONLINEAR NUMERICAL MANIFOLD METHOD * LUO Shao-ming ( ~ , ~ ] ) ,
ZHANG Xiang-wei ( ~ l ~ J ) ,
CAI Yong-chang ( ~ . 7 ~ ) (Mechatronic Engineering Department, Engineering College, Shantou University, Shantou 515063, P R China) (Paper t~om ZHANG Xiang-wei, Member of Editorial Committee, AMM) Abstract: The physical-cover-oriented variational principle of nonlinear numerical manifold method(NNMM) for the analysis of plastical p r o b l e ~ is put forward according to the displacement model and the characters of numerical manifold method ( N M M ) . The theoretical calculating formulations and the controlling equation of NNMM are derived. As an example, the plate with a hole in the center is calcaulated and the results show that the solution precision and efficiency of NNMM are agreeable. K e y words: variational principle ; numerical manifold method; nonfinear analysis ; plastical
flow CLC n u m b e r : O176
D o c u m e n t code: A
Introduction The finite cover techniques, similar to the conception of f'mite cover used in manifold analysis of modem mathematics, are introduced into Numerical Manifold method ( N M M ) and the finite covers consist of mathematical covers and physical covers which can be separated. The NMM is considered as an all-new and prosperous numerical analysis method that can deal with continuous and discontinuous mechanical problems uniformly. But up to now, the theoretical system and applications of NMM is confined to the linear analysis [1] or can only deal with some special problems f~-] . In engineering practices it is nonlinear (geometry or material nonlinear) problems that often occur. In order to make NMM to be widely used in engineering fields, it is urgent to set up the theoretical system of Nonlinear Numerical Manifold Method ( N N M M ) . In this paper, the physical-cover-oriented variational principles of Nonlinear Numerical Manifold Method ( N N M M )
for the analysis of nonlinear problems is put forward according to the
displacement model and the characteristics of Numerical Manifold Method.
The theoretical
R e c e i v e d date: 2000-03-09 ; R e v i s e d date: 2000-08-29
Foundation item: the Natural Science Foundation of Guangdong Province (994396) Biographies: LUO Shao-ming (1966 ~ ), Associate Professor, Doctor; ZHANG Xiang-wei (1950 ~ ), Professor, Doctor, suppervisor of Ph D Candidate, President of Shantou University 1401
1402
LUO Shao-ming, ZHANG Xiang-wei and CAI Yong-chang
calculating formulations and the controlling equations of NNMM are derived. As an example, the numerical analysis results are shown. The calculating results show that NNMM is a high efficiency numerical analysis method with high solution precision. By further expanding, the NNMM can be widely used in practical engineering fields.
1
The Basic Equations of Plastic Flow Theory a n d V a r i a t i o n a l
Principles
With Euler description, continuous body is in equilibrium under static condition at lime t. Suppose the stress state and loading history are known. The external force increments are d/5~ on stress boundary /", and displacement increments are dgi on displacement boundary /'=. All the equations are linear when these increments are infinitesimal. The basic equations are as follows: 1 ) Equilibrium Equations : da~, i = O;
(1)
2) Strain-Stress Relations :
1
deq = ~-(du;,j + duj,i);
(2)
3) The relation of stress increments daij and strain increments de~ is linear; 4) Stress Boundary Conditions:
da~inj = dp~ 5) Displacement Boundary Conditions : du~ = d~,
(in F , ) ;
(3)
(in F , ) .
(4)
The principle of minimum potential energy of plastic flow theory can be described as[3] : With all dui and deij which are smooth enough and satisfy Eqs. (2) and ( 4 ) , the du i which makes the following functional
Hap = f a [ A ( d e q ) - d F i d u i ] d a - f
d;iduidV
r
(5)
minimum must be the exact solution of displacement increments. For stress-hardening materials, the strain-stress relation is d%. - ( 1 -~_2 V)dakkS~ + ~dcr} + a " H o~a--~--f aii df,
(6)
increment strain density function A (de//) is G d" 3a,,. fe d ke="t a-~ko~ A(de//) - 6(1 -E 2u) dekkdeu + Gde'kt de~z
a"
3f 3f E ~ + aaq atTij
1 +
~
(7)
For ideal plastic materials, the strain-stress relation is [4]
de~ -
lim
( 1 - 2v). ~ d~ 3"E aakkoii+~+,= df
~--o,d/'--o~ - --
d~ > O.
i n c r e m e n t strain density function A (deii) is
3; H ~-e--d;t, a~Tq
(8) (9)
Variational Principles of Nonlinear Manifold Method E
1403
G( ~3-~---fde.= / 2 a amrt
A ( d s q ) - 6(1 - 2 u ) dqkdeu + Gde'kzde'kt - a"
8f
]
8f
(10)
3aq 3aq
2
The Variational
Principles
of Nonlinear
Numerical
Manifold
Method
Dissect the solution domain with N manifold element and each element is the intersection of some physical cover Is] . On the border of two adjacent elements, the displacement is continuous, and the potential energy functional (5) can be written as
~)df'}
iZZp = EN {fr~ (A(')(deij)-d~'idu~'))df2-fr'")d/51du~ m~l
(11)
i
the uppercase m stands for element number and ~
stands for summation over elements.
In Numerical Manifold Method, the element is the intersection of some physical cover. Suppose element m is the intersection of physical cover U,(1), U,(z),"', U,(q), then the displacement model of two dimensional problem can be expressed through weighted function w,(,) ( x , y ) dul(x,y)}
duz(x ,y)
=
E
,
,.,
we(r)
IdUl.(.)(.,y) } = t dua,(r) ( X , y )
[ Te(r)j(x,Y) ]{ ADe(r)J } = t=l q
]=I
~,_~[ Td,) ( x, y) ]{ AD4,)
},
(12)
t=l
where
[ Ti ( x , Y ) ] = ( Tn Tiz q,l(x,y) t2,1(x,y) [ Til(x'Y)]
9
[
r~z(x,y)
[r0(x,r)]
r,.)=
- -
tl,z(x, y)
tl,3(x, y)
""
t2,2(x,y)
t2,3(x,y)
...
tl,z~(x,Y)] tz,z=(x,y) = (13)
'
tj,2(x,y)
= [tj,x(x,y)
and {ADd, )} = l a d n
tI,2.-l(x,Y) t,.,z~-l(x,y)
Adp_ ...
tj,3(x,y)
"'"
tj,a._l(X, y)
tj,2.(x,y)]
Ad~ IT is the generalized displacement increment.
By variation with Eq. ( 11 ), we have N[f
[ oa(,,)~de~ ")
)
m=l
fr,,.,d'i ~,~jET,,a]{SAD-}dP}, p q(,~)
(14)
q ( m ) stands for that manifold element m is the intersection of q physical covers. As for3A/3deq, it is symmetrical with i, j , then we have
f t~ adcq OA(,,),~u~q "~
=
f o 3de aA((n~)O)(lujiq'
By using Green equation, Eq. (15) can be rewritten as
"
(15)
LU0 Shao-ming, ZHANG Xiang-wei and CAI Yong-chang
1404
fn
~A(') 3dr i
~8A(,~) -('~)
~][ T~,,]ISAD,,}dF -
[ r~, ] {~AD~,}dO. .
(16)
q(*~)
Consider the displacement continuity in adjacent elements m and m', we have
~1I;p =
~,
O
-kad~/,)]
fr,.,[.~ - 3 A(.0 ( ' ) n (j r " ) ,
ad~ ij
-dY,
,j
]~[r..]I~Z~O.}dt1+ r
d/~ilE[Tm;]{SAD,~}dp}+ q(.)
3A (') ~_j[ , . , ( ~ n ~
(.,)
+
(~,)jF
aA ('') ~n}")I~[T.~,]{3AD..}dF.
(17)
q(,a)
{3AD~ } are independent variable in/2~ and on/'<'~) a n d / ' ( ~ ' ) ; so, with condition ~n~p --- 0 we can get the following results: 1) On each physical cover, the equilibrium equation of Nonlinear Numerical Manifold Method is
( aA~') I
I r a q i + dF;[Tm,] = 0;
(18)
2) On boundary with given stress of each physical cover 9~(.,) ade~ J [T.,,] - dp~[T.,~] = 0;
(19)
3) On the border of two adjacent manifold elements ~(~') a A ( ' ) (')[T,,..] + ,,zx ad~p )~j ad~p'~ rt(m, J ) [r.;]
_- 0.
(20)
This is the variational principle of Nonlinear Numerical Manifold Method based on plastic flow theory.
3
The Controlling Equation of N o n l i n e a r N ! ~ m e r i e a l M a n i f o l d Method
Manifold element is def'med as the intersection of physical cover. Let manifold element ra be the intersection of physical covers U, Cx) , U,(2) ,..', U,(q)
du(')r
= [du~:)
Ldu~.,) l =
du'(-')]'
l
~.[T~(x,y)]{aD.},
with { & D,, } is generalized displacement increment, [ T= ( x, y ) ] is defined by Eq. ( 13 ). Stress increment and strain increment relation can be written as
de. = B " A D ~ ,
(22)
with &D~, = (AD,(1)
AD,(2 )
"'"
AD,(~)) T
Variational Principles of Nonlinear Manifold Method
1405
is generalized displacement increment vector;
B"* = (Bda)
B.(~)
9 -.
B,(q)
)
is strain matrix, its components are
I [Bi] =
3tlA Ox
8ti,2 8x
0 t 1,2m 9x
~t2,1
at2, 2
t2,2m
8y
3y
0tl,1
0t2,1
0t2,2
0tl,2
+ Ox
8y
(23)
~ tl,2m ~t~ ~m 8y + ~
Oy + Ox
The increment stress and strain can be written as
Atr =
D,p An
= DepB"AD,. .
(24)
Substituting the above equations into Eq. ( 1 1 ) , we can get
nap =
ADr~K
- A D ~ A R (')
,
(25)
m=l
with
K(~) ep
f
O
BT D,pBdI2,
(26)
ZXR" _- f~ [ r ~ ( x , y ) ] ' d F d a
+
fr:Cr(x,y)3Tdpdr.
(27)
K(~) is the stiffness matrix of Nonlinear Numerical Manifold Method, AR (m) is the load vector ep of Nonlinear Numerical Manifold Method and LADY 2 - - - " - 'K(m) e AD~ is the strain energy density of manifold element m . Furthermore. the global equation can be assembled with Eq. (27)
17~p = 1 A D T K,pAD - A D T A R , N
(28)
N
K,p = ~ K ! ;
) , AR :
m=l
~_j~AR (m) ,
(29)
m~l
with A D , K,p , A R are global generalized displacement vector, global stiffness matrix and global load vector. With minimum condition of variation ~//a~ = 0, we can get the following equation:
K,,AD = A R .
(30)
This is the controlling equation of Nonlinear Numerical Manifold Method. 4
Numerical
Example
As an example, the square plate with a hole in the center to which the distributed tension stress s = 140 MPa are applied uniformly at the left and right sides along the x axis direction. The plate is stress-hardening material with elastical module E" = 2.1 x 106 MPa, H = 3 x 103 MPa, Poisson rado/z = 0 . 3 , elastical limitation a, = 170 MPa. The problem is a plane stress problem. By considering the symmetric, a quarter of the structures is taken into account and the manifold elements and nodes of physical covers are shown in Fig. i . There are 40 mnnlfold elements and 54 nodes of physical covers. The first-order Lagrange interpolating functions are used in cover nodes.
LU0 Shao-ming, ZHANG Xiang-wei and CAI Yong-ehang
1406
I
3.5 -- analytical solution 3.0
i
2.5 X i
---.--4
z
~" 2.0 1.5
i
2
Fig. 1
Meshes of the plate with a hold
Fig.2
t
3 y/m
3
4
5
a, at section x = 0
The comparison of analytical solution a, with ax of section x = 0 calculated by N N M M is shown in Fig. 2. It can be seen from Fig. 2 that in manifold method with first-order cover functions used in physical covers, even if the simple 4-node quadrilateral manifold elements shown in Fig. 1, the calculating results are still in good agreement with those of theoretical analysis. But in finite element method ( F E M ) the same good agreement can only be reached by using multi-node high order isoparametcr elements.
This shows that Nonlinear Numerical
Manifold Method is an effective and prosperous numerical analysis method for nonlinear problems. 5
Conclusions Numerical Nanifold Method is a all-new numerical analysis method.
In this paper,
the
Nonlinear Numerical Manifold Method variational principles based on plastical flow theory and the corresponding calculating formula are put forward by introducing the variational method. It is an expansion to the Nonlinear Numerical Manifold Method theory and can be expected to be widely used in engineering fields to deal with nonlinear problems.
References : [. 1 ] [ 2 ]
[ 3 ] [4 ] [ 5 ]
WANG Zhi-yin, LI Yun-peng. Manifold method in analysis of large deformation for rock[J]. Chinese Journal of Geotechnical Engineering, 1998,20(1) :33 ~ 36. (in Chinese) ZHU Yi-wen, ZENG You-lin, CHEN Ming-xiang. Incremental manifold method in analysis of large deformation of rock[ J ] . Chinese Journal of Rock Mechanics and Engineering, 1999,18 (1) : 1 ~ 5. (in Chinese) Kyuichiro Washizu. Variational Methods in Elasticity and Plasticity[ M ] . LAO Liang, HAO Songlin transl. Beijing : Science Press, 1984,30 - 56. (Chinese version) ZHANG Ru-qing. Variational Principle of Solid Mechanics and Its Applications[ M ] . Chongqing: Chongqing University Press, 1991,29 ~ 45. (in Chinese) Sill Gen-hua. Numerical Manifold Method and Discontinuous Deform Analysis [ M ] . PEI Jue-min transl. Beijing: Tsinghua University Press, 1997,3 ~ 22. (Chinese version)