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Approximate criteria for ductile porous materials having a Green type matrix: Application to double porous media
Computational Materials Science 62 (2012) 189–194
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Approximate criteria for ductile porous materials having a Green type matrix: Application to double porous media W.Q. Shen a, J.F. Shao a, L. Dormieux b, D. Kondo c,⇑ a
Laboratoire de Mécanique de Lille-UMR 8107 CNRS, USTL, 59655 Villeneuve d’Ascq, France Unité de Recherche Navier, UMR 8205 CNRS, 6/8 Avenue B. Pascal, 77455 Champs sur Marne, France c Institut D’Alembert, UMR 7190 CNRS, UPMC (Paris 6), 75005 Paris, France b
a r t i c l e
i n f o
Article history: Received 16 April 2012 Received in revised form 4 May 2012 Accepted 6 May 2012 Available online 19 June 2012 Keywords: Porous materials Ductile Macroscopic yield function Plastic compressibility Gurson-type models Double porous media Green materials
a b s t r a c t In the framework of limit analysis theory, we derive closed-form expressions of approximate criteria for ductile porous materials whose plastically compressible matrix obeys to an elliptic criterion. The general methodology is based on limit analysis of a hollow sphere subjected to a uniform strain rate boundary conditions. We first consider a porous medium with a Green type matrix and establish the corresponding macroscopic yield function. Then, the obtained results are used in order to investigate double porous materials whose solid phase at the microscale (the smallest scale) obeys a von Mises criterion. The results are assessed by comparing them with numerical data, and with recently published results. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Since the pioneering work of Gurson [9], limit analysis-based modeling of the ductile behavior of porous media have been the subject of intense research. Among existing contributions, mention can be done of works concerning with voids shape effects (see for instance [5,6,21,22]) or with matrix plastic anisotropy [15,1,14,18,11]. For a complete review, the reader may refer to the recent review by [2]. In contrast to the above cited works which all considered that the matrix is incompressible, few researches have been carried out for porous materials with plastic compressible matrix. In the context of limit analysis-based developments, Lee and Oung [13] took into account the plastic compressibility of the matrix by considering that its behavior is described by a Mises–Schleicher criterion, while studies by [10,8] or [26] consider that the matrix behavior obeys a Drucker–Prager criterion.1 The present study is devoted to the investigation of porous media whose solid matrix obeys to an elliptic criterion, expressed as a quadratic function of the first invariant of stress and of the von Mises equivalent stress. A typical example of such a criterion is
⇑ Corresponding author. E-mail address: [email protected] (D. Kondo). Note also the contribution by Maghous et al. [16] who implemented and extended the so-called modified secant moduli method (equivalent to the variational approach proposed by Ponte Castañeda [23]) for a Drucker–Prager matrix. 1
0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.05.021
that proposed by Green [7] (see also [12]) and more recently considered by [3]. It is interesting to note that elliptic criteria2 may also result by applying a nonlinear homogenization approach to porous materials having a Von Mises matrix (see [23] or [17]). The objective of the present study is to establish macroscopic criteria for ductile porous materials having a Green type matrix. The paper is organized as follows. In Section 2, we present and apply Hill–Mandel homogenization to derive the plastic potential for a porous material having a matrix obeying to an elliptic criterion in the form (1). A closed-form expression of the criterion is derived (see Section 2.2). The results are then assessed and discussed by comparing them with numerical data, and with results available in the literature. The proposed development is particularly interesting for some porous geomaterials or polymers. Moreover, for various cases of practical interest, including application to metals, the porosity can be found at two scales which will be denoted microscale and mesoscale in the following. For this class of double porous media for which two populations of voids exist at two well separated scales, Vincent et al. [27] recently established original results by performing two step homogenization. For the transition from microscale to mesoscale, these authors considered a von Mises solid phase and adopted a Gurson criterion. Their final result is given in the form of a macroscopic criterion which was obtained after several approximations. In this paper, we develop criteria for 2
The elliptic criterion has been used by [25] in the context of ductile porous media.
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porous media with 2 populations of porosities by adopting instead of Gurson criterion, the newly developed criterion presented in Section 2. 2. Approximate macroscopic criterion for a porous medium having a Green type matrix 2.1. General methodology of limit analysis theory applied to a porous medium with the Green matrix Let us first consider a hollow sphere whose internal and external radii are respectively denoted a and b; the porosity corresponding to this hollow sphere is then f = a3/b3. Uniform strain rate boundary conditions are applied on the outer surface of the hollow sphere: v = D.x for x = ber. The quantity D represents the applied uniform macroscopic strain rate. The matrix is isotropic and rigid perfect plastic. It is assumed that the matrix obeys an elliptic criterion of the form:
a
/ðrÞ ¼ br2eq þ ðtr rÞ2 r20 6 0 2
ð1Þ
where a and b are positive material constants (Green-type criterion); In Eq. (1) r denotes the microscopic stress tensor and r0 is related to the shear strength of the matrix. r2eq ¼ 32 r0 : r0 with r0 representing the deviatoric part of the stress tensor. Note that Eq. (1) can be also put in the following form:
/ðrÞ ¼
3 r : M : r r20 6 0 2
ð2Þ
with M ¼ aJ þ bK, in which J and K are the standard projectors of isotropic fourth-order symmetric tensors having both major and minor symmetries. These projectors are expressed as J ¼ 13 1 1, K ¼ I J where 1; I are the second-order and the symmetric fourth-order unit tensors, respectively. Components of I are Iijkl ¼ 12 ðdik djl þ dil djk Þ in which d denotes the well-known Kronecker’s symbol. The plastic compressibility of the matrix lies in the presence of the hydrostatic component of the stress in Eq. (2). Application of the normality rule gives the local strain rate:
_ d¼K
@/ _M:r ¼ 3K @r
ð3Þ
2 _ 2 r2 d : H : d ¼ 4K 0 3
ð4Þ
v K:A jXj
pðdÞdV ¼ inf
X x
r0
v K:A jXj
Z
~eq dV d
ð8Þ
X x
P(D) represents the macroscopic dissipation, X denotes the volume of the unit cell, jXj = 4pb3/3, where as x denotes the volume of the void, jxj = 4pa3/3. The infimum in (8) is taken over all kinematically admissible (K.A) velocity fields, v. As classically, the limit stress states at the macroscopic scale are shown to be of the form: R¼
@P @D
ð9Þ
2.2. The considered velocity field and the corresponding macroscopic yield function Following [27], the velocity field in the matrix is chosen in the form: 3
v ¼ Ax þ
b ðDm AÞ er þ D0 :x r2
ð10Þ
with Dm ¼ 13 trD and D0 the deviatoric part of the macroscopic strain rate D. The homogeneous field Ax is used to introduce a compressible component in the velocity field v; the two remaining terms (inspired from the Gurson velocity field) are kinematically admissible with (D A1). More precisely, the second term in (10) corresponds to the expansion of the cavity and the outer volume, while the third one describes the shape change of the cavity and of the outer volume without change of volume. Hence, for any value of A, the whole velocity field v complies with the uniform strain rate D applied to the hollow sphere. The strain rate in the solid matrix can be obtained from Eq. (10); in spherical coordinates, one obtains: 3
E
E
d ¼ A1 þ d ; d ¼ D0 þ
b ðDm AÞ ½1 3er er r3
ð11Þ
To compute the local dissipation in the solid matrix, one needs ~eq . To this end, substituting (11) in Eq. (6) provides: to calculate d
~2 ¼ 2 A2 ð1 : H : 1Þ þ 4 Að1 : H : dE Þ þ dE 2 ; with dE 2 ¼ 2 dE d eq eq eq 3 3 3
ð5Þ
where the scalar deq is the equivalent strain rate, and dm the volumetric strain rate, defined as:
deq ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d : K : d; 3
:H:d
ð12Þ 3
E
Taking into account the identities 1 : H : 1 ¼ a ; 1 : H : d ¼ 0, one obtains:
" #2 3 D2eq 4 b3 ðDm AÞ 0 2A2 4 b ðDm AÞ 2 ~ deq ¼ þ þ D : ½1 3er er þ r3 r3 a b b 3b ð13Þ
K_ can then be written as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ~eq d:H:d d 1 2dm deq 3 K_ ¼ ¼ þ ¼ 2r0 2r0 a b 2r 0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d : H : d; 3
Z
1
E
_ is the local plastic multiplier whose expression will be prowhere K vided later. Let H be the inverse of M; it follows that H ¼ M1 ¼ a1 J þ 1b K. The plastic multiplier is then derived from (3) as follows:
~eq ¼ d
R : D 6 PðDÞ ¼ inf
dm ¼
1 trd 3
ð6Þ
It follows that the local plastic dissipation p(d) in the matrix takes the form:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2dm deq ~eq pðdÞ ¼ r : d ¼ 3K_ r : M : r ¼ r0 þ ¼ r0 d a b
ð7Þ
For the uniform strain rate boundary conditions, considered in the present study, the following inequality holds for all macroscopic stress R and macroscopic strain rate D [20,4]:
Due to the presence of the scalar A (which remains unknown in the definition of the velocity field), the macroscopic dissipation, P(D), is computed owing to a minimization procedure with respect to A:
h
i
e ðD; AÞ ; with P e ðD; AÞ ¼ PðDÞ ¼ min P A
r0 jXj
Z
~eq dV d
ð14Þ
X x
Obviously, the determination of the macroscopic criterion ~eq . In order to obtain a closedrequires to compute the integral of d form expression, it is convenient to apply the following inequality ~2 [9]: to d eq
Z
Xx
~eq dV ¼ d
Z
X x
6 4p
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~2 ðr; h; uÞdV d eq
Z
b a
1=2 ~2 i r 2 hd dr eq SðrÞ
ð15Þ
~2 i where SðrÞ is the sphere of radius r and hd eq SðrÞ is the average of ~2 ðr; h; uÞ over all the orientations: d eq
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~2 i hd eq SðrÞ ¼
2A2
a
þ
" 3 #2 D2eq 4 b ðDm AÞ þ 3 b r b
ð16Þ
for which h1 3er er iSðrÞ ¼ 0 has been used. This eventually yields an upper bound of the macroscopic dissipation (see Eq. (14)) by computing:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #2 u 2 3 u2A D2eq r 4 b ðDm AÞ 0 t e ðD; AÞ ¼ 4pr 2 dr P þ þ 3 r X a a b b Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f du ¼ r0 M 2 þ N2 u2 2 u 1 " #1f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M2 þ N2 u2 uN ¼ r0 N arcsinh u M Z
b
e ðD; AÞ, then The condition (20) ensuring the minimum of P reads:
e 2A @ P e 2 @P pffiffiffi ¼ 0 @M aM @N b
ð25Þ
Combining (24) and (25) yields:
e @P 3aRm M ¼ 2A @M
ð26Þ
and then, with the help of (23):
ð17Þ
e @P ¼ @M
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 2 aR þ bR2eq 2 m
ð27Þ
1
2
D2
where M2 ¼ 2Aa þ beq , N2 ¼ 4b ðDm AÞ2 and the change of variable 3 u ¼ br3 has been introduced. e ðD; AÞ over the unAs mentioned before, one has to minimize P known variable A (see Eq. (14)) and to determine the macroscopic plastic criterion by taking advantage of the approximate exprese ðD; AÞ. In practice, rather than treating these two steps sion of P successively, the computation of the criterion may be done by addressing them simultaneously (cf. [19]). It comes:
R¼
e ðD; AÞ e ðD; AÞ @P @P ; with ¼0 @D @A
ð18Þ
To solve (18), it is convenient to make the following change of vare ðD; AÞ ¼ P e ðM; NÞ with M and N introduced before. The maciable: P roscopic stress tensor R reads then:
R¼
e e @M @ P e @N @P @P þ ¼ : : @D @M @D @N @D
ð19Þ
with the condition of minimization with respect to A (second relation in (18)):
e @M @ P e @N @P þ ¼0 : : @M @A @N @A
ð20Þ
Finally, from (21) we can get the expressions of RA and RB:
RA ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 9 2 3 b aRm þ bR2eq ; RB ¼ Rm 2 2
ð28Þ
Putting this result in (22) yields the searched closed-form expression of the following macroscopic criterion of the porous medium having a matrix obeying to the Green criterion (1):
b
R2eq
r20
þ
pffiffiffi 9a R2m 3 b Rm 1 f2 ¼ 0 þ 2f cosh 2 r0 2 r20
ð29Þ
Note that the presence of the term R2m constitutes the macroscopic counterpart of the local plastic compressibility and the main difference with the Gurson criterion. It is worth noticing that the particular case of a = 0 and b = 1 (von Mises solid matrix) readily yields the Gurson criterion of the porous material. 3. Application to porous materials with two populations of voids
Similarly to the approach used by Gurson, one can then establish the following relationships:
3.1. The problem statement
2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 e @P N2 N2 RA ¼ ¼ r0 4 1 þ 2 f 2 þ 2 5 @M M M e @P N N RB ¼ ¼ r0 arcsinhð Þ arcsinh fM M @N
We aim now at validating the macroscopic criterion given by Eq. (29) by applying it to a material containing two populations of spherical cavities at two different scales (see Fig. 1). The voids in the two populations are assumed to be spherical and isotropically distributed. Let us recall the notation jXj which denotes the volume of the representative elementary volume. Following [28], we also denote by xb the domain occupied by the cavities at the smallest scale (microscale) and xe the one occupied by the voids at the upper scale (mesoscale). With these notations, the porosities
ð21Þ
which define the parametric form of the searched macroscopic yield function. N By eliminating the parameter M in these two relationships, one gets:
RA
2
r0
þ 2f cosh
RB
r0
1 f2 ¼ 0
ð22Þ
which is of Gurson [9] type with appropriate quantities RA et RB. Compared to criteria proposed by [28] for the same class of porous materials, differences provided by (22) mainly lie in the definition of RA and RB which have to be now explicited. To this end, noticing that M depends only on D and A, while N is a function of Dm and A, it comes:
Rm ¼
e e e e 1 @P 2 @P @P 2D0 @ P ¼ pffiffiffi R0 ¼ 0 ¼ 3 @Dm 3 b @N 3bM @M @D
Mesoscale porosity
ð23Þ Microscale porosity
in which:
pffiffiffi e @P 3 b Rm ¼ @N 2
ð24Þ
Fig. 1. Double porous media with homogeneous boundary strain rate.
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Fig. 2. Yield criterion [24]: Cross: upper bound ‘‘4.15’’ proposed by Vincent [28]; Square: numerical simulations by finite elements; Circle: value constraint Rrm0 with the hydrostatic loading; Dashed line: approximate criterion of Vincent [27]; Solid line: criterion (33). fb is the microscale porosity while fe denotes the mesoscale one.
Fig. 3. Yield criterion [17]: Cross: upper bound ‘‘4.15’’ proposed by Vincent [28]; Square: numerical simulations by finite elements; Circle: value constraint Rrm0 with the hydrostatic loading; Dashed line: approximate criterion of Vincent [27]; Solid line: criterion (36). fb is the microscale porosity while fe denotes the mesoscale one.
homogenization techniques (see, e.g. [24]) which lead to an elliptic criterion. This is required in order to be able to apply the results established in Section 2. Two slightly different elliptic criteria of the porous matrix at mesoscale will be considered in the following. 3.2. Case of a matrix described by Ponte Castañeda [23] criterion The porosity at the microscale being denoted fb, the elliptic criterion established by [23], by considering a variational approach, is:
2 2 2 req 9 rm 2 1 þ fb þ fb ð1 fb Þ ¼ 0 3 4 r0 r0
ð31Þ
This criterion can be expressed in the form of Eq. (1) by setting: Fig. A.1. Cell of a double porous medium subjected to an hydrostatic loading.
fb, fe, corresponding respectively to the micro and meso scales, and the total porosity f can be expressed as:
fb ¼
j xb j ; j X xe j
fe ¼
j xe j ; jXj
f ¼
jxe j þ jxb j ¼ fe þ fb ð1 fe Þ jXj
ð30Þ
The assumption of a separation between the two scales of the voids allows to perform a two-steps homogenization. For the first step, from micro to mesoscale, we adopt variational nonlinear
a¼
fb
b¼
; 2ð1 fb Þ2
1 þ 23 fb
ð32Þ
ð1 fb Þ2
Denoting by fe being the porosity at the mesoscale, the macroscopic criterion of the double porous medium, derived in the previous section (see Eq. (29)), reads then: 2 1 þ 23 fb Req
ð1 fb Þ2
r20
1
fe2
þ
9f b
R2m
4ð1 fb Þ2
r20
¼0
3 þ 2f e cosh 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 þ 23 fb Rm ð1 fb Þ2
r0 ð33Þ
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W.Q. Shen et al. / Computational Materials Science 62 (2012) 189–194
Fig. A.2. The comparison between the exact value Rrm0 and the value obtained by the criterion (29) with different a and b (2D) in the case of hydrostatic loading, f = 0.1. Blue: obtained by (29); Red: obtained by (43). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In the case of hydrostatic loading, it can be noted that the value of Rrm0 , obtained from Eq. (33) is very close to the exact solution that we established by studying a perfectly rigid plastic hollow sphere having a matrix obeying to the elliptic criterion (1) (the proof is shown in Appendix A). To assess the capabilities of the developed criterion (33), we compare it to the criterion proposed by [28] (see also Vincent et al. [27]) in which, interestingly, the homogenization from micro to mesoscale was achieved by considering a Gurson type matrix. Fig. 2 shows that in the case fb = 0.01, fe = 0.01, the surface predicted by the criterion (33) is outside the upper bound (the results of [28]), and outside the numerical available results, for large stress triaxialities; the predicted yield surface is very close to the other two for low stress triaxialities. In the case fb = 0.1, fe = 0.01, with low stress triaxiality, the surface predicted by the new criterion (33) is inside the upper bound, while it is outside when the stress triaxiality is large. In the case fb = 0.01 and fe = 0.1, then fb = 0.1 and fe = 0.1, the different results appear more in agreement. 3.3. Case of a matrix described by the criterion of Michel and Suquet [17] Another elliptic type criterion of porous media proposed by Michel and Suquet [17] has the following expression:
2 2 2 2 req 9 1 fb rm 2 1 þ fb þ ð1 f b Þ ¼ 0 3 4 lnðfb Þ r0 r0
ð34Þ
which corresponds to a criterion of the general form (1) in which:
a¼
1fb lnðfb Þ
2 2
2ð1 fb Þ
;
b¼
1 þ 23 fb
ð35Þ
ð1 fb Þ2
Making use of this estimate, it follows that the criterion for a double porous material is: Fig. A.3. The comparison between the exact value Rrm0 and the value obtained by the criterion (29) with different a and b (2D) in the case of hydrostatic loading, f = 0.01. Blue solid line: obtained by (29); Red dashed line: obtained by (43). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2 1 þ 23 fb Req
ð1 fb Þ2 1
r20 fe2
þ
¼0
2 1fb 9 lnðf R2m bÞ 4ð1 fb Þ2
r20
3 þ 2f e cosh 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 þ 23 fb Rm ð1 fb Þ2
r0 ð36Þ
The comparison of the yield locus according to criterion (36) in this case with that provided by [28] (semi-analytical expression and numerical simulations by finite elements) is shown on Fig. 3. We note that in the case fb = 0.01, fe = 0.01; fb = 0.1, fe = 0.01, and fb = 0.1, fe = 0.1, our results from (36) are inside the upper bound. In the case fb = 0.01, fe = 0.1, with large stress triaxiality, our results are still inside the upper bound, while at low stress triaxiality they coincide well with the upper bound proposed by [28]. A qualitative agreement with numerical data is noted. 4. Conclusion
Fig. A.4. The comparison between the exact value Rrm0 and the value obtained by the criterion (29) with different a and b (3D) in the case of hydrostatic loading, f = 0.1. Blue surface: obtained by (29); Red surface: obtained by (43). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The present study concerns the development of a micromechanical approach for modeling ductile porous media made up of a matrix obeying an elliptic criterion. Based on limit analysis theory, we first derive a closed-form criterion for a porous material with Green type matrix containing spherical voids. By considering a two steps homogenization procedure, we then used this criterion to derive the macroscopic yield function of double porous media whose solid phase at microscale obeys to a von Mises criterion. To this end, two elliptic-type criteria one by [23] and the other by [17] have been considered in the first homogenization step. The derived closed-form expressions for the criteria obtained for the double porous media have been assessed through comparison with those established by [27].
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Further extensions of the present study will consist in considering two populations of saturated pores (at micro and meso scales) with different pore pressures. Appendix A. Solution of hollow sphere with Green type matrix, subjected to hydrostatic loadings Let us consider a hollow sphere made up of a rigid perfectly plastic matrix and subjected to a hydrostatic loading (see Fig. A.1). The internal and external radi are respectively denoted a and b. This hollow sphere is subjected to a uniform hydrostatic loading at external boundary, i.e. r n = Rmn with Rm P 0. The matrix is still assumed to obey to an elliptic Green [7] type criterion:
3 r : ðaJ þ bKÞ : r r20 ¼ 0 2
ð37Þ
Due to the spherical symmetry of the problem, the microscopic stress tensor is reduced to the following components: rrr, rhh ruu. It follows that the equilibrium equations give:
drrr rhh rrr ¼2 dr r
ð38Þ
The boundary conditions read:
rrr ðaÞ ¼ 0; rrr ðbÞ ¼ Rm
ð39Þ 2
Moreover, we have the equivalent stress r ¼ ðrhh rrr Þ and the local hydrostatic stress rm ¼ 13 ðrrr þ 2rhh Þ ¼ rrr 23 ðrrr rhh Þ, hence the criterion is written: 2 eq
9 bðrrr rhh Þ2 þ a 2
2
2 3
rrr ðrrr rhh Þ r20 ¼ 0
ð40Þ
According to the Eq. (40), we can get the expression of rrr rhh:
rrr rhh ¼ r0
6a rrrr0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 8a þ 4b 18ab rrr2 0
ð41Þ
4a þ 2b
With this relationship, the equilibrium Eq. (38) can be expressed as:
d rrrr0 dr
þ
6a rrrr0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 8a þ 4b 18ab rrr2 0
ð2a þ bÞr
¼0
ð42Þ
By solving (42), we can get the expression of rrrr0 in the implicit form:
1 0 pffiffiffiffiffiffiffiffiffi r 3 rffiffiffiffiffiffi rr 2b a C B a 2b r0 ffiC arctan B 2ln 3 þ A þ ln ðQ Þ ¼ 0 @3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r a r 8a þ 4b 18ba rrr2
ð43Þ
0
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi r2 4a 3b 2a rrrr0 þ 2b þ 2a 8a þ 4b 18ba rrr2 3 a rrrr0 þ 2 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ¼ 2 : pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi r2 r2 3 a rrrr0 2 4a þ 3b 2a rrrr0 þ 2b þ 2a 8a þ 4b 18ba rrr2 2 þ 9a rrr2 0
0
Note that when a ? 0 and b = 1 (case of von Mises matrix), we get from (43) the expected exact solution at the boundary r ¼ b : Rrm0 ¼ 23 lnðf Þ. Figs. A.2, A.3 compare the prediction obtained from (29) with the exact values provided by (43) for different porosities. It is observed that the two results are very closed, the approximated value being always bigger than the exact one (see also this in the Fig. A.4). References [1] A.A. Benzerga, J. Besson, European Journal of Mechanics A/Solids 20 (2001) 397–434. [2] A.A. Benzerga, J.B. Leblond, Advances in Applied Mechanics (2010). [3] J. Besson, C. Guillemer-Neel, Mechanics of Materials 35 (2003) 1–18. [4] P. De Buhan, A fundamental approach to the yield design of reinforced soil structures – Chap. 2: yield design homogenization theory for periodic media, Thèse d’état, Université Pierre et Marie Curie, Paris VI, France, 1986. [5] M. Garajeu, P. Suquet, Journal of Mechanics and Physics of Solids 45 (1997) 873–902. [6] M. Gologanu, J.B. Leblond, G. Perrin, J. Devaux, Recent extensions of gurson’s model for porous ductile metals, in: P. Suquet (Ed.), Continuum Micromechanics, Springer-Verlag, 1997. [7] R.J. Green, International Journal of Mechanical and Science 14 (1972) 215–224. [8] J. Faleskog, T.F. Guo, C.F. Shih, Journal of Mechanics and Physics of Solids 56 (2008) 2188–2212. [9] A.L. Gurson, Journal of Engineering Materials and Technology 99 (1977) 2–15. [10] H.Y. Jeong, International Journal of Solids Structures 39 (2002) 1385–1403. [11] S.M. Keralavarma, A.A. Benzerga, Journal of the Mechanics and Physics of Solids 58 (2010) 874–901. [12] H.A. Kuhn, C.L. Downey, International Journal of Powder Metallurgy 7 (1971) 15–25. [13] J.H. Lee, J. Oung, Journal of Applied Mechanics 67 (2000) 288–297. [14] K. Liao, Computers and Structures 82 (2004) 2573–2583. [15] J. Pan, K. Liao, S. Tang, Mechanics of Materials 26 (1997) 213–226. [16] S. Maghous, L. Dormieux, J.F. BarthTlTmy, European Journal of Mechanics A/ Solid 28 (2009) 179–188. [17] J-C. Michel, P. Suquet, Journal of the Mechanics and Physics of Solids 40 (1992) 783–812. [18] V. Monchiet, O. Cazacu, E. Charkaluk, D. Kondo, International Journal of Plasticity 24 (2008) 1158–1189. [19] Vincent Monchiet, Contributions a la modTlisation micromTcanique de l’endommagement et de la fatigue des mTtaux ductiles, Thèse, UniversitT des sciences et Technologies de Lille, France, 2006. [20] P. Suquet, Homogenization Techniques for Composite Media, Elements of Homogenization for Inelastic Solid Mechanics, in: E. Sanchez-Palencia (Ed.), Springer, Verlag, 1985, pp. 193–278. [21] J.W. Pardoen, T. Hutchinson, Journal of the Mechanics and Physics of Solids 48 (2000) 2467–2512. [22] J.W. Pardoen, T. Hutchinson, Acta Mater 51 (2003) 133–148. [23] P. Ponte Castaneda, Journal of the Mechanics and Physics of Solids 39 (1991) 45–71. [24] P. Ponte Castaneda, P. Suquet, Nonlinear composites, Advances in Applied Mechanics, vol. 34, Academic Press, New York, 1998, pp. 171–302. [25] S. Shima, M. Oyane, International Journal of Mechanical Science 18 (1976) 285–291. [26] M. Trillat, J. Pastor, European Journal of Mechanics 24 (2005) 800–819. [27] P.-G. Vincent, Y. Monerie, P. Suquet, Computes Rendus Mecanique 336 (2008) 245–259. [28] Pierre-Guy Vincent, ModTlisation micromTcanique de la croissance et de la percolation de pores sous pression dans une matrice cTramique a haute tempTrature, Thèse, UniversitT d’aix-marseille1, Provence, 2007.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Approximate criteria for ductile porous materials having a Green type matrix: Application to double porous media W.Q. Shen a, J.F. Shao a, L. Dormieux b, D. Kondo c,⇑ a
Laboratoire de Mécanique de Lille-UMR 8107 CNRS, USTL, 59655 Villeneuve d’Ascq, France Unité de Recherche Navier, UMR 8205 CNRS, 6/8 Avenue B. Pascal, 77455 Champs sur Marne, France c Institut D’Alembert, UMR 7190 CNRS, UPMC (Paris 6), 75005 Paris, France b
a r t i c l e
i n f o
Article history: Received 16 April 2012 Received in revised form 4 May 2012 Accepted 6 May 2012 Available online 19 June 2012 Keywords: Porous materials Ductile Macroscopic yield function Plastic compressibility Gurson-type models Double porous media Green materials
a b s t r a c t In the framework of limit analysis theory, we derive closed-form expressions of approximate criteria for ductile porous materials whose plastically compressible matrix obeys to an elliptic criterion. The general methodology is based on limit analysis of a hollow sphere subjected to a uniform strain rate boundary conditions. We first consider a porous medium with a Green type matrix and establish the corresponding macroscopic yield function. Then, the obtained results are used in order to investigate double porous materials whose solid phase at the microscale (the smallest scale) obeys a von Mises criterion. The results are assessed by comparing them with numerical data, and with recently published results. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Since the pioneering work of Gurson [9], limit analysis-based modeling of the ductile behavior of porous media have been the subject of intense research. Among existing contributions, mention can be done of works concerning with voids shape effects (see for instance [5,6,21,22]) or with matrix plastic anisotropy [15,1,14,18,11]. For a complete review, the reader may refer to the recent review by [2]. In contrast to the above cited works which all considered that the matrix is incompressible, few researches have been carried out for porous materials with plastic compressible matrix. In the context of limit analysis-based developments, Lee and Oung [13] took into account the plastic compressibility of the matrix by considering that its behavior is described by a Mises–Schleicher criterion, while studies by [10,8] or [26] consider that the matrix behavior obeys a Drucker–Prager criterion.1 The present study is devoted to the investigation of porous media whose solid matrix obeys to an elliptic criterion, expressed as a quadratic function of the first invariant of stress and of the von Mises equivalent stress. A typical example of such a criterion is
⇑ Corresponding author. E-mail address: [email protected] (D. Kondo). Note also the contribution by Maghous et al. [16] who implemented and extended the so-called modified secant moduli method (equivalent to the variational approach proposed by Ponte Castañeda [23]) for a Drucker–Prager matrix. 1
0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.05.021
that proposed by Green [7] (see also [12]) and more recently considered by [3]. It is interesting to note that elliptic criteria2 may also result by applying a nonlinear homogenization approach to porous materials having a Von Mises matrix (see [23] or [17]). The objective of the present study is to establish macroscopic criteria for ductile porous materials having a Green type matrix. The paper is organized as follows. In Section 2, we present and apply Hill–Mandel homogenization to derive the plastic potential for a porous material having a matrix obeying to an elliptic criterion in the form (1). A closed-form expression of the criterion is derived (see Section 2.2). The results are then assessed and discussed by comparing them with numerical data, and with results available in the literature. The proposed development is particularly interesting for some porous geomaterials or polymers. Moreover, for various cases of practical interest, including application to metals, the porosity can be found at two scales which will be denoted microscale and mesoscale in the following. For this class of double porous media for which two populations of voids exist at two well separated scales, Vincent et al. [27] recently established original results by performing two step homogenization. For the transition from microscale to mesoscale, these authors considered a von Mises solid phase and adopted a Gurson criterion. Their final result is given in the form of a macroscopic criterion which was obtained after several approximations. In this paper, we develop criteria for 2
The elliptic criterion has been used by [25] in the context of ductile porous media.
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porous media with 2 populations of porosities by adopting instead of Gurson criterion, the newly developed criterion presented in Section 2. 2. Approximate macroscopic criterion for a porous medium having a Green type matrix 2.1. General methodology of limit analysis theory applied to a porous medium with the Green matrix Let us first consider a hollow sphere whose internal and external radii are respectively denoted a and b; the porosity corresponding to this hollow sphere is then f = a3/b3. Uniform strain rate boundary conditions are applied on the outer surface of the hollow sphere: v = D.x for x = ber. The quantity D represents the applied uniform macroscopic strain rate. The matrix is isotropic and rigid perfect plastic. It is assumed that the matrix obeys an elliptic criterion of the form:
a
/ðrÞ ¼ br2eq þ ðtr rÞ2 r20 6 0 2
ð1Þ
where a and b are positive material constants (Green-type criterion); In Eq. (1) r denotes the microscopic stress tensor and r0 is related to the shear strength of the matrix. r2eq ¼ 32 r0 : r0 with r0 representing the deviatoric part of the stress tensor. Note that Eq. (1) can be also put in the following form:
/ðrÞ ¼
3 r : M : r r20 6 0 2
ð2Þ
with M ¼ aJ þ bK, in which J and K are the standard projectors of isotropic fourth-order symmetric tensors having both major and minor symmetries. These projectors are expressed as J ¼ 13 1 1, K ¼ I J where 1; I are the second-order and the symmetric fourth-order unit tensors, respectively. Components of I are Iijkl ¼ 12 ðdik djl þ dil djk Þ in which d denotes the well-known Kronecker’s symbol. The plastic compressibility of the matrix lies in the presence of the hydrostatic component of the stress in Eq. (2). Application of the normality rule gives the local strain rate:
_ d¼K
@/ _M:r ¼ 3K @r
ð3Þ
2 _ 2 r2 d : H : d ¼ 4K 0 3
ð4Þ
v K:A jXj
pðdÞdV ¼ inf
X x
r0
v K:A jXj
Z
~eq dV d
ð8Þ
X x
P(D) represents the macroscopic dissipation, X denotes the volume of the unit cell, jXj = 4pb3/3, where as x denotes the volume of the void, jxj = 4pa3/3. The infimum in (8) is taken over all kinematically admissible (K.A) velocity fields, v. As classically, the limit stress states at the macroscopic scale are shown to be of the form: R¼
@P @D
ð9Þ
2.2. The considered velocity field and the corresponding macroscopic yield function Following [27], the velocity field in the matrix is chosen in the form: 3
v ¼ Ax þ
b ðDm AÞ er þ D0 :x r2
ð10Þ
with Dm ¼ 13 trD and D0 the deviatoric part of the macroscopic strain rate D. The homogeneous field Ax is used to introduce a compressible component in the velocity field v; the two remaining terms (inspired from the Gurson velocity field) are kinematically admissible with (D A1). More precisely, the second term in (10) corresponds to the expansion of the cavity and the outer volume, while the third one describes the shape change of the cavity and of the outer volume without change of volume. Hence, for any value of A, the whole velocity field v complies with the uniform strain rate D applied to the hollow sphere. The strain rate in the solid matrix can be obtained from Eq. (10); in spherical coordinates, one obtains: 3
E
E
d ¼ A1 þ d ; d ¼ D0 þ
b ðDm AÞ ½1 3er er r3
ð11Þ
To compute the local dissipation in the solid matrix, one needs ~eq . To this end, substituting (11) in Eq. (6) provides: to calculate d
~2 ¼ 2 A2 ð1 : H : 1Þ þ 4 Að1 : H : dE Þ þ dE 2 ; with dE 2 ¼ 2 dE d eq eq eq 3 3 3
ð5Þ
where the scalar deq is the equivalent strain rate, and dm the volumetric strain rate, defined as:
deq ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d : K : d; 3
:H:d
ð12Þ 3
E
Taking into account the identities 1 : H : 1 ¼ a ; 1 : H : d ¼ 0, one obtains:
" #2 3 D2eq 4 b3 ðDm AÞ 0 2A2 4 b ðDm AÞ 2 ~ deq ¼ þ þ D : ½1 3er er þ r3 r3 a b b 3b ð13Þ
K_ can then be written as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ~eq d:H:d d 1 2dm deq 3 K_ ¼ ¼ þ ¼ 2r0 2r0 a b 2r 0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d : H : d; 3
Z
1
E
_ is the local plastic multiplier whose expression will be prowhere K vided later. Let H be the inverse of M; it follows that H ¼ M1 ¼ a1 J þ 1b K. The plastic multiplier is then derived from (3) as follows:
~eq ¼ d
R : D 6 PðDÞ ¼ inf
dm ¼
1 trd 3
ð6Þ
It follows that the local plastic dissipation p(d) in the matrix takes the form:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2dm deq ~eq pðdÞ ¼ r : d ¼ 3K_ r : M : r ¼ r0 þ ¼ r0 d a b
ð7Þ
For the uniform strain rate boundary conditions, considered in the present study, the following inequality holds for all macroscopic stress R and macroscopic strain rate D [20,4]:
Due to the presence of the scalar A (which remains unknown in the definition of the velocity field), the macroscopic dissipation, P(D), is computed owing to a minimization procedure with respect to A:
h
i
e ðD; AÞ ; with P e ðD; AÞ ¼ PðDÞ ¼ min P A
r0 jXj
Z
~eq dV d
ð14Þ
X x
Obviously, the determination of the macroscopic criterion ~eq . In order to obtain a closedrequires to compute the integral of d form expression, it is convenient to apply the following inequality ~2 [9]: to d eq
Z
Xx
~eq dV ¼ d
Z
X x
6 4p
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~2 ðr; h; uÞdV d eq
Z
b a
1=2 ~2 i r 2 hd dr eq SðrÞ
ð15Þ
~2 i where SðrÞ is the sphere of radius r and hd eq SðrÞ is the average of ~2 ðr; h; uÞ over all the orientations: d eq
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W.Q. Shen et al. / Computational Materials Science 62 (2012) 189–194
~2 i hd eq SðrÞ ¼
2A2
a
þ
" 3 #2 D2eq 4 b ðDm AÞ þ 3 b r b
ð16Þ
for which h1 3er er iSðrÞ ¼ 0 has been used. This eventually yields an upper bound of the macroscopic dissipation (see Eq. (14)) by computing:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #2 u 2 3 u2A D2eq r 4 b ðDm AÞ 0 t e ðD; AÞ ¼ 4pr 2 dr P þ þ 3 r X a a b b Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f du ¼ r0 M 2 þ N2 u2 2 u 1 " #1f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M2 þ N2 u2 uN ¼ r0 N arcsinh u M Z
b
e ðD; AÞ, then The condition (20) ensuring the minimum of P reads:
e 2A @ P e 2 @P pffiffiffi ¼ 0 @M aM @N b
ð25Þ
Combining (24) and (25) yields:
e @P 3aRm M ¼ 2A @M
ð26Þ
and then, with the help of (23):
ð17Þ
e @P ¼ @M
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 2 aR þ bR2eq 2 m
ð27Þ
1
2
D2
where M2 ¼ 2Aa þ beq , N2 ¼ 4b ðDm AÞ2 and the change of variable 3 u ¼ br3 has been introduced. e ðD; AÞ over the unAs mentioned before, one has to minimize P known variable A (see Eq. (14)) and to determine the macroscopic plastic criterion by taking advantage of the approximate exprese ðD; AÞ. In practice, rather than treating these two steps sion of P successively, the computation of the criterion may be done by addressing them simultaneously (cf. [19]). It comes:
R¼
e ðD; AÞ e ðD; AÞ @P @P ; with ¼0 @D @A
ð18Þ
To solve (18), it is convenient to make the following change of vare ðD; AÞ ¼ P e ðM; NÞ with M and N introduced before. The maciable: P roscopic stress tensor R reads then:
R¼
e e @M @ P e @N @P @P þ ¼ : : @D @M @D @N @D
ð19Þ
with the condition of minimization with respect to A (second relation in (18)):
e @M @ P e @N @P þ ¼0 : : @M @A @N @A
ð20Þ
Finally, from (21) we can get the expressions of RA and RB:
RA ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 9 2 3 b aRm þ bR2eq ; RB ¼ Rm 2 2
ð28Þ
Putting this result in (22) yields the searched closed-form expression of the following macroscopic criterion of the porous medium having a matrix obeying to the Green criterion (1):
b
R2eq
r20
þ
pffiffiffi 9a R2m 3 b Rm 1 f2 ¼ 0 þ 2f cosh 2 r0 2 r20
ð29Þ
Note that the presence of the term R2m constitutes the macroscopic counterpart of the local plastic compressibility and the main difference with the Gurson criterion. It is worth noticing that the particular case of a = 0 and b = 1 (von Mises solid matrix) readily yields the Gurson criterion of the porous material. 3. Application to porous materials with two populations of voids
Similarly to the approach used by Gurson, one can then establish the following relationships:
3.1. The problem statement
2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 e @P N2 N2 RA ¼ ¼ r0 4 1 þ 2 f 2 þ 2 5 @M M M e @P N N RB ¼ ¼ r0 arcsinhð Þ arcsinh fM M @N
We aim now at validating the macroscopic criterion given by Eq. (29) by applying it to a material containing two populations of spherical cavities at two different scales (see Fig. 1). The voids in the two populations are assumed to be spherical and isotropically distributed. Let us recall the notation jXj which denotes the volume of the representative elementary volume. Following [28], we also denote by xb the domain occupied by the cavities at the smallest scale (microscale) and xe the one occupied by the voids at the upper scale (mesoscale). With these notations, the porosities
ð21Þ
which define the parametric form of the searched macroscopic yield function. N By eliminating the parameter M in these two relationships, one gets:
RA
2
r0
þ 2f cosh
RB
r0
1 f2 ¼ 0
ð22Þ
which is of Gurson [9] type with appropriate quantities RA et RB. Compared to criteria proposed by [28] for the same class of porous materials, differences provided by (22) mainly lie in the definition of RA and RB which have to be now explicited. To this end, noticing that M depends only on D and A, while N is a function of Dm and A, it comes:
Rm ¼
e e e e 1 @P 2 @P @P 2D0 @ P ¼ pffiffiffi R0 ¼ 0 ¼ 3 @Dm 3 b @N 3bM @M @D
Mesoscale porosity
ð23Þ Microscale porosity
in which:
pffiffiffi e @P 3 b Rm ¼ @N 2
ð24Þ
Fig. 1. Double porous media with homogeneous boundary strain rate.
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W.Q. Shen et al. / Computational Materials Science 62 (2012) 189–194
Fig. 2. Yield criterion [24]: Cross: upper bound ‘‘4.15’’ proposed by Vincent [28]; Square: numerical simulations by finite elements; Circle: value constraint Rrm0 with the hydrostatic loading; Dashed line: approximate criterion of Vincent [27]; Solid line: criterion (33). fb is the microscale porosity while fe denotes the mesoscale one.
Fig. 3. Yield criterion [17]: Cross: upper bound ‘‘4.15’’ proposed by Vincent [28]; Square: numerical simulations by finite elements; Circle: value constraint Rrm0 with the hydrostatic loading; Dashed line: approximate criterion of Vincent [27]; Solid line: criterion (36). fb is the microscale porosity while fe denotes the mesoscale one.
homogenization techniques (see, e.g. [24]) which lead to an elliptic criterion. This is required in order to be able to apply the results established in Section 2. Two slightly different elliptic criteria of the porous matrix at mesoscale will be considered in the following. 3.2. Case of a matrix described by Ponte Castañeda [23] criterion The porosity at the microscale being denoted fb, the elliptic criterion established by [23], by considering a variational approach, is:
2 2 2 req 9 rm 2 1 þ fb þ fb ð1 fb Þ ¼ 0 3 4 r0 r0
ð31Þ
This criterion can be expressed in the form of Eq. (1) by setting: Fig. A.1. Cell of a double porous medium subjected to an hydrostatic loading.
fb, fe, corresponding respectively to the micro and meso scales, and the total porosity f can be expressed as:
fb ¼
j xb j ; j X xe j
fe ¼
j xe j ; jXj
f ¼
jxe j þ jxb j ¼ fe þ fb ð1 fe Þ jXj
ð30Þ
The assumption of a separation between the two scales of the voids allows to perform a two-steps homogenization. For the first step, from micro to mesoscale, we adopt variational nonlinear
a¼
fb
b¼
; 2ð1 fb Þ2
1 þ 23 fb
ð32Þ
ð1 fb Þ2
Denoting by fe being the porosity at the mesoscale, the macroscopic criterion of the double porous medium, derived in the previous section (see Eq. (29)), reads then: 2 1 þ 23 fb Req
ð1 fb Þ2
r20
1
fe2
þ
9f b
R2m
4ð1 fb Þ2
r20
¼0
3 þ 2f e cosh 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 þ 23 fb Rm ð1 fb Þ2
r0 ð33Þ
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W.Q. Shen et al. / Computational Materials Science 62 (2012) 189–194
Fig. A.2. The comparison between the exact value Rrm0 and the value obtained by the criterion (29) with different a and b (2D) in the case of hydrostatic loading, f = 0.1. Blue: obtained by (29); Red: obtained by (43). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In the case of hydrostatic loading, it can be noted that the value of Rrm0 , obtained from Eq. (33) is very close to the exact solution that we established by studying a perfectly rigid plastic hollow sphere having a matrix obeying to the elliptic criterion (1) (the proof is shown in Appendix A). To assess the capabilities of the developed criterion (33), we compare it to the criterion proposed by [28] (see also Vincent et al. [27]) in which, interestingly, the homogenization from micro to mesoscale was achieved by considering a Gurson type matrix. Fig. 2 shows that in the case fb = 0.01, fe = 0.01, the surface predicted by the criterion (33) is outside the upper bound (the results of [28]), and outside the numerical available results, for large stress triaxialities; the predicted yield surface is very close to the other two for low stress triaxialities. In the case fb = 0.1, fe = 0.01, with low stress triaxiality, the surface predicted by the new criterion (33) is inside the upper bound, while it is outside when the stress triaxiality is large. In the case fb = 0.01 and fe = 0.1, then fb = 0.1 and fe = 0.1, the different results appear more in agreement. 3.3. Case of a matrix described by the criterion of Michel and Suquet [17] Another elliptic type criterion of porous media proposed by Michel and Suquet [17] has the following expression:
2 2 2 2 req 9 1 fb rm 2 1 þ fb þ ð1 f b Þ ¼ 0 3 4 lnðfb Þ r0 r0
ð34Þ
which corresponds to a criterion of the general form (1) in which:
a¼
1fb lnðfb Þ
2 2
2ð1 fb Þ
;
b¼
1 þ 23 fb
ð35Þ
ð1 fb Þ2
Making use of this estimate, it follows that the criterion for a double porous material is: Fig. A.3. The comparison between the exact value Rrm0 and the value obtained by the criterion (29) with different a and b (2D) in the case of hydrostatic loading, f = 0.01. Blue solid line: obtained by (29); Red dashed line: obtained by (43). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2 1 þ 23 fb Req
ð1 fb Þ2 1
r20 fe2
þ
¼0
2 1fb 9 lnðf R2m bÞ 4ð1 fb Þ2
r20
3 þ 2f e cosh 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 þ 23 fb Rm ð1 fb Þ2
r0 ð36Þ
The comparison of the yield locus according to criterion (36) in this case with that provided by [28] (semi-analytical expression and numerical simulations by finite elements) is shown on Fig. 3. We note that in the case fb = 0.01, fe = 0.01; fb = 0.1, fe = 0.01, and fb = 0.1, fe = 0.1, our results from (36) are inside the upper bound. In the case fb = 0.01, fe = 0.1, with large stress triaxiality, our results are still inside the upper bound, while at low stress triaxiality they coincide well with the upper bound proposed by [28]. A qualitative agreement with numerical data is noted. 4. Conclusion
Fig. A.4. The comparison between the exact value Rrm0 and the value obtained by the criterion (29) with different a and b (3D) in the case of hydrostatic loading, f = 0.1. Blue surface: obtained by (29); Red surface: obtained by (43). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The present study concerns the development of a micromechanical approach for modeling ductile porous media made up of a matrix obeying an elliptic criterion. Based on limit analysis theory, we first derive a closed-form criterion for a porous material with Green type matrix containing spherical voids. By considering a two steps homogenization procedure, we then used this criterion to derive the macroscopic yield function of double porous media whose solid phase at microscale obeys to a von Mises criterion. To this end, two elliptic-type criteria one by [23] and the other by [17] have been considered in the first homogenization step. The derived closed-form expressions for the criteria obtained for the double porous media have been assessed through comparison with those established by [27].
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Further extensions of the present study will consist in considering two populations of saturated pores (at micro and meso scales) with different pore pressures. Appendix A. Solution of hollow sphere with Green type matrix, subjected to hydrostatic loadings Let us consider a hollow sphere made up of a rigid perfectly plastic matrix and subjected to a hydrostatic loading (see Fig. A.1). The internal and external radi are respectively denoted a and b. This hollow sphere is subjected to a uniform hydrostatic loading at external boundary, i.e. r n = Rmn with Rm P 0. The matrix is still assumed to obey to an elliptic Green [7] type criterion:
3 r : ðaJ þ bKÞ : r r20 ¼ 0 2
ð37Þ
Due to the spherical symmetry of the problem, the microscopic stress tensor is reduced to the following components: rrr, rhh ruu. It follows that the equilibrium equations give:
drrr rhh rrr ¼2 dr r
ð38Þ
The boundary conditions read:
rrr ðaÞ ¼ 0; rrr ðbÞ ¼ Rm
ð39Þ 2
Moreover, we have the equivalent stress r ¼ ðrhh rrr Þ and the local hydrostatic stress rm ¼ 13 ðrrr þ 2rhh Þ ¼ rrr 23 ðrrr rhh Þ, hence the criterion is written: 2 eq
9 bðrrr rhh Þ2 þ a 2
2
2 3
rrr ðrrr rhh Þ r20 ¼ 0
ð40Þ
According to the Eq. (40), we can get the expression of rrr rhh:
rrr rhh ¼ r0
6a rrrr0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 8a þ 4b 18ab rrr2 0
ð41Þ
4a þ 2b
With this relationship, the equilibrium Eq. (38) can be expressed as:
d rrrr0 dr
þ
6a rrrr0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 8a þ 4b 18ab rrr2 0
ð2a þ bÞr
¼0
ð42Þ
By solving (42), we can get the expression of rrrr0 in the implicit form:
1 0 pffiffiffiffiffiffiffiffiffi r 3 rffiffiffiffiffiffi rr 2b a C B a 2b r0 ffiC arctan B 2ln 3 þ A þ ln ðQ Þ ¼ 0 @3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r a r 8a þ 4b 18ba rrr2
ð43Þ
0
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi r2 4a 3b 2a rrrr0 þ 2b þ 2a 8a þ 4b 18ba rrr2 3 a rrrr0 þ 2 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ¼ 2 : pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi r2 r2 3 a rrrr0 2 4a þ 3b 2a rrrr0 þ 2b þ 2a 8a þ 4b 18ba rrr2 2 þ 9a rrr2 0
0
Note that when a ? 0 and b = 1 (case of von Mises matrix), we get from (43) the expected exact solution at the boundary r ¼ b : Rrm0 ¼ 23 lnðf Þ. Figs. A.2, A.3 compare the prediction obtained from (29) with the exact values provided by (43) for different porosities. It is observed that the two results are very closed, the approximated value being always bigger than the exact one (see also this in the Fig. A.4). References [1] A.A. Benzerga, J. Besson, European Journal of Mechanics A/Solids 20 (2001) 397–434. [2] A.A. Benzerga, J.B. Leblond, Advances in Applied Mechanics (2010). [3] J. Besson, C. Guillemer-Neel, Mechanics of Materials 35 (2003) 1–18. [4] P. De Buhan, A fundamental approach to the yield design of reinforced soil structures – Chap. 2: yield design homogenization theory for periodic media, Thèse d’état, Université Pierre et Marie Curie, Paris VI, France, 1986. [5] M. Garajeu, P. Suquet, Journal of Mechanics and Physics of Solids 45 (1997) 873–902. [6] M. Gologanu, J.B. Leblond, G. Perrin, J. Devaux, Recent extensions of gurson’s model for porous ductile metals, in: P. Suquet (Ed.), Continuum Micromechanics, Springer-Verlag, 1997. [7] R.J. Green, International Journal of Mechanical and Science 14 (1972) 215–224. [8] J. Faleskog, T.F. Guo, C.F. Shih, Journal of Mechanics and Physics of Solids 56 (2008) 2188–2212. [9] A.L. Gurson, Journal of Engineering Materials and Technology 99 (1977) 2–15. [10] H.Y. Jeong, International Journal of Solids Structures 39 (2002) 1385–1403. [11] S.M. Keralavarma, A.A. Benzerga, Journal of the Mechanics and Physics of Solids 58 (2010) 874–901. [12] H.A. Kuhn, C.L. Downey, International Journal of Powder Metallurgy 7 (1971) 15–25. [13] J.H. Lee, J. Oung, Journal of Applied Mechanics 67 (2000) 288–297. [14] K. Liao, Computers and Structures 82 (2004) 2573–2583. [15] J. Pan, K. Liao, S. Tang, Mechanics of Materials 26 (1997) 213–226. [16] S. Maghous, L. Dormieux, J.F. BarthTlTmy, European Journal of Mechanics A/ Solid 28 (2009) 179–188. [17] J-C. Michel, P. Suquet, Journal of the Mechanics and Physics of Solids 40 (1992) 783–812. [18] V. Monchiet, O. Cazacu, E. Charkaluk, D. Kondo, International Journal of Plasticity 24 (2008) 1158–1189. [19] Vincent Monchiet, Contributions a la modTlisation micromTcanique de l’endommagement et de la fatigue des mTtaux ductiles, Thèse, UniversitT des sciences et Technologies de Lille, France, 2006. [20] P. Suquet, Homogenization Techniques for Composite Media, Elements of Homogenization for Inelastic Solid Mechanics, in: E. Sanchez-Palencia (Ed.), Springer, Verlag, 1985, pp. 193–278. [21] J.W. Pardoen, T. Hutchinson, Journal of the Mechanics and Physics of Solids 48 (2000) 2467–2512. [22] J.W. Pardoen, T. Hutchinson, Acta Mater 51 (2003) 133–148. [23] P. Ponte Castaneda, Journal of the Mechanics and Physics of Solids 39 (1991) 45–71. [24] P. Ponte Castaneda, P. Suquet, Nonlinear composites, Advances in Applied Mechanics, vol. 34, Academic Press, New York, 1998, pp. 171–302. [25] S. Shima, M. Oyane, International Journal of Mechanical Science 18 (1976) 285–291. [26] M. Trillat, J. Pastor, European Journal of Mechanics 24 (2005) 800–819. [27] P.-G. Vincent, Y. Monerie, P. Suquet, Computes Rendus Mecanique 336 (2008) 245–259. [28] Pierre-Guy Vincent, ModTlisation micromTcanique de la croissance et de la percolation de pores sous pression dans une matrice cTramique a haute tempTrature, Thèse, UniversitT d’aix-marseille1, Provence, 2007.