Constitutive modeling and finite element methods for TRIP ... - CiteSeerX

Comput. Methods Appl. Mech. Engrg. 195 (2006) 5094–5114 www.elsevier.com/locate/cma

Constitutive modeling and finite element methods for TRIP steels I. Papatriantafillou, M. Agoras, N. Aravas *, G. Haidemenopoulos Department of Mechanical and Industrial Engineering, University of Thessaly, Pedion Areos, Volos 38334, Greece Received 28 February 2005; received in revised form 23 September 2005; accepted 27 September 2005

Abstract A constitutive model that describes the mechanical behavior of steels exhibiting ‘‘TRansformation Induced Plasticity’’ (TRIP) during martensitic transformation is presented. Multiphase TRIP steels are considered as composite materials with a ferritic matrix containing bainite and retained austenite, which gradually transforms into martensite. The effective properties and overall behavior of TRIP steels are determined by using homogenization techniques for nonlinear composites. A methodology for the numerical integration of the resulting elastoplastic constitutive equations in the context of the finite element method is developed and the constitutive model is implemented in a general-purpose finite element program. The model is calibrated by using experimental data of uniaxial tension tests in TRIP steels. The problem of necking of a bar in uniaxial tension is studied in detail. The constitutive model is used also for the calculation of ‘‘forming limit diagrams’’ for sheets made of TRIP steels; it is found that the TRIP phenomenon increases the strain at which local necking results from a gradual localization of the strains at an initial thickness imperfection in the sheet.  2005 Elsevier B.V. All rights reserved. Keywords: TRIP steels; Plasticity; Finite element methods

1. Introduction The TRansformation Induced Plasticity (TRIP) phenomenon has led to the development of a new generation of multiphase low-alloy steels that exhibit an enhanced combination of strength and ductility satisfying the requirements of automotive industry for good formability and high-strength. The excellent combination of mechanical properties of TRIP steels is attributed to their complex microstructure; proper heat treatment produces steels consisting of dispersed retained austenite and bainite in a ferritic matrix. The retained austenite is metastable at room temperature and, under the effect of stress and/or plastic deformation, transforms to martensite. In particular, at temperatures just above the so-called Ms temperature, transformation can be induced via stress-assisted nucleation on the same sites which trigger the spontaneous transformation on cooling, but now assisted by the thermodynamic effect of applied stress [28]. Above a temperature M rs , the transformation stress exceeds the flow stress of the parent phase and transformation is preceded by significant plastic deformation; this is known as strain-induced nucleation and involves the production of new nucleation sites by plastic deformation [27]. Finally, above a temperature Md, no transformation is observed prior to fracture. In the present paper we focus our attention to the temperature range above M rs and below Md, where the dominant nucleation mechanism is strain-induced. Several authors have developed constitutive models for the mechanical behavior of dual-phase TRIP steels [22,23,34, 8–11,45,5–7,3,4,12,14]. Herein, we build on the work of Stringfellow et al. [34] and develop constitutive equations for four-phase TRIP steels. In particular, we consider TRIP steels that consist of a ferritic matrix with dispersed bainite and austenite, which transforms gradually into martensite as the material deforms plastically. *

Corresponding author. Tel.: +30 2421 074002; fax: +30 2421 074009. E-mail address: [email protected] (N. Aravas).

0045-7825/$ - see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2005.09.026

I. Papatriantafillou et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 5094–5114

5095

TRIP steels are essentially composite materials with evolving volume fractions of the individual phases. A constitutive model for TRIP steels is developed in this paper. The total strain is assumed to be the sum of elastic, plastic and transformation parts. The plastic part is determined by using homogenization techniques for nonlinear composites that have been developed recently by Ponte Castan˜eda, Suquet and co-workers [31,32,35–37]. The transformation strain rate has both deviatoric and volumetric parts and is proportional to the rate of change of the volume fraction of martensite. The evolution of martensite due to martensitic transformation is described by a kinetic model, which takes into account temperature, plastic strain and stress state. The model proposed considers the different hardening behavior of the individual phases and estimates the different levels of strain accumulated in the constituent phases. A methodology for the numerical integration of the resulting nonlinear constitutive equations is developed and implemented in a general-purpose finite element program (ABAQUS). Experimental data from interrupted tensile tests in specific TRIP steel are used for the calibration of the constitutive model. The model is used together with the finite element method for the analysis of necking in uniaxial tension. The constitutive model is used also for the calculation of ‘‘forming limit diagrams’’ for sheets made of TRIP steels. Standard notation is used throughout. Boldface symbols denote tensors the orders of which are indicated by the context. All tensor components are written with respect to a fixed Cartesian coordinate system with base vectors ei (i = 1, 2, 3), and the summation convention is used for repeated Latin indices, unless otherwise indicated. The prefice det indicates the determinant, a superscript T the transpose, a superposed dot the material time derivative, and the subscripts s and a the symmetric and anti-symmetric parts of a second order tensor. Let a, b be vectors, A, B second-order tensors, and C a fourth-order tensor; the following products are used in the text (a b)ij = ai bj, A : B = Aij Bij, (A Æ B)ij = Aik Bkj, (A B)ijkl = Aij Bkl, (C : A)ij = Cijkl Akl, and (C : D)ijkl = Cijpq Dpqkl. The inverse C1 of a fourth-order tensor C that has the ‘‘minor’’ symmetries Cijkl = Cjikl = Cijlk is defined so that C : C1 = C1 : C = I, where I is the symmetric fourth-order identity tensor with Cartesian components Iijkl = (dik djl + dil djk)/2, dij being the Kronecker delta.

2. Description of the constitutive model In this section, a constitutive model that describes the mechanical behavior of TRIP steels is developed. TRIP steels are essentially composite materials with evolving volume fractions of the individual phases. In particular, we consider four-phase TRIP steels that consist of a ferritic matrix with dispersed bainite and austenite, which transforms gradually into martensite as the material deforms plastically. The following labels are used for the individual phases: (1) or (m) for martensite, (2) or (a) for austenite, (3) for bainite, and (4) for ferrite. No restriction is placed on the magnitude of the deformation and appropriate ‘‘finite strain’’ cosntitutive equations are developed. An important aspect of the martensitic transformation is the strain softening which occurs due to the strain associated with the transformation process. This strain softening is accounted for by introducing in the constitutive model an additional deformation rate that is proportional to the rate of increase of the volume fraction of martensite. The total deformation rate is written as the sum of the elastic, plastic and transformation parts: D ¼ De þ Dp þ DTRIP .

ð1Þ

In the following we discuss in detail the constitutive equations for the individual parts of D, and the evolution equations of the volume fraction of the phases. 2.1. Elastic constitutive relations The elastic properties of all four phases are essentially the same and the TRIP steel can be viewed as homogeneous in the elastic region. Standard isotropic linear hypoelasticity is assumed and the constitutive equation for De is written as r

De ¼ M e : r

or

r

r ¼ Le : D e ;

ð2Þ

r

where r is the Jaumann derivative of the stress tensor r, M e is the elastic compliance tensor defined as Me ¼

1 1 K þ J; 2l 3j

Le ¼ M e

1

¼ 2lK þ 3jJ;

1 J ¼ dd; 3

K ¼ I  J;

ð3Þ

l and j denote the elastic shear and bulk moduli, d and I the second- and symmetric fourth-order identity tensors, with Cartesian components dij (the Kronecker delta) and Iijkl = (dik djl + dil djk)/2.

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2.2. Constitutive equations for Dp The TRIP steel is viewed as a four-phase composite in which the isotropic, viscoplastic phases are distributed statistically uniformly and isotropically. The phases are assumed to be plastically incompressible and the corresponding constitutive equations are written in terms of a ‘‘viscous potentials’’ W(r) = W(r)(req): DpðrÞ ¼

oWðrÞ 1 ¼ _ pðrÞ N ¼ hðrÞ s; 2 or

ð4Þ

with 3 N¼ s; 2req

_ pðrÞ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pðrÞ oWðrÞ D : DpðrÞ ¼ ; 3 oreq

hðrÞ ¼

3_ pðrÞ 3 oWðrÞ ¼ ; req oreq req

ð5Þ

where Dp(r) is the plastic part of the deformation rate tensor of phase r (r = 1, 2, 3, 4), s is the deviatoric part of the stress (r) tensor,phffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 1/l(r) is the viscous shear compliance of phase r that depends on the von Mises equivalent stress req ¼ ð3=2Þs : s, l(r)(req) is the viscous shear modulus, and pðrÞ is the equivalent plastic strain. Our goal is to determine the corresponding viscoplastic constitutive equation for the ‘‘composite’’ TRIP steel, i.e., an equation of the form Dp = Dp(r). Ponte Castan˜eda [31,32] developed a variational procedure for the determination of the effective properties of such nonlinear composites. Suquet [35–37] showed that a certain variation of the ‘‘secant method’’ (the so-called ‘‘modified secant method’’) is identical to the variational procedure of Ponte Castan˜eda. According to these theories, the constitutive equation for the homogenized TRIP steel is of the form Dp ¼ M hom : r ¼

1 s; 2lhom

M hom ðreq Þ ¼

1 K; 2lhom ðreq Þ

ð6Þ

where M hom is the effective viscous compliance tensor that depends on req, and lhom = 1/hhom is the overall viscous shear modulus of the homogenized medium. According to the modified secant method, the procedure for the determination of hhom is as follows. Let h denote the collection h  (h(1), h(2), h(3), h(4)). Then, for a given macroscopic stress state r, the modified secant method involves three steps: • A linear theory that provides and expression for M hom ðhÞ ¼ 12 hhom ðhÞK and its partial derivatives with respect to h(r). • The solution of 2 · 4 = 8 nonlinear equations for the eight unknowns ðhðrÞ ; rðrÞ eq Þ: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 oWðrÞ 1 ohhom ðrÞ ; r ¼ r ; ð7Þ hðrÞ ¼ ðrÞ eq eq ðrÞ cðrÞ ohðrÞ req oreq P4 where c(r) is the volume fraction of phase r ð r¼1 cðrÞ ¼ 1Þ. hom • Once the above eight nonlinear equations are solved for ðhðrÞ ; rðrÞ ðhÞ ¼ 12 hhom ðhÞK is found, and the overall plaseq Þ, M tic deformation rate is determined form the equation 1 Dp ¼ M hom ðhÞ : r ¼ hhom ðhÞs. 2

ð8Þ

We use the above modified secant method procedure in order to determine hhom(req). In particular, the individual phases are assumed to be ‘‘power law’’ materials with viscous potentials of the form ðrÞ

_0 rðrÞ req y  W ðreq Þ ¼ ðrÞ m þ 1 rðrÞ y

!mðrÞ þ1

ðrÞ

ð9Þ

;

so that _ pðrÞ



¼

ðrÞ _ 0

req ðrÞ

ry

!mðrÞ and

h

ðrÞ

ðrÞ

¼

ðrÞ

req

ðrÞ

ry

3_0 ry

ðrÞ

!mðrÞ 1 ;

ð10Þ

where ryðrÞ is a reference stress, _ 0 a reference strain rate, and m(r) the strain-rate sensitivity exponent. In the limit as m(r) ! 1, the response of the phases is rate-independent perfectly-plastic with a yield stress ryðrÞ . Since all phases are assumed to be distributed uniformly in the ferritic matrix, the following Hashin–Strikman linear estimate is used for the determination of hhom (see for example [46, p. 673])

I. Papatriantafillou et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 5094–5114

0 1 hhom

4 BX ¼B @ r¼1

cðrÞ

10

5097

11

4 CBX C cðrÞ CB C . 1 3 A@ r¼1 1 3 A þ þ hðrÞ 2hð4Þ hðrÞ 2hð4Þ

hðrÞ

ð11Þ

The last expression defines the required linear-theory expression for M hom ðhÞ ¼ 12 hhom ðhÞK in the modified secant procedure. The details of the calculation of hhom are given in Appendix A. 2.3. Constitutive equations for the transformation part DTRIP We follow Stringfellow et al. [34] and assume that DTRIP can be written as 1 DTRIP ¼ Af_ N þ _ pv d; 3

ð12Þ

is the transformation dilawhere A is a dimensionless constant, f = c(1) is the volume fraction of martensite, and _ pv ¼ DTRIP kk tation rate. Note that Dpkk ¼ 0, so that _ pv is the total non-elastic dilatational rate. The transformation deformation rate given in (12) consists of a deviatoric term that models the transformation shape strain and a dilational term accounting for the volume change associated with the transformation. The parameter A depends on the von Mises equivalent stress req and is defined by the equation [34] req ð13Þ A ¼ A0 þ A1  ; sa where A0 and A1 are dimensionless constants, and sa is a reference austenitic stress. The transformation dilatation rate _ pv is determined as follows. When martensitic transformation occurs, a volume (dV(a)) of austenite is transformed to martensite with corresponding volume dV(m). The relative volume change Dv associated with the transformation is dV ðmÞ  ðdV ðaÞ Þ dV ðmÞ þ dV ðaÞ ; ¼ ðaÞ dV ðaÞ ðdV Þ which implies that V_ ðmÞ dV ðmÞ or V_ ðaÞ ¼  . dV ðaÞ ¼  1 þ Dv 1 þ Dv Dv ¼

ð14Þ

ð15Þ

We consider next a representative volume element (RVE) of the TRIP steel with total volume V, and let V(m) = V(1), V(a) = V(2), V(3) and V(4) denote the corresponding volumes of martensite, austenite, bainite and ferrite in the RVE, so that V = V(m) + V(a) + V(3) + V(4). Since the volume of bainite and ferrite do not change when martensitic transformation occurs ðV_ ð3Þ ¼ V_ ð4Þ ¼ 0Þ, we have that V_ ¼ V_ ðmÞ þ V_ ðaÞ , which, in view of (15b), can be written as V_ ¼

Dv _ ðmÞ V . 1 þ Dv

ð16Þ

Since changes in volume due to elastic strains are small and fully recoverable, it is assumed that changes in dilatation are due to non-elastic volumetric deformation rates only. Since Dpkk ¼ 0, this implies that _ pv ¼ V_ =V , which in view of the last equation, can be written as _ pv ¼

Dv V_ ðmÞ 1 þ Dv V

or

V_ ðmÞ 1 þ Dv p ¼ _ . V Dv v

ð17Þ

Also, the definition f = V(m)/V implies that V_ ðmÞ V_ f . f_ ¼ V V

ð18Þ

Using (17) and the definition V_ =V ¼ _ pv in the last equation, we conclude that 1  ð1  f ÞDv p _ v f_ ¼ Dv

or

_ pv ¼

Dv f_ . 1  ð1  f ÞDv

ð19Þ

Le´al [21] found experimentally that Dv ’ 0.02 to 0.05 in austenitic steels. Therefore, the last equation can be written as _ pv ’ Dv f_ .

ð20Þ

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Finally, Eq. (12) for DTRIP can be written in the form   1 TRIP D ¼ AN þ Dv d f_ . 3

ð21Þ

2.4. Evolution of the volume fraction of the phases Stringfellow et al. [34] extended the one-dimensional model of Olson and Cohen [27] for the kinetics of martensitic transformation and developed a general model that takes into account the effects of stress triaxiality on the process. The model of Olson and Cohen [27] is based on the observation that strain-induced nucleation occurs predominately at shear-band intersections. Stringfellow et al. [34] derive an evolution equation for the volume fraction of martensite f = c(1) of the form _ f_ ¼ cðaÞ ðAf _ pðaÞ þ Bf RÞ; (a)

ð22Þ

(2)

where c = c is the volume fraction of austenite, pðaÞ ¼ pð2Þ is the representative equivalent plastic strain in the austenite, and R is the stress triaxiality defined as the ratio of the hydrostatic stress p = rkk/3 to the von Mises equivalent stress req, i.e., R = p/req. The parameters Af and Bf that enter the evolution equation of f are defined as [34] Af ¼ ab0 rð1  fsb Þðfsb Þ

r1

"

P;

 2 # g2 1 g  g r _ Bf ¼ pffiffiffiffiffiffi b0 ðfsb Þ exp  H ðRÞ; 2 sg 2psg

ð23Þ ð24Þ

where H is the Heaviside unit step function, fsb is the volume fraction of shear bands in the austenite defined by [27]   ð25Þ fsb ¼ 1  exp apðaÞ ; a is a constant that represents the rate of shear band formation dfsb =dpðaÞ in the austenite at low strains, P(g) is the probability that a shear band will act as a nucleation site for martensite and is defined as "  2 # Z g 1 1 g0  g P ðgÞ ¼ pffiffiffiffiffiffi exp  ð26Þ dg0 ; 2 sg 2psg 1 g is a normalized thermodynamic driving force defined as gðH; RÞ ¼ g0  g1 H þ g2 R;

ð27Þ

(g0, g1, g2) are positive dimensionless constants, H is a normalized temperature, which is related to temperature T according to the equation T  M rs;ut HðT Þ ¼ ; M rs;ut 6 T 6 M d;ut ; ð28Þ M d;ut  M rs;ut and ðM rs;ut ; M d;ut Þ are the absolute ðM rs ; M d Þ temperatures for uniaxial tension. We discuss next the evolution equations for c(a), c(3) and c(4). Starting with the definition c(3) = V(3)/V, we find ð3Þ c_ ¼ cð3Þ V_ =V . Taking into account that V_ =V ¼ _ pv ’ Dv f_ , we conclude that c_ ð3Þ ¼ cð3Þ Dv f_ .

ð29Þ

Similarly, we find c_ ð4Þ ¼ cð4Þ Dv f_ .

ð30Þ

Finally, since f + c(a) + c(3) + c(4) = 1, we find c_ ðaÞ ¼ ðf_ þ c_ ð3Þ þ c_ ð4Þ Þ, so that c_ ðaÞ ¼ ½1  ðcð3Þ þ cð4Þ ÞDv f_ .

ð31Þ

2.5. Summary of constitutive equations for TRIP steels The constitutive model developed in the previous sections can be summarized as follows: D ¼ De þ Din ; r

De ¼ M e : r

or

r

r ¼ Le : D e ;

ð32Þ ð33Þ

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1 Din ¼ Dp þ DTRIP ¼ _ q N þ _ pv d; 3

5099

ð34Þ

where N¼

3 s; 2req

1 _ q ¼ req hhom þ Af_ ; 3

_ _ pv ¼ Din kk ¼ Dv f ;

ð35Þ

and the evolution of the state variables is defined by the following equations: _ f_ ¼ c_ ð1Þ ¼ cðaÞ ðAf _ pðaÞ þ Bf RÞ; c_ ðaÞ ¼ c_ ð2Þ ¼ ðf_ þ c_ ð3Þ þ c_ ð4Þ Þ; c_

ð3Þ

c_

ð4Þ

ð36Þ ð37Þ

¼ c Dv f_ ; ¼ cð4Þ Dv f_ . ð3Þ

ð38Þ ð39Þ

In the equations stated above, the quantities hhom, A, Af and Bf have the following functional dependencies: hhom ¼ hhom ðreq ; cðrÞ Þ;

A ¼ Aðreq Þ;

Af ¼ Af ðpðaÞ ; RÞ

_ Bf ¼ Bf ðpðaÞ ; R; RÞ.

ð40Þ

We conclude this section with an alternative presentation of the constitutive equations. The form of the constitutive equations given below is used for the calculation of the ‘‘forming limit diagrams’’ for TRIP steels presented in Section 5.2. Substituting the evolution equation for f_ into the expressions (35b and c) and using (32)–(34), we can write the total deformation rate in the form 1 r _ D ¼ M e : r þ req hhom N þ S þ RY; 3 where

  1 pðaÞ _ S ¼ c Af  AN þ Dv d 3 ðaÞ

and

ð41Þ   1 Y ¼ c Bf AN þ Dv d . 3 ðaÞ

ð42Þ

Using the definition of triaxiality R = p/req, we find 1 p_  Rr_ eq r R_ ¼ ¼ ðd  RNÞ : r . req req

ð43Þ

Finally, substituting the above expression for R_ into (41) we find r

D ¼ M : r þQ

or

r

r ¼ L : D þ P;

ð44Þ

where M ¼ Me þ

1 Yðd  RNÞ; req

1 Q ¼ req hhom N þ S; 3

ð45Þ

and L ¼ M 1 ;

P ¼ M 1 : Q.

ð46Þ

The quantity P on the right hand side of (44b) is an ‘‘initial stress’’-type term that arises in the constitutive equations of rate-dependent solids. We note also that, if R_ 6 0, then M ¼ M e and L ¼ Le . 2.6. Hardening of the phases As mentioned in Section 2.2, the nonlinear behavior of the phases is described by a viscoplastic constitutive equation of the form !mðrÞ ðrÞ ðrÞ req pðrÞ pðrÞ pðrÞ _ D ¼ _ N; ¼ _ 0 ; ð47Þ ðrÞ ry ðrÞ

(r) ðrÞ where ð_0 ; rðrÞ ! 1, the above equations correspond to a rate-independent perfectlyy ; m Þ are material constants. As m ðrÞ plastic material with a yield stress ry . The constitutive equations for the TRIP steel are used to carry out finite element calculations and the solution is developed incrementally as discussed in Section 3 below. The representative equivalent plastic strain rate _ pðrÞ in each phase is

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determined in the homogenization process (see Appendix A), and this allows for the determination of the total plastic strain pðrÞ . A large value is used for the strain rate sensitivity exponents m(r), so that the material is essentially rate-indepðrÞ pendent and rðrÞ . y plays the role of the flow stress, which depends, in general, on  In order to account for the hardening of the phases and, at the same time, be consistent with the assumption used in the derivation of the model, within each increment ryðrÞ takes a constant value, which equals the value at the start of the increment ryðrÞ ðpðrÞ n Þ for r = 2, 3, 4, i.e., for the austenitic, bainitic and ferritic phases. For the martensitic phase, the following methodology is used for the calculation of rðmÞ y . Stringfellow et al. [34] point out that a unit of martensite which forms at a given austenite plastic strain inherits the strain-hardened dislocation structure of its parent austenite. Therefore, each newformed unit of martensite is initially harder than the initial hardness of any previously-formed martensite and substantially harder than the initial undeformed austenite. To account for this effect, we follow the suggestion of Stringfellow et al. [34] and use the following formula for the evaluation of the constant value of rðmÞ over each increment y fn Df pðmÞ rðmÞ ¼ H ðnþ1 Þ þ H ðnpðaÞ Þ; ð48Þ y fnþ1 fnþ1 ¼ H ðpðmÞ Þ is the hardening function of martensite, and fn and fn+1 = fn + Df the volume fraction of martensite where rðmÞ y at the start and at the end of the increment. 3. Numerical implementation of the constitutive model In a finite element environment, the solution is developed incrementally and the constitutive equations are integrated at the element Gauss integration points. Let F denote the deformation gradient tensor and s the collection of state variables q ¼ ðcðrÞ ; pðrÞ Þ, r = 1, 2, 3, 4. At a given Gauss point, the solution (Fn, rn, qn) at time tn as well as the deformation gradient Fn+1 at time tn+1 are known, and the problem is to determine (rn+1, qn+1). We use an algorithm for the numerical integration of the constitutive equations similar to that developed by Aravas [1] for pressure-dependent plasticity models. In the following, we present first an algorithm that can be used in three-dimensional, plane strain, and axisymmetric problems; the special case of plane stress conditions is discussed in a separate section that follows. 3.1. General algorithm The time variation of the deformation gradient F during the time increment [tn, tn+1] can be written as FðtÞ ¼ DFðtÞ  Fn ¼ RðtÞ  UðtÞ  Fn ;

ð49Þ

tn 6 t 6 tnþ1 ;

where R(t) and U(t) are the rotation and right stretch tensors associated with DF(t). The corresponding deformation rate D(t) and spin W(t) tensors are given by _  DF1 ðtÞ ; _  F1 ðtÞ ¼ ½DFðtÞ DðtÞ  ½FðtÞ s s

ð50Þ

_  F1 ðtÞ ¼ ½DFðtÞ _  DF1 ðtÞ ; WðtÞ  ½FðtÞ a a

ð51Þ

and where the subscripts s and a denote the symmetric and anti-symmetric parts, respectively. If it is assumed that the Lagrangian triad associated with DF(t) (i.e., the eigenvectors of U(t)) remains fixed over the time interval (tn, tn+1), it can be shown readily that _  RT ðtÞ; DðtÞ ¼ RðtÞ  EðtÞ

_ WðtÞ ¼ RðtÞ  RT ðtÞ;

r ^_ ðtÞ  RT ðtÞ; rðtÞ ¼ RðtÞ  r

ð52Þ

where E(t) = ln U(t) is the logarithmic strain relative to the configuration at the start of the increment, and ^ðtÞ ¼ RT ðtÞ  rðtÞ  RðtÞ. r It is noted that at the start of the increment (t = tn) DFn ¼ Rn ¼ Un ¼ d;

^n ¼ rn ; r

and

En ¼ 0;

ð53Þ

whereas at the end of the increment (t = tn+1) DFnþ1 ¼ Fnþ1  F1 n ¼ Rnþ1  Unþ1 ¼ known;

and

Enþ1 ¼ ln Unþ1 ¼ known.

ð54Þ

Then, the constitutive equations of the model can be written in the form E_ ¼ E_ e þ E_ in ; ^_ ¼ Le : E_ e ; r

ð55Þ ð56Þ

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^ þ 1 _ p d; E_ in ¼ _ q N 3 v

5101

ð57Þ

where ^ ¼ 3 r ^; N 2req

1 _ q ¼ req hhom þ Af_ ; 3

_ pv ¼ Dv f_ .

ð58Þ

The evolution equations of the volume fractions of the phases can be written as _ f_ ¼ cðaÞ ðAf _ pðaÞ þ Bf RÞ;

ð59Þ

c_ ðaÞ ¼ ðf_ þ c_ ð3Þ þ c_ ð4Þ Þ;

ð60Þ

cð3Þ _ pv ;

ð61Þ

c_ ð4Þ ¼ cð4Þ _ pv .

ð62Þ

c_

ð3Þ

¼

Eqs. (55), (56), (58c), (60)–(62) can be integrated exactly as follows: DE ¼ DEe þ DEin ;

ð63Þ

^nþ1 ¼ rn þ Le : DEe ¼ rn þ Le : ðDE  DEin Þ ¼ r ^e  Le : DEin ; r

ð64Þ

Dpv

ð65Þ

¼ Dv Df ;

ðaÞ

ð3Þ

ð4Þ

cnþ1 ¼ 1  ðfnþ1 þ cnþ1 þ cnþ1 Þ;

ð66Þ

ð3Þ

p cnþ1 ¼ cð3Þ n expðDv Þ;

ð67Þ

ð4Þ cnþ1

ð68Þ

p ¼ cð4Þ n expðDv Þ; e

e

^ ¼ rn þ L : DE is the known ‘‘elastic predictor’’, and the notation DA = An+1An has been used. where r The remaining Eqs. (57), (58b) and (59) are intergrated numerically by using a backward-Euler scheme: 1 ^ nþ1 ; DEin ¼ Dpv d þ Dq N 3 1 Dq ¼ req jnþ1 hhom nþ1 þ Anþ1 Df ; 3   ðaÞ pðaÞ Df ¼ cnþ1 Af jnþ1_ nþ1 Dt þ Bf jnþ1 DR ;

ð69Þ ð70Þ ð71Þ

where DR ¼

pnþ1 p  n . req jnþ1 req jn

ð72Þ

pðaÞ

_ We recall that hhom nþ1 and nþ1 are determined by the homogenization procedure described in Section 2.2. Next, we substitute (69) into (64) and set Le ¼ 2lK þ 3jJ to find ^ nþ1  jDp d. ^nþ1 ¼ r ^e  2lDq N r v

ð73Þ

^nþ1 is The deviatoric part of r ^ nþ1 ; ^snþ1 ¼ ^se  2lDq N

ð74Þ

^ nþ1 ¼ ð1:5=req j Þ^snþ1 into the last equation and solve ^ . If we set N where ^s is the deviatoric part of the elastic predictor r nþ1 for ^snþ1 , we reach the conclusion that ^snþ1 and ^se are co-linear. Therefore, e

e

3 3 ^snþ1 ¼ e ^se ¼ known; ð75Þ 2req jnþ1 2req pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^e . where reeq ¼ ð3=2Þ^se : ^se is the von Mises equivalent stress of r ^ ¼ r : N ¼ req Þ and the pressure axis Next, we project the stress tensor defined in (73) onto the deviatoric plane ð^ r:N ðð1=3Þ^ r : d ¼ ð1=3Þr : d ¼ pÞ to find ^ nþ1 ¼ N

req jnþ1 ¼ reeq  3lDq ; e

pnþ1 ¼ p 

jDpv ;

where pe ¼ ð1=3Þ^ re : d.

ð76Þ ð77Þ

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We choose Dq and Df as the primary unknowns and treat (70) and (71) as the basic equations in which reqjn+1, pn+1, ðaÞ ð3Þ ð4Þ Dpv ; cnþ1 ; cnþ1 and cnþ1 are defined by Eqs. (76), (77) and (65)–(68). Newton’s method is used for the numerical solution of ð3Þ ð4Þ ðaÞ (70) and (71). Once Dq and Df are found, reqjn+1, pn+1, Dpv ; cnþ1 ; cnþ1 and cnþ1 are determined from (76), (77) and (65)– ^nþ1 and rn+1 are computed from (73) and rnþ1 ¼ Rnþ1  r ^nþ1  RTnþ1 ; which completes the integration process. (68). Finally, r 3.2. Plane stress problems In this section we discuss the application of the previous algorithm to problems of plane stress. In such problems, the out-of-plane strain components are not defined kinematically and some modifications to the method described in the previous section are needed. We consider a thin plane disc of uniform thickness loaded in its plane and the plane X3 = 0 is taken to coincide with the mean plane of the disc. The in-plane displacements and deformation gradient fields are assumed to be of the form 2

u1 ¼ u1 ðX 1 ; X 2 Þ;

u2 ¼ u2 ðX 1 ; X 2 Þ

and

F 11 6 ½F  ¼ 4 F 21 0

F 12 F 22 0

The stress tensor is also assumed to be of the form 2 3 r11 r12 0 6 7 ½r ¼ 4 r21 r22 0 5; 0 0 0

3 0 7 0 5. F 33

ð78Þ

ð79Þ

where the components rab = rab(X1, X2) (a, b = 1, 2) are averaged through the thickness. Eqs. (78) are consistent provided that u3 = u3(X3). However, in finite strain problems, when the in-plane displacement field is inhomogeneous, the out-of-plane displacement and the corresponding thickness variation will be functions of (X1, X2). Then, the question arises as to whether the plane stress conditions are maintained as the disc deforms. The conditions under which the plane stress assumption is accurate have been studied in detail by Hutchinson and Needleman [20] for certain problems (see also [26]). For the rest of this section, we assume that, as the material deforms, the resulting thickness variation is insignificant, so that the plane stress assumption is valid and Eqs. (78) and (79) hold to a good approximation. Referring to the methodology described in the previous section, we note that the deformation gradient associated with the increment under consideration can be written as 2 3 DF 11 DF 12 0 6 7 ½DF nþ1  ¼ 4 DF 21 DF 22 ð80Þ 0 5; 0

0

Dk3

where DF ab are the known in-plane components, and Dk3 the unknown out-of plane component, which is determined from the condition r33jn+1 = 0. We write also 2 3 2 3  11 U  12 U 0 cos  h  sin h 0 6 7 6 7  22 ½U nþ1  ¼ 4 U ð81Þ U 0 5; ½Rnþ1  ¼ 4 sin h cos h 0 5; 21 0 and

2

 11 DE 6  ½DE ¼ 4 DE12 0

0

 12 DE  22 DE 0

Dk3

0

0

0

1

3

7 0 5; DE3

ð82Þ

where ‘‘bared’’ quantities are known and DE3 = ln(Dk3) is the unknown out-of-plane component of DE determined from the condition r33jn+1 = 0. Next, we write  þ DE3 a; where a ¼ e3 e3 ; DE ¼ DE ð83Þ  is the known in-plane part of DE. Also, using (81b) we conclude that ei are the unit base vectors, and DE ^33 jnþ1 ¼ r33 jnþ1 ¼ 0. r

ð84Þ

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5103

The equations of the problem are now written as DE ¼ DEe þ DEin ;

ð85Þ in

e

e

  L : ðDE  DE3 aÞ; ^nþ1 ¼ r r p Dv ¼ Dv Df ; ðaÞ

ð3Þ

ð86Þ ð87Þ

ð4Þ

cnþ1 ¼ 1  ðfnþ1 þ cnþ1 þ cnþ1 Þ;

ð88Þ

ð3Þ cnþ1 ð4Þ cnþ1

ð89Þ

p ¼ cð3Þ n expðDv Þ;

¼

cð4Þ n

expðDpv Þ;

ð90Þ

 is the known elastic predictor that corresponds to the known part of DE, and e ¼ rn þ Le : DE where r 1 ^ nþ1 ; DEin ¼ Dpv d þ Dq N 3 1 Dq ¼ req jnþ1 hhom nþ1 þ Anþ1 Df ; 3   ðaÞ pðaÞ Df ¼ cnþ1 Af jnþ1_ nþ1 Dt þ Bf jnþ1 DR ;

ð91Þ ð92Þ ð93Þ

where DR ¼

pnþ1 p  n . req jnþ1 req jn

ð94Þ

We substitute (91) in (86) and set a = a 0 + (1/3)d and L ¼ 2lK þ 3jJ to find ^ nþ1  DE3 a0 Þ  jðDp  DE3 Þd; ^nþ1 ¼ r e  2lðDq N r v

ð95Þ

^nþ1 are: where a 0 = (1/3)(e1e1e2e2 + 2e3e3) is the deviatoric part of a. The deviatoric and spherical parts of r ^ nþ1  DE3 a0 Þ; ^snþ1 ¼ se  2lðDq N pnþ1 ¼ pe  jðDpv  DE3 Þ.

ð96Þ ð97Þ

^ nþ1 ¼ ð1:5=req j Þ^snþ1 in (96) and solve for ^snþ1 to find Next, we substitute N nþ1 ^snþ1 ¼

1 ðse þ 2lDE3 a0 Þ. 1 þ 3Dq ðl=req jnþ1 Þ

ð98Þ

The last equation shows that ^snþ1 and se are not co-linear in this case. Then, we use the last equation to calculate reqjn+1: 3 3=2 2 ðreq jnþ1 Þ ¼ ^snþ1 : ^snþ1 ¼ ðse þ 2lDE3 a0 Þ : ðse þ 2lDE3 a0 Þ. 2 2 ½1 þ 3Dq ðl=req jnþ1 Þ

ð99Þ

If we take into account that a 0 : a 0 = 2/3 and se : a0 ¼ se : a ¼ se33 , we conclude that 2

ðreq jnþ1 Þ ¼

1 ½1 þ 3Dq ðl=req jnþ1 Þ

2

ð re2 se33 DE3 þ 4l2 DE23 Þ; eq þ 6l

ð100Þ

which we solve for reqjn+1: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 req jnþ1 ¼ 3lDq þ r se33 DE3 þ 4l2 DE23 ; ð101Þ eq þ 6l ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p eeq ¼ ð3=2Þse : se . where r ^33 jnþ1 ¼ 0 can be written as ^s33 jnþ1 þ pnþ1 ¼ 0, which, in view of (98), can be written as The plane stress condition r   1 4 e s þ lDE3 þ pnþ1 ¼ 0. ð102Þ 1 þ 3Dq ðl=req jnþ1 Þ 33 3 We choose Dq, Df and DE3 as the primary unknowns and treat the following as the basic equations: 1 Dq ¼ req jnþ1 hhom nþ1 þ Anþ1 Df ; 3 ðaÞ pðaÞ Df ¼ cnþ1 ðAf jnþ1_ nþ1 Dt þ Bf jnþ1 DRÞ;   1 4

se33 þ lDE3 þ pnþ1 ¼ 0; 3 1 þ 3Dq l=req jnþ1

ð103Þ ð104Þ ð105Þ

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_ pðaÞ in which reqjn+1, pn+1, c(3), c(4) and c(a) are defined by Eqs. (101), (97) and (88)–(90). We note also that hhom nþ1 and nþ1 are determined by using the homogenization technique described in Section 2.2. Newton’s method is used for the solution of (103)–(105). Once Dq, Df and DE3 are found, reqjn+1, pn+1, c(3), c(4) and c(a) are defined by Eqs. (101), (97) and (88)–(90). ^nþ1 and rn+1 are computed from (95) and rnþ1 ¼ Rnþ1  r ^nþ1  RTnþ1 ; which completes the integration process. Finally, r 4. Calibration of the model The hardening curve ry ðpðaÞ Þ of the austenitic phase is taken from the experimental data of Naturani et al. [25]. The hardening curves of the other three phases are taken form a relatively recent research report [38]. The corresponding mathematical expressions for room temperature are ry ðpðaÞ Þ ¼ 300 þ 500ðpðaÞ Þ pð3Þ

ry ð

0:25

pð3Þ 0:25

Þ ¼ 810 þ 753ð

Þ

; ;

ry ðpðmÞ Þ ¼ 1200 þ 1025ðpðmÞ Þ pð4Þ

ry ð

pð4Þ 0:47

Þ ¼ 290 þ 690ð

Þ

0:13

ð106Þ

;

ð107Þ

;

where the flow stresses are given in MPa. A series of uniaxial tension tests were carried out on an experimental TRIP steel. The chemical composition of the TRIP steel is shown in Table 1, and a schematic representation of its heat treatment is shown in Fig. 1 [29]. The tensile test specimens were designed according to DIN EN 10002, with a measuring range of 50 mm. At the end of the heat treatment, a triple-phase microstructure of 50% ferrite, 38% bainite and 12% retained austenite is obtained. The retained austenite transforms into martensite as the material deforms plastically. The tensile tests were interrupted at different stages and the amount of martensite was measured. The experimental data presented below show that an amount of martensite appears at ‘‘zero plastic strain’’; this is due to the fact that stress-assisted transformation takes place before macroscopic yielding occurs in the TRIP steel. The constitutive model developed in the present work accounts only for plastic strain-induced transformation; in order to account for the aforementioned early transformation, and based on the experimental data, we use in our calculations the following initial volume fractions for the individual phases: f0 = 0.017, ðaÞ ð3Þ ð4Þ c0 ¼ 0:12  f0 ¼ 0:103, c0 ¼ 0:38 and c0 ¼ 0:50. In order to calibrate the model, we solve the uniaxial tension problem numerically. The constitutive model presented in the previous sections is implemented in the ABAQUS general-purpose finite element program [15]. This code provides a general interface so that a particular constitutive model can be introduced as a ‘‘user subroutine’’ (UMAT). The integration of the elastoplastic equations is carried out using the algorithm presented in Section 3.1. The finite element formulation is based on the weak form of the momentum balance, the solution is carried out incrementally, and the discretized nonlinear equations are solved using Newton’s method. The Jacobian of the equilibrium Newton-loop requires the so-called ‘‘linearization moduli’’ of the algorithm, which are reported in [30]. The uniaxial tension problem is solved in ABAQUS by using one four-node isoparametric axisymmetric finite element. The element (CAX4H in ABAQUS) uses the so-called B-bar method for the numerical evaluation of the element stiffness Table 1 Chemical composition of experimental TRIP steel (wt%) C

Mn

Si

Al

P

0.2

1.4

0.5

0.75

0.04

Microstructure Intercritical annealing Low Coiling Temp.

Temperature

50% 50% A F Cooling 20 °C/s

12% A

Bainite Isothermal Transformation 400 °C 200 s

38%

B

50% F

Cooling 10 °C/s

Heating 100 °C/s

Time

Fig. 1. Schematic representation of heat treatment for the experimental TRIP steel (A: austenite, B: bainite, F: ferrite).

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5105

Table 2 Constants of the kinetic model used to fit the experimental data g0

g1

g2

g

sg

A0

A1

sa (MPa)

a

b0

2

3400

4.7

493

3230

292

0.012

0.057

496

8.7

1.8

VOLUME FRACTION OF MARTENSITE [%]

r

12 10 8 6 4 2 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

NOMINAL STRAIN

Fig. 2. Predicted f– curve together with the experimental data (dark triangles).

NOMINAL STRESS [MPa]

800

600

400

200

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

NOMINAL STRAIN

Fig. 3. Predicted r– curve (dark line) together with the experimental data (lighter line).

matrix [17,18]. The problem is solved incrementally and the end displacement is increased gradually at a nominal strain rate of 104 s1 until a final elongation of 30% is reached. The values E = 220 GPa and m = 0.3 for the elastic Young’s modulus and Poisson’s ratio are used in the calculations. The reference strain rates and strain-rate sensitivity exponents are assumed to take the same value for all phases: ðrÞ _ 0 ¼ 104 s1 and m(r) = 60 (r = 1, 2, 3, 4). The relative volume change associated with the martensitic transformation is taken to be Dv = 0.02, and the values Md,ut = 80 C and M rs;ut ¼ 15 C are used. The calculations are carried out for room temperature, i.e., T = 23 C. The values of the parameters that enter the transformation kinetics model are chosen so that the predictions of the model agree with the f curves determined experimentally, where  is the axial nominal strain. These values are shown in Table 2. Figs. 2 and 3 show the predicted f and r curves together with the experimental data, where r is the nominal stress. The model predictions fit the data reasonably well. 5. Applications 5.1. Necking of a bar In this section, we use the constitutive model for TRIP steels developed in Section 2 to study the development of a neck in a tension specimen. From a mathematical point of view, necking is a bifurcation of the uniform solution to the problem.

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It is well known that bifurcations cannot take place at realistic strain levels when elastic–viscoplastic (as opposed to rateindependent elastic–plastic) constitutive models are used (e.g., [41]). Therefore, in viscoplastic problems, bifurcation analyses are replaced usually by studies of the sensitivity of the solution to small imperfections of some kind. Such an imperfection approach is used here in order to study the development of a neck in a tension specimen. We consider a cylindrical specimen with aspect ratio L0/R0 = 3, where 2L0 is its initial length and R0 its initial radius. We introduce the cylindrical system shown in Fig. 4 and identify each material particle in the specimen by its position vector X = (r, z) in the undeformed configuration. Because of symmetry, only one half of the cylindrical specimen corresponding to z P 0 is analyzed. The problem is solved by using the ABAQUS general-purpose finite element program. The finite element mesh used is shown in Fig. 4; it consists of 675 CAX4H isoparametric axisymmetric elements in a 15 · 45 grid. In order to promote necking, a geometric imperfection of the following form is introduced pz RðzÞ ¼ R0  nR0 cos ; ð108Þ 2L0 where R(z) is the perturbed radius of the specimen and the value n = 0.005 is used. All nodes along the midplane z = 0 are constrained to move only in the radial direction, and all nodes along the z-axis are constrained to have zero radial displacement. The deformation is driven by the uniform prescribed end-displacement in the z-direction on the shear-free top end and the lateral surface is kept traction free. The material constants reported in Section 4 are used in the calculations. The initial volume fractions of the four phases ðaÞ ð3Þ ð4Þ are assumed to be f0 = 0.017, c0 ¼ 0:103, c0 ¼ 0:38 and c0 ¼ 0:50. For comparison purposes, a separate set of calculations is carried out for a non-transforming steel that consists of the ð3Þ three phases, i.e., retained austenite, bainite and ferrite with constant volume fractions f0 = 0, c(a) = 0.12, c0 ¼ 0:38 and ð4Þ c0 ¼ 0:50. Fig. 5 shows the stress–strain curves for both the transforming and non-transforming steels. The arrows on the curves indicate the points of maximum load. For the TRIP steel the maximum nominal stress of 732 MPa occurs at a nominal

Fig. 4. Finite element mesh.

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NOMINAL STRESS [MPa]

800

600

400

200

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

NOMINAL STRAIN

Fig. 5. Stress–strain curves for a TRIP steel (dark line) and a non-transforming steel (lighter line).

strain of 0.263; the corresponding values for the non-transforming steel are 677 MPa and 0.208. Fig. 5 shows that the TRIP effect hardens the material and increases substantially the range of uniform elongation. Fig. 6 shows the deformed configurations for the transforming and non-transforming materials at a nominal strain level of 50%. The formation of martensite stabilizes the neck and leads to its propagation down the length of the specimen. At the nominal strain of 50%, the non-transforming material exhibits a 41.3% reduction of its minimum cross section; the corresponding reduction for the TRIP steel is 31.4%. Fig. 7 shows contours of the hydrostatic stress p = rkk/3 for the TRIP and non-transforming materials at a nominal strain of 50%. The maximum value of p occurs at the center of the neck and is higher in the non-transforming specimen. There are two main reasons for this: (a) the reduction of the minimum cross section is higher in the non-transforming case, thus causing a higher level of hydrostatic stress [2], and (b) the transformation of austenite to martensite is accompanied by a local increase in volume which, in turn, lowers the local hydrostatic tension. The corresponding contours of the triaxiality ratio R = p/req are similar to those of the hydrostatic stress, with R taking higher values in the non-transforming case. It should be noted that in the case of uniform elongation R = 1/3 everywhere, whereas for the necking problem the maximum R-value occurs at the center of the neck with Rmax = 0.520 for the TRIP and Rmax = 0.646 for the non-transforming steel at a nominal strain of 50%.

Fig. 6. Deformed configurations at a nominal strain of 50%: (a) TRIP, (b) non-transforming steel.

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Fig. 7. Contours of hydrostatic stress p = rkk/3 in MPa at a nominal strain of 50%: (a) TRIP, (b) non-transforming steel.

5.2. Forming limit diagrams We consider a sheet made of TRIP steel that is deformed uniformly on its plane in such a way that the in-plane principal strain increments increase in proportion. We study the possibility of the formation of a neck in the form of a narrow straight band (Fig. 8) and construct the corresponding ‘‘forming limit diagram’’. The problem of necking in biaxially stretched viscoplastic sheets has been studied by Hutchinson and Neal [19], Tvergaard [39,40] and Needleman and Tvergaard [26]. As discussed in Section 2.5, the constitutive equations of the TRIP steel can be written in the form r

r ¼ L : D þ P.

ð109Þ

It turns out that the formulation of the problem is simpler, if one works with ‘‘nominal’’ quantities. Let t = J F1 Æ r be the nominal (1st Piola–Kirchhoff) stress tensor, where J = detF; then, it can be shown readily that r t_ ¼ J F1  ðr þDkk r  r  D  D  r þ r  LT Þ;

ð110Þ

where L = D + W is the velocity gradient. Substituting (109) into the last equation, we find t_ ¼ C : F_ T þ B;

ð111Þ

N ψ X2 X1

Fig. 8. Narrow band in biaxially stretched sheet.

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5109

where 1 C ijkl ¼ JF 1 im F kn ðLmjnl þ V mjnl Þ;

B ¼ J F1  P;

ð112Þ

and

1 rik djl  dik rjl  ril djk  dil rjk þ rij dkl . ð113Þ 2 In the following, we examine the possibility of the formation of a neck as shown in Fig. 8; both inside and outside the neck a state of uniform plane stress is assumed. Since the in-plane displacements are continuous, their spatial derivatives parallel to the neck remain uniform. The only discontinuities in the displacement gradient are restricted kinematically to the form [13,16,33]  oua ð114Þ ¼ Ga N b ; oX b V ijkl ¼

where X is the position vector of a material point in the undeformed configuration, [ ] denotes the difference (jump) of the field within the neck and the field outside the neck, N is the unit vector normal to the neck in the undeformed configuration, and G is the jump in the normal derivative of u, i.e., [ou/oN]  [(ou/oX) Æ N] = G. The vector G takes a constant value within the neck and depends on the imposed uniform deformation gradient outside the neck; a method for the determination of G is discussed in the following. In Eq. (114) and for the rest of this section, Greek indices take values in the range (1, 2), where X1–X2 is the plane of the sheet. In view of (114), the in-plane components of the uniform deformation gradient inside the neck take the form F bab ¼ F ab þ Ga N b ;

ð115Þ

where the superscript b denotes quantities within the neck, and quantities with no superscript correspond to the uniform field outside the neck. The corresponding matrix form of the deformation gradients is 2 3 2 3 k1 þ G 1 N 1 G1 N 2 0 k1 0 0 6 7 6 7 k2 þ G2 N 2 0 5. ½F  ¼ 4 0 k2 0 5 and ½F b  ¼ 4 G2 N 1 ð116Þ b 0 0 k3 0 0 k3 The plane stress condition r33 = 0 implies that t33 = 0 as well. Using the equation t_33 ¼ 0 to solve for F_ 33 , we arrive at the in-plane constitutive relations needed for the sheet necking analysis ^ abcd F_ dc þ B ^ ab ; t_ab ¼ C

ð117Þ

where the plane stress moduli and initial stress term are given in terms of their three-dimensional counterparts by ^ abcd ¼ C abcd  C ab33 C 33cd C C 3333

and

^ ab ¼ Bab  C ab33 B33 . B C 3333

ð118Þ

Equilibrium across the neck requires that T a  HN b tba ¼ H b N b tbba  T ba ;

ð119Þ

where Hb and H, respectively, denote the initial thickness of the sheet inside and outside the neck. The rate form of this equilibrium relation is T_ a ¼ HN b t_ba ¼ H b N b t_bba ¼ T_ ba .

ð120Þ b

Substituting (117) into the last equation and using (116) for F and F , we obtain Aab G_ b ¼ ba ;

ð121Þ

where Aab

^ cadb N d ¼ N cC

 and

ba ¼ N b

 H ^ H ^ b b ^ _ ^ C bacd  C bacd F dc  b Bba  Bba . Hb H

ð122Þ

In a perfect sheet (Hb = H), before necking occurs we have that Cb = C and Bb = B, the right hand side of (121) van_ ¼ 0Þ until at some stage the determinant of the coefficient matrix [A] ishes, and the deformation remains homogeneous ðG vanishes, and this is the condition of a local necking bifurcation. In most rate-dependent plasticity models the instantaneous moduli L are the elastic moduli Le . In the constitutive model for TRIP steels presented in Section 2, the components Lijkl are of order ‘‘elastic modulus’’, and L ¼ Le when R_ 6 0. In such cases, the first bifurcation point in a perfect specimen (Hb = H) is that predicted by linear hypoelasticity and occurs at unrealistically large strains. In the context of a

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rate sensitivity formulation the most important feature is the very strong sensitivity to small imperfections, whereas bifurcation plays no central role [41–44]. Therefore, in order to carry out a necking analysis for a rate dependent solid it is always necessary to introduce some small inhomogeneity (imperfection). We follow the approach of Marciniak and Kuzynski [24], known as the ‘‘M–K’’ model, in which the sheet is assumed to contain a small initial inhomogeneity and necking results from a gradual localization of the strains at the inhomogeneity. The inhomogeneity is in the form of straight narrow band (neck) of reduced thickness Hb < H (Fig. 8). Both inside and outside the band a state of uniform plane stress is assumed, and the analysis consists in determining the uniform state of deformation inside the band that is consistent kinematically and statically with the prescribed uniform state outside the band [39,40,26]. In the presence of an initial thickness imperfection, the right hand side of (121) does not vanish and these equations provide a system that defines the two unknowns G_ 1 and G_ 2 . Given the initial sheet thickness inside and outside the band and the imposed uniform deformation history outside the band, Eqs. (121) are solved incrementally _ to obtain the deformation history inside the band. Localization is said to occur when the ratio of some scalar for DG ¼ GDt measure of the amount of incremental straining inside the band to the corresponding value outside the band becomes unbounded. In our calculations, an initial thickness imperfection is introduced and the deformation gradient outside the band F is prescribed in such a way that the corresponding principal logarithmic strains 1 and 2 outside the band increase in proportion, i.e. (d2/d1 = 2/1 = q = const.). The uniform solution outside the band is determined incrementally by using the plane stress algorithm presented in Section 3.2. At the end of every increment, Eq. (121) is used to determine DG and this defines the corresponding deformation gradient inside the band Fb. Then, the uniform solution inside the band is determined by using again the algorithm presented in Section 3.2. Necking localization is assumed to occur, when the ratio of some scalar measure of the amount of incremental straining inside the band to the corresponding value outside the band becomes very large; in particular, the calculations are terminated when either one of the two conditions jDG1j/Dk1 > 30 or jDG2j/Dk1 > 30 is satisfied. In order to improve the accuracy of the calculations and be able to use increments of reasonable size, ‘‘equilibrium correction’’ is used in (121), i.e., instead of the rate of equilibrium Eq. (120), we use equilibrium itself: Tnþ1 ¼ Tbnþ1 ; where the subscript n + 1 denotes values at the end of the increment. Then, we set Tnþ1 in the last equation to find Aab jn DGb ¼ ba jn Dt þ

1 ðTn  Tbn Þ ; Hb

ð123Þ b b _ ’ Tn þ Tn Dt and Tnþ1 ’ Tn þ T_ bn Dt ð124Þ

which is used for the determination of DG instead of (121). In the last equation, Dt is the time increment, the subscript n denotes values at the start of the increment, and the last term on the right hand side accounts for any unbalanced forces at the end of the previous increment. The material constants reported in Section 4 are used in the calculations. The initial volume fractions of the four phases ðaÞ ð3Þ ð4Þ in the TRIP steel are assumed to be f0 = 0.017, c0 ¼ 0:103, c0 ¼ 0:38 and c0 ¼ 0:50. For comparison purposes, a separate set of calculations is carried out for a non-transforming steel that consists of the three phases, i.e., retained austenite, bainite and ferrite with constant volume fractions f = 0, c(a) = 0.12, c(3) = 0.38 and c(4) = 0.50. In all cases, a constant logarithmic strain rate outside the band _ 1 ¼ 104 s1 is imposed. For every value of q = d2/d1, we set N ¼ cos We1 þ sin We2 , where W is the angle of inclination of the band relative to the X1-axis in the undeformed configuration, and calculations are carried out for values of W that cover the range 0 6 W 6 90; the critical value Wcr for each q is determined as that giving the minimum localization strain. Fig. 9 shows ‘‘forming limit curves’’ obtained for imposed proportional straining q for two different values of the initial thickness imperfection, namely Hb/H = 0.999 and Hb/H = 0.99. In particular, the curves in Fig. 9 show the values of cr cr 1 ¼ cr 1 and 2 ¼ q1 at which either of the conditions jDG1j/Dk1 > 30 or jDG2j/Dk1 > 30 is satisfied for different values of q. The two solid curves correspond to the TRIP steel, whereas the dashed curves are for the non-transforming steel. The TRIP phenomenon increases the necking localization strains. In particular, for an initial thickness imperfection of Hb/H = 0.99 and q = 0 (plane strain), the critical strain cr 1 increases from 0.2145 for the non-transforming steel to 0.2541 for the TRIP steel; the corresponding values of cr for Hb/H = 0.999 and q = 0 are 0.3179 for the non-transforming 1 steel and 0.3567 for the TRIP steel. A comparison of the model predictions with available experimental data is also presented in Fig. 9. The experimental points on the forming limit diagram were determined for the same TRIP steel used for the calibration of the model (Table 1). An ‘‘Erichsen’’ universal sheet metal testing machine was employed for the experiments. A hemispherical punch with a diameter of 50 mm was used and the punch velocity was set to 1 mm/s [29]. The agreement between the model prediction and the experimental data is reasonable. It should be noted also that an initial thickness inhomogeneity is one of the options to be used in the M–K model. Alternatively, a geometrically perfect sheet can be used and the initial inhomogeneity

I. Papatriantafillou et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 5094–5114

5111

0.7 0.6

Hb = 0.999 H

TRIP

Hb = 0.99 H

TRIP

0.5 0.4

ε1

cr

0.3 0.2 0.1 0 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ε2

cr

Fig. 9. Forming limit curves for two different values of initial thickness inhomogeneities Hb/H = 0.999 and 0.99. The solid lines correspond to the TRIP steel, whereas the dashed lines are for a non-transforming steel. The dark triangles are experimental data.

1.7 1.6 1.5 1.4 b

ε I 1.3 ε1

Hb = 0.99 H

1.2

Hb = 0.999 H

1.1 1.0 0.9 0.00

0.05

0.10

0.15

0.20

ε1

0.25

0.30

0.35

0.40

Fig. 10. Growth of maximum principal logarithmic strain inside the band bI for the TRIP steel and q = 0.

1.02 1.00 0.98 0.96 0.94 Hb = 0.99 H

h 0.92 h 0.90 b

Hb = 0.999 H

0.88 0.86 0.84 0.82 0.80 0.00

0.05

0.10

0.15

0.20

ε1

0.25

0.30

0.35

0.40

Fig. 11. Evolution of band thickness hb for the TRIP steel and q = 0. In the figure a capital H denotes the initial thickness, whereas a lower case h indicates the deformed thickness.

can be in the form of band of ‘‘softer’’ material. Further work in this direction is now underway in order to better understand the predictions of the model vs. the experimental data.

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The necking development, leading to the forming limit curves of Fig. 9, is illustrated in Fig. 10, where the ratio of the maximum principal logarithmic strain inside ðbI Þ and outside the neck (1) are plotted versus 1 for the TRIP steel for q = 0. b Fig. 11 shows the corresponding evolution of current (deformed) thickness of the band hb ¼ H b e3 normalized by the current thickness outside the band h ¼ He3 as 1 increases, where 3 and b3 is the out-of-plane normal logarithmic strain outside and within the band respectively. Acknowledgements This work was carried out while the authors were supported by the European Coal and Steel Community (ECSC) through the 7210-PR-370 Project. Appendix A. Calculation of hhom In the following, we present the details of the calculation of hhom for the four-phase TRIP steel, for given composition c and macroscopic stress req, by using the modified secant procedure described in Section 2.2. Eqs. (7) are written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrÞ 1 ohhom 3_0  ðrÞ ðrÞ mðrÞ 1 ðrÞ ðrÞ ðrÞ  b ¼ req ðhÞr ; h ¼ req ry ; r ¼ 1; 2; 3; 4; ð125Þ req eq ðrÞ cðrÞ ohðrÞ ry (r)

where the b(r)’s are ‘‘stress concentration factors’’. The effective viscous compliance is defined by Eq. (11): 0 10 11 cðrÞ 4 4 BX CBX C 1 cðrÞ hðrÞ CB C . ¼B hom @ A @ A 1 3 1 3 h r¼1 r¼1 þ þ hðrÞ 2hð4Þ hðrÞ 2hð4Þ

ð126Þ

Eqs. (125) provide 8 nonlinear equations for the determination of ðhðrÞ ; rðrÞ eq Þ. We discuss next the solution of this nonlinear system and reduce the problem to the solution of a system of 3 nonlinear equations. We define the ratios x(r)  h(r)/h(4) for r = 1, 2, 3 and use Eqs. (125) to find x

ðrÞ

¼E

ðrÞ

bðrÞ b

mðrÞ 1

ð4Þ ð4Þ m 1

ðrÞ

where E ðreq Þ ¼

;

ðrÞ _ 0 ryð4Þ ð4Þ _ 0

ðrÞ

ry

mð4Þ ðrÞ mð4Þ

mðrÞ

rmeq

for r ¼ 1; 2; 3.

Using the definitions of b(r) in (125a), namely sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ohhom ðrÞ ; b ¼ cðrÞ ohðrÞ

ð127Þ

ð128Þ

and the definition of hhom given in (126), we can express the factors b(r) in terms of x  (x(1), x(2), x(3)) as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4PðxÞ þ 6AðxÞ ðrÞ ; for r ¼ 1; 2; 3; b ¼ ð2 þ 3xðrÞ ÞPðxÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 BðxÞPðxÞ  AðxÞCðxÞ ; bð4Þ ¼ PðxÞ cð4Þ

ð129Þ ð130Þ

where AðxÞ ¼

4 X 2cðrÞ xðrÞ ; 2 þ 3xðrÞ r¼1

BðxÞ ¼ 

3 X r¼1

r¼1

4cðrÞ xðrÞ ð2 þ

4 X

PðxÞ ¼

2 3xðrÞ Þ

;

2cðrÞ ; 2 þ 3xðrÞ

CðxÞ ¼ 

with xð4Þ ¼ 1;

3 2cð4Þ X 4cðrÞ  . ðrÞ 2 5 r¼1 ð2 þ 3x Þ

ð131Þ ð132Þ

We consider x  (x(1), x(2), x(3)) as the basic unknowns, and note that (129) and (130) define the b(r)’s in terms of x. The basic equations that define x are (127), which can be written as mðrÞ 1

ðrÞ

G ðxÞ  x

ðrÞ

ðrÞ

 E ðreq Þ

½bðrÞ ðxÞ

½bð4Þ ðxÞ

mð4Þ 1

¼ 0;

r ¼ 1; 2; 3.

ð133Þ

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5113

The above system of nonlinear equations is solved numerically for x  (x(1), x(2), x(3)) by using a Newton scheme. Once x is hom found, the b(r)’s are determined from (129) and (130), ðhðrÞ ; rðrÞ from (126). eq Þ from (125), and h It should be noted that in the process of homogenization one can determine also the representative equivalent plastic strain rates _ pðrÞ in the phases by using the equations !mðrÞ rðrÞ eq ðrÞ pðrÞ _ ¼ _ 0 . ð134Þ ðrÞ ry Also, the corresponding macroscopic equation for the homogenized four-phase TRIP steel is 1 Dp ¼ hhom s ¼ _ p N; 2

N¼

3 s; 2req

1 _ p ¼ req hhom . 3

ð135Þ

Remark. If all strain rate sensitivity exponents are equal, i.e., m(1) = m(2) = m(3) = m(4)  m, then the E(r)’s and the solution ðrÞ x are independent of the macroscopic stress req. In that case, the b(r)’s are also independent of req, the req ’s are (r) hom hom m1 proportional to req, and the h ’s and h are proportional to req . Therefore, h can be written as m1 hhom ¼ 3Breq ;

ð136Þ (r)

where B is a function of c , 3 m1 Dp ¼ Breq s ¼ _ p N; 2

ðrÞ _ 0

and ryðrÞ . Then, the constitutive equation of the homogenized TRIP steel takes the form

where _ p ¼ Brmeq ;

ð137Þ

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