Non-quadratic Kelvin modes based plasticity criteria for anisotropic materials

International Journal of Plasticity 27 (2011) 328–351

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Non-quadratic Kelvin modes based plasticity criteria for anisotropic materials Rodrigue Desmorat a,*, Roxane Marull a,b a b

LMT-Cachan, ENS Cachan/UPMC/CNRS/PRES UniverSud Paris, 61 avenue du président Wilson, 94235 Cachan Cedex, France Snecma Villaroche, Rond-point René Ravaud - Réau, 77550 Moissy-Cramayel, France

a r t i c l e

i n f o

Article history: Received 16 October 2009 Received in final revised form 20 May 2010 Available online 10 June 2010 Keywords: Kelvin modes Yield criterion Anisotropic materials Cubic symmetry CMSX2 superalloy

a b s t r a c t Novel (non-quadratic) plasticity criteria based on Kelvin modes are formulated here for anisotropic materials. As an example, such a macroscopic criterion is applied with success to the case of FCC nickel-base single crystals. Indeed, relying on the cubic symmetry of the material, the Kelvin decomposition of elasticity tensor easily allows for the definition of an objective and loading independent criterion. The criterion identification is performed from different loading cases for CMSX2 single crystal superalloy. Tension–torsion yield surfaces at room temperature and yield stress dependence on crystal orientation are modeled. The Kelvin modes based criterion is compared to experimental data, to Hill and Barlat and coworkers macroscopic criteria and to Schmid law predictions. The results show that a simple three-parameter yield function built thanks to von Mises equivalent Kelvin stresses accounts for a satisfying plasticity criterion for such alloys. Non-quadratic norm kka plasticity framework is addressed. Intrinsic generalizations of Hershey–Hosford criterion are proposed for cubic material symmetry. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Plasticity of anisotropic materials is often difficult to model. It has nevertheless an important role in forming processes (Hill, 1950; Barlat et al., 1991, 1997; Karafillis and Boyce, 1993; Lademo et al., 1999; Yoon et al., 2000; Bron and Besson, 2004), when dealing with failure modes (Wu et al., 2003; Boumaiza et al., 2006) or for specific materials such as aeronautics single crystal superalloys (Nouailhas and Culie, 1991; Zamiri et al., 2007; Zamiri and Pourboghrat, 2010). In macroscopic models, the plastic behavior is described by means of a convex yield surface that evolves during plastic deformation. The first anisotropic yield function was proposed by Hill (1948) for orthotropic materials. From the beginning, Hill has pointed out the limitations of his famous criterion and has extended it into a more general power function of degree n in plane stress conditions (Hill, 1950),

f ¼

X

Apqr rp11 rq22 rr12  1 < 0 ! elasticity

ð1Þ

pþqþr6n

where the powers p, q, r are positive or zero integers, with p + q + r 6 n and with the necessary existence of r even when 1 and 2 are the principal axes of orthotropy. Non-quadratic yield functions have recently regained interest (Aretz et al., 2007; Banabic et al., 2003; Soare et al., 2008), for instance when tension–compression asymmetry has to be taken into account (Cazacu and Barlat, 2004; Hu, 2005). * Corresponding author. E-mail address: [email protected] (R. Desmorat). 0749-6419/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2010.06.003

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329

Barlat et al. (1991) and coworkers approach has proved efficient when modeling complex plastic anisotropy, of aluminum alloys for example. It introduces one (or more) linearly transformed stress deviator s ¼ s0 ¼ L : r in existing isotropic criterion functions fiso – often based on Hershey–Hosford yield criterion allowing for continuous transition from von Mises to Tresca criterion (Hershey, 1954; Hosford, 1972),

f ðrÞ ¼ fiso ðL : r0 Þ < 0 ! elasticity

ð2Þ

The identification of the linear transformation L (of sometimes several) is still not an easy task in the general case. On the other hand, elasticity anisotropy has been extensively studied, but some modeling tools have not so much been put into light. For instance, Kelvin decomposition of the Hooke tensor has been rediscovered only 25 years ago (Rychlewski, 1984; Mehrabadi and Cowin, 1990). Apart from elasticity, its applications are not numerous but concern damage modeling (Biegler and Mehrabadi, 1995; François, 1995) and strength criteria (Arramon et al., 2000) of initially anisotropic materials. One proposes in present work to use the information given by Kelvin decomposition to derive a family of macroscopic yield criteria based on so-called Kelvin modes. General anisotropic plasticity can be described thanks to this approach, but as an application one mainly focuses here on the ability of the modeling to deal with FCC single crystal plasticity, where macroscopic criteria usually encounter difficulties (Nouailhas and Cailletaud, 1995). Superalloy CMSX2 is considered, either at room temperature or at 650 °C. Tension–torsion loading surfaces and yield stress vs crystal orientation curves will be modeled (Sections 5 and 6). Finally, an intrinsic formulation of non-quadratic norm kka plasticity will be given. 2. Kelvin decomposition in elasticity and for plasticity coupling As just mentioned, Kelvin spectral formulation of elasticity (Thomson (Lord Kelvin), 1856, 1878) has been rediscovered quite recently. Its applications have naturally concerned linear anisotropic elasticity but also materials with asymmetric elasticity in tension and in compression (Desmorat and Duvaut, 2003), multiaxial strength criteria (Arramon et al., 2000), creep (Bertram and Olschewski, 1996; Mahnken, 2002) and Continuum Damage Mechanics (Biegler and Mehrabadi, 1995; François, 1995; Desmorat, 2000, 2009). As pointed out by Cowin et al. (1991), it is possible to use such a formulation as a tool to derive yield criteria and plasticity theories for initially anisotropic materials, as it naturally extends the concept of deviatoric and hydrostatic stresses to anisotropy. 2.1. Kelvin decomposition of elasticity tensor The elasticity fourth rank tensor E (resp. its Voigt matricial representation [E]) has eigenvalues K(I) and corresponding second rank symmetric eigentensors e(I) (resp. eigenvectors ^eðIÞ ) are solutions of the eigenproblem

E : eðIÞ ¼ KðIÞ eðIÞ ; ½E^eðIÞ ¼ KðIÞ ^eðIÞ ;

eðIÞ : eðJÞ ¼ dIJ

ð3Þ

^eðIÞ  ^eðJÞ ¼ dIJ

with dIJ the Kronecker symbol. The couples ðKðIÞ ; eðIÞ or ^ eðIÞ Þ are the Kelvin modes, the eigenvalues K(I) are the Kelvin moduli (Rychlewski, 1984). These moduli are at most six (in the general anisotropy case for instance), they are only two in case of isotropic elasticity (3K and 2G if K and G are the bulk and shear moduli). One will find in Mehrabadi and Cowin (1990) work the expressions of Kelvin moduli and modes for the principal material symmetries. Kelvin decomposition of Hooke tensor is then simply the following re-writing (either in a tensorial or in a matricial one),

E¼

6 X

KðIÞ eðIÞ  eðIÞ

()

½E ¼

I¼1

6 X

KðIÞ ^eðIÞ ð^eðIÞ ÞT

ð4Þ

I¼1

with T meaning the transpose. There is always a family of six orthogonal eigentensors e(I) but some eigenvalues can be multiple (therefore the eigentensors family not unique), depending on the material symmetry. The terms of identical moduli K(I) = KK can be advantageously grouped (François, 1995),

E¼

N66 X

KK P K

PK ¼

K¼1

X ðIÞ

eðIÞ  eðIÞ

ð5Þ

K

I=K ¼K

with N the number of different Kelvin moduli. The projectors PK are unique for a given elasticity tensor E. The compliance tensor is

S ¼ E1 ¼

N66 X 1 K¼1

KK

PK

ð6Þ

and the elastic energy density reads

we ¼

N 1 1 1X 1 K :E:¼ r:S:r¼ r : rK 2 2 2 K¼1 KK

ð7Þ

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with  and r the strain and stress tensors and where

rK ¼ PK : r

ð8Þ K

For a given material symmetry, last equation defines in a unique and objective manner the Kelvin stress r as the projection of the stress tensor on the Kth Kelvin mode. If the same projection is made for the strain, i.e. K ¼ PK : , the elasticity law r ¼ E :  is equivalent to

rK ¼ KK K 8K

ð9Þ

2.2. Yield criterion defined in terms of Kelvin stresses A criterion function f(r) defines the elasticity domain by f < 0 and is often used as a potential to derive, by normality, the plasticity evolution laws. In case of isotropy f is usually expressed in terms of stress invariants. The anisotropic case is more difficult (Hill, 1950, 1979; Lemaitre et al., 2009; Barlat et al., 2005). A possibility that has not much been studied is to use Kelvin stresses and to define the criterion function as a function of the rK,

f ðrÞ ¼ f ðrK¼1 ; . . . ; rK¼N Þ

ð10Þ

Such a formulation is objective (thanks to the objective definition of Kelvin stresses). Global convexity is gained from ensuring the convexity with respect to each Kelvin stress. In the isotropic case, the two Kelvin stresses associated with the Kelvin moduli 3K = E/(1  2m) and 2G = E/(1 + m) (with E and m the Young modulus and the Poisson ratio) are respectively the hydrostatic stress,

1 3

rH ¼ trr 1

ð11Þ

with 1 the second order unit tensor, and the deviatoric stress

r0 ¼ r  rH

ð12Þ

Defining thus a plasticity criterion based on Kelvin modes gives back the well known feature that f is a function of the deviatoric and hydrostatic stresses when isotropy is considered. 2.2.1. Yield criteria for general anisotropy General anisotropy but also orthotropy are too complex to be apprehended by the Kelvin stresses concept. Indeed the calculation of the rK is not an easy task in these cases, and having to deal with six Kelvin stresses will make it difficult to carry out such an approach in any case. Several anisotropic formulations for Kelvin modes based yield criteria can however be proposed. Note that crystallographic considerations tend to invalidate quadratic criteria for anisotropic materials (Hosford, 1996). Also, a generalized yield criterion involving different stresses exponents were formulated by Hill, as Eq. (1), in order to model the earing of deep-drawn cups in plane stress. With this in mind, different stresses exponents nK 6 n are introduced next as material parameters. A formulation of the Kelvin modes based criterion homogeneous to MPan (i.e. rn) can thus be

f ¼

N X

 n =2 cK rK : rK K  1

ð13Þ

K¼1

with eventually different values nK 6 n for the considered exponents. The constants cK P 0 are also material parameters. Equally large values of nK = n ? 1 give back Arramon et al. (2000) criterion of anisotropic material strength governed by the maximum Kelvin stress. Note that this criterion cannot lead to incompressible plasticity (by normality) if the hydrostatic stress rH is not a Kelvin mode. But if rH = rK=1 is an eigenmode of elasticity tensor E, set as Kelvin mode 1, i.e. if1 E1111 + E1122 + E1133 = E2211 + E2222 + E2233 = E3311 + E3322 + E3333, plastic incompressibility is simply gained by setting cK=1 = 0. Incompressible plasticity is often assumed. It can be gained – still by normality – from an anisotropic criterion rewritten as (constants cK may differ from those of Eq. (13))

f ¼

N X

 n =2 cK ðrK Þ0 : ðrK Þ0 K  1

ð14Þ

K¼1

where ()0 stands for the deviatoric part. Note that the matricial norm kk2 = (() : ())1/2 = (tr()2)1/2 has been used in previous expressions. Following Hershey (1954) and Hosford (1972), the norm kka = (trjja)1/a can be used instead of kk2 in either Eq. (13) or Eq. (14), with for example for the incompressible case, 1

Condition satisfied in case of isotropy and of cubic symmetry.

R. Desmorat, R. Marull / International Journal of Plasticity 27 (2011) 328–351

f ¼

N X

N   X  n a nK =aK cK ðrK Þ0 aK  1 ¼ cK trðrK Þ0  K 1 K

K¼1

331

ð15Þ

K¼1

If the exponents nK are all equal to n, last equation defines an effective stress as introduced by Karafillis and Boyce (1993) and Bron and Besson (2004),

"

r ¼

N X

  a n=aK cK trðrK Þ0  K

#1=n ð16Þ

K¼1

The use of norm kka for full plasticity coupling (allowing continuous transition von Mises ? Tresca), and the corresponding definition of an intrinsic expression of the normality rule is detailed in Section 8. Last, Kelvin decomposition may be used as a guide to identify Barlat transformation tensors, for instance by looking for their projections on Kelvin modes. In other words tensors L can be set as

L¼

N X

aK PK

ð17Þ

K¼1

with PK the Kelvin projectors defined by Eq. (5), and where parameters aK are the unknowns. There are a maximum of 6 parameters aK (the same number as for classical Hill criterion or as for Barlat initial criterion). All the criteria should be confronted to experimental data in order to conclude about their validity. One chooses next to focus on the proposed modeling ability in the not so simple case of single crystal superalloys elasto-plasticity which exhibits cubic material symmetry.

2.2.2. Application to cubic symmetry Cubic material symmetry, encountered for instance in FCC single crystals alloys, allows for an easy Kelvin decomposition of the elasticity tensor and ends up to simple Kelvin stresses. Kelvin decomposition has besides been applied by Bertram and Olschewski (1996) to SRR99 FCC single crystal elasticity and creep modeling. In this cubic case, if stresses and strains are rewritten (thanks to Voigt notation)



pffiffiffi

pffiffiffi

pffiffiffi

r^ ¼ r11 ; r22 ; r33 ; 2r23 ; 2r31 ; 2r12 

^ ¼ 11 ; 22 ; 33 ;

T ð18Þ

pffiffiffi pffiffiffi pffiffiffi T 223 ; 231 ; 212

the elasticity law r ¼ E :  takes the canonical form (in the natural anisotropy basis),

2

r^ ¼ ½E^ ½E1

1 E

 mE  mE

6 m 1 6  E E  mE 6 6m m 1 6 E E ¼6 E 6 0 0 0 6 6 0 0 0 4 0 0 0

0

0

0

0

0

0

1 2G

0

0

1 2G

0

0

0

3

7 07 7 07 7 7 07 7 07 5

ð19Þ

1 2G

with E the Young modulus, m the Poisson ratio and G the shear modulus (different from E/2(1 + m)). There are N = 3 Kelvin moduli,

E 1  2m E ¼ 1þm ¼ Kð6Þ ¼ 2G

KK¼1 ¼ Kð1Þ ¼ 3K ¼ KK¼2 ¼ Kð2Þ ¼ Kð3Þ KK¼3 ¼ Kð4Þ ¼ Kð5Þ

Still in the material natural anisotropy basis the corresponding eigentensors are

ð20Þ

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2

p1ffiffi 3

0

0

3

7 6 p1ffiffi 07 eð1Þ ¼ 6 5 40 3 0 0 p1ffiffi3 2 1 3 pffiffi 0 0 2 6 7 ð2Þ e ¼ 4 0  p1ffiffi 0 5; 2 0 2

0

6 eð4Þ ¼ 4 p1ffiffi

2

0

0 p1ffiffi 2

0

0 3

0

7 0 5;

0

0

2

p1ffiffi 6

0

0

3

7 6 p1ffiffi 0 7 eð3Þ ¼ 6 5 40 6 0 0  p2ffiffi6 2 3 2 0 0 0 0 0 6 0 0 p1ffiffi 7 6 ð5Þ ð6Þ e ¼40 0 e ¼4 2 5; p1ffiffi 0 0 p1ffiffi2 0 2

ð21Þ 3 p1ffiffi 2 7 05 0

defining the projectors – renamed PH ; Pd ; Pd – in an objective and unique manner,

PK¼1 ¼ PH ¼ eð1Þ  eð1Þ PK¼2 ¼ Pd ¼ eð2Þ  eð2Þ þ eð3Þ  eð3Þ P

K¼3

d

ð4Þ

¼P ¼e

e

ð4Þ

ð5Þ

þe

e

ð5Þ

ð22Þ ð6Þ

þe

e

ð6Þ

Kelvin stresses rK ¼ PK : r are in fact easily gained in the cubic symmetry (natural) basis:  rK=1 = rH is the hydrostatic stress 13 trr 1 (associated with the Kelvin modulus 3K),  rK=2 = rd is the diagonal part of the deviatoric stress in natural anisotropy basis (associated with the Kelvin modulus E/(1 + m)),  rK¼3 ¼ rd is the out of diagonal deviatoric tensor in this same basis (associated with the Kelvin modulus 2G). Two deviatoric stresses, rd and rd , are then naturally introduced from Kelvin analysis with

r0 ¼ rd þ rd and r ¼ rd þ rd þ rH

ð23Þ

Yield functions for materials with cubic symmetry are finally set as

f ¼ f ðrd ; rd ; rH Þ

ð24Þ d

d

The hydrostatic stress is usually dropped off for incompressible materials with then f ¼ f ðr ; r Þ. Applying here the yield criterion defined in previous section for the general anisotropy (either Eq. (14) or Eq. (13) with c1 = 0 as the hydrostatic stress is the first Kelvin mode) leads then to

 n3 =2  n2 =2 f ¼ c2 rd : rd þ c3 rd : rd 1

ð25Þ

3. Link with existing criteria for material cubic symmetry The novel criterion (25) which is formulated at macroscopic level has to be compared to existing macroscopic yield criteria formulated for cubic symmetry. Three criteria are chosen for this purpose: Hill criterion first, which is a classical and easy to carry out anisotropic criterion; then Cazacu and Barlat (2003) invariants based criterion and finally Barlat yield function (Barlat et al., 1991, 2005) since it is more flexible and since it has proved efficient for more general anisotropy. Kelvin modes decomposition and the decomposition of the stress tensor into r ¼ rd þ rd þ rH will be found very useful to rewrite other existing criteria in some particular cases. As a first example, Hill yield criterion function for cubic symmetry, with F and L as material parameters,

h i   f ¼ F ðr11  r22 Þ2 þ ðr22  r33 Þ2 þ ðr33  r11 Þ2 þ 2L r212 þ r223 þ r231  1

ð26Þ

is simply the case n2 = n3 = 2 of criterion (25),

f ¼ 3F rd : rd þ L rd : rd  1

ð27Þ

Second, Cazacu and Barlat (2003) applied the theory of representation to extend Drucker (1949) isotropic criterion function of the third invariant to material anisotropy cases among which the cubic symmetry. Let us recall that second and third invariants of the stress deviator are classically defined as

J2 ¼

1 2 1 r ¼ r0 : r0 ; 3 eq 2

J 3 ¼ detðr0 Þ

and that the yield criterion proposed by Drucker (1949) is

ð28Þ

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R. Desmorat, R. Marull / International Journal of Plasticity 27 (2011) 328–351 2

f ¼ J 32  cJ 23  k

ð29Þ

with c and k as material parameters. Cazacu and Barlat (2003) have reformulated it f ¼ metry case using generalized invariants defined in the natural anisotropy basis as

J C2 ¼

ðJ C2 Þ3



cðJ C3 Þ2

2

 k in the cubic sym-

i m1 h ðr11  r22 Þ2 þ ðr22  r33 Þ2 þ ðr33  r11 Þ2 þ m2 ðr212 þ r223 þ r231 Þ 6

ð30Þ

and

J C3 ¼

4 2 1 p ðr3 þ r322 þ r333 Þ  p1 r233 ðr11 þ r22 Þ þ r222 ðr11 þ r33 Þ þ r211 ðr22 þ r33 Þ þ p1 r11 r22 r33 27 1 11 9 9

p3 2 2 2 þ 2p2 r12 r23 r13  r ð2r33  r11  r22 Þ þ r13 ð2r22  r11  r33 Þ þ r23 ð2r11  r22  r33 Þ 3 12

ð31Þ

with m1, m2, p1, p2, p3 as material parameters. Kelvin decomposition for the cubic symmetry case allows for the definition of two stress deviators, rd and rd . To consider them leads to an intrinsic re-writing of the invariants J C2 and J C3 as

J C2 ¼

m1 d m r : rd þ 2 rd : rd 2 2

ð32Þ

  J C3 ¼ p1 detðrd Þ þ p2 detðrd Þ þ p3 tr rd  rd  rd

Last, Barlat and coworkers Yld91 and Yld2004-18p yield functions (Barlat et al., 1991, 2005) involve several linear transformations Li or [Li] that can in fact be defined as a combination of Kelvin projectors for material cubic symmetry. For cubic symmetry, each transformation is indeed defined by (with bi and ei as material parameters)

2

2bi

bi

bi

0

0

0

6 b 6 i 6 bi 16 i ½L  ¼ 6 36 6 0 6 4 0

2bi bi

bi 2bi

0 0

0 0

0 0

0

0

3ei

0

0

0

0

0

3ei

0

0

0

0

3

7 7 7 7 7 0 7 7 7 0 5

ð33Þ

3ei

and is applied to the stress tensor in order to define the deviatoric transformed stresses si ¼ Li : r (to be put in the yield function in place of r0 , see Eq. (55)). One can notice that

Li : r ¼ bi rd þ ei rd

ð34Þ d

d

This defines each linear transformation, as announced, as a linear combination of Kelvin projectors P and P (22):

Li ¼ bi Pd þ ei Pd

ð35Þ

Existing macroscopic criteria based on invariants or linear transformations can thus be expressed thanks to Kelvin decomposition elements (Kelvin stresses or Kelvin projectors). 4. FCC superalloys family Nickel-base single crystal superalloys, such as AM1, AM3, CMSX2, CMSX4, PWA1480 or René N4, are nowadays widely used in aircraft engines, especially for turbine blades, due to their excellent high-temperature mechanical properties. The directed growth of a unique crystal (Durand-Charre, 1997), improving mechanical properties, leads to an anisotropic mechanical behavior that makes all material properties strongly orientation-dependent. Usual macroscopic yield criteria such as Hill (1948) classical criterion fail to describe this orientation effect (Bande and Nemes, 2005) and a yield criterion based on a crystallographic scale study using Schmid law seems therefore unavoidable to take this specificity into account (Schmid and Boas, 1935; Sheh and Stouffer, 1990; Méric et al., 1991; Nouailhas and Cailletaud, 1995; Marchal, 2006a). The single crystal CMSX2 is a two-phase material composed by a faced centered cubic (FCC) c matrix and c0 coherent L12 precipitates (see Fig. 2). Concerning the material deformation mechanisms, it is first of all important to distinguish octahedral and ‘‘cubic” slips activated in relative phenomena (cf. Fig. 1). Octahedral slip is classically observed in single crystals and is responsible for their plasticity. ‘‘Cubic” or more precisely pseudo-cubic slip, often consisting in a kind of zig-zag slip on octahedral planes in matrix channels (Bettge and Oesterle, 1999; Marchal, 2006b), is not possible in all single crystals but is observed in FCC alloys with L12 precipitates (like CMSX2). Such a pseudo-cubic slip is moreover made easier by some addition elements in these superalloys such as Mo, Nb, Ta, Ti and W (Marchal, 2006b). Predominance of a slip family on the other is conditioned by temperature and crystal orientation. Single crystal nickel-base superalloys usually behave in a same manner between 20 °C and 700 °C. In this temperature range, the deformation mechanisms are governed by (Marchal, 2006b):

334

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Fig. 1. Octahedral and pseudo-cubic slips, after Poubanne (1989).

Fig. 2. AM1 microstructure (Kaminski, 2007) and schematic two-phase alloy microstructure (Poubanne, 1989).

 octahedral slips in h0 0 1i and h1 2 3i family directions, at resolved shear stress  pseudo-cubic slips in h1 1 1i family directions, at resolved shear stress sccub ,  octahedral and pseudo-cubic slips in h0 1 1i family directions.

scoct ,

Hanriot (1993) observes a predominant octahedral slip with a minoritary but existing pseudo-cubic slip in h0 1 1i directions in an AM1 sample under cyclic loading at 650 °C. According to Oesterle et al. (2000) and Hanriot (1993), pseudo-cubic slip appears mostly in the h1 1 1i direction and the researchers agree with the thermal activation of this phenomenon around 700 °C (Schmid and Boas, 1935; Brien, 1995; Marchal, 2006b), i.e. will not be considered next as the presented applications are at room temperature and at 650 °C. Schmid criterion disagreement with experimental observations is sometimes observed when c0 cube L12 coherent precipitates are subjected to shear mechanism, that is not taken into account in Schmid law (for instance at room temperature). Critical resolved shear stress orientation dependence and tension–compression asymmetry are especially poorly modeled because of this phenomenon (Poubanne, 1989). Schmid criterion has nevertheless proved efficient in component design and remains a reference in single crystals modeling. Tension-shear elasticity domain as well as orientation dependence of mechanical properties in FCC nickel-base single crystal are most often expressed thanks to crystallographic considerations (Sheh and Stouffer, 1990; Méric et al., 1991). More advanced phenomenological criteria can however be used, as studied next. Bande and Nemes (2005) propose a ‘‘combined approach” introducing a crystallographic based factor in the phenomenological Hill yield criterion in order to model the effect of orientation on the yield stress and therefore to obtain an efficient hybrid criterion. But the proposed criterion, defining the elasticity domain, seems not so easy to extend to full plasticity coupling.

R. Desmorat, R. Marull / International Journal of Plasticity 27 (2011) 328–351

335

Fig. 3. Location of the strain gages on the thin tube specimen (left) and [1 0 0] and [1 1 0] CMSX2 tension–torsion yield loci at room temperature (right), after Nouailhas and Cailletaud (1995).

5. Tension–torsion yield surfaces of FCC single crystals Nouailhas and Cailletaud (1995) have tested h0 0 1i oriented CMSX2 samples in tension–torsion. They have studied them in two referentials or bases (Fig. 3):  a first one which corresponds to crystallographic axis ([1 0 0], [010], [0 0 1]), i.e. a strain gage was stucked in [1 0 0] secondary orientation. To this referential, that matches with the material natural anisotropy basis and then makes the writing of constitutive equations easy, corresponds the expression of the stress tensor

0

0 0

0

1

C r½100 ¼ B @0 0 s A 0 s r

ð36Þ

 1 0, [0 0 1]), i.e. strain gage set in [1 1 0] secondary  a second one rotated by 45° around [0 0 1] direction ([1 1 0], ½1 orientation. In order to focus on similitudes and/or differences between criteria, one will focus next on the modeling in the [1 0 0] secondary orientation. All discussed criteria will be identified on (Nouailhas and Cailletaud, 1995) [1 0 0] experimental data (except Schmid criterion whose parameters are known from their work). 5.1. Yield surface from Schmid criterion Nouailhas and Cailletaud (1995) note that Schmid criterion is qualitatively satisfied when plotting the CMSX2 superalloy yield surface in [1 0 0] at room temperature but does not fit experimental points (circles in Fig. 3) for states of stresses close to pure torsion (Fig. 3). They explain this phenomenon by the complex stress distribution that occurs in the tubes tested in tension–torsion and the need of finite element computations for a more accurate modeling. Critical resolved shear stresses were taken as scoct =375 MPa and sccub =490 MPa. 5.2. Yield surface from existing macroscopic criteria Having in mind that Schmid law limitations are due to complex crystallographic phenomena, let us see how well macroscopic criteria manage to model the single crystal tension–torsion behavior. 5.2.1. Hill criterion In the case of cubic symmetry, Hill criterion (26) only involves parameters F and L. Pure tension and pure shear loading cases allow for their identification as F ¼ 1=2r2y and L ¼ 1=2s2y , where ry and sy respectively are the yield stresses in pure tension and pure torsion. Hill yield locus in tension–torsion is the ellipse 2Fr2 + 2Ls2 = 1, with s = r23. Tension–torsion yield

336

R. Desmorat, R. Marull / International Journal of Plasticity 27 (2011) 328–351

Fig. 4. Tension–torsion yield loci obtained with Hill (F = 6.14  107 MPa2, L = 1.78  106 MPa2) and Barlat Yld91 (b = 1, e = 0.928, a = 8) criteria.

Fig. 5.

rdeq vs rdeq tension–torsion loading surface and Kelvin modes based plasticity criterion (CMSX2 at room temperature).

locus obtained thanks to this criterion is plotted in Figs. 4 and 7 (with for ry = 900 MPa and sy = 530 MPa for CMSX2 at room temperature).

5.2.2. Barlat criterion Barlat et al. (1991) Yld91 yield criterion is applied to the studied loading case. Yield locus is gained as

f ¼



1 2

1=a

ry

jS1  S2 ja þ jS2  S3 ja þ jS3  S1 ja

1=a

1¼0

ð37Þ

where the Si are the eigenvalues of transformed stress s defined by s ¼ L : r and where exponent a is chosen equal to 8 because of the FCC crystallographic symmetry of the alloy (Hosford, 1996; Cazacu and Barlat, 2003). As mentioned in Section 3, the cubic material symmetry makes linear transformation L depends on two parameters b and e (Eq. (33)). Here again, pure tension and pure shear cases allow for the parameters identification. Firstly, pure tension r = diag[r, 0, 0] gives

R. Desmorat, R. Marull / International Journal of Plasticity 27 (2011) 328–351

02

br

3

B s¼@ 0

1

0

0

 13 br

0

0

 13 br

0

337

2 C A and S1 ¼ br; 3

1 S2 ¼ S3 ¼  b r 3

ð38Þ

and f ¼ 0 () 2jbrja ¼ 2ray . So that from r = ry, one simply has b = 1. Parameter e is identified in the pure shear case:

0

s 0

0

1

0

0

es 0

0

0 0

C B r¼B @ s 0 0 A so that s ¼ @ es 0 0 0

1

C 0A 0

ð39Þ

and then S1 = 0, S2 = es, S3 = es. The criterion f ¼ 0 () 2jesja þ j2esja ¼ 2ray (with here a = 8) gives

e¼



ry 1 sy 1 þ 27

1=8 ð40Þ

with s = sy = 530 MPa the known yield shear stress for CMSX2 at room temperature. The corresponding tension–torsion yield locus is plotted in Fig. 4. It appears that the loci resulting from Hill and Barlat criteria are very close from each other. Both Hill and Barlat criteria give quite satisfying results for this tensile-shear loading case. The consideration of a second linear transformation does not improve much the result in this case. 5.3. Yield locus from Kelvin modes based criterion Kelvin decomposition applied to cubic symmetry materials has exhibited three objective Kelvin stresses that can be used as a basis of a novel criterion whatever the loading and the crystal orientation. The proposed criterion (Eq. (25)) can be rewritten as

f ¼

2n2 =2 c2 

where ðÞeq d eq

3

rdeq

n2 =2

n2

þ

2n3 =2 c3  3

n3 =2

rdeq

n3

160

ð41Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 32 ðÞ0 : ðÞ0 stands for von Mises norm and where



d

r ¼ r

 eq

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 d ¼ r : rd ; 2

d eq



d

r ¼ r

 eq

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 d ¼ r : rd 2

ð42Þ

It is possible to replot experimental points from Nouailhas and Cailletaud (1995) in a new diagram with set of axes ðrdeq ; rdeq Þ. The stress tensor in present tension–torsion case is expressed as Eq. (36) in natural anisotropy basis. Deviatoric stress tensors rd and rd are easily gained as < 1 0 0 > axes match the material natural anisotropy basis. According to Section 2.2.2, the Kelvin stresses are the hydrostatic stress

0

1

r 0 0 1 C rH ¼ B @0 r 0A 3 0 0 r

ð43Þ

and the deviatoric stresses

0

 13 r

rd ¼ B @ 0 0

0

0

 13 r 0

0

1

0 0 0

1

B C C 0 A and rd ¼ @ 0 0 s A r 0 s 0

ð44Þ

2 3

The corresponding equivalent stresses are therefore

pffiffiffi

pffiffiffi

rdeq ¼ jrj ¼ jr33 j and rdeq ¼ 3jsj ¼ 3jr23 j

ð45Þ

The locus of experimental points in the ðr r diagram can be approximated by a degree n polynomial. Zero-slope condition for rdeq ¼ 0, gained from experiments, makes the polynomial-provided relation linking rdeq and rdeq quite simple. Whatever the chosen exponent n for the power law, we have indeed d eq ;

"

rdeq ¼ r y 1 

rdeq ry

d eq Þ

!n #

where two yield stresses are introduced as material parameters, d  ry is p the ffiffiffi value of req at the elasticity pffiffiffi limit in pure tension (s = 0),   ry ¼ 3sy is the value of rdeq ¼ 3s at the elasticity limit in pure torsion (r = 0).

ð46Þ

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R. Desmorat, R. Marull / International Journal of Plasticity 27 (2011) 328–351 Table 1 Kelvin modes based plasticity criterion parameters of CMSX2 superalloy. pffiffiffi

Temperature (°C)

ry (MPa)

r y ¼ 3sy (MPa)

n

20 650

900 1030

920 955

5 2

Fig. 6. Yield locus from Kelvin modes based plasticity criterion identified on [1 0 0] experimental data (CMSX2 at room temperature, Schmid criterion reported in dot line).

Fig. 5 shows the interpolation polynomials for n = 3–5. Doing so, we have in fact established a novel non-quadratic yield criterion, which is convex. The corresponding elasticity domain is given by

f ¼

rdeq ry

!n þ

rdeq 1<0 r y

ð47Þ

This formulation is a particular case of the yield criterion defined for general anisotropy (Eq. (14)) applied to cubic symmetry (Eq. (41)) with  n2 = n and 2n2 =2 c2 =3n2 =2 ¼ 1=rny ,  y.  n3 = 1 and 21=2 c3 =31=2 ¼ 1=r Thanks to this expression and to relationships between Kelvin equivalent stresses and the stress components r23 and r33, the CMSX2 tension–torsion yield locus f = 0 is replotted in the shear stress-tensile stress plane at room temperature for the set of parameters given in Table 1 (Fig. 6). We have indeed

jr33 j ¼ jrj ¼ rdeq

1 and jr23 j ¼ jsj ¼ pffiffiffi rdeq 3

ð48Þ

Recall that the equivalent stresses rdeq and rdeq are framework independent as they are defined with respect to the natural anisotropy basis (and not to the working basis). The criterion (47) identified on [1 0 0] experimental data is plotted and compared to Schmid criterion in Fig. 6. The yield locus shape is fully satisfying. It presents 2 angular points (due to the non-quadratic feature and exponent n) instead of 6 and no straight lines like Schmid criterion does, what seems more realistic and more efficient from a computational point of view. For completeness, note that Kelvin modes based criterion can also be directly identified on Schmid law predictions (the corresponding parameters are often known whereas tension–torsion experimental data are rarely available). Corresponding methodology and results are given in Appendix A.

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339

Fig. 7. Yield surfaces from Hill and one-transformation Barlat Yld91 criteria and from Kelvin modes based criterion identified on experimental data (CMSX2 at room temperature).

Let us also compare the Kelvin modes based criterion with Hill and Barlat Yld91 (one-transformation) macroscopic criteria. Fig. 7 shows the yield loci in the shear-tension plane obtained from the three criteria identified on the same [1 0 0] experimental data. Although the results are quite similar, the Kelvin decomposition based criterion seems to better fit the experimental points. This is mainly because of the locus angular points around pure tension and its quite large horizontal flat part around pure shear (this is due to the non-quadratic feature n = 5 of novel criterion function (47)). Remark. The adjective ‘‘non-quadratic” has been used for yield criterion (47) – but also for Hill (1950) criterion, Eq. (1) – in order to point out that the stresses do not all act at the power 2 as they do in quadratic von Mises and classical Hill (1948) criteria. Another meaning for ‘‘non-quadratic” concerns the mathematical norm used for defining invariants and the introduction of parameter a larger than 2 (as in Hershey–Hosford and Barlat criteria, Eqs. (63) and (49), as in Eqs. (15) and (16)). It is possible to introduce such a parameter within Kelvin modes based criteria, setting for instance for cubic materials and in natural anisotropy basis

f ¼

n=a pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 1 a a a r  r j þ j r  r j þ j r  r j þ r212 þ r223 þ r231  1 6 0 j 33 22 11 33 22 11 rny 2 r y

ð49Þ

This defines a cubic yield criterion. The case a = 2 recovers the novel criterion function (47). More details on the corresponding mathematical framework will be given in Section 8.

Fig. 8. h misorientation definition.

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6. Tensile yield stress orientation dependence of FCC single crystals In previous section, all calculations have been led in a same crystallographic orientation, h1 0 0i. Let us now focus on how these criteria behave in uniaxial tension but with crystallographic misorientation h (angle with respect to [0 0 1]-direction of Fig. 8) such as

0

0

r ¼ rN  N ¼ rB @0

0 2

sin h

0 sin h cos h

0

1

C sin h cos h A ¼ rR

ð50Þ

cos2 h

where r is the tensile stress, R = N  N, with Req = 1 and with

0

1 0 B C N ¼ @ sin h A cos h

ð51Þ

the loading direction. 6.1. Yield stress orientation dependence from Schmid criterion The effect of a misoriented stress from [0 0 1] to [0 1 1] on the yield conditions is studied first for Schmid criterion. One considers CMSX2 superalloy at 650 °C, for which yield stress orientation dependence experimental data are available and Schmid analysis parameters are known (scoct ¼ 420 MPa and sccub ¼ 390 MPa, Hoinard et al., 1995). The authors assume that cubic slip has no impact on yield stress rYield for this orientation range and at this temperature so that it reads

rYield ðhÞ ¼

scoct maxðR : ms Þ

ð52Þ

where ms ¼ 12 ðms  ns þ ns  ms Þ is the orientation tensor of octahedral slip system s, with ns the normal of the slip plane and ms the slip direction of the system, with the max taken among all octahedral slip systems. Yield stress obtained from Schmid criterion for CMSX2 reasonably fits experimental points (Hoinard et al., 1995) in the orientation range 0–30° (Fig. 9) but the strong increase in rYield(h) in the range 30–45° is problematic, first because of the yield stress overestimation, second because of the fact that the curve has necessarily to be symmetric from the vertical axis h = 45° (to the range 45–90°). This symmetry leads indeed to an angular point at h = 45° in Schmid analysis (if the whole 0– 90° range is considered). Schmid law limitations for this loading case can be explained by the dependence of critical resolved shear stress on orientation – phenomenon observed by Poubanne (1989) and Marchal (2006b) – that is not taken into account in the modeling.

Fig. 9. CMSX2 yield stress orientation dependence from Schmid law at 650 °C and symmetry axis.

R. Desmorat, R. Marull / International Journal of Plasticity 27 (2011) 328–351

341

Fig. 10. Yield stress orientation dependence rYield(h) from Hill criterion and one- and two-transformation Barlat Yld91 and Yld2004-18p criteria (CMSX2 at 650 °C, a = 8). Both Hill and Barlat criteria identified from pure tension and pure shear yield stresses.

Fig. 11. Yield stress orientation dependence rYield(h) from Hill criterion, one- and two-transformation Barlat Yld91 and Yld2004-18p criteria and Schmid criterion (CMSX2 at 650 °C, a = 8). Both Hill and Barlat criteria identified from [0 0 1] and [0 1 1] tensile yield stresses.

6.2. Yield stress from existing macroscopic criteria 6.2.1. Hill criterion In this loading case, r = rYield(h) is the tensile yield stress, classically given by

1

rYield ðhÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

ð53Þ

2F þ ð2L  6FÞ sin h cos2 h

for Hill criterion (26). Two strategies are conceivable to identify Hill parameters. The first one is the same as for previous study of tension–torsion yield surfaces, i.e. uses the [1 0 0] yield stress values in pure tension ry and pure shear sy at 650 °C. These values are obtained from Schmid analysis through the knowledge of scoct and sccub with then

F ¼ 4:71  107 MPa2

L ¼ 1:65  106 MPa2

The corresponding curve is plotted in Fig. 10. It appears obviously that the result is unsatisfying for Hill criterion. The curves with Barlat criteria are commented in Section 6.2.2.

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Table 2 One- and two-transformation Barlat parameters (CMSX2 at 650 °C). Criterion and data used for identification

ry (MPa)

b1

e1

b2

e2

a

1L, 2L, 1L, 2L,

1030 1030 1030 1030

1 0.8 1 0

1.018 0 1.195 1.6

– 1.174 – 1.554

– 1.777 – 0.03

8 8 8 8

pure tension and pure torsion (Fig. 10) pure tension and pure torsion (Fig. 10) 0° and 45° yield stresses (Fig. 11) 0° and 45° yield stresses (Figs. 11 and 14)

A second strategy consists in identifying parameter F thanks to the yield stress measured in crystallographic direction [1 0 0], i.e. at h = 0°, and parameter L thanks to the yield stress in [1 1 0] direction, i.e. at h = 45°. This case corresponds indeed to the relationship

L¼

2

F

r2½110

ð54Þ

This second strategy allows for the obtention of the curves shown in Fig. 11 with (still at 650 °C)

F ¼ 4:71  107 MPa2

L ¼ 2:03  106 MPa2

The result is much better since the curve passes through experimental points. 6.2.2. Barlat criterion As for Hill criterion, two strategies are possible that lead to similar conclusions (see Figs. 10 and 11, parameters from Table 2). The initial approach consists in applying only one linear transformation (1L Yld91 yield function). To consider two linear transformations gives more flexibility (2L Yld2004-18p yield function) since one-transformation Barlat criterion is not sufficient to recover the non-monotonic shape of CMSX2 yield stress orientation dependence expected from Schmid analysis (also reported in Fig. 11). The two-transformation Barlat et al. (2005) criterion Yld2004-18p reads

f ¼

" X

1 41=a ry

#1=a jSi  Sj ja ð1Þ

ð2Þ

160

ð55Þ

i;j

ðkÞ

with Si the eigenvalues of transformed stress sk ¼ Lk : r. Here again, the curves obtained by identifying the parameters thanks to Schmid analysis – therefore to shear data – are less satisfying. The direct identification on the oriented tensile test data proposes however a quite good modeling of the orientation effect when using two linear transformations. 6.3. Yield stress from Kelvin modes based criterion Let us build again a graphical representation of yield loci in the single plane ðrdeq ; rdeq Þ for different loading cases at 650 °C, therefore with experimental points of different nature:  points from yield surface in tension–torsion oriented [0 0 1] locus (tension–torsion points obtained from Schmid law),  additional points from yield stress dependence on crystal orientation; the experimental data come from (Hoinard et al., 1995) and both representations by Schmid criterion and by the Kelvin modes based plasticity criterion (Eq. (46)) are given. In the case of yield stress orientation dependence, the Kelvin stresses are in natural anisotropy basis (see Section 2.2.2)

0

1

r 0 0 1 C rH ¼ B @0 r 0A 3 0 0 r 0

 13

rd ¼ rB @ 0 0

0 0

rd ¼ rB @0 0

0

ð56Þ 0

1

C 2 A 0 sin h  13 0 cos2 h  13 1 0 0 C 0 sin h cos h A sin h cos h 0

and one has then, with

req ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrdeq Þ2 þ ðrdeq Þ2 ¼ rYield ðhÞ,

ð57Þ

ð58Þ

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Fig. 12.

343

rdeq vs rdeq for different loading types and novel criterion (46) (CMSX2 at 650 °C).

Fig. 13. Yield stress orientation dependence rYield(h) from Schmid and novel Kelvin modes based criterion (CMSX2 at 650 °C).

d eq

r

rffiffiffi 3 ¼ rYield ðhÞ sinð2hÞ and 4

d eq

r

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 ¼ rYield ðhÞ 1  sin ð2hÞ 4

ð59Þ

Experimental points and Schmid criterion for misorientation are plotted thanks to these relationships in Fig. 12 (rYield(h) experimentally known) while the tension–torsion curve obtained with Schmid criterion is plotted in the same plane thanks to Eq. (48) established in Section 5.3. Experimental misorientation and tension–torsion curves are found close from each other so that it is possible to approximate them all thanks to a unique power function (novel criterion, Eq. (46)). The corresponding material parameters are given in Table 1.  y ; nÞ represents misoriented tension as well as The fact that the power law (46) with a unique set of parameters ðry ; r tension–torsion is an important result: this validates the use of Kelvin stresses (and their invariants) determined from the

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Fig. 14. Yield stress orientation dependence rYield(h) from Hill, Barlat (two transformations), Schmid and novel Kelvin modes based criterion (CMSX2 at 650 °C). Hill and Barlat criteria identified from [0 0 1] and [0 1 1] yield stresses.

analysis of the elasticity tensor in order to define a general (macroscopic) yield criterion for cubic FCC single crystals. The application to cubic symmetry (Eq. (41)) of the general proposed criterion (Eq. (14)) is fully validated. The yield stress orientation dependence curve rYield(h) is then plotted in Figs. 13 and 14. These results confirm that the novel Kelvin modes based criterion leads to a more physical curve shape in the range 30– 45° (to 60° then) than the one obtained from Schmid law. A vanishing slope is obtained at the position h = 45° of the vertical symmetry axis. 6.4. Discussion on the macroscopic criteria misorientation responses Fig. 10 has shown that the introduction of a second linear transformation in Barlat criterion is unavoidable to get better results. Barlat et al. (1991) Yld91 criterion identified on tension–torsion Schmid analysis remains unsufficient. To identify Hill and Barlat parameters on oriented tensile test data strongly improves the rYield(h) misorientation result, especially for Barlat criterion whose response becomes very close from the ones obtained with Schmid analysis and with the Kelvin modes based criterion (see Fig. 14), except for the zero slope at h = 0. The now well known flexibility related to the introduction of a second linear transformation is once more illustrated. Nevertheless, the lack of data between 5° and 30° makes it difficult to conclude on which would be the best criterion to model the orientation effect on the yield stress. The parameters identification is however made very simple for the novel Kelvin modes based criterion by the introduction of the new rdeq vs rdeq graph. Despite the lack of experimental data, Kelvin modes based criterion provides a satisfying modeling of the orientation effect. 7. Comparison of the different criteria on the different loading cases Schmid crystallographic criterion provides satisfying results that can however be improved by the use of Barlat (twotransformations) or Kelvin modes based criteria. In the tension-shear stress plane, the shape of Schmid law resulting yield loci is its main drawback because straight slopes and angular points tend to go counter to physical considerations and because they lead to numerical difficulties. On the other hand, the studied macroscopic criteria can be divided into two categories.  Hill and Yld91 one-transformation Barlat criteria, which are very easy to identify and to carry out, do not properly model the misorientation curve.  The Yld2004-18p (2L) criterion and the novel Kelvin modes based criterion are very efficient whatever the loading case but their identification and implementation may be not as straightforward as for previous criteria.  y ; n can however be easily gained either from a tension–torsion test or Kelvin modes based criterion parameters ry ; r from Schmid analysis. It can be rewritten in terms of critical resolved shear stresses (see Appendix A). One has indeed

pffiffiffi

pffiffiffi

ry ¼ 6scoct and r y ¼ 6sccub and then,

ð60Þ

R. Desmorat, R. Marull / International Journal of Plasticity 27 (2011) 328–351

345

Fig. 15. Tension–torsion yield loci for Schmid and Kelvin modes based criteria and for Hill and Barlat Yld2004-18p criteria identified from pure tension and pure shear yield stresses (CMSX2 at 650 °C). Corresponding misorientation curves are those of Figs. 10 and 13.

Fig. 16. Tension–torsion yield loci for Schmid and Kelvin modes based criteria and for Hill and Barlat Yld2004-18p criteria identified from [0 0 1] and [0 1 1] yield stresses (CMSX2 at 650 °C). Corresponding misorientation curves are those of Figs. 14 and 13.

f ¼

rd

pffiffiffieq 6scoct

!n

rdeq

þ pffiffiffi 1 6sccub

ð61Þ

It is then quite easy to identify it on the basis of Schmid analysis. A main advantage of this novel criterion is that it can simply model both shear-tension and misorientation results with the same set of material parameters. The Yld2004-18p criterion has more parameters (ry, b1, b2, e1, e2, a) so that the knowledge of the shear stress only provides a relationship between material parameters. Moreover, the results with Barlat et al. (2005) criterion strongly depend on the data used for the identification (the shear yield stress or the [0 1 1] tensile yield stress in previous examples). This induces the existence of two different sets of parameters at a given temperature (see Table 2). One notices indeed in Figs. 15 and 10 on the one hand and in Figs. 16 and 14 on the other hand that pure shear yield stress and [0 1 1] tensile yield stress is not modeled by a single set of material parameters with such a criterion. It is nevertheless delicate to conclude about Barlat criterion and Kelvin modes based criterion achievements since misoriented tensile response is most often the determinant loading case for design and since the corresponding experimental points are not numerous.

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Fig. 17. rdeq vs rdeq curves for both tension–torsion and misoriented tension loading cases (CMSX2 at 650 °C). (a) Hill criterion, (b) Barlat et al. (2005) Yld2004-18p criterion and (c) Kelvin modes based criterion.

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347

Last, plotting the macroscopic criteria in the new rdeq vs rdeq diagram (Figs. 17 and 12) shows that Hill and Barlat and coworkers criteria exhibit a horizontal tangent at rdeq ¼ 0 and a vertical tangent at rdeq ¼ 0 so that the corresponding curves do not pass easily through experimental points (this partially explains why a single set of material parameters for these criteria is not valid for every loading case). At the opposite, the angular point and the inclined (non-vertical) tangent at rdeq ¼ 0 for the novel Kelvin decomposition based criterion allow for a better fitting of experimental data. Note that this tangent is due to n > 1 and that it is the same as for Schmid criterion. 8. Intrinsic formulation of plasticity with non-quadratic norm kka Last, let us address the mathematical formulation of plasticity with non-quadratic norm kka = (tr()a)1/a generalizing quadratic ‘‘norm 2” kk2 = (tr()2)1/2. Recall that von Mises stress (and J2 invariant) are defined by use of norm kk2 as

req ¼

pffiffiffiffiffiffiffi 3J 2 ¼

rffiffiffi 3 0 kr k2 2

ð62Þ

8.1. Intrinsic writing for Hershey–Hosford based criteria In order to give an intrinsic formulation of principal stresses written Hershey–Hosford criterion (Hershey, 1954; Hosford, 1972)

fHH ¼

1 2

1=a

ry



1=a jr1  r2 ja þ jr2  r3 ja þ jr3  r1 ja 1¼0

ð63Þ

using norm kka it is necessary to first point that the von Mises norm of the deviatoric transformed tensor m ¼ m0 ¼ M : r ¼ M : r0 such as

0 B m¼B @

p1ffiffi ð 3

r33  r22 Þ r12 r13

r12 p1ffiffi ð 3

r11  r33 Þ r23

r13 r23 p1ffiffi ð 3

r22  r11 Þ

1 C C A

ð64Þ

is equal to von Mises norm of stress tensor r,

req ¼ meq ¼ ðM : rÞeq ¼

rffiffiffi 3 kM : rk2 2

ð65Þ

The matricial representation of non-symmetric tensor M is:

2

0  p1ffiffi3 p1ffiffi3 6 1 6 pffiffi3 0  p1ffiffi3 6 6 p1ffiffi p1ffiffi 6 3 0 3 ½M ¼ 6 6 0 0 0 6 6 4 0 0 0 0

0

0

0

0

0

3

7 07 7 7 0 0 07 7 1 0 07 7 7 0 1 05 0

0

0

ð66Þ

0 1

An intrinsic (strictly convex if a > 1) generalization of Hershey–Hosford criterion in any framework can then be proposed as

fHH ¼

pffiffiffi 3 21=a ry

kM : rka  1

ð67Þ

which recovers for a diagonal stress tensor diag(r1, r2, r3)

     

 r3  r2 a r1  r3 a r2  r1 a 1=a kM : rka ¼  pffiffiffi  þ  pffiffiffi  þ  pffiffiffi  3 3 3

ð68Þ

For materials exhibiting cubic symmetry, the splitting r ¼ rd þ rd þ rH into a deviatoric diagonal stress, a deviatoric nondiagonal stress and a hydrostatic stress (in natural anisotropy basis, see Section 2.2.2) allows to define a convex effective  such as a cubic extension to full 3D stress states of Hershey–Hosford criterion is stress r

"

f ¼

r 3a=2 ¼ 1 r kM : rd kaa þ Ckrd kaa ry 2

#1=a

with C a positive parameter and a > 1. With then in natural anisotropy (cubic) basis

ð69Þ

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r ¼

 a 1=a  1   jr33  r22 ja þ jr11  r33 ja þ jr22  r11 ja þ Ctr rd  2

ð70Þ

Another intrinsic generalization adapted to single crystals, generalizing both Hershey–Hosford criterion and the novel Kelvin modes based criterion Eq. (47) to cubic symmetry, consider usual ‘‘norm 2” krd k2 for the shear stresses so that

" pffiffiffi #n  rdeq 3  d  f ¼ 1=a M:r a þ 1 r y 2 ry

ð71Þ

It recovers criterion (49) when expressed in natural anisotropy basis. M is the non-symmetric transformation tensor (66). Yld91 Barlat et al. (1991) anisotropic criterion based on isotropic fHH criterion function is then either (one-transformation L into Eq. (67), c being a material parameter)

f ¼ ckL : M : rka  1 or (two-transformations L

ð1Þ

ð72Þ ð2Þ

and L

into Eq. (69))

a i1=a h a    f ¼ Lð1Þ : M : rd a þ Lð2Þ : rd  1 a

ð73Þ

Again, convexity is obtained with a > 1 (see Bron and Besson, 2004; Cazacu et al., 2006). 8.2. Normality rule for non-quadratic kka-plasticity The advantage of intrinsic expressions such as Eq. (67), (69) or (71) is that they allow for a general writing of the normality rule, using the derivative

@kTka jTja2  T ¼ @T kTka1 a

ð74Þ

of the norm kTka with respect to the second order tensor T (with ðjTja2  TÞij ¼ jTja2 ik T kj , see Appendices B and C). For instance in case of criterion (67),

h i pffiffiffi M : jM : rja2  ðM : rÞ 3 @f _ p ¼ k_ HH ¼ k_ 1=a @r kM : rka1 2 ry a

ð75Þ

Note that tensors jTja2 and T commute, as do tensors jM : rja2 and ðM : rÞ. 9. Conclusion A novel non-quadratic formulation of plasticity criteria, based on Kelvin modes, has been proposed in order to model the elasticity domain of anisotropic materials from a macroscopic point of view. The criteria identification seems difficult to carry out in the case of general anisotropy. But it can easily be performed for cubic materials such as single crystal nickel-base superalloys. The criterion has been successfully applied to CMSX2 superalloy where Hill criterion, Barlat one-transformation criterion and even Schmid law encounters difficulties to fully describe both the tensile-shear behavior and the tensile yield stress orientation dependence. Two-transformation Barlat criterion allows for a good modeling of the tensile yield stress orientation dependence but at the cost of additional parameters and still with difficulties to represent the pure shear yield stress. With the proposed Kelvin modes based criterion (47), the results from both studied loading cases, tension–torsion as well as misoriented tension, are in agreement with experimental yielding data. A representation of different loading types in a unique and newly defined ðrdeq ; rdeq Þ stress plane has proved helpful for the modeling and has pointed out the limitations of Hill and Barlat criteria. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Applications to single crystal are usually made with the usual quadratic norm of a tensor ðÞ : ðÞ ¼ k  k2 . The definition of ‘‘less rounded” or more angular criteria (for instance in the biaxal (r11, r22) plane) may straightforwardly be defined by replacing in the proposed criteria the usual norm kk2 by the norm kka. The adequate mathematical framework has been presented. Intrinsic generalizations of isotropic Hershey–Hosford criterion to cubic material symmetry have been proposed. A Kelvin modes based criterion is naturally a basis to build a complete plasticity model using normality rule

 ¼ e þ p e ¼ E1 : r @f  ¼ k_ @r _p

ð76Þ

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349

Fig. 18. [1 0 0] and [1 1 0] yield surfaces from Kelvin modes identified on Schmid law (CMSX2 at room temperature).

_ ¼ 0; k_ P 0; f 6 0 in plasticity or from a viscosity law in with k_ a multiplier gained from the Kuhn–Tucker conditions kf viscoplasticity (Lemaitre et al., 2009). Such a plasticity model will be developed in a further study. Last, the tension–compression yielding asymmetry is another significative limitation of Schmid criterion. Two types of such a behavior can be observed – dependent or independent on the hydrostatic pressure – and classically modeled in the isotropic case by the introduction within the criterion function f of either the first invariant of the stress tensor or the third invariant of the stress deviator. Such recent macroscopic anisotropic modeling (including cubic symmetry) independent from hydrostatic pressure can be found in Hu (2005) and Cazacu et al. (2006). For hydrostatic pressure sensitive materials (Spitzig et al., 1976) the yielding asymmetry can be gained from the Kelvin modes based yield criterion by keeping the dependence on first stress invariant, setting for example, still for cubic symmetry,

f ¼

rdeq þ ktrr ry ð1 þ kÞ

!n þ

 r rdeq þ ktr  1 r y ð1 þ kÞ

ð77Þ

 as additional Drucker–Prager parameters. with k and k Acknowledgement The authors thank Snecma (Safran group) that supported this study. Appendix A. Cross identification of the Kelvin modes based criterion from Schmid parameters Fig. 18 shows the Kelvin modes based criterion identified directly on Schmid law predictions (with n = 4 instead of 5). The material parameters for CMSX2 superalloy at room temperature are then:

pffiffiffi

pffiffiffi

ry ¼ 6scoct ¼ 920 MPa; r y ¼ 6sccub ¼ 1200 MPa; n ¼ 4 Near pure tension, the [1 0 0] experimental points are quite good fitted. The curves obtained from Kelvin modes based criterion tangent Schmid yield locus in pure torsion. Appendix B. Absolute value and power a of a symmetric tensor In previous equations, but also in Eq. (15) and (16), the absolute value jTj and the power jTja (in terms of principal components) of a symmetric second order tensor T have been used (with a = a  2). If eigenvalues and eigenvectors of T are the couples (kI, nI), with knIk2 = 1, they are defined as

jTj ¼

3 X I¼1

jkI jnI  nI

jTja ¼

3 X I¼1

jkI ja nI  nI

ð78Þ

350

R. Desmorat, R. Marull / International Journal of Plasticity 27 (2011) 328–351

They may also be calculated as follows, 1. make T diagonal as Tdiag = P1 TP, with P the change of basis matrix, 2. take either the absolute and/or the power a of the principal values, defining jTdiagj or jTdiagja, 3. turn back the tensor in the initial basis, jTj = PjTdiagjP1 or jTja = PjTdiagja P1. Appendix C. Calculation of differentials and derivatives of trjTja and of kTka = (trjTja)1/a Still with (kI, nI) (and knIk2 = 1) the eigenvalues and eigenvectors of symmetric tensor T, one has

trjTja ¼

3 X

jkI ja trðnI  nI Þ ¼

3 X

jkI ja

ð79Þ

jkI ja2 kI dkI

ð80Þ

I¼1

I¼1

and so

dðtrjTja Þ ¼

3 X

djkI ja ¼ a

I¼1

3 X I¼1

kI and nI being solution of the eigenvalue problem TnI = kInI,

dT  nI þ T  dnI ¼ dkI nI þ kI dnI

ð81Þ

so that nIdT nI + nITd nI = dkInInI + kInIdnI with nIdnI = 0 due to knIk2 = 1. Also with nIT dnI = kInId nI = 0 this gives

ðnI  nI Þ : dT ¼ dkI nI  nI

)

dkI ¼ ðnI  nI Þ : dT

ð82Þ

which put into Eq. (80) ends up to

dðtrjTja Þ ¼ a

3 X

h i jkI ja2 kI ðnI  nI Þ : dT ¼ ajTja2 T : dT

ð83Þ

I¼1

or

@trjTja ¼ ajTja2 T @T

ð84Þ

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