1979 Finite Element Method Applied
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Transcript of 1979 Finite Element Method Applied
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THE FINITE ELEMENT METHOD APPLIED TO
LARGE ELASTO-PLASTIC DEFORMATIONS
OF SOLIDS
George Z. Voyiadjis
Department of Civil Engineering, University of Petroleum and Minerals,
Dhahran, SaudiArabia
;\;J)I~.r ..)iJ~W4.i..,.,'J..I~."s~;1I.;.J'1~WIIf Js,~ .:-jJl ( , ~)(5.r)J ! l,..,..: .s .. )
J...,1)J.l(Finite Element)~>~ i ~I I~ ..)J - ~~'11 Ifa~ .;.J~)J.l W~I ;\;J..ull. .s.f-l,1 ~~'11 ~l:--..)iJ~WI~~ .F - . . ) ~l:JI ~~'1IJ 'J h-Ij'1\
The constitutive equations describing the behavior of large elasto-plastic defor-
mations of solids have been proposed in a previous paper [ 1 1 .The problem of analysisof displacements, stresses and strains in elements made of this material subject to
arbitrarily large deformations under the conditions of plane strain has been formu-
lated in terms of the finite element method.
0377-9211/79/0104-0041$01.00
1979 by The University of Petroleum and Minerals
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THE FINITE ELEMENT METHOD APPLIED TO
LARGE ELASTO-PLASTIC DEFORMATIONS
OF SOLIDS
In this paper the finite element method is used in thecase of problems involving large elasto-plastic de-
formations of materials. Such materials display aninitial linear elasticity which is followed, at increasing
loading, by plastic strains with no rate, or viscous
effects. The topics included in this paper are:
I. The finite element formulation of the analysis of
two-dimensional problems.
2. Methods of solution of the equations resulting from
the use of the finite element method.
In this paper, the problem of analysis oflarge elastic-
plastic deformations will be formulated in terms of the
finite element method. Since the basic procedures of
the finite element method are described in numerous
papers and monographs (see, for example Reference
[2]), only the aspects related to the present problem
will be emphasized in this paper.
The fundamental equation in the following argu-ments is the principle oj virtual work (Equation (2.26),
Reference [1]:
L 8eTs dV - L 8uT fdV- I s 8uT P dS =0 (1)which is the condition of equilibrium in terms of the
displacement field u (x, y). In the finite element method,
the displacement field u is approx'imated by a discrete
model which contains a finite number of independent
nodal displacements. For this purpose, the regions of
integration in Equation (I) are divided into a finitenumber of subregions, or elements. Within each ele-
ment, the displacement field is approximated by known
functions which are continuous across the element
boundaries. For a nonlinear elastic body, for which the
stress vector is aJunction of the displacement gradients,Equation (l) results in a system of nonlinear algebraicequations. In the present problem of an elastic-plastic
body, the stress s is a Junctional of the displacement
gradients, Equation (l) leads to a system ofJunctional
equations for the nodal displacements.
Figure 1 shows a typical element mesh in an arbitrary
body with m elements and n nodal points. The compo-
nents of displacement at a node iare Ui and Vi' The
Figure 1, Typical Element Mesh; Notation for Global
Nodal Displacements
nodal displacement vector for the body, or the system,
is the 2n X 1 matrix q whose elements are
A typical element with its nodal displacements is
shown in Figure 2. The nodal displacement vector Qk
of the element k is
Since, at the corresponding nodes, the nodal dis-
placements of an element are identical with the nodal
displacements of the body, it is evident that
where the form of T k is implied by the definitions
ofqandql stated by Equations (3) and (4), respectively.
The displacement field within theelementkis approx-imated by linear functions of x andy,
Typical Triangular Element; Notation for
Element Displacements
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1u = - [ (a, + blx + c,y )u, + (a 2 + b 2x + C 2Y)U2
2,1
+ (a 3 + b 3x + C 3Y )U3]
1v =2 ,1 [ ( a l + blx + c l y ) v \ + (a 2 + b 2x + C2Y )V2
+ (a 3 + b 3x + C3Y )V3]
where
XI Y I
2,1 = 2 X (area of triangle) = det X 2 Y 2
X 3 Y 3
al =X2Y3 - X3Y 2 , bl =Y 2 - Y3, C, =X3 - X2
with a 2, b 2, C2, a 3, 3, and C3 obtained by cyclic per-mutation of indices.
In matrix notation, with
Uk = [~)b
N, = (a, + blX + c,y)/2,1, N2 = ... etc.
1 =[ 6 ~ JN k = (I N " 1N 2, 1N 3 ) '
Equation (6) becomes
Uk=Nkqk'
In view of Equation (5),
Uk=NkTkq
The components of strain are computed from thecomponents of displacement according to equation (8).
[ 11. The result is
where the matrix Bk corresponds to the linear terms
in the strain-displacement relations, while BZ takes
into account the nonlinear terms. It can be easily
verified that the matrices Bk a nd BZ are shown by
Equations (10) and (11), respectively.
1 [ bl 0 b2 0 b3 0J2,1 0 CI 0 C2 0 C3
CI b l C2 b 2 C3 b )
BZ= FkG kHk
(10)
(11)
[
b OlO b,O b 2 0 b 2 0 b 3 0 b 3 0 ]
C, 0 C, 0 C2 0 C2 0 C3 0 C3
C, b l CI b , C2 b 2 C2 b 2 C3 b 3 C3 b 3
u , 0 0 0
0 0 u , 0
0 V I 0 0
0 0 0 VI
U2 0 0 0
[ ~
0 b 2 0 b )
g JGk= 0 0 U2 0 Hk= bl 0 b 2 0
0 V2 0 0 CI 0 C2 0 C3
0 0 0 V2 0 C1 0 C2 0 C3
U3 0 0 0
0 0 U3 0
0 V3 0 0
0 0 0 V 3
In the following, the differential dek will be needed
rather than the strain e k itself. Differentiation of (9)
yields
consequently,
dek=(Bk + BZ) dqk=Bk dqk
dek=BkT k dq. (13)
From the stress-strain relation Equation (3.23),
Reference [ 1] ;
Sk=L DkBkdqkSubstitution of (8) and (13) into (1) results
k=m
8qTITif Bis,dVk~ 1 V k
k~m
-8qTITif NifdVk~ 1 V k
- 8qTI Tif Nip dS =0 (16)k Sk
The above equation is valid for any virtual 8q T;there-fore
'Im Ti f BiskdV-1mTiFk- kfn TiP k= Ok~ I V k k~ I k- I
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and the last summation in Equations (16) and (17)
extends only over the boundary elements.
Equation (17) is a nonlinear functional equation in q,
because Bk isa function of qk and, thus, of q,and s is afunctional of q as shown by(15).
The first term on the left-hand side of Equation (17)
will be denoted by Q (q); the second and the third terms
combined, by R. Thus, Equation (17) can be written as
Q;(q) = R i, i = 1,2, ... , 2n (18b)
The numerical solution of (18) or (19) will require the
derivatives
8Q i = ' Kij i = 1,2, ... , 2 n8q . 1 2 2
J J=" ... ,n
Differentiation of Equation (17)yields
k~m
dQ =ITJf (dBJ Sk +BJ dsk) dV (20)k~ I V
k
From (12), (10) and (11),
- 1dBk=dBk"=-
4 ,1 2
d( : ~) d( ~ ; ) 0 0
o 0 d (~: )d (: ; )d(~:) d(:;) d(:~) dG:)
[ b'0 b2 0 b3
~;]. 0 b, o b2 0c1 0 C2 0 c30 c1 0 c2 0 C3
and
dlF Sk =Ck dqk
where
[
bl 0 b2 0 b3 0J. 0 bl 0 b2 0 b]
C1 0 C2 0 C3 0
o C1 0 C2 0 C3
The first term in Equation (20) isthus,
L kdB! Sk dV=K~dq (23)
is the so-called initial stress, or geometric, stiffness
matrix of the element k. The second term in Equation
(20) can be written, with (14) as
f BJDkBk dqkdV =Kk dqkVk (25)
where
Kk =f
B"TDkB"dV + f (B"TDkB;;
Vk Vk
+ B;;TDkB" + B;;TDkB;;T)dV (26)
The matrix Kk willbecalled the material stiffness matrix
of the element k. (In the problems of nonlinear elasti-
city, the matrix Kk is known as the elastic stiffness
matrix.)
With (23),(25), and (5),(20) becomes
k=m
dQ =ITJ (KZ+ Kd Tk dq (27)k~1
and, hence, the matrix K, whose elements are K ij =
oQ) oqj, is
k=m
K =ITJ(K~ + Kk) Tk (28)k~ I
Kk =K~+ Kk
(21) is known as the tangent stiffness matrix of the element k.The matrix K defined byEquation (28) is theglobal tan-
gent stiffness matrix.
bl 0 C1 0
0 bl 0 C,Ck= , b2 0 C2 0
0 b2 0 C2b3 0 C3 0
0 b3 0 C3
The functional equations (18)are solved by applying
incremental steps ofloading and performing iterations
within each increment. The computational procedure
is then as follows:
Let q(n)and R(n)be the nodal displacements and the
right-hand side of (18),respectively, at the end of the
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nth load increment, for which
Q(q(n)) _ R(n) =O .
Let LlR be the next load increment, and let Llq be the
corresponding nodal displacement increment; they
satisfy
Q(q(n) + Llq) - (R(n)+ LlR) = O. (30)
The first approximation for Llq will be obtained by
taking the first term of the Taylor expansion of Q at
q(n), i.e., by replacing Equation (30) with
Q(q(n)) + Ko Llq - (R(n)+ LlR) =0
or, in view of (29),
KoLlq - LlR = 0
K
o=(~~) q~q(n)
is the tangent stiffness matrix at q = q(n) (the tangent
stiffness matrices have been discussed in Section 2;
see Equations (19), (27), and (28)).
From (31), the first approximation of Llq follows as
Llq(l) =K.;-ILlR.
If the above approximate value is substituted into
Equation (30), we have
Q(q(n) + Llq(I)) - (R(n)+ LlR) =-lJ'. (32)
The next correction to Llq is obtained by expanding Q
at q =q(n)+ Llq(I)' which results in
Q(q(n) + Llq(I)) + K(I)Llq - (R(n) + LlR) = 0
or, with (32) in
K(I)Llq =lJ'. (33)
where K(j) is the tangent stiffness at q =q(n)+ Llq(l)'From (33),
Llq =Llq(l) + Llq(2)' (35)
The ith correction to Llq follows, thus, from
KU-I) = K(q(n) + Llq(l) + + Llqu-I) (37)
lJ'U-l) = Q(q(n) + Llq(l) + + Llq(i-l))- (R(n)+ LlR) (38)
The value of Llq after i corrections is
Llq =Llq(l) + Llq(2)+ . .. + LlqU) (39)
The above procedure is formally identical with the
known Newton-Raphson method for the solution
of a system of nonlinear algebraic equations. The
convergence of this method for a system of nonlinearalgebraic equations has been widely discussed (see, for
example, [3]).
It should be kept in mind, however, that the equa-
tions of the present problem (Equations (18)) are not
algebraic equations in q. Consequently, many pro-
perties of the Newton-Raphson method, especially
those related to the nature ofthe approximation and to
the convergence of the process, are not transferable
to the case of Equations (18). The differences between
the Newton-Raphson method applied to a system of
nonlinear algebraic equations (e.g. of a nonlinear
elastic problem) and the system offunctional equations(as in the present problem) is illustrated in Figures
3a,b.
In the case of algebraic equations (Figure 3a), after
the first value Llq(l) has been found, the correspondiag
(b) Q
R (n +1)
Figure 3. Iteration Processfor (a) A Nonlinear ElasticSolid andfor (b) An Elasto-Plastic solid
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value of Q(q(n) + ,1q(J)) (point A) and the slope K1 for
the next step can be computed exactly. The process
is known to converge to the exact solution [31.
In the case of functional equations (Figure 3b), from
the value ,1q(I), the value of Q(q(n) + ,1q(I)) can bedetermined only approximately, i.e., thepointAcannot
be located exactly on the true curve Q(q), and the slopeKj for the next step is also only approximate. Con-
sequently, the convergence theorems of the Newton-
Raphson method, developed for systems of algebraic
equations, are not directly applicable to the present
problem. The same objections apply to the modified
Newton-Raphson method (in which Ki
is replaced by
the matrix Ko in every step of iteration) as well as toother iterative procedures developed for the finite
element analysis of elasto-plastic solids, such as the
'initial strain' method [4)and the 'initial stress' method[51.The treatment of the incremental Equations (31)
as a system of ordinary differential equations (proposedin Reference [9)) and the application of higher-order
numerical integration methods does not remove thedifficulties outlined above.
In spite of the lack of a formal proof of convergence,
the method described in this section has been shown to
yield results of remarkable accuracy. References[6-8) contain a detailed discussion and numerical
results which prove, at least heuristically, the validity
of the procedure.
The problem of analysis of displacements, stresses,
and strains in elements made of elasto-plastic metals,
subjected to arbitrarily large deformations under the
conditions of plane strain has been formulated in
terms of the finite element method. The resulting
system of integral equations for the nodal displace-
ment can be solved by using a combination of Euler's
forward integration with the Newton-Raphson itera-
tion at each step.
The work described in this paper seems to demon-
strate the feasibility of the theory of plasticity oflarge
deformations and the finite element technique in solv-
ing complex problems (i.e., any shape and any deform-
ation) of the mechanics of elasto-plastic metals. At the
same time, certain topics have been exposed as requir-
ing further extensive investigations. They are
1. Efficient methods of solution of the integral
equations resulting from the application of the
finite element technique to the problems of stressand strain analysis. Specially, reduction of
computer time and clarification of the nature of
convergence appear to be of utmost urgency.
2. Criteria for selection of optimal types of finite
elements. Clearly, the experience accumulated in
linearly elastic problems is not directly transferable
to the present problems.
REFERENCES
[I] G. Z. Voyiadjis, 'Constitutive equations for work-hardening elasto-plastic materials', The Arabian Jour-nal for Science and Engineering, 3, No. I (1977),pp.
19-27.[2] O.C. Zienkiewicz, The Finite Element Method in Engin-
eering Science. McGraw-Hill, 1971.[3] J. M. Ortega and W. C. Rheinboldt, Iterative Solution
of Nonlinear Equations in Several Variables.AcademicPress, 1970.
[4] J. H. Argyris, 'Elasto-plastic matrix displacementanalysis of three-dimentional continua', Journal of the
Royal Aeronautical Society,69 (1965), pp. 633-635.[5] O.C.Zienkiewicz, S. Valliappan and I.P. King,'Elasto--
plastic solutions of engineering problems; "InitialStress"; Finite Element Approach', InternationalJour-nal for Numerical Methods in Engineering, 1 (1969) pp.75-100.
[6] P. V. Marcal, 'A comparative study of numerical
method of elasto-plastic analysis', AIAA Journal, 6(1968), pp. 157-158.
[7] J. H. Argyris and D. W. Scharpf, 'Methods of elasto-plastic analysis', lSD, ISSe Symposium on Finite Ele-ment Technology,Stuttgart (1969).
[8] O. C. Zienkiewicz and G. C. Nayak, 'Elasto-plasticstress analysis. A generalization for various constitutiverelations including strain softening', International
Journal Numerical Methods in Engineering,S (1972), pp.113-135.
[9] Z. P. Bazant, 'Matrix differential equations and higher-order numerical methods for problems of non-linearcreep, viscoelasticity and elasto-plasticity', Inter-national Journal for Numerical Methods in Engineering,4 (1972), pp. 11-15.