Constitutive equations for porous solids with matrix behaviour dependent on the second and third stress invariants

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Constitutive equations for porous solids with matrix behaviour dependent on the second and third stress invariants TagedPD1X XA. BenallalD2X X* TagedPLMT, ENS Paris-Saclay/CNRS/Universite
Paris-saclay, 61 Avenue du Pre
sident Wilson, Cachan Cedex, F 94235, France

TAGEDPA R T I C L E

I N F O

Article History: Available online xxx TagedPKeywords: Porous materials Effective yield criterion Lode angle

TAGEDPA B S T R A C T

Constitutive equations are developed for voided materials and ductile fracture taking into account possible effects of Lode angle in the yielding behaviour of the matrix. The Gurson criterion (Gurson, 1977) [4] is generalized to such circumstances. A semi-closed form expression , similar to the Gurson criterion is obtained for the effective yield criterion for the porous solid and involves four different functions , all dependent on the macroscopic stress triaxiality and Lode angle but are not generally available in closed form. In parallel, a parametric representation of the effective yield criterion is provided which allows for the derivation of closed form results for pure shear stress states and also at very high stress triaxialities. In the former case corresponding to a zero macroscopic mean stress, the contour of the yield domain in the p-plane has exactly the shape of the yield surface of the matrix in the deviatoric plane but a size reduced by a factor 1
f ; with f the porosity of the voided material. In the latter, effective yield stresses for the voided material are slightly different from the Gurson result and found to be set by the yield stress at a microscopic stress Lode angle p3 for very high positive triaxiality and by the yield stress at a microscopic stress Lode angle 0 for very high negative triaxiality. Various numerical results are furnished to illustrate all the obtained results. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction TagedPAn important issue in material mechanics is the development and validation of accurate macroscopic constitutive equations for engineering materials allowing good design and in service predictions for structural components and in particular for impact loadings. This is a difficult task and very often the needed constitutive equations are rather obtained in an ad hoc, sometimes empirical and in many cases in a phenomenological way. In these ad-hoc and phenomenologically developed constitutive equations, the microstructure and physical mechanisms responsible for the behaviour and fracture of these materials are usually not taken into account. Translation of this microstructure and mechanisms information to the macroscopic level can be done through homogenization and scale transitions in the spirit of McClintock [1], Rice and Tracey [3] approaches for void growth and fracture and by Gurson [4] for yielding of porous materials. The three contributions are so important that they are still currently in use today. All three contributions were undertaken with a yielding of the matrix obeying a plastic behaviour governed by the von Mises yield criterion. Traditionally, the von Mises criterion is the most utilised criterion because of its mathematically simple form. *

Corresponding author. E-mail address: [email protected]

TagedPThere are situations where the von Mises matrix behaviour seems insufficient for reproducing the experimental observations. For instance, Ohashi and Tokuda [5] obtained detailed information about the plastic behaviour of real materials by precise measurement of plastic deformation of thin-walled tubular specimens of initially-isotropic mild steel under combined loading of torsion and axial force. They used trajectories consisting of two straight lines at a constant rate of the effective strain. From these experimental results, they found that the effect of the third invariant of the strain tensor appeared even for proportional deformation consisting of torsion and axial force. Further, they observed the effective stress to drop suddenly with increasing effective strain and that coaxiality between the stress deviator and the plastic strain increment tensor to be seriously disturbed just after the corner of the strain trajectory. These local disturbances are recovered along the second branch of the trajectory. In another important experimental investigation, Rousset [6] measured precisely subsequent yield surfaces for an 2024 aluminium alloy and observed that even in the simpler case of proportional loadings, subsequent yield surfaces are distorded with a corner forming in the loading direction and a flattening in the opposite one. Also, with the use of polycrystal theory of plasticity it was found (see e.g. Hershey [15] and Hosford [16]) that the yield surfaces for fcc-metals do not have the elliptical form described by the von Mises criterion. In another context, the forming limit diagrams are seen to be significantly dependent on the yield surface [9]. Many

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Please cite this article as: A. Benallal, Constitutive equations for porous solids with matrix behaviour dependent on the second and third stress invariants, International Journal of Impact Engineering (2017), http://dx.doi.org/10.1016/j.ijimpeng.2017.05.004

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TagedPaluminium alloys for instance exhibit significant anisotropy in strength, plastic flow and ductility. The use of Hill’s original anisotropic criterion [10] (based on the von Mises criterion) has been shown unsuitable for f.c.c metals or for materials exhibiting low r values (where r is the ratio of the width to the thickness strain under uniaxial tension). Experimental evidence shows here that the biaxial flow stress in aluminum alloys is larger than the uniaxial flow stress [7,8] whereas Hill’s theory [10] predicts the contrary. A number of theoretical, experimental and numerical investigations exist in the literature with the objective of a better description of yielding of isotropic and anisotropic materials. Thus, for isotropic f.c.c, materials, Hershey [15] and Hosford [16] have proposed the same equation for the description of the yield surface and this will be used in the sequel of this paper. This equation has also been generalized by Logan and Hosford [13] for anisotropic materials. Another criterion was proposed by Hill [11] while other improved yield criteria have been proposed by Barlat et al. [12]. Lademo [17] investigated several of these yield criteria and clearly demonstrated the need of a more complex yield behaviour for aluminium alloys. He also shows that the contours of shear stress change their shape for increasing values of shear stress and this calls for a coupling between the shear and normal stress components in the equation for the yield criterion. To close this paragraph, one can conclude that independently from all the criteria sketched above, and for incompressible plasticity, all involves in a way or another the third stress (Lode angle) invariant even in the isotropic case. This is the subject of this paper aimed at deriving macroscopic constitutive equations for voided materials the matrix of which has a yielding behaviour dependent on both the second and third stress invariants. The general yield function considered herein encompasses most of the usual criteria. The derivation of the effective yield criterion for the porous solid is carried out in the framework of the Gurson approach and the analysis will be limited here to isotropic materials. TagedPEffects of the Lode angle in the Gurson approach appear at two different levels. Beside the fact that the matrix behaviour is dependent on the Lode angle, it also enters in the homogenization process as the stress state in the representative volume cell (a hollow sphere) is heterogeneous. In a recent paper the author and coworkers [24] assessed the effects of the third stress invariant in the yielding of ductile porous solids arising from the later effect by considering a von Mises yielding behaviour for the matrix. This was done by simply avoiding the approximation used by Gurson [4] and considering the full expression of the microscopic dissipation. For small porosities encountered on ductile fracture of metals, observed changes and roles of the Lode angle are found rather small although from the qualitative point of view, non-symmetry of the yield locus and changes on its curvature are observed. However, some changes were found in the intermediate regime of triaxialities and a careful inspection of these changes are seen to be second order effects (of the triaxiality) rather than direct effects of the Lode angle in the yielding of porous materials which only arise at third order of Gurson m parameter. The coming analysis will consider all these aspects. TagedPOther situations calling for more complex behaviour (either plasticity or fracture) are a number of experimental observations on failure under low or negative triaxialities (McClintock [2], Johnson and Cook [18], Bao and Wierzbicki [14], Barsoum and Faleskog [19] and Fourmeau et al. [23]) . Shear-dominated stress states such as plugging failure in projectile penetration are other examples [20] and many others can be found in the above references. Nahshon and Hutchinson [27] amended for instance the Gurson model in a phenomenological way by making the evolution of the porosity also dependent on the third invariant of the stress. A number of other experimental, theoretical and numerical studies have emerged since on the comprehension and the modelling of ductile fracture at low triaxialities and in particular on the introduction of the third

iTagedP nvariant of stress in constitutive equations. The Lode angle effects have been also included in [31] and [32] and studied by Danas and Ponte Castaneda [21,22] in an alternative approach to limit analysis of unit cell and based on second order variational homogenization techniques. TagedPThe outline of the paper is as follows. In Section 2, we set the notations used throughout the paper. The constitutive equations of the matrix that we have in mind are described in details in Section 3. Section 4 describes the derivation of the parametric representation of the effective yield surface of the voided material when the Gurson trial velocity field is used. This parametric form is used in Section 5 for various numerical simulations and to obtain some closed form results for hydrostatic and pure loadings. In Section 6 we give a mathematical semi-explicit expression for the equation of the yield domain fully including effects of the Lode angle. Throughout the paper, the results are illustrated using two different yielding behaviour for the matrix. 2. Notations TagedPThe paper is concerned with the effective behaviour of porous ductile materials described by a representative volume element V containing voids and the rest occupied by a matrix the constitutive behaviour of which is considered here as incompressible, isotropic and rigid-plastic. In all the paper, s and e_ denote the microscopic stress and strain rate in the matrix while the macroscopic stress and _ The latter are defined by strain rate are called respectively S and E. Z 1 S ¼ V ¼ s dV ð1Þ V V 1 E_ ¼ < e_ > V ¼ V

Z V

e_ dV

ð2Þ

where the operator < ¢ > V refers to averaging over the volume V of the representative volume element. TagedPThe invariants of the microscopic stress tensor are the mean stress s m, the von Mises equivalent stress s eq and the stress Lode angle v defined respectively by ! rffiffiffiffiffiffiffiffiffiffiffiffi 1 3 1 27 det s ð3Þ sij sij and v ¼ arccos s m ¼ s ii ; s eq ¼ 3 2 3 2 s 3eq where s is the microscopic stress deviator and repeated summation is used. The same invariants will be used for the macroscopic stress S and are given by 0 1 rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 1 27 det S A S S and Q ¼ arccos@ ð4Þ Sm ¼ Skk ; Seq ¼ 3 2 ij ij 3 2 S3 eq

S is the stress deviator and Q the Lode angle of the macroscopic stress. For isotropic materials considered here, the investigation range of both v and Q can be limited 0  Q  p3 . The ordered macroscopic principal stresses are denoted by S1  S2  S3. Beside the Lode angle Q, other equivalent measures can be used to describe effects of the third stress invariant of the macroscopic stress, such as the Lode parameter given by L¼

2S2
S1
S3 S
Sm pffiffiffi  p ¼3 2 ¼ 3 tan Q
6 S1
S3 S1
S3

ð5Þ

taking values in the range
1  L  1. TagedPFig. 1(a) shows the effective yield domain obtained by Gurson in the principal stress coordinate system (S1, S2, S3). The Lode angle Q is best represented in the octahedral plane (see Lubliner [26]) 0 where Q is (taken here) as the angle between the projection S1 of the maximum principal direction on the octahedral plane and the 0 stress deviator component S on this plane depicted in Fig. 1(b) showing a section of the yield surface. It is usually convenient to

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Fig. 1. (a) The Gurson yield domain in the principal stress frame (S1, S2, S3). (b) A section of the yield domain on a plane to the hydrostatic axis (octahedral plane  perpendicular  2 S3 or p-plane) and representation of the Lode angle Q. (c) Lines of constant Lode angle and Lode parameter in the plane S ; [27]. The tension and compression meridians are S1 S1 shown in red and in blue respectively while the dashed line represents pure shear loadings. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 0

agedPiT ntroduce the coordinate systems e1 along S1 and e2 directly orthogonal to it in the plane such that rffiffiffi 2S
S
S3 2  S1 ¼ 1 pffiffiffi2 Seq cosQ; and ¼ 3 6 ð6Þ rffiffiffi S
S 2  S2 ¼ 3pffiffiffi 2 ¼ Seq sinQ 3 2 As underlined earlier, the value for the Lode angle Q is thus varied only between 0 and p3 .The lines of constant Lode angle (or Lode parameter) in the principal stress space are depicted in Fig. 1(c) borrowed from Nashon and Hutchinson [27].The two meridian planes corresponding to the two extreme values of the Lode angle define all axisymmetric loadings (lines AB and AD of Fig. 1(c) and are called the tensile and compressive meridians, respectively. The tensile meridian is so named because uniaxial tension is one of the load cases which corresponds to a Lode angle of 0. The compressive meridian is given that name because uniaxial compression corresponds to a Lode angle of p3 . The meridian Q ¼ p6 , line AC in Fig. 1(c) represent a combination of a shear stress and a hydrostatic stress. TagedPFor the most general isotropic behaviour, the yield locus has a three-fold symmetry. Note however that the yield surface can have a six-fold symmetry in the case when it is symmetric with respect to the origin. Finally, the Lode angle Q allows to write the stress deviator (in its principal frame) as 0 1 cosQ 0  0  2p B C B 0 C 0 cos Q
2 0 C 3 ð7Þ S ¼ Seq B B C   3 @ 2p A 0 0 cos Q þ 3 and therefore the principal deviatoric stresses as 0

2 3

0

2 3



S1 ¼ Seq cosQ; S2 ¼ Seq cos Q


 2p ; 3

0



2 3

S3 ¼ Seq cos Q þ

 2p 3

ð8Þ We will also need the invariants of the microscopic and macroscopic strain rate tensor e_ and E_ tensors defined similarly as those of the macroscopic stress (3) and (4) by ! rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 4det e_ _ m ¼ _ ii ; _ eq ¼ e_ : e_ and z ¼ arccos ð9Þ 3 3 3 _ 3eq and 1 _ E_ m ¼ trE; 3

E_ eq ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 _0 _0 E :E 3

0 1 _0 1 4det E A and h ¼ arccos@ 3 3 E_ eq

TagedPHere again, the respective Lode angles z and h of the microscopic  and macroscopic strain rates lie in the range 0; p3 . The macroscopic strain rate deviator takes the form in its principal frame 0 1 cosh 0 0   B C 2p B 0 C 0 cos h
0 B C _ 3 E_ ¼ E_ eq B ð11Þ C ¼ Eeq e0   B C @ 2p A 0 0 cos h þ 3 3. Constitutive relations for the matrix TagedPIn all the paper, the constitutive behaviour of the matrix of the porous solid is considered as incompressible, isotropic and rigidplastic. The following section makes more precise this constitutive behaviour. The 0 and 00 denote respectively the first and second derivatives of a scalar function. 3.1. Yield function and flow rule TagedPThe yielding of the matrix is described by a yield function f positive and homogeneous of degree one in the stress in the form f ðs Þ ¼ s eq gðvÞ

ð12Þ

where v is the Lode angle of the microscopic stress tensor defined in (3). The yield domain is therefore defined by the relation

s eq gðvÞ
s 0  0

ð13Þ

Function g(v), which describes possible effects of the third invariant of stress on yielding can be normalized in a number of ways (for instance by gð0Þ ¼ 1 in which case s 0 is the yield limit in uniaxial tension). An important requirement for the yield surface is its convexity. Relation (13) represents a cylindrical surface the axis of which is the hydrostatic axis in the principal stress frame. Therefore its convexity is governed by the convexity of its section in the deviatoric plane. In a polar reference system in this deviatoric plane, the equation of the yield surface is s eq ðvÞ ¼ gðsv0 Þ ¼ rðvÞ. The convexity condition is therefore r2 þ 2ðr0 Þ2
rr00  0 and consequently g must satisfy the following condition gðvÞ þ g00 ðvÞ  0

ð14Þ

The yield surface is also considered smooth in all the paper. In this case, using normality as the flow rule, the strain rate is obtained by ð10Þ

e_ ¼ l

@f @s

ð15Þ

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TagedPwhere

zTagedP ¼ zðe_ Þ is dependent only on the strain rate e_ . It is obtained by

@f 3 s @v ¼ gðvÞ þ s eq g0 ðvÞ s eq @s @s 2

ð16Þ

and the gradient of the Lode angle with respect to the stress is given by ! " # @v 9 s:s 2 1 s 1
cos3 ¼

v ð17Þ 2s eq sin3v 3 @s s eq s 2eq 9 p We note that the gradient @@v s is always singular at v ¼ 0 and v ¼ 3 (axisymmetric states of stress) so that the gradient of the yield function with respect to the stress is so unless p ¼0 ð18Þ g0 ð0Þ ¼ g0 3

conditions that we will assume throughout the paper. When these conditions are not met, the yield surface has a vertex (or an edge) at these two locations. The analysis in the whole paper will be limited to smooth yield surfaces for the matrix. To this end, we assume beside (18), that the function   g is itself continuously differentiable in the whole range 0; p3 . Two situations arise in these conditions. Due to isotropy of the matrix behaviour, the yield surface has either a six or three fold symmetries as already mentioned above.  In the latter case, function g is either increasing or decreasing in 0; p3 and is stationary at the ends of this interval as indicated by (18). In the former, besides stationarity of g at these ends, g i is also stationary at another location in the interval, say at   an intermediate value k 2 0; p3 where g0 ðkÞ ¼ 0. Consequently g is either first increasing in [0, k] and decreasing on [0, k] or viceversa. Two examples are given later to illustrate these possibilities. TagedPThe effects of the Lode angle on yielding is two fold: on one hand it reduces or increases the effective stress for yielding through the function g (see (13) and in the other hand, it brings a new component of the strain rate that is not directed by the stress as can be seen from (16). We note that this last component is always deviatoric and orthogonal to the stress deviator in the deviatoric plane. 3.2. The dissipation function pðe_ Þ T he maximum dissipation function that we denote p is important agedPT in the derivation of the effective behaviour as we will seen in the next section. For a complete view of what follows, the interested reader can refer to Salen¸c on [30]. The maximum dissipation is defined by

zðvÞ ¼ v þ arctan

g0 ðvÞ gðvÞ

ð23Þ

and the function Z(z) is obtained from z(v) by a symmetry with respect to the bissector in the plane (z, v). By the fact that Z(z) is the inverse function of z(v), we also have   g0 ZðzÞ 0 GðzÞ ð24Þ G ðz Þ ¼
gðZðzÞÞ Let us remark that when s  is axisymmetric (v ¼ 0 or v ¼ p=3), so is e_ . Therefore, by construction, the function Z(z) satisfies Zð0Þ ¼ 0 and Zðp=3Þ ¼ p=3

ð25Þ

The function G(z) defined in (22) plays the same role as g(v) for the yield function and satisfies, using relations (18) and (24) G0 ð0Þ ¼ G0 ðp=3Þ ¼ 0

ð26Þ

3.3. Two examples TagedPWe give here two examples of currently used yield functions illustrating the properties described in the above section and that will be used in our coming numerical simulations agedPT The Hershey
Dalgreen
Hosford yield surface [15,16], currently used for a better description of yielding and forming of aluminium alloys [17]. This corresponds with m  1 and with s 1  s 2  s 3 denoting the ordered microscopic principal stresses, to the yield function

m1 1 fðs 1
s 2 Þm þ ðs 2
s 3 Þm þ ðs 1
s 3 Þm g ¼ s eq gðvÞ f ðs Þ ¼ 2 ð27Þ which can be written in the form (12) with the choice

      2 1 2p m 2p cosv
cos v
þ cos v
gðvÞ ¼ 3 2 3 3  m   m m1 2p 2p
cos v þ þ cosv
cos v þ 3 3

ð28Þ

ð22Þ

Functions g(v) given by (28) are depicted in Fig. 2(a) and their corresponding yield surfaces are displayed in Fig. 2(b) for m ¼ 2; m ¼ 8; m ¼ 20 and m ¼ 800. The dots in Fig. 2(a) represent the Tresca criterion corresponding to m ¼ 1 and formula (29) and are seen to almost coincide with the curve associated to the high value m ¼ 800 as expected from the fact that the Tresca criterion is also obtained in the limit m ! 1. For the same values of m, Fig. 3 shows the corresponding z(v) functions given in (23) (continuous lines) and their inverses Z(z) (dashed lines) and relating the microscopic stress and strain rate Lode angles v and z while in Fig. 4, we plot function G(z) defining the maximum dissipation in (22). It is easily checked now that as required, conditions (18) are indeed satisfied for all finite m 6¼ 1. Moreover m ¼ 2 and m ¼ 4 correspond to the von Mises criterion while the Tresca yield function is obtained in the limiting casesm !  1 or m ! 1. For the Tresca criterion, one gets in the range 0; p3     2 2p 2 p cosv
cos v þ ¼ pffiffiffi cos v
ð29Þ gðvÞ ¼ 3 3 6 3

where Z(z) is the inverse function of z(v). z(v) here is the Lode angle of the microscopic strain rate e_ associated by the flow rule to a microscopic stress with Lode angle v and it is understood that

so that g0 ð0Þ ¼ p1ffiffi3 and g0 p3 ¼
p1ffiffi3 and therefore violating conditions (18) as expected. In this case, one also founds that for Tresca criterion, z(v) and Z(z) are step functions while G(z) converges (in a suitable sense) when m ! 1 to the function

pðe_ Þ ¼ sup s : e_ s 2C

ð19Þ

where C ¼ fs =f ðs Þ  0g and is obtained in closed form in [24] for bounded C. The details are not repeated here and due to convexity of the yield function f , the function pðe_ Þ is obtained for every e_ p ðe_ Þ if Trðe_ Þ ¼ 0 pðe_ Þ ¼ g ð20Þ þ1 if Trðe_ Þ 6¼ 0

pg ðe_ Þ is obtained by seeking all stress states s  lying on the yield surface, i.e. f ðs Þ ¼ s 0 and satisfying e_ ¼ l

@f  ðs Þ @s

ð21Þ

for some positive scalar λ. We found then

_ eq _ pg ðe_ Þ ¼ s 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     2 ¼ s 0 eq GðzÞ 2 0 g ZðzÞ þ g ZðzÞ

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Fig. 2. The Hershey
Dalgreen
Hosford yield domain. (a) Function g(v) and (b) Representation in the deviatoric plane for different values of the parameter m. The dots in (a) correspond to m ¼ 1; m ! 1 and to Eq. (29).

Fig. 3. The Hershey
Dalgreen
Hosford yield domain. Associated z(v) (continuous lines) and Z(z) (dashed lines). The dashed lines are symmetric of the continuous lines with respect to the bissector of the plane. Observe also that the Tresca criterion corresponds in the limit to step functions in this plane.

8 p > < cosz if z  ; 6 G 1 ðzÞ ¼   p > : cos z
p if z  : 3 6

ð30Þ

TagedP The Willam and Warnkle [29] yield surface is given by the function gðvÞ ¼



2 4ð1
b Þcos2 p3
v þ ð2b
1Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi



2 2 2ð1
b Þcos p3
v þ ðð2b
1Þ 4ð1
b Þcos2 p3
v þ bð5b
4Þ

ð31Þ This gives in a continuous way in the deviatoric plane yield surfaces going from von Mises (b ¼ 1) to a triangular shape for b ¼ 0:5. Functions g(v) given by (31) are depicted in Fig. 5(a) and their corresponding yield surfaces are displayed in Fig. 5(b) for b ¼ 1; b ¼ 0:8; b ¼ 0:6 and b ¼ 0:500. For the same values of b, Fig. 6 shows the corresponding z function given in (23) (continuous lines) and its inverse Z (dashed lines) and relating the microscopic stress and strain rate

LTagedP ode angles v and z while in Fig. 6, we plot function G(z) defining the maximum dissipation in (22). As shown in [29], the elliptical dependence provides convex cross-sections   p for

all 0values of 1 b 2 12 ; 1 . For b ¼ ; from (31), gð v Þ ¼ 2cos
v and g ðp=3Þ ¼ 0 2 3 pffiffiffi but g0 ð0Þ ¼ 3 6¼ 0. Here again, conditions (18) are satisfied except when b ¼ 12. Further, In this case also, one founds that for b ¼ 12 ; z(v) and Z(z) are step functions. 4. Effective behaviour for voided materials with matrix yielding dependent on lode angle TagedPLet us consider a representative volume element V of a porous ductile material containing voids occupying the domain v and the rest filled by the matrix. The constitutive behaviour of this matrix is assumed to be rigid-plastic with a microscopic yield domain defined in Section 3. In his work, Gurson [4] proposed an approximate yield criterion for voided materials using a limit analysis approach of a hollow sphere cell. More specifically, he used a simple

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Fig. 4. The Hershey
Dalgreen
Hosford yield domain. Associated function G(z). Observe here that function G has a limit when m ! 1 given by Eq. (30).

Fig. 5. The Willam
Warnke yield domain. (a) Function g(v) and (b) Representation in the deviatoric plane.

TagedPincompresssible rigid-plastic constitutive behaviour for the matrix satisfying the von Mises criterion. To obtain the approximate yield criterion, Gurson used in the upper bound theorem of limit analysis _ satisfying compatibility and boundary a particular trial field e_ ðEÞ conditions corresponding to prescribed macroscopic rates of deformation E_ at the boundary of the hollow sphere. By bounding the macroscopic dissipation from above, Gurson was able to derive upper bounds to the macroscopic stresses required to sustain plastic flow and these upper bound macroscopic stresses for the considered cell geometry and a for a range of macroscopic deformation rates allow to construct an upper bound yield locus for the porous material. These stresses are defined by @P S¼ _ ð32Þ @E _ is the upper bound (to the exact macroscopic dissipawhere PðEÞ

tion) obtained by the trial kinematically admissible velocity field Z   _ ¼ 1 p e_ ðEÞ _ dV PðEÞ ð33Þ V V

and pðe_ Þ is the microscopic dissipation defined in Section 3. The same approach with the same trial velocity field is followed here but with the matrix behaviour governed by the constitutive equations described in Section 3. TagedPWith the representative cell of Gurson [4], i.e a hollow sphere cell with external radius b and a void with radius a, denoting the

3

TagedP orosity by f ¼ ab3 ; one uses a spherical coordinate spherical system p 3 (r, u , ’) and the variable change l ¼ br3 ; with dV ¼ sin’dud’; so that any integral over the volume of the cell V is obtained by Z Z 1Z f 1 1 dldV ð:ÞdV ¼ ð:Þ ð34Þ V V 4p 1 V l2 4.1. Stress invariants and some general consequences _ (33) depends on the TagedPWhen the macroscopic dissipation PðEÞ 0 _ we three invariants E_ m ; E_ eq ; det E_ of the macroscopic strain rate E; have shown (see [24]) that relation (32) leads to

Sm ¼

1 @P 3 @E_ m

ð35Þ

1 20 3 _ E_ 1 E_ 5 2 @P E_ 0 4 @P 4@E: 1 A S ¼ _ _
_
1
cos3h 2 3 @Eeq Eeq 3Eeq sin3h @h 2 2 E_ eq E_ 0

ð36Þ

eq

We note here that (36) leads to a singularity for h ¼ 0 and h ¼ p3 @P unless @@P h jh¼0 ¼ @h jh¼p3 ¼ 0. This is actually the case with the assumptions adopted here, the microscopic dissipation pðe_ Þ being non singular, the trial field being linear in terms of the macroscopic strain rate so that the stress is continuous and the macroscopic dissipation is a smooth function at h ¼ 0 and h ¼ p3 . This gives in passing h ¼ 0 ! Q ¼ 0 and h ¼ p3 ! Q ¼ p3 .

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7

Fig. 6. The Willam
Warnke yield domain. Associated z(v) and Z(z). The dashed lines are symmetric of the continuous lines with respect to the bissector of the plane. Observe also that for b ! 12 step functions are obtained.

TagedPThe second of these relations gives rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 0 Seq ¼ S : S ¼ U2 þ V 2 2



 2  2 0 U U
3 V 2 cos3h
V V 2
3 U 2 sin3h det S ¼ 27

TagedPwhere E_ 0 and E_ m are the deviatoric and volumetric components of ð37Þ ð38Þ

where we have set @P U¼ @E_ eq

the macroscopic strain rate while er is the unit vector in the radial 3 direction, 1 the second order unit tensor and l ¼ br3 . For this field, we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð44Þ _ eq ¼ E_ eq 1
4mlH þ 4H2 l2 and its associated Lode angle z reads (using relation (9))

1 @P and V ¼ E_ eq @h

ð39Þ

TagedPAlternatively, (37) and (38) allow to calculate the Lode angle Q of the macroscopic stress S as



U U 2
3 V 2 cos3h
V V 2
3 U 2 sin3h cos3Q ¼ 3=2 ðU 2 þ V 2 Þ ¼ cos3ðh þ ξ Þ

ð40Þ

using the angle ξ defined by

1 @P V V 2
3 U2 V E_ @h tan3ξ ¼ or tanξ ¼ ¼ eq@P 2 2 U U ðU
3 V Þ _

ð41Þ

4det e_ cos3h þ 6ð1
2dÞlH þ 12ml H2
8l H3 ¼ 2 3 _ 3eq ð1
4mlH þ 4H2 l Þ2 2

cos3z ¼

ð45Þ

where we have introduced the ratio H of the volumetric to the effective macroscopic strain rates (strain rate triaxiality) H¼

E_ m E_ eq

ð46Þ

and the parameters m and d are given by

 0 eT ¢ E_ ¢ er E_ rr 1 pffiffiffi 1 2
3 sinhcos2u sin f þ coshð3cos2f þ 1Þ m¼ r _ ¼ ¼ _ 2 Eeq Eeq 2 ð47Þ

@Eeq

leading to

3

0

U ð42Þ Q ¼ h þ ξ ¼ h þ arctan V ξ is then the angle between the macroscopic stress deviator S and the macroscopic strain rate deviator E0 in the deviatoric plane (The same plane is used to represent both quantities). 4.2. The trial field and the macroscopic dissipation TagedPThe strain rate associated to the trial velocity field used by Rice and Tracey [3] and Gurson [4] is given by 0 e_ ¼ E_ þ lE_ m ð1
3 er  er Þ ð43Þ

eTr ¢ ðE_ Þ2 ¢ er E_ eq    1  pffiffiffi 2
2 3 sinð2hÞcosð2uÞ sin ðfÞ þ cosð2hÞ 3cosð2fÞ þ 1 þ 4 ¼ 8 ð48Þ

d¼

4.3. Parametric representation of the yield criterion TagedPSubstituting the expression of the microscopic dissipation

_ (33), and p given by (22) in the macroscopic dissipation PðEÞ after some algebraic manipulations, relations (35) and (39) give

Z 1Z GðzÞ½2lH
m dldV f qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 V 1
4mlH þ 4H2 l2 l 2  3    0 2 2 2 2 2 2 2 Z 1Z 1 1 g ZðzÞ GðzÞ 6d
16H l þ 4Hlm þ 2 þ 4H l m þ 12H l þ cosð3hÞð2Hl
mÞ
6Hlm
17 dldV f 4 5  2 6p 1 V sin3z gðZðzÞÞ l 2 4H2 l
4Hlm þ 1

Sm 1 ¼ s 0 6p

ð49Þ

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Z 1Z GðzÞ½1
2mlH dldV f qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 1 V 1
4mlH þ 4H2 l2 l 2        3 3

0 2 2 2 Z 1Z s0 f 1 g Zðz Þ Gðz Þ 6H 2Hl
4H l ðm
1Þm
2Hl m þ 3 þ 2m þ m þ 2d 2Hlð4Hl
mÞ
1 þ cosð3hÞðm
2HlÞ þ 1 7 dldV 4 5  2 4p 1 V sin3z gðZðzÞÞ l2 2pl 4HlðHl
mÞ þ 1 U

1 ¼ s 0 4p

ð50Þ

 @m Z 1Z GðzÞ
V s0H f dldV @h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ s 0 2p 1 V l 2 1
4mlH þ 4H2 l  2   3   @d @m  2 2 0 Z 1Z cosð3 4H l
4H lm þ 1 sinð3 h Þ þ 4H l l h Þ þ 4H l ð
3 d þ H lm þ 2Þ
2H 7 dldV s0 f 1 g ZðzÞ GðzÞ 6 @h @h 6 7  2 5 4p 1 V sin3z gðZðzÞÞ 4 2 l2 4H2 l
4HlmðhÞ þ 1

ð51Þ

TagedPRelations (49), (50) and (51) when combined to (37) and (40) give a parametric representation of the approximate yield domain of the porous solid in terms of the two parameters H and h describing the _ Elimination of these two parameters was macroscopic strain rate E. not possible and the yield function G(Seq, Sm, Q, f) is thus defined by this parametric representation 8 > < Sm ¼ s 0 Sm ðH; h; f Þ GðSeq ; Sm ; Q; f Þ ¼ 0 , ð52Þ Seq ¼ s 0 Seq ðH; h; f Þ > : Q ¼ Sd ðH; h; f Þ

TagedP5.1.2. Pure hydrostatic loadings TagedPWhen H ! þ 1 ; i.e. for purely hydrostatic macroscopic strain rates, the macroscopic stress is also purely hydrostatic and one gets

In the absence of Lode angle effects in the yield behaviour of the matrix and after integration with respect to λ, the results obtained in [24] are recovered.

One also obtains

5. Properties and shape of the effective yield surface of the voided material TagedPRelations (49), (50) and (51) allow to obtain numerically the seeked approximate effective yield domain for arbitrary matrix yielding function given by (12) and satisfying the conditions required in Section 3. Though implicite, these relations also permit to obtain some closed-form results. These are summarized below in particular for hydrostatic and shear loadings before giving some numerical results in Section 5.2. In Section 6, we tempt a semi-closed form representation of the approximate effective yield surface. 5.1. Closed-form results results T he details of the results are given in [25]. We only sketch the agedPT results here.

Sm
23 ln f ¼ s0 g p3

A similar result is obtained when H !
1 . In this case, one finds

Sm 23 ln f ¼ s 0 gð0Þ

ð57Þ

V !0

ð58Þ

so that Seq ¼ 0. The same result is obtained when H !
1 . TagedPIn both cases, the result slightly differs from the result given by Gurson [4] by the term appearing in the denominator and due to the dependence of the yield surface of the matrix on the third invariant of stress. Observe here that for high positive stress triaxialities (H ! 1), the macroscopic yield stress is set by the microscopic yield stress in compression while for high negative stress triaxialities (H !
1 ), the macroscopic yield stress is set by the microscopic yield stress in tension. This result is a Lode angle effect and can be directly recovered by solving directly the problem of a hollow sphere constituted of a matrix with yielding behaviour as used here and subjected to a uniform macroscopic pressure Sm (see [25]). The problem for a hollow sphere can also be analysed for a more general constitutive relation for the matrix. For instance , for a matrix with yielding governed by f ðs Þ ¼ s eq gðvÞ þ 3as m  k

Sm 1
f g ¼ s0 3a

Q ¼ ZðhÞ

g¼

Seq gðQÞ ¼ ð1
f Þs 0

ð54Þ

which is up to the size reduction factor 1
f exactly the equation of the yield surface of the matrix (see (13)).

ð59Þ

one finds (see [25])

with

and that the equation of the yield contour in the p-plane is given by

ð56Þ

U!0

TagedP5.1.1. Shear loadings TagedPWhen H ! 0, corresponding to pure shear results with Sm ¼ 0; it is found that ð53Þ

ð55Þ

2a  2a þ sgnðSm Þgð13 arccos½
sgnðSm Þ

ð60Þ

ð61Þ

2a while for negative mean For positive mean stresses g ¼ 2aþg ðp3 Þ 2a and this gives the macroscopic yield stresses stresses g ¼ 2a
gð0Þ for all materials with this type of yielding. Further, a limit process allows to obtain these macroscopic yield stresses for the nonsmooth

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TagedPcases such as the Tresca criterion, the Mohr-Coulomb criterion but also for a fully triangular yield shape.

5.2. Numerical simulations TagedPEffective yield domains are obtained here numerically for both examples given in Section 3. The integrals in (49), (50) and (51) were computed numerically using a simple trapezoidal quadrature rule with a fine discretization. The range of the angle f was subdivised into 50 intervals while those of the angle u and λ were subdivised into 30 intervals each. Fig. 7 shows general views of the full yield domains obtained for a porosity f ¼ 0:01 obtained for three different yielding criteria for the matrix: von Mises on the left, Hershey
Dalgreen
Hosford yield criterion with m ¼ 20 in the middle and finally the Willam
Warnke yield criterion for the matrix with b ¼ 0:6 on the right. We emphasize that for the von Mises yield criterion, the full dissipation was considered which leads to a slightly different result from the Gurson criterion. While this figure does not clearly show all the differences between the three effective yield criteria, it does show one particular such difference, the size and the non symmetry of the yield domain along the hydrostatic axis in the case of the Willam
Warnke case. Indeed, as expected from the closed form results (55) and (56), the effective yield stresses for pure hydrostatic loadings are different in the negative and positive sides. For a better general view, the same plot is limited in Fig. 8 only to the range S3  0. This allows in particular to see the contours of cross sections along the plane S3 ¼ 0 shown in back continuous lines. Better views for the Hershey
Dalgreen
Hosford yield criterion with m ¼ 20 (left) and for the Willam
Warnke yield criterion for the matrix with b ¼ 0:6 (right) are shown in Fig. 9 in the frame   (S1 ; S2 ; Sm ) (see (6)).

9

TagedPMore detailed results are discussed now for the Hershey
Dalgreen
Hosford yield criterion with m ¼ 20 and for the Willam
Warnke yield criterion for the matrix with b ¼ 0:6. They represent sections of the effective yield domains in octahedral planes corresponding to different macroscopic mean stresses. TagedPFig. 10 displays the results for Hershey
Dalgreen
Hosford yield   criterion with m ¼ 20 in the deviatoric planes in the frame (S1 ; S2 ). The figure presents different contours of the effective yield criterion of the voided material obtained for different mean stresses Sm/s 0. The largest contour is associated to a zero mean stress (p-plane) and its shape is exactly that of the matrix yield surface as indicated by the closed-form result (54) while its size is indeed reduced by the factor 1
f according to the same result. As the mean stress Sm is increased in the positive direction, the shape of the effective yield domain is seen to get more and more rounded up but still keeping the same shape and the six-fold symmetry of the matrix yield surface. For larger magnitude of the mean stress or traixialities, the effective yield domain is seen to change progressively its shape to a rounded triangular shape with a three-fold symmetry. Increasing further the triaxiality, the effective yield domain is observed to get more and more rounded up to an ”almost” circular shape. The same scenario is observed when starting from zero triaxiality and decreasing the mean stress. However, the contours obtained for negative and positive mean stresses with the same absolute value differ from each other but are symmetric with respect to the origin. This symmetry can be explained by the symmetries of the matrix yield surface. TagedPFig. 11displays exactly the same results for the Willam
Warnke yield surface for the matrix with b ¼ 0:60. Here again, the sharp triangular shape that we observe for a zero mean stress and complying with (54) is rounded up more and more with increasing triaxialities in the positive side and with decreasing mean stresses in the

Fig. 7. General views of the simulated effective yield doamin for a porosity f ¼ 0:01 and for von Mises yield criterion (left), Hershey
Dalgreen
Hosford yield criterion with m ¼ 20 (middle) and for the Willam
Warnke yield criterion for the matrix with b ¼ 0:6 (right).

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Fig. 8. (a) The yield domain in the principal stress frame. (b) A section of the yield domain on a plane perpendicular to the hydrostatic axis (octahedral plane or p-plane) and representation of the Lode angle Q.

Fig. 9. General views of the simulated effective yield doamin for a porosity f ¼ 0:01. (a) Hershey
Dalgreen
Hosford yield criterion for the matrix with m ¼ 20. (b) Willam
Warnke yield criterion for the matrix with b ¼ 0:6.

TagedPnegative side. The contours maintain their triangular shape and symmetries untill very high (positive or negative) triaxialities where they are seen to take an “almost” circular shape. In contrast to Fig. 10, there is no symmetry with respect to the Sm ¼ 0 plane and the yielding of the voided material for positive and negative

tTagedP riaxialities is somewhat different. Not only the effective yield stresses for hydrostatic stresses are different in the positive and negative sides and set by the yield stresses of the matrix at Lode angles 0 and p3 ; the shapes of the domain are also different for negative and positive triaxialities.

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11

Fig. 10. Effective yield domain for the Hershey
Dahlgren
Hosford yield criterion with m D 20. Cross sections of the yield domain by planes perpendicular to the hydrostatic axis   (octahedral plane) plotted in the frame (S1 ; S2 ) as defined by relations (6).

Fig. 11. Effective yield domain for the Willam
Warnke yield criterion with b ¼ 0:60. Cross sections of the yield domain by planes perpendicular to the hydrostatic axis (octahe  dral plane plotted in the frame (S1 ; S2 ) as defined by relations (6).

TagedPThe effective yielding of the voided material inherits somewhat along the hydrostatic axis the characteristics of the yielding behaviour of the matrix in the deviatoric plane and leads to strong couplings between hydrostatic and Lode angle effects. In this setting, it would be interesting to analyse these couplings for a matrix

tTagedP he yielding of which is dependent also on the mean stress of the microscopic stress. This is out of the scope of this paper. TagedPFig. 12 for the Hershey
Dalgreen
Hosford yield criterion with m ¼ 20 and Fig. 13 for the Willam
Warnke yield criterion for the matrix with b ¼ 0:6 show the shapes of the effective yielding

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Fig. 12. Macroscopic effective stress versus macroscopic mean stress for different meridians for of the effective yield domain and a porosity f ¼ 0:01. Hershey
Dalgreen
Hosford yield criterion for the matrix with m ¼ 20.

Fig. 13. Macroscopic effective stress versus macroscopic mean stress for different meridians for of the effective yield domain and a porosity f ¼ 0:01. Willam-Warnke yield criterion for the matrix with b ¼ 0:6.

TagedPdomains along the hydrostatic axis. These are presented for different meridians (constant Lode angle Q) of the yield domain. They are given for the tension meridian (Q ¼ 0) corresponding to the line AB in Fig. 1, the compression meridian (Q ¼ p3 Þ) corresponding to the line AD in Fig. 1 and the meridian (Q ¼ p3 ) corresponding to the line AC in Fig. 1 representing states of stress of pure shear plus hydrostatic stress. An extra meridian (Q ¼ p6 ) is shown for the Willam
Warnke case.

6. Equation of the effective yield surface of voided materials with matrix yielding behaviour dependent on the second and third stress invariants TagedPThe former parametric representation provides the exact prediction of the yield criterion with the Gurson-Rice trial field. It has allowed to obtain closed-form results for the particular situations of pure hydrostatic loadings and shear loadings and can be used for

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TagedPnumerical simulations with arbitrary yield surfaces in the form (13). For engineering purposes, one however needs simpler expressions for the effective yield domain. This is the purpose of this this section where we explore the possibility of obtaining a semi-explicit equation for the effective domain. TagedPThe idea of deriving this expression is to relate our formulation to the Gurson one [4]. We recall here the macroscopic dissipation P used here and the one P G used by Gurson and given respectively by

P ¼ PðE_ eq ; H; h; f Þ ¼

s 0 E_ eq 4p

Z 1 Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f dl 2 1
4mlH þ 4H2 l GðzÞ 2 dV

l

V

ð62Þ

TagedP



S ¼

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rf 1 G G 2 1 þ 4H2 l dl ¼ Seq þ 3HSm 1 2

l

G

13

ð74Þ

G

agedPIT ndex G in Sm and Seq in formulae (69) and (70) stands for Gurson to emphasize that these were the expressions obtained in his model for the hydrostatic and equivalent stresses respectively. Of utmost importance is the relation between these two quantities, namely the Gurson yield criterion, i.e. ! G G ðSeq Þ2 3Sm þ 2f cosh ð75Þ ¼ 1 þ f2 2s 0 s 20

where we emphasize that the microscopic Lode angle z(h, H, λ, u, f) appearing in (62) is actually space dependent but also on h and H and is constant in the following three situations

To obtain this result, Gurson [4] eliminated H from (72) and (73). Sm is directly related TagedPNow the macroscopic stress triaxiality T ¼ S eq E_ m to the strain rate triaxiality H ¼ E_ and the Lode angle h of the maceq roscopic strain rate. This relation is obtained from (69) and (70) as   @R @R T G ðHÞ þ 13 @H R þ H @H  ð76Þ T¼ cosξ ¼ TðH; h; f Þ 2 @R G R
H @RÞ @H
3H @H T ðHÞÞ

p zðh; 1 ; l; u; fÞ ¼ ; zðh; 0; l; u; fÞ ¼ h and zðh;
1 ; l; u; fÞ

where TG(H) is the tress triaxiality in the Gurson model given by

1

Z 1 Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f s E_ 2 dl 1 þ 4H2 l 2 dV PG ¼ PG ðE_ eq ; H; f Þ ¼ 0 eq 4p 1 V l

ð63Þ

3

¼0

ð64Þ

and we define the ratio of these two dissipations by RðH; h; f Þ ¼

P PG

ð65Þ

We note, using (64), that Rðh; H; f Þ satisfies p ; Rðh; 0Þ ¼ GðhÞ and Rðh;
1 Þ ¼ Gð0Þ Rðh; 1 Þ ¼ G 3

ð66Þ

Other important properties of Rðh; 1 Þ are that lim

@R

H ! 0 @H

¼0

ð67Þ

and @R @R ¼ lim ¼0 lim H ! 1 @H H !
1 @H

Sm S 1 @R  ðh; HÞS ¼ Rðh; HÞ þ s0 s 0 3 @H G eq

ð68Þ

ð69Þ

S Seq 1 @P @R  ðh; HÞS ¼ ¼ Rðh; HÞ
H cosξ @H s 0 @E_ eq s0 s0

ð70Þ

V¼

Seq 1 @P @R  ¼ ðh; HÞS ¼ sinξ s0 E_ eq @h @h

ð71Þ

where

S ¼ s0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! Z 1 2 f 2Hdl 2 2H þ f 2 þ 4H2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 3 2 f ð2H þ 1 þ 4H2 1 þ 4H2 l

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Seq R1 xdx ¼ f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 þ 4H2
f 2 þ 4H2 s0 x2 þ 4H2

reads after substitution of (70) and (71) in (42) 2 3 Rð1 þ 3T G ðHÞ 4 5 ¼ QðH; h; f Þ  Q ¼ h þ arctan  @R @R G R
H @H T ðHÞ
3H2 @H

Q

ð77Þ

The two relations (76) and (77) are invertible. Indeed, the associated Jacobian determinant is always nonzero when the dissipation P is strictly convexe. This is too long to be reported here and we assume here the strict convexity of the macroscopic dissipation. Through the implicit function theorem, one obtains H and h solely as functions of T , Q and f acting as a parameter in (76) and (77). and h ¼ hðT; Q; f Þ

ð78Þ

and allow after substitution to write the terms appearing in (69), (70) and (71) as functions of T, Q and f as Rðh; HÞ ¼ RðT; Q; f Þ

ð79Þ

1 @R  ðh; HÞS ¼ QðT; Q; f Þ 3 @H

ð80Þ

@R  ðh; HÞS ¼ PðT; Q; f Þ @H

ð81Þ

H

Relations (69) and (70) are thus rewritten in the form G

Sm Sm
Q ¼ Rs 0 s0

ð82Þ

G

U¼

G m

G

Sm G . In the other hand, the macroscopic stress Lode angle Seq

H ¼ HðT; Q; f Þ

The dissipation P is therefore written as RðH; hÞPG . Before proceeding, we make here an important remark. In writing the dissipation P in this last form, we have introduced no approximation (assuming Rð H; hÞ is computed exactly). Therefore the derivation that follows will still lead to an upper bound of the exact effective yield domain. Further, this upper bound is convex, this last point following from the Gurson analysis itself and the choice of a linear trial velocity field in terms of the macroscopic strain rate. TagedPWith the dissipation P written as RðH; hÞPG from relation (65), relations (35) and (39) give now G m

T G ðHÞ ¼

ð72Þ

G

ð73Þ

Seq cosξSeq þ P ¼ Rs 0 s0

ð83Þ

and upon final substitution of these two last expressions in (75), one obtains a representation of the yield domain as   ðaSeq þ PÞ2 Sm
Q þ 2f cosh ð84Þ ¼ 1 þ f2 2 Rs 0 R2 s 0 where we have omitted for clarity the dependence of P, Q and R on the stress triaxiality T, the Lode angle Q and the porosity f. TagedPThis is the general equation of the effective yield domain of a voided material the matrix of which has a yielding behaviour dependent on the second and third stress invariants in the form (13). It has formally the same global expression as the Gurson model but involves four different functions a, R, P and Q all dependent on the stress triaxiality T, the Lode angle Q and the porosity f. These

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Fig. 14. Hershey
Dahlgren
Hosford matrix yield criterion with m ¼ 20 and for a porosity f ¼ 0:01. Top figures: Tridimensional representations of the functions RðH; hÞ (left) and R(T, Q) (right). As expected, RðH; hÞ goes towards Gð0Þ ¼ Gðp=3Þ ¼ 1 when H ! §1 and takes the value G(h) shown in Fig. 4 for the same conditions when H ¼ 0. Bottom: Tridimensional representations of the functions a(T, Q) (left), P(T, Q) (centre) and Q(T, Q) (right).

TagedPfunctions are in general not available in closed-form but are however easily obtained numerically. They can be tabulated and only necessitate the knowledge of the function RðH; hÞ.

TagedPAs an illustration, these functions are determined here for the Hershey
Dahlgren
Hosford matrix yield criterion with m ¼ 20 and a porosity f ¼ 0:01 after calculating the function RðH; hÞ. This is

Fig. 15. Hershey
Dahlgren
Hosford matrix yield criterion with m ¼ 20 and for a porosity f ¼ 0:01. Contours of sections of the effective yield domain for different mean stresses   Sm plotted in the frame (S1 ; S2 ) as defined by relations (6). Thin continuous lines: parametric representation (49), (50) and (51). Thick dashed lines: Eq. (84).

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15

Fig. 16. Willam
Warnke yield criterion for the matrix with b ¼ 0:6 and for a porosity f ¼ 0:01. Contours of sections of the effective yield domain for different mean stresses Sm   plotted in the frame (S1 ; S2 ) as defined by relations (6). Thin continuous lines: parametric representation (49), (50) and (51). Thick dashed lines: Eq. (84).

TagedPobtained by computing by the simple trapezoidal quadrature rule the dissipations P and P G given in (62) and (63). TagedPFig. 14 shows the function RðH; hÞ in the top left picture while the top right one represents the function R(T, Q). The bottom pictures show from left to right the three functions a, P and Q. TagedPThis section is closed by a final comparison of the effective yield domain obtained by the parametric representation (49), (50) and (51) and the effective yield domain obtained directly with Eq. (84) after calculating numerically a, P and Q. This is given here for the Hershey
Dahlgren
Hosford matrix yield criterion with m ¼ 20 and the Willam
Warnke yield criterion for the matrix with b ¼ 0:6; a porosity f ¼ 0:01 for both criteria. The comparison is shown in Figs. 15 and 16. These figures show different contours of the effective yield criterion of the voided material in the deviatoric plane for different mean stresses Sm/s 0, positive and negative. The thin continuous lines correspond to the parametric representation and the thick dashed lines to Eq. (84) and one can see that the agreement is very good. 7. Conclusion TagedPWe have developed micromechanically based constitutive equations for isotropic porous materials when the yielding behaviour of the matrix is dependent on both the equivalent stress and the Lode angle. Using the Gurson approach with the same trial velocity field allowed to derive in this case an upper bound for the effective yield criterion of the porous material. This was obtained first in a parametric form that was used to obtain some closed-form results in the case of hydrostatiic and pure shear loadings. In the latter case, the shape of the yield criterion for zero mean stress is found to be exactly the one of the

TagedP atrix while its size is reduced by the factor1
f ; f being the m porosity. In the former, the results show that the effective yield stress for hydrostatic loadings is affected by Lode angle effects. For negative high stress triaxialities, it is set by the yield stress of s0 the matrix in the tension meridian gð0Þ ; while for positive large triaxialities, it is defined by the yield stresses of the matrix the compression meridian gsp0 . In a second step, we have provided a ð3Þ semi-explicite expression for the effective yield domain which has formally the same form as the Gurson model but involves four different functions dependent on the macroscopic stress triaxiality and Lode angle. These functions are involved in a similar way as Tvergaard parameters [28] but play another role of representing effects of Lode angle. The equation for the yield criterion is simple but the four functions are generally not available in closed-form. However they can obtained numerically and tabulated. One should point out here, that the approach used here can be used in a more general context of the matrix yielding behaviour if one is able to obtain a rigorous upper bound for the macroscopic dissipation. However other issues remain to be studied. TagedPWe have considered in this paper only smooth yield surfaces for the matrix. It is possible to include the singular case at the expense of more complex developments but many of the usual singular criteria can be analysed with our approach by a limiting process. We have seen here that the Tresca criterion can be studied in the framework of the Hershey
Dalgren
Hosford yield criterion by taking large values for the parameter m. The other limitation of the presentation here is the isotropy of the material. Its extension to anisotropic porous materials is surely worth for practical purposes. Finally, use of the developed constitutive equations for localization and fracture of ductile materials need to be undertaken. These issues are under investigation.

Please cite this article as: A. Benallal, Constitutive equations for porous solids with matrix behaviour dependent on the second and third stress invariants, International Journal of Impact Engineering (2017), http://dx.doi.org/10.1016/j.ijimpeng.2017.05.004

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Please cite this article as: A. Benallal, Constitutive equations for porous solids with matrix behaviour dependent on the second and third stress invariants, International Journal of Impact Engineering (2017), http://dx.doi.org/10.1016/j.ijimpeng.2017.05.004