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ACTA MECHANICA SINICA (English Series), Vo1.18, No.0, December 2002
The Chinese Society of Theoretical and Applied Mechanics
Chinese Journal of Mechanics Press, Beijing, China
Allerton Press, INC., New York, U.S.A.
ISSN 05677718
A COMBINED PARAMETRIC QUADRATIC PROGRAMMING
AND PRECISE INTEGRATION METHOD BASED DYNAMIC
ANALYSIS OF ELASTICPLASTIC HARDENING/SOFTENING
PROBLEMS*
Zhang Hongwu (~) Zhang Xinwei (~)
(State Key Laboratory of Structural Analysis and Industrial Equipment, Department of
Engineering Mechanics, Dalian University of Technology, Dalian 116024, China)
ABSTRACT: The objective of the paper is to develop a new algorithm for numerical
solution of dynamic elasticplastic strain hardening/softening problems. The gradient
dependent model is adopted in the numerical model to overcome the result mesh
sensitivity problem in the dynamic strain softening or strain localization analysis.
The equations for the dynamic elasticplastic problems are derived in terms of the
parametric variational principle, which is valid for associated, nonassociated and
strain softening plastic constitutive models in the finite element analysis. The precise
integration method, which has been widely used for discretization in time domain of
the linear problems, is introduced for the solution of dynamic nonlinear equations.
The new algorithm proposed is based on the combination of the parametric quadratic
programming method and the precise integration method and has all the advantages
in both of the algorithms. Results of numerical examples demonstrate not only the
validity, but also the advantages of the algorithm proposed for the numerical solution
of nonlinear dynamic problems.
KEY WORDS: precise integration method, parametric quadratic programming
method, strain localization, strain softening, dynamic response
1 INTRODUCTION
Several kinds of time integration method for structural dynamic analysis in time domain
have been developed in the past decades. Recently, Zhong [1] proposed a precise integration
method, which has many advantages such as absolute stability, zeroamplitude rate of decay,
zeroperiod specific elongation and nonoverstep properties. This method has been used
successfully in many problems of mechanics such as linear dynamic and heat conduction
problems [2'3]. The discussion of the method was given recently by Zhang and Zhong [4]
where the optimum parameter selection is suggested.
A large number of engineering materials including metals, concrete, soil, rock, poly
mers and microelectronic structures show a reduction of loadcarrying capacity with the
Received 5 February 2001, revised 5 November 2001
* The project supported by the National Key Basic Research Special Foundation (G1999032805), the
National Natural Science Foundation of China (19872016, 50178016, 19832010) and the Foundation for
University Key Teacher by the Ministry of Education of China
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Voi.18, No.6 Zhang & Zhang: Dynamic Analysis of Plastic Hardening/Softening Problems 639
increasing of loading and deformations. This is called the strain softening or localization [5,6] .
Numerical simulation of such kind of nonlinear problems generally is not so easy as that for
linear dynamic problems [7] .
It has been proved mathematically that the mesh sensitivity is observed when the finite
element method is used in the analysis of above softening materials Is,a]. To overcome the
mesh sensitivity problem, the ratedependent model has been introduced in the constitutive
equations [1~ . The higher order time derivative terms prevent the field equations from
becoming elliptic and the problem from being wellposed. Dispersive waves and implicit
length scale enable the ratedependent continuum capture the localization of deformation.
On the other hand, Cosserat constitutive model is based on the idea of a macrostructure
subdivided into microelements [121 . A length scale is introduced by a finite size of the micro
element. As far as a two dimensional problem is concerned,, the rotational degree is added
to the translation degrees as well as the couple stresses and generalized curvatures. The
shortcoming of this model is its failure to treat the compressive waves. As a recent work, it
has been pointed out that a meshless method can also be used in the treatment of fracture
problems in plastic softening form [13].
The gradient dependent model, according to the nonlocal theory, is based on the in
clusion of higherorder spatial derivatives in the constitutive equations [7,14,15]. The gradient
model reflects the fact that the interaction between the microstructural deformations in the
localization area is nonlocal. The yield function of the gradient dependent model depends
not only upon the hardening/softening parameter but also upon the Laplacian. The use of
a higherorder gradient dependent model can result in a wellposed set of partial differential
equations. Furthermore, the gradient dependent model explicitly incorporates an internal
length scale. From a dispersion analysis it can be found that the continuum model is capable
of transforming a travelling wave into a stationary localization wave.
In this paper, the parametic quadratic programming method [16'17] combined with
the precise integration method is adopted to solve the dynamic elasticplastic harden
ing/softening problems: Through the parametric quadratic programming method, the gradi
ent dependent model can be implemented easily by a simple modification of the enhancement
matrix in the governing equations for the solution of the classical elasticplastic problems.
By introducing the mathematical programming method, it avoids the iteration process such
as that used in the classic methods. Secondly, it breaks away the constraint of the Drucker
hypothesis in the plastic flow theory. It can be used to solve the nonassociate flow and
the nonnormal flow problems. With the parametric quadratic programming method, the
numerical stability is well reserved and the numerical precision is enhanced. The new algo
rithm developed in this paper keeps all the advantages in both of the parametric quadratic
programming method and the precise integration method for the solution of the dynamic
strain hardening/softening problems.
2 FORMULATION OF THE GRADIENT DEPENDENT MODEL
In the conventional plastic theory, the yield function f depends on the parameters of
stress (r, plastic strain e p and some inner variables. Without loss, of generalities, the isotropic
hardening/softening materials are considered here, and the yield function can be defined as
f : f (o', 6 p, t~) (i)
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640 ACTA MECHANICA SINICA (English Series) 2002
where ~ is the softening parameter in the constitutive model.
The nonlocal theory is introduced in the gradient dependent model. In a given point
x of the material, we can obtain the nonlocal inner variable &(x) by average weight to the
inner variable in the domain V, i.e.
~(x) = g(~)~(x + ~)dV V g(~)dV = 1 (2)
where g(~) is the weight function.
By Taylor expansion to the local variable, we have
~x x) 1 c':9~2(x) 2
1 v%;3(x) 3
+3!
1 o':ge;4(x) 4
+4! +"
(3)
Substituting Eq. (3) into Eq.(2) with the consideration of the isotropic hypothesis of material,
we have
I~ ~ Ig ] CIV2Ig ] C2~4~ ~ "'"
(4)
In a secondorder approximation, the yield function can be written as
f = / (5)
It can be seen that the main difference between the conventional plastic theory and the
gradient dependent plasticity model is the introduction of the gradient of softening/hardening
parameters in the yield function. Therefore, in the gradient dependent plasticity, the yield
status of a material point is not only related with its own plastic parameters as in the
conventional plasticity but also under the influence of the plastic parameters in the neigh
boring region. The size of the influence region is determined by the internal length scale
in the gradient dependent model. According to the gradientdependent model, the plastic
deformation in a point will be extended to a certain region.
Based on the gradientdependent model, the elasticplastic constitutive equations can
be defined as follows
(ogT
d~r = D (de  de p) de v = \ff~//A (6)
f(o',z p,~,V2~) <0 A= { >0 when f=O
when f<0

=0 (7)
where )~ is the plastic flow multiplier; ~ is the hardening/softening parameter.
of
Without loss of generality, we assume d~ = hA, e = h0~~ , in which, h is the
hardening/softening modulus such as defined in the conventional plasticity theory.
By Taylor expansion of Eq.(7), we obtain
f0 + Wde MA + ~V2A < 0 A > 0 (8)
where
W= D M=
\~] + h (9)
in which a is generally defined as the parameter of the gradient dependent model or the
internal length scale parameter.
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Vol.I8, No.6 Zhang & Zhang: Dynamic Analysis of Plastic Hardening/Softening Problems 641
3 PARAMETRIC
DEPENDENT
VARIATIONAL
MODEL
PRINCIPLE FOR THE GRADIENT
The parametric minimum potential energy principle based on the gradientdependent
model for the dynamic nonlinear problems can be expressed as follows: For all of the pos
sible incremental displacement solutions which satisfy the straindisplacement relations and
displacement boundary conditions, the exact solution minimizes the potential energy of the
system
t2 { 1 dbidui] ds
H = ft 1 ~ [~ditiP diti l
 ~dui,jDijklduk,l + )~iRkuduk,l +
/rp dp~duidF }dt
(10)
under the control of the system state equations
f0 + Wdr  MA + eV2A + v = 0
ATv = 0 ~,, v > 0 (11)
bi is the body force, Rkzm = Oag~_mDijkl. Thus, the dynamic problems are changed
c,o~ij
where
into
rain. H[)~()] (12)
s.t. f (du,)., V2A) + • = 0 )~Wv = 0 )~, ~, > 0 (13)
where ~, is the parametric variable which does not take part in the variation process and
controls the system state varying between elastic and plastic ones. v is the slack vector,
du is the incremental displacement vector. Equation (t3) is the system control equation
derived from the constitutive relations.
From the Parametric Variational Principle as stated above and the spatial discretization
technique used in the finite element method, we can obtain the finite element equations of
the nonlinear dynamic problem as
Mdik + Kdu  )~ = dP
Cdu  U)~  d + v = 0 (14)
ATv = 0
A, v _~ 0
where du is the incremental displacement vector, and
M=fpNTNd9 K=/BTDBd9 dP=/rpNTdpdF
(15)
are the mass matrix, stiffness matrix and load vector, respectively. These matrices and the
vector have the same meaning as those derived in the conventional finite element method.
The new matrices and vectors generated by the Parametric Variational Principle are
= f N~ D e Og~ ,~()
L
J.
f
~'fi~, '~~ C = jn WzN2dl2
(16)
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ACTA MECHANICA SINICA (English Series) 2002
U= /~ (MN~ +~.V2NA) ds d=/~ fod~
(17)
where 4i is the plastic potential matrix which represents the plastic potential of the system, C
is the constrained matrix which represents the constraint status, 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. ~ = C T. d and • are the
constraint and slack vectors, A is the parametric vector, which is the plastic flow parameter.
It can be seen clearly that the introducing of the gradientdependent model only adds
the gradient item into the hardening matrix. In the parametric quadratic programming
algorithm, we need use a quadratic interpolation to A.
If the damping effect is considered, the damping matrix needs to be added in the
dynamic equation, we have
Mdi~ + Gdiz + Kdu  A~ = dP (18)
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 control equa
tion results in a standard quadratic programming problem which can easily be solved by
many methods such as the Wolf method and Lemke method.
4 PRECISE INTEGRATION METHOD IN TIME DOMAIN
The precise integration method is a new algorithm for numerical solution of differential
equations. The algorithm has the absolute stability, zeroamplititude rate of decay, zero
period specific elongation and nonoverstep properties. We extend the method here to the
numerical solution of dynamic elasticplastic problems.
Considering the dynamic equation
Mi~ + Ku  A~ = F
(19)
and combining it with the identity equation {/~} = {~}, we obtain the following differential
equation
= HV + r + ~*A*
{o} {0}
V= u r= M_IF(t )
[ o o'] o,[o
H= _M_IK
(20)
where
(21)
o]
M 1~ 0
The homogeneous solution of Eq.(20) is
y(t) =
(22)
where
T(T) = exp(H x T) (23)
In the integration step t C [t}, t}+l], T = t t}, C is a constant vector and determined
by the initial conditions of the incremental step.
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Vol.18, No.6 Zhang & Zhang: Dynamic Analysis of Plastic Hardening/Softening Problems 643
Assuming that the special solution Vp(t) of Eq.(20) has been calculated, t = tk is the
initial time of the current step, then we can obtain the integration constant C. The general
solution of Eq.(20) is
V(t)  T(~) [V (ti()  Vp(tK)] 4 Vp(t) (24)
The special solution of Eq.(20) is
Vp(t) = H 1 [r0 4 Hlrl 4 rl (t tk)] 
H 1 [~*A~ 4 HI~*A; 4 ~*A~ (t  tk)] (25)
where ro, rl, A~, 3~ are constants.
Substituting Eq.(25) into Eq.(24) and let AA* = A~ (t  tk), ~ = t  tk, the general
solution is obtained
V (tk+l)= [T~H1HI"*AA .1~_ HI'*AA*]+T~HI"*A~+ V (tk+l) (26)
where
V(tk+l)=T[V(tk)+Hl(ro+H*rl)]Hl[ro+Hlrl+rlr]
(27)
T~ = T  I (28)
This is the integration scheme of the precise integration method for the solution of the
dynamic elasticplastic problems. When the structure is in the linear status, the parametric
variable X is zero. Then the above integration scheme (26) will be reduced to Vk+l. This
is just the linear solution such as shown in [1, 4].
From Eq.(26), we can give the incremental solution of the Eq.(20)
[ ]
/',v(tk+~) = g(tk+~)  v(tk) = T~H~//~*A,V 1  H~O*A,V
T
+
TaHI~*)~; 4 ~/" (tk+l)  V (tk) (29)
In the precise integration method, the dimension of the status vector Vk+l is doubled
so that the dynamic equation can be changed into one order differential equation. Therefore,
the dimension of the system control equation needs also to be doubled
C*AV (tk+l)  U*AA*  d* 4 v* = 0
[0 c o]
(30)
where
have
o]
d ={0
Substituting the incremental general solution (29) into the system control Eq.(30), we
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where
ACTA MECHANICA SINICA (English Series)
d  C* [~zk+ 1  Vk]  C*TaHI~*,~
2002
(32)
C*TaHIO*A; = (33c)
For the numerical implementation, the above equation can be rewritten in the following
form
{o}+ [ }
IV O] { ('U,k§ Uk) [
(34)
5 NUMERICAL EXAMPLES
Example 1.
vertical bars is 1 m. The cross sections of all bars are 1.0 • 10 3 m 2. The material parameters
are E  210 GPa, p = 7 800 kg/m 3. The load applied on the truss joint is 2.0 • 10 3 N as
shown in Fig.1. We consider here the single break hardening material strainstress relation
as plotted in Fig.2. The yield stress is 4.0 • 10 6 Pa. Because the free vibration period of
the structure is 0.013 35 s, the length of time step are, respectively, selected as 3.0 x 10 4 s,
1.5 • 104s and 0.75 • 10 4 s. The corresponding numbers of the time step are 30, 60 and
120. With the precise integration method, the displacement in 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 shown in Figs.3 and 4 where the value of total time is the same
but divided into different time steps. It can be seen that the results of the precise integration
method are less dependent on the length of time step than those of the Newmark method.
One truss structure is shown in Fig.1. The length of top, bottom and
Fig.1 Twodimensional truss model
I
E' = E/IO
0
Fig.2 Strainstress relationship
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Vol.18, No.6 Zhang & Zhang: Dynamic Analysis of Plastic Hardening/Softening Problems 645
0.0E+00
2.0E05
4.0E05
6.0E05
E 8.0E05
~oEo4
o.
:~  1.2E04
1,4E04
1.6E04
J
0.000
/ / \/
0.002 0.004
t/s
0.006 0.008
~step=30 ~ step=60 +step=120j
P
f
O.OE+O0,
2.0E05
4.0E=05
~ 8.oEo5
] ~ r
\ / i
/
~" 1.2E04
~
1.4E04
1.6E04
k
~ ~ ~,~,~e w
0.000 0.002 0.004
t/s
0.006 0.008
I~ step=30 ~ step=60 ~ step=1201
Fig.3 Results obtained by the Precise
Integration method with different
time steps (Total time0.009 s)
Fig.4 Results obtained by the Newmark
method with different time steps
(Total time=0.009 s)
It can be found that the results of the Newmark method with 180 steps is rather close to
those obtained by the precise integration method with 30 time steps. This means that the
precise integration method is better with respect to the computation accuracy. For the
computation efficiency, the Precise Integration method with 120 time steps uses 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 close to the Newmark method with
180 steps and the computational time is reduced greatly.
Example 2. One dimensional bar in the tension and strain softening state. The
strain softening problem is sketched in Figs.5~7. Load: q0 = 0.75 N. Material: E  20 GPa,
p = 2000.0kg/m 3, h = 2.0GPa. The yield stress is 2.0MPa. 8 = 5 x 104 N.
L = 100 mm
i T
A  1 mme F(t)
Fig.5 One dimensional bar in the tension and softening state
F(t)
" t
Fig.6 Loadtime relationship Fig.7 Stressstrain relationship
The problem is computed with the gradient dependent model. The bar is divided into
20, 40, 80 and 160 elements, respectively. Parameters h = 2.0 GPa, ~ = 5 x 104 N and the
length of time step= 1.5 x 107s. The strain localization along the bar at t  1.8 x 105s
is given in Fig.8. It can be seen that the width of the plastic zone keeps constant with
different meshes. The material internal length scale parameter l = ~5/h = 5 mm. The
corresponding width of the localization zone is 15.7 mm, which is just the half of 27rl.
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646 ACTA MECHANICA SINICA (English Series) 2002
In Fig.9, the strain localization results along the bar with different internal length scale
parameters are given. The values of ~ are 1.25 • 104 N, 2.5 x 104 N, 5.0 • 104 N, 10.0 • 104 N,
20.0 • 104 N, respectively.
7.0E04 ~
6.0E04
5.0E04
1.0E03
~ 20 elements
~ 40 elements
~80
elements
8.0E04
.E
~
I[~
~
4.0E04 ~ 6.0E04
3.0E04 2.0E04 ~ II __ 160 el .... ts __ 4.0E04
1.0E04  __ , 2.0E04
0.0E+00 0.0E+00
0 10 20 30 40 50
x/mm
60 70 80 90 100 0
Fig.8 Strain localization along the bar ob
tained by the different meshes with
the gradient dependent model
Fig.9
l=2.50 rnm
l=3.54 rnm
/=5.00 mm
l=7.07 mrn
l=10.0mm
10 20 30 40 50 60 70 80 90 100
x/mm
Numerical results with different in
ternal length scale parameters
Example 3. Impact test of a doublenotched specimen. The geometry of the problem
is shown in Fig.10. The loadtime relation is plotted in Fig.ll. Parameters: q0 = 3.5 •
106N/m, to = 3.5 • 105s. E = 40.7GPa, p = 2350.0kg/m 3, h = 2.5GPa. The yield
stress of the material is 4.0 GPa. ~ = 5 • 104 N. Two kinds of finite element meshes are
adopted in the computation.
90m 7 10mmI
IS mm
I" 60ram "1
~ F(t)
Fig.10 Impact test of a double
notched specimen
F(t)
Fig.ll Loadtime relationship
The structural deformation results at time 5.0 x 10 5 s with different meshes based on
the gradient dependent model are given in Figs.12(a) and (b). The results of the coarse mesh
is close to the fine mesh's, especially in the notched area. This shows again the advantages
of the gradient dependent model.
In Figs.13 and 14, the axial strains in the center section of the specimen are given with
different meshes based on the conventional constitutive model and the gradient dependent
model. The mesh dependence in the conventional constitutive model is observed. On the
contrary, the results of the gradient dependent model show the expansion of the localization
area.
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Vol.18, No.6 Zhang & Zhang: Dynamic Analysis of Plastic Hardening/Softening Problems 647
I ....
(a) Coarse mesh (b) Fine mesh
Fig.12 Results of structural deformation based on
the gradient dependent model
5.0E+00 5.0E01
^
4.0E+00
000
i ~ ~.SE+00
2.0E+00
1.5E+00
4.0E01
0 0
 
,
~ 2.0E01
I.OE+O0
5.0E01
O.OE+O0
I.OEOI ~ ~ ~
  !~1 [l_k. [ _ [~
O.OE+O0
0 i0 20 30 40 50 60 70
y/ram
80 90 0 I0 20 30 40 50 60 70 80 90
y/into
Fig.13 Axial strains in the center section
of the specimen based on the con
ventional constitutive model, mesh
1: fine mesh, mesh 2: coarse mesh
Fig.14 Axial strains in the center section
of the specimen based on the gradi
ent dependent model, mesh 1: fine
mesh, mesh 2: coarse mesh
6 CONCLUSIONS
This paper described a new algorithm for numerical simulation of elasticplastic strain
hardening/softening problems. The gradient dependent model based on the nonlocal the
ory was adopted to overcome the mesh dependent problem in the analysis of the dynamic
strain softening problem. For the numerical analysis, the parametric variational principle is
adopted which makes the gradient dependent model easily implemented in the algorithm.
Furthermore, a parametric quadratic programming algorithm combined with the precise in
tegration methods in time domain is derived and turns the problem into a standard linear
complementary problem. Numerical examples are given and demonstrate the validity and
efficiency of the theory and algorithm presented in this paper.
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