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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL.
40, 1909—1922 (1997)
AN UPPER BOUND FINITE ELEMENT PROCEDURE FOR
SOLVING LARGE PLANE STRAIN DEFORMATION
CHUNG-LI HWAN
Department of Mechanical Engineering, Feng Chia ºniversity, 100 ¼enhwa Road, Seatwen, ¹aichung, ¹aiwan, R.O.C.
SUMMARY
A unique and robust upper bound finite element procedure is developed for the analysis of large plastic
deformation problems under plane strain condition. It can consistently treat problems with isotropic strain
varying materials. It can also effectively solve problems with any initial ‘guessed’ velocity field, even from an
random number generator. To explore and demonstrate the capability of this new approach, strip tension
and plane strain compression problems are solved. For validation, the computed results are compared with
existing analytical or experimental solutions in good agreement. The phenomenon of shear band formation
can be simulated and, as expected, is found to develop more distinctly in strain softening materials than in
perfectly plastic and strain hardening materials. ( 1997 by John Wiley & Sons, Ltd.
KEY WORDS :
upper bound; finite element; plane strain; large deformation
INTRODUCTION
Large plastic deformation problems of metal can generally be solved by two kinds of formulation,
known as solid formulation and flow formulation.1 In the analysis of metal-forming problems,
elastic strain of a deformed metal can be neglected because it is usually very small as compared to
plastic strain. The neglect of elastic strain can simplify the formulation by treating the deforming
solid as a non-Newtonian fluid. Formulation based on this simplification is known as flow
formulation.2~5 It can provide most important information accurately at a fraction of the
computing cost of solid formulation. However, flow formulation approach cannot predict the
complete stress—strain history, which can only be obtained by using solid formulation in terms of
finite strain elastoplastic incremental analysis.6~10
Classical limit analysis11,12 belonging to the category of flow formulation, it involves two
principles which lead to both the lower bound and the upper bound approaches. The former
predicts a load which is less than or equal to the exact limit load needed to enable the incipience of
plastic flow of metals. The solution from the lower bound approach must satisfy equilibrium
equations, a suitable yield criterion, and static boundary conditions. On the other hand, the upper
bound approach provides a load which is at least equal to or greater than the exact limit load, and
solves for a kinematically admissible velocity field which must satisfy kinematic boundary
conditions.
Like the classical approach,11,12 modern limit analysis uses a pair of related formulations to
bound the exact solutions from above and below. But it is more mature in theory and methodology. It uses an inequality form of constitutive relation and establishes a duality relation that
equates the least upper bound to the greatest lower bound.13,14 It applies computational
CCC 0029—5981/97/101909—14$17.50
( 1997 by John Wiley & Sons, Ltd.
Received 15 May 1995
Revised 8 July 1996
1910
C.-L. HWAN
optimization techniques to approach the corresponding maximum and minimum solutions,15
sometimes simultaneously. The rigorous convex analysis16 and computational optimization
techniques help to put modern limit analysis on a solid mathematical foundation.
Based on the recent advances in limit analysis mentioned above, static and kinematic solutions for many complex problems17~19 have been obtained with high degree of accuracy and
certainty of convergence. A kinematic solution can be interpreted either as a steady large
deformation flow solution from an Eulerian view point or an instantaneous velocity field in
a Lagrangian co-ordinate system. The latter interpretation enables the integration of velocity
field in a small time step to provide a corresponding displacement field, which in turn updates the
configuration of the deforming body and the computational grid system. A subsequent limit
problem is then solved for this new configuration. This updating process is repeated to form
a sequence leading to the solution of a large deformation problem and is hereafter called
sequential limit analysis.
Since the von Mises yield criterion is employed and expressed in an inequality form, stresses do
not appear in this upper bound formulation. As a result, neither complicated stress updating6~8
nor rigid zone treatment1 is needed in sequential limit analysis. The effect of material nonlinearity is incorporated in the analysis by using a yield criterion that varies step-by-step locally
with deformation history. Using such a stepwise model, not only strain hardening material but
also perfect plastic and strain softening materials can be solved consistently. Another major
advantage of sequential limit analysis is the global stability of computation even in the case of
increasing deformation under decreasing load, such as necking.
Like most flow formulations, sequential limit analysis cannot provide certain information, such
as elastic strain, residual stress and spring-back after load removal, which can only be obtained by
an elastoplastic incremental analysis. However, sequential limit analysis can provide most
important information with reasonable computing cost.
Using an inequality to express the von Mises yield criterion, we shall first present the primal
(lower bound) formulation for a solid undergoing large plastic deformation under plane strain
condition. Through use of the weak equilibrium statement, the general divergence theorem, the
constant shear parameter model for interface friction1 and a generalized Hölder inequality,20 we
then arrive at the dual (upper bound) formulation. A duality theorem is then stated to equate the
greatest lower bound to the least upper bound. Numerical procedure for the upper bound
formulation is then presented. Finally, examples related to strip tension and plane strain
compression tests are computed for demonstration.
LOWER BOUND FORMULATION
Lower bound problems arise naturally thus their mathematical formulation is called the primal.
Using equilibrium equations, static boundary conditions and a yield criterion, a lower bound
problem under the plane strain condition can be posed as follows.
P! pnn dS
maximize j"
1
subject to + · p"0
in D
p · n"0
on !
0
EpE )k'(e )
V
1
in D
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
(1)
( 1997 by John Wiley & Sons, Ltd.
FINITE ELEMENT PROCEDURE FOR LARGE PLANE STRAIN DEFORMATION
1911
where + · denotes the two-dimensional divergence operator, p is the symmetric 2]2 stress matrix
function, 0 is the two-dimensional null vector, D is the two-dimensional domain, ! is the traction
0
free boundary and the integral condition j": p dS represents the limit load applied on the
!1 nn
boundary ! by the applied normal stress p . In this study, j is the tensile load for the tension
1
nn
problem or the press load for the compression problem.
In equation (1), the von Mises yield criterion is generally modified to take the effect of strain
hardening or softening into account. EpE is the norm notation used to denote the von Mises
V
yield function of stress matrix function p. k is the yield stress in shear determined experimentally.
'(e ) is a strain hardening or softening function of e , which is the accumulated equivalent plastic
1
1
strain in a local sense. For a perfectly plastic material, '(e )"1.
1
DUALITY AND UPPER BOUND FORMULATION
The exact upper bound formulation under the plane strain condition can be derived through an
upper bounding process. Using the virtual work concept, the equilibrium equation in problem (1)
is satisfied in a weak sense such that
PPD u · (+ · p)dA"0
(2)
for all u"Mu(x, y), v(x, y)N3K, where u and v are components of the velocity vector u in
both x and y directions, respectively. The set K is the space of all kinematically admissible velocity functions, u, which satisfy the incompressibility and kinematic boundary
conditions.
After using a general divergence theorem and a sequence of mathematical manipulation,21
equation (2) gives
PPDp : eR dA#mk P! D u4 D dS
j"
(3)
&
where eR is the symmetric strain rate matrix, u is the tangential component of velocity vector along
4
the frictional boundary ! and 0)m)1 is the lubrication factor commonly used to model the
&
induced shear stress from friction, q "!mk sign (u ).
4
4
Consequently, the lower bound formulation defined by equation (1) can be restated as
maximize
j
PPDp:eR dA#mkP! Du4 D dS
subject to j"
&
+ · u"0
(4)
EpE )k'(e )
V
1
kinematic boundary conditions on LD
where + · u"Lu/Lx#Lv/Ly"0 denotes the incompressibility condition which is required in the
study of metal plasticity and LD is the total boundary of the domain D.
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
1912
C.-L. HWAN
Using a generalized Hölder inequality,20 Dp : eR D)EpE EeR E , and the von Mises yield criterion
V
"
defined in equation (1), j expressed by equation (3) is bounded such that
PPD p : eR dA#mk P! Du4 D dS
)
PPD EpEV EeR E" dA#mkP! Du4 D dS
)k
PPD '(e1)EeR E " dA#mkP! Du4 D dS
j"
&
&
&
"jM (u)
(5)
where jM (u) is an upper bound functional and EeR E is the dual norm of the strain rate matrix
"
function eR , which can be expressed, in terms of its components, as
EeR E "J(eR !eR )2#4eR 2
xy
"
xx
yy
SA
"
B A
B
Lu Lv 2
Lu Lv 2
#
!
#
Lx Ly
Ly Lx
(6)
through use of the flow rule associated with von Mises yield criterion.
Once having obtained the upper bound functional, the upper bound problem can be mathematically defined as the dual one which seeks the smallest value of jM (u) where u3K. The dual
problem is a constrained minimization one which can be posed as
minimize
jM
PPD '(e1 )SA Lx!Ly B #A Ly#LxB dx dy
#mk
P! Du4 D dS
Lu
subject to jM "k
Lv 2
Lu
Lv 2
(7)
&
Lu Lv
# "0
Lx Ly
kinematic boundary conditions on LD
Because the inequality remaining between both the upper bound and the lower bound
functionals is sharp (equality inclusive) as shown by equation (5), the upper bound problem yields
a weak duality relation to the lower bound problem. The weak duality relation establishes the
equivalence of the least upper bound to the greatest lower bound. This is mathematically known
as a saddle-point problem which can be stated as
maximize j(p)"j*"minimize jM (u)
(8)
where the exact solution j* can be obtained by either maximizing a lower bound functional j or
minimizing an upper bound functional jM . The minimizing of an upper bound functional jM is called
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
( 1997 by John Wiley & Sons, Ltd.
FINITE ELEMENT PROCEDURE FOR LARGE PLANE STRAIN DEFORMATION
1913
the upper bound approach in which the duality condition (equality) can be obtained by a proper
choice of K and a minimizer u3K.
NUMERICAL IMPLEMENTATION
To solve numerically the upper bound problem given by equation (7), the following procedure
is employed. First, through the use of the penalty function procedure22 to treat the incompressibility constraint, the constrained minimization problem is transformed into an unconstrained minimization one. Since a standard non-linear programming method is not applicable
to solve the unconstrained minimization problem which involves a functional with square
root integrand, a combined smoothing successive approximation method4 is then applied to solve
the unconstrained minimization problem iteratively by a sequence of quadratic programming
problems.
A quadratic programming problem in each iteration is then discretized using both the fournode quadrilateral element for the subdomains and the two-node line element for the frictional
subboundaries. Finally, by assembling all the element stiffness matrices and element velocity
vectors into their corresponding global positions,21 the approximate functional in the nth
iteration, jM , becomes a quadratic form in matrix notation as
n
jM "qT K q
n
n n n
(9)
where q is the global velocity vector in the nth iteration and K is the global stiffness matrix which
n
n
is symmetric, banded and positive definite.
Having obtained the quadratic functional through use of the finite element discretization, the
approximate problem in the nth iteration can be stated as
minimize jM
n
subject to jM "qT K q
n n n
n
(10)
kinematic boundary conditions on LD
which can be solved for the velocity vector q in terms of the optimality condition.
n
Since the procedure is iterative, a suitable criterion is needed to terminate the iteration.
A convergence criterion used in this study is the error norm of the velocity vectors, which is
defined as E "Eq !q
E /Eq
E . This norm is used to calculate the deviation measured in
u
n
n~1 2
n~1 2
the Euclidean space in terms of the calculated velocity vectors between two successive iterations
and required to decrease from iteration to iteration. When it is smaller than 10~5, the solution is
regarded as good enough and the iteration stops. The ‘converged’ jM and q are interpreted as the
n
n
limit load, jR, and the corresponding global velocity vector, qR, for the original unconstrained
minimization problem.
Both geometry and material property for the successive configuration are updated by using the
velocity field obtained from the current configuration. Geometry is updated by multiplying the
cnodal velocity vector with a small pseudo-time increment, dt, which is a scale factor related to
the step size, to obtain a small increment of the displacement vector at each node. The
displacement increment is then used to update the current configuration and computational grid
system. A sequence of updating leads to large deformation.
Material property for the new configuration is updated as follows. The velocity field is first used
to calculate the equivalent strain rate, which is defined as eN "J2/3(eR eR )1@2, at each Gaussian
ij ij
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
1914
C.-L. HWAN
integration point. The calculated equivalent strain rate is then multiplied by dt to yield a
small increment of equivalent plastic strain. The equivalent plastic strain increment is then
used to update the accumulated equivalent plastic strain at each Gaussian integration point,
which in turn gives the updated yield stress by an interpolation on a chosen material property
curve.
To proceed the analysis of large deformation, the pseudo-time increment, dt, must be carefully
selected. In general, dt, must be chosen small enough such that dtEMº*NE is much smaller than
=
¸, where ¸ is a characteristic length of the problem. This condition validates the small deformation theory. Computationally, the choice of dt may also be subject to other factors, such as the
time necessary for the next free node to contact the die or tool surface, the desired maximum
strain-increment, and the error in the volume constancy.1
RESULT AND DISCUSSION
Strip tension problems
Tensile specimens have been known to exhibit localization of plastic flow into neck-shaped
regions near the stage of maximum load in tensile tests for many years. Even when great care is
taken to achieve uniformity of material properties and symmetry of loading, the formation of
necking has been unavoidable. We wish to simulate this well-known necking phenomenon
computationally as the first test case for sequential limit analysis.
As shown in Figure 1, a subdomain of the strip, D , is chosen as the computational domain
abcd
with initial dimensions ¸]H and an aspect ratio ¸/H"6. Such a subdomain is discretized by
the finite element mesh system with 341 nodes and 300 elements. As the strip elongates, a neck is
expected to form at boundary a—d where symmetric boundary conditions and denser grids are
specified. Symmetric conditions are also specified for boundary a—b. Shear-free conditions are
specified for the right end b—c, where the tensile force, f, is applied and stress-free conditions are
specified for boundary c—d.
Using a perfectly plastic material in the analysis, an initial test yields the following results. Let
2h denote the neck thickness, d denote the displacement at the right end and k be the yield stress
in shear. The thickness ratio, h/H, and the normalized applied load, ""f/4Hk, are plotted
against d/¸ in Figure 2 in solid and dash curves, respectively. Since elastic strain is neglected and
perfectly plastic material is used, necking begins at the start of the deformation and the load
decreases with reduced neck thickness as expected. If the tensile stress in the neck cross-section
were uniformly equal to 2k as a state of uniaxial stress would be, we would have f /2h"2k and
the curves f/4Hk and h/H in Figure 2 would then coincide. The fact that the load curve is slightly
above the thickness curve requires an explanation. The reason is that the biaxial stress states near
the surface (point d) and the triaxial stress states near centre (point a) allow the tensile stress in the
Figure 1. Schematic diagram of a strip in tension
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
( 1997 by John Wiley & Sons, Ltd.
FINITE ELEMENT PROCEDURE FOR LARGE PLANE STRAIN DEFORMATION
1915
Figure 2. " and h/H as functions of d/¸ for a strip in tension
cross-section to be higher than 2k according to the von Mises yield criterion. The tensile stress is
maximum at the centre of the neck.23
A sequence of grids and equivalent plastic strain contours for the above problem are shown in
Figure 3. The top one is for the undeformed configuration (d/¸"0). The second one denotes
a deformed configuration (d/¸"0·025) in which the contour values of equivalent plastic strain
are 0·03, 0·06, 0·09, 0·12, 0·15 and 0·18, respectively. The next one depicts the deformed configuration (d/¸"0·05) with contour values 0·15, 0·3, 0·45 and 0·6. The fourth one is another deformed
configuration (d/¸"0·075) where strain contour values are 0·25, 0·5, 0·75, 1·0 and 1·25. The last
deformed configuration (d/¸"0·1) shows strain contours with values 0·4, 0·8, 1·2 and 1·6. Since
the largest contour value in all these configurations is at the centre of the neck and increases with
the deformation, it is evident that the phenomenon of strain localization is closely related to the
development of necking.
To investigate the effect of material property on the development of necking, a mathematical
model of strain varying materials, '(e )"p /p "1#Men , is used in the analysis. p is the
1
0
1
: 0
initial equivalent yield stress which is equal to 2k in plane strain condition, p is a subsequent
:
equivalent yield stress and e is the accumulated equivalent plastic strain. Choosing n equal to 1,
1
such a model describes a class of materials with slope M. The ones with positive slope M correspond
to linear hardening materials. On the other hand, those with negative slope M correspond to
linear softening materials. Perfectly plastic materials are modelled by choosing M equal to zero.
The normalized applied loads, ""f/4Hk, are plotted against d/¸ in Figure 4 for a class of
materials with different slope M. It is shown in Figure 4 that the phenomenon of necking develops
more quickly for softening materials than for perfectly plastic solids. The development of necking
in hardening materials is slower than in perfect plastic solids and the onset of necking may even
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
1916
C.-L. HWAN
Figure 3. Grids and equivalent plastic strain contours for a strip in tension
Figure 4. " as functions of d/¸ for materials with slope M in plane strain tension
be retarded due to the effect of strong strain hardening. For the same end displacement, d/¸,
softening materials tend to yield at the lower tensile load while hardening materials act to yield at
the higher tensile load as compared to perfectly plastic materials.
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
( 1997 by John Wiley & Sons, Ltd.
FINITE ELEMENT PROCEDURE FOR LARGE PLANE STRAIN DEFORMATION
1917
Figure 5. Grids and equivalent plastic strain contours for materials with slopes equal to !1, !0·5, 0, 0·5, 0·75,
respectively, in plane strain tension
To further investigate the influence of material property on the development of necking, the
deformed grids and equivalent plastic strain contours for five kinds of materials at the same end
displacement, d/¸"0·0375, are plotted in Figure 5. The top one is for M equal to !1; the second
one is for M equal to !0·5; the third one is for M equal to 0 (perfectly plastic materials); the
fourth one is for M equal to 0·5; the last one is for M equal to 0·75. Because of strain softening,
shear band localization appears distinctly in both softening materials (M"!1 and M"!0·5).
On the other hand, the effect of strain hardening tends to retard strain localization and yield
a more uniform deformation in the specimen before necking as illustrated by the last two plots of
Figure 5.
Block compression problems
Because necking in a tensile specimen destroys the homogeneous strain/stress field, thus
rendering the data useless beyond the point, compression tests are commonly used to determine
the flow curve up to high strains. However, there still exist problems associated with compression;
tests as well. Short specimens must be used to avoid buckling. The end of the specimen in contact
with the press heads must be lubricated to allow free-side expansion. The lubrication is never
perfect in reality. Poor lubrication will cause the mid-section of the specimen to bulge. By bulging,
the material closer to the centre moves outward faster than the material near the platens. This
again destroys uniaxiality. We wish to simulate this well-known bulging phenomenon computationally as the second test case for sequential limit analysis.
To facilitate the study of compression problems under the plane strain condition, a block
compressed by a pair of flat rigid platens with a compression force per unit thickness, f , is
schematically shown in Figure 6. A subdomain of the block, D , is chosen as the computational
abcd
domain with initial dimensions ¼]H and an aspect ratio ¼/H"1. Such a subdomain is
discretized by the finite element mesh system with 289 nodes and 256 elements. Symmetric
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
1918
C.-L. HWAN
Figure 6. Schematic diagram of a block in compression
Figure 7. " as a function of 1!h/H for perfectly plastic materials under plane strain compression
boundary conditions are specified at both boundaries a—b and a—d. Stress-free conditions are
specified for the boundary b—c. Mixed boundary conditions are specified for boundary c—d by
imposing displacement boundary conditions along the travelling direction of press heads and
frictional shear stresses along the interface between press heads and the workpiece.
Employing the shear parameter friction model, plane strain compression tests are simulated
under two extreme lubrication conditions (m"0 for perfect lubrication; m"1 for no lubrication)
for perfectly plastic materials. Let the distance between the press heads, 2h (0(h )H), be the
control parameter for the compression sequences and k denote the yield stress in shear. The
computed results are shown in Figure 7, where the normalized applied load, ""f /4¼k, is
plotted as a function of (1!h/H) for the cases m"0 and 1. The results for are verified by the
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
( 1997 by John Wiley & Sons, Ltd.
FINITE ELEMENT PROCEDURE FOR LARGE PLANE STRAIN DEFORMATION
1919
simple exact solution f"4¼Hk/h. As expected, the load—displacement curve for m"1 becomes
gradually higher than that for m"0 as deformation continues. The reason is that the effect of
friction have gradually caused severe distortion along the diagonals of the block, which requires
higher plastic work than that for a uniform deformation without friction.
In the simulation of plane strain compression of perfectly plastic materials under no lubrication, the occurrence of severe mesh distortion in the computational domain can cause deterioration of the computational results and even prohibit further analysis at some step. To ensure the
quality of our computed results and to continue on the analysis until h/H"0·5, mesh rezoning24
was performed at 1!h/H"0·3. Grids and equivalent plastic strain contours before and after
mesh rezoning are shown in Figure 8 for comparison.
Grids and equivalent plastic strain contours for m"1 at six different states of deformation
(1!h/H"0, 0·1, 0·2, 0·3, 0·4, 0·5) are depicted in Figure 9 for illustration. These plots show that
a slight bulge occurs during the deforming process together with a severe distortion along the
Figure 8. Grids and equivalent plastic strain contours before and after mesh rezoning
Figure 9. Grids and equivalent plastic strain contours for perfectly plastic materials under plane strain compression at
1!h/H"0, 0·1, 0·2, 0·3, 0·4 and 0·5, respectively
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
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C.-L. HWAN
diagonal line a —c. These plots also show that the stress-free edge near the corner of the block
progressively folds itself over and comes into contact with the press head due to large rotation.
Computationally, the phenomenon of contact cannot be treated in a continuous sense as it must
be physically. As a result of using a discrete contact model in computer program, several
discontinuities exists in the dotted curve in Figure 7.
To investigate the effect of strain hardening on strain localization and the formation of bulging,
aluminum is chosen for the simulation of block compression because it exhibits significant strain
hardening. The mechanical behaviour of aluminum is expressed by /(e )"p /p "(1#16·4e )0>25,
1
: 0
1
which is a mathematical model that relates equivalent yield stress, p , to the accumulated
:
equivalent plastic strain, e , for quasi-static compression.25 p is the annealed yield stress of
1
0
106 Mpa. The computed results are shown in Figure 10, where the normalized applied load,
""f/4¼k, is plotted as a function of (1!h/H) for the cases m"0 and 1. Due to the effect of
strain hardening, the computed applied loads for aluminum are higher than those for perfectly
plastic materials as shown in Figure 7.
Grids and equivalent plastic strain contours for aluminum under plane strain compression
with no lubrication are depicted in Figure 11 for illustration. These plots show that a more
significant bulging and a more uniform strain distribution occurs during the deforming process
than in the case of perfectly plastic materials. This is because strain hardening can effectively
prevent a concentration of strain from occurring and result in a more uniform strain distribution
and a more significant bulging.26 Because of a more uniform strain distribution, the simulation
can be performed up to more than 50 per cent deformation without severe mesh distortion.
Because of large rotation and friction, the maximum accumulated equivalent plastic strain for
each deformed configuration is found near the original corner point c.
Figure 10. " as a function of 1!h/H for aluminum under plane strain compression
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
( 1997 by John Wiley & Sons, Ltd.
FINITE ELEMENT PROCEDURE FOR LARGE PLANE STRAIN DEFORMATION
1921
Figure 11. Grids and equivalent plastic strain contours for aluminum under plane strain compression at 1!h/H"0, 0·1,
0·2, 0·3, 0·4 and 0·5, respectively
CONCLUSION
A new approach, sequential limit analysis, has been presented in this study for solving large
plastic deformation problems under plane strain condition. This method is an extension of
a modern limit analysis. It is not only mathematically concise in formulation but it is also
computationally easy to implement. Both plane strain tension and compression problems have
been studied by this method. The formation of shear band localization can be simulated in plane
strain tension and found to develop more distinctly in strain softening materials than
in both perfectly plastic and strain hardening materials. As a result, necking occurs earlier and
develops quicker in strain softening materials. In plane strain compression of perfectly plastic
materials, the formation of shear band may eventually lead to severe mesh distortion, which
requires the use of mesh rezoning if further computation is required. Because of the effect of strain
hardening, no severe mesh distortion occurs in the compression of aluminum and the analysis can
be performed up to more than 50 per cent deformation without mesh rezoning.
The present analysis is done without the consideration of formability, which is important in
practical metal-forming problems and can be evaluated if an adequate ductile fracture model is
chosen.
REFERENCES
1. S. Kobayashi, S. I. Oh and T. Atlan, Metal Forming and ¹he Finite Element Method, Oxford University Press,
New York, 1989.
2. C. H. Lee and S. Kobayashi, ‘New solution to rigid plastic deformation problems using matrix method’, Journal of
Engineering for Industry, 95, 865—873 (1973).
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
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C.-L. HWAN
3. O. C. Zienkiewicz and P. N. Godbole, ‘Flow of plastic and visco-plastic solids with special reference to extrusion and
forming processes’, Int. j. numer. methods eng., 8, 3—16 (1974).
4. W. H. Yang, ‘A variational principle for plastic flows’, Proc. 9th º.S. National Congress of Applied Mechanics, Cornell
University, ASME, 1982.
5. O. C. Zienkiewicz, ‘Flow formulations for numerical solutions of forming processes’, in J. F. T. Pittman et al. (eds.),
Numerical Analysis of Forming Processes, Wiley, New York, 1984.
6. E. H. Lee, ‘Elastic—plastic deformation at finite strain’, J. Appl. Mech., 36, 1—6 (1969).
7. H. D. Hibbit, P. V. Marcal and J. R. Rice, ‘A finite element formulation for problems of large strain and large
displacement’, Int. J. Solids Struct., 6, 1069—1086 (1970).
8. R. M. McMeeking and J. R. Rice, ‘Finite element formulation for problems of large elastic—plastic deformation’, Int.
J. Solids Struct., 11, 601—616 (1975).
9. A. S. Wifi, ‘An incremental complete solution of the stretch forming and deep drawing of a circular blank using
a hemispherical punch’, Int. J. Mech. Sci., 18, 23—31 (1976).
10. E. H. Lee, R. L. Mallett and W. H. Yang, ‘Stress and deformation analysis of the metal extrusion process’,
Comput. Methods Appl. Mech. Eng., 10, 339-353 (1977).
11. W. Prager and P. G. Jr. Hodge, ¹heory of Perfectly Plastic Solids, Chapman & Hall, London, 1951.
12. B. M. Fraeijs de Veubeke, ‘Upper and lower bounds in matrix structural analysis’, in B. M. Fraeijs de Veubeke (ed.),
Matrix Methods of Structural Analysis, Pergamon Press, Oxford, 1964.
13. R. Temam and F. Demengel, ‘Duality and limit analysis in plasticity’, in W. H. Yang (ed.), ¹opics in Plasticity, AM
Press, Ann Arbor, MI, 1991.
14. W. H. Yang, ‘A duality theorem for plastic plates’, Acta Mech., 69, 177—193 (1987).
15. D. G. Luenberger, ¸inear and Nonlinear Programming, Addison-Wesley, Reading, MA, 1984.
16. R. T. Rockaffellar, Convex Analysis, Princeton University Press, Princeton, 1970.
17. H. Huh, ‘Limit analysis in plane stress’, Ph.D. Dissertation, The University of Michigan, Ann Arbor, 1986.
18. K. H. Liu, ‘Limit analysis of plane strain extrusions’, Ph.D. Dissertation, The University of Michigan, Ann Arbor,
1986.
19. T. Tyan, ‘Simulation and optimization for metal forming processes: cutting, rolling and extrusion’, Ph.D. Dissertation,
The University of Michigan, Ann Arbor, 1990.
20. W. H. Yang, ‘On generalized Hölder inequality’, Nonlinear Anal., 16, 489—498 (1991).
21. C. L. Hwan, ‘Large plastic deformations by sequential limit analysis: a finite element approach with applications in
metal forming’, Ph.D. Dissertation, The University of Michigan, Ann Arbor, 1992.
22. O. C. Zienkiewicz and P. N. Godbole, ‘A penalty function approach to problems of plastic flow of metals with large
surface deformations’, J. Strain Anal., 10, 180—185 (1975).
23. H. L. Morrison and O. Richmond, ‘Large deformation of notched perfectly plastic tensile bars’, J. Appl. Mech,
971—977 (1972).
24. Jung-ho Cheng and N. Kikuchi, ‘A mesh rezoning technique for finite element simulation of metal forming processes’,
Int. j. numer. methods eng., 23, 219—228 (1986)
25. W. Johnson and P. B. Mellor, Engineering Plasticity, Ellis Horwood, Chichester, U.K., 1983.
26. R. Houlston, G. W. Vickers and D. L. Anderson, ‘A finite element and experimental study of rigid—plastic
compression’, Int. j. numer. methods eng., 23, 1407—1437 (1986).
.
INT. J. NUMER. METHODS ENG., VOL. 40: 1909—1922 (1997)
( 1997 by John Wiley & Sons, Ltd.
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