anisotropic and asymmetrical yielding and its distorted-ijp2016.pdf

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Anisotropic and asymmetrical yielding and its distortedevolution: Modeling and applications

H. Li a, *, X. Hu a,b, H. Yang a, **, L. Li a

a State Key Laboratory of Solidication Processing, School of Materials Science & Engineering, Northwestern Polytechnical University,

Xi'an, 710072, Chinab Automobile Steel Research Institute, Research Institute of Baosteel, Shanghai, 201900, China

a r t i c l e i n f o

Article history:

Received 6 October 2015

Received in revised form 1 February 2016

Available online 19 March 2016

Keywords:

A. Yield condition

B. Anisotropic material

B. Constitutive behaviour

C. Finite elements

C. Numerical algorithms

a b s t r a c t

Characterizing the anisotropy/asymmetry-induced distortional yielding and subsequent

evolution is still a challenge for potential usages of hard-to-deform materials. From

perspective of multiple mechanisms, two types of yield functions are classied, viz., the

principal shear stress-based models (SSM) and the stress invariants-based models (SIM);

then a unied continuum-based discontinuous (CBD) framework is constructed, in which

SSM and SIM are introduced to capture the distorted shape of the yielding, and an inter-

polation approach is adopted to smoothly present the nonlinear evolution of the distorted

plasticity in the full stress space. Taking the CPB06 (Cazacu et al., 2006) and Yoon's criteria

(Yoon et al., 2014) as typical SSM and SIM, the CBD framework is implemented in the

explicit 3D-FE platform for practical usages by combining implicit algorithm and inter-

polation approach, and the Nelder-Mead (N-M) method and the genetic algorithm (GA)

approach are evaluated for calibrating of CBD related to convergence, overlapping and

accuracy. The evaluation proves that the GA-based method is suitable for CBD, and the SIM

seems to be feasible for embedding into the CBD framework because of its solid physical

basis and numerical robustness. Taking high strength titanium alloy tube (HSTT) as a case,

the distorted plasticity evolution of the HSTT with six typical initial textures are charac-

terized, then the correlations among initial textures, distorted behaviors and inhomoge-

neous deformation are quantitatively established to improve the multi-defect constrained

formability in uniaxial tension/compression and mandrel bending.

2016 Elsevier Ltd. All rights reserved.

1. Introduction

The urgent needs for lightweight and high-performance components in many industries require the precision forming of

hard-to-deform materials with complex structures. The precision forming specic to these types of components depends onaccurate and efcient modeling of their plastic behaviors under complex loading conditions. While coordinated by multiple

mechanisms such as twinning and the Non-Schmid effect (Patra et al., 2014; Kabirian et al., 2015; Tuninetti et al., 2015), many

hard-to-deform materials, not only HCP structured polycrystalline aggregates but also some BCC or even FCC structured ones,

tend to present pronounced anisotropy/asymmetry behaviors. In particular, the microstructure variation during the

* Corresponding author. Tel./fax:86 29 88495632.** Corresponding author. Tel./fax:86 29 88495632.

E-mail addresses: [email protected](H. Li),[email protected](H. Yang).

Contents lists available atScienceDirect

International Journal of Plasticity

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j p l a s

http://dx.doi.org/10.1016/j.ijplas.2016.03.0020749-6419/2016 Elsevier Ltd. All rights reserved.

International Journal of Plasticity 82 (2016) 127e158


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successive deformation induces the distorted evolution of yield loci in sizes and shapes. Additionally, the forming of these

components generally involves complex loadings such as tension, compression, torsion, internal pressure or their combi-nations. The inherent unique behaviors and external boundary conditions easily induce inhomogeneous deformation and

further result in the dramatic evolution of strong texture reorientation and much distorted plasticity, which may affect theformability of these materials. Thus, to excavate the forming potentials of these high strength and lightweight materials, it is

imperative and fundamental to construct suitable constitutive models to describe the distortional yielding and subsequent

evolution and then to evaluate their applicability in practical processes (Gawad et al., 2015).

Based on understanding of the unique plastic behaviors such as anisotropy, to advance the constitutive modeling of newlydeveloped materials is the ultimate frontiers (Chaboche, 2008; Banabic, 2010; Horstemeyer and Bammann, 2010; Lee and

Barlat, 2014; Chang and Kochmann, 2015; Smith et al., 2015). The multi-scale modeling blueprint prevails for thoroughlycharacterizing the materials' constitutive features at the atomistic scope, meso scope and continuum scale (McDowell, 2010;

Zhang et al., 2014). Atomistic-scale modeling is used to reveal underlying mechanisms, such as non-Schmid effects ( Bassani

and Racherla, 2011). Mesoscopic modeling, seen in the Taylor-Bishop-Hill polycrystal model and the Visco-plastic self-

consistent (VPSC) crystal plasticity model, is used to relate the macroscale plastic deformation to the mesoscale micro-structures (Agnew and Duygulu, 2005; Graff et al., 2007; Choi et al., 2009; Kabirian et al., 2015; Patra et al., 2014; Cyr et al.,

2015), and both anisotropy and asymmetry can be predicted. However, the intensive computation costs strongly limit theapplications of these approaches in practice. Thus, developing the constitutive formulations at the macroscopic level is the

preferable way to achieve accurate and efcient simulation of complex forming processes (Lee and Barlat, 2014).

At the macro scale, regarding the yield criteria,ow rules and hardening laws, many continuum-based constitutive models

have been proposed and numerically implemented into FE platforms (Banabic, 2010; Xiao et al., 2012; Lee and Barlat, 2014).To characterize the texture-induced anisotropy, as shown in Fig.1, many anisotropic models have been proposed. To cover the

abnormalanisotropy of aluminum alloys, several anisotropic yield functions have been developed, extended and applied;the typical ones include Karallis-Boyce model, YLD91, YLD96, YLD2000-2d, YLD2004-18p, Banabic model, the homogeneous

polynomials (Karallis and Boyce, 1993; Barlat et al., 1991, 1997, 2003, 2005; Banabic, 2010; Soare et al., 2008; Bron and

Besson, 2004; Iadicola et al., 2008). In addition to the anisotropy, as shown inFig. 1, asymmetry yield is observed for HCP

structured materials and even BCC ones. The root cause is still under intensive exploration, but includes possibilities such as

porosity deformation, the polar nature of twinning and non-Schmid law (Cazacu and Stewart, 2009; Bassani and Racherla,2011; Mohr et al., 2013). Several efforts have been undertaken to describe the yield asymmetry aside from plastic anisot-

ropy (Cazacu and Barlat, 2004; Cazacu et al., 2006, 2010; Plunkett et al., 2008; Cazacu and Stewart, 2009; Ghaffari et al., 2014;Tuninetti et al., 2015).

The above studies focus on describing the initial anisotropy or asymmetry behaviors for certain materials, and the evo-

lution of the yield surface is largely described using combinations of isotropic and kinematic hardening laws ( Wegener and

Schlegel, 1996; Lee et al., 2008; Choi and Pan, 2009). Due to the interaction of multiple deformation mechanisms associatedwith complex microstructures, the distorted yielding and nonlinear hardening in full stress states during deformation have

been frequently observed (Barlat et al., 2005; Choi et al., 2009; Khan et al., 2009). Within the Mises criterion framework,Franois (2001)introduced a distorted stressto replace the usual stress deviator to obtain the egg-shapedyield surface for

an aluminum alloy used for both proportional and non-proportional tension-torsion loading paths. Taking combined

isotropic, kinematic and distortional hardening into account,Shutov and Ihlemann (2012)proposed a rheological model to

describe the distortion of the yield surface for an annealed aluminum alloy. By introducing three material parameters, amodied Franois model (2001) based on egg-shapedsubsequent yield surfaces has been developed to describe the change

in the shape of the yield surface of the 1100 Al ( Yue et al., 2014). Until now, modeling distorted plasticity and its evolution in

full stress space still remains a challenge for practical metal forming.This study focuses on accurately and efciently modeling the distorted plasticity and its evolution of hard-to-deform

materials for practical usage. First, we conduct a critical review of the methodologies for developing macroscopic

Fig. 1. Distorted behaviors induced by anisotropy and asymmetry.

H. Li et al. / International Journal of Plasticity 82 (2016) 127e158128


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constitutive models. Second, a unied continuum-based discontinuous (CBD) framework is constructed to characterize the

initial distorted yielding and subsequent evolution in a full stress space. Third, the above discontinuous models arenumerically implemented in a commercial 3D-FE platform, and the calibration methods for the CBD are discussed. Next,

several CBD models are evaluated regarding capturing capabilities on anisotropy/asymmetry coupled distorted behaviors.Then, because titanium tubes currently present the most potential for bleedingtransformation in many industries, such as

aerospace, nuclear, chemical and healthcare (Yang et al., 2012; Banerjee and Williams, 2013), taking high-strength titanium

alloy tubes (HSTT) of Ti-3Al-2.5V (SAE, 2010) as the case, the evaluation and application of the above CBD constitutive models

are conducted, and new transferable knowledge about the correlations among initial textures, distorted behaviors andinhomogeneous deformation is quantitatively claried during several practical processes such as uniaxial tension/

compression and mandrel bending.

2. Critical review of developing continuum-based constitutive models

From viewpoint of underlying physical mechanisms, the most typical constitutive models are revisited to nd a meth-

odology for constructing a continuum-based model with sound physical meaning and numerical robustness to describedistorted plasticity and its evolution in hard-to-deform materials for practical usage.

2.1. Physical deconstruction of the yield functions

The yield criterion for any form of the equations can be physically decomposed. The shear stress is the basic element for all

yield criteria. This behavior is based on the observation that plastic strain occurs by crystallographic gliding under shear stresswhen the maximum shear stress reaches a critical value. The earliest proposed Tresca criterion (also called the maximum

shear stress criterion) was established according to the above theory. By adding another two principal shear stresses, the

Mises criterion was constructed to extend the single shear stress-based Tresca model to the full shear stress-based model forsmooth and convex description of the yield surface. By introducing the anisotropy parameters, the most widely used

anisotropic yield criterion, Hill'48 quadratic yield function (Hill, 1948), was constructed to describe the anisotropy of thematerial. It is worth noting that the Mises and Hill'48 criteria can be reformulated as the form of the second invariant of the

stress tensorJ2. From the mechanism of Schmid glide, the above full shear stress-based model (SSM) is equivalent to the J2-

based model. In light of the physical meanings of principal shear stresses and the stress invariants, most of the advanced yield

functions have been proposed considering various micro mechanisms-induced behaviors. The developed yield criteria can becalled the SSM or the stress invariants-based criterion (SIM).

2.1.1. Shear stress-based criterion (SSM)

By replacing the xed exponent 2 with a variable exponent a, Hershey (1954) introduced a non-quadratic formu-lation, in which the principal shear stresses are the major elements. Hosford (1972) extended Hershey's model

allowing for a continuous transition between Mises and Tresca formulations.Barlat et al. (1991)rewrote the Hosford

criterion in a form containing the deviator principal stresses. Karallis and Boyce (1993) further generalized theHosford criterion.

By introducing the anisotropy constants, the above isotropy criteria were further developed to describe the anisotropy. Todescribe the anomalous behavior of aluminum alloys, the Hill'48 criterionwas extended to a non-quadratic function (Hill,

1979).Hosford (1979)generalized his own isotropy criterion to anisotropy yielding. Barlat et al. (1997)proposed a more

general anisotropy expression of the yield function introduced by himself in 1991. The YLD2004-18p was constructed with

the capability to predict the occurrence of six and eight ears in cup drawing processes ( Barlat et al., 2005). In addition to the above anisotropy behaviors, when describing the asymmetry in HCP alloys, Cazacu and Barlat (2004)

proposed an isotropic yield function, as shown in Eq. (1), by introducing an asymmetry factor k. Next, via the lineartransformation of the stress tensor, as shown in Eq.(2), this isotropic criterion was extended to an anisotropic formulation

CPB06, which can describe both the anisotropy and the asymmetry (Cazacu et al., 2006). By adding another lineartransformation, the CPB06ex2 yield criterion (Plunkett et al., 2008) was constructed for a more accurate description of the

plastic ow and the anisotropy during both tension and compression.

fs jjS1jkS1ja jjS2jkS2j

a jjS3jkS3ja taY0 (1)

whereS1,S2,S3are the principal values of the stress deviator, a is a positive integer, a1.

fs X

1

k$X1

aX

2

k$X2

aX

3

k$X3

a taY0 (2)

whereL is a constant fourth-order tensor, and P1,P2,P3are the principal values of. P L[s].

H. Li et al. / International Journal of Plasticity 82 (2016) 127e158 129


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2.1.2. Stress invariants-based criterion (SIM)

C By adding the third invariants of the deviatoric stress tensor, Drucker (1949) proposed an isotropy criterion to representthe experimental data located between the Tresca and Mises yield surfaces. By introducing the rst invariant of the

Cauchy stress, the Drucker-Prager yield criterion (1952) was developed to represent the plastic deformation of soils,polymers, foams and other pressure-dependent materials. Based on Hill'48 (1948) criterion and the Drucker-Prager

model (1952),Liu et al. (1997)proposed an asymmetric yield function for plastically orthotropic materials.

C Based on the theory of the representation of tensor functions, Cazacu et al. (2001) developed a method for the

generalization of the invariants of the stress deviators J2 and J3, and proposed an extension of the Drucker isotropicyield criterion to orthotropy by replacing J2and J3with J02 andJ

03 .

C To capture the asymmetry specic to alloys with HCP structures,Cazacu and Barlat (2004)proposed an isotropic yieldfunction in the form of Eq. (3). Similarly, using the linear transformation approach and replacing the Cauchy stress with

the L[s], the above isotropic and asymmetry criterion was extended to an anisotropic formulation, as shown in Eq.(4)(Nixon et al., 2010). In theI1-J2-J3framework, the most widely used Gurson-Tvergaard-Needleman porous plasticity

was extended to include the effects of hydrostatic stress and the third invariant of stress on the matrix materials (Gaoet al., 2011). Independently, usingJ2andJ3,Khan and Yu (2012)proposed a yield criterion with product formulation as

shown in Eq.(5)to describe both the anisotropy and asymmetry of Ti-6Al-4V metals, and J3was implicitly included bythe Lode parameterx. Considering the full stress invariants, I1,J2andJ3, and assuming a linear dependence of yielding

on the rst invariant, as shown in Eq. (6), Yoon et al. (2014) proposed an orthotropic yield model to describe the

anisotropy and asymmetry of pressure sensitive metals.

fs J2s3=2

c$J3s t

3

Y0 (3)

wherec is the asymmetry coefcient and expressed in terms of uniaxial yield stresses intension sTand compression sCas

c3ffiffiffi3

ps3T s3C=2s3T s3C.

fs hJ2

i3=2

c$J3

t3Y0 (4)

f fs$gs 1 (5)

wheref(s) refers to Hill'48 criterion,g(s)ec(x 1),xcos3q 27J3=23J23=2.

fs bI1

24J2A!!

3=2

J3B!35

1=3

tY

0 (6)

where J2A is the second stress invariant of the transformed stress tensor ofA, J3B is the third stress invariant ofanother transformed stress tensor ofB; A LAs, B LBs, LA and LB are two distinct fourth-order linear trans-formation tensors (Barlat et al., 1991).

2.2. Explicit and implicit modeling evolution of plasticity

Given the yield criteria, accurately and efciently modeling the evolution of the yield loci is another imperative issue that

depends on the ow rules and hardening laws.

2.2.1. Flow rules

To describe the relationship between the applied stress and the plastic strain increment, the concept of plastic po-tential was proposed. If the plastic potential is different from the yield function, g s f, this ow rule is called non-

associated ow rule (non-AFR). Otherwise, when g f, this ow rule is called associated ow rule (AFR, also callednormality rule). Whether to use the AFR or non-AFR depends on the distinguishable physical meanings of the consequent

yield functions and plastic potential. In crystalline plasticity, the yield function depends on the resolved shear stress onseveral intersecting slip planes, while the ow potential depends on which slip plane activates. In this case, the derivative

of the consequent yield functions with respect to the stress determines the value of the plastic strain increment, while thederivative of the plastic potential determines the ow direction, which is the direction of the plastic strain increment.

Thus, according to the different physical meanings, the selection ofow rules relies on whether the stress states have the

same effects on hardening and plastic ow.

For most metallic materials, the AFR has been conrmed to be accurate when establishing a constitutive relationship.However, non-AFR must be used for pressure sensitive materials (Spitzig et al., 1975; Stoughton and Yoon, 2004) because of

the different effects of pressure stress on the yielding behaviors and theow direction. The most general way to construct the



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plastic potential is to follow the form of the corresponding consequent yielding functions; the different material constants

should be calibrated by considering the experimentally obtained plastic ow characteristics as comprehensively as possible.However, the preferable way to describe the plastic ow is to directly use the non-AFR.

2.2.2. Hardening laws

Compared with the initial yielding formulation, the subsequent yield locus may evolve with an obvious change in sizes,

shapes or even locations along the straining (Ishikawa, 1997). Eq. (7) explicitly presents the general consequent yield

equations, in which the evolution models of the internal variables, such as aij, Aij and R, should be constructed from themeasurements for the descriptions of the expansion, translation and distortion of the yield loci.

Fsij;aij;Aij; R

0 (7)

whereaij is the back stress related to the changes in the loading paths,Aij is the coefcient matrix describing the anisotropy or

asymmetry behaviors, and R is the hardening stress.In most cases, the isotropic hardening models, such as the Hollomon and Swift power laws, J-C model, KHL model ( Khan

et al., 2009) and physical model (Haddadi et al., 2006), are used to describe the proportional expansion of the yield surface in

whichaijequals zero, Aijis constant andR is generally related to the effective plastic strain, temperature and strain rate.

By introducing the back stress, the kinematic hardening is used to describe the special phenomena upon reversal loadingsuch as the Bauschinger effect, smooth elastoplastic transient behavior, permanent softening and stagnation behavior

(Bruschi et al., 2014). The kinematic hardening model describes the yield surface translating by using aijin the stress spacewithout changing the form or size, in whichAijandRare constant. Several kinematic hardening models have been developed

such as the linear kinematic hardening laws (Prager, 1956; Ziegler, 1959) and nonlinear kinematic hardening laws (Armstrongand Frederick, 1966; Chaboche, 1986; Yoshida and Uemori, 2002; Lee et al., 2008; Choi and Pan, 2009; Xiao et al., 2012; Mohr

et al., 2013). Because reverse loading-related phenomena usually occur during slight plastic deformations, the kinematichardening models were used for modeling forming processes with small plastic strains.

The other non-isotropic hardening rules, such as the distorted hardening models, in which Aij is a nonlinear function ofeffective plastic strain, are used to represent the irregular uniform evolution of the yield surfaces because of variations in

microstructure, such as dislocation and texture (Abedrabbo et al., 2006; Gao et al., 2011).Yeganeh (2007)simultaneously

considered the deformation-induced anisotropy, kinematic and isotropic hardening to develop a constitutive model incor-

porating the yield surface distortion. Treating the coefcient of the J3 term as a function of accumulated equivalent plasticstrain, the distortion hardening was introduced in the asymmetry yielding function (Zhai et al., 2014). Regarding the three

different deformation modes and considering the evolution of the anisotropy coefcients and asymmetry parameters with

the locally accumulated plastic strains, the consequent yielding loci were established based on the CPB06ex2 yield function(Muhammad et al., 2015).

It is noted that, for some hard-to-deform materials, such as titanium and magnesium alloys, the multiple mechanisms ofslipping and twinning cause more severe non-uniform evolution of the yield surface shape with plastic deformation. Thismakes the curve tting unsuitable or even unable to obtain the coefcient matrix in the explicit hardening law as in Eq. (7).

Plunkett et al. (2006) uses the CPB06 yield function and interpolation technique to capture the anisotropy evolution as a result

of the evolving textures in high-purity zirconium metals. The nonlinear hardening laws are implicitly included by thedifferent yield loci.

Fig. 2. Unied CBD constitutive framework for describing the evolution of plasticity.



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3. Unied continuum-based discontinuous (CBD) framework

Based on the above brief reference on the development of methodologies for continuum-based constitutive models, it

becomes possible to collect ideas for constructing a feasible constitutive framework to model distorted plasticity and the

evolution of hard-to-deform materials in the full stress space for practical usage.

As shown inFig. 2, considering the difculty that the explicit hardening law has in capturing the irregular evolution ofyield loci, especially at large deformation, a unied CBD framework is constructed to discontinuously describe the distorted

plasticity and its evolution in the full stress space. The distorted shapes of the yielding at different incremental deformation

stages can be described by the SSM, the SIM or the combined SSM SIM; an interpolation approach is adopted to smoothlypresent the nonlinear evolution of distorted plasticity in the full stress space based on the established individual yield loci.

The newly developed CPB06 model (Cazacu et al., 2006) and Yoon's model (Yoon et al., 2014) can be considered as two typical

full SSM and full SIM that describe the anisotropy/asymmetry-induced distorted yield loci under different strains. The non-AFR is used, and the hardening laws are implicitly included in the CBD constitutive framework. The major features of the CBD

constitutive model are described below.

As shown in Fig. 3, from the perspective of the physical meaning,J2 can be used to describe the reversible shear mechanismthat obeys Schmid law,J3can be used to characterize the irreversible (directional) deformation modes, such as twinning and

non-Schmid effects, andI1has the capability to reect the effects of the pressure stress. While, principal shear stress cannot

represent the irreversible mechanisms to reect the asymmetry. As mentioned in Section 2, the effects of principal shearstress on yielding are equivalent to those of J2. Thus, compared with the SSM, the SIM presents a more sound physical

background as mentioned in Section2.

For SIM, within theI1-J2-J3plasticity model framework, through two different fourth-order linear transformation tensorsLA and LB to the stress invariants, any anisotropy and asymmetry induced distortion plasticity may be conveniently

achieved without considering the convex any further as shown in Fig. 3. For cold rolled sheet metals, the pressure

insensitivity can be assumed for yielding and hardening. Thus, in Eq. (6), the material constant b is set to zero, and this SIM

can be reduced to the following form:

fs

J2

A!!

3=2

J3

B!

t3Y0 (8)

where the associated linear transformations on the stress for a 3D case are

LA

26666664

a3a2=3 a3=3 a2=3 0 0 0a3=3 a1a3=3 a1=3 0 0 0a2=3 a1=3 a1a2=3 0 0 0

0 0 0 a4 0 00 0 0 0 a5 00 0 0 0 0 a6

37777775

Fig. 3. Schematics of the SIM in different coordinates reecting multiple physical mechanisms.



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LB

26666664

b3b2=3 b3=3 b2=3 0 0 0b3=3 b1b3=3 b1=3 0 0 0b2=3 b1=3 b1b2=3 0 0 0

0 0 0 b4 0 00 0 0 0 b5 00 0 0 0 0 b6

37777775

For SSM as in Eq. (2), aside from the linear transformation Lin Eq.(9), the asymmetry behaviors may be described byadding an asymmetry factor k or even more linear transformations such as CPB06exn. When dealing with CPB06 model, to

satisfy the convex, a 1 and 1 k 1, and for the asymmetry behaviors, the related material factors should meet specialconstraints as shown in Eq. (10). This formula causes more difculty in the numerical implementation such as the deri-

vation and calibration of material parameters.

L CT (9)

C

26666664

C11 C12 C13 0 0 0C12 C22 C23 0 0 0C

13 C

23 C

33 0 0 0

0 0 0 C44 0 00 0 0 0 C55 00 0 0 0 0 C66

37777775 T

26666664

2=3 1=3 1=3 0 0 01=3 2=3 1=3 0 0 0

1=

3

1=

3 2=

3 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

37777775

sT

sC

jF1j kF1

a jF2j kF2a jF3j kF3

a

jF1jkF1a jF2jkF2

a jF3jkF3a

1=a(10)

whereF1, F2and F3can be found inAppendix A.

This CBD framework has high exibility in selecting the yield functions. As shown inFig. 2, for this method, only limitedindividual yield loci at different intervals of straining need to be established to describe the irregular evolution of the

subsequent yielding. The sole SSM or SIM can be reduced and replaced by a mixed pattern of the different models, such as

Hill'48 Yoon and Hill'48 CPB06exn. At the small strain stages, the Hill'48 yield equations can be constructed to describethe evolution of anisotropy, and the SIM or SSM can be used to capture the distortional hardening in latter deformation

stages. Additionally, in light of the correlation between dominant deformation mechanisms and yielding behaviors, theabove model is conveniently reduced to only anisotropy or asymmetry forms if necessary.

Although several experimental procedures are available for testing materials under different stress states, it is not alwayspossible to probe them all. As mentioned in the Introduction, if the experimental data are not available for a given strain

path, the VPSC crystal plasticity framework can be used to replace the missing experimental data for material calibration ofseveral yield loci (Gawad et al., 2015).

4. Numerical implementation of CBD constitutive models

Taking the CPB06exn and Yoon's criterion as the typical SSM or SIM, the above CBD framework is numerically imple-

mented into the explicit 3D-FE platform for practical usages by combining the implicit algorithm and interpolation approach.

4.1. Implicit integration algorithm for stress updating

A general constitutive equation adapted to small strain and rate-independent elastoplasticity is used, which is reduced

from the general formula of the nite deformationelastoplasticity (Simo and Hughes, 1998; Belytschko et al., 2014):8>>>>>>>>>:

_s C : _e C : _ _p

_p _lrs;q_q _lhs;q_f fs : _s fq, _q 0_l 0; f 0; _lf 0

(11)



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whereCis the fourth-order elastic modulus tensor,fs andfq are the derivation of the yield function, _l is the plastic multiplier, r

is the direction of the plastic ow (the AFR is assumed, i.e., r~ fsalong the normal direction of the current yield surface instress space),q is the internal variable (such as effective plastic strain), h is the gradient of the internal variable, and the last

equation represents the loading-unloading condition, the linear elastic behavior is assumed.The magnitude of the plastic multiplier _lcan be determined by Eq. (12):

_l fs :C : _

fq,h fs :C :r (12)

For a strain-driven FE formula, at the beginning of increment step n 1,n; pn;qn and strain increment DDt_ aregiven. The integration algorithm is used to getn1; pn1;qn1to meet consistency condition. Accordingly, the stress rate,plastic strain rate and gradient of an internal variable can be calculated.

To overcome the shortcomings of the explicit algorithm, such as the rst order forward Euler integration method, theimplicit integration one can be derived based on Euler's backward integration and return mapping technique ( Simo and

Hughes, 1998). Eq.(13)shows the integration equations. Compared with the explicit algorithm, the implicit one is uncon-

ditionally stable. The updating variables are determined using the results of the last increment. This avoids some impractical

values resulting in pseudo-unloading.

8>>>>>>>:

n1 n D

pn1

pn Dln1rn1

qn1 qn Dln1hn1sn1 C :

n1

pn1

fn1 fsn1;qn1 0

(13)

whereDlnDt_ln.A numerical scheme based on the elastic trial stress, plastic corrector and returning mapping technique is applied. Fig. 4

shows the updating algorithm with a geometric interpretation.

First, we obtain the plastic strain increment:

Dpn1

pn1

pn Dln1rn1 (14)

By substituting Eq.(14)into Eq.(13), we can get Eq.(15)

Fig. 4. Stress updating algorithm using the closest projective method.



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sn1 C :n1

pn D

pn1

C :

n Dn

pn D

pn1

snC : Dn C : D

pn1 s

trialn1 Dln1C :rn1

(15)

where strialn1 snC :Dn is the elastic trial stress tensor,Dln 1C:rn 1 is the plastic correction along the plastic owdirection of increment step n 1. The state of elastic prediction is driven by the increment of total strain, and the plasticcorrection process is driven by the plastic multiplier _l.

In the elastic prediction process, from Eq.(15), we obtain

Dsn1 C : Dpn1 Dln1C :rn1 (16)

During the plastic correction process, the total strain is constant, and the linearization relates to the increment of theplastic multiplierDl. The Newton iteration is used to solve the nonlinear algebraic equations. As Simo and Hughes (1998)

observed, the process of the Newton iteration based on the classication of linear equations is essentially the same as theclosest point projection. In the rest of this paper, we will omit the corner mark n1 for the increment of stress and time inthe equations, so all the values are the (n1)th increment step unless specically illustrated. We rewrite Eq.(13)in theform of Eq.(17).

8:ak C1 : Dsk Dl

kDrk dl

krk 0bk Dqk Dl

kDhk dl

khk 0

fk fks

: Dsk f

kq $Dq

k 0

(18)

where

Drk rks

: Dsk r

kq $Dq

k;Dhk hks

: Dsk hkq $Dq

k (19)

The corner mark sand q denote the derivative.

From Eq.(18), we can get Ds(k),Dq(k) anddl(k) simultaneously.

By substituting Eq.(19)into Eq.(18), we obtain

hAk

i1DskDqk

n~a

ko

dlkn

~rko

(20)

where

hAk

i1C

1

Dlrs

DlrqDlhs I Dlhq

k;n

~ako

a

k

bk

;n

~rko

r

k

hk

(21)

The stress and internal variables can be determined asDs

k

Dqk

hAk

in~ako

dlkhAk

in~rko

(22)

Substituting this into Eq.(18), dl(k) can be obtained

dlk

fk vfkAk~ak

vfkAk~rk

(23)

where we use the mark:



10/32

vf fs fq

(24)

Lastly, obtain the updating values of plastic strain, internal variables and plastic multiplier.

8


11/32

sxx TCxyqcos2 q; syy TCxyqsin

2 q; sxy TCxyqsinqcosq (30)

The uniaxial tensile and compressive yield stress in the x-y plane are

Txyq tY3K021 K022 3=2

2K00

1K0021 K

0022

1=3

(31)

Cxyq tY

3K021 K

022

3=2 2K

00

1

K

0021 K

0022

1=3(32)

where

K01

a2cos2 q a1sin

2 q.

6; K02

ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiha2 2a3cos

2 q a1 2a3sin2 qi.

62

a4sin qcosq2

r

K00

1

b2cos2 q b1sin

2q.

6; K00

2

ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffihb2 2b3cos

2 q b1 2b3sin2qi.

62

b4sin qcosq2

r

Similarly, Txz(q) andCxz(q) are the uniaxial tensile and compressive yield stresses in the x-z plane with an angle ofq fromthe transverse direction:

Txzq tY

3M021 M

022

3=2 2M

00

1

M

0021 M

0022

1=3(33)

Cxzq tY

3M021 M

022

3=2 2M

00

1

M

0021 M

0022

1=3(34)

where

M0

1

a1cos2 q a3sin2 q.6; M02

ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiha1 2a2cos2 q a3 2a2sin2 qi.62 a5sin qcosq2rM

00

1

b1cos2 q b3sin

2 q.

6; M00

2

ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffihb1 2b2cos

2 q b3 2b2sin2 qi.

62

b5sin qcosq2

r

Similarly, Tyz(q) andCyz(q) are the uniaxial tensile and compressive yield stresses in the y-z plane with an angle ofq from

the vertical direction:

Tyzq tY

3N021 N

022

3=2 2N

00

1

N

0021 N

0022

1=3(35)

Cyzq tY

3N021 N

022

3=2 2N

00

1

N

0021 N

0022

1=3(36)

where

N01

a3cos2 q a2sin

2 q.

6; N02

ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffih2a1a3cos

2 q 2a1a2sin2 qi.

62

a6sin qcosq2

r

N00

1

b3cos2 q b2sin

2 q.

6; N00

2

ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffih2b1b3cos

2 q 2b1b2sin2 qi.

62

b6sin qcosq2

r

Tension and compression yield stress for SSM (CPB06)-based CBD

Similarly, in the x-y plane:

Txyq tYjK1j kK1a jK2j kK2a jK3j kK3a1=a (37)



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Cxyq tY

jK1jkK1a jK2jkK2

a jK3jkK3a1=a (38)

with

K1

KxxKyy

ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiKxxKyy

2 4K2xy

q 2; K2

KxxKyy

ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiKxxKyy

2 4K2xy

q 2; K3 Kzz

where

Kxx F1cos2 q J1sin2 q; Kyy F2cos2 q J2sin2 q

Kzz F3cos2 q J3sin

2 q; Kxy C44sin qcosq

J1 2

3C12

1

3C11

1

3C13;J2

2

3C22

1

3C12

1

3C23;J3

2

3C23

1

3C13

1

3C33

In the x-z plane:

Txzq tY

jM1jkM1a jM2jkM2

a jM3jkM3a1=a (39)

Cxzq tY

jM1j kM1a jM2j kM2

a jM3j kM3a1=a (40)

with

M1

MxxMzz

ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiMxxMzz

2 4M2xz

q 2; M2

MxxMzz

ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiMxxMzz

2 4M2xz

q 2; M3 Myy

where

Mxx F1sin2 q P1cos

2 q; Myy F2sin2 q P2cos

2 q

Mzz F3sin2 q P3cos

2 q; Mxz C55sin qcosq

P12

3C13

1

3C11

1

3C12;P2

2

3C23

1

3C12

1

3C22;P3

2

3C33

1

3C13

1

3C32

In the y-z plane:

Tyzq tY

jN1jkN1a jN2jkN2a jN3jkN3a

1=

a

(41)

Cyzq tY

jN1j kN1a jN2j kN2

a jN3j kN3a1=a (42)

with

N1

NyyNzz

ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiNyyNzz

2 4N2yz

q 2; N2

NyyNzz

ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiNyyNzz

2 4N2yz

q 2; N3 Nxx

where

Nxx J1cos2 q P1sin

2 q; Nyy J2cos2 q P2sin

2 q

Nzz J3cos2 q P3sin

2 q; Nyz C66sin qcosq

Plastic ow for the CBD constitutive models

For both the SIM and SSM-based CBD models, according to the AFR, the anisotropic exponent Rqis obtained in the x-yplane:

Rq dq90+

dzz

vF

vsxxsin2 q

vF

vsyycos2 q

vF

vsxysinqcosq

vF

vsxx

vF

vsyy

(43)

Error function

For calibration purposes, using the above equations, an error function as shown in Eq. (44)is constructed based on pre-

dicted tensile stress ratios and plastic ow ratios. The coefcients related to the anisotropy/asymmetry parameters are ob-

tained by minimizing the error function.



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ErrorX

i

s

expi

.s

prei

12

X

j

R

expj

.R

prej

12

(44)

where the superscript preand exprepresent the predicted value and the experimental data, respectively. This objectivefunction is constructed based on the Weight-Sum approach where the weight factors for all experimental values are set

to 1.

The SIM-based model (Yoon's model), as shown in Eq.(8), involves 12 coefcients in the fourth-order tensorsLA

andLB

inthe full stress space. Among them, eight coefcients (a1, a2, a3, a4, b1, b2, b3, b4) are related to the in-plane plastic deformation

behavior. They are calibrated using four uniaxial tensile yield stresses in the rolling direction (RD), the direction at a 45angleto the RD, the transverse direction (TD) and the equibiaxial tension, denoted as T0, Txy45, T90 and Tb, and four uniaxialcompressive yield stresses along the same orientations denoted as C0, Cxy45, C90and Cb. With the material constants identied

above, the proposed yield function could describe the plastic behavior of metals under the x-y plane stress condition.

However, it cannot be used to model anisotropic/asymmetric plastic deformation under 3D loading because the through-

thickness plastic behavior related parameters a5, a6, b5 and b6are not fully calibrated yet. These material constants arecomputed based on the yield stress and plastic ow data in the x-z and y-z planes. Here, we use the uniaxial tensile and

compressive yield stresses in the x-z and y-z planes along a 45angle from the rolling direction denoted as Tzx45,Tyz45, Czx45andCyz45.

For the SSM (CPB06)-based model, as shown in Eqs.(2) and (9), nine independent anisotropy related parameters and one

asymmetry factork should be calibrated. Here, eight yield stresses in the x-y plane, T0,Txy45,T90,Tb,C0,Cxy45,C90andCb, and

two yield stresses in the x-z plane and the y-z plane,T

zx45andT

yz45, are employed.

5.2. Optimization algorithms

To minimize the above objective function Eq. (44), an efcient optimization algorithm should be used to identify the

parameters combined with the available experimental data. The iterative and heuristic methods are two currently used

optimization approaches. The typical iterative algorithms include the simplex method, the penalty function method, thegeneralized Lagrange multiplier method and gradient-based algorithms. The typical heuristic methods are the intelligent

algorithms, such as the genetic algorithm (GA), the simulation annealing algorithm, the ant colony algorithm, the particleswarm algorithm and the immune algorithm. Although the iterative algorithms usually have solid mathematical foundations,

the complicated derivation is difcult to solve in many problems. Even if the derivation can be obtained, an incorrect choice of

initial values may lead to severe local minima. Relatively, the heuristic algorithms depend less on the properties of the

objective functions and can provide approximate solutions with better robust stabilization despite the lack of strict math-

ematical descriptions and expensive in terms of CPU time. Due to the high nonlinearity of the above objective function,instead of the iterative methods, two types of heuristic algorithms, adv., the Nelder-Mead (N-M) method (Nelder and Mead,1965; Mathews and Fink, 2004) and GA (Lin and Yang, 1999), are selected as the candidates, and a comparison study is

conducted to develop a suitable calibration method.

The N-M method is a technique for minimizing a nonlinear objective function ofNvariables in an N-dimensional space

without constraints (Nelder and Mead,1965). This method generates a new test position by extrapolating the behavior of theobjective function measured at each test point arranged as a simplex, which is a special polytope ofN 1 vertices (P0, P1, Pn)inNdimensions. The algorithm identies the point of the greatest function value and replaces this point with a better onewith a smaller function value, which is obtained through reection, expansion, contraction and reduction according to the

evaluation of the function value of every point to get a new simplex. This allows the constantly updated simplex to shrink to

the optimum solution. The original points of the simplex are used to dene a set of ablique axes with co-ordinates xi, then the

points may be taken as a N(N1)Hessian matrix.GA mimics the natural selection process, such as inheritance, selection, mutation and crossover ( Lin and Yang, 1999). The

quality of the GA search is governed by genetic representation, population size, population initialization,

tness functions, thenumber of generations and the probabilities and operators of selection, crossover and mutation. For the GA method, theoptions used for this research are as follows: representation type is double vector; population size is 1000; tness scaling

option is ranked; the algorithm adopts stochastic uniform method to select parents based on the scaled values calculated by

the rank scaling function; ve of the selected parents will be elite; other than elite children, 95% will be produced through

scatter recombination and 5% will be produced by mutation; number of generations is 500.

6. Results and discussions

By taking uniaxial tension/compression and mandrel bending of the HSTT as the case, the above CBD constitutive

models have been identied, implemented, evaluated and nally applied to establish the quantitative correlationsamong initial textures, distorted behaviors and inhomogeneous deformation as well as multi-defect constrained

formability.



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6.1. Case material and forming procedures

As previously mentioned, cold rolling may produce titanium tubes with strong variations in their initial textures, which

causes great uctuations of their mechanical properties, and in turn, affects the formability. The preliminary study has

conrmed that the irregular yielding and nonlinear hardening behaviors along monotonic loadings are pronounced. How-ever, due to the hollow structure's limitation of tubular materials, generally only the axial tension along the rolling direction

can be used to identify the plastic response, and the Hill'48 criterion is often employed with the assumption of normal

anisotropy and planar isotropy (Dick and Korkolis, 2015). Compared with sheet metals, the modeling of tubular materials lagsfar behind (Kuwabara, 2007); this is especially true for titanium tubes. Subjected to the complex loading conditions such as

bending, inhomogeneous tension/compression deformation and multiple forming defects may occur (Yang et al., 2012),which are closely related to the constitutive characteristics (Corona et al., 2006; Cazacu et al., 2013). Here, HSTTof Ti-3Al-2.5 V

(SAE, 2010) is taken as a case material. The initial texture of the as-received HSTT is the near radial crystallographicreorientation.

Three types of experiments are conducted for the HSTT, viz., a uniaxial tension test, a uniaxial compression test andmandrel bending. As shown inFig. 5, for the tension test, the tube is clamped and tensioned by inserting a tube plug into a

piece of tube specimen. The compression test is performed with the tubular specimen. Fig. 6shows that, upon bending, non-

uniform tension and compression inevitably occur at the extrados and intrados of the bent tube to accommodate the bending

deformations, viz., wall thinning, wall thickening, cross-section attening and springback, which affects the service perfor-mance. Thus, mandrel bending should be an ideal process to assess the CBD constitutive models. The bending radius Rd is

2.0D, the bending angle is 135, the relative pushing speedVp/Vis 100%; The mandrel diameter dis designed as 9.94 mm, themandrel extension lengthe is assigned as 2 mm.

6.2. Suitable identication methods for CBD constitutive models

According to the equivalent work principle, and by taking the uniaxial tension data along the RD as a reference, as seen in

Table 1, the yield stresses of HSTT for the individual equivalent plastic strains (0.2%, 2.5%, 5%, 10%, 20%and 30%) under differentpaths are calculated. By comparing the calibration capability of N-M method and GA method regarding convergence, over-

lapping and accuracy, the suitable method for the identication of numerous parameters of the CBD framework is developed,

viz., the GA-based method, and some key factors such as initial values are provided. The Yoon's criterion is considered here as

35180

50

12

0.9

Tube Tube plug 12

(a) (b)

Fig. 5. Dimensions of the specimens: (a) Tension; (b) Compression.

O

Rd

Mandrel shank

Tangent point

Bend die

Tube

Mandrel ball

Clamp die

Wiper die

Pressure diee

D d

Mandrel extension length Vp/V

A

A

t

D'

t'

t'

A-A

F

F

t t

Extrados of tube Intrados of tube

Fig. 6. Mandrel bending principle and non-uniform tension and compression deformation.



15/32

a typical SIM to be incorporated into the CBD constitutive framework for the identication procedure. Tables 2 and 3 show thenally calibrated material parameters for each level of plastic strain by the N-M method and the GA method, respectively. The

discontinuous yield loci can then be depicted as shown in Fig. 7. The predicted yield loci obtained by both optimization

methods pass through all the corresponding data points.In the N-M method, the successful calibration is very sensitive to the initial parameters of the coordinates for 13 points

associating with the Yoon's criterion-based SIM. By using a 13 12 matrix as the initial values at the beginning of theoptimization, the material parameters of the CBD constitutive model corresponding to the rst strain increment of 0.2% areiteratively obtained. Then, the above rst set of parameters is used to obtain the material parameters of the yield equationcorresponding to the next level of the equivalent strain such as 2.5%. The material parameters under the other strain levels are

calculated to calibrate the entire discontinuous model. We found that, in the N-M method, the non-convergence problem islikely to occur, especially for initial small strains. It is difcult to determine the initial values of the 13 points using the N-M

method. Finally, the 13 12 matrix (1,0,0,, 0), (0, 1,0,,0),, (0,0,0,,1), (1,1,1,,1) is determined as the initial values toobtain the material parameters listed in Table 2. Unfortunately, as shown in Fig. 7(a), even the convergence problem is solved,

another abnormal phenomenon, viz., the overlapping of the neighboring yield loci under certain levels of equivalent plasticstrains, usually occurs, especially at the larger strain stages with much more pronounced distortional hardening. This

Table 1

Yield stress for individual equivalent plastic stain under different strain paths (MPa).

p 0.2% 2.5% 5.0% 10% 20% 30%

T0 739 846 886 928 974 1002

T90 877 989 1021 1041 1069 1097

Tb 992 1085 1122 1151 1166 1179

C0 725 828 862 907 1006 1070

C90 828 948 992 1050 1154 1221

Cb 988 1071 1110 1165 1244 1298Txy45 869 983 1019 1061 1098 1127

Tyz45 978 1075 1108 1133 1151 1154

Tzx45 978 1081 1144 1236 1302 1344

Cxy45 861 976 1015 1046 1078 1104

Cyz45 945 1034 1072 1115 1210 1264

Czx45 983 1084 1123 1166 1198 1214

Table 2

Material coefcients under various levels of equivalent plastic strain by the N-M.

p 0.2% 2.5% 5.0% 10% 20% 30%

a1 0.5888 0.6344 0.6485 0.6671 0.6943 0.7048

a2 0.8944 0.9258 0.9310 0.9290 0.9187 0.9151

a3 1.1206 1.0933 1.0941 1.0914 1.0488 1.0210

a4 0.8877 0.8881 0.8955 0.9058 0.9212 0.9251

a5 0.7122 0.7475 0.7455 0.7324 0.7469 0.7592

a6 0.6706 0.7200 0.7348 0.7537 0.7633 0.7740

b1 1.2593 1.2488 1.2146 0.7363 0.7194 0.8097b2 0.0164 0.0464 0.0552 0.0024 0.3132 0.3872b3 1.0448 1.0937 1.1230 0.8085 1.1534 1.2951

b4 0.0081 0.0066 0.0044 0.0001 0.0640 0.0283b5 0.0439 0.0417 0.0061 1.5968 0.7237 0.7725b6 0.3225 0.3541 0.3341 0.2809 0.0088 0.0167

Table 3

Material coefcients under various levels of equivalent plastic strain by the GA.

p 0.2% 2.5% 5.0% 10% 20% 30%

a1 0.5866 0.6347 0.6501 0.6669 0.6969 0.7077

a2 0.8950 0.9256 0.9293 0.9294 0.9181 0.9158

a3 1.1218 1.0928 1.0951 1.0917 1.0478 1.0195

a4 0.8878 0.8878 0.8957 0.9059 0.9214 0.9246

a5 0.7124 0.7478 0.7450 0.7323 0.7465 0.7581a6 0.6697 0.7203 0.7356 0.7535 0.7657 0.7810b1 0.2825 0.3159 0.2452 0.2162 1.2161 1.3460b2 0.1627 0.2655 0.3226 0.4649 0.2336 0.3246b3 0.8807 0.6909 0.6838 0.4684 0.8255 0.9434b4 0.3120 0.2582 0.2150 0.5014 0.2489 0.2250b5

0.1241 0.1216

0.2951 0.8570

0.7660 0.8365

b6 0.3456 0.4046 0.3837 0.3042 0.4533 0.5638



16/32

indicates that the N-M based method may converge to non-stationary points on problems that are related to the selection of

the initial values; thus, the reliability of the material coefcients optimized by the N-M method remains uncertain.The GA-based method is relatively more robust and exible. In essence, the convergence result can be obtained if the

appropriate initial values are determined as mentioned in Section 5 for the GA-related algorithm. Similarly, the initial pa-

rameters are used to obtain the convergence results for the small strain. Then, to solve the overlapping problem, the

Fig. 7. Calibrated yield loci using different calibration methods for the as-received tube: (a) N-M method; (b) GA method.



17/32

previously obtained parameters are used as a reference to obtain the material parameters for the next level of equivalent

plastic strain. This method can provide a better chance to converge to the global minimum and completely overcome thedifculty of choosing correct starting values for the constants in traditional optimization techniques.

Because both the non-convergence for each yield locus and the overlapping of the neighboring calibrated yield loci can be

solved, the above GA-based method is used to identify the material parameters in this study to describe strong anisotropy andasymmetry behaviors. In particular, the material parameters obtained using N-M based method and the GA-based method are

used in the 3D-FE simulation of the HSTT bending. Fig. 8shows that there are obvious discrepancies between the prediction

results by the N-M based method and the GA-based one.

6.3. Preferable yield functions for CBD constitutive model

As mentioned in Section 3, the CBD methodology has highexibility in selecting the yield functions for each level of plastic

strain, and the CPB06exn, Yoon's model and the mixed model of Hill'48Yoon are taken as the typical SSM and SIM can-didates to be implemented into the CBD framework. The type of yield criterion preferable for describing the evolution of the

distortional hardening should be conrmed. Other typical criteria, viz., the Mises, normal anisotropy Hill'48 and full Hill'48

function, are also considered into the CBD for comparison.

6.3.1. Yield loci evolution by different yield functions

Based onFig. 7(b), the normalized yield loci are presented. Fig. 9shows the dramatic change in the yield shape withdeformation for the as-received HSTT in either normal stress plane or shear stress plane. Fig.10 shows the predicted yield loci

0 20 40 60 80 100 120 140

2

4

6

8

10

12

14

16

18

20

Thinning (N-M)

Thinning (GA)

Wallchanging

degree(%)

Angles from initial bending plane to tangent section

Thickening (N-M)

Thickening (GA)

2

4

6

8

10

Flattening (N-M)

Flattening (GA)Sectionflatteningdegree(%)

Fig. 8. Comparison of the predicted bending deformations with the material parameters calibrated using the N-M and GA methods.

Fig. 9. Normalized yield loci for the as-received tubes.



18/32

for the as-received tube constructed using the CPB06-based SSM, Mises, normal anisotropy Hill'48 and full Hill'48 yieldfunctions. For the full Hill'48 criterion, the parameters at different interval strains are calibrated using the GA method based

on Table 1. For the normal anisotropy Hill'48, the constant R of 1.51 is directly calculated using uniaxial tension tests along the

axial direction of the tube.FromFig. 10, the greatest discrepancy between the calibrated yield loci and the experimental data can be found for the

Mises criterion and the normal anisotropy Hill'48 criterion. For the full Hill'48 yield criterion, the deformation behaviors can

be accurately depicted within the plastic strain of less than 10%, and further distorted yield behaviors cannot be described dueto its obvious asymmetry in the quadratic equation. The CPB06 presents a more accurate description than the full Hill'48 yield

equation. However, there is a certain discrepancy against the experimental data in the biaxial tension/compression region

with the equivalent plastic strain of larger than 10%. As shown in Fig. 7(b), the Yoon's criterion-based SIM provides the mostaccurate description of the anisotropic/asymmetry-induced distortional plasticity and evolution.

Additionally, as shown inFig. 11, by comparing the yield surfaces constructed by the different yield functions, it is found

that, with small deformations of less than 10%, the yield loci by all three yield functions, i.e., the full Hill'48, CPB06-based SSM

and Yoon's criteria-based SIM, are the same, while with plastic deformations of larger than 10%, the yield loci shapes by theabove three yield criteria vary greatly, although they pass through nearly all of the experimental data.

6.3.2. Prediction capability of inhomogeneous deformation

The above calibrated yield loci are introduced into the CBD framework. Considering the multiple forming indexes, a

thorough comparison of the predictions by the different CBD constitutive models is conducted against the experimental

results. The detailed FE modeling issues for mandrel bending can be found in the literature (Li et al., 2012). In the FE models,the Young's modulus of the HSTT equals 102.662 GPa.

Fig. 12shows the comparison of the predicted stressestrain curves with the experimental ones for uniaxial tension/

compression. The predictions by the SSM (CPB06) and SIM (Yoon's criterion)-based CBD models agree with the experimentaldata of uniaxial loading. However, for each forming index of the tube bending, the predictions by the different CBD models

vary greatly.Fig. 13shows that, as for the plastic strain distribution by the Mises criterion, hardly any plastic deformation

occurs near the neutral layer of the tube, which causes the largest springback angle, as shown in Fig. 14(a), Whereas for the

Fig. 10. Comparison of the yield loci evolutions for the as-received tube using the different yield functions: (a) CPB06; (b) full Hill'48; (c) normal anisotropy

Hill'48; (d) Mises.



19/32

Yoon's criterion-based SIM, CPB06-based SSM, Hill'48Yoon's mixed criterion and Hill'48 model, the predicted elastic strainhas a lower share of the total strain, resulting in a smaller springback. However, the predictions of the Hill'48 and the Mises

criteria are much closer to the experimental ones. The reason for this interesting result may be attributed to the assumptions

of both linear elastic behavior and isotropy plastic behavior for the Mises criterion, which will be further conrmed in futureresearch. It has been observed that the nonlinear elastic recovery phenomenon is obvious for HSTT during unloading, In this

study, we focus on modeling of anisotropy and asymmetric induced plasticity and revealing the non-ignorable effects of the

distorted yielding on springback.Fig. 14(b) shows that the cross-section attening degrees predicted by Yoon's criterion, CPB06, Hill'48 Yoon mixed

criterion and full Hill'48 based-CBD models present remarkable uctuations along the entire bending region. Yoon's criterion,

Hill'48Yoon mixed criterion and full Hill'48-based CBD models produce the largest values of the cross-section attening,which is much closer to the experimental results. Fig. 14(c) and (d) show that the similar tendencies of the wall thickening andthe wall thinning degree are predicted by all CBD constitutive models. Except for the Mises criterion and CPB06-based CBD,

both the wall thinning and wall thickening predictions by the other models all agree with the experimental results. Insummary, the Yoon's criterion and Hill'48 Yoon mixed criterion-based CBD models can provide the most accurate pre-dictions of the above forming indexes with the exception of the springback.

As mentioned in Section6.3.1, the CPB06 presents an accurate description of the distortional hardening. However, when

the discontinuous yield loci by the CPB06 are implemented into the CBD framework, the unequal deformation during bendingcannot be accurately predicted. Why does this interesting phenomenon happen? It is known that the prediction accuracy of

the constitutive model depends on whether the local deformation characteristics under special stress states closely related tothe forming indexes are captured by the proposed model or not.

Here, the stress states along the inner and outer crest line of tube and geometrical center line of the tube are depicted as

shown inFig. 15. It is found that there simultaneously exist distinguishable tension and compression regions during tube

bending. The forming quality is closely related to these local stress states. Especially, as shown inFig. 6, because the cross-

Fig. 11. Comparison of the yield surfaces constructed by the different yield functions: (a) 10%; (b) 30%.



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section attening is subjected to the resultant force of tension and compression stress, it is considered to be more sensitive tothe distorted plasticity and evolution of the tubular materials.

As shown in Fig. 11, with the small strain of less than 10%, the shapes of all three yield loci coincide, while the shapes of the

three yield loci gradually become different as the plastic deformation increases. At the strain of 30%, the distinction is mostobvious in all four quadrants. This distinction in yield loci shapes maybe a nontrivial reasonwhy the predictions by the Yoon's

criterion and Hill'48-based CBD models are in good agreement with the experimental data, whereas the predictions by the

CPB06-based CBD have an apparent discrepancy.It is known that the prediction accuracy of the plastic deformation for the constitutive model is determined by the reliable

computation of both the yielding and the plastic ow direction. Because the AFR rule is used, the plastic ow direction is

directly related to the yield function. As shown in Fig. 16, the abovementioned shape discrepancy of the CPB06-based CBD

thus results in obvious different predictions from the ones using Yoon's criterion and the Hill'48-based CBD models. Ac-cording to Eqs. (A-5) and (A-8), the r and rs at different strain increments can be calculated based on the established

discontinuous yield functions. Here, considering the stress state ofs

[s0

,s0

,0,0,0,0]1, the detailed rand rs with p of 30% arecalculated as below for different yield models including the CPB06, Yoon's criterion and Hill'48. It is found that both values oftherandrsare different for the three yield functions. It is noted that the spatial allocations of the randrsfor the CPB06 are

most dramatic.

r

8>>>>>>>>>>>:

0:7880:212

1000

9>>>>>>=>>>>>>;

s0; rs

26666664

0:388 0:030 0:418 0 0 00:030 0:082 0:112 0 0 0

0:418 0:112 0:530 0 0 00 0 0 1:537 0 00 0 0 0 0:952 00 0 0 0 0 1:807

37777775s0 CPB06 (45)

r

8>>>>>>>>>>>:

0:4890:511

1000

9>>>>>>=>>>>>>;

s0; rs

26666664

0:326 0:170 0:162 0 0 00:170 0:332 0:168 0 0 0

0:162 0:168 0:342 0 0 00 0 0 0:726 0 00 0 0 0 0:892 00 0 0 0 0 0:491

37777775s0 Yoon0s criterion (46)

r

8>>>>>>>>>>>:

0:6870:313

1000

9>>>>>>=>>>>>>;

s0; rs

26666664

0:420 0:240 0:180 0 0 00:240 0:320 0:080 0 0 00:180 0:180 0:260 0 0 0

0 0 0 1:010 0 00 0 0 0 0:900 00 0 0 0 0 0:390

37777775s0 Hill

048 (47)

As mentioned in Section3, the accuracy of CPB06 in describing the anisotropy and asymmetry depends on the careful

selection of the values ofa, kand the linear transformation matrix. In this study, the CPB06 yield function is used with an aof

0.00 0.05 0.10 0.15 0.20

700

750

800

850

900

950

1000

1050

Exp (compression)

Exp (tension)

SIM (Yoon's model)-based CBD (compression)

SIM (Yoon's model)-based CBD (tension)

SIM (CPB06)-based CBD (compression)

SIM (CPB06)-based CBD (tension)

Truestress(MPa)

Plastic strain

Fig. 12. Comparison between numerical and experimental stressestrain curves.



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2. In the report (Cazacu et al., 2006), it was suggested that, by modifying the value ofa, the plastic deformation of differenttypes of materials can be described. As shown in Fig. 17, for different values ofa, the yield loci of CPB06 all pass through the six

experimental data. However, the shape of the yield loci becomes very distorted. This uncertainty of the yield surfaces by the

CPB06 can be further improved by increasing the linear transformation matrix as CPB06ex2, CPB06ex3 or CPB06exn (Plunkettet al., 2008). However, it is unfortunate that the initial physical meaning may be lost away, more uncertainties may be

induced, and the total number of the involved material parameters increases dramatically, which causes the yield function to

be unfeasible for analyzing the practical forming processes.In addition to the prediction accuracy, the computation cost is also evaluated. The simulations in the paper are all per-

formed on the personal computer with Inter(R) Core (TM) i3-3220 CPU, 3.30 GHz and 3.47 GB memory. The actual bending

Fig. 13. Comparison of effective plastic strain predicted by the different constitutive models.

Exp SIM(Yoon) Hill+YoonSSM(CPB06) Hill'48 Hill '48R Mises

0

1

2

3

4

5

6

7

8=

0 20 40 60 80 100 120 140

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

= 100%D D

DD

Sectionflatteningdegree(%)


Exp

SIM(Yoon's model)

Hill'48+SIM(Yoon's model)

SSM(CPB06)

Hill'48

Hill'48R

Mises

0 20 40 60 80 100 120 140

0

5

10

15

20

25

= 100%t t

tt


Wallthickeningdegree(%)

Exp

SIM(Yoon's model)


SSM(CPB06)

Hill'48

Hill'48R

Mises

0 20 40 60 80 100 120 140

0

5

10

15

20

25

= 100%t ttt

Wallthinningdegree(%)


Exp

SIM(Yoon's model)


SSM(CPB06)

Hill48

Hill48R

Mises

(a) (b)

(c) (d)

Fig. 14.Comparison of bending deformations for different constitutive models: (a) springback angle; (b) wall thickening; (c) wall thinning; (d) section attening.



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time is 3.5 s, and the computation times for the SIM (Yoon's model), Hill'48Yoon mixed criterion, SSM (CPB06) and Hill'48-based CBD models are 6.5 h, 6.3h, 10.1 h and 4.3 h, respectively. Additionally, Yoon's yield equation associated with a Swift

hardening model is established as the case of the traditional continuous model with the assumption of isotropic hardening. Asshown inFig. 18, the expected discrepancy is observed for different forming indexes. However, the computation cost for the

continuous model is 6.2 h, and the difference in the computation efciency is minor. Thus, the computation efciency is

validated. This further indicates that the CBD constitutive framework is suitable for practical usages in complex forming

processes.In summary, from the perspective of capturing capabilities and efciencies of these models, we can conclude that the CBD

framework provides a practical model for the characterization of the anisotropy and asymmetry related inhomogeneousdeformations. The Yoon's criterion-based SIM proves to be preferable when embedded into the CBD framework for the

prediction of the distorted plasticity-induced unequal deformation. In the following, we used the SIM-based CBD constitutive

model to explore the role of distorted plasticity in the formability.

6.4. Application of the CBD constitutive models

Because strong texture variations of the HSTT mayoccur during fabrication processing, it is fundamental to have an insight

into which type of textures can achieve improved formability for subsequent processes. By taking the SIM (Yoon's criterion)-based CBD constitutive models as the bridge, the quantitative correlations among the initial textures, distorted behaviors and

inhomogeneous deformation as well as multi-defect constrained formability are established under uniaxial tension/

compression as well as mandrel bending.

Fig. 15. Stress states of the bending tube.

Fig. 16. Schematic of the different plastic ow for various yield surfaces.



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6.4.1. 3D characterization of tubes with different initial textures

In addition to the near radial texture of the as-received tubes, another ve typical initial textures of the HSTT are

considered, viz., rolling texture, tangential texture, radial texture, bimodal texture and random texture. Here, the distortedplasticity and its evolution in HSTT with these initial textures are characterized by SIM (Yoon's criterion)-based CBD in 3D

space, as shown inFig. 19. The detail calibrated parameters by GA-based method are shown in Appendix B.It can be seen that the diverse yielding behaviors and their evolution features are presented for the tubes with different

initial textures. The yield loci for the random texture are an elliptical shape with sound symmetry and isotropy, which can be

depicted exactly using the Mises yield function. Similar to the as-received tube, with the exception of random texture, theyield loci present a nonlinear and distorted evolution in shapes. Of these initial textures, the yield loci for the bimodal textureare most distorted with the highest strength under in-plane equibiaxial loading. The yield loci for the as-received texture

seem more similar to the combinations of the loci for the radial texture and the tangent texture.

6.4.2. Role of distorted plasticity in uniaxial tension/compression

The detailed tension and compression processes are mentioned in Section 6.1. For the uniaxial tension, as shown in Fig. 20,

theow stress and elongation of the HSTT with different initial textures differ greatly from each another. The tube with therolling texture has the best tensile properties for both strength and elongation. The reason is attributed to the parallel

relationship between the grain orientation (c-axis of the grain) and the loading direction, which causes increased defor-mation resistance along the axial direction and more uniform deformation. In contrast, for the tube with an initial bimodal

texture, due to harder deformation resistance along both the normal and tangent directions compared to that along the axial

direction, the deformation along the axial direction cannot be effectively accommodated by the other deformation sources.

Fig. 17. The yield surfaces calibrated by CPB06-based SSM at different a.

0 20 40 60 80 100 120 140

0

2

4

6

8

10

12

14

16

18

20

Thickening (continuous)

Thickening (discontinuous)

Thinning (continuous)

Thinning (discontinuous)

Wallchangingdegree(%)


2

4

6

8

10

12

14

Flattening (continuous)

Flattening (discontinuous)

Sectionflatteningdegree(%)

Fig. 18. Comparison of the bending deformations predicted using the different material models.



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This causes decreased strength and poorer formability with uniaxial tension along axial direction. For the tubes with initialradial and tangent textures, the worst elongation is observed, in which necking occurs after a short stretching.

For uniaxial compression with 30% height reduction, as shown in Fig. 21, different deformation behaviors occurs. For the

tubes with initial rolling, random and tangential textures, severe inhomogeneous deformation, viz., the cross sectiondistortion of the ring specimens, is observed. The reason is that the deformation resistance along the thickness direction of

the tube is less than the resistance along the axial direction. However, as previously mentioned, the tubes with initial rolling,

random and tangential textures all present preferential behaviors under tension conditions. For the tubular materials withinitial bimodal, radial and as-received textures, homogenous deformation can be observed with only slight attening phe-

nomena. The reason is attributed to the sufcient deformation coordination capability along the thickness direction of the

tube.

6.4.3. Role of distorted plasticity in bending formability

The details of the mandrel bending are mentioned in Section 6.1. Fig. 22 represents the distribution contours of the

equivalent plastic strain for bent tubes with different initial textures. The distribution of plastic strain in tubes with bimodaland as-received textures is more homogeneous and thus has a greater share of the total strain. As a result, the springback

angles of the tubes with bimodal textures and as-received textures are the lowest as shown in Fig. 23(a). The springback of the

tubes with bimodal textures is approximately 50% less than that of the as-received tube and 74% less than that of the tubes

Fig. 19. Calibrated yield loci from SIM-based CBD model with different initial textures: (a) rolling texture; (b) tangent texture; (c) radial texture; (d) bimodal

texture; (e) random texture.



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with an initial rolling texture, which present the largest springback angle. Regarding the cross-section attening, as shown inFig. 23(b), there are obvious uctuations along the entire bending regions for tubes with initial bimodal textures and as-received textures, adv., apparent minimal value and maximal value are observed. However, these deformation characteris-

tics for the tubes with other textures show little change along the bending regions. Because of difcult deformation along the

normal direction of tube coordinated by a harder pyramidal slip and compression twinning, both the wall thinning degreeand the wall thickening degree of the tubes with the initial bimodal texture and the as-received texture are much smaller than

those with the other textures, as shown in Fig. 23(c) and (d).

The comparison of the above forming indexes has claried that the initial bimodal texture can signicantly improve thebending formability, i.e., reduced cross-section attening, wall thinning/thickening and springback. This means that a higher

formability with smaller bending radius can be achieved if the near bimodal texture was tailored during cold rolling by

dedicatedly coordinating the plastic ows along the axial, normal and hoop directions.

7. Conclusions and remarks

This study focuses on accurately and efciently modeling of anisotropy/asymmetry-induced distorted plasticity andnonlinear evolutions of hard-to-deform materials. The constructed CBD framework is numerically implemented, fully eval-

uated and practically applied to provide transferable knowledge of the correlations among initial textures, distorted plasticity,inhomogeneous deformation and multi-defect constrained formability. The conclusions and remarks are as follows:

(1) We try to divide the existing yield criteria into two categories from the viewpoint of the physical meanings, viz., the

principal shear stress-based models (SSM) and the stress invariants-based models (SIM). Considering the multi-mechanism coordinated distortional yielding and nonlinear evolution of the materials, a unied CBD constitutive

framework is constructed, in which both the discontinuous SSM, the SIM or their combinations are used to characterize

the distorted shape of the yielding, and an interpolation approach is adopted to present the nonlinear evolution of thedistorted plasticity in the full stress space. The microstructure evolutions (such as texture variation) caused nonlinear

hardening and ow rules are implicitly included in this discontinuous framework.

(2) Taking CPB06 and Yoon's criterion into the CBD constitutive framework, both the SSM- and SIM-based CBD models are

successfully numerically implemented into the explicit 3D-FE platform by combining implicit algorithm and inter-polation approach to simulate complex forming processes. Via a comparison of the N-M and GA methods in terms of

convergence, overlapping and accuracy, the GA-based approach is proposed to accurately and efciently calibrate thenumerous parameters of the CBD framework, and some key factors such as the initial values are provided. Regarding

the capturing capability of anisotropy/asymmetry coupled distorted behaviors, the SIM model seems to be preferable

for embedding into the CBD framework because of its sound physical basis and numerical robustness.

(3) Taking the uniaxial tension/compression and mandrel bending as the cases, the distorted plasticity and evolution ofthe HSTT with six typical initial textures are characterized. The correlations among the initial textures, distorted

behaviors and inhomogeneous deformation are quantitatively established to improve the formability of the hard-to-

deform materials. Different texture characteristics and plastic deformation behaviors are required to improve theformability in different forming processes. The anisotropy/asymmetry characteristics specic to HSTT greatly affect the

inhomogeneous deformation and related defects. For tension, the tubes with rolling textures present the best tensile

properties for strength and elongation. For compression, a more uniform deformation is observed for the tubes with

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.1750

200

400

600

800

1000

1200

As-recieved

Rolling

Tangential

Radial

Bimodal

Random

Truestress(MPa)

True strain

Fig. 20. True stressestrain curves of tubes with different initial textures during uniaxial tension.



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the as-received texture and bimodal texture. For complex bending, the bimodal texture is identied as most preferablefor the best bendability in terms of cross-section attening, wall thickening, wall thinning and springback.

(4) The currently developed materials with more complex microstructures may cause the transition from elasticity to

plasticity to be more distorted in the full stress space, particularly accompanying with interaction of the multiplecomplex mechanisms, such as reversible/irreversible, hardening/softening and cyclic/monotonic behaviors. This sit-

uation demands further extensive evaluation, extension and application of the present CBD constitutive framework for

describing greater nonlinear plastic behaviors in the case of complex forming conditions with spatial, abrupt strain

path changes or thermal-mechanical coupling loadings, such as cold/warm sheet/bulk sheet forming. In any event,

Fig. 21. Compression deformation of tubes with different initial textures: (a) cross section; (b) half of the ring specimen.



27/32

efciently calibrated and robustly implemented constitutive models with fewer parameters, greater accuracy andsound physical backgrounds are required when analyzing practical forming processes which are subjected to complex

loading boundaries.

Fig. 22. Comparison of effective plastic strain for tubes with different initial textures.

As-receivedRolling Tangential Radial Bimodal Random0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 1400.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

As-recieved

Rolling

Tangential

Radial

Bimodal

RandomSectionflattening

degree(%)

Angles from initial bending plane to tangent section ( )

0 20 40 60 80 100 120 140

0

5

10

15

20

25

30

As-received

RollingTangential

Radial

Bimodal

Random

Wallthick

eningdegree(%)


0 20 40 60 80 100 120 1400

5

10

15

20

25

30As-received

Rolling

Tangential

Radial

Bimodal

Random

Wallthinningdegree(%)


(a) (b)

(c) (d)

Fig. 23. Comparisons of the bending deformations: (a) springback angle; (b) section attening; (c) wall thickening; (d) wall thinning.


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