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Plastic yield surfaces of anisotropic porous materials in terms of effective electric conductivities
Mechanics of Materials 38 (2006) 908–923 www.elsevier.com/locate/mechmat
Plastic yield surfaces of anisotropic porous materials in terms of effective electric conductivities Igor Sevostianov a, Mark Kachanov a
b,*
Department of Mechanical Engineering, New Mexico State University, P.O. Box 30001, Las Cruces, NM 88003, USA b Department of Mechanical Engineering, Tufts University, 204 Anderson Hall, Medford, MA 02155, USA Received 10 December 2004
Abstract Plastic yield surface of a porous ductile material with generally anisotropic porous space is constructed in terms of the effective conductivities. Such a cross-property connection is possible provided hardening of the bulk material is negligible and porosity does not exceed 0.15–0.17. The following results are utilized: (1) computational studies of Zohdi et al. [Zohdi, T., Kachanov, M., Sevostianov, I., 2002. A microscale numerical analysis of a plastic flow in a porous material. International Journal of Plasticity 18, 1649–1659] show that, if the mentioned conditions are met, local ‘‘pockets’’ of plasticity remain isolated and well contained in the elastic field, resulting in approximately linear stress–strain curve, almost up to the yield point; (2) elasticity–conductivity cross-property connections for porous materials that were given in the explicit form by Sevostianov and Kachanov [Sevostianov, I., Kachanov, M., 2002. Explicit cross-property correlations for anisotropic two-phase composite materials. Journal of the Mechanics and Physics of Solids 50, 253–282]. 2005 Elsevier Ltd. All rights reserved. Keywords: Plasticity; Porosity; Yield; Conductivity; Porous material
1. Introduction Yield stress of a porous metal is not a precisely defined point, but represents a certain approximation of reality—small ‘‘pockets’’ of inelasticity may develop even at low applied stresses. We consider cases when the yield surface can be clearly identified. Experimental evidence (Wang et al., 1996; da Silva
and Ramesh, 1997), as well as theoretical analyses (Sevostianov and Kachanov, 2001) and computational studies (Zohdi et al., 2002) show (Fig. 1) that this is the case, provided:
*
Corresponding author. Tel.: +1 617 627 3318; fax: +1 617 627 3058. E-mail addresses: [email protected] (I. Sevostianov), [email protected] (M. Kachanov). 0167-6636/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2005.06.023
(A) Hardening of the bulk (‘‘dense’’) material is negligible. More precisely, the bulk material must have a clearly identifiable yield point followed by an approximately horizontal plateau (this does not exclude a non-horizontal curve at later stages of loading). Aluminum and steel are examples. (B) Volume fraction of pores (porosity p) does not exceed a certain critical level beyond which
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Fig. 1. Elastic–plastic behavior of porous metals. (a) Computational simulation (Zohdi et al., 2002) for porous aluminum at three volume fractions of ellipsoidal pores; experimental data for dense and porous Ti–6Al–4V under (b) uniaxial tension and (c) shear (da Silva and Ramesh, 1997).
the stress–strain curve starts to lose a clearly identifiable yield point. Computational studies of Zohdi et al. (2002) show that, in the case of oblate pores with aspect ratio 1/3, this level is reached at p 0.15 and, for the spherical pores, at p 0.17. If the above conditions are met, the overall stress–strain curve remains approximately linear almost up to the yield point, with the slope determined by effective elastic properties of the porous material. The underlying microscale mechanism, made explicit in the mentioned computational studies, is as follows. Local plastic ‘‘pockets’’ remain small and well contained in the surrounding elastic field (the latter is not significantly perturbed by them) until, within a narrow stress interval—that can be idealized as the yield point—a substantial volume fraction of the matrix plasticizes. At, or
near, the yield point, the field of stress deviator is almost uniform. Remark. The described scenario—that differs from the one of plastic localization—is based on the essential assumption that the bulk material does not harden so that small plastic ‘‘pockets’’ blunt stress concentrations, making further propagation of the ‘‘pockets’’ difficult (as illustrated by computational simulation of Zohdi et al., 2002). Otherwise, the mentioned stress blunting is less pronounced, and this may lead to localization of plastic deformation. Yet another restriction is that pores of ‘‘extreme’’ eccentricities (spheroids with aspect ratios smaller than 0.3 or larger than 4) should not be present in significant numbers (Zohdi, 2004). Experimental data on plastic yield at different porosities indicate that the macroscopic strain at the yield point—defined as the intersection point
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of the elastic line and the horizontal plastic plateau—remains approximately constant (independent of porosity). This observation (that was given a microscale interpretation by Zohdi and Kachanov, 2005), together with linearity of the stress–strain curve almost up to the yield point, makes it possible to express yield surface in the stress space in terms of the effective elastic constants and yield stress of the bulk material. This leads, in turn, to construction of the yield surface in terms of those characteristics of porous space that determine the effective elasticity—appropriate scalar or tensorial measures of pore concentration, shapes and orientations (Sevostianov and Kachanov, 2001). However, such constructions require the microscale information that may not be readily available, such as the orientation distribution of pores (for example, the extent of orientation scatter about a certain preferential orientation). It would be desirable, therefore, to directly link the yield surface to effective conductivities—quantities that are easily measured. We make use of elasticity–conductivity connection for porous materials that was given, in the explicit form, by Kachanov et al. (2001), Sevostianov and Kachanov (2002) and verified experimentally, for metal materials, by Sevostianov et al. (2002a,b). The conductivity can be either the electric or the thermal one; from the practical standpoint, the electrical ones are of interest since they are easier to measure. This connection leads to explicit expressions of the yield surface in terms of effective conductivities. Such expressions—given in the text to follow— still contain some microstructural information, namely, parameters that reflect the distribution of pore shapes. However, the sensitivity to pore shapes is relatively mild. We emphasize that no information on the orientation distribution of pores or on the overall porosity is needed. Thus, if the conductivity data are available, construction of the yield surface requires only limited (and rather approximate) microscale information. The sensitivity to pore shapes is further reduced if pore aspect ratios are not identical but have a ‘‘scatter’’ about a certain point. We discuss the role of the scatter, modeling it by Gaussian distribution. The results are derived for the spheroidal pores. Although the actual pore shapes may or may not resemble spheroids, this restriction is not as severe as it seems. Indeed, many of the shape ‘‘irregularities’’ (like ‘‘jaggedness’’ of pore boundaries) are
either unimportant for the effective properties or affect the conductive/elastic properties in a similar way, so that the elasticity–conductivity connections—on which the present work is based—are not affected (see Kachanov and Sevostianov (2005) for a discussion in detail). A more restrictive limitation is that the aspect ratios of pores should be between 0.3 and 4, i.e. pores of ‘‘extreme’’ eccentricities should not be present in significant numbers. We focus on the case of transverse isotropy that appears to be the main observed type of plastic anisotropy. It covers parallel pore orientations (that may be randomly perturbed) and spheroidal pores with axes lying in a plane (that may have some random orientation scatter about the plane). We also discuss the case of isotropy that covers randomly oriented spheroids and spheres. We assume that the bulk (‘‘dense’’) material is isotropic and its properties (elasticity, conductivity and the yield stress) are known.
2. Background results on anisotropic porous materials We overview relevant results on the effective elastic/conductive properties and plastic yield of porous materials. These results hinge on the proper quantitative characterization of the porous space. This means identifying parameters of pore concentration that incorporate contributions of individual pores in accordance with their actual contributions to the physical property considered. For mixtures of pores of diverse shapes and orientations, such parameters are non-trivial, and they are, generally, different for different physical properties. Importantly, for the effective elasticity and effective conductivity, the proper microstructural parameters turn out to be largely similar, giving rise to the cross-property connection. 2.1. Effective elasticity Effective elasticity of a porous material determines the slope of the overall stress–strain curve that remains approximately linear almost up to the yield point. Together with the observation that the strain at yield is approximately independent of porosity (and, thus, the same as for the bulk material), this leads to expression of the yield surface in the stress space in terms of effective elastic constants and yield stress of the bulk material.
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We define fourth rank compliance contribution tensor H(N) of Nth pore by the extra overall strain ðN Þ ðN Þ due to its presence: Deij ¼ H ijkl rkl , where rkl is the macroscopic stress. H-tensors were calculated for the ellipsoidal pores, in terms of EshelbyÕs tensor, by Kachanov et al. (1994) (see, also, Kanaun (1983) where related results in a different form were given). The sum ^ ¼ H
1 X ðN Þ H 1p N
ð2:1Þ
gives the collective effect of all pores on the overall elastic compliances in the approximation of MoriTanakaÕ scheme, MTS (1973) that has been reformulated in the form used here by Benveniste (1986). This scheme places each pore into the average, over the solid phase, stress; then multiplier (1 p)1 accounts for interactions between pores. Its omission corresponds to the non-interaction approximation. Remark. Approximate schemes other than MTS can be incorporated into our analysis. However, at porosities not exceeding 0.15–0.17, sensitivity of the results to the choice of a particular scheme is relatively low. MTS appears to be the most appropriate one, since the field of stress deviator is almost uniform near the yield point (Zohdi et al., 2002). Effective compliances can be represented as a ^ ijkl where S 0 are compliances sum S ijkl ¼ S 0ijkl þ H ijkl of the bulk material, therefore for the elastic potential (in stresses) we have f ðrÞ ¼ f0 þ Df ;
ð2:2Þ
where f0(r) = (1/2E0)[(1 + m0)rijrji m0(rkk)2] is the potential in absence of pores and ^ : r ¼ ð1=2ÞH ^ ijkl rij rkl Df ¼ ð1=2Þr : H
ð2:3Þ
is the change due to pores (see, for example, Kachanov et al., 1994). The structure of Df implies that ^ is the general proper microfourth rank tensor H structural parameter for the effective elastic properties (it represents individual pores according to their contributions to the overall compliance). For multiple pores of general spheroidal shapes, H-tensor of an individual pore (with unit vector n along its axis) that enter the sum (2.1), have the form
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V 1 ðh1 h2 =2ÞII þ h2 trJ þ ð2h3 þ h2 2h1 Þ H¼ 2 V 1 ðInn þ nnI Þ þ ðh5 2h2 ÞðJ nn þ nn JÞ 2 þ ðh6 þ h1 þ h2 =2 2h3 h5 Þnnnn ; ð2:4Þ where V* is the pore volume, V is the reference volume; I and J are unit tensors of the second and fourth ranks (Iij = dij, 2Jijkl = dikdlj + dildkj); and coefficients hi are given by Eq. (A2.1). Hereafter, dyadic notations are used, so that, for example, (Inn)ijkl = dijnknl. Strictly speaking, for a general mixture of multiple pores of diverse aspect ratios ^ obtained by summation and orientations, tensor H, (2.1), possesses the general anisotropy of the fourth rank elasticity tensor. In the case of randomly oriented pores ^ ¼ H
p ð6h1 2h2 þ 8h3 h5 þ h6 ÞII 15ð1 pÞ p ð2h1 þ 11h2 4h3 þ 8h5 þ 2h6 ÞJ. þ 30ð1 pÞ ð2:5Þ
However, for the spheroidal pores, an important ^ can be approximately simplification is possible: H expressed, with good accuracy, in terms of the symmetric second rank tensor 1 X x¼ ðV nnÞðkÞ ð2:6Þ V k (V k is kth pore volume and nk is a unit vector of its axis), in the sense that Df can be expressed with good accuracy in terms of x: h i p 2 b1 ðtr rÞ þ b2 trðr rÞ Df 2E0 ð1 pÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} isotropic terms
1 ½b3 r r þ b4 ðtr rÞr : x þ ð2:7Þ 2E0 ð1 pÞ ^ can be approximated by the or, alternatively, H expression ^ 1 p ðb1 II þ b2 JÞ H E0 1 p 1 1 ½b3 ðx J þ J xÞ þ b4 ðxI þ IxÞ; þ 2E0 1 p ð2:8Þ where factors bi, that reflect the distribution of pore aspect ratios, are given by (A2.4).
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Such a reduction, beyond a non-trivial and somewhat counterintuitive implication for the overall anisotropy (approximate orthotropy, coaxial to x, for any orientation distribution of pores), leads to explicit elasticity–conductivity linkage, as discussed in the text to follow. In the isotropic case (random orientations of spheroids), x = (p/3)I and ^ 1 p ½ðb1 þ b3 =3ÞII þ ðb2 þ b4 =3ÞJ. H E0 1 p
ð2:9Þ
This expression is a special case of the approximate formula (2.8) (whereas formula (2.5) is exact). 2.2. Conductivity We define symmetric second rank resistivity contribution tensor of a pore HR by DG ¼ H R U;
ð2:10Þ
where G is the far-field temperature gradient and U is the heat flux vector per unit volume. Tensor HR can be expressed in terms of EshelbyÕs tensor for conductivity. In the case of the spheroidal pore, HR ¼
V 1 ðD1 I þ D2 nnÞ; V k0
sor K—is given in terms of the same tensor x, defined by ((2.6), that enters the effective elasticity: k 0 K 1 I ¼
1 ðd 1 pI þ d 2 xÞ. 1p
ð2:12Þ
Here, k0 is the conductivity of the bulk material and multiplier (1 p)1 accounts for interactions between pores in the framework of MTS. Factors d1,2 reflect the pore aspect ratios distribution and are given by (A2.4). In the case of spheres, d1 = 3/2 and d2 = 0. In the case of overall isotropy (randomly oriented spheroids), 1 k p ðd 1 þ d 2 =3Þ . ¼ 1þ ð2:13Þ k0 1p Our analysis is limited to porosities not exceeding 0.15–0.17 (otherwise, a clearly identifiable yield point ceases to exist). Together with restriction on aspect ratio 0.3 < c < 4, this translates into a limitation on the possible range of combinations of effective conductivities k1/k0, k3/k0 (Fig. 2). In the isotropic case of random pore orientations, this implies k/k0 > 0.691 for c = 0.3, k/k0 > 0.730 for c = 1 (spherical pores) and k/k0 > 0.714 for c = 4.
ð2:11Þ
where dimensionless factors D1,2 are elementary functions of the aspect ratio, see (A2.5). In the case of multiple pores, the effective resistivity tensor K1—an inverse of the conductivity ten-
2.3. Elasticity–conductivity connection Both effective resistivity tensor K1 and effective elastic compliance tensor S can be expressed in terms of tensor x. This leads to the explicit
Fig. 2. (a) Possible combinations of principal conductivities k1 = k2 and k3 that correspond to aspect ratios 0.3 < c < 4. (b) Range of functional dependencies of k1 on porosity p for transversely isotropic and isotropic cases.
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elastic-conductive connection between changes in elastic compliances and changes in resistivity due to pores provided pore aspect ratios are not correlated with either pore volumes or pore orientations (pore volumes and pore orientations may be correlated). E0 ðS S 0 Þ ¼ ðC 1 II þ C 2 JÞ k 0 trðK 1 Þ 3 þ C 3 ðk 0 K 1 IÞI þ Iðk 0 K 1 IÞ þ C 4 ðk 0 K 1 IÞ J þ J ðk 0 K 1 IÞ . ð2:14Þ This connection is approximate; its accuracy is generally good and depends on PoissonÕs ratio m0 and pore aspect ratios (Sevostianov and Kachanov, 2002). Within this approximation, the (orthotropic) elastic properties are coaxial with the principal axes of K. Dimensionless coefficients C14 are shape factors that reflect the pore aspect ratio distribution (see (A2.3)). Their presence is due to differences between the effects of pore shapes on the overall elasticity and overall conductivity (otherwise, the cross-property connection would have been fully independent of pore shapes). However, the sensitivity of C14 to pore shapes is relatively mild, see the discussion to follow. In the case of transverse isotropy (x1x2 is the isotropy plane) (2.14) takes the form
G0 G13 k 0 k 11 k 0 k 33 ¼ 2C 2 þ ðC 2 þ C 4 Þ ; G13 k 11 k 33 E0 k 0 k 11 k 0 k 33 þ ðC 1 þ 2C 3 Þ ; m0 m31 ¼ 2C 1 E3 k 11 k 33 ð2:15Þ
2ð1 þ m0 Þ
(of the first three relations, only two are independent) where Ei is the effective YoungÕs modulus in the xi direction and Gij, mij are the effective shear moduli and PoissonÕs ratios in planes xixj. In the case of overall isotropy, the relations above reduce to the following ones: E0 E b1 þ b2 þ b3 þ b4 k 0 k ¼3 ; E k 3d 1 þ d 2 m0 kð3d 1 þ d 2 Þ þ 3ðk 0 kÞðb1 þ b3 Þ m¼ . kð3d 1 þ d 2 Þ þ 3ðk 0 kÞðb1 þ b2 þ b3 þ b4 Þ ð2:16Þ The isotropy takes place in one of the two cases: (A) spherical pores, and (B) randomly oriented spheroids. Whereas in case (A) the cross-property connections (2.16) are exact, in case (B) they are approximate, since they are based on approximate ^ in terms of second representation (2.8) of tensor H rank tensor x. However, in case (B), the exact cross-property connection can be established independently, by utilizing the exact result (2.5) for effective elasticity:
E0 E h1 þ 7h2 =10 þ 6h3 =5 þ 3h5 =5 k 0 k ¼ ; E 3d 1 þ d 2 k m0 kð3d 1 þ d 2 Þ ðk 0 kÞð6h1 =5 2h2 =5 þ 8h3 =5 h5 =5Þ m¼ . kð3d 1 þ d 2 Þ þ ðk 0 kÞð7h1 =5 7h2 =10 2h2 =5 þ 6h3 =5 þ 3h5 =5 þ 2h6 =5Þ
E0 E1 k 0 k 11 ¼ 2ðC 1 þ C 2 þ C 3 þ C 4 Þ E1 k 11 k 0 k 33 þ ðC 1 þ C 2 Þ ; k 33 G0 G12 k 0 k 11 k 0 k 33 2ð1 þ m0 Þ ¼ ð2C 2 þ C 4 Þ þ C2 ; G12 k 11 k 33 E0 k 0 k 11 k 0 k 33 þ C1 ; m0 m21 ¼ 2ðC 1 þ C 3 Þ E2 k 11 k 33 E0 E3 k 0 k 11 ¼ 2ðC 1 þ C 2 Þ E3 k 11 k 0 k 33 þ ðC 1 þ C 2 þ 2C 3 þ 2C 4 Þ ; k 33
913
ð2:17Þ
2.4. On the sensitivity of elasticity–conductivity connection to pore shape distribution The said sensitivity is of importance for the following reason. Utilization of cross-property connection (2.14) requires knowledge of C14, i.e. information on the pore aspect ratio distribution. The sensitivity of C14 to the pore aspect ratio distribution determines the required accuracy of this microscale information. The sensitivity to c is, obviously, maximal when all the pores have the same aspect ratios. It is reduced when c represents the maximum point of a certain distribution over aspect ratios. In the text
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to follow, we assume normal (Gaussian) distribution over c ! 2 1 ðc c0 Þ /ðcÞ ¼ pffiffiffiffiffiffi exp ð2:18Þ 2r2 r 2p with parameter r characterizing sharpness of the peak at c0; in the case of identical shapes, /(c) is a delta function, d(c c0). In the text to follow, we will always truncate Gaussian distribution to the interval of aspect ratios 0.3 < c < 4 where our basic model remains valid. Factors C14 and aij entering the elasticity–conductivity and the conductivityplasticity connections, will then be functions of c0 and r. Fig. 3 shows the sensitivity of C14 to pore aspect ratios c, in the relevant range 0.3 < c < 4.0. It is relatively mild even in the case of identical shapes (Fig. 3a). In the case of normal distribution with maximum at c0, the sensitivity to c0 is quite low for parameter r = 1 (Fig. 3b) and almost vanishes at r = 2 (Fig. 3c). Therefore, the required information on the pore shape distribution can be rather
imprecise. In cases of substantial ‘‘scatter’’ in pore shapes (r > 1), factors C14 can practically be treated as constants. 2.5. Plastic yield In the case of transverse isotropy (x1x2 is the isotropy plane), that appears to be the most common type of plastic anisotropy, the yield condition of a porous material has the form: 2
2s ¼ A1 ðrkk Þ2 þ A2 sij sji |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} isotropic part
þ A3 ðrkk Þr33 þ A4 r3j rj3 þ A5 r233 ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð2:19Þ
anisotropic part
where s* is the yield stress of the bulk material and sij = rij (rkk/3)dij is the stress deviator. Porosity makes the yield condition sensitive to the first stress invariant rkk. In the case of isotropy, this sensitivity was discussed by Skorokhod (1965), Green (1972) and Shima and Oyane (1976), among others.
Fig. 3. Sensitivity of factors C14 entering the elasticity–conductivity connection (2.11) to pore aspect ratios, distributed by normal law (d), at different values of parameter r: (a) r ! 0 (identical aspect ratios), (b) r = 1, (c) r = 2.
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Dimensionless factors A15 depend on the porous space geometry; they were given in terms of appropriate parameters of pore concentration, orientations and shapes by Sevostianov and Kachanov (2001). Remark. Our analysis for the case of transverse isotropy can be extended to the case of general orthotropy. 3. Plastic yield in terms of effective conductivities Combining expressions for plastic yield factors Ai in terms of effective elastic constants and the elasticity–conductivity linkage, the plastic yield condition can be given in terms of the effective conductivities. 3.1. Plastic yield factors Ai in terms of conductivities The most straightforward way to carry out the required calculations (that involve fourth rank tensorial inversions) is to use the technique of ‘‘standard’’ tensorial bases due to Kunin (1983) and Walpole (1984), see Appendix A. Any trans^ in particuversely isotropic fourth rank tensor—H, lar—is represented as a linear combination of six base tensors T(m): ^ ¼ H
6 X
^ hm T ðmÞ .
ð3:1Þ
m¼1
^m are expressed in terms As soon as coefficients h ^ ijkl (formulas (A1.2)), finding the product of H HijmnHnmkl and the inverse H 1 ijkl become standard operations. Thus, the elasticity–conductivity con^ yield the following nection (2.14) and S S 0 ¼ H expressions of ^ hm in terms of effectiveconductivities k11, k33: ^h1 E0 ¼ ð2C 1 þ C 2 þ 2C 3 þ C 4 Þ k 0 k 11 k 11 þ ðC 1 þ C 2 =2Þ
k 0 k 33 ; k 33
^h2 E0 ¼ 2ðC 2 þ C 4 Þ k 0 k 11 þ C 2 k 0 k 33 ; k 11 k 33 ^h3 E0 ¼ ^h4 E0 ¼ ð2C 1 þ C 3 Þ k 0 k 11 þ ðC 1 þ C 3 Þ k 0 k 33 ; k 11 k 33 ^h5 E0 ¼ 2ð2C 2 þ C 4 Þ k 0 k 11 þ 2ðC 2 þ C 4 Þ k 0 k 33 ; k 11 k 33
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k 0 k 11 ^ h6 E0 ¼ 2ðC 1 þ C 2 Þ k 11 þ ðC 1 þ C 2 þ 2C 3 þ 2C 4 Þ
k 0 k 33 ; k 33
ð3:2Þ where shape-dependent factors C14 are given by (A2.3). Plastic yield factors Ai can be expressed in terms of ^hm as follows (see Sevostianov and Kachanov (2001); two misprints are corrected here): 2G0 6ð1 m0 Þ^ A1 ¼ h1 6m0 ^ h2 h3 ð1 þ m0 Þ^ 3ð1 þ m0 Þ i 2G2 h ^2 ^2 ^2 þ 0 12h 1 þ 6h3 h2 ; 3 2 h2 þ 4G20 ^ h2 ; A2 ¼ 1 þ 4G0 ^ i 4G0 h ^ A3 ¼ 2h1 þ ð1 þ m0 Þ^ h2 þ ð2 þ m0 Þ^ h3 m0 ^ h6 1 þ m0 h i ^1 h ^3 h ^2 þ h ^2 2h ^2 ; ^3 þ 2h ^6 4h þ 4G2 4h 0
A4 ¼
2 4G0 ð^ h5 2^ h2 Þ þ 2G20 ð^ h5
1
2
3
2 4^ h2 Þ;
h i A5 ¼ 2G0 2^ h1 þ ^ h2 4^ h3 2^ h5 þ 2^ h6 h 2 i 2 2 2 2 þ 2G20 4^ h1 þ ^ h2 þ 6^ h3 ^ h5 þ 2^ h6 4^ h3 ð2^ h1 þ ^ h6 Þ ;
ð3:3Þ
where G0 is the shear modulus of the bulk material. Substituting (3.2) into (3.3) yields the main result of the present work—plastic yield factors Ai in terms of conductivities:
2 k 0 k 11 k 0 k 33 k 0 k 11 þ a12 þ a13 k 11 k 33 k 11
2 k 0 k 33 k 0 k 11 k 0 k 33 þ a14 þ a15 ; k 33 k 11 k 33
2 k 0 k 11 k 0 k 33 k 0 k 11 A2 ¼ 1 þ a21 þ a22 þ a23 k 11 k 33 k 11
2 k 0 k 33 k 0 k 11 k 0 k 33 þ a24 þ a25 ; k 33 k 11 k 33
2 k 0 k 11 k 0 k 33 k 0 k 11 A3 ¼ a31 þ a32 þ a33 k 11 k 33 k 11
2 k 0 k 33 k 0 k 11 k 0 k 33 þ a34 þ a35 ; k 33 k 11 k 33
2 k 0 k 11 k 0 k 33 k 0 k 11 A4 ¼ a41 þ a42 þ a43 k 11 k 33 k 11
2 k 0 k 33 k 0 k 11 k 0 k 33 þ a44 þ a45 ; k 33 k 11 k 33 A1 ¼ a11
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2 k 0 k 11 k 0 k 33 k 0 k 11 þ a52 þ a53 k 11 k 33 k 11
2 k 0 k 33 k 0 k 11 k 0 k 33 þ a54 þ a55 . k 33 k 11 k 33
A5 ¼ a51
ð3:4Þ Dimensionless coefficients aij are not sensitive to either pore orientation distribution or porosity, but they do depend on the distribution of pore aspect ratios (via C14), as illustrated by Fig. 4. Note that these relations are explicit and contain no adjustable parameters.
The utility of plasticity–conductivity connections (3.4) is as follows. Without them, Ai can be expressed in terms of coefficients ^hm —characteristics of the pore space. However, these characteristics require knowledge not only of pore shapes, but of the orientation distribution and porosity as well. Such information may not be readily available. Utilization of the cross-property connection makes this information unnecessary. Indeed, if the conductivities are known, the only microscale information needed in order to construct the yield surface is the distribution over pore shapes.
Fig. 4. Coefficients aij entering the plasticity–conductivity connection (3.4) as functions of pore aspect ratios (identical aspect ratios).
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The required accuracy of this information depends on the sensitivity of the plasticity–conductivity connection to pore shapes. This issue is discussed below, in the case of identical aspect ratios and in the more realistic case of Gaussian distribution over the aspect ratios. 3.2. Sensitivity of the plasticity–conductivity connection to pore aspect ratios
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to pore shapes is low, even in the case of identical aspect ratios. It is even lower if a certain scatter over the aspect ratios is present. Remark. Factors A1 and A5 are much smaller than A2, A3, A4. It appears that these two terms in the yield condition can be neglected, at least, in the considered range of aspect ratios 0.3 < c < 4. 4. Cases of overall isotropy
The sensitivity of coefficients aij to pore shapes in the range of aspect ratios 0.3 < c < 4.0 is seen from Fig. 4. Fig. 4 assumes that all pores have identical aspect ratios. A more direct examination of the sensitivity of plastic yield factors Ai to pore shapes is presented in Figs. 5 and 6. They show the dependence of Ai on k11/k0 at several fixed values of k33/k0, for the oblate and prolate pores of identical aspect ratios, respectively. (The range of conductivities covered in these figures corresponds to the relevant intervals of porosity and aspect ratios, see Fig. 2). The curves of Figs. 5 and 6 are quite close. Thus, the sensitivity
In the case of overall isotropy, yield condition (2.3) reduces to the form 2
2
2s ¼ A1 ðrkk Þ þ A2 sij sji ;
ð4:1Þ
where the first term reflects sensitivity to the average hydrostatic stress (note that, usually, A1 A2). Of six coefficients ^hm , only three—^h1 , ^h2 and ^h3 ¼ ^h1 ^h2 =2—enter formulas for A1, A2, and they are given in terms of effective isotropic conductivity k as follows: E0 ^h1 ¼ aðk 0 kÞ=k, E0 ^h2 ¼ bðk 0 kÞ=k, so that the plastic yield factors, expressed in terms of k are
Fig. 5. Plastic yield factors A15 as functions of conductivity k1/k0 at several values of k3/k0. All pores have aspect ratio c = 0.4.
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Fig. 6. Plastic yield factors A15 as functions of conductivity k1/k0 at several values of k3/k0. All pores have aspect ratio c = 3.
k0 k k 3ð1 þ m0 Þ
2 1 2 k0 k þ ð6a bÞ ; 2 k 3ð1 þ m0 Þ
2 1 k0 k A2 ¼ 1 þ b ; 1 þ m0 k
A1 ¼
2ð1 2m0 Þ 2
ity/conductivity connections (2.17). This would yield the same formulas (4.2) but with different coefficients a, b:
½6a b
ð4:2Þ
where a, b reflect the aspect ratio distribution; they are given by a ¼ 3C 1 þ ð3=2ÞC 2 þ 2C 3 þ C 4 ; b ¼ 3C 2 þ 2C 4 .
ð4:3Þ
Relations (4.2) are illustrated in Fig. 7 (identical aspect ratios are assumed). Formulas (4.3) for a, b follow from results for the transversely isotropic case, in the special case of isotropy. They reflect the approximations involved in deriving the elasticity–conductivity connection (2.16), in particular, the approximate character of (2.8) and (2.9). However, the case of isotropy can be analyzed independently, using the exact elastic-
a 12q1 28q3 þ 13q6 16q2 þ 3q5 þ ¼ ; E0 120ðq1 q6 q23 Þ 60q2 q5 b 4q þ 4q3 þ q6 32q2 þ 11q5 þ ¼ 1 ; E0 30q2 q5 60ðq1 q6 q23 Þ
ð4:4Þ
where coefficients qi are given in Appendix B. Relations (4.4) are illustrated in Fig. 8 (identical aspect ratios are assumed). Comparison of the last two figures shows that exact relations (4.4) produce a lower sensitivity to pore aspect ratios. This indicates that the approximations involved in deriving the general anisotropic plasticity–conductivity connection tend to exaggerate the sensitivity to pore shapes. We now consider an important for applications case of normal distribution of aspect ratios that has maximum at c0 = 1 (sphere). Fig. 9 compares the extreme case of parameter r = 5 (an almost uniform distribution over aspect ratios in the interval
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Fig. 7. Case of overall isotropy. Plastic yield factors A1,2 as functions of conductivities, based on approximate relations (4.3) (identical pore aspect ratios).
Fig. 8. Case of overall isotropy. Plastic yield factors A1,2 as functions of conductivities, based on exact relations (4.4) (identical pore aspect ratios).
0.3 < c < 4) with the opposite extreme case when all the pores are spherical (r = 0), for which 2 2
ð1 m0 Þð1 2m0 Þ k 0 k ð1 m0 Þ k0 k þ ; A1 ¼ 2 2 k k 12ð1 þ m0 Þ 3ð1 þ m0 Þ
2 5ð1 m0 Þ k 0 k A2 ¼ 1 þ . 7 5m0 k ð4:5Þ The curves corresponding to the two cases are very close. This means that, as long as the maximum point of normal distribution is c0 = 1 (sphere), the
effect of the shape ‘‘scatter’’ is negligible. In such cases, therefore, no microscale information is needed at all, and plastic yield factors A1, A2 are given by (4.5). 5. Discussion and conclusions In their earlier work, the authors constructed yield surfaces of porous metals in terms of the porous space geometry—more precisely, in terms of those parameters of pore distribution that determine the effective elasticity. However, the microscale
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Fig. 9. Case of overall isotropy. Plastic yield factors A1,2 as functions of conductivities. Comparison of the case of spherical pores with the case of normal distribution over pore aspect ratios, c0 = 1 (sphere) being the maximum point.
information contained in these parameters may not be readily available, particularly the information on pore orientations. We suggest a methodology to construct anisotropic yield surfaces in terms of electric conductivities—quantities that are easily measured. It is, obviously, required that the yield point is identifiable, i.e. is not ‘‘fuzzy’’ beyond recognition. This translates into two requirements: • The bulk (‘‘dense’’) material experiences a negligible hardening (more precisely, it should have a clearly identifiable yield point followed by an approximately horizontal plateau; this does not exclude a non-horizontal curve at later stages of loading). Aluminum and steel are examples. • Porosity does not exceed 0.15–0.17. The formulas that relate the plastic yield to conductivities do not contain any information on either pore orientations or the overall porosity. They do depend, though, on the distribution over pore aspect ratios. However, the sensitivity to this factor is relatively mild, even in the case of identical aspect ratios, as seen from comparison of Figs. 5 and 6. If some scatter in pore shapes is present (modeled by Gaussian distribution over aspect ratios), the sensitivity to the average aspect ratio is reduced even further. Therefore, the required knowledge of the
aspect ratio distribution can be rather imprecise; moreover, if pore shapes are known to have a sufficient scatter, no microscale information is needed at all. We also further examined the elasticity–conductivity connection (on which the present results are based), from the point of view of their sensitivity to the pore shape distribution. It is found that, whereas the sensitivity is noticeable in the case of identical pore aspect ratios, a scatter in pore shapes reduces this sensitivity substantially (Fig. 3). The case of overall isotropy (randomly oriented spheroids) is considered. It is found that, in the important for applications case when the distribution of pore aspect ratios has maximum at c0 = 1 (sphere), the effect of the shape ‘‘scatter’’ is negligible and the plastic yield condition is practically the same as in the case when all pores are spherical. In this case, therefore, no microscale information is needed at all, in order to utilize the plasticity–conductivity connection. The results apply to an arbitrary mixture of spheroidal pores (diverse aspect ratios and arbitrary orientation distribution), under the condition that the distributions over aspect ratios and over orientations are not statistically correlated. The restriction of spheroidal shapes can actually be relaxed; for example, moderate ‘‘jaggedness’’ of pore boundaries and some other shape ‘‘irregularities’’ can be
I. Sevostianov, M. Kachanov / Mechanics of Materials 38 (2006) 908–923
ignored (they are either unimportant for the properties considered, or their effects on plastic yield and conductivity are similar and, hence, do not affect the link between the two). A more restrictive limitation is that geometric ‘‘extremes’’—strongly oblate or strongly prolate pores (aspect ratios smaller than 0.3 or larger than 4)—are not present in significant numbers. The present work focuses on the cases of overall transverse isotropy (that appears to be the main case of plastic anisotropy) and isotropy. However, the same line of derivation can be applied to a more general case of overall orthotropy.
coefficients of this representation are expressed in terms of components Wijkl as follows: w1 ¼ ðW1111 þ W1122 Þ=2; w3 ¼ W1133 ;
ð1Þ
T ijkl ¼ ðhik hlj þ hil hkj hij hkl Þ=2; ð4Þ T ijkl
¼ hij mk ml ;
¼
¼ mi mj hkl ;
ð6Þ
T ijkl ¼ mi mj mk ml ; ðA1:1Þ where hij = dij mimj and m = miei is a unit vector of the axis of transverse isotropy. We assume that x1x2 is the isotropy plane. Then base tensors T(i) have the following non-zero components: ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
T 1111 ¼ T 2222 ¼ T 1122 ¼ T 2211 ¼ 1; ð2Þ
¼ ¼ ¼
6 X
ðiÞ
ðA1:3Þ
si T ijkl
has the following coefficients si in this basis: s1 ¼
1m ; 4Gð1 þ mÞ
s5 ¼
2 ; G
s6 ¼
s2 ¼
1 ; 2G
s3 ¼ s4 ¼
m ; 2Gð1 þ mÞ
1 . 2Gð1 þ mÞ
ðA1:4Þ
The table of multiplication of the base tensors has the form:
¼ 1=2;
ð3Þ T 1133
¼
ð3Þ T 2233
¼ 1;
ð6Þ
¼ ¼
¼ ¼
ð5Þ T 2332 ð5Þ T 3131
T1
T2
T3
T4
T5
T6
2T1 0 0 2T4 0 0
0 T2 0 0 0 0
2T3 0 0 2T6 0 0
0 0 T1 0 0 T4
0 0 0 0 T5/2 0
0 0 T3 0 0 T6
Representing an arbitrary fourth rank symmetric tensors X and Y in this tensorial basis X¼
6 X
X k T ðkÞ ;
Y¼
k¼1
6 X
Y k T ðkÞ
ðA1:5Þ
k¼1
we readily obtain the inverse of X and a product of X and Y as follows.
¼ 1; T 3333 ¼ 1; ð5Þ T 1331 ð5Þ T 3223
T1 T2 T3 T4 T5 T6
ð2Þ
T 1212 ¼ T 2121 ¼ T 1221 ¼ T 2112 ¼ T 1111 ¼ T 2222 ¼ 1=2; ¼
ðA1:2Þ
i¼1
ð5Þ
ð2Þ T 2211 ð4Þ T 3322 ð5Þ T 2323 ð5Þ T 3113
w5 ¼ 4W1313 ;
For example, elastic compliance tensor of the isotropic material 1 1 m ðdik djl þ dil dkl Þ dij dkl S ijkl ¼ 2G 2 1þm
T ijkl ¼ ðhik ml mj þ hil mk mj þ hjk ml mi þ hjl mk mi Þ=4;
ð2Þ T 1122 ð4Þ T 3311 ð5Þ T 1313
w4 ¼ W3311 ;
ð2Þ
T ijkl ¼ hij hkl ; ð3Þ T ijkl
w2 ¼ 2W1212 ;
w6 ¼ W3333 .
Appendix A. A technique of analytic inversion and multiplication of fourth rank tensors We outline a convenient technique of analytic inversion and multiplication of fourth rank tensors, due to Kunin (1983) and Walpole (1984). It is based on expressing tensors in ‘‘standard’’ tensorial bases. In the case of transversely isotropic symmetry, the following basis of six tensors T(m) is convenient:
921
ð5Þ
(a) Inverse tensor:
¼ T 3232 ¼ 1=4. ðA1:8Þ
For an arbitrary transversely isotropic fourth rank tensor W represented in this basis X Wijkl ¼ wm T mijkl
X 1 ¼
X 6 ð1Þ 1 ð2Þ X 3 ð3Þ X 4 ð4Þ T þ T T T X2 2D D D 4 ð5Þ 2X 1 ð6Þ T ; þ T þ ðA1:6Þ X5 D
where D = 2(X1X6 X3X4).
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(b) Product X : Y (tensor with ijkl components XijmnYmnkl) is by X : Y ¼ ð2X 1 Y 1 þ X 3 Y 4 ÞT ð1Þ þ X 2 Y 2 T ð2Þ þ ð2X 1 Y 3 þ X 3 Y 6 ÞT ð3Þ þ ð2X 4 Y 1 þ X 6 Y 4 ÞT
ð4Þ
1 þ X 5 Y 5 T ð5Þ 2
þ ðX 6 Y 6 þ 2X 4 Y 3 ÞT ð6Þ .
where Z bi ¼
We consider a spheroidal pore with x3 being the symmetry axis and c the aspect ratio. We express tensor Hijkl of this pore in terms of base tensors ð16Þ T ijkl of the standard basis given in P6tensorial ðmÞ Appendix A: H ijkl ¼ m¼1 hm T ijkl . Coefficients hi were given by Sevostianov and Kachanov (1999) for an inclusion that has arbitrary elastic properties. For a pore, they reduce to the following ones:
V q6 V 1 ; h2 ¼ ; 2 V 4ðq1 q6 q3 Þ V q2 V q3 V 4 h3 ¼ h4 ¼ ; h ¼ ; 5 V 2ðq1 q6 q23 Þ V q5 V q1 h6 ¼ ; ðA2:1Þ V ðq1 q6 q23 Þ
h1 ¼
di ¼
0
1
Di ðcÞ/ðcÞ dc 0
and /(c) is the shape distribution density (specialized as Gaussian function in the main text), i.e. /(c)dc is the volume fraction of pores with aspect ratios in the interval (c, c + dc). Dimensionless functions Bi(c) and Di(c) are given by 1
D1 ¼ ð1 f0 ðcÞÞ ; D2 ¼ ð1 3f 0 ðcÞÞ=2f 0 ðcÞ½1 f0 ðcÞ; B1 ¼ E0 ð^h1 ^h2 =2Þ; B2 ¼ E0 ^h2 ; B3 ¼ E0 ð2^h3 þ ^h2 2^h1 Þ;
B4 ¼ E0 ð^h5 2^h2 Þ ðA2:5Þ
and coefficients ^hi are obtained from hi as follows: ^h1 ¼ h1 ð1 d sign h1 Þ; ^h2 ¼ h2 ð1 d sign h2 Þ; ^h3 ¼ h3 ð1 þ d sign h3 Þ; ^h5 ¼ h5 ð1 þ d sign h5 Þ;
where
q3 ¼ q4 ¼ 2G0 ½ð2j 1Þf0 þ 2f 1 ; q5 ¼ 4G0 ½f0 þ 4f 1 ; q6 ¼ 8G0 ½jf0 f1 ;
d¼
1 ; 2ð1 m0 Þ jc2
f0 ¼
c 1
c
h6 þ h1 þ h2 =2 2h3 h5 . jh6 j þ jh1 j þ jh2 j=2 þ 2jh3 j þ jhj5
ðA2:6Þ
ðA2:7Þ
In the case of identical aspect ratios c = c0, the shape distribution density is a delta-function: /(c) = d(c c0) and integrals (A2.4) reduce to Bi(c0) and Di(c0), correspondingly.
c2 ð1 gÞ ; 2ðc2 1Þ
½ð2c2 þ 1Þg 3; 2 4ðc2 1Þ 8 pffiffiffiffiffiffiffi 1c2 > 1 > arctan ; oblate shape; < cpffiffiffiffiffiffiffi c 2 1c g¼ pffiffiffiffiffiffiffi > cþ c2 1 > : p1ffiffiffiffiffiffiffi ln pffiffiffiffiffiffiffi ; prolate shape: 2 2 2c
ðA2:3Þ
ðA2:4Þ
q1 ¼ G0 ½4j 1 2ð3j 1Þf0 2f 1 ; q2 ¼ 2G0 ½1 ð2 jÞf0 f1 ;
f1 ¼
Bi ðcÞ/ðcÞ dc;
Z
^h6 ¼ h6 ð1 d sign h6 Þ;
where
j¼
1
ðA1:7Þ
Appendix B. Tensor H of a spheroidal pore. Pore shape factors C14
b1 d 2 2b3 d 1 b2 d 2 2b4 d 1 ; C2 ¼ ; d 2 ðd 2 þ 3d 1 Þ d 2 ðd 2 þ 3d 1 Þ b3 b4 C3 ¼ ; C4 ¼ ; d2 d2
C1 ¼
References
c 1
ðA2:2Þ For a sphere, f0 = 1/3 and f1 = j/15 = 1/(30(1 m0)). Coefficients C14 entering (2.11) are related to coefficients h16 as follows:
Benveniste, Y., 1986. On the Mori–Tanaka method for cracked solids. Mechanics Research Communications 13 (4), 193–201. da Silva, M.G., Ramesh, K.T., 1997. The rate-dependent deformation and localization of fully dense and porous Ti– 6Al–4V. Materials Science and Engineering A232, 11–22. Green, R.J., 1972. A plasticity theory for porous solids. International Journal of Mechanical Sciences 14, 215–224. Kachanov, M., Sevostianov, I., 2005. On quantitative characterization of microstructures and effective properties. International Journal of Solids and Structures 42, 309–336.
I. Sevostianov, M. Kachanov / Mechanics of Materials 38 (2006) 908–923 Kachanov, M., Tsukrov, I., Shafiro, B., 1994. Effective moduli of solids with cavities of various shapes. Applied Mechanics Reviews 47 (1), S151–S174. Kachanov, M., Sevostianov, I., Shafiro, B., 2001. Explicit crossproperty correlations for porous materials with anisotropic microstructures. Journal of the Mechanics and Physics of Solids 49, 1–25. Kanaun, S., 1983. Elastic medium with random field of inhomogeneities Chapter 7 in book. In: Kunin, I.A. (Ed.), Elastic Media with Microstructure, vol. II. Springer-Verlag, Berlin, pp. 165–228. Kunin, I.A., 1983. Elastic Media with Microstructure. SpringerVerlag, Berlin. Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica 21, 571–574. Sevostianov, I., Kachanov, M., 1999. Compliance tensor of ellipsoidal inclusions. International Journal of Fracture 96, L3–L7. Sevostianov, I., Kachanov, M., 2001. On the yield condition for anisotropic porous materials. Materials Science and Engineering A 313, 1–15. Sevostianov, I., Kachanov, M., 2002. Explicit cross-property correlations for anisotropic two-phase composite materials. Journal of the Mechanics and Physics of Solids 50, 253–282.
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Sevostianov, I., Bogarapu, M., Tabakov, P., 2002a. Correlation between elastic and electric properties for cyclically loaded metals. International Journal of Fracture 115, L15–L20. Sevostianov, I., Kova´cˇik, J., Simancˇ´ık, F., 2002b. Cross-property correlation for metal foams: theory and experiment. International Journal of Fracture 114, L23–L28. Shima, S., Oyane, M., 1976. Plasticity theory for porous metals. International Journal of Mechanical Sciences 18, 285–291. Skorokhod, V., 1965. Mean square stress and strain rate for porous materials. Soviet Powder Metallurgy and Metal Ceramics 12, 31–35. Walpole, L.J., 1984. Fourth-rank tensors of the thirty-two crystal classes: multiplication tables. Proceedings of the Royal Society of London (A) 391, 149–179. Wang, Z.C., Davies, T.J., Ridley, N., Ogwu, A.A., 1996. Superplasticity of ceramic materials—II. Effect of initial porosity and doping on the superplastic behaviour of alumina. Acta Materialia 44, 4301–4309. Zohdi, T., 2004. Private communication. Zohdi, T., Kachanov, M., 2005. A note on the micromechanics of plastic yield of porous solids. International Journal of Fracture 133 (3), L31–L35. Zohdi, T., Kachanov, M., Sevostianov, I., 2002. A microscale numerical analysis of a plastic flow in a porous material. International Journal of Plasticity 18, 1649–1659.
Plastic yield surfaces of anisotropic porous materials in terms of effective electric conductivities Igor Sevostianov a, Mark Kachanov a
b,*
Department of Mechanical Engineering, New Mexico State University, P.O. Box 30001, Las Cruces, NM 88003, USA b Department of Mechanical Engineering, Tufts University, 204 Anderson Hall, Medford, MA 02155, USA Received 10 December 2004
Abstract Plastic yield surface of a porous ductile material with generally anisotropic porous space is constructed in terms of the effective conductivities. Such a cross-property connection is possible provided hardening of the bulk material is negligible and porosity does not exceed 0.15–0.17. The following results are utilized: (1) computational studies of Zohdi et al. [Zohdi, T., Kachanov, M., Sevostianov, I., 2002. A microscale numerical analysis of a plastic flow in a porous material. International Journal of Plasticity 18, 1649–1659] show that, if the mentioned conditions are met, local ‘‘pockets’’ of plasticity remain isolated and well contained in the elastic field, resulting in approximately linear stress–strain curve, almost up to the yield point; (2) elasticity–conductivity cross-property connections for porous materials that were given in the explicit form by Sevostianov and Kachanov [Sevostianov, I., Kachanov, M., 2002. Explicit cross-property correlations for anisotropic two-phase composite materials. Journal of the Mechanics and Physics of Solids 50, 253–282]. 2005 Elsevier Ltd. All rights reserved. Keywords: Plasticity; Porosity; Yield; Conductivity; Porous material
1. Introduction Yield stress of a porous metal is not a precisely defined point, but represents a certain approximation of reality—small ‘‘pockets’’ of inelasticity may develop even at low applied stresses. We consider cases when the yield surface can be clearly identified. Experimental evidence (Wang et al., 1996; da Silva
and Ramesh, 1997), as well as theoretical analyses (Sevostianov and Kachanov, 2001) and computational studies (Zohdi et al., 2002) show (Fig. 1) that this is the case, provided:
*
Corresponding author. Tel.: +1 617 627 3318; fax: +1 617 627 3058. E-mail addresses: [email protected] (I. Sevostianov), [email protected] (M. Kachanov). 0167-6636/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2005.06.023
(A) Hardening of the bulk (‘‘dense’’) material is negligible. More precisely, the bulk material must have a clearly identifiable yield point followed by an approximately horizontal plateau (this does not exclude a non-horizontal curve at later stages of loading). Aluminum and steel are examples. (B) Volume fraction of pores (porosity p) does not exceed a certain critical level beyond which
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909
Fig. 1. Elastic–plastic behavior of porous metals. (a) Computational simulation (Zohdi et al., 2002) for porous aluminum at three volume fractions of ellipsoidal pores; experimental data for dense and porous Ti–6Al–4V under (b) uniaxial tension and (c) shear (da Silva and Ramesh, 1997).
the stress–strain curve starts to lose a clearly identifiable yield point. Computational studies of Zohdi et al. (2002) show that, in the case of oblate pores with aspect ratio 1/3, this level is reached at p 0.15 and, for the spherical pores, at p 0.17. If the above conditions are met, the overall stress–strain curve remains approximately linear almost up to the yield point, with the slope determined by effective elastic properties of the porous material. The underlying microscale mechanism, made explicit in the mentioned computational studies, is as follows. Local plastic ‘‘pockets’’ remain small and well contained in the surrounding elastic field (the latter is not significantly perturbed by them) until, within a narrow stress interval—that can be idealized as the yield point—a substantial volume fraction of the matrix plasticizes. At, or
near, the yield point, the field of stress deviator is almost uniform. Remark. The described scenario—that differs from the one of plastic localization—is based on the essential assumption that the bulk material does not harden so that small plastic ‘‘pockets’’ blunt stress concentrations, making further propagation of the ‘‘pockets’’ difficult (as illustrated by computational simulation of Zohdi et al., 2002). Otherwise, the mentioned stress blunting is less pronounced, and this may lead to localization of plastic deformation. Yet another restriction is that pores of ‘‘extreme’’ eccentricities (spheroids with aspect ratios smaller than 0.3 or larger than 4) should not be present in significant numbers (Zohdi, 2004). Experimental data on plastic yield at different porosities indicate that the macroscopic strain at the yield point—defined as the intersection point
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of the elastic line and the horizontal plastic plateau—remains approximately constant (independent of porosity). This observation (that was given a microscale interpretation by Zohdi and Kachanov, 2005), together with linearity of the stress–strain curve almost up to the yield point, makes it possible to express yield surface in the stress space in terms of the effective elastic constants and yield stress of the bulk material. This leads, in turn, to construction of the yield surface in terms of those characteristics of porous space that determine the effective elasticity—appropriate scalar or tensorial measures of pore concentration, shapes and orientations (Sevostianov and Kachanov, 2001). However, such constructions require the microscale information that may not be readily available, such as the orientation distribution of pores (for example, the extent of orientation scatter about a certain preferential orientation). It would be desirable, therefore, to directly link the yield surface to effective conductivities—quantities that are easily measured. We make use of elasticity–conductivity connection for porous materials that was given, in the explicit form, by Kachanov et al. (2001), Sevostianov and Kachanov (2002) and verified experimentally, for metal materials, by Sevostianov et al. (2002a,b). The conductivity can be either the electric or the thermal one; from the practical standpoint, the electrical ones are of interest since they are easier to measure. This connection leads to explicit expressions of the yield surface in terms of effective conductivities. Such expressions—given in the text to follow— still contain some microstructural information, namely, parameters that reflect the distribution of pore shapes. However, the sensitivity to pore shapes is relatively mild. We emphasize that no information on the orientation distribution of pores or on the overall porosity is needed. Thus, if the conductivity data are available, construction of the yield surface requires only limited (and rather approximate) microscale information. The sensitivity to pore shapes is further reduced if pore aspect ratios are not identical but have a ‘‘scatter’’ about a certain point. We discuss the role of the scatter, modeling it by Gaussian distribution. The results are derived for the spheroidal pores. Although the actual pore shapes may or may not resemble spheroids, this restriction is not as severe as it seems. Indeed, many of the shape ‘‘irregularities’’ (like ‘‘jaggedness’’ of pore boundaries) are
either unimportant for the effective properties or affect the conductive/elastic properties in a similar way, so that the elasticity–conductivity connections—on which the present work is based—are not affected (see Kachanov and Sevostianov (2005) for a discussion in detail). A more restrictive limitation is that the aspect ratios of pores should be between 0.3 and 4, i.e. pores of ‘‘extreme’’ eccentricities should not be present in significant numbers. We focus on the case of transverse isotropy that appears to be the main observed type of plastic anisotropy. It covers parallel pore orientations (that may be randomly perturbed) and spheroidal pores with axes lying in a plane (that may have some random orientation scatter about the plane). We also discuss the case of isotropy that covers randomly oriented spheroids and spheres. We assume that the bulk (‘‘dense’’) material is isotropic and its properties (elasticity, conductivity and the yield stress) are known.
2. Background results on anisotropic porous materials We overview relevant results on the effective elastic/conductive properties and plastic yield of porous materials. These results hinge on the proper quantitative characterization of the porous space. This means identifying parameters of pore concentration that incorporate contributions of individual pores in accordance with their actual contributions to the physical property considered. For mixtures of pores of diverse shapes and orientations, such parameters are non-trivial, and they are, generally, different for different physical properties. Importantly, for the effective elasticity and effective conductivity, the proper microstructural parameters turn out to be largely similar, giving rise to the cross-property connection. 2.1. Effective elasticity Effective elasticity of a porous material determines the slope of the overall stress–strain curve that remains approximately linear almost up to the yield point. Together with the observation that the strain at yield is approximately independent of porosity (and, thus, the same as for the bulk material), this leads to expression of the yield surface in the stress space in terms of effective elastic constants and yield stress of the bulk material.
I. Sevostianov, M. Kachanov / Mechanics of Materials 38 (2006) 908–923
We define fourth rank compliance contribution tensor H(N) of Nth pore by the extra overall strain ðN Þ ðN Þ due to its presence: Deij ¼ H ijkl rkl , where rkl is the macroscopic stress. H-tensors were calculated for the ellipsoidal pores, in terms of EshelbyÕs tensor, by Kachanov et al. (1994) (see, also, Kanaun (1983) where related results in a different form were given). The sum ^ ¼ H
1 X ðN Þ H 1p N
ð2:1Þ
gives the collective effect of all pores on the overall elastic compliances in the approximation of MoriTanakaÕ scheme, MTS (1973) that has been reformulated in the form used here by Benveniste (1986). This scheme places each pore into the average, over the solid phase, stress; then multiplier (1 p)1 accounts for interactions between pores. Its omission corresponds to the non-interaction approximation. Remark. Approximate schemes other than MTS can be incorporated into our analysis. However, at porosities not exceeding 0.15–0.17, sensitivity of the results to the choice of a particular scheme is relatively low. MTS appears to be the most appropriate one, since the field of stress deviator is almost uniform near the yield point (Zohdi et al., 2002). Effective compliances can be represented as a ^ ijkl where S 0 are compliances sum S ijkl ¼ S 0ijkl þ H ijkl of the bulk material, therefore for the elastic potential (in stresses) we have f ðrÞ ¼ f0 þ Df ;
ð2:2Þ
where f0(r) = (1/2E0)[(1 + m0)rijrji m0(rkk)2] is the potential in absence of pores and ^ : r ¼ ð1=2ÞH ^ ijkl rij rkl Df ¼ ð1=2Þr : H
ð2:3Þ
is the change due to pores (see, for example, Kachanov et al., 1994). The structure of Df implies that ^ is the general proper microfourth rank tensor H structural parameter for the effective elastic properties (it represents individual pores according to their contributions to the overall compliance). For multiple pores of general spheroidal shapes, H-tensor of an individual pore (with unit vector n along its axis) that enter the sum (2.1), have the form
911
V 1 ðh1 h2 =2ÞII þ h2 trJ þ ð2h3 þ h2 2h1 Þ H¼ 2 V 1 ðInn þ nnI Þ þ ðh5 2h2 ÞðJ nn þ nn JÞ 2 þ ðh6 þ h1 þ h2 =2 2h3 h5 Þnnnn ; ð2:4Þ where V* is the pore volume, V is the reference volume; I and J are unit tensors of the second and fourth ranks (Iij = dij, 2Jijkl = dikdlj + dildkj); and coefficients hi are given by Eq. (A2.1). Hereafter, dyadic notations are used, so that, for example, (Inn)ijkl = dijnknl. Strictly speaking, for a general mixture of multiple pores of diverse aspect ratios ^ obtained by summation and orientations, tensor H, (2.1), possesses the general anisotropy of the fourth rank elasticity tensor. In the case of randomly oriented pores ^ ¼ H
p ð6h1 2h2 þ 8h3 h5 þ h6 ÞII 15ð1 pÞ p ð2h1 þ 11h2 4h3 þ 8h5 þ 2h6 ÞJ. þ 30ð1 pÞ ð2:5Þ
However, for the spheroidal pores, an important ^ can be approximately simplification is possible: H expressed, with good accuracy, in terms of the symmetric second rank tensor 1 X x¼ ðV nnÞðkÞ ð2:6Þ V k (V k is kth pore volume and nk is a unit vector of its axis), in the sense that Df can be expressed with good accuracy in terms of x: h i p 2 b1 ðtr rÞ þ b2 trðr rÞ Df 2E0 ð1 pÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} isotropic terms
1 ½b3 r r þ b4 ðtr rÞr : x þ ð2:7Þ 2E0 ð1 pÞ ^ can be approximated by the or, alternatively, H expression ^ 1 p ðb1 II þ b2 JÞ H E0 1 p 1 1 ½b3 ðx J þ J xÞ þ b4 ðxI þ IxÞ; þ 2E0 1 p ð2:8Þ where factors bi, that reflect the distribution of pore aspect ratios, are given by (A2.4).
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Such a reduction, beyond a non-trivial and somewhat counterintuitive implication for the overall anisotropy (approximate orthotropy, coaxial to x, for any orientation distribution of pores), leads to explicit elasticity–conductivity linkage, as discussed in the text to follow. In the isotropic case (random orientations of spheroids), x = (p/3)I and ^ 1 p ½ðb1 þ b3 =3ÞII þ ðb2 þ b4 =3ÞJ. H E0 1 p
ð2:9Þ
This expression is a special case of the approximate formula (2.8) (whereas formula (2.5) is exact). 2.2. Conductivity We define symmetric second rank resistivity contribution tensor of a pore HR by DG ¼ H R U;
ð2:10Þ
where G is the far-field temperature gradient and U is the heat flux vector per unit volume. Tensor HR can be expressed in terms of EshelbyÕs tensor for conductivity. In the case of the spheroidal pore, HR ¼
V 1 ðD1 I þ D2 nnÞ; V k0
sor K—is given in terms of the same tensor x, defined by ((2.6), that enters the effective elasticity: k 0 K 1 I ¼
1 ðd 1 pI þ d 2 xÞ. 1p
ð2:12Þ
Here, k0 is the conductivity of the bulk material and multiplier (1 p)1 accounts for interactions between pores in the framework of MTS. Factors d1,2 reflect the pore aspect ratios distribution and are given by (A2.4). In the case of spheres, d1 = 3/2 and d2 = 0. In the case of overall isotropy (randomly oriented spheroids), 1 k p ðd 1 þ d 2 =3Þ . ¼ 1þ ð2:13Þ k0 1p Our analysis is limited to porosities not exceeding 0.15–0.17 (otherwise, a clearly identifiable yield point ceases to exist). Together with restriction on aspect ratio 0.3 < c < 4, this translates into a limitation on the possible range of combinations of effective conductivities k1/k0, k3/k0 (Fig. 2). In the isotropic case of random pore orientations, this implies k/k0 > 0.691 for c = 0.3, k/k0 > 0.730 for c = 1 (spherical pores) and k/k0 > 0.714 for c = 4.
ð2:11Þ
where dimensionless factors D1,2 are elementary functions of the aspect ratio, see (A2.5). In the case of multiple pores, the effective resistivity tensor K1—an inverse of the conductivity ten-
2.3. Elasticity–conductivity connection Both effective resistivity tensor K1 and effective elastic compliance tensor S can be expressed in terms of tensor x. This leads to the explicit
Fig. 2. (a) Possible combinations of principal conductivities k1 = k2 and k3 that correspond to aspect ratios 0.3 < c < 4. (b) Range of functional dependencies of k1 on porosity p for transversely isotropic and isotropic cases.
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elastic-conductive connection between changes in elastic compliances and changes in resistivity due to pores provided pore aspect ratios are not correlated with either pore volumes or pore orientations (pore volumes and pore orientations may be correlated). E0 ðS S 0 Þ ¼ ðC 1 II þ C 2 JÞ k 0 trðK 1 Þ 3 þ C 3 ðk 0 K 1 IÞI þ Iðk 0 K 1 IÞ þ C 4 ðk 0 K 1 IÞ J þ J ðk 0 K 1 IÞ . ð2:14Þ This connection is approximate; its accuracy is generally good and depends on PoissonÕs ratio m0 and pore aspect ratios (Sevostianov and Kachanov, 2002). Within this approximation, the (orthotropic) elastic properties are coaxial with the principal axes of K. Dimensionless coefficients C14 are shape factors that reflect the pore aspect ratio distribution (see (A2.3)). Their presence is due to differences between the effects of pore shapes on the overall elasticity and overall conductivity (otherwise, the cross-property connection would have been fully independent of pore shapes). However, the sensitivity of C14 to pore shapes is relatively mild, see the discussion to follow. In the case of transverse isotropy (x1x2 is the isotropy plane) (2.14) takes the form
G0 G13 k 0 k 11 k 0 k 33 ¼ 2C 2 þ ðC 2 þ C 4 Þ ; G13 k 11 k 33 E0 k 0 k 11 k 0 k 33 þ ðC 1 þ 2C 3 Þ ; m0 m31 ¼ 2C 1 E3 k 11 k 33 ð2:15Þ
2ð1 þ m0 Þ
(of the first three relations, only two are independent) where Ei is the effective YoungÕs modulus in the xi direction and Gij, mij are the effective shear moduli and PoissonÕs ratios in planes xixj. In the case of overall isotropy, the relations above reduce to the following ones: E0 E b1 þ b2 þ b3 þ b4 k 0 k ¼3 ; E k 3d 1 þ d 2 m0 kð3d 1 þ d 2 Þ þ 3ðk 0 kÞðb1 þ b3 Þ m¼ . kð3d 1 þ d 2 Þ þ 3ðk 0 kÞðb1 þ b2 þ b3 þ b4 Þ ð2:16Þ The isotropy takes place in one of the two cases: (A) spherical pores, and (B) randomly oriented spheroids. Whereas in case (A) the cross-property connections (2.16) are exact, in case (B) they are approximate, since they are based on approximate ^ in terms of second representation (2.8) of tensor H rank tensor x. However, in case (B), the exact cross-property connection can be established independently, by utilizing the exact result (2.5) for effective elasticity:
E0 E h1 þ 7h2 =10 þ 6h3 =5 þ 3h5 =5 k 0 k ¼ ; E 3d 1 þ d 2 k m0 kð3d 1 þ d 2 Þ ðk 0 kÞð6h1 =5 2h2 =5 þ 8h3 =5 h5 =5Þ m¼ . kð3d 1 þ d 2 Þ þ ðk 0 kÞð7h1 =5 7h2 =10 2h2 =5 þ 6h3 =5 þ 3h5 =5 þ 2h6 =5Þ
E0 E1 k 0 k 11 ¼ 2ðC 1 þ C 2 þ C 3 þ C 4 Þ E1 k 11 k 0 k 33 þ ðC 1 þ C 2 Þ ; k 33 G0 G12 k 0 k 11 k 0 k 33 2ð1 þ m0 Þ ¼ ð2C 2 þ C 4 Þ þ C2 ; G12 k 11 k 33 E0 k 0 k 11 k 0 k 33 þ C1 ; m0 m21 ¼ 2ðC 1 þ C 3 Þ E2 k 11 k 33 E0 E3 k 0 k 11 ¼ 2ðC 1 þ C 2 Þ E3 k 11 k 0 k 33 þ ðC 1 þ C 2 þ 2C 3 þ 2C 4 Þ ; k 33
913
ð2:17Þ
2.4. On the sensitivity of elasticity–conductivity connection to pore shape distribution The said sensitivity is of importance for the following reason. Utilization of cross-property connection (2.14) requires knowledge of C14, i.e. information on the pore aspect ratio distribution. The sensitivity of C14 to the pore aspect ratio distribution determines the required accuracy of this microscale information. The sensitivity to c is, obviously, maximal when all the pores have the same aspect ratios. It is reduced when c represents the maximum point of a certain distribution over aspect ratios. In the text
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to follow, we assume normal (Gaussian) distribution over c ! 2 1 ðc c0 Þ /ðcÞ ¼ pffiffiffiffiffiffi exp ð2:18Þ 2r2 r 2p with parameter r characterizing sharpness of the peak at c0; in the case of identical shapes, /(c) is a delta function, d(c c0). In the text to follow, we will always truncate Gaussian distribution to the interval of aspect ratios 0.3 < c < 4 where our basic model remains valid. Factors C14 and aij entering the elasticity–conductivity and the conductivityplasticity connections, will then be functions of c0 and r. Fig. 3 shows the sensitivity of C14 to pore aspect ratios c, in the relevant range 0.3 < c < 4.0. It is relatively mild even in the case of identical shapes (Fig. 3a). In the case of normal distribution with maximum at c0, the sensitivity to c0 is quite low for parameter r = 1 (Fig. 3b) and almost vanishes at r = 2 (Fig. 3c). Therefore, the required information on the pore shape distribution can be rather
imprecise. In cases of substantial ‘‘scatter’’ in pore shapes (r > 1), factors C14 can practically be treated as constants. 2.5. Plastic yield In the case of transverse isotropy (x1x2 is the isotropy plane), that appears to be the most common type of plastic anisotropy, the yield condition of a porous material has the form: 2
2s ¼ A1 ðrkk Þ2 þ A2 sij sji |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} isotropic part
þ A3 ðrkk Þr33 þ A4 r3j rj3 þ A5 r233 ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð2:19Þ
anisotropic part
where s* is the yield stress of the bulk material and sij = rij (rkk/3)dij is the stress deviator. Porosity makes the yield condition sensitive to the first stress invariant rkk. In the case of isotropy, this sensitivity was discussed by Skorokhod (1965), Green (1972) and Shima and Oyane (1976), among others.
Fig. 3. Sensitivity of factors C14 entering the elasticity–conductivity connection (2.11) to pore aspect ratios, distributed by normal law (d), at different values of parameter r: (a) r ! 0 (identical aspect ratios), (b) r = 1, (c) r = 2.
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Dimensionless factors A15 depend on the porous space geometry; they were given in terms of appropriate parameters of pore concentration, orientations and shapes by Sevostianov and Kachanov (2001). Remark. Our analysis for the case of transverse isotropy can be extended to the case of general orthotropy. 3. Plastic yield in terms of effective conductivities Combining expressions for plastic yield factors Ai in terms of effective elastic constants and the elasticity–conductivity linkage, the plastic yield condition can be given in terms of the effective conductivities. 3.1. Plastic yield factors Ai in terms of conductivities The most straightforward way to carry out the required calculations (that involve fourth rank tensorial inversions) is to use the technique of ‘‘standard’’ tensorial bases due to Kunin (1983) and Walpole (1984), see Appendix A. Any trans^ in particuversely isotropic fourth rank tensor—H, lar—is represented as a linear combination of six base tensors T(m): ^ ¼ H
6 X
^ hm T ðmÞ .
ð3:1Þ
m¼1
^m are expressed in terms As soon as coefficients h ^ ijkl (formulas (A1.2)), finding the product of H HijmnHnmkl and the inverse H 1 ijkl become standard operations. Thus, the elasticity–conductivity con^ yield the following nection (2.14) and S S 0 ¼ H expressions of ^ hm in terms of effectiveconductivities k11, k33: ^h1 E0 ¼ ð2C 1 þ C 2 þ 2C 3 þ C 4 Þ k 0 k 11 k 11 þ ðC 1 þ C 2 =2Þ
k 0 k 33 ; k 33
^h2 E0 ¼ 2ðC 2 þ C 4 Þ k 0 k 11 þ C 2 k 0 k 33 ; k 11 k 33 ^h3 E0 ¼ ^h4 E0 ¼ ð2C 1 þ C 3 Þ k 0 k 11 þ ðC 1 þ C 3 Þ k 0 k 33 ; k 11 k 33 ^h5 E0 ¼ 2ð2C 2 þ C 4 Þ k 0 k 11 þ 2ðC 2 þ C 4 Þ k 0 k 33 ; k 11 k 33
915
k 0 k 11 ^ h6 E0 ¼ 2ðC 1 þ C 2 Þ k 11 þ ðC 1 þ C 2 þ 2C 3 þ 2C 4 Þ
k 0 k 33 ; k 33
ð3:2Þ where shape-dependent factors C14 are given by (A2.3). Plastic yield factors Ai can be expressed in terms of ^hm as follows (see Sevostianov and Kachanov (2001); two misprints are corrected here): 2G0 6ð1 m0 Þ^ A1 ¼ h1 6m0 ^ h2 h3 ð1 þ m0 Þ^ 3ð1 þ m0 Þ i 2G2 h ^2 ^2 ^2 þ 0 12h 1 þ 6h3 h2 ; 3 2 h2 þ 4G20 ^ h2 ; A2 ¼ 1 þ 4G0 ^ i 4G0 h ^ A3 ¼ 2h1 þ ð1 þ m0 Þ^ h2 þ ð2 þ m0 Þ^ h3 m0 ^ h6 1 þ m0 h i ^1 h ^3 h ^2 þ h ^2 2h ^2 ; ^3 þ 2h ^6 4h þ 4G2 4h 0
A4 ¼
2 4G0 ð^ h5 2^ h2 Þ þ 2G20 ð^ h5
1
2
3
2 4^ h2 Þ;
h i A5 ¼ 2G0 2^ h1 þ ^ h2 4^ h3 2^ h5 þ 2^ h6 h 2 i 2 2 2 2 þ 2G20 4^ h1 þ ^ h2 þ 6^ h3 ^ h5 þ 2^ h6 4^ h3 ð2^ h1 þ ^ h6 Þ ;
ð3:3Þ
where G0 is the shear modulus of the bulk material. Substituting (3.2) into (3.3) yields the main result of the present work—plastic yield factors Ai in terms of conductivities:
2 k 0 k 11 k 0 k 33 k 0 k 11 þ a12 þ a13 k 11 k 33 k 11
2 k 0 k 33 k 0 k 11 k 0 k 33 þ a14 þ a15 ; k 33 k 11 k 33
2 k 0 k 11 k 0 k 33 k 0 k 11 A2 ¼ 1 þ a21 þ a22 þ a23 k 11 k 33 k 11
2 k 0 k 33 k 0 k 11 k 0 k 33 þ a24 þ a25 ; k 33 k 11 k 33
2 k 0 k 11 k 0 k 33 k 0 k 11 A3 ¼ a31 þ a32 þ a33 k 11 k 33 k 11
2 k 0 k 33 k 0 k 11 k 0 k 33 þ a34 þ a35 ; k 33 k 11 k 33
2 k 0 k 11 k 0 k 33 k 0 k 11 A4 ¼ a41 þ a42 þ a43 k 11 k 33 k 11
2 k 0 k 33 k 0 k 11 k 0 k 33 þ a44 þ a45 ; k 33 k 11 k 33 A1 ¼ a11
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2 k 0 k 11 k 0 k 33 k 0 k 11 þ a52 þ a53 k 11 k 33 k 11
2 k 0 k 33 k 0 k 11 k 0 k 33 þ a54 þ a55 . k 33 k 11 k 33
A5 ¼ a51
ð3:4Þ Dimensionless coefficients aij are not sensitive to either pore orientation distribution or porosity, but they do depend on the distribution of pore aspect ratios (via C14), as illustrated by Fig. 4. Note that these relations are explicit and contain no adjustable parameters.
The utility of plasticity–conductivity connections (3.4) is as follows. Without them, Ai can be expressed in terms of coefficients ^hm —characteristics of the pore space. However, these characteristics require knowledge not only of pore shapes, but of the orientation distribution and porosity as well. Such information may not be readily available. Utilization of the cross-property connection makes this information unnecessary. Indeed, if the conductivities are known, the only microscale information needed in order to construct the yield surface is the distribution over pore shapes.
Fig. 4. Coefficients aij entering the plasticity–conductivity connection (3.4) as functions of pore aspect ratios (identical aspect ratios).
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The required accuracy of this information depends on the sensitivity of the plasticity–conductivity connection to pore shapes. This issue is discussed below, in the case of identical aspect ratios and in the more realistic case of Gaussian distribution over the aspect ratios. 3.2. Sensitivity of the plasticity–conductivity connection to pore aspect ratios
917
to pore shapes is low, even in the case of identical aspect ratios. It is even lower if a certain scatter over the aspect ratios is present. Remark. Factors A1 and A5 are much smaller than A2, A3, A4. It appears that these two terms in the yield condition can be neglected, at least, in the considered range of aspect ratios 0.3 < c < 4. 4. Cases of overall isotropy
The sensitivity of coefficients aij to pore shapes in the range of aspect ratios 0.3 < c < 4.0 is seen from Fig. 4. Fig. 4 assumes that all pores have identical aspect ratios. A more direct examination of the sensitivity of plastic yield factors Ai to pore shapes is presented in Figs. 5 and 6. They show the dependence of Ai on k11/k0 at several fixed values of k33/k0, for the oblate and prolate pores of identical aspect ratios, respectively. (The range of conductivities covered in these figures corresponds to the relevant intervals of porosity and aspect ratios, see Fig. 2). The curves of Figs. 5 and 6 are quite close. Thus, the sensitivity
In the case of overall isotropy, yield condition (2.3) reduces to the form 2
2
2s ¼ A1 ðrkk Þ þ A2 sij sji ;
ð4:1Þ
where the first term reflects sensitivity to the average hydrostatic stress (note that, usually, A1 A2). Of six coefficients ^hm , only three—^h1 , ^h2 and ^h3 ¼ ^h1 ^h2 =2—enter formulas for A1, A2, and they are given in terms of effective isotropic conductivity k as follows: E0 ^h1 ¼ aðk 0 kÞ=k, E0 ^h2 ¼ bðk 0 kÞ=k, so that the plastic yield factors, expressed in terms of k are
Fig. 5. Plastic yield factors A15 as functions of conductivity k1/k0 at several values of k3/k0. All pores have aspect ratio c = 0.4.
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Fig. 6. Plastic yield factors A15 as functions of conductivity k1/k0 at several values of k3/k0. All pores have aspect ratio c = 3.
k0 k k 3ð1 þ m0 Þ
2 1 2 k0 k þ ð6a bÞ ; 2 k 3ð1 þ m0 Þ
2 1 k0 k A2 ¼ 1 þ b ; 1 þ m0 k
A1 ¼
2ð1 2m0 Þ 2
ity/conductivity connections (2.17). This would yield the same formulas (4.2) but with different coefficients a, b:
½6a b
ð4:2Þ
where a, b reflect the aspect ratio distribution; they are given by a ¼ 3C 1 þ ð3=2ÞC 2 þ 2C 3 þ C 4 ; b ¼ 3C 2 þ 2C 4 .
ð4:3Þ
Relations (4.2) are illustrated in Fig. 7 (identical aspect ratios are assumed). Formulas (4.3) for a, b follow from results for the transversely isotropic case, in the special case of isotropy. They reflect the approximations involved in deriving the elasticity–conductivity connection (2.16), in particular, the approximate character of (2.8) and (2.9). However, the case of isotropy can be analyzed independently, using the exact elastic-
a 12q1 28q3 þ 13q6 16q2 þ 3q5 þ ¼ ; E0 120ðq1 q6 q23 Þ 60q2 q5 b 4q þ 4q3 þ q6 32q2 þ 11q5 þ ¼ 1 ; E0 30q2 q5 60ðq1 q6 q23 Þ
ð4:4Þ
where coefficients qi are given in Appendix B. Relations (4.4) are illustrated in Fig. 8 (identical aspect ratios are assumed). Comparison of the last two figures shows that exact relations (4.4) produce a lower sensitivity to pore aspect ratios. This indicates that the approximations involved in deriving the general anisotropic plasticity–conductivity connection tend to exaggerate the sensitivity to pore shapes. We now consider an important for applications case of normal distribution of aspect ratios that has maximum at c0 = 1 (sphere). Fig. 9 compares the extreme case of parameter r = 5 (an almost uniform distribution over aspect ratios in the interval
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919
Fig. 7. Case of overall isotropy. Plastic yield factors A1,2 as functions of conductivities, based on approximate relations (4.3) (identical pore aspect ratios).
Fig. 8. Case of overall isotropy. Plastic yield factors A1,2 as functions of conductivities, based on exact relations (4.4) (identical pore aspect ratios).
0.3 < c < 4) with the opposite extreme case when all the pores are spherical (r = 0), for which 2 2
ð1 m0 Þð1 2m0 Þ k 0 k ð1 m0 Þ k0 k þ ; A1 ¼ 2 2 k k 12ð1 þ m0 Þ 3ð1 þ m0 Þ
2 5ð1 m0 Þ k 0 k A2 ¼ 1 þ . 7 5m0 k ð4:5Þ The curves corresponding to the two cases are very close. This means that, as long as the maximum point of normal distribution is c0 = 1 (sphere), the
effect of the shape ‘‘scatter’’ is negligible. In such cases, therefore, no microscale information is needed at all, and plastic yield factors A1, A2 are given by (4.5). 5. Discussion and conclusions In their earlier work, the authors constructed yield surfaces of porous metals in terms of the porous space geometry—more precisely, in terms of those parameters of pore distribution that determine the effective elasticity. However, the microscale
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Fig. 9. Case of overall isotropy. Plastic yield factors A1,2 as functions of conductivities. Comparison of the case of spherical pores with the case of normal distribution over pore aspect ratios, c0 = 1 (sphere) being the maximum point.
information contained in these parameters may not be readily available, particularly the information on pore orientations. We suggest a methodology to construct anisotropic yield surfaces in terms of electric conductivities—quantities that are easily measured. It is, obviously, required that the yield point is identifiable, i.e. is not ‘‘fuzzy’’ beyond recognition. This translates into two requirements: • The bulk (‘‘dense’’) material experiences a negligible hardening (more precisely, it should have a clearly identifiable yield point followed by an approximately horizontal plateau; this does not exclude a non-horizontal curve at later stages of loading). Aluminum and steel are examples. • Porosity does not exceed 0.15–0.17. The formulas that relate the plastic yield to conductivities do not contain any information on either pore orientations or the overall porosity. They do depend, though, on the distribution over pore aspect ratios. However, the sensitivity to this factor is relatively mild, even in the case of identical aspect ratios, as seen from comparison of Figs. 5 and 6. If some scatter in pore shapes is present (modeled by Gaussian distribution over aspect ratios), the sensitivity to the average aspect ratio is reduced even further. Therefore, the required knowledge of the
aspect ratio distribution can be rather imprecise; moreover, if pore shapes are known to have a sufficient scatter, no microscale information is needed at all. We also further examined the elasticity–conductivity connection (on which the present results are based), from the point of view of their sensitivity to the pore shape distribution. It is found that, whereas the sensitivity is noticeable in the case of identical pore aspect ratios, a scatter in pore shapes reduces this sensitivity substantially (Fig. 3). The case of overall isotropy (randomly oriented spheroids) is considered. It is found that, in the important for applications case when the distribution of pore aspect ratios has maximum at c0 = 1 (sphere), the effect of the shape ‘‘scatter’’ is negligible and the plastic yield condition is practically the same as in the case when all pores are spherical. In this case, therefore, no microscale information is needed at all, in order to utilize the plasticity–conductivity connection. The results apply to an arbitrary mixture of spheroidal pores (diverse aspect ratios and arbitrary orientation distribution), under the condition that the distributions over aspect ratios and over orientations are not statistically correlated. The restriction of spheroidal shapes can actually be relaxed; for example, moderate ‘‘jaggedness’’ of pore boundaries and some other shape ‘‘irregularities’’ can be
I. Sevostianov, M. Kachanov / Mechanics of Materials 38 (2006) 908–923
ignored (they are either unimportant for the properties considered, or their effects on plastic yield and conductivity are similar and, hence, do not affect the link between the two). A more restrictive limitation is that geometric ‘‘extremes’’—strongly oblate or strongly prolate pores (aspect ratios smaller than 0.3 or larger than 4)—are not present in significant numbers. The present work focuses on the cases of overall transverse isotropy (that appears to be the main case of plastic anisotropy) and isotropy. However, the same line of derivation can be applied to a more general case of overall orthotropy.
coefficients of this representation are expressed in terms of components Wijkl as follows: w1 ¼ ðW1111 þ W1122 Þ=2; w3 ¼ W1133 ;
ð1Þ
T ijkl ¼ ðhik hlj þ hil hkj hij hkl Þ=2; ð4Þ T ijkl
¼ hij mk ml ;
¼
¼ mi mj hkl ;
ð6Þ
T ijkl ¼ mi mj mk ml ; ðA1:1Þ where hij = dij mimj and m = miei is a unit vector of the axis of transverse isotropy. We assume that x1x2 is the isotropy plane. Then base tensors T(i) have the following non-zero components: ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð2Þ
ð2Þ
ð2Þ
ð2Þ
T 1111 ¼ T 2222 ¼ T 1122 ¼ T 2211 ¼ 1; ð2Þ
¼ ¼ ¼
6 X
ðiÞ
ðA1:3Þ
si T ijkl
has the following coefficients si in this basis: s1 ¼
1m ; 4Gð1 þ mÞ
s5 ¼
2 ; G
s6 ¼
s2 ¼
1 ; 2G
s3 ¼ s4 ¼
m ; 2Gð1 þ mÞ
1 . 2Gð1 þ mÞ
ðA1:4Þ
The table of multiplication of the base tensors has the form:
¼ 1=2;
ð3Þ T 1133
¼
ð3Þ T 2233
¼ 1;
ð6Þ
¼ ¼
¼ ¼
ð5Þ T 2332 ð5Þ T 3131
T1
T2
T3
T4
T5
T6
2T1 0 0 2T4 0 0
0 T2 0 0 0 0
2T3 0 0 2T6 0 0
0 0 T1 0 0 T4
0 0 0 0 T5/2 0
0 0 T3 0 0 T6
Representing an arbitrary fourth rank symmetric tensors X and Y in this tensorial basis X¼
6 X
X k T ðkÞ ;
Y¼
k¼1
6 X
Y k T ðkÞ
ðA1:5Þ
k¼1
we readily obtain the inverse of X and a product of X and Y as follows.
¼ 1; T 3333 ¼ 1; ð5Þ T 1331 ð5Þ T 3223
T1 T2 T3 T4 T5 T6
ð2Þ
T 1212 ¼ T 2121 ¼ T 1221 ¼ T 2112 ¼ T 1111 ¼ T 2222 ¼ 1=2; ¼
ðA1:2Þ
i¼1
ð5Þ
ð2Þ T 2211 ð4Þ T 3322 ð5Þ T 2323 ð5Þ T 3113
w5 ¼ 4W1313 ;
For example, elastic compliance tensor of the isotropic material 1 1 m ðdik djl þ dil dkl Þ dij dkl S ijkl ¼ 2G 2 1þm
T ijkl ¼ ðhik ml mj þ hil mk mj þ hjk ml mi þ hjl mk mi Þ=4;
ð2Þ T 1122 ð4Þ T 3311 ð5Þ T 1313
w4 ¼ W3311 ;
ð2Þ
T ijkl ¼ hij hkl ; ð3Þ T ijkl
w2 ¼ 2W1212 ;
w6 ¼ W3333 .
Appendix A. A technique of analytic inversion and multiplication of fourth rank tensors We outline a convenient technique of analytic inversion and multiplication of fourth rank tensors, due to Kunin (1983) and Walpole (1984). It is based on expressing tensors in ‘‘standard’’ tensorial bases. In the case of transversely isotropic symmetry, the following basis of six tensors T(m) is convenient:
921
ð5Þ
(a) Inverse tensor:
¼ T 3232 ¼ 1=4. ðA1:8Þ
For an arbitrary transversely isotropic fourth rank tensor W represented in this basis X Wijkl ¼ wm T mijkl
X 1 ¼
X 6 ð1Þ 1 ð2Þ X 3 ð3Þ X 4 ð4Þ T þ T T T X2 2D D D 4 ð5Þ 2X 1 ð6Þ T ; þ T þ ðA1:6Þ X5 D
where D = 2(X1X6 X3X4).
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(b) Product X : Y (tensor with ijkl components XijmnYmnkl) is by X : Y ¼ ð2X 1 Y 1 þ X 3 Y 4 ÞT ð1Þ þ X 2 Y 2 T ð2Þ þ ð2X 1 Y 3 þ X 3 Y 6 ÞT ð3Þ þ ð2X 4 Y 1 þ X 6 Y 4 ÞT
ð4Þ
1 þ X 5 Y 5 T ð5Þ 2
þ ðX 6 Y 6 þ 2X 4 Y 3 ÞT ð6Þ .
where Z bi ¼
We consider a spheroidal pore with x3 being the symmetry axis and c the aspect ratio. We express tensor Hijkl of this pore in terms of base tensors ð16Þ T ijkl of the standard basis given in P6tensorial ðmÞ Appendix A: H ijkl ¼ m¼1 hm T ijkl . Coefficients hi were given by Sevostianov and Kachanov (1999) for an inclusion that has arbitrary elastic properties. For a pore, they reduce to the following ones:
V q6 V 1 ; h2 ¼ ; 2 V 4ðq1 q6 q3 Þ V q2 V q3 V 4 h3 ¼ h4 ¼ ; h ¼ ; 5 V 2ðq1 q6 q23 Þ V q5 V q1 h6 ¼ ; ðA2:1Þ V ðq1 q6 q23 Þ
h1 ¼
di ¼
0
1
Di ðcÞ/ðcÞ dc 0
and /(c) is the shape distribution density (specialized as Gaussian function in the main text), i.e. /(c)dc is the volume fraction of pores with aspect ratios in the interval (c, c + dc). Dimensionless functions Bi(c) and Di(c) are given by 1
D1 ¼ ð1 f0 ðcÞÞ ; D2 ¼ ð1 3f 0 ðcÞÞ=2f 0 ðcÞ½1 f0 ðcÞ; B1 ¼ E0 ð^h1 ^h2 =2Þ; B2 ¼ E0 ^h2 ; B3 ¼ E0 ð2^h3 þ ^h2 2^h1 Þ;
B4 ¼ E0 ð^h5 2^h2 Þ ðA2:5Þ
and coefficients ^hi are obtained from hi as follows: ^h1 ¼ h1 ð1 d sign h1 Þ; ^h2 ¼ h2 ð1 d sign h2 Þ; ^h3 ¼ h3 ð1 þ d sign h3 Þ; ^h5 ¼ h5 ð1 þ d sign h5 Þ;
where
q3 ¼ q4 ¼ 2G0 ½ð2j 1Þf0 þ 2f 1 ; q5 ¼ 4G0 ½f0 þ 4f 1 ; q6 ¼ 8G0 ½jf0 f1 ;
d¼
1 ; 2ð1 m0 Þ jc2
f0 ¼
c 1
c
h6 þ h1 þ h2 =2 2h3 h5 . jh6 j þ jh1 j þ jh2 j=2 þ 2jh3 j þ jhj5
ðA2:6Þ
ðA2:7Þ
In the case of identical aspect ratios c = c0, the shape distribution density is a delta-function: /(c) = d(c c0) and integrals (A2.4) reduce to Bi(c0) and Di(c0), correspondingly.
c2 ð1 gÞ ; 2ðc2 1Þ
½ð2c2 þ 1Þg 3; 2 4ðc2 1Þ 8 pffiffiffiffiffiffiffi 1c2 > 1 > arctan ; oblate shape; < cpffiffiffiffiffiffiffi c 2 1c g¼ pffiffiffiffiffiffiffi > cþ c2 1 > : p1ffiffiffiffiffiffiffi ln pffiffiffiffiffiffiffi ; prolate shape: 2 2 2c
ðA2:3Þ
ðA2:4Þ
q1 ¼ G0 ½4j 1 2ð3j 1Þf0 2f 1 ; q2 ¼ 2G0 ½1 ð2 jÞf0 f1 ;
f1 ¼
Bi ðcÞ/ðcÞ dc;
Z
^h6 ¼ h6 ð1 d sign h6 Þ;
where
j¼
1
ðA1:7Þ
Appendix B. Tensor H of a spheroidal pore. Pore shape factors C14
b1 d 2 2b3 d 1 b2 d 2 2b4 d 1 ; C2 ¼ ; d 2 ðd 2 þ 3d 1 Þ d 2 ðd 2 þ 3d 1 Þ b3 b4 C3 ¼ ; C4 ¼ ; d2 d2
C1 ¼
References
c 1
ðA2:2Þ For a sphere, f0 = 1/3 and f1 = j/15 = 1/(30(1 m0)). Coefficients C14 entering (2.11) are related to coefficients h16 as follows:
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