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Anisotropic ductile fracture Part II: theory A.A. Benzerga a

a,*

, J. Besson b, A. Pineau

b

Department of Aerospace Engineering, Texas A&M University, 3141 TAMU, College Station, TX 77843-3141, USA b Ecole des Mines de Paris, Centre des Materiaux, UMR CNRS 7633, BP 87, F91003 Evry Cedex, France Received 18 December 2003; received in revised form 8 June 2004; accepted 11 June 2004 Available online 15 July 2004

Abstract A theory of anisotropic ductile fracture is outlined and applied to predict failure in a low alloy steel. The theory accounts for initial anisotropy and microstructure evolution (plastic anisotropy, porosity, void shape, orientation and spacing) and is supplemented by a recent micromechanical model of void-coalescence. A rate-dependent version of the theory is employed to solve boundary value problems. The application to the studied steel relies on material parameters inferred from quantitative metallography measurements. The quantitative prediction of damage accumulation and crack initiation in notched bars is achieved without any adjustable factor and is discussed under various stress states and loading orientations. Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Alloys; Anisotropic plasticity; Fracture; Void coalescence; Micromechanics

1. Introduction In this article we address the issue of the quantitative prediction of ductile fracture. To that end, an anisotropic theory is presented, which is based on the mechanics of porous plastic materials. The theory is applied to model the anisotropic failure of a low alloy medium strength steel characterized in Part I of this work [1]. The aim of the prediction is precisely to be quantitative, that is matching experiments without ﬁtting parameters. So far, modeling based on micromechanics has essentially involved isotropic models [2,3] and has been successful to some extent [4–9]. While many of the qualitative aspects of ductile fracture have been explained by isotropic approaches, quantitative predictions are still a challenge. The isotropic model, e.g. [3], accounts for pressure-sensitivity through a mechanism of dilational plasticity, which is (i) homogeneous in the *

Corresponding author. Tel.: +1-979-845-1602; fax: +1-979-8456051. E-mail address: [email protected] (A.A. Benzerga).

elementary volume and (ii) neglects microstructure evolution, namely plastic anisotropy and void shape. It is now established that a competing localized dilational mechanism of plasticity, associated with void coalescence, needs to be accounted for; see [10–13]. It is also established that the concept of pressure-sensitivity needs reﬁnement through the incorporation of void shape effects [14–17]. Earlier attempts to incorporate void shape eﬀects in micromechanical models, e.g. [18], were of an empirical character and, as noted by their authors, were restrictive. A theory that incorporates the void shape eﬀect is necessarily anisotropic. But anisotropy in ductile materials enters in many ways. In initially isotropic materials, it enters through the deformation-induced evolution of microstructural features, which can be individual (grain shape, pore shape and size) or collective (texture and spatial distribution of pores). But, of course, any of these features can be initially anisotropic as a result of the metal-working or pre-deformation history so that directionality arises in most mechanical properties of the material. Modeling fracture anisotropy is largely unexplored. Yet both toughness and ductility of many

1359-6454/$30.00 Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.06.019

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structural alloys depend on the loading orientation (for steel see [1] and references therein; for aluminum and titanium alloys see [19,20]). Anisotropy of fracture properties in ductile materials is usually attributed to the shape of second-phase particles or inclusions such as manganese sulﬁde in steel. This eﬀect was analyzed in [16] using a model for porous materials incorporating void shape eﬀects [15]. Anisotropy in materials with only equiaxed particles [21] suggests, however, that the particle spatial distribution and/or material’s texture may also be of signiﬁcance. These two eﬀects were theoretically investigated in [17] and [22,23], respectively. It was conﬁrmed that any anisotropy in void distribution has an inﬂuence on void coalescence, not on void growth, with a possible net eﬀect on fracture toughness [17]. Also, in [23] it was found that the eﬀect of plastic anisotropy can explain the diﬀerence in ductility between steel and aluminum alloys when other known eﬀects are kept the same. Therefore, a complete theory of ductile fracture should take account of all the material aspects listed above: inclusion and void shape, void distribution anisotropy and plastic anisotropy as a macroscopic eﬀect of material texture or grain elongation. So far each aspect was analyzed separately [16,17,23]. Here the three features are assembled in a general approach based on the two-mechanism plasticity model [11,12] with an explicit illustration of how the approach can be implemented and assessed against experiments.

2. Formulation of the theory The weak form of the principal of virtual work is written as Z Z Z S : dE dV ¼ T du dS þ f du dV ; ð1Þ V

S

V

with 1 T F FI ; ð2Þ 2 is the symmetric second Piola–Kirchoﬀ stress where S tensor, E is the Green–Lagrange strain, F is the defor mation gradient, R is Cauchy stress, J ¼ detðF Þ, T and f are, respectively, the surface tractions and body forces if any, u is the displacement vector and V and S are the volume and surface of the body in the reference conﬁguration. An updated Lagrangian formulation is used [24] which employs objective space frames with the reference conﬁguration being either chosen at the beginning of the increment or at the end of the increment. Unless otherwise stated, the latter option has been adopted in most solutions here so that the stress measure S reduces to the Cauchy stress. S ¼ J F1 R FT ;

E¼

The constitutive framework is that of a progressively cavitating anisotropic viscoplastic solid initiated in previous studies [16,17]. The formulation was detailed in [25] but for completeness it is outlined here. In the objective (polar or co-rotational) frame, the strain rate e , and a tensor is written as the sum of an elastic part, D p viscoplastic part, D . Assuming small elastic strains and isotropic elasticity, a hypo-elastic law is expressed using the rotated stress P De ¼ C1 : P_ ;

ð3Þ

P ¼ J XT R X;

ð4Þ

where C is the rotated tensor of elastic moduli. If the co rotational frame is used then the rotation tensor X is identiﬁed with the spin Q (skew-symmetric part of the velocity gradient) so that Jaumann rate of R is used. If the polar frame is used then X is identiﬁed with the rotation R resulting from the polar decomposition of the deformation gradient F and the Green–Naghdi rate of R ðpÞ is used. The viscoplastic part of the strain rate, D , is obtained by normality from the gauge function: ðpÞ; / ¼ rH r

ð5Þ

is the matrix yield stress, p the eﬀective plastic where r strain and rH is an eﬀective matrix stress which is implicitly deﬁned through an equation of the type FðR ; f ; S; ez ; H; rH Þ ¼ 0 with f the porosity, S the shape parameter (logarithm of the void aspect ratio W ), ez the void axis and H Hill’s fourth-rank tensor. The potential F admits two diﬀerent expressions, FðcÞ and FðcþÞ , prior to and after coalescence, respectively. The ﬂow potential prior to coalescence is given by [15,25] ! 3R0 : H : R0 :R jA ðcÞ F ðR; f ; S; H; rH Þ ¼ þ 2qw f cosh h rH 2r2H 1 q2w f 2 ¼ 0;

ð6Þ

0

where ðÞ refers to the deviator, h is a factor calculated using Hill coeﬃcients, expressed in the basis ½e ¼ ðeL ; eT ; eS Þ pointing onto the principal directions of orthotropy [22,23], as 1=2 2 hL þ hT þ hS 1 1 1 1 h¼ þ ; þ þ 5 hL hT þ hT hS þ hS hL 5 hTS hLS hLT ð7Þ is the void anisotropy tensor expressed in the and A basis ½e0 ¼ ðex ; ey ; ez Þ associated with the void A ¼ a2 ðex ex þ ey ey Þ þ ð1 2a2 Þez ez ;

ð8Þ

with a2 in (8) and j in (6) being scalar functions of both f and S (Table 1). The coeﬃcient qw is void-shape dependent and was determined by Gologanu [15] to ﬁt unit-cell results: qw ¼ 1 þ ðq 1Þ= cosh S;

ð9Þ

A.A. Benzerga et al. / Acta Materialia 52 (2004) 4639–4650 Table 1 Coeﬃcients used in Eqs. (6), (8) and (18), analytical expressions and particular values Coeﬃcient

eðSÞ Eðf ; SÞ jðf ; SÞ a2 ðf ; SÞ a1 ðSÞ aG 1 ðSÞ

Prolate cavity (S P 0)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 expð2SÞ E3 e3 ¼f 1 E2 1 e2 1 1 1 pﬃﬃﬃ e pﬃﬃﬃ þ ð 3 2Þ ln E 3 ln f 1 þ E2 3 þ E4 1 1 e2 tanh1 e 2 2e 2e3 1 3 e2

Limit cases Sphere (S 0)

Cylinder (S ! 1)

0

1

0

1

3 2

pﬃﬃﬃ 3

1 3 1 3 1 3

1 2 1 2 1 2

For an arbitrary void shape (spheroid, cone, . . .), v is exactly related to the void spacing ratio, k, through a shape factor c as (see Fig. 1) (

1=3 EZ ¼ ez ; 3c Wf k ð12Þ v¼ f 1=3 k EZ ? ez : W 3c W k In [11] k ¼ 1=3 was used but reﬁnements based on unitcell calculations suggested that k ¼ 1 is more accurate. For an arbitrary loading orientation v admits a more complex expression but k is always deﬁned as the ratio of the Z-spacing to the average radial spacing. The function cðvÞ was introduced in [12] to represent the actual non-spheroidal void shapes observed during coalescence (see e.g. Fig. 9(b) in [1]), with c v

where q ¼ 1:6 is the value taken by qw for a spherical void. In general, the hI coeﬃcients in (7) are taken to be function of the eﬀective plastic strain p. The ﬂow potential after the onset of coalescence is given by [11,12]: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3=2R0 : H : R0 I : R 1 ðcþÞ F ðR; v; S; H; rH Þ ¼ þ 2 rH rH 3 ð1 v2 ÞCf ðv; SÞ 6 0; ð10Þ 2 where v is the ligament size ratio deﬁned in the loading frame ½E ¼ ðEX ; EY ; EZ Þ (see Fig. 1) and Cf is given by 2 pﬃﬃﬃﬃﬃﬃﬃ v1 1 þ 1:3 v1 ; Cf ðv; SÞ ¼ 0:1 2 1 2 W þ 0:1v þ 0:02v W ¼ eS :

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ð11Þ

c

1vc

c

The constants cc ¼ 1=2 and cf correspond to the shapes at incipient coalescence (spheroid) and at complete coalescence (often a cone so that cf ¼ 1), respectively. As v ! 1 the material loses all stress carrying capacity. vc is the value of the ligament size ratio at the onset of coalescence, which occurs when FðcÞ ¼ FðcþÞ ¼ 0. vc is not an adjustable factor. Both forms (6) and (10) of the plastic potential F H deﬁne rH with the remarkable property or :R ¼ rH so oR that, assuming equality of macroscopic plastic work rate and matrix dissipation, the viscoplastic strain rate is written as 1 oF oF ðpÞ D ¼ ð1 f Þp_ : ð14Þ orH oR

The strain rate eﬀect is accounted for through a Norton law for the eﬀective plastic strain p !m rH r p_ ¼ ð15Þ K with the uniaxial stress–strain relation having the form

p p=1 ðpÞ ¼ rL 1 þ þ Q 1 e r ; ð16Þ 0

Fig. 1. Representative volume element of a transversely isotropic material subject to a loading with a major normal stress along Z. (a) Loading parallel to the voids EZ kez . (b) Loading perpendicular to the voids EZ ? ez . Bottom views are in the plane of coalescence normal to EZ .

where K and m are material dependent constants, rL is the longitudinal yield stress and Q, 0 and 1 are material constants. In the limit of a rate independent material , F 6 0 deﬁnes a with rH replaced everywhere by r convex yield domain. Examples of the yield surfaces deﬁned by (6) and (10) are shown in Fig. 2 assuming plastic isotropy (H ¼ I ) for a particular axisymmetric loading path. The evolution laws of the microstructural variables prior to coalescence are given by f_ ¼ ð1 f ÞI : DðpÞ ;

ð17Þ

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Fig. 2. Yield surfaces for axisymmetric loadings resulting from the intersection of the yield domains deﬁned by (6) and (10) in the rate-independent limit: ð Þ initial surface corresponding to f ¼ 0:00075 and W ¼ 15; ( ) at the onset of coalescence with f ¼ 0:04, W ¼ 5 and v ¼ 0:34; (– –) during coalescence with f ¼ 0:15, W ¼ 2:2 and v ¼ 0:75.

2 4 G pﬃﬃﬃ _S ¼ 3 1 þ 9 T þ T ð1 f Þ2 a1 a1 D0ðpÞ zz 2 2 2 1 3a1 1 3a1 þ ð18Þ þ 3a2 1 I : DðpÞ ; f where the term between ½ in (18) is the result of ﬁts to unit-cell calculations, T is the stress triaxiality ratio (i.e. ratio of the hydrostatic stress, 1=3I : R , to the mises eﬀective stress) and a1 and aG are given in Table 1. 1 After the onset of coalescence the relevant variables are k, v and W ; their rates are given by [12] 3 0 k_ ¼ kDzzðpÞ ; ð19Þ 2 which holds regardless of the spatial distribution of voids, 3 k 3c v 0 ð20Þ 1 DzzðpÞ þ c_ ; v_ ¼ 4 W v2 2c which results from plastic incompressibility of the matrix material, and 9k c W 0 _ W ¼ 1 2 DzzðpÞ c_ : ð21Þ 4v v 2c Assuming that voids rotate with the material, the evolution of void orientation is given by _ X T ez ; e_ z ¼ X

ð22Þ

where X is the rotation used in (4) so that (22) follows from the objective frame description. If the co-rotational _ XT is simply the spin W. In space frame is used then X this case the rotation is determined by integration of _ ¼ W Q with the initial condition Q ¼ I . ExperiQ t 0 mental evidence supports the general form (22) if the loading axes ½E are initially aligned with the void axes

½e0 as in Fig. 1; otherwise (22) is not valid because the void spin is then diﬀerent from the material spin as evidenced through careful experimentation [11]. The formulation above has been implemented in the ﬁnite element code ZeBuLoN, which is designed as in [26] so that the constitutive behavior is, to a large extent, independent from equilibrium and kinematics thanks to the use of objective space frames. This enabled us to test diﬀerent options. For instance, formulations with the Green–Naghdi and Jaumann rates of Cauchy stress were employed, each involving a diﬀerent specialization of the rotation rate in (22). In the case where the load axis is initially parallel to a principal orientation (here L or T), the diﬀerence was found to be negligible. Thus, the calculations analyzed here were carried out using the Jaumann rate. A fully implicit time integration procedure was used for the local behavior in conjunction with an iterative Newton–Raphson method. The consistent tangent matrix was computed as detailed in [25].

3. Predictive modeling of fracture We now illustrate how an approach based on the constitutive modeling of Section 2 can be implemented to predict fracture anisotropy in round notched steel bars. The three directions of orthotropy L, T and S correspond to the rolling, transverse and short transverse orientations, respectively. The reader is referred to [1] for the experimental counterpart. Assessment against experiments is carried over global (force, strain to failure) as well as local (porosity at incipient coalescence) quantities. 3.1. Finite element model and input parameters 3D calculations were carried out using geometries corresponding to the three tested notched bars. For brevity most results are shown for bars with a shallow notch (f ¼ 10) or bars with a sharp notch (f ¼ 2). As in [1] the parameter f refers to ten times the ratio of notch radius to the minimal section diameter U0 . The meshes used have 20-node quadratic sub-integrated quadrilaterals. Flat elements are used in the minimal section which have dimensions 325 325 100 lm3 . The initial height of 100 lm is low enough to avoid numerical errors associated with the presence of overly elongated elements at the onset of fracture due to the large deformations involved. The mesh is designed symmetrically with respect to the two principal directions of any cross-section. Because the analyses are restricted to loadings along principal axes, only one eighth of a bar is meshed. The boundary conditions are speciﬁed as follows. Lateral surfaces are free of normal tractions and symmetry conditions are enforced. A uniform displacement U is prescribed on the top surface.

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second by transforming each MnS plate-like particle into an equivalent spheroid with axis ez ¼ eL (Fig. 3(c)). This is in keeping with the observation in [1] that the initially 3D voids evolve rapidly into spheroids as the void growth rate is much larger in the short transverse direction (see Figs. 8 and 13 in [1]). The values used for the elastic properties are E ¼ 210 GPa and m ¼ 0:3. Also, the coeﬃcients in (16) are calculated as rL ¼ 426 MPa, 0 ¼ 1:78, Q ¼ 0:38 and 1 ¼ 0:06. These values are the outcome of a standard optimization procedure that provides the best ﬁt to the experimental uniaxial stress–strain curve obtained in tension along the L axis. Consistently, Hill coeﬃcients in (7) and (6) are given in terms of the ratios hI =hL , which were previously determined using tension and compression tests [25] as 0.917, 1.333, 1.354, 1.200 and 1.135 for I ¼ T, S, TS, SL and LT, respectively. Coeﬃcient hL is determined by the requirement that the equivalent strain rate be equal to the logarithmic strain rate in tension along L. For details on the identiﬁcation procedure see [23]. To focus the analysis on the low strainrate regime of the experiments, the parameters K and m in (15) were given the values 1 MPa and 5, respectively, so that strain-rate sensitivity is negligible in all

The initial porosity is identiﬁed with the particle volume fraction. This follows from the fact that void nucleation is complete at strains that are low in comparison with strains at incipient coalescence [1,11,27]. The focus here is on loadings along the transverse, T, or rolling, L, orientations so that EZ ¼ eT (T-loading) or EZ ¼ eL (L-loading), respectively. In all cases the common axis of the elongated voids is ez eL . The model presented in Section 2 is transversely isotropic in the absence of plastic anisotropy, but the real microstructure is 3D. Each population of particles is characterized by seven average parameters: volume fraction, fv , aspect ratios, W I , and spacing ratios, kI ; see Table 2 where the data of [1] is recollected. Hence an approximation is made whereby an ‘‘equivalent initial microstructure’’ is deﬁned. If the particles are assumed to be uniformly distributed and their shapes not to depart too much from the average shape then the products of their aspect ratios and their spacing ratios are each close to 1 and the microstructure is then described by ﬁve independent parameters. This number is further reduced to three, ﬁrst by transforming each unit-cell whose relative dimensions are the spacing ratios into a conﬁguration dependent cylinder (Figs. 3(a) and (b)), Table 2 3D inclusion characteristics of the studied steel Population

MnS Equiaxed All

Volume fraction

Aspect ratios

Spacing ratios

fv ð%Þ

Vv ð%Þ

L W

T W

S W

kL

kT

kS

0.024 0.032 0.056

0.047 – 0.075

4.8 1 –

1/28 1 –

8.0 1 –

0.63 0.96 0.79

0.84 0.98 0.92

1.89 1.06 1.37

Ratios are deﬁned using a circular permutation rule (L, T, S). Vv is the value inferred from chemical analysis. The value of fv includes results on ﬁve sections [11].

Fig. 3. Schematic for the treatment of initial anisotropy. The geometry of Fig. 1 is obtained in two steps. Step 1: the unit cell of the particulate aggregate is approximated by a cylinder whose axis is the loading axis: (a) major stress parallel to L; (b) major stress parallel to T. Step 2: the void nucleated on an MnS inclusion is approximated by an equivalent spheroid as shown in (c).

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calculations reported in following sections. For simplicity, a constant shape factor has been assumed so that c_ ¼ 0 in (20) and (21). The value of c, which does not aﬀect the prediction of failure initiation, is taken to be 0.8 in L-loading and 0.5 in T-loading. Such values are consistent with the elongated voids developing into conical shapes during coalescence while ﬂat voids keep ellipsoidal shapes (see e.g. Fig. 9(b) and (c) in [1]). Also, to simplify the ﬁnal stages of failure it was assumed that the stress carrying capacity vanishes when the ligament area is only 10% of the total area (i.e. v 0:83). This agrees well with the coalescence process being terminated by micro-crack linkage before impingement. 3.2. Salient features of the theory A typical force (P ) versus diameter-reduction (DU) response is given in Fig. 4(a) for a bar with f ¼ 10

subject to transverse loading. Various stages of the post-coalescence regime are highlighted in (b) and (c). Diameter reduction is given along both the rolling direction, DUL , and the through-thickness direction, DUS , as an evidence for the anisotropy of deformation. Moreover, Fig. 4(b) shows the anisotropy of damage accumulation, here measured by the number of elements that undergo the post-coalescence behavior, while Fig. 4(c) shows the subsequent anisotropy in crack growth. Before the stage marked A in Fig. 4(a) the plastic ﬂow of the material at the current loading point is normal to the smooth yield surface (shown in dashed lines in Fig. 2) with the evolution of the microstructure being entirely deﬁned by (17) and (18) for the porosity and void aspect ratio, respectively. Indeed, as long as the material lies on the pre-coalescence yield surface there is no eﬀect of the spacing ratio k. The evolution of mi-

Fig. 4. (a) Typical curve force vs. diameter reduction displaying several stages of fracture and anisotropy denoted A–E. Case of transverse loading (EZ ¼ eT ) with f ¼ 10, f0 ¼ 0:00075, S0 ¼ 3 and k0 ¼ 1. (b) Elements undergoing coalescence are painted black. (c) Broken elements are painted black. A: Onset of coalescence; B: crack initiation; C: anisotropy in extent of coalescence and crack growth; D: last element along L undergoing coalescence; E: last element along L broken.

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crostructural variables and stress triaxiality ratio at the center of the bar is given in Fig. 5 (solid lines correspond to the case f ¼ 10 discussed here). Rapid void growth occurs at the center of the bar and is accompanied with a steady decrease in the aspect ratio, which, under T-loading, corresponds to the void opening up. Correspondingly, the ligament size ratio, v, increases exponentially from a relatively high initial value (v0 0:1) that reﬂects an unfavorable loading conﬁguration; see Fig. 5(c). This increase in v (i.e. decrease of local ligament area) strongly aﬀects the geometric factor Cf in (11). When for the ﬁrst time the combined decrease of both ligament area and Cf outweighs the increase in the local axial stress the mode of deformation shifts toward the coalescence mode so that for a small increase of deformation the material point now lies on the planar part of the post-coalescence yield surface; see Fig. 2. This shift in the deformation mode ﬁrst occurs at the center of the bar as shown in Fig. 4(b) at stage A. As a consequence the global force drops quite abruptly although there is no crack yet in the specimen. When a crack has nucleated at stage B (see Fig. 4(c)) the number of elements undergoing coalescence has very much increased. Coalescence extension as

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well as crack growth are both anisotropic as can be seen in Figs. 4(b) and (c). The crack clearly advances faster in the L direction than in the S direction. This is attributed to plastic anisotropy, not to the fact that voids are longer along L in the plane of coalescence. Fig. 6 shows contours of porosity and void aspect ratio in the two principal meridian planes (planes containing the loading direction L and either T or S). The microscopic eﬀect of plastic anisotropy consists of a faster development of porosity along T than along S. The number of Gauss points undergoing coalescence at the stage shown in Fig. 6 is 5 along T versus 3 along S. This eﬀect is amenable to the sole coupling between damage and plastic anisotropy before the onset of coalescence. After the onset of coalescence at the center of the bar, the anisotropy of crack initiation is enhanced by the anisotropy in void distribution as well. As seen in Fig. 7 void distribution is inhomogeneous and anisotropic. Higher values of the spacing ratio k are reached along T. Large values of k correspond to the voids being more closely spaced along the radial direction than along the axial direction, which precipitates further coalescence because the ligament is then relatively smaller.

Fig. 5. Eﬀect of notch geometry in T-loading on the evolution of (a) stress triaxiality, (b) ligament size ratio, (c) void aspect ratio and (d) porosity at the center of a notched bar with f0 ¼ 0:00075, S0 ¼ 3 and k0 ¼ 1:0.

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Fig. 7. Contours of the void spacing ratio, k, showing the anisotropy in void distribution in L-loading (EZ ¼ eL ) with f ¼ 10, f0 ¼ 0:00075, S0 ¼ 0 and k0 ¼ 1:5.

age data in Table 2 and the standard deviation an estimate for W0 is 15.0 ( 5), which leads to the range of values S0 ¼ 2:35–3 for MnS inclusions. Of course S0 0 for the equiaxed particles. 2. The approximate cylindrical unit cell is parallel to the major normal stress and has aspect ratio k0X called the spacing ratio. For the two loading conﬁgurations qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Fig. 6. Eﬀect of plastic anisotropy on bar deformation and void growth under L-loading with f ¼ 10, f0 ¼ 0:00075, S0 ¼ 0 and k0 ¼ 1:5. (a) Cross-section in L–T plane, (b) cross-section in L–S plane.

3.3. Comparison with experiments 3.3.1. Initial microstructure A quantitative comparison with experiments is made using the equivalent initial microstructure shown in Fig. 3 (also refer to Section 3.1). The three microstructural input parameters, S0 , k0 and f0 , are determined as follows: 1. The aspect ratio of MnS inclusions is taken as the geometric mean of the length-to-width and length-ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ to-thickness aspect ratios, that is: W0 W S =W T where bars were dropped for clarity. Using the aver-

S T of interest qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃhere we have: k0L k =k and L S k0T k =k . Using the average values of kI reported in Table 2, the estimates used are k0L ¼ 1:5 and k0T ¼ 0:6 for MnS inclusions and 1.04 and 0.95, respectively, for equiaxed particles. 3. The initial porosity, f0 , is taken to be of the order of the volume fraction of active nucleation sites. An upper limit for f0 is the sum of fv ðMnSÞ and fv ðOxideÞ (see Table 2). This upper value is taken to be 0.00075. The lower limit is taken to be fv ðMnSÞ so that f0 ¼ 0:0004. Furthermore, sensitivity analyses to initial conditions, within the ranges mentioned above, have revealed that S0 and to a lesser extent k0 are more critical than f0 . Relevant initial conditions can be discussed and several hypotheses formulated depending on the contribution of the equiaxed particles to the fracture process. The calculations performed in the case of a transverse loading were pursued until the force completely dropped to zero whereas those performed in the case of an L-loading were terminated after some crack propagation has taken place but before the complete vanishing of the global force. This is so because the ductility in the L-direction is so high that the initially ﬂat elements elongate too much for the results to be accurate after extensive crack growth.

3.3.2. Strains to failure We ﬁrst discuss the prediction of failure in the case of L-loading where the major normal traction is parallel to

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the void axis. Fig. 8 shows the comparison with representative experimental data. The scatter of the measured diameter reductions at crack initiation is indicated by horizontal bars. Two sets of initial microstructural variables were used leading to two diﬀerent predictions. In both sets f0 ¼ 0:00075 and k0 ¼ 1:5 but S0 ¼ 1:95 in one set and S0 ¼ 0 in the other. The case S0 ¼ 0 was investigated because at low stress triaxiality crack initiation is indeed caused by the initially equiaxed voids; e.g. see Fig. 9(a) in [1]. If the initial void aspect ratio has the intermediate value S0 ¼ 1:95 between S0 (MnS) and S0 (Oxide) the model provides an excellent agreement with experiments in the case f ¼ 2 (Fig. 8(b)) but it predicts too ductile a response in the case f ¼ 10, Fig. 8(a) (dotted line). On the other hand, the prediction is excellent for f ¼ 10 if equiaxed voids are the only voids involved in the coalescence process (S0 ¼ 0), Fig. 8(a) (solid line). To investigate this, Fig. 9 shows the evolution of the microstructural variables (f , S and v) at the center of each bar. First consider the low triaxiality case (f ¼ 10).

Fig. 8. Experimental and calculated load vs. diameter reduction curves using f0 ¼ 0:00075 and k0 ¼ 1:5 for L-loading and (a) f ¼ 10 with S0 ¼ 1:95 (prediction 1) and S0 ¼ 0 (prediction 2); (b) f ¼ 2 (prediction 1 only).

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For comparison, the results of the reference calculation using the value S0 ¼ 1:95 are also shown (dotted lines). In this case Fig. 9(b) clearly shows that long voids elongate further, which excludes internal necking coalescence. In experiments, such voids may rather be involved in a necklace-like coalescence, which only elongates the void so that the necking mode of

Fig. 9. Eﬀect of notch geometry in L-loading on the evolution of (a) ligament size ratio, (b) void aspect ratio and (c) porosity at the center of a notched bar with f0 ¼ 0:00075, k0 ¼ 1:5 and S0 ¼ 0 for the f ¼ 10 bar and S0 ¼ 1:95 for the f ¼ 2 bar. The dotted lines are for the reference calculation with f ¼ 10 and S0 ¼ 1:95.

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coalescence becomes less likely. Based on the result obtained using S0 ¼ 0 (solid line) along with experimental facts, we conclude that failure initiation under L-loading at low stress triaxiality is controlled by the coalescence of initially equiaxed voids. Note that using S0 ¼ 0 above while using for f0 and k0 values that also account for MnS inclusions is justiﬁed based on (i) dilatancy associated with the growth of elongated voids aﬀects the global response; (ii) the void spatial distribution as measured by k0 is aﬀected by both populations of voids. At higher stress triaxiality (case of f ¼ 2) the situation is diﬀerent. A scenario is conﬁrmed whereby the elongated voids actively contribute to the coalescence process, in keeping with experiments [1]. This is so because, as deformation proceeds, relative void elongation is diminished as seen in Fig. 9(b) (dashed line). This decrease in the void aspect ratio is due to the fact that radial void growth is faster than the axial growth even though the major normal traction is axial (notice the high rate of v in Fig. 9(a)). With both populations of voids involved, the value S0 ¼ 1:95 is found here to best represent the extent to which each population contributes to crack initiation in the f ¼ 2 bar. Next we discuss the prediction of failure in the case of T-loading. Here the major normal traction is perpendicular to the void axis. Thus axisymmetry is broken. For that reason, the calculations here were carried out using two variants for the deﬁnition of ligament size ratio: k ¼ 1=3 in Eq. (12)2 as in [11] or k ¼ 1. The results shown in Fig. 10(a) using k ¼ 1=3, f0 ¼ 0:0004, S0 ¼ 2:35 and k0 ¼ 0:6 clearly overestimate ductility (dashed lines). Recognizing both the deleterious eﬀect of MnS inclusions under T-loading and the contribution of equiaxed voids, the prediction is signiﬁcantly improved using f0 ¼ 0:00075, S0 ¼ 3 and k0 ¼ 1 (solid line). The improvement here is mainly due to the value of k0 and not to the larger value of S0 ¼ 3 because, for very elongated shapes, the pre-coalescence model is insensitive to S0 given the saturation of the coeﬃcient a2 in Eq. (8) [11,25]. Using the value k ¼ 1 inferred from unit-cell calculations leads to better results as shown below. The average strain at crack initiation, c ¼ ln U20 = ðUJ UK Þ, is a measure of ductility that incorporates plastic anisotropy. Fig. 11 summarizes the predictions of ductility under both loading orientations (with k ¼ 1 under transverse loading). The predicted strains at crack initiation fall within the experimental scatter and, for Tloading, are slightly above the average experimental values. The fracture anisotropy that is well rendered here is associated with initial anisotropy for f ¼ 2 and with the presence of two populations of voids for f ¼ 10. The diﬀerence with respect to initial values between the two orientations is mainly due to k0 being conﬁguration dependent. Also the current theory does not explicitly

Fig. 10. Experimental and calculated load vs. diameter reduction curves for T-loading and (a) f ¼ 10 with f0 ¼ 0:0004, k0 ¼ 0:6 and S0 ¼ 2:35 (prediction 1) and f0 ¼ 0:00075, k0 ¼ 1 and S0 ¼ 3 (prediction 2); (b) f ¼ 2 (prediction 2 only).

Fig. 11. Comparison between measured and predicted average strains to failure initiation in notched bars for two loading orientations.

distinguish between the two populations of voids. Therefore, there is certainly room for further improvement.

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3.3.3. Local porosities In making comparisons at the microscopic level, the interest is in the evolution of all microstructural variables. The emphasis is laid, however, on local porosities because isotropic models fail to predict these whenever they fairly predict the macroscopic fracture strains. The predicted evolution of the void aspect ratio, S ¼ ln W , and the ligament size ratio, v, was shown in Figs. 5 and 9 for loading along T and L, respectively. Only the void aspect ratio can be directly related to the void growth ratios measured in the companion paper [1] by noting that under L-loading W =W0 is simply the relative axial to lateral void growth ratio, cL =cS . Therefore, if void growth is predominantly extensional (cL > cS ), as in the f ¼ 10 bar (low triaxiality ratio T), then S must be an increasing function of strain. Conversely, if void growth is mostly dilational (cL < cS ) then S must be a decreasing function. Comparison between Fig. 13 in [1] (maximum values) and the present Fig. 9(b) shows that the model not only accounts for that qualitative trend but also picks up the transition from extensional to dilational void growth for elongated cavities. Under T-loading, assuming a spheroidal shape for the voids at the initiation of macroscopic failure and noting that W =W0 is identiﬁed with the measured pﬃﬃﬃﬃﬃﬃﬃﬃﬃ cL = cT cS , it is also easy to check that the void aspect ratio (as deﬁned here in the local frame) must decrease, irrespective of the stress state. This is precisely what is seen in Fig. 5(c) above, consistent with the experimental trends. Next we proceed to the comparison of the predicted porosities at the onset of coalescence, fc , with the local void area fractions measured in Part I. As noted there, measured porosities comprised between the local average, f , and the local maximum, fmax , should be representative of near-coalescence states. This type of detailed comparison is only possible for an L oriented bar with f ¼ 4 and a T oriented bar with f ¼ 10, for which local measurements were carried out. Fig. 12 shows the predicted porosity, fc (open symbols), the local porosities, f and fmax when available, along with the average void area fraction, fa , deﬁned in [1] as the total void area over the total area analyzed. In Fig. 12 error bars shown for fa include the accuracy of measuring void dimensions and total analyzed area [1] while those shown for f and fmax indicate the standard error with 95% conﬁdence intervals. In particular, the error in estimating fmax is computed using all cells having porosity larger than 0.02. Under L-loading, the predicted fc matches quite well the maximum local porosity, fmax , whereas, under Tloading, the predicted fc falls between f and fmax . Both predictions are remarkable not only because they fall within the relevant experimental range but also because each prediction approximately corresponds to the local maximum in the frequency distribution of the measured

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Fig. 12. Measured and predicted porosities at incipient coalescence in notched bars. (a) L orientation. (b) T orientation. fmax : largest local porosity; f : local average; fa : global average.

void area fractions (see Fig. 15 in Part I). Also, under L-loading fc is found to increase with enhancing stress triaxiality and so does the measured average porosity, but to a lesser extent. This trend of an increasing fc with increasing stress triaxiality, within the range considered here, is consistent with previous theoretical predictions [12,17]. It is worth recalling that the measured values correspond to elongated voids. Thus, for consistency, the fc values in Fig. 12 were determined for the same set of initial microstructural variables. In particular, the value corresponding to f ¼ 10 does not correspond to an actual onset of coalescence; it is the value that porosity would have taken at failure initiation in experiments if the elongated cavities were involved in that initiation. For the T orientation case, the porosity at coalescence is not found to vary much within the range of stress triaxiality explored here. However, because of the heuristic character of our extension of the coalescence criterion to non-axisymmetric conﬁgurations, that model in particular is open to improvements.

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In all loading conﬁgurations considered here, the values of fc are an outcome of the calculations and not an input. Quantitative comparisons based on isotropic models usually require that critical porosities be adjusted so that the macroscopic failure strains can be fairly predicted. It should be emphasized that commonly used values for the critical porosity, compared with our fc , fall in the range 0.001–0.003 for an initial porosity, f0 , of 0.00075 and can be even smaller for lower values of f0 . Such ‘‘critical porosities’’ are one order of magnitude lower than the values measured in [1] and, for the L orientation case, they are of the order of the global average porosity, fa . But what is critical to the onset of failure in notched bars is the local porosity, not fa .

4. Conclusions In this paper, a general theory of ductile fracture has been proposed and its application to engineering alloys illustrated. The theory ﬁrst accounts for matrix anisotropy which may result from texture development during the metal working process. Next, the theory accounts for morphological anisotropy which can be initial, due to non-spherical second-phase particles, or induced, due to the evolution of void shape. The theory also accounts for the whole void-coalescence process through a micromechanical model. Accounting for that process leads to an eﬀective yield surface which displays regions of extreme curvature. One outcome of using the developed constitutive relations is a loading response that includes the transition from a pre-coalescence stage to a postcoalescence stage without using adjustable factors. The predictive approach pursued in this study relies on: (i) a careful characterization of the microstructure in the undeformed state using standard quantitative metallography techniques; (ii) recording the mechanical response of the material under various stress states and orientations with a special attention to plastic anisotropy; (iii) a ﬁnite-element implementation of the constitutive relations relevant to anisotropy and to diﬀerent modes of dilational deformation; (iv) a complementary collection of data related to the microstructure in the deformed state with the aim of assessing the approach in an unbiased way. The calculations here quantitatively reproduce the behavior seen in notched bars in remarkable detail. Moreover, the accuracy of the predictions has been achieved under the following conditions: (i) no adjustable parameter has been used (the counterpart for that is the collection of a large amount of data on the initial microstructure); (ii) the quality of the prediction is obtained using a very low value for the initial porosity, which is inferred from measurements; (iii) predictions of global failure strains and local

porosities at incipient simultaneously.

coalescence

are

achieved

Acknowledgements Financial support from Gaz de France (Direction de la Recherche; Program managers R. Batisse and M. Zarea) is greatly acknowledged. References [1] Benzerga AA, Besson J, Pineau A. Anisotropic ductile fracture. Part I: experiments. Acta Mater 2004. doi:10.1016/j.actamat. 2004.06.020, this issue. [2] Rice JR, Tracey DM. J Mech Phys Solids 1969;17:201–17. [3] Gurson AL. J Eng Mater Technol 1977;99:2–15. [4] Beremin FM. In: Nemat-Nasser S, editor. Three-dimensional constitutive relations of damage and fracture. New York: Pergamon Press; 1981. [5] Needleman A, Tvergaard V. J Mech Phys Solids 1984;32:461–90. [6] Tvergaard V. Adv Appl Mech 1990;27:83–151. [7] Pineau A. In: Argon AS, editor. Topics in fracture and fatigue. Berlin: Spinger–Verlag; 1992. [8] Zhang ZL, Niemi E. Fat Frac Eng Mater Struct 1994;17:695–707. [9] Besson J, Steglich D, Brocks W. Int J Solids Struct 2001;38:8259– 84. [10] Gologanu M. Etude de quelques problemes de rupture ductile des metaux. PhD thesis, Universite Paris 6, 1997. [11] Benzerga AA. Rupture ductile des t^ oles anisotropes. PhD thesis, Ecole Nationale Superieure des Mines de Paris, 2000. [12] Benzerga AA. J Mech Phys Solids 2002;50:1331–62. [13] Pardoen T, Hutchinson JW. Acta Mater 2003;51:133–48. [14] Gologanu M, Leblond J-B, Devaux J. J Mech Phys Solids 1993;41(11):1723–54. [15] Gologanu M, Leblond J-B, Perrin G, Devaux J. In: Suquet P, editor. Continuum micromechanics. Berlin: Springer-Verlag; 1995. p. 61–130. [16] Benzerga AA, Besson J, Batisse R, Pineau A. In: Brown MW, de los Rios ER, Miller KJ, editors. 12th European Conference on Fracture. ESIS, European Group on Fracture Publication; September 1998. [17] Benzerga AA, Besson J, Pineau A. J Eng Mater Technol 1999;121:221–9. [18] Becker R, Smelser RE, Richmond O. J Mech Phys Solids 1989;37(1):111–29. [19] Forsyth PJE, Stubbington CA. Met Technol 1975;2:158–77. [20] Agarwal H, Gokhale AM, Graham S, Horstemeyer MF. Metall Mater Trans A 2002;33:3443–8. [21] Deshpande NU, Gokhale AM, Denzer DK, Liu J. Metall Mater Trans A 1998;29:1191–201. [22] Benzerga AA, Besson J, Pineau A. In: Peseux B, Aubry D, Pelle JP, Touratier M, editors. Actes du 3eme Colloque National en Calcul des Structures. Presses Academiques de l’Ouest; 20–23 May 1997. [23] Benzerga AA, Besson J. Eur J Mech 2001;20(3):397–434. [24] Ladeveze P. Sur la theorie de la plasticite en grandes deformations. Technical Report 9; LMT Ecole Normale Superieure; Cachan, France; 1980. [25] Benzerga AA, Besson J, Batisse R, Pineau A. Modelling Simul Mater Sci Eng 2002;10:73–102. [26] Besson J, Foerch R. Comput Methods Appl Mech Eng 1997;142:165–87. [27] Beremin FM. Met Trans A 1981;12:723–31.

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