# A new algorithm for numerical solution of dynamic elastic–plastic hardening and softening problems

**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.

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**ABSTRACT:**In this chapter, a second-order scheme of precise time-step integration (PTI) method is introduced for dynamic analysis with respect to long-term integration and transient responses while spatial discretization is realized with the differential quadrature method. Rather than transforming into first-order equations, a recursive scheme is presented in detail for direct solution of the homogeneous part of second-order differential and algebraic equations. The sine and cosine matrices involved in the scheme are calculated using the so-called N 2 algorithm, and the corresponding particular solution is also presented where the excitation vector is approximated by the truncated Taylor series. The performance and numerical behaviors of the second-order scheme of the PTI method are tested by a series of numerical examples in comparison with the first-order scheme or with the traditional time-marching Newmark-β method as the reference. The issue of spurious high-frequency responses resulting from spatial discretization for shock-excited structural dynamic analysis is also studied in the framework of the second-order PTI method. The effects of spatial discretization, numerical damping and time step on solution accuracy are explored by analyzing longitudinal vibrations of a shock-excited rod with rectangular, half-triangular and Heaviside step impact. - [Show abstract] [Hide abstract]

**ABSTRACT:**A fast precise integration method (FPIM) is proposed for solving structural dynamics problems. It is based on the original precise integration method (PIM) that utilizes the sparse nature of the system matrices and especially the physical features found in structural dynamics problems. A physical interpretation of the matrix exponential is given, which leads to an efficient algorithm for both its evaluation and subsequently the solution of large-scale structural dynamics problems. The proposed algorithm is accurate, efficient and requires less computer storage than previous techniques.Structural Engineering & Mechanics 07/2012; 43(1). · 0.80 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper presents a simulation study of the free flexural vibration behavior of non-uniform taper bars of circular and rectangular cross-section under body force loading due to gravity. The loading is controlled statically to take the bar to its post-elastic state so as to predict its dynamic behavior in the presence of plastic deformation. Hence the analysis is carried out in two parts; first the static problem under axial gravity loading is solved, then the dynamic problem is solved in this loaded condition. Appropriate variational method is employed to derive the set of governing equations for both the problems. The formulation is based on unknown displacement field which is approximated by finite linear combinations of orthogonal admissible functions. The present method is validated successfully with a well-known finite element package. Results are presented to investigate the effect of shape and size on the dynamic behavior of non-uniform taper bars. The study can be extended to study the post-elastic dynamic behavior of other related problems such as rotating beams and rotating disks.Applied Mathematical Modelling 01/2009; 33(11):4163-4183. · 2.16 Impact Factor

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.

1740

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

1742

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.

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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=30 Step=60 Step=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

0.00 0.02 0.040.06 0.080.100.12

t(s)

Displacement (mm)

Step=30Step=60Step=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

0.000.020.04 0.06

t(s)

0.08 0.100.12

Displacement (mm)

Step=30Step=60 Step=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

0.00 0.02 0.04 0.06

t(s)

0.08 0.10 0.12

Displacement (mm)

Precise-Step =30 Newmark-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.

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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)

50 60708090 100

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

0 10 203040 5060 7080 90100

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

0 10203040

x(mm)

50 6070 8090 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

20 30405060 7080

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

0 1020

30 405060 70 80

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.

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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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