a semi-analytical integration method for j2 flow theory of plasticity with linear isotropic...

Comput. Methods Appl. Mech. Engrg. 198 (2009) 2151–2166

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Comput. Methods Appl. Mech. Engrg.

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A semi-analytical integration method for J2 flow theory of plasticitywith linear isotropic hardening

László Szabó *

Department of Applied Mechanics, Budapest University of Technology and Economics, H-1111 Budapest, M}uegyetem rkp. 5, Hungary

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 September 2008Received in revised form 26 January 2009Accepted 7 February 2009Available online 20 February 2009

Keywords:von Mises elastoplasticityLinear isotropic hardeningExact time integrationConsistent tangent modulus

0045-7825/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cma.2009.02.007

* Tel.: +36 1 463 1369; fax: +36 1 4633471.E-mail address: [email protected]

In this paper, an exact time integration scheme is presented for von Mises elastoplasticity with linear iso-tropic hardening at small deformations. The method is based on the constant strain rate assumption,which is widely accepted in displacement based finite element applications. The deviatoric form of therate equation, using the method proposed by Krieg and Krieg in [R.D. Krieg, D.B. Krieg, Accuracies ofnumerical solution methods for the elastic-perfectly plastic model, J. Press. Vess. Technol. Trans. ASME99 (1977) 510–515], is rewritten to a system of nonlinear ODEs with two scalar functions: the radiusof the Mises circle and an angle obtained by using the scalar product of the stress deviator and the strainrate. The time integration of these functions gives an exact solution in implicit form using the incompletebeta function. In addition, by the exact linearization of this new semi-analytical solution a stress updatingalgorithm and the consistent tangent operator are derived. The numerical performance and accuracy ofthe proposed method are illustrated on numerical examples. Moreover, a comparison of the proposedmethod with the well-known radial return method is included.

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1. Introduction

The Prandtl–Reuss constitutive model or the so-called J2-flowtheory (i.e., elastoplastic model based on the von Mises yield func-tion, associated flow rule and elastic-perfectly plastic, or isotropichardening and/or kinematic hardening) have recently become verypopular in computational elastoplasticity. In its classical form, thematerial is characterized by an elastic-perfectly plastic or an iso-tropic strain hardening model. In the last 50 years, a large numberof contributions were produced in order to extend the model to in-clude complex behavior of materials (e.g., linear and nonlinearkinematic hardening, anisotropy, softening, strain rate dependenceand damage). It is not the purpose of this paper to provide a de-tailed review of these extensions, but rather to concentrate onthe simple elastic–plastic J2-flow theory with von Mises yield func-tion and linear isotropic hardening at small deformations. For an indepth treatment of plasticity we refer to the literature, e.g., Hill[16], Lubarda [25], Nemat-Nasser [28], Khan and Huang [19], Zycz-kowski [46], Lubliner [26], Armero [3] and Chaboche [7], amongothers.

Most rate-independent plastic models are formulated in termsof rate-type constitutive equations. In this case, the finite elementanalysis of elastoplastic problems requires, among others, the inte-

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gration of the constitutive relations. Thus, the choice of the integra-tion algorithm has considerable influence on the efficiency,accuracy and convergence of the elastoplastic incremental solu-tion. Consequently, an extensive benchmarking is needed tounderstand the characteristics of the algorithm including, forexample, the associated convergence rates, accuracy and stability.

Within the stress updating procedure, the integration of elasto-plastic constitutive equations maybe performed either by numeri-cally or using exact integration. Over the last three decades, severalnumerical integration methods have already been proposed. It isalso not intent of this paper to outline of these works. The inter-ested reader can find more details on numerical integration tech-niques in the books of Simo and Hughes [35], Han and Reddy[15], Belytschko et al. [5], Nemat-Nasser [28], de Souza Netoet al. [9], among others.

Exact integration methods or closed-form solutions are avail-able for a limited cases such as the J2-flow theory. Examples ofsuch treatments include those of Krieg and Krieg [20], Reuss [30],Gratacos et al. [14], Hong and Liu [18], Ristinmaa and Trying[33], and Wei et al. [42], which are based on the use elastic-per-fectly plastic model, and the time integration of constitutive equa-tion are given in exact form using the constant strain rateassumption. Closed-form analytical solutions for linear kinematichardening model (e.g. the Prager–Ziegler rule) has also been pre-sented, for example, Wang and Chang [40], Yoder and Whirley[45], Ristinmaa and Trying [33], Romanshencko et al. [34], andSzabó and Kovács [38].

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2152 L. Szabó / Comput. Methods Appl. Mech. Engrg. 198 (2009) 2151–2166

In the context of linear isotropic hardening, there are severalapproaches available in the literature, e.g. Yoder and Whirley[45], Chan [8], Wang and Chang [39,41], Ristinmaa and Trying[33], Krieg and Xu [21], Liu [22] and Mukherjee and Liu [27]. How-ever, as opposite to the perfectly plastic and kinematic hardeningmodels, no attempts have been made to provide an exact solutionfor the linear isotropic hardening model. Although, the motion ofthe Mises circle in the deviatoric stress space is quite simple,namely, it expands uniformly (or isotropically), the function whichdescribes the time dependence of this change is still unknown.

Using a similar method introduced by Krieg and Krieg [20], thepresent work aims at obtaining the analytical solution for timeintegration of Prandtl–Reuss equation with linear isotropic hard-ening, assuming constant strain rate. The solution strategy is basedon the pioneering work of Krieg and Krieg [20], namely, the devia-toric form of the rate equation is rewritten as a set of nonlinear or-dinary differential equation with two scalar functions. These arethe radius of the Mises circle, RðtÞ, and the angle, w(t), definedin the scalar product of the stress deviator and the strain rate.Then, the time integration of w gives an exact solution in implicitform, t ¼ f ðwÞ through the use of the incomplete beta function. Inaddition, an extension of the new method for the Drucker–Pragermodel is also presented. Moreover, by the exact linearization ofthe new semi-analytical solution a stress updating algorithm andthe consistent tangent operator are derived. The proposed methodis implemented in the finite element code ABAQUS [1] using theuser defined material modul (UMAT).

The paper is organized as follows: At the end of this Section thenotation used throughout the paper is introduced. In Section 2, theconstitutive relations of J2 flow theory of elastoplasticity with lin-ear isotropic hardening are briefly summarized. In Section 3, an ex-act solution of the time integration of rate constitutive equations ispresented. Based on this analytical solution, in Section 4, the stressupdating algorithm and the expressions for the consistent tangentmodulus are obtained in explicit forms ready for finite elementapplications. In Section 5, some numerical examples are presentedto assess the performance of the proposed method. Moreover, theaccuracy of the algorithm is compared to the standard radial returnmethod. In addition, a new analytical solution for constant stressrate assumption is given in this paper and used in the numericalexamples.

Regarding notation, tensors are denoted by bold-face charac-ters, the order of which is indicated in the text. The tensor productis denoted by �, and the following symbolic operations apply:a : b ¼ aijbij; and ðC : aÞij ¼ Cijklakl, with the summation over re-peated indices. The symbol kak ¼

ffiffiffiffiffiffiffiffiffiffia : ap

is used to denote a normof second-order tensor a. The superposed dot denotes the materialtime derivative or rate, the superscripts �1 denotes inverse, andthe prefix tr indicates the trace. Furthermore, the second-orderand the fourth-order identity tensors are denoted by d and I,respectively.

2. Constitutive relations for J2 flow theory of plasticity at smalldeformation

In this section, the constitutive equations of the commonly usedJ2 flow theory of plasticity with linear isotropic hardening arebriefly reviewed. This outlined follows the treatments of Simoand Hughes [35] and Auricchio and Beirão da Veiga [4].

The rate of change of the strain measure decomposes into anelastic and a plastic contribution, namely

_e ¼ _ee þ _ep; ð1Þ

and for the case of linear elasticity, the stress rate tensor is relatedto the elastic strain rate as

_r ¼ De : _ee; ð2Þ

where De is the fourth-order elasticity tensor. In what follows, lin-ear isotropic elasticity is assumed, so that

De ¼ 2GTþ Kd� d; ð3Þ

where T ¼ I� 13 d� d is the fourth-order deviatoric operator tensor,

and G and K are the shear and bulk moduli, respectively.The Huber–von Mises yield condition is defined by

Fðr;RÞ ¼ ksk � RðcÞ 6 0; ð4Þ

where s ¼ r� 13 trrd is the deviatoric stress, and the function RðcÞ

defines the hardening law in terms of a scalar plastic state variable,c.

For the linear isotropic hardening, the function RðcÞ takes theform

R ¼ R0 þ hc; ð5Þ

where R0 is a material constant related to the initial value yieldstress R0 ¼

ffiffiffiffiffiffiffiffi2=3

pry, and h ¼ 2H=3; where H is the constant plastic

hardening modulus.The associated flow rule for the plastic strain rate, using (4), is

given by

_ep ¼ _ksksk ; ð6Þ

where _k is a plastic multiplier.Note that the scalar plastic state variable c is defined by

c ¼Z t

0

_kdt: ð7Þ

The loading/unloading conditions can be expressed in Kuhn–Tuckerform as

_k P 0; Fðr;RÞ 6 0; _kFðr;RÞ ¼ 0: ð8Þ

The plastic multiplier _k; using the plastic consistency condition_F ¼ 0; and the Eqs. (1)–(7), takes the form

_k ¼ 2Gs : _eRð2Gþ hÞ : ð9Þ

Finally, combining (2), (6) and (9), the elastoplastic constitutiverelations may be expressed by

_r ¼ Dep : _e; Dep ¼ De � 4G2

R2ð2Gþ hÞs� s; ð10Þ

where Dep is the so-called elastoplastic, or continuum tangent mod-ulus tensor.

The constitutive equation of elastoplasticity defined above canbe separated into a deviatoric and hydrostatic parts as follows

_s ¼ 2G _e� 2G

R2ð1þ nÞsðs : _eÞ; tr _r ¼ 3Ktr _e; ð11Þ

where _e ¼ _e� 13 tr _ed is deviatoric strain rate, and n ¼ h=ð2GÞ ¼

H=ð3GÞ is a dimensionless constant parameter.Finally, the rate of the radius of the yield surface, from (5), (7)

and (9), can be expressed by

_R ¼ 2GaR

s : _e; ð12Þ

where

a ¼ n1þ n

� h2Gþ h

� H3Gþ H

: ð13Þ

L. Szabó / Comput. Methods Appl. Mech. Engrg. 198 (2009) 2151–2166 2153

3. A semi-analytical integration method for linear hardeningelastoplasticity with constant strain rate assumption

3.1. von Mises plasticity model with linear isotropic hardening

In this section, we consider the time integration of the constitu-tive Eq. (11) with the constant strain rate assumption.

Let ½0; t� be the time interval, over which integration of (11) isperformed. We assume that the solution is known at the initialstate, t ¼ 0, and we consider the following strain history,eðtÞ ¼ t _e, with _e ¼ constant.

Define the following inner product

s : _e ¼ kskk _ek cos w � Rk _ek cos w; ð14Þ

where w is defined according to Krieg and Krieg [20] as the anglebetween the deviatoric stress tensor and the deviatoric strainrate.

Then, by taking the dot product of the constitutive relation (11)1

with _e and from (12) and (14), we obtain the following system ofnonlinear ordinary differential equations

_wsin w

þ 2Gk _ekR¼ 0; ð15Þ

_R� 2Gk _eka cos w ¼ 0 ð16Þ

with the initial conditions Rð0Þ ¼ Rn and wð0Þ ¼ wn. ð0 < wn 6 p=2Þ.The solution of the system differential equations (15) and (16)

can be obtained by using the following two steps.

Step 1. First, the parameter R as the function of w is to be deter-mined. Using Eqs. (15) and (16) with the relationship_R ¼ ðdR=dwÞ _w, we obtainZ R

Rn

dRR¼ �a

Z w

wn

dwtan w

: ð17Þ

Integrating (17) yields

RðwÞ ¼ Rnsin wn

sin w

� �a

: ð18Þ

Note that this expression has been obtained previously by Yoderand Whirley [45] and Ristinmaa and Tryding [33].

Step 2. To compute the function wðtÞ, we need to integrate (15)with respect to time. Substituting (18) into (15), weobtain

2Gk _ekRn sina wn

¼ �ðsin wÞ�ð1þaÞ dwdt; ð19Þ

which was previously presented a slightly different form by Risti-nmaa and Tryding [33].

Then, integrating Eq. (19) over the time interval ½0; t�with initialcondition wð0Þ ¼ wn, we get

2Gk _ekRn sina wn

Z t

0dt ¼ �

Z w

wn

ðsin wÞ� 1það Þ dw; ð20Þ

and finally we arrive to the formula

4Gk _ektRn sina wn

¼ B cos2 w;12;� a

2

� �� B cos2 wn;

12;� a

2

� �: ð21Þ

Here B½x; m;l� is the incomplete beta function defined by the expres-sion (see [2,37]):

B½x; m;l� ¼Z x

0sm�1ð1� sÞl�1ds; 0 6 x < 1: ð22Þ

While best known from its application in statistics, it is alsowidely used in many other fields. Dutka [11] provides a his-tory of the development and numerical evaluation of thisfunction.

It is obvious by observing the above formula that explicitexpression for w for a given time, t; cannot be determined. How-ever, Eq. (21) can be easily solved for the angle w by an appropriateiterative method.

An alternative method to solve (21) for w is based on the workof Dominici [10]. This method gives an algorithm to compute theseries expansion for the inverse of the incomplete beta function.The discussion and numerical implementation of this algorithmis summarized in the Appendix A.

Finally, the deviatoric stress tensor (similarly to the solutionfor the elastic-perfectly plastic model see for example [14,20,22,32,42]), can be determined by a linear combination of thedeviatoric stress, sn, and deviatoric strain rate as

sðtÞ ¼ R sin wRn sin wn

� �sn þ

R sinðwn � wÞk _ek sin wn

� �_e: ð23Þ

Note that for radial (proportional) loading wn ¼ 0; and the angle wremains zero under the time steps. In this case, the stress deviatoris proportional to sn and the solution is identical to that obtainedby the radial return mapping method.

Remark 1. It should be noted that in a plastic loading step, whenthe angle wðtÞ tends to zero, from (18), the radius of the Misescircle tends to infinity. Because limx!1þB½x; m;�l� ¼ 1;l > 0, from(21)

limw!0þ

B cos2 w;12;� a

2

� �¼ 1

it is apparent that wðtÞ ¼ 0 cannot be reached for finite times t P tn;

and proportionality between sðtÞ and 2G _eðt � tnÞ only occurs at infi-nite value of R. Consequently, for large time increment (although,the value of w may becomes very small) the value of R remainsfinite.

Remark 2. For the elastic-perfectly plastic case, n ¼ 0; a ¼ 0;Rn � R0, and the incomplete beta function reduces to (see Spanierand Oldham [37]. p. 575)

B x;12;0

� �¼ 2 tanh�1 ffiffiffi

xp� �

: ð24Þ

Then, the Eq. (21) can be restated in the simplified form

2Gk _ektR0

¼ tanh�1ðcos wÞ � tanh�1ðcos wnÞ; ð25Þ

which can be solved for cos w yielding

cos w ¼tanh

2Gk _ektR0

� �þ cos wn

1þ tanh2Gk _ekt

R0

� �cos wn

: ð26Þ

Finally, it follows from (26) that

tanw2

� �¼ exp �2Gk _ekt

R0

� �tan

wn

2

� �; ð27Þ

which is the formula found in Krieg and Krieg [20], Ristinmaa andTryding [33], Gratacos et al. [14] and Wei et al. [42], and Yoderand Whirley [45].


3.2. An extension of the semi-analytical integration method to theDrucker–Prager plasticity model with linear hardening

The main objective of this section is to present an exact integra-tion to the Drucker–Prager model based on the new methodproposed in this paper. The numerical integration method for thismodel have been fairly extensively worked out by, for example,Loret and Prevost [24], Hjia et al. [17], Liu [23], Rezaiee-Pajandand Nasirai [31] and Genna and Pandolfi [12], de Souza Netoet al. [9]. Loret and Prevost [24], using Krieg’s method, obtainedan exact solution for the non-associative Drucker–Prager modelwith isotropic hardening. However, they used a numerical integra-tion scheme to obtain the time function of angle the wðtÞ. Thepresent work aims at obtaining an explicit expression for thisfunction. Here, the notation similar to Loret and Prevost [24] isfollowed.

The constitutive equations are defined by Eqs. (1)–(3) with thefollowing yield function

Fðr;jÞ 1ffiffiffi2p Sþ

~a3

trr� c 6 0; ð28Þ

where S ¼ ksk, and ea, c are material parameters, implicit functionsof the hardening parameter j:

The plastic strain rate is defined as a non-associated flow rule

_ep ¼ Ksffiffiffi2p

Sþeb3

d

!; ð29Þ

where eb is a material parameter.The hardening rule is defined by the evolution of the hardening

parameter j as

_j ¼ �Keh oFðr; jÞoj

� ��1

; ð30Þ

where eh is the plastic hardening modulus. In what follows we as-sume that eh is constant.

The deviatoric and the hydrostatic parts of the constitutiveequations, using Eqs. (1)–(3), (29) and (30) with the plastic consis-tency condition _F ¼ 0; can be derived as

_s ¼ 2G _e� 2 G2eHS2s s : _eþ

eaKStr _effiffiffi2p

G

� �;

tr _r ¼ 3Ktr _e� 3ffiffiffi2p ebKGeHS

s : _eþeaKStr _effiffiffi

2p

G

� �; ð31Þ

where

eH ¼ Gþ eaebK þ eh: ð32Þ

Note that when the parameters ea ¼ eb ¼ 0 and eh ¼ h=3; the equa-tions defined above reduce to the Eq. (11).

We apply Krieg’s method described in the previous section tothe time integration of the set of constitutive Eq. (31) with the con-stant strain rate assumption, Accordingly, we finally obtain

_wsin w

þ 2Gk _ekS¼ 0; ð33Þ

_S� 2Gk _ek ðmþ n� 2Þ2

cos wþm� n2

� �¼ 0; ð34Þ

where

m ¼ 2� GeH ð1þ VÞ < 2; n ¼ 2� GeH ð1� VÞ > 0; ð35Þ

and

V ¼eaKtr _effiffiffi2p

Gk _ek: ð36Þ

The parameter V defined above characterizes the ratio of dilata-tional and deviatoric deformations.

The solution of the system of differential Eqs. (33) and (34) fol-lows the same steps already discussed in Section 3.1. According tostep 1, first, the parameter S in terms of the angle w may be de-rived. From expressions (33) and (34), eliminating the 2Gk _ek term,yields the following result:Z S

Sn

dSS¼ 1

2

Z w

wn

2�m� ntan w

þ n�msin w

� �dw: ð37Þ

We may integrate this equation to get

SðwÞ ¼ Snsin 1

2 w

sin 12 wn

1�m

cos 12 w

cos 12 wn

1�n

: ð38Þ

This is the same expression as derived by Loret and Prevost [24].Note that because the angles w and wn are given in the interval½0;p� and wn P w, the absolute value in (38) is unnecessary.

The next task is to determine the time function of angle w. Com-bining (33) and (38), we find that

4Gk _ekSn sinm�1 1

2 wn cosn�1 12 wn

Z t

0dt

¼ �Z w

wn

sin12

w

� ��m

cos12

w

� ��n

dw: ð39Þ

Integrating (39) yields

4Gk _ekt

Sn sinm�1 12

wn cosn�1 12

wn

¼ B cos2 12

w;1� n

2;1�m

2

� �� B cos2 1

2wn ;

1� n2

;1�m

2

� �:

ð40Þ

We have thus obtained the basic formulation for the angle w in im-plicit form using the incomplete beta function. It should be pointedout that the integral expressions (20) or (39) have been defined anddiscussed by several authors [8,17,21,24,33,39–41,45], however,these integrals cannot be obtained in closed-form. Here, this prob-lem is solved and this result can be extended to the combined linearisotropic and linear kinematic hardening model which will be pre-sented in a separate article.

Finally, by a similar reasoning as before, the time function of thedeviatoric stress tensor is computed by

s tð Þ ¼ S sin wSn sin wn

� �sn þ

S sinðwn � wÞk _ek sin wn

� �_e: ð41Þ

It is worth emphasizing that the present solution presented above isrestricted to the integration of Eq. (39). A detailed discussions onthe Drucker–Prager model based on this integration method (forexample, an investigation of the range of parameters n and m, def-inition of the hardening modulus, singularity at the apex of thecone, contact stress state, stress updating and consistent tangent)are to be considered in future work.

Finally, we note that the method presented above can beadapted to a modified Drucker–Prager yield function of whichthe hydrostatic tensile apex has been removed by the use of hyper-bolic meridians.

4. Stress updating and consistent tangent modulus

4.1. Stress updating algorithm

The semi-analytical integration method described in the pre-ceding section gives the exact stress response when the strain in-put is given by a rectilinear strain path. This assumption is

a

b

Fig. 1. Schematic illustration of the angle w and stress update in the deviatoricprincipal stress space; ðaÞ elastic to plastic step, ðbÞ plastic to plastic step.


widely accepted in the displacement based finite element applica-tion. In this section, following the standard methodology used forthe elastic-perfectly plastic case (see e.g. Gratacos et al. [14], Risti-nmaa [32], Wei et al. [42]), we present a stress updating procedurebased on the new integration method introduced in this paper. Theconsiderations are hereafter restricted to the Mises plasticitymodel.

Within a typical time increment Dt ¼ ½tn; tnþ1�; it is assumedthat at the time tn the strain, en and the stress, rn are known, andthe main goal of the procedure presented below is to determinethe stress at time tnþ1 for a given strain increment, De.

From the input parameters: sn ¼ rn � 13 trrnd;De ¼ Den � 1

3 trDend;G;h;R0 and cn, first the trial deviatoric elastic stress iscalculated

strialnþ1 ¼ sn þ 2GDe: ð42Þ

Then, using the yield condition

Fðs;RÞ ¼ ksk � ðR0 þ hcÞ 6 0 ð43Þ

with the quantities R0; kstrialnþ1k; ksnk and Rn ¼ R0 þ cnh; the following

cases are considered.

Case A : Full elastic step (elastic to elastic)When the conditions

Fðsn;R0Þ < 0 and Fðstrialnþ1;R0Þ 6 0 ð44Þ

are fulfilled, the increment is elastic, and the new deviatoric stressstate at time tnþ1 is

snþ1 ¼ sn þ 2GDe: ð45Þ

Case B : Elastic-elastoplastic transition (elastic to plastic)When the stress values at time tn are associated to theelastic state and the trial stress state located outside onthe current yield surface, namely

Fðsn;R0Þ < 0 and Fðstrialnþ1;R0Þ > 0; ð46Þ

then, the deviatoric strain increment divided into a purely elasticpart, kDe, and an elastoplastic part, ð1� kÞDe. For the strainincrement, kDe, the solution is obtained by an elastic step and thestress state lies on the yield surface. This contact stress state isgiven by

s0 ¼ sn þ 2GkDe; ð47Þ

where the parameter k, using the yield condition s0 : s0 ¼ R20, can be

calculated in the following closed-form

k ¼�sn : Deþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsn : DeÞ2 þ kDek2ðR2

0 � sn : snÞq

2GkDek2 : ð48Þ

We note that the condition (46) implies that 0 < k < 1:Next, basedon stress state s0 and De, the following nonlinear equation deter-mines the angle wnþ1

gðwnþ1Þ : ¼ B cos2 wnþ1;12;� a

2

� �� B cos2 w0;

12;� a

2

� �� 2g sin�a w0 ¼ 0; ð49Þ

where

g ¼ 2Gð1� kÞkDekR0

; cos w0 ¼s0 : DeR0kDek : ð50Þ

The details of the solution of (49) for wnþ1 are discussed in theAppendix A.The deviatoric stress, snþ1 remains in the plane definedby ðs0;DeÞ. Thus, the deviatoric stress snþ1, in view of (23), may beexpressed as a linear combination of s0 and De :

snþ1 ¼ as0 þ bDe; ð51Þ

where

a ¼ Rnþ1 sin wnþ1

R0 sin w0; b ¼ Rnþ1

kDeksinðw0 � wnþ1Þ

sin w0; ð52Þ

and

Rnþ1 ¼ R0sin w0

sin wnþ1

� �a

: ð53Þ

Case C : Full plastic step (plastic to plastic)When the stress state at time tn is plastic and the currentradius of the Mises circle, Rn ¼ R0 þ hcn, and the trialdeviatoric stress imply that

Fðsn;RnÞ ¼ 0 and Fðstrialnþ1;RnÞ > 0: ð54Þ

Then, the new stress state becomes also plastic. The stress updatingcan be calculated according to Case B with s0 � sn;w0 � wn;R0 � Rn

and k ¼ 0. The Case B and Case C are illustrated in Fig. 1. The Fig. 1ashows an elastic-elastoplastic transition step, while a purely plasticloading step is shown in the Fig. 1b, respectively.Case D: Elastic unloading step (plastic to elastic)

When the stress state at the time step tn is plastic and

Fðsn;RnÞ ¼ 0 and Fðstrialnþ1;RnÞ < 0; ð55Þ

then the new stress state at time tnþ1 becomes elastic, and it can becalculated according to Case A.Case E: Elastic unloading and plastic loading step (plastic to elastic

to plastic)


When the stress state at the time step tn is plastic and

F sn;Rnð Þ ¼ 0 and Fðstrialnþ1;RnÞ > 0 and

cos wn ¼sn : DeRnkDek < 0 wn >

p2

�; ð56Þ

then the loading step is partial elastic then plastic again. In this case,there is a transition from the plastic to plastic state through theelastic region. The stress values at time tn are associated to the plas-tic state and the trial stress state located outside on the currentyield surface. The deviatoric strain increment divided into a purelyelastic part, ekDe, and an elastoplastic part, ð1� ekÞDe, where the fac-tor ek can be calculated by

ek ¼ Rn cosðp� wnÞGkDek : ð57Þ

The new contact stress state is given by

esn ¼ sn þ 2GekDe; kesnk ¼ Rn; ð58Þ

and the initial value of angel w is defined by

cos ewn ¼esn : DeRnkDek : ð59Þ

The new stress state can be calculated similar to Case B withs0 � esn;w0 � ewn;R0 � Rn and k ¼ ek.

The new stress, using the updated deviatoric stress and (11),can be calculated for the all cases as follows:

rnþ1 ¼ snþ1 þ13

trrnþ1d ¼ snþ1 þ Ktrenþ1d: ð60Þ

4.2. Numerical procedure for the piecewise linear hardening model

In this section, we will extend the stress updating algorithm todeal with a piecewise linear approximation of the hardening curve.For simplicity, we restrict our attention to Case C, so that initial andfinal stress states are both plastic. The input data aresn;De;Rn; cn;wn and the piecewise hardening curve is given bythe following non-standard form

ð61Þ

where N is the number of segments. Note that, the parameter setdefined by (61), can be easily computed from the standard formof the piecewise linearized hardening curve given by the ðri; ep

i Þpairs.

In the first step, the location of the initial stress state, Rn has tobe found. Assume now that Ri 6 Rn < Riþ1 which means the radiusof the Mises circle associated to the current stress at time tn is lo-cated in the ith segment. By using (53), the angle w at the end ofthis segment, can be calcultated as

wiþ1 ¼ arcsin sin wnRn

Riþ1

� �1ai

" #; ð62Þ

where the parameter ai is calculated from the plastic hardeningmodulus, hi corresponding to the ith segment.

Then, from (49), a trial value of factor k is derived as

k1 ¼Rn sinai wn

2GkDek B cos2 wiþ1;12;� ai

2

� �� B cos2 wn;

12;� ai

2

� �� ; ð63Þ

which represents the length of kDek between Rn and Riþ1. When theparameter k1 P 1, the increment is completed within this segmentand stress updating can be calculated according to Case B withs0 � sn;w0 � wn;R0 � Rn and k ¼ 0: However, for k1 < 1, the incre-ment passes this segment and the examination expands on the seg-ment following on the row. Accordingly, the angle wiþ2 associated tothe end of the segment iþ 2 is computed from the expression

wiþ2 ¼ arcsin sin wiþ1Riþ1

Riþ2

� � 1aiþ1

" #ð64Þ

for which the factor k2 is computed to be

k2 ¼Riþ1 sinaiþ1 wiþ1

1� k1ð Þ2GkDek

� B cos2 wiþ2;12;� aiþ1

2

� �� B cos2 wiþ1;

12;� aiþ1

2

� �� : ð65Þ

Then, if k2 P 1, the increment is terminated in the segment iþ 1,and the stress updating can be evaluated as before. Otherwise pro-ceed, set iþ 3, and proceed to the next segment. Assume that thegiven increment passes through the segments i; iþ 1; . . . ;p, andthe new stress state is located in segment pþ 1: Until segment p,the quantities wi;wiþ1; . . . ;wp;wpþ1 and k1; k2; . . . ; kpþ1�i, are deter-mined by the procedure defined above. Angle wnþ1, based on stressstate spþ1 and De, can be calculated from the following nonlinearequation

gðwnþ1;wpþ1; apþ1;gÞ :¼ B cos2 wnþ1;12;� apþ1

2

� �� B cos2 wpþ1;

12;� apþ1

2

� �� 2g sin�apþ1 wpþ1 ¼ 0;

where

g ¼ 2Gð1� kÞkDekRpþ1

; apþ1 ¼hpþ1

2Gþ hpþ1: ð66Þ

Here, the factor k is defined by k ¼ kpð1� kp�1

ð1� . . .� k3ð1� k2ð1� k1ÞÞ . . .ÞÞ or in a more compact form as

k ¼ �Xpþ1�i

q¼1

Ypþ1�i

r¼q

ð�1Þ2rþ1kr: ð67Þ

Finally, the radius of the Mises circle at time tnþ1 is given by

Rnþ1 ¼ Rpþ1sin wpþ1

sin wnþ1

� �apþ1

; ð68Þ

and the new deviatoric stress state is expressed as

snþ1 ¼Rnþ1 sin wnþ1

Rn sin wnsn þ

Rnþ1 sinðwn � wnþ1ÞkDek sin wn

De: ð69Þ

Note that in this method presented above the implicit form of thetime function of w, (21) is especially beneficial. A summary of thealgorithm is detailed in Table 1.

4.3. Consistent tangent modulus

In this section, we discuss the linearization of the constitutiveintegration algorithm, leading to the so-called consistent or algo-rithmic tangent modulus. Displacement based finite elementmethods for elastoplastic problems are generally based on New-ton–Raphson techniques. It is well known that to maintain thequadratic rate of asymptotic convergence for Newton’s method,the tangent operator must be consistent with the numerical algo-rithm employed to integrate the plasticity rate equations [35,36].The closed-form expression for the consistent elastoplastic tangentmodulus tensor will now be given for the semi-analytical integra-

Table 1Algorithm for stress updating method with the piecewise linear hardening.

1 Input variables: sn;Rn;De; kDek;Wn;G and YIELD ¼ ½ðR1; h1Þ; ðR2;h2Þ; . . . ; ðRN ; hNÞ2.a IF RN P Rn THEN

The initial and final stress state are located in the last segment

Calculate g ¼ 2GkDekRn

; n ¼ hN

2G; aN ¼

nN

1þ nNand determine the angle Wnþ1 from

gðWnþ1;Wn;g; aNÞ ¼ 0

Calculate the output variable

Rnþ1 ¼ Rnsin Wn

sin Wnþ1

� �aN

; snþ1 ¼Rnþ1 sin Wnþ1

Rn sin Wnsn þ

Rnþ1 sinðWn �Wnþ1ÞkDek sin Wn

De

EXITELSE3.a DO i ¼ 1;N � 1

Get Ri ¼ YIELDði;1Þ;Riþ1 ¼ YIELDðiþ 1;1Þ;hi ¼ YIELDði;2Þ4.a IF Ri 6 Rn < Riþ1 THEN

The initial stress states is located in the segment i, Initialize kT ¼ 05.a DO j ¼ i;N � 1

Rjþ1 ¼ YIELDðjþ 1;1Þ;hj ¼ YIELDðj;2ÞIF i ¼ j THEN

Rj ¼ Rn;Wj ¼ Wn

ELSERj ¼ YIELDðj;1Þ;Wj ¼ Wjþ1

ENDIFCalculate nj ¼

hj

2G; aj ¼

nj

1þ nj;Wjþ1 ¼ arc sin sin Wj

Rj

Rjþ1

� � 1aj

" #,

kj ¼Rj sinaj Wj

2Gð1� kT ÞkDek B cos2 Wjþ1;12;�

aj

2

� �� B cos2 Wj;

12;�

aj

2

� �� 6.a IF kj P 1 THEN

The final stress state is located in the segment j

Calculate g ¼ 2Gð1�kT ÞkDekRj

and determine the angle Wnþ1 from

gðWnþ1;Wj;g; ajÞ ¼ 0Calculate the output variable

Rnþ1 ¼ Rjsin Wj

sin Wnþ1

� �aj

; snþ1 ¼ Rnþ1 sin Wnþ1Rn sin Wn

sn þRnþ1 sinðWn �Wnþ1ÞkDek sin Wn

De

EXIT6.b ENDIF

Calculate kT ¼ kjð1� kT Þ5.b END DO

The final stress state is located in the last segment

Calculate g ¼ 2Gð1� kT ÞkDekRN

; aN ¼hN

2Gþ hNand determine the angle

Wnþ1 from gðWnþ1;Wjþ1;g; aNÞ ¼ 0Calculate the output variable

Rnþ1 ¼ RNsin Wjþ1

sin Wnþ1

� �aN

; snþ1 ¼Rnþ1 sin Wnþ1

Rn sin Wnsn þ

Rnþ1 sinðWn �Wnþ1ÞkDek sin Wn

De

EXIT4.b ENDIF3.b END DO2.b ENDIF


tion algorithm presented in this paper. The method employed hereis based on the results for elastic-perfectly plastic case presentedby Ristinmaa [32] and Wei et al. [42].

By differentiating the stress with respect to the strain at timetnþ1, we obtain the formula

drnþ1 ¼ornþ1

oenþ1: de � ornþ1

oDe: de ¼ Dc : de; ð70Þ

where Dc is the consistent elastoplastic tangent modulus tensor,which, using (60), can be separated into a deviatoric and a spherical(hydrostatic) part, namely,

Dc ¼ornþ1

oDe¼ osnþ1

oDeoDeoDeþ K

otrenþ1

oDed ¼ osnþ1

oDe: Tþ Kd� d: ð71Þ

The deviatoric part, osnþ1=oDe can be computed from (51), yielding

osnþ1

oDe¼ a

os0

oDeþ s0 �

oaoDeþ De� ob

oDeþ bI: ð72Þ

The quantities os0=oDe, oa=oDe and ob=oDe, using (47), (49), (52)and (52), can be easily calculated, and we arrive the followingexpression for the consistent elastoplastic tangent moduli tensor:

Dc ¼ a1s0 � s0 þ a2s0 �Deþ a3De� s0 þ a4De�Deþ ð2Gkaþ bÞTþ Kd� d;

ð73Þ

where

a1 ¼ að1� aÞ B2

tan wnþ1� A2

tan w0

� �; a2 ¼ að1� aÞ B1

tan wnþ1� A1

tan w0

� �;

a3 ¼ A2C1 � B2C2 �2Gka

R0kDek cos w0; a4 ¼ A1C1 � B1C2 �

b

kDek2 ;

9>>>=>>>; ð74Þ

and the parameters Ai;Bi and Ci are defined by

A1 ¼1

kDek2 tan w0

� 2GkR0kDek sin w0

; A2 ¼ � kDekR0 cos w0

A1;

A3 ¼ R0 sin wnþ1Rnþ1

; A4 ¼ 1þag cos w0sin w0

;

B1 ¼ A3 A1A4 � gkDek2

�; B2 ¼ A3 A2A4 � 2Gk

R20 cos w0

�;

C1 ¼ab

tan w0þ Rnþ1 sin wnþ1

kDek sin2 w0

; C2 ¼ abtan wnþ1

þ Rnþ1 cosðw0�wnþ1ÞkDek sin w0

:

9>>>>>>>>>>=>>>>>>>>>>;ð75Þ

It is noted that the derived consistent (algorithmic) tangent opera-tor in Eq. (73) is not symmetric. The loss of symmetry is caused bythe fact that a2 – a3.

5. Numerical examples

In this section, four numerical examples are presented to inves-tigate the effectiveness and accuracy of the semi-analytical inte-gration method presented in this paper. In the first twoexamples, the input load histories are described by a non-propor-tional strain path with plane strain condition. In the third, one athin-walled cylindrical tube subjected to combined loads of ten-sion and torsion is examined by finite element method with onethree dimensional element. The finite element calculations are per-formed by finite element code ABAQUS [1] through the user sub-routine UMAT (some informations can be found in Appendix C).In the fourth example, a thin rectangular strip with circular hole,subject to uniaxial extension in a plane strain state is analysed.In these calculations, the radial return method built-in ABAQUSand the new integration method implemented in UMAT will becompared.

5.1. A non-proportional linear strain path

In this example, the accuracy of the proposed algorithm isinvestigated by a non-proportional linear strain path. Moreover,for comparison purposes, the results associated to the standard ra-dial return method is also presented.

5.1.1. Problem descriptionThe loading history is piecewise rectilinear in the tension and

shear and the strain path is given by the following straincomponents

ðe11; c12Þ :¼ ðe0; 0ÞA ! ðe; 0ÞB ! ðe;3eÞC ! �e;72e

� �D

! ð�3e;�eÞE ! e;�72e

� �F! 11

4e;0

� �G; ð76Þ

where e ¼ 5� 10�3 and the other components are zero. The graph-ical representation of this strain path is given in Fig. 2a, whereas theFig. 2b shows the strain path on the ðe11; c12Þ plane.

Assuming the plane strain case, the corresponding stress state isgiven as


r ¼r11 s12 0s21 r22 00 0 r33

264375: ð77Þ

Because in the plane strain component e22 equals to zero so thatr22 ¼ r33:

Consider the time interval t 2 ½t0;6�, and assume that initialstress state at t ¼ t0 corresponds to the elastic limit. Then, the ini-tial strains are defined as

e11ðt0Þ � e0 ¼ry

2G; c12ðt0Þ ¼ 0; ð78Þ

and the initial stress components are given by

r11ðt0Þ ¼1� m

1� 2mry; r22ðt0Þ ¼ r33ðt0Þ ¼

m1� 2m

ry;

s12 t0ð Þ ¼ 0: ð79Þ

It should be noted that in the whole strain path given here the out-put stress states remain plastic, i.e. between t0 and t ¼ 6 purelyplastic loading occurs.

5.1.2. Reference solutionFor the problem considered here, the constitutive relation,

using the Eq. (10), is defined by

_r11 ¼ Dep11

_e11 þ Dep14

_c12;

_r22 ¼ _r33 ¼ Dep21

_e11 þ Dep24

_c12;

_s12 ¼ Dep41

_e11 þ Dep44

_c12;

9>=>; ð80Þ

where

a

b

Fig. 2. Strain path history and exact stress response in Example 5.1; (a) input: time hiðe11; c12Þ strain space, (c) results: time function of the r11 and s12 stress components, (d

Dep11 ¼2G

1�m1�2m

�2ðr11�r22Þ2

3r2ð1þnÞ

!; Dep

21 ¼2Gm

1�2mþðr11�r22Þ2

3r2ð1þnÞ

!;

Dep44 ¼G 1� 3 s2

12

r2ð1þnÞ

� �; Dep

14 ¼Dep41 ¼�2Dep

24 ¼2Gs12ðr22�r11Þ

r2ð1þnÞ ;

9>>>>=>>>>; ð81Þ

and

r2 ¼ ðr11 � r22Þ2 þ 3s212: ð82Þ

The reference solution for the strain path defined by (76), is per-formed by MATHEMATICA (Wolfram Research, Inc. 2005 [44]), usingthe NDSolve procedures with high accuracy condition (Method ->‘‘StiffnessSwitching”).

5.1.3. Numerical resultsIn the numerical calculation, the whole strain path divided into

six time steps associated with points A–G. The numerical materialproperties employed are listed in Table 2.

The calculation is performed by the following initial values:e0 ¼ 1:3125� 10�3, r11ðt0Þ ¼ 385 ½MPa�, r22ðt0Þ ¼ r33ðt0Þ ¼175 ½MPa�, s12ðt0Þ ¼ 0. Solving the system of nonlinear ODEs de-fined above, the computed stress responses for the given strainpath can be seen in Fig. 2c and d. The results obtained by thenew integration method are summarized in Table 3.

The errors listed in this table are calculated by

da ¼ absaref � a�

aref

� �� 100;

where a� is one of the stress components evaluated by the new inte-gration method presented in this paper or radial return method,respectively, whereas aref is the reference solution for the samestress components obtained by the method mentioned above.

c

d

story of the prescribed e11 and c12 strain components, (b) input: strain path in the) results: stress output in the ðr11; s12Þ stress space.

Table 2Material properties for Examples 5.1.

Shear modulus G ¼ 8� 104 ½MPa�Poisson ratio v ¼ 0:3125Initial yield stress Gy ¼ 210 ½MPa�Linear hardening modulus H ¼ 104 ½MPa�

time [sec]

stre

ss r

elat

ive

erro

rs[%

]

10steps

50steps

200steps

200 steps

spets05

spets01

radial return method

new method

Fig. 3. Stress relative errors for 20, 50 and 200 time steps by the new and the radialreturn methods in Example 5.2.


To evaluate the accuracy of the proposed method a comparisonis made with standard radial return integration algorithm, and theresults are summarized in Table 4.

It is concluded that the new semi-analytical method, as ex-pected, is more accurate than the radial return method. For largetime step the results based on the new integration method canbe practically regarded as exact. However, for the same time step,the radial return method yields a large relative errors, especially inthe shear stress component.

5.2. A non-proportional nonlinear strain path

Attention is next focused on a nonlinear strain path, which is gi-ven by the following strain components

e11ðtÞ ¼ e0 þ 5ðsin tÞ � 10�3; c12ðtÞ ¼ 20ð1� cos tÞ � 10�3; ð83Þ

and the other components are zero. The stress state is defined by(77) as previously.

Consider the time interval t 2 ½0;2p� and assume that initialstress state corresponds to the elastic limit. The same materialproperties and the initial values have been used as in the Example5.1. The reference (used also for the ‘‘exact”) solution for the strainpath defined by (83), is performed also by MATHEMATICA solvingEq. (80).

Define the following relative error

dr ¼krref � r�kkrrefk � 100; ð84Þ

Table 3Strain path coordinates and the calculated stress components with relative errors obtaine

Time [s] Point label Strain components [�10�3] Stress components

e11 c12 r11 Erro

New semi-analytical integration method presented in this papert0 A 1.3125 0 385 01 B 5 0 1089.07 5:302 C 5 15 933.505 5:453 D �5 17.5 �1173.34 1:164 E �15 �5 �2953.07 1:545 F 5 �17.5 1304. 66 1:746 G 13.75 0 2822.54 6:65

Table 4Stress components and relative errors calculated by the radial return method in Example

Time [s] Point label Stress components [MPa]

r11 Error dr11

Radial return methodt0 A 385 01 B 1089.077 5:3089� 10�8

2 C 956.26 2.43763 D �1156.989 1.39364 E �2978.373 0.85675 F 1265.2 3.02476 G 2884.602 2.1987

where r� is the stress evaluated by the new integration method pre-sented in this paper or by the radial return method, whereas rref isthe ‘‘exact” solution obtained by the method mentioned in Section5.1.2. Here the stress tensor r is fulfilled by the computed stresscomponents according to (77). Fig. 3 compares the relative errors,dr, calculated by the new and the radial return methods for threedifferent time steps in logarithmic scale. It is seen from the figurethat the relative error shows a significant differences between thenew and the radial return method when the time step increasing.It is interesting to note that the new method proposed in this paperprovides a more accurate results for 50 time steps than the radialreturn method for 200 time steps.

The rate of convergence can be investigated, for example, by thetotal stress error measure considered previously by Auricchio andBeirão da Veiga [4] a slightly different form, which is defined by

TNr ¼

Dt2p

XN

n¼1

krrefn � r�nkkrref

n k� 100 � 1

N

XN

n¼1

drð Þn: ð85Þ

d by the new integration method in Example 5.1.

[MPa]

r dr11 r22 ¼ r33 Error dr22 s12 Error ds12

175 0 0 089� 10�8 855.467 6:5321� 10�8 0 054� 10�5 933.247 2:5064� 10�5 179.092 4:5724� 10�6

63� 10�5 �813.33 1:6882� 10�5 47.7131 1:2486� 10�5

25� 10�5 �2723.46 7:5719� 10�6 �258.033 2:5213� 10�5

98� 10�5 747.669 3:3671� 10�5 �175.286 1:6065� 10�5

48� 10�6 2438.73 7:7311� 10�6 357.981 1:9996� 10�8

5.1.

r22 ¼ r33 Error dr22 s12 Error ds12

175 0 0 0855.47 6:5321� 10�8 0 0921.87 1.2991 176.664 1.3557�821.506 1.0052 80.713 69.1623�2710.813 0.4644 �237.673 7.8906767.4 2.6391 �210.103 19.86242407.699 1.2724 299.012 16.4726

Table 5Stress total errors for different number of steps in Example 5.2.

No. of steps N Stress total errors TNr

New method Radial return method

10 4.22987 8.0282420 1.07909 4.2411750 0.174087 1.70428

100 0.0436293 0.85637200 0.0111058 0.42965400 0.00304728 0.215351

[Mpa]


The results are listed in Table 5 and plotted in Fig. 4. This figureshows the stress total errors versus the number of time steps in log-arithmic scale. In the case of radial return method the slope repre-sents a linear convergency, whereas the new method provides aquadratic one.

5.3. Non-proportional linear stress paths

The third example demonstrates the accuracy and convergenceof the proposed method in finite element analysis performing theso-called one element test. In this example, a classical problem,namely a thin-walled cylindrical tube which is subjected to axialtension and torsion will be analyzed. Because of the stress distribu-tion in the tube, remote from ends, is constant everywhere, and theproblem can be modeled by one element.

A similar example, using single one plane element, was treatedby Nyssen [29] and Wissmann and Hauck [43] (with isotropiclinear hardening), Genna [13], Szabó and Kovács [38] (with linearkinematic hardening), Benallal et al. [6], (with nonlinear isotropicand kinematic hardening including damage effect). Wei et al.[42] examined the same problem by using 8-node trilinear isopara-metric brick element. They have tested a consistently linearizedexact stress update algorithm for non-hardening model.

5.3.1. Problem descriptionThe non-proportional loading path in the two dimensional

ðr; sÞ stress space is plotted in Fig. 5 and the corresponding stresscoordinates are given in Table 6.

Consider a cube of unite volume subject to axial load and shear.The finite element calculations were performed by using adisplacement based 8-node trilinear isoparametric brick elementwith 2 � 2 � 2 Gauss integration (C3D8 element in ABAQUS code).The material properties used are given in Table 2 except theplastic hardening modulus considered is 5 � 103 ½MPa�. The ABAQUSinput file to this example can be downloaded from http://www.mm.bme.hu/~szabol/semianalinteg/example3.zip.

step number N


new method

rorrelatotsserts

TN

1

2

1

1

Fig. 4. Stress total errors versus number of time steps in Example 5.2.

5.3.2. Exact solutionThe analytical solution of the problem considered here is corre-

sponds to the time integration of the constitutive Eq. (11) with theconstant stress rate assumption. For a special load path (the tube isfirst stressed in tension to the yielding and is then twisted underconstant axial stress) Hill [16] gives the exact solution for the axialand shear strains with linear isotropic hardening. Here, the loadpath is more complicated, and we need to derive an exact solutionfor general case.

The constitutive relation, (11) for the constant stress rates canbe integrated exactly and solution for the deviatoric strain is givenin Appendix B. In the example investigated here, for a given stressincrements Dr ¼ Dt _c and Ds ¼ Dt _s with the initial values rn andsn at t ¼ tn, the strain increments D�ðtÞ and DcðtÞ, using the new re-sults presented in the Appendix B, can be calculated by

D�ðtÞ ¼ Dr 1Hþ 1

2Gð1þ mÞ

� �þ p 2

ffiffiffi3p

DrA� 3DsB �

; ð86Þ

DcðtÞ ¼ Ds 3Hþ 1

G

� �þ 3pð2DsAþ DrBÞ; ð87Þ

where

p ¼ snDr� rnDs2HðDr2 þ 3Ds2Þ ; ð88Þ

and

A ¼ arctan

ffiffiffi3p

rnDs� snDrð Þrn ðrn þ DrÞ þ 3snðsn þ DsÞ

" #;

B ¼ lnðrn þ DrÞ2 þ 3ðsn þ DsÞ2

r2n þ 3s2

n

" #: ð89Þ

For the loading history (see Fig. 5), using (86) and (87), the exacttime functions of the normal and shear strains can be calculated,and some reference values at the points labeled with A0;A;B andC are given in Table 6.

[Mpa]

Fig. 5. Loading path in the two dimensional ðr; sÞ stress space in Example 5.3.

Table 6Stress path coordinates and the corresponding exact strain values in Example 5.3.

Stress components Exact strain components

r [MPa] s [MPa] e c

A0 105 105 5� 10�4 1:3125� 10�3

A 150 150 9:71428571� 10�3 2:8875� 10�2

B 330 90 2:09690338� 10�2 4:12499200� 10�2

C 410 0 3:02022687� 10�2 4:24493367� 10�2

http://www.mm.bme.hu/~szabol/semianalinteg/example3.zip



It must be noted. In the example analyzed here is associated toplane stress state. Thus, because of the presented integrationmethod valid (applicable) for three dimensional or plane strain

c

b

a

Fig. 6. Relative errors in the normal and shear strain components for differentincrement numbers in Example 5.3.; (a) 2 increments, (b) 8 increments, (c) 16increments.

cases, the problem cannot be modeled by two dimensional ele-ment by a strain driven finite element calculation.

5.3.3. Numerical results and discussionThe numerical calculations are performed by the UMAT subrou-

tine based on the new semi-analytical integration method and bythe radial return method built-in ABAQUS. Because of the load pathuntil the point A is proportional, thus the numerical solution coin-cides with the exact one, our analysis is focused on the part of theload path from A to C. In the first calculation, the load path be-tween A and C is divided by 2, 8 and 16 increments, and the rela-tive errors of the axial and shear strains are plotted in Fig. 6. Theerrors are defined in the following form:

de ¼ abseex � ecal

eex

� �� 100; dc ¼ abs

cex � ccal

cex

� �� 100;

where ecal and ccal are the strain components calculated by the finiteelement method based on the new integration method presented inthis paper or on the radial return method, respectively, whereas eex

and cex are the exact solutions.Next, the relative errors of shear strain at the end of load path

(see point C in Fig. 5) are calculated for different number of incre-ments. The results is presented in Fig. 7 in logarithmic scale, andalso summarized in Table 7.

The new method shows a quadratic convergence rate while theradial return method provides linear one as seen in Table 7.

5.4. An initial boundary value problem: extension of a strip with acircular hole

5.4.1. Problem descriptionA thin perforated plate under plane strain subject to stretching

along its longitudinal axis is analysed in this example. This type ofproblem has been studied by many researchers, for example,Auricchio and Beirão da Veiga [4], de Souza Neto et al. [9], Kriegand Xu [21], Ristinmaa [32], Simo and Hughes [35].

Geometry, boundary condition and mesh of the problem areshown in Fig. 8. The strip has dimension 36 ½mm� � 20 ½mm� witha circular hole in the center (R ¼ 5 ½mm�) and thickness equal to1 ½mm�. Due to symmetry reason, only one quarter of geometry ismodeled. The numerical calculations are performed with eightnoded three dimensional linear brick element (C3D8). The mesh,seen in Fig. 11b, consists of 1474 elements with 3128 nodes.Through the thickness, only one element layer was employed.The strip is subjected the following boundary conditions: on thetop the displacement is described in z-direction umax ¼ 1 ½mm�)and kinematic constraints applied along the symmetry edges.Plane strain state is assumed. The material parameters are Young’s

increment number

gCtniopta

srorreevitaler

]%[

new method


Fig. 7. Relative error of shear strain versus increment numbers at the end of loadpath (point C).

Table 7Relative errors and convergence rates of the shear strain at point C for the new semi-analytical and the radial return methods.

Increment number New method Radial return method

dc [%] convergence rate dc [%] Convergence rate

Error dc [%] at point C and convergence rate10 0.954019 – 3.683921 –

20 0.259165 lnð0:954019Þ�lnð0:259165Þlnð20Þ�lnð10Þ ¼ 1:880 1.903175 lnð3:683921Þ�lnðl:903175Þ

lnð20Þ�lnð10Þ ¼ 0:953

50 0.042540 lnð0:259165Þ�lnð0:042540Þlnð50Þ�lnð20Þ ¼ 1:972 0.775881 lnð1:903175Þ�lnð0:775881Þ

lnð50Þ�lnð20Þ ¼ 0:989


lnð100Þ�lnð50Þ ¼ 0:991


lnð200Þ�lnð100Þ ¼ 0:996


lnð400Þ�lnð200Þ ¼ 0:998

y

x

b

a

Fig. 8. Strip with a circular hole in Example 5.4.; (a) geometry and boundarycondition, (b) finite element mesh.

radial return methodnew method

increment number

uAtniopta

srorreevitaler

]%[

Fig. 9. Relative errors of displacement uA versus number of increments.

increment number

petse

mitrep

emit

UPC


new method

Fig. 10. Average CPU time of the new and redial return method in Example 5.4.


modulus E ¼ 2� 105 ½MPa�, Poisson ratio m ¼ 0:3, the yield stressry ¼ 150 ½MPa�, and constant plastic hardening modulusH ¼ 4� 104 ½MPa�.

The problem is analysed using the radial return method (built-in ABAQUS) and the new integration method implemented into theUMAT. The convergence criterion in the equilibrium Newton itera-tion was adjusted to Ra

n ¼ 5� 10�8, while the ABAQUS default va-lue is Ra

n ¼ 5� 10�3: The ABAQUS input file to this example canbe downloaded from http://www.mm.bme.hu/~szabol/semianalin-teg/example4.zip.

Table 8Displacement uA; relative errors du; CPU time and total iteration number versus number o

Time step New semi-analytical integration method

Number Dt [s] (�10�3) uA [mm] Error dU [%] (�10�4) CPU [s] Total iteratio

50 20 0.463798 107.602 161.7 179100 10 0.463760 25.445 286.0 313200 5 0.463751 7.3531 522.1 535300 1/3 0.463750 3.1483 722.2 722400 2.5 0.463749 1.7682 952.8 907

5.4.2. Numerical resultsIn the calculation, the displacement uA of point A (see Fig. 8a)

along the x direction was monitored, and five differenttime steps were considered: Dt ¼ 0:02 ½s�;Dt ¼ 0:01 ½s�;Dt ¼

f increments in Example 5.4.

Radial return method built in ABAQUS

n number uA [mm] Error dU [%] (�10�4) CPU [s] Total iteration number

0.463904 336.174 138.3 1590.463815 144.259 243.1 2780.463779 66.631 450.1 4720.463768 42.911 642.8 6680.463763 32.130 855.7 846



increment number

rebmun

noitaretilatotpets

emit

rep new methodradial return method

Fig. 11. Average number of total iteration for the new and radial return method inExample 5.4.


0:005 ½s�;Dt ¼ 1=300 ½s�;Dt ¼ 0:0025 ½s�. The results obtained aresummarized in Table 8.

The relative error is calculated according to

du ¼ absuref

A � uA

urefA

� �� 100;

where the displacement value urefA is the reference solution evalu-

ated with the proposed integration method using a very fine timestep (Dt ¼ 0:0005 ½s�, 2000 increments, uref

A ¼ 0:4637481 ½mm�).The relative errors for different increments (time steps) are shownin Fig. 9 in logarithmic scale.

The Fig. 9 and Table 8 clearly demonstrate that the proposedmethod converges quadratically at the global level. The used CPUtime and the total iteration numbers of the calculations with theproposed method (UMAT) and the radial return method built-inABAQUS are also listed in Table 8, and their average values areplotted on Figs. 10 and 11 in logarithmic scale.

These numerical results demonstrate that the semi-analyticalintegration method has better convergence and accuracy proper-ties than the radial return method, although no significant differ-ences regarding to the iteration numbers and CPU times. In

Fig. 12. MATHEMATICA modules to the calculation of a

addition, we remark that in the calculations based on the newmethod the tangent stiffness is unsymmetric.

6. Conclusion

In the present paper, a semi-analytical integration method forvon Mises elastoplasticity with linear isotropic hardening has beenpresented under the constant strain rate assumption. In the newsolution, the time function of the angle w (defined in the scalarproduct of the stress deviator and the strain rate) is expressed inimplicit form through the use of the incomplete beta function. Toovercome the difficulties due to the implicit nature of the solution,for a given time (strain) increment, the method proposed by Dom-inici [10] has been applied to derive an expression for the angle w.In addition, it was shown that the method presented in this paperis also applicable to the Drucker–Prager model.

Sufficient details of the stress updating algorithm and the gen-eration of a consistent tangent operator were presented. Thesealgorithms are implemented in finite element code ABAQUSthrough the user subroutine UMAT. Accuracy and efficiency ofthe present method are demonstrated on numerical examples.

The numerical algorithms presented in this paper are valid forlinear isotropic hardening. However, they maybe extendable fornonlinear isotropic hardening models, i.e., when the function RðcÞis nonlinear (e.g., a saturation hardening term of the exponentialtype, or simple power hardening), using piecewise linearizationtype arguments. This consideration was illustrated in Section 3.2.Finally, it is important to note that the method can also be ex-tended to combined linear isotropic and linear kinematic harden-ing models.

Acknowledgement

This research has been supported by the National Developmentand Research Foundation, Hungary (under Contract: OTKA, K72572). This support is gratefully acknowledged.

ngle wnþ1 based on the nested derivative method.


Appendix A. A solution algorithm to determine the angle wnþ1:

In this section, an algorithm is presented to calculate the anglewnþ1 defined in the nonlinear equation appearing in (49)

gðwnþ1Þ : ¼ B cos2 wnþ1;12;� a

2

� �� B cos2 w0;

12;� a

2

� �� 2g sin�a w0 ¼ 0; ðA:1Þ

where

g ¼ 2Gð1� kÞkDekR0

; cos w0 ¼s0 : DeR0kDek : ðA:2Þ

For large strain increment or when parameter a is very small, theangle wnþ1 tends to zero and the function B½cos2 wnþ1;

12 ;� a

2� ! 1.In this case, it is better to use the symmetry relation

B cos2 wnþ1;12;� a

2

� �¼

ffiffiffiffipp

C � a2

� C 1�a

2

� � B sin2 wnþ1;�a2;12

� �; ðA:3Þ

where C½x� is the gamma function. Then, the Eq. (A.1) can be rewrit-ten as

B sin2 wnþ1;�a2;12

� �¼ zða;g;w0Þ; ðA:4Þ

where

zða;g;w0Þ � z ¼ffiffiffiffipp

C � a2

� C 1�a

2

� � B cos2 w0;12;� a

2

� �� 2g sin�a w0:

ðA:5Þ

To obtain the angle wnþ1, the expression (A.4) may be formally in-verted to give

wnþ1 ¼ arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB�1 z;� a

2;12

� �s: ðA:6Þ

This result may indeed be useful in computer environments whereinternal representations of inverse incomplete beta functions areavailable. Alternatively, if the incomplete beta function is available(e.g., MATHEMATICA, MATLAB), one can obtain the inverse byemploying a standard iterative method such as the Newton or thebisection methods.

Here we present a method proposed by Dominici [10], whichgives an algorithm to compute the series expansion for the inverseof a given function, hðxÞ, and it is especially powerful when thefunction has the form

hðxÞ ¼Z x

0

1f ðxÞ dx; ðA:7Þ

and f ðxÞ is some function simpler than hðxÞ.Following Dominici [10], we define the function f ðxÞ as the reci-

procal of the argument of the integral in (22) as follows:

f ðxÞ ¼ x1�mð1� xÞ1�l: ðA:8Þ

In addition, we consider Dm½f �ðxÞ the mth nested derivative of thefunction f ðxÞ, defined by the following recursion (see Dominici[10]):

D0½f �ðxÞ ¼ 1;

Dm½f �ðxÞ ¼ ddx½f ðxÞ �Dm�1½f �ðxÞ�; m P 1:

ðA:9Þ

Then, the inverse of the incomplete beta function can be computedby the following series expansion

B�1½z; m;l� ¼ bþ f ðbÞXmP1

Dn�1½f �ðbÞ ðz� z0Þm

m!; ðA:10Þ

where z0 ¼ B½b; m;l�, with f ðbÞ– 0;1 and jz� z0j < e, for somee > 0. Here 0 < b < 1 is an arbitrary constant.

The inverse of the incomplete beta function in (A.6), using themethod defined above can be calculated as

sin2 wnþ1 ¼ B�1 z;� a2;12

� �¼ sin2 ~wþ gð~wÞ

XmP1

Dm�1½g�ð~wÞ ðz�~zÞm

m!; ðA:11Þ

where ~w is an approximation of wnþ1, with ~z ¼ B sin2 ~w;� a2 ;

12

h i, and

the function gðyÞ is defined as

gðyÞ ¼ cos y sin2þa y: ðA:12Þ

The mth nested derivatives of the function gðyÞ defined by the fol-lowing recursion:

D0½g�ðyÞ ¼ 1;

Dm½g�ðyÞ ¼ 1sin 2y

ddy½gðyÞ �Dm�1½g�ðyÞ�; m P 1:

ðA:13Þ

We note that, if ~w is a good estimate of the angle wnþ1, then the ser-ies expression in (A.11) converges quite rapidly.

From the input parameters R0; n; ða ¼ n=ð1þ nÞÞ;g and w0; theangle wnþ1 can be derived as follows.

First, an approximation value of the radius of the Mises circle,using the radial return method may be calculated

eR ¼ R01þ n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2g cos w0 þ g2

p1þ n

: ðA:14Þ

Then, from the expression (18) a good estimate for the value of wnþ1

is derived as

sin ~w ¼ R0eR� �1

a

sin w0: ðA:15Þ

With ~w the parameter ~z is calculated

~z ¼ B sin2 ~w;� a2;12

� �: ðA:16Þ

From (A.5) and (A.11)–(A.13) the quantity B�1 z;� a2 ;

12

� can be calcu-

lated, and finally, using (A.6), the angle wnþ1 can be determined.The parameter range of practical interest is defined as:

0 < w0 6p2 ;0 < n 6 0:9 and 0 < g 6 20 with kDek 6 10�2 and

0 < 2G=R0 6 200. We note that within this parameter range the firstseven terms of the nested derivatives given in (A.13) the maximumrelative error less than 10�8: This algorithm is summarized in Box 1.

Input variables: G;H; s0;De and kStep 1: Calculate n ¼ H

3G ; a ¼n

1þn ; kR0k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffis0 : s0p

; kDek ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDe : Dep

;w0 ¼ arccos s0 :DeR0kDek and g ¼ 2Gð1�kÞkDek

R0

Step 2: Compute ~w ¼ arcsin 1þn

1þnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ2g cos w0þg2p

� �1a

sin w0

" #

Step 3: Compute ~z ¼ B½sin2 ~w;� a2 ;

12� and z ¼

ffiffiffipp

C �a2½ �

C 1�a2½ ��

B cos2 w0;12 ;� a

2

� � 2g sin�a w0

Step 4: Compute u ¼ sin2 ~wþ cos ~w sin2þa ~wP8

m¼1Q m�1ð~wÞðz�~zÞm

m!

Step 5: wnþ1 ¼ arcsinffiffiffiup


The parameters Q i; i 2 ½0;7� in Box 1. are defined as Q 0 ¼ 1 and

Q mðyÞ ¼1

sin 2yd

dy½ðcos y sin2þa yÞ � Q m�1ðyÞ�; m P 1: ðA:17Þ

For the sake of completeness the MATHEMATICA modules for thecalculation of wnþ1 are given in Fig. 12.

Appendix B. Time integration of constitutive equations with theconstant stress rate assumption

In this section, the exact time integration of constitutive Eq. (18)with constant stress rate assumption is presented. This result pro-vides an analytical solution for the time history of deviatoric strain,eðtÞ, when the load path is linear (piecewise rectilinear) in thestress space and linear isotropic hardening materials areconsidered.

We start our developments by inverting the deviatoric rate con-stitutive equation. The inverse of (11)1 is defined by therelationship

_e ¼ 12G

_sþ 1

2GR2nsðs : _sÞ: ðB:1Þ

When the deviatoric stress rate, _s; is constant during the time stepDt; then, the deviatoric stress can be expressed in the time intervalt 2 ½tn; tn þ Dt� as

sðtÞ ¼ sn þ Dt _s � sn þ Ds; ðB:2Þ

where sn ¼ sðtnÞ:Substituting sðtÞ defined by (B.2) into (B.1), then integrating, we

get

De ¼ 14Gn

2ð1þ nÞDsþ ln½r2 þ 2r cos xn þ 1� sn �cos xn

rDs

�� 2 sin xn

rarctan

r sinxn

1þ r cos xn

� �Ds

� �; ðB:3Þ

where

Rn ¼ ksnk; r ¼ kDskRn

; cos xn ¼sn : DsRnkDsk : ðB:4Þ

In the case of radial loading, xn ¼ 0, and the deviatoric strain incre-ment can be calculated in the following simple form

De ¼ 1þ n2Gn

Ds: ðB:5Þ

Note that, the expression (B.3) gives a reference solution to theanalysis of a numerical integration method. Furthermore, it pro-vides, for example, the results presented by Hill for thin-walledtube in torsion and extension (see, p. 73 in [16]), and Lubliner forthin-walled tube in torsion and extension or pressure and axialforce loading (see, pp. 165–170 in [26]).

Appendix C. UMAT subroutine based on the new semi-analyticalintegration method

The complete Fortran source code can be downloaded fromhttp://www.mm.bme.hu/~szabol/semianalinteg/umatsemi.zip. Notethat the UMAT subroutine contains a part which calculates theincomplete beta function. These subroutines are BETAINC, BETAand GAMMA downloaded from http://jin.ece.uiuc.edu/routines/mincob.for. In the procedure used in these subroutines, the calcu-lation of the incomplete beta function, B½x; m;l� requires that m > 0and l > 0: The incomplete beta function with the given argumentsdefined in this paper can be easily transformed to the appropriateform (see Spanier and Oldham [37]. p. 576.) as

B cos2 x;12;�a

2

� �¼ 1�1

a

� �B cos2 x;

12;1�a

2

� �þ2

acosxsin�a x; ðC:1Þ

B sin2 x;�a2;12

� �¼ 1�1

a

� �B sin2 x;1�a

2;12

� ��2

acosxsin�a x: ðC:2Þ

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a semi-analytical integration method for j2 flow theory of plasticity with linear isotropic...

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