Page 1
A new algorithm for numerical solution of dynamic
elastic–plastic hardening and softening problems
Hongwu Zhanga,*, Xinwei Zhanga, Jiun-Shyan Chenb
aState Key Laboratory of Structural Analysis and Industrial Equipment, Department of Engineering Mechanics,
Dalian University of Technology, Dalian 116024, PR China
bDepartment of Civil and Environment Engineering, University of California, Los Angeles, Los Angeles, CA 90095-1593, USA
Received 25 September 2002; accepted 4 March 2003
Abstract
The objective of this paper is to develop a new algorithm for numerical solution of dynamic elastic–plastic strain
hardening/softening problems, particularly for the implementation of the gradient dependent model used in solving
strain softening problems. The new algorithm for the solution of dynamic elastic–plastic problems is derived based on
the parametric variational principle. The gradient dependent model is employed in the numerical model to overcome the
mesh-sensitivity difficulty in dynamic strain softening or strain localization analysis. The precise integration method,
which has been used for the solution of linear problems, is adopted and improved for the solution of dynamic non-
linear equations. The new algorithm is proposed by taking the advantages of the parametric quadratic programming
method and the precise integration method. Results of numerical examples demonstrate the validity and the advantages
of the proposed algorithm.
? 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Elasto-plasticity; Precise integration method; Parametric quadratic programming method; Gradient dependent model;
Dynamic response
1. Introduction
Traditionally, incremental iteration method is widely
used in non-linear problems. However, it often faces the
problem of low convergent speed, especially for the
super-non-linear problems such as post-buckling and
strain softening problems. Based on the parametric
variational principle [23,26], the parametric quadratic
programming method was developed as an effective
way to solve the non-linear problems. By introducing
mathematical programming method, the parametric
quadratic programming method avoids the iteration
procedures. For elastic–plastic problem, this method
avoids the limitation of the Drucker hypothesis. It can
also be applied to the non-associated plastic constitutive
model, non-normal sliding and strain softening problems
[20,23].
In the past several decades, many kinds of time in-
tegration methods (see [1,2]) have been proposed. Re-
cently, Zhong [25] proposed a precise integration
method, which has many advantages such as absolute
stability, zero-amplitude rate of decay, zero-period
specific elongation and non-overstep properties. This
method has been used successfully in many linear dy-
namic problems [9,12] and heat conduction [21] prob-
lems. The discussion of the method was recently given
by Zhang and Zhong [22] where the optimum parame-
ters selection was suggested. In this paper, the para-
metric variational principle is generalized for the
dynamic analysis of the elastic–plastic strain hardening/
softening problems. The parametric quadratic pro-
gramming method combined with the precise integration
method is adopted to solve the dynamic elastic–plastic
hardening/softening problems.
*Corresponding author. Tel./fax: +86-411-4708769.
E-mail address: zhanghw@dlut.edu.cn (H. Zhang).
0045-7949/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0045-7949(03)00167-6
Computers and Structures 81 (2003) 1739–1749
www.elsevier.com/locate/compstruc
Page 2
A large number of engineering materials can be
classified as softening materials in which localized large
strain are observed. This is called the strain softening or
localization problem [11,17]. This phenomenon often
acts as a precursor of structural failure and has a de-
terminant effect on the deformation of the structure.
Numerical simulation of such kind of non-linear prob-
lems is generally more difficult than that of the linear
dynamic problems [3]. It has been proved mathemati-
cally that the mesh sensitivity problem is exists when the
finite element method is carried out in the analysis of the
softening materials [13,16]. The dependence on the dis-
cretization is not only with respect to mesh refinement
but also with respect to mesh alignment. If the classical
constitutive model is adopted directly in these softening
materials, the initial value problem becomes ill-posed
and cannot discribe the underlying physics properly.
So far, the following three methods are the most ef-
fective in overcoming the above mesh sensitivity prob-
lem: rate-dependent model ([10,15,19], etc.), Cosserat
constitutive model ([14], etc.), non-local and gradient
model [3,6,7]. A reproducing kernel regularization
method [26] has been proposed as a generalization of
non-local and gradient models without the need of ad-
ditional boundary conditions. In this paper, the gradient
model is used to overcome the mesh sensitivity problem.
As the Laplacian of the hardening/softening parameter
is embedded in the material constitutive equations, we
obtain not the algebra equations but the differential
equations from the constitutive equations. This re-
markably increases the complexity in the numerical
implementation [6] of the algorithm. de Borst and
Muhlhaus [6] solved the problem successfully by intro-
duction of unknown quantities (displacement and plastic
multiplier) in each node. One important work in this
paper is to develop a new algorithm for the implemen-
tation of the gradient dependent model in the finite
element analysis.
Section 2 of this paper discribes the gradient depen-
dent model and the corresponding formulation. Section
3 presents the parametric variational principle and the
parametric quadratic programming method for the
gradient dependent model in the dynamic analysis. In
Section 4, the Newmark integration method is adopted
in the discretization of time domain. In Section 5, the
precise integration method is introduced in the numeri-
cal solution of the dynamic elastic–plastic hardening/
softening equations. Finally, numerical examples are
given to verify the theory and algorithm proposed in this
paper.
2. Formulation of the gradient dependent model
In the conventional plasticity theory, the yield func-
tion f depends on the parameters of stress r, plastic
strain epand some internal variables. Without loss of
generalities, the isotropic hardening/softening materials
is considered here, and the yield function can be defined
as
f ¼ fðr;ep;jÞð1Þ
In gradient dependent model, the yield function can be
written as
f ¼ fðr;ep;j;r2jÞð2Þ
where j is the hardening/softening parameter in the
constitutive model.
It can be seen that the main difference between the
conventional plasticity theory and the gradient depen-
dent plasticity model is the introducing of the gradient
of softening/hardening parameters in the yield function.
Therefore, in gradient dependent plasticity, the yield
status of a material point is not only related to its own
plastic parameters but also under the influence of the
plastic parameters in the neighboring region. The size of
the influence region is determined by the internal length
scale in the gradient dependent model. According to the
gradient dependent model, the plastic deformation in a
point will expand to a certain region.
Based on the gradient dependent model, the elastic–
plastic constitutive equations can be defined as follows
dr ¼ Dðde ? depÞ;
dep¼
og
or
??
k
ð3Þ
fðr;ep;j;r2jÞ60;
k ¼
P0
¼ 0
when f ¼ 0
when f < 0
?
ð4Þ
where k is the plastic flow multiplier; g is the plastic
potential surface. The Von-Mises yield function f is
adopted in the numerical examples which will be shown
in Section 6.
Without loss of generality, we assume dj ¼ hk,
? c c ¼ hðof=ðor2jÞÞ, and h is the hardening/softening
modulus.
Applying Taylor expansion to Eq. (4), we obtain the
consistent equation
f0þ Wde ? Mk þ? c cr2k60;
kP0
ð5Þ
where
W ¼
of
or
?
??T
og
or
D;
M ¼ W
?
?
of
oep
??T
og
or
??T
þof
ojh
ð6Þ
in which ? c c is the parameter of the gradient dependent
model or the internal length scale parameter.
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H. Zhang et al. / Computers and Structures 81 (2003) 1739–1749
Page 3
3. Parametric variational principle for elastic–plastic
problems with the gradient dependent model
3.1. Parametric variational principle
The parametric variational principle is the applica-
tion of optimum system control theory to the unspeci-
fied boundary value problems in continuum mechanics.
In this principle, the constitutive relations of the physical
phenomena are taken into account by means of the se-
lection of some proper state and control variables such
as those used in a control system. Thus the parametric
variational method can be used for solution of the
problems where the conventional variational principles
are not being successful, and it also simplifies the solu-
tion process.
The parametric variational principle contains two
kinds of state variables in its variational function. The
first kind are variables such as the displacements of the
structure which will take part in the variation process,
whereas the other one, such as the plastic multipliers is
taken as a control variable during the variational pro-
cess, and is determined by the minimization/maximiza-
tion of the variational function. On the other hand, the
material constitutive relations work just as a control
system in the boundary value problems during the whole
variational process. The differences between the para-
metric variational principle and the conventional varia-
tional principle can be shown by the following figure:
δΠ 0
=
Conventional variational
principle
δΠ [λ( )]=0
Constitutive
control system
Parametric variational
principle
.
The parametric minimum potential energy principle
based on the gradient dependent model for the dynamic
non-linear problems can be discribed as follows: for all
of the possible incremental displacement solutions which
satisfy the strain–displacement relations and displace-
ment boundary conditions, the exact solution minimizes
the potential energy of the system
Zt2
P ¼
t1
Z
X
1
2d_ u uiqd_ u ui
?
(
?1
2dui;jDijklduk;l
þ kiRkliduk;lþ dbidui
?
dX ?
Z
Cp
d? p piduidC
)
dt
ð7Þ
at the control of the system state equations
f0þ Wde ? Mk þ? c cr2k þ t ¼ 0;
kTt ¼ 0;
k;tP0
ð8Þ
Here, biis the body force, Rklm¼ ðogm=orijÞDijkl, and t is
the slack vector. This problem can be stated as
min:
P½kð?Þ?
fðdu;k;r2kÞ þ t ¼ 0;
ð9Þ
s:t:
kTt ¼ 0;
k;tP0
ð10Þ
where k is the parametric variable which does not take
part in the variation process but controls the system
state varying between elastic and plastic ones. du is the
incremental displacement vector, and dp is the load
vector. Eq. (10) is the system control equation derived
from the constitutive relations.
3.2. Parametric quadratic programming method
To solve the non-linear problem, the general algo-
rithm is to take the linearization of the non-linear
equations in conjunction with the incremental iteration.
In contrast, the parametric quadratic programming
method adopts the algorithm of the programming the-
ory in the algebra solution without iteration processes.
A detail discription and summary of the discretization
procedure can be found in the work by Zhang et al. [24].
The discretized finite element equations of the non-linear
dynamic problem can be expressed as
Md€ u u þ Kdu ? kU ¼ dP
Cdu ? Uk ? d þ t ¼ 0
kTt ¼ 0;
where
k;tP0
8
:
<
ð11Þ
M ¼
Z
Z
X
qNTN dX;
K ¼
Z
X
BTDBdX;
dP ¼
Cp
NTd? p pdC
ð12Þ
are the mass matrix, stiffness matrix and load vector,
respectively. These matrices and vector have the same
meaning as those of the conventional finite element
method. The new matrices and vectors generated by the
parametric variational principle are
U ¼
Z
X
Nu
L;jDe
ijkl
ogb
orij
dX;
C ¼
Z
X
WklbNu
m;ldX
ð13Þ
U ¼
Z
XðMNk
mþ? c cr2NkÞdX;
d ¼ ?
Z
X
f0dX
ð14Þ
where U is the plastic potential matrix which represents
the plastic potential of the system, C is the constrained
matrix which represents the constraint status, and U is
the hardening matrix which indicates the hardening
status. For the associated flow rule, the plastic potential
matrix is the transposition of the constrained matrix, i.e.
U ¼ CT. d and t are the constraint and slack vectors,
and k is the parametric vector whose physical meaning is
the plastic flow parameter.
It can be seen clearly that the introduction of the
gradient dependent model only adds the gradient item
H. Zhang et al. / Computers and Structures 81 (2003) 1739–1749
1741
Page 4
into the hardening matrix. In the parametric quadratic
programming algorithm, quadratic interpolation for the
parametric k is needed.
If the damping effect is considered, the damping
matrix needs to be added in the dynamic equation, we
have
Md€ u u þ Gd_ u u þ Kdu ? kU ¼ dP
ð15Þ
where G is the damping matrix.
The incremental displacement can be solved with the
discretization of the dynamic equation in time domain.
Substituting the incremental displacement into the con-
trol equation results in a quadratic programming prob-
lem which can be solved by many methods such as the
Wolf method and Lemke method [4,8].
4. Discretization in time domain with the Newmark time
integration algorithm
Discretization in time domain of dynamic equation
(15) is carried out by means of the Newmark scheme at
first. The algorithm can be expressed as
_ u utþDt¼ _ u utþ ½ð1 ? cÞ€ u utþ c€ u utþDt?Dt
utþDt¼ utþ _ u utDt þ ½ð0:5 ? bÞ€ u utþ b€ u utþDt?Dt2
?
ð16Þ
where c, b are the integration parameters of the New-
mark scheme.
From Eq. (16) the incremental formulations of the
Newmark scheme can be obtained by the following ex-
pressions
d_ u u ¼ ½ð1 ? cÞ€ u utþ c€ u utþDt?Dt
du ¼ _ u utDt þ ½ð0:5 ? bÞ€ u utþ b€ u utþDt?Dt2
?
ð17Þ
Then the incremental velocity and acceleration are of the
following formulations
D€ u u ¼
D_ u u ¼ € u utDt þ
1
bDt2½Du ? _ u utDt ? 0:5Dt2€ u ut?
c
bDt½Du ? Dt_ u ut? 0:5Dt2€ u ut?
(
ð18Þ
Substituting Eq. (18) into (11), we obtain
KDu ? kU ¼ F
ð19Þ
where
K ¼ K þ
F ¼ DF þ M
1
bDt2M þ
h
c
bDtG
1
bDt_ u utþ1
2b€ u ut
i
þ G
c
b_ u utþ
c
2b? 1
??
Dt€ u ut
hi
(
ð20Þ
are the effective stiffness matrix and effective load vector
respectively.
The dynamic elastic–plastic problem is now changed
into the following quadratic programming problem
Kdu ? kU ¼ F
Cdu ? Uk ? d þ t ¼ 0
kTt ¼ 0;
which is a generalized formulation of a linear comple-
mentary problem.
k;tP0
8
:
<
ð21Þ
Remark 1. What difference between the algorithm de-
veloped here and the conventional iteration method is
that the iteration progress in the new algorithm is per-
formed by the solution of the parametric programming
problem (21). For instance, in the well known Lemke?s
algorithm, the base/pivot exchange is generally needed
so that the complementary conditions can be satisfied.
The physical meaning of this base/pivot exchange is just
like the re-calculation of the element stress state after the
residual force is computed. This is the reason why the
important iteration procedure and consistent tangent
matrix (see [2,18]) are not calculated explicitly in the
method proposed here.
5. Precise integration method in time domain
5.1. General scheme of the precise integration method for
solution of dynamic elastic–plastic problems
The precise integration method is a new algorithm
for numerical solution of differential equations. It has
the absolute stability, zero-amplitude rate of decay,
zero-period specific elongation and non-overstep prop-
erties. We extend this method here to the numerical
solution of dynamic elastic–plastic problems.
Considering the dynamic equation
M€ u u þ Ku ? kU ¼ F
and combining with the identical equation f_ u ug ¼ f_ u ug,
we obtain the following differential equation
ð22Þ
_V V ¼ HV þ r þ U?k?
where
ð23Þ
V ¼
u
_ u u
?
? ?
;
H ¼
0I
?M?1K
;
k?¼
0
k
?
?
?
;
r ¼
0
M?1FðtÞ
??
;
U?¼
00
M?1U
0
0
? ?
ð24Þ
The homogeneous solution of Eq. (23) is
VðtÞ ¼ ½TðsÞ?C
where
ð25Þ
½TðsÞ? ¼ expð½H? ? sÞ
In the integrative step t 2 ½tk;tkþ1?, s ¼ t ? tk, C is a
constant vector and decided by the initial conditions of
the incremental step.
ð26Þ
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H. Zhang et al. / Computers and Structures 81 (2003) 1739–1749
Page 5
The key step in Eq. (26) is the evaluation of the ex-
ponential matrix T. Precise integration method provides
a scheme (2Nalgorithm) to compute the exponential
matrix precisely. At first, the evaluation of the expo-
nential matrix of (26) is evaluated as
T ¼ eA¼ ðeA0Þm
ð27Þ
where
A0¼ A=m;
A ¼ Hs;
m ¼ 2N
N is a numerical parameter, and a constant value N ¼ 20
was proposed in Zhong [25]. Then eA0is evaluated by the
power series (pP1)
eA0ffi
X
p
i¼0
A0i
i!¼ I þ Ta;
Ta¼
X
p
i¼1
A0i
i!
ð28Þ
From Eqs. (27) and (28), the computation matrix T can
be furthermore expressed as
T ¼ eA¼ ðI þ TaÞ2N¼ ðI þ TaÞ2N?1ðI þ TaÞ2N?1
¼ ðI þ 2Taþ T2
Note that in the precise integration method the unit
matrix I should not be directly included in the compu-
tation of (29), as to reduce the round off error. The
following procedures are recommended
aÞ2N?1
ð29Þ
Step I:
Ta( 2Taþ T2
and after N times loop, the exponential matrix will be
obtained by
a
ð30Þ
Step II:
T ¼ eA¼ I þ Ta
The particular solution of the Eq. (23) is
ð31Þ
VPðtÞ ¼ ?H?1½r0þ H?1r1þ r1ðt ? tkÞ?
? H?1½U?k?
0þ H?1U?k?
1þ U?k?
1ðt ? tkÞ?ð32Þ
The general solution of Eq. (23) is
VðtÞ ¼ ½TðsÞ?ðVðtkÞ ? VPðtkÞÞ þ VPðtÞ
where r0, r1, k0and k1are constants.
Substituting Eq. (32) into (33) and letting Dk?¼
k?
ð33Þ
1ðt ? tkÞ, s ¼ t ? tk, the general solution is obtained
Vðtkþ1Þ ¼ TaH?1H?1U?Dk?1
þ TaH?1U?k?
s
?
? H?1U?Dk?
?
0þ Vðtkþ1Þð34Þ
where
Vðtkþ1Þ ¼ T½VðtkÞ þ H?1ðr0þ H?1r1Þ?
? H?1½r0þ H?1r1þ r1s?ð35Þ
Ta¼ T ? I
when the structure is linear elastic, the parametric vari-
able k is zero. Then the above integrative scheme (34)
will reduce to Vkþ1which is the linear solution shown in
Zhong [25] and Zhang and Zhong [22].
ð36Þ
5.2. Solution technique of the method proposed
From Eq. (34), the incremental solution of the
problem (23) is
DVðtkþ1Þ ¼ Vðtkþ1Þ ? VðtkÞ
¼ TaH?1H?1U?Dk?1
þ TaH?1U?k?
In the precise integration method, the dimension of the
status vector Vkþ1 is doubled so that the dynamic
equation can be reduced into first order differential
equation. It is therefore that the dimension of the system
control equation needs also to be doubled
s
?
? H?1U?Dk?
?
0þ Vðtkþ1Þ ? VðtkÞð37Þ
C?DVðtkþ1Þ ? U?Dk?? d?þ t?¼ 0
where
ð38Þ
C?¼
C
0
?
0
0
??
;
U?¼
U
0
t
0
0
0
??
;
d?¼
d
0
? ?
;
Dk?¼
Dk
0
?
;
t?¼
? ?
ð39Þ
Substituting the incremental general solution (37) into
the system control equation (38), we have
t?þ C?TaH?1H?1U?1
¼ d ? C?½Vkþ1? Vk? ? C?TaH?1U?k?
where
s
?
? H?1U?
?
Dk?? U?Dk?
0
ð40Þ
C?TaH?1H?1U?1
s
?K?1U
?
? H?1U?
?
¼
?C
Ta12
s? I
?
0
??
ð41aÞ
Vðtkþ1Þ ? VðtkÞ ¼
? u uðtkþ1Þ ? uðtkÞ
?_ u u _ u uðtkþ1Þ ? _ u uðtkÞ
??
ð41bÞ
C?TaH?1U?k?
0¼
?CTa11K?1Uk0
0
??
ð41cÞ
For the numerical implementation, the above equation
can be rewritten as the following form
H. Zhang et al. / Computers and Structures 81 (2003) 1739–1749
1743
Page 6
t
0
? ?
þ
?C
Ta12
s? I
?
?
?K?1UDk
?
?
0
??
?
UDk
0
??
¼
d
0
? ?
?
?
C
0
0
0
ð? u ukþ1? ukÞ
ð?_ u u _ u ukþ1? _ u ukÞ
??
?
?CTa11K?1Uk0
0
ð42Þ
5.3. Implementation of the algorithm
The implementation of the algorithm proposed can
be concluded briefly as what follows:
(1) Forming the matrices H, H?1and T, the load vectors
r0and r1, generating the sub-matrices Ta11, Ta12and
Ta22of the matrix T.
(2) Time integration for each time step.
(1) Forming element and global plastic potential
matrix U, constraint matrix C, hardening matrix
U and constraint vector d.
(2) Calculating Vðtkþ1Þ, which corresponds to the
elastic solution.
(3) Calculating DVkþ1from Eq. (41b) and generating
the incremental displacement vector ? u uðtkþ1Þ?
uðtkÞ.
(4) Calculating the coefficient matrices in Eq. (42),
substituting the incremental displacement vector
? u uðtkþ1Þ ? uðtkÞ into Eq. (42), and taking compu-
tation of items at the right side.
(5) Solving quadratic programming problem (42)
and obtaining the incremental parametric vari-
able Dk.
(6) Substituting the incremental parametric variable
Dk into Eq. (34), obtaining the status variable
and displacement vectors of the current step.
(7) If the total integration steps have not been fin-
ished, then returning to step (1) and going on
the computation of the next time step. Other-
wise, the computation is completed.
(3) Stop.
Remark 2. It is worthwhile to take an analysis and
comparison about the time cost and operation number
of matrix products used respectively in the precise and
Newmark integration algorithms. It should be noticed
that the matrices H, T and K used in the precise inte-
gration method are constant during the integration
process step by step. So some matrices in Eq. (42) can be
calculated at the first time integration step and can be
used directly in the steps after that. Comparing with the
operation number used in the Newmark method, the key
part in the precise integration method is in the compu-
tation of (41b) where the incremental displacement and
velocity are performed. So the operation in Eq. (35) is
rather important for the computation cost of the algo-
rithm. For a very general load case, i.e. that the r0and r1
are time dependent, the operation number of matrix
product will be increased by a factor of 4. In this way,
the computation cost will be greater than that used by
the Newmark method. However, for some special load
cases, such as when r0and r1are constant, the operation
number will be the same as that of the Newmark
method. On the other hand, due to the high accuracy of
the precise integration method, generally, the time step
size can be greater than that permitted for Newmark
method. Thus, we can conclude that the precise inte-
gration method is more effective particularly when the
high accuracy results are needed in the numerical sim-
ulations. The disadvantages of the method are that for a
general case of the problem, the time cost can be larger
than that needed when the Newmark method is used.
Remark 3. It will be also noticed from above discription
that the computation cost for the programming method
developed here are almost the same for the solution of
the plastic models with or without gradient dependent
item. This is due to the fact that only the matrix U in Eq.
(14) is modified when the gradient dependent model is
adopted. This additional calculation is only performed
on the element level when the element matrix U is gen-
erated.
6. Numerical examples
Example 6.1 (One truss structure is showed in Fig. 1).
The length of top, bottom and vertical trusses is 1 m.
The cross sections of all trusses are 1.0?10?3m2. The
material parameters are E ¼ 210 GPa; q ¼ 7800 kg/m3.
The load applied on the truss joint is 2.0?103N as
shown in Fig. 1. We consider here the simple hardening
material strain–stress relation as plotted in Fig. 2. The
yield stress is 4.0?106Pa. Because the first order of free
vibration period of the structure is 0.01335 s, the length
of time step are respectively selected as 3.0?10?4,
1.5?10?4and 0.75?10?4s. The corresponding num-
bers of the time step are 30, 60 and 120. With the precise
integration method, the displacement of Y direction of
the point where the load is applied is calculated in the
situations of the elasticity and plasticity respectively.
The displacement results are showed in Figs. 3–5 where
the value of total time is the same but divided into dif-
X
Y
Fig. 1. Two-dimensional truss model.
1744
H. Zhang et al. / Computers and Structures 81 (2003) 1739–1749
Page 7
ferent time steps. It can be seen that the results of the
precise integration method are less dependent on the
length of time step than that of the Newmark method.
As it is showed in Fig. 5, the results of the Newmark
method with 180 steps is rather closed to that obtained
by the precise integration method with 30 time steps.
This means the precise integration method has more
advantage on the computation accuracy. For the com-
putation efficacy, the precise integration method with
120 time steps has the same computational time as the
Newmark method with 180 steps. If we use the precise
integration method with 30 steps, the computational
accuracy is closed to the Newmark method with 180
steps and the computational time is reduced remarkably.
Example 6.2 (Dynamic response of an elastic–plastic
cantilever plate). It is shown in Fig. 6 that a cantilever
plate works with a jump load on the top surface. The
curve of strain–stress relationship is the same as shown
in Fig. 2. Von-Mises constitutive model is adopted. The
material parameters are: Young?s modulus E ¼ 20:5
GPa, mass density q ¼ 7:8 ? 103kg/m3, Poisson ratio
t ¼ 0:3, plastic yield stress is 20 GPa, plate thickness is
5 mm. 30, 60 and 120 time steps are calculated and
the lengths of time step are respectively selected as
4.0?10?3, 2.0?10?3and 1.0?10?3s. The displacement
with time variation in vertical direction of point A is
shown respectively in Figs. 7–9. Fig. 7 shows the com-
parison among the results obtained by the precise inte-
gration method with different lengths of integration time
step. Fig. 8 further gives the corresponding results ob-
tained by the Newmark integration method. Fig. 9
shows the comparison between the results obtained re-
spectively by the precise integration method with large
time step and the Newmark one with small time step.
0
ε
σ
E
E ’=1/10E
Fig. 2. Strain–stress relationships for two-dimensional truss
and elastic–plastic cantilever plate.
-1.6E-04
-1.4E-04
-1.2E-04
-1.0E-04
-8.0E-05
-6.0E-05
-4.0E-05
-2.0E-05
0.0E +00
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
t(s)
Displacement
Step=30
Step=60
Step=120
Fig. 3. Results obtained by the precise integration method with
different lengths of time step (total time¼0.009 s).
-1.6E-04
-1.4E-04
-1.2E-04
-1.0E-04
-8.0E-05
-6.0E-05
-4.0E-05
-2.0E-05
0.0E +00
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
t(s)
Displacement (m)
Step=30Step=60Step=120
Fig. 4. Results obtained by the Newmark method with different
lengths of time step (total time¼0.009 s).
-1.6E-04
-1.4E-04
-1.2E-04
-1.0E-04
-8.0E-05
-6.0E-05
-4.0E-05
-2.0E-05
0.0E+00
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
t(s)
Displacement(m)
Precise-Step=30Newmark-Step=180
Fig. 5. Comparison between the results obtained by the precise
integration method with 30 time steps and the Newmark
method with 180 time steps.
150mm
50mm
A
Fig. 6. An elastic–plastic cantilever plate.
H. Zhang et al. / Computers and Structures 81 (2003) 1739–1749
1745
Page 8
Obviously, it can be observed that the results with the
precise integration method are not so sensitivity to the
length of time step as those obtained by the Newmark
algorithm. With a large length of time step, the precise
integration method can obtain high accuracy results as
those obtain by Newmark integration method with a
small length of time step. This is the same as that ob-
tained in the previous example.
Example 6.3 (One-dimensional bar in the tension and
strain softening state). The strain softening problem is
sketched in Figs. 10–12. Load: q0¼ 0:75 N; material:
E ¼ 20 GPa, q ¼ 2000:0 kg/m3, h ¼ ?2:0 GPa. The yield
stress in Von-Mises plasticity model is 2.0 MPa;
? c c ¼ 5 ? 104N.
The mesh sensitivity problem can be checked easily
by the results obtained by this example. The bar is di-
vided into 10, 20 and 40 elements respectively. In Fig. 13
the strain results at t ¼ 4:2 ? 10?5s with the different
meshes are plotted after the dynamic wave reflects from
the left boundary. Mesh sensitivity results are obvious:
strain localization and the width of the localization zone
decreases when more elements are used.
The problem is now computed with the gradient de-
pendent model. The bar is divided into 20, 40, 80 and
160 elements respectively. Parameters h ¼ ?2:0 GPa,
? c c ¼ 5 ? 104N and the length of time step¼1.5?10?7s.
The strain localization along the bar at t ¼ 1:8 ? 10?5s
is given in Fig. 14. It can be seen that the width of the
plastic zone keeps constant with the different mesh.
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.000.020.040.060.080.100.12
t(s)
Displacement (mm)
Step=30Step=60 Step=120
Fig. 7. Comparison among results obtained by the precise in-
tegration method with different lengths of time step.
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.00 0.020.040.06
t(s)
0.080.100.12
Displacement (mm)
Step=30Step=60Step=120
Fig. 8. Comparison among results obtained by the Newmark
integration method with different lengths of time step.
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
0.000.02 0.040.06
t(s)
0.080.10 0.12
Displacement (mm)
Precise-Step =30Newmark-Step =120
Fig. 9. Comparison between results obtained by the precise and
Newmark integration methods with large and small lengths of
time step.
L=100mm
F(t)
A=lmm2
Fig. 10. One-dimensional bar in the tension and softening state.
t
F(t)
q0
Fig. 11. Load–time relationship for one-dimensional bar.
ε
σ
E
h
Fig. 12. Stress–strain relationship for one-dimensional bar.
1746
H. Zhang et al. / Computers and Structures 81 (2003) 1739–1749
Page 9
From l ¼
parameter l ¼ 5 mm. The corresponding width of the
localization zone is 15.7 mm which is just the half value
of 2pl. The similar results have been also announced in
the work by de Borst and Muhlhaus [6].
In Fig. 15, the strain localization results along the bar
with the different internal length scale parameters are
given. The values of ? c c are 1.25?104, 2.5?104, 5.0?104,
10.0?104, 20.0?104N respectively, and the corre-
sponding internal length scale values obtained are
ffiffiffiffiffiffiffiffiffiffiffi
?? c c=h
p
, the material internal length scale
shown in Fig. 16 from 2.50 to 10.0 mm. The numerical
results are rather closed to the theory solutions with the
different internal length scale parameters.
Example 6.4 (Impact test of a double-notched specimen).
The geometry of the problem is shown in Fig. 16. The
load–time relation is plotted in Fig. 17. Von-Mises
plasticity model is used and the material parameters are:
q0¼ 3:5 ? 106N/m, t0¼ 3:5 ? 10?5s; E ¼ 40:7 GPa,
q ¼ 2350:0 kg/m3, h ¼ ?2:5 GPa. The yield stress of the
material is 4.0 GPa; ? c c ¼ 5 ? 104N. Two kinds of finite
element meshes are adopted in the computation.
In Fig. 18a and b, the results of structural deforma-
tion at time 5.0?10?5s with different meshes based on
the conventional constitutive relation are given. It can
be seen clearly that the differences between the results of
these two meshes, especially in the notched area where
the material changes into softening. The structural de-
formation results at time 5.0?10?5s with different
meshes based on the gradient dependent model are given
in Fig. 19a and b. The results of the coarse mesh is
closed to the fine mesh?s, especially in the notched area.
This embodies again the advantages of the gradient
dependent model.
0.0E+00
1.0E-04
2.0E-04
3.0E-04
4.0E-04
5.0E-04
6.0E-04
7.0E-04
8.0E-04
0 10203040
x(mm)
506070 8090100
Strain
20 Elements
40 Elements
80 Elements
160 Elements
Fig. 14. Strain localization along the bar obtained by the dif-
ferent meshes with the gradient dependent model.
0.0E+00
2.0E-03
4.0E-03
6.0E-03
8.0E-03
1.0E-02
1.2E-02
1.4E-02
1.6E-02
1.8E-02
2.0E-02
010 2030 405060708090 100
x(mm)
Plastic Strain
10 Elements
20 Elements
40 Elements
Fig. 13. Mesh sensitivity problem in one-dimensional case.
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
010203040
x(mm)
5060708090 100
Strain
l=2.50mm
l=3.54mm
l=5.00mm
l=7.07mm
l=10.0mm
Fig. 15. Numerical results with different internal length scale
parameters.
90mm
45mm
x
y
60mm
10mm
10mm
F(t)
Fig. 16. Impact test of a double-notched specimen.
t
F(t)
q0
t0
Fig. 17. Load–time relationship for impact test of a double-
notched specimen.
H. Zhang et al. / Computers and Structures 81 (2003) 1739–1749
1747
Page 10
In Figs. 20 and 21, the axial strains in the center
section of the specimen are given with the different me-
shes based on the conventional constitutive model and
the gradient dependent model. The mesh dependence
existing in the conventional constitutive model is ob-
served. On the contrary, the results of the gradient de-
pendent model possesses well property which embodies
the expanding of the localization area.
Remark 4. It has been pointed that in elastoplastic dy-
namics with softening, the contractivity of perturbation
(algorithmic stability) is crucial. Algorithms uncondi-
tionally stable for non-softening constitutive models
become ‘‘conditionally stable’’ (or even unconditionally
unstable) in the presence of softening [5]. However, this
phenomenon does not occur in the computation of the
numerical examples with softening discribed above, and
the algorithm presents still unconditional stable such as
that used for non-softening consititutive models. This
shows on the other hand the advantages of the algo-
rithm developed in this paper.
7. Conclusions
What presented above discribed a new algorithm for
numerical simulation of elastic–plastic strain hardening/
softening problems. The gradient dependent model
based on the non-local theory was adopted to overcome
the mesh dependent problem in the analysis of the dy-
namic strain softening problem. For the numerical
analysis, the parametric variational principle is adopted
which makes the gradient dependent model be easily
implemented in the algorithm. Furthermore, a para-
metric quadratic programming algorithm combined
with both the Newmark and the precise integration
methods in time domain is derived and changes the
problem into a linear complementary problem. Numer-
Fig. 18. Results of structural deformation with the conven-
tional constitutive model. (a) Coarse mesh and (b) fine mesh.
Fig. 19. Results of structural deformation based on the gradi-
ent dependent model. (a) Coarse mesh and (b) fine mesh.
0.0E+00
0
5.0E-01
1.0E+00
1.5E+00
2.0E+00
2.5E+00
3.0E+00
3.5E+00
4.0E+00
4.5E+00
5.0E+00
90
y(mm)
Strain
Mesh 1
Mesh 2
10
2030 4050 607080
Fig. 20. Axial strain in the center section of the specimen based
on the conventional constitutive model. Mesh 1: fine mesh,
Mesh 2: coarse mesh.
0.0E+00
010 20
30405060 7080
90
1.0E-01
2.0E-01
3.0E-01
4.0E-01
5.0E-01
y(mm)
Strain
Mesh 1
Mesh 2
Fig. 21. Axial strain in the center section of the specimen based
on the gradient dependent model. Mesh 1: fine mesh, Mesh 2:
coarse mesh.
1748
H. Zhang et al. / Computers and Structures 81 (2003) 1739–1749
Page 11
ical examples are given and the results demonstrate the
validity and efficiency of the theory and algorithm pre-
sented in this paper.
Acknowledgements
The financial supports from the National Key Basic
Research Special Foundation (G1999032805), the Sci-
entific Fund for National Outstanding Youth of China,
the National Natural Science Foundation of China
(10225212, 50178016, 19872016) and the Foundation for
University Key Teacher by the Ministry of Education of
China are greatly acknowledged.
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