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On the steady-state creep of microcellular metals
Scripta Materialia 57 (2007) 33–36 www.elsevier.com/locate/scriptamat
On the steady-state creep of microcellular metals Randoald Mueller, Se´bastien Soubielle, Russell Goodall, Fre´de´ric Diologent and Andreas Mortensen* Ecole Polytechnique Fe´de´rale de Lausanne (EPFL), Laboratory for Mechanical Metallurgy, CH-1015 Lausanne, Switzerland Received 24 January 2007; accepted 7 March 2007 Available online 5 April 2007
A variational estimate for non-linear composite deformation is specialized to provide a simple estimate of uniaxial steady-state creep rates in microcellular metals deforming by power-law creep knowing their linear modulus. Resulting expressions compare well with previous theory and data. 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Aluminium foams; Plastic deformation; Creep; Mean-field analysis; Variational estimate
In a recent publication [1] we adapted and simplified analytical approaches that extend models for linear elastic composite deformation to non-linear materials (e.g. [2–10]). These enable the prediction of monotonic uniaxial plastic flow curves in power-law microcellular metals knowing their linear elastic modulus. It is well-known that, under certain simple and frequently obeyed conditions, solutions for elastic deformation and viscous flow of isotropic materials are interchangeable, passage from one to the other simply involving an exchange of strain and displacement in the elasticity solution with strain-rate and velocity in the viscous creep solution, respectively [11]. Expressions for power-law uniaxial flow that were given for microcellular metals in Ref. [1] are, therefore, directly adaptable for the prediction of the rate of steady-state creep displayed by microcellular solids made of a bulk material that deforms by power-law creep. The purpose of this note is to give a simple and user-friendly derivation of the relevant equations and discuss their predictive power and accuracy. Consider an isotropic microcellular or porous material of relative density Vm; for simplicity we call this hereafter a ‘‘foam’’. It can be made of any material that deforms by time-independent power-law plasticity, or alternatively deforms at steady-state by power-law creep; for simplicity in nomenclature we call this base material a ‘‘metal’’ in what follows.
If we assume that, when the foam deforms by linear elastic deformation, the metal from which it is made behaves as if it were incompressible, the Young’s modulus of the foam, Ef, is a simple linear function of the Young’s modulus, E0, of the metal from which it is made
* Corresponding author. Tel.: +41 21 693 29 12; fax: +41 21 693 46 64; e-mail: andreas.mortensen@epfl.ch
ðreq Þ2 ¼
Ef ¼ F E0
ð1Þ
F is a function of Vm that depends on the foam inner architecture (or its ‘‘mesostructure’’, meaning the shape of its internal free surface). Having made the assumption that, within the foam, the metal deforms without changing volume, expressions for F that derive from classical mean-field theory are simplified by taking the limit where the metal bulk modulus Km tends to infinity [1]. We now assume that the metal deforms in uniaxial tension according to the simple power-law r ¼ c ðeÞn
ð2Þ
where r denotes stress, e is either the strain or, for steady-state creep, the strain-rate, and c and n are constants. For uniaxial deformation, stress and strain or strain-rate are obtained by first estimating the average second-order moment of the equivalent stress field, req, in a secant linear comparison composite according to oð1=Ef Þ 1 2 r oð1=E0 Þ V m f
1359-6462/$ - see front matter 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2007.03.013
with req ¼
rffiffiffiffiffiffiffiffiffiffiffiffi D E r2eq
ð3Þ
34
R. Mueller et al. / Scripta Materialia 57 (2007) 33–36
where rf is the uniaxial stress applied to the foam (Eq. (74) of [8] written for an incompressible solid phase). Inserting Eq. (1), this simply becomes pffiffiffiffiffiffiffiffiffiffi rf ¼ V mF ð4Þ req
If the foam deforms by bending of struts, as is characteristic of low-density open-cell foams, then a = 2 and one obtains
The secant formulation can then be written as
This can be compared with the result derived using engineering beam analysis by Andrews et al. [12]
Ems
req ¼ eeq
ð5Þ
and rf Ef ¼ ð6Þ ef where Ems is the secant (Young’s) modulus of the metal, Ef is that of the foam and ef is the strain or strain-rate of the foam. In uniaxial tension, combining Eqs. (1), (5) and (6), one obtains rf req ¼ F ef eeq
ð7Þ
Inserting Eqs. (2) and (4) and rearranging, one concludes that the foam deforms uniaxially according to the same power-law as the matrix 1n C 1þn ¼ F 2 V m2 ð8Þ rf ¼ C ðef Þn with c In creep deformation it is customary to write the steadystate (secondary) creep law in the form (e.g. [12]) N r with e_ 0 ¼ K expðQ=RT Þ ð9Þ e_ ¼ e_ 0 r0 1=N
. From Eq. (8) Evidently N = n1, and c ¼ r0 ð_e0 Þ the creep rate of the foam is then N 1þN rf ðN 1Þ with r0f ¼ r0 F ð 2N Þ V m 2N e_ f ¼ e_ 0f r0f and e_ 0f ¼ e_ 0 ð10Þ For each linear elasticity model determining the Young’s modulus of a foam, it is then relatively straightforward to insert the suitable value for F in Eq. (8) or Eq. (10) to derive the creep rate of the foam knowing the power-law creep constants for the metal; relevant expressions are given in Ref. [1]. In the particular case where the Young’s modulus of the foam is related to that of the metal by a simple power-law, F ¼ A ðV m Þ
a
ð11Þ
where A can differ from unity if data are only fitted over a limited range of relative density, Eq. (10) specializes to 1þN ðN ð1þaÞþa1Þ r0f ¼ r0 Að 2N Þ V m 2N
ð12Þ
The case a = 1, meaning Ef/Em = F = A Æ Vm, corresponds to deformation of a foam that is effectively governed by the uniaxial straining of metal occupying a volume fraction A within the foam (as in the solid strut unit cell of Ref. [13]). One would then expect that the non-linear stress constant be r0f ¼ r0 A V m which is consistent with Eq. (12).
ð13Þ
1þN ð1þ3N Þ r0f ¼ r0 Að 2N Þ V m 2N
1 N þ 2 ðN Þ N ð1þ3N Þ V m 2N r0f ¼ r0 C4 C 5 ð2N þ 1Þ
ð14Þ
ð15Þ
where C4 and C5 are two constants, estimated by comparison of Eq. (15) with data for Young’s modulus and for the yield stress of foams [12]. The dependence on Vm is the same in the two expressions. The proportionality factor, on the other hand, differs in two respects: (i) there is a difference in its dependence on N; this is a result of the fact that N-dependent factors emerge from differential calculus operations in the beam analysis; and (ii) the Andrews et al. equation contains two constants. These are then fitted independently by analogy with elastic (N = 1) and perfectly plastic (N tending to infinity) metal deformation laws [12]. In Eq. (14), on the other hand, there is only one constant. The reason is that, for consistency in the approach, compatible constants must appear in expressions giving the modulus and the yield stress of a foam made of an elastic perfectly plastic metal (Eq. (8) with n = 0). If we adapt the two constants C4 and C5 for consisp tency in this p regard, then we must have C4 = 2/ A, and C5 = 1/(2 A) (Eq. (8) with n = 0 and Eq. (11) with a = 2). Eq. (15) then becomes r0f ¼ r0 ðN þ 2ÞðN Þ 1
1þN N N1 ð1þ3N Þ 2 N Að 2N Þ V m 2N ð2N þ 1Þ
ð16Þ
now showing the same dependence as Eq. (14) on both A and Vm. The ratio R between the two predictions (Eq. (16) divided by Eq. (14)) is then simply R ¼ ðN þ 2ÞðN Þ 1
N N 1 2N ð2N þ 1Þ
ð17Þ
This is plotted in Figure 1. As seen, R remains close to one for all N values between 1 and 20, deviating by at most 17% from unity (and by less at high N values, where the rate of variation of strain-rate with stress is highest). The two approaches are therefore consistent provided the yield stress and the modulus are evaluated consistently with one another. Another simple test of Eq. (10) can be obtained by considering low-density foams made of linked straight struts. For such foams, the engineering beam analysis of Andrews, Gibson and Ashby is expected to apply relatively well, meaning that the elastic modulus and creep rate of these materials should rank roughly as the resistance to elastic and non-linear bending, respectively, of individual beams subjected to bending (only roughly: even foams made of straight struts also see strut torsion in addition to bending; e.g. Ref. [14]). Now, significant variations in the stiffness and flow stress of such foams can be produced by hollowing out the foam arms, keeping the relative density constant. Predictions of Eq. (10)
R. Mueller et al. / Scripta Materialia 57 (2007) 33–36
Figure 1. The variation with N of the ratio, R, between the value of r0f for a foam where the Young’s modulus is related to that of the metal by a simple power-law determined by the current analysis and that determined by the engineering beam analysis of Andrews et al. [12] (Eq. (16) divided by Eq. (14)).
can thus be compared with estimations of engineering beam theory, from which Eq. (15) was derived. Equations for the deformation of hollow beams are given in Refs. [15,16]. Focusing on struts with a square cross-section, engineering beam analysis leads to the conclusion that a hollow in the beam, the width of which is a fraction f of the total beam width, produces (all else being constant) at any given Vm an increase in Ef/Em by a factor KE ¼
1 þ f2 1 f2
ð18Þ
and in C/c by a factor Kr ¼
1 f 3þn ð1 f 2 Þ
3þn 2
Eq. (8), on the other hand, states that 1þn 1þn 1 þ f2 2 2 Kr KE ¼ 1 f2
35
Figure 2. Variation in the factor describing the increase in C/c in foams with hollow struts, Kr (plotted on a logarithmic scale), with the width, f, of the hollow in the beam. Equations derived by the current approach and engineering beam analysis are plotted for various values of N.
ined; these are N = 4, Q = 176 kJ mol1. The value of K = 5.06 · 1020 s1 may be estimated using the data they report and the value of r0 (=G) = 22.02 GPa for pure aluminium given in Ref. [17]. Values for Young’s modulus of these foams can be estimated by fitting experimental data given in the literature for such foams, [18–21]. yielding F ¼ 2:39 V 2:37 m Figure 3 compares the experimental data found by Andrews et al. for the steady-state creep rate of Duocel foam at 275 C under a stress of 0.42 MPa (their Fig. 11) with (i) the predictions of the model proposed by Andrews et al. (Eq. 8 of Ref. [12]) and (ii) Eq. (10). The ordinate axis on Figure 3 normalizes for temperature
ð19Þ
ð20Þ
The two predictions are compared in Figure 2 for various values of N = 1/n; as seen, the agreement between flow stress values predicted by the two models for a given strain or strain-rate is again within around 10%. Comparison of predictions of Eq. (10) with experimental data on the steady-state creep rate of porous or microcellular metals requires also knowing the elastic modulus of the same porous material (because F must be known, Eq. (1)). This can be done for the data of Andrews et al. on the open-cell aluminium foam produced by ERG and known as Duocel foam, using which steady-state strain-rates were measured for a variety of applied stresses and temperatures, for foams with three different Vm, namely 0.06, 0.09 and 0.14 [12]. Creep constants of the alloy from which this foam was made (6101-T6 aluminium alloy) were also measured by Andrews et al. in the temperature range exam-
Figure 3. Comparison of experimental ([12]) data for the steady-state creep rate for open-celled aluminium foam (Duocel) of various densities at 275 C under a stress of 0.42 MPa, with the predictions of the model proposed by Andrews et al. [12] and the predictions of Eq. (10) in this work. The ordinate axis compensates for the effect of varying temperature and applied stress.
36
R. Mueller et al. / Scripta Materialia 57 (2007) 33–36
and stress. It is seen that the results of the calculations using Eq. (10) (based on the elastic properties of the foam) compare well with the actual creep data recorded for this material, the prediction being somewhat closer to the data than that from the model of Andrews et al. In short, the variational estimates of Ponte Castan˜eda, Suquet et al. can be used to derive simple expressions for the non-linear plastic or creep deformation of highly porous metals knowing their linear elastic modulus; however, it must be kept in mind that there are limitations to equations presented here. (i) In the formulation of the equations, a small deformation is assumed; foams deform with large local strains, which cause even foams of linear elastic metal to behave non-linearly at higher strain [14,22–24], cause the relative density to evolve during deformation, and finally cause the foam to become anisotropic after a significant cell shape change. So in all rigour equations given here apply only for the early stages of deformation. (ii) The fact that isotropic foam properties were assumed from the onset also means that the analysis does not apply to anisotropic foams, including the cubic or hexagonal symmetry structures most often assumed in numerical simulations of foam mechanical behaviour. (iii) Foams often show behaviour governed by local instability events, such as buckling of beams or membranes (see Ref. [25] for an illustration). In particular, in stretching-dominated foams of high beam coordination, ‘‘yield’’ is indeed often dominated by buckling rather than stable plastic flow [26]. For such cases what precedes is also clearly invalid. (iv) Errors may be introduced due to the nature of creep behaviour. Steady-state creep is assumed: primary creep may dominate initial (low strain) foam creep data. Also, in creep, one must keep in mind that a single steady-state deformation law is assumed. Local peak stresses outside its range of validity may cause shifts in behaviour, for example by power-law breakdown at high strain-rates, as proposed by Wanner et al. [27,28], or at low applied stress (and high temperature) by diffusion creep, such as reported by Goussery [29]. Finally, one must also keep in mind that the predictive precision in terms of strain-rate at a given stress of any power-law creep model is limited when the creep exponent N is high. For instance, a 10% error in the average stress will result in errors in the creep rate by factors of 1.6 and 2.6 with N = 5 and 10, respectively. In summary, simplifying Ponte Castan˜eda and Suquet’s variational estimates for non-linear composite deformation yields a simple expression for the uniaxial steady-state flow rate of power-law creeping metal foams knowing the foam elastic modulus as a fraction of that of the metal. The estimate is shown to be consistent with the data of Andrews et al. for open-celled aluminium foam, and with the conclusions of other models in the literature for the steady-state power-law creep of isotropic microcellular materials.
This research was supported by the Swiss National Science Foundation, project 200020-100179. [1] J.F. Despois, R. Mueller, A. Mortensen, Acta Mater. 54 (16) (2006) 4129–4142. [2] P. Ponte Castan˜eda, J. Mech. Phys. Solids 39 (1) (1991) 45. [3] D.R.S. Talbot, J.R. Willis, Int. J. Solids Struct. 29 (14–15) (1992) 1981. [4] J.C. Michel, P.M. Suquet, J. Mech. Phys. Solids 40 (4) (1992) 783–812. [5] P.M. Suquet, J. Mech. Phys. Solids 41 (6) (1993) 981– 1002. [6] P.M. Suquet, Comptes Rendus de l’Acade´mie des Sciences de Paris 320 (IIb) (1995) 563–571. [7] P. Ponte Castan˜eda, in: P.M. Suquet (Ed.), Continuum Micromechanics, Springer-Verlag, New York, 1997, pp. 131–195. [8] P.M. Suquet, in: P.M. Suquet (Ed.), Continuum Micromechanics, Springer-Verlag, New York, 1997, pp. 187– 264. [9] P. Ponte Castan˜eda, P.M. Suquet, Adv. Appl. Mech. 34 (1998) 171–302. [10] J.R. Willis, Eur. J. Mech. A/Solids 19 (2000) S165–S184. [11] N.J. Hoff, Quart. Appl. Math. 12 (1) (1954) 49–55. [12] E.W. Andrews, L.J. Gibson, M.F. Ashby, Acta Mater. 47 (10) (1999) 2853–2863. [13] A.M. Hodge, D.C. Dunand, Intermetallics 9 (7) (2001) 581. [14] H.X. Zhu, N.J. Mills, J.F. Knott, J. Mech. Phys. Solids 45 (11–12) (1997) 1875. [15] H.S. Zurob, Y. Bre´chet, J. Mater. Sci. 40 (22) (2005) 5893. [16] E.W. Andrews, Mater. Lett. 60 (2006) 618–620. [17] H.J. Frost, M.F. Ashby, Deformation Mechanism Maps, Pergamon Press, Oxford, UK, 1982, pp. 1281–1288. [18] H. Bart-Smith, A.F. Bastawros, D.R. Mumm, A.G. Evans, D.J. Sypeck, H.N.G. Wadley, Acta Mater. 46 (10) (1998) 3583–3592. [19] T.G. Nieh, N.H. Kinney, J. Wadsworth, A.J.C. Ladd, Scripta Mater. 38 (10) (1998) 1487–1494. [20] E.W. Andrews, W. Sanders, L.J. Gibson, Mater. Sci. Eng. A270 (1999) 113–124. [21] E.W. Andrews, G. Gioux, P. Onck, L.J. Gibson, Int. J. Mech. Sci. 43 (2001) 701–713. [22] A.M. Kraynik et al., Foams and Emulsions, Proceedings NATO Advanced Study Institute on Foams, Emulsions and Cellular Materials, Cargese, Corsica, France, Kluwer Academic Publishers, Dordrecht, 1999. [23] W.E. Warren, A.M. Kraynik, J. Appl. Mech. 58 (1991) 376–381. [24] H.X. Zhu, A.H. Windle, Acta Mater. 50 (5) (2002) 1041. [25] A. Marmottant, J.F. Despois, L. Salvo, E. Maire, M. Bornert, A. Mortensen, 4th International Conferences on Porous Metals and Metal Foaming Technology (Metfoam 2005), Kyoto, Japan, Japan Institute of Metals, Sendai, 2005. [26] V.S. Deshpande, N.A. Fleck, M.F. Ashby, J. Mech. Phys. Solids 49 (2001) 1747–1769. [27] P. Zhang, M. Haag, O. Kraft, A. Wanner, E. Arzt, Philos. Mag. A 82 (16) (2002) 2895. [28] M. Haag, A. Wanner, H. Clemens, P. Zhang, O. Kraft, E. Arzt, Metall. Mater. Trans. 34A (12) (2003) 2809– 2817. [29] V. Goussery, PhD Thesis, Ecole Nationale Supe´rieure des Mines de Paris, 2004.
On the steady-state creep of microcellular metals Randoald Mueller, Se´bastien Soubielle, Russell Goodall, Fre´de´ric Diologent and Andreas Mortensen* Ecole Polytechnique Fe´de´rale de Lausanne (EPFL), Laboratory for Mechanical Metallurgy, CH-1015 Lausanne, Switzerland Received 24 January 2007; accepted 7 March 2007 Available online 5 April 2007
A variational estimate for non-linear composite deformation is specialized to provide a simple estimate of uniaxial steady-state creep rates in microcellular metals deforming by power-law creep knowing their linear modulus. Resulting expressions compare well with previous theory and data. 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Aluminium foams; Plastic deformation; Creep; Mean-field analysis; Variational estimate
In a recent publication [1] we adapted and simplified analytical approaches that extend models for linear elastic composite deformation to non-linear materials (e.g. [2–10]). These enable the prediction of monotonic uniaxial plastic flow curves in power-law microcellular metals knowing their linear elastic modulus. It is well-known that, under certain simple and frequently obeyed conditions, solutions for elastic deformation and viscous flow of isotropic materials are interchangeable, passage from one to the other simply involving an exchange of strain and displacement in the elasticity solution with strain-rate and velocity in the viscous creep solution, respectively [11]. Expressions for power-law uniaxial flow that were given for microcellular metals in Ref. [1] are, therefore, directly adaptable for the prediction of the rate of steady-state creep displayed by microcellular solids made of a bulk material that deforms by power-law creep. The purpose of this note is to give a simple and user-friendly derivation of the relevant equations and discuss their predictive power and accuracy. Consider an isotropic microcellular or porous material of relative density Vm; for simplicity we call this hereafter a ‘‘foam’’. It can be made of any material that deforms by time-independent power-law plasticity, or alternatively deforms at steady-state by power-law creep; for simplicity in nomenclature we call this base material a ‘‘metal’’ in what follows.
If we assume that, when the foam deforms by linear elastic deformation, the metal from which it is made behaves as if it were incompressible, the Young’s modulus of the foam, Ef, is a simple linear function of the Young’s modulus, E0, of the metal from which it is made
* Corresponding author. Tel.: +41 21 693 29 12; fax: +41 21 693 46 64; e-mail: andreas.mortensen@epfl.ch
ðreq Þ2 ¼
Ef ¼ F E0
ð1Þ
F is a function of Vm that depends on the foam inner architecture (or its ‘‘mesostructure’’, meaning the shape of its internal free surface). Having made the assumption that, within the foam, the metal deforms without changing volume, expressions for F that derive from classical mean-field theory are simplified by taking the limit where the metal bulk modulus Km tends to infinity [1]. We now assume that the metal deforms in uniaxial tension according to the simple power-law r ¼ c ðeÞn
ð2Þ
where r denotes stress, e is either the strain or, for steady-state creep, the strain-rate, and c and n are constants. For uniaxial deformation, stress and strain or strain-rate are obtained by first estimating the average second-order moment of the equivalent stress field, req, in a secant linear comparison composite according to oð1=Ef Þ 1 2 r oð1=E0 Þ V m f
1359-6462/$ - see front matter 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2007.03.013
with req ¼
rffiffiffiffiffiffiffiffiffiffiffiffi D E r2eq
ð3Þ
34
R. Mueller et al. / Scripta Materialia 57 (2007) 33–36
where rf is the uniaxial stress applied to the foam (Eq. (74) of [8] written for an incompressible solid phase). Inserting Eq. (1), this simply becomes pffiffiffiffiffiffiffiffiffiffi rf ¼ V mF ð4Þ req
If the foam deforms by bending of struts, as is characteristic of low-density open-cell foams, then a = 2 and one obtains
The secant formulation can then be written as
This can be compared with the result derived using engineering beam analysis by Andrews et al. [12]
Ems
req ¼ eeq
ð5Þ
and rf Ef ¼ ð6Þ ef where Ems is the secant (Young’s) modulus of the metal, Ef is that of the foam and ef is the strain or strain-rate of the foam. In uniaxial tension, combining Eqs. (1), (5) and (6), one obtains rf req ¼ F ef eeq
ð7Þ
Inserting Eqs. (2) and (4) and rearranging, one concludes that the foam deforms uniaxially according to the same power-law as the matrix 1n C 1þn ¼ F 2 V m2 ð8Þ rf ¼ C ðef Þn with c In creep deformation it is customary to write the steadystate (secondary) creep law in the form (e.g. [12]) N r with e_ 0 ¼ K expðQ=RT Þ ð9Þ e_ ¼ e_ 0 r0 1=N
. From Eq. (8) Evidently N = n1, and c ¼ r0 ð_e0 Þ the creep rate of the foam is then N 1þN rf ðN 1Þ with r0f ¼ r0 F ð 2N Þ V m 2N e_ f ¼ e_ 0f r0f and e_ 0f ¼ e_ 0 ð10Þ For each linear elasticity model determining the Young’s modulus of a foam, it is then relatively straightforward to insert the suitable value for F in Eq. (8) or Eq. (10) to derive the creep rate of the foam knowing the power-law creep constants for the metal; relevant expressions are given in Ref. [1]. In the particular case where the Young’s modulus of the foam is related to that of the metal by a simple power-law, F ¼ A ðV m Þ
a
ð11Þ
where A can differ from unity if data are only fitted over a limited range of relative density, Eq. (10) specializes to 1þN ðN ð1þaÞþa1Þ r0f ¼ r0 Að 2N Þ V m 2N
ð12Þ
The case a = 1, meaning Ef/Em = F = A Æ Vm, corresponds to deformation of a foam that is effectively governed by the uniaxial straining of metal occupying a volume fraction A within the foam (as in the solid strut unit cell of Ref. [13]). One would then expect that the non-linear stress constant be r0f ¼ r0 A V m which is consistent with Eq. (12).
ð13Þ
1þN ð1þ3N Þ r0f ¼ r0 Að 2N Þ V m 2N
1 N þ 2 ðN Þ N ð1þ3N Þ V m 2N r0f ¼ r0 C4 C 5 ð2N þ 1Þ
ð14Þ
ð15Þ
where C4 and C5 are two constants, estimated by comparison of Eq. (15) with data for Young’s modulus and for the yield stress of foams [12]. The dependence on Vm is the same in the two expressions. The proportionality factor, on the other hand, differs in two respects: (i) there is a difference in its dependence on N; this is a result of the fact that N-dependent factors emerge from differential calculus operations in the beam analysis; and (ii) the Andrews et al. equation contains two constants. These are then fitted independently by analogy with elastic (N = 1) and perfectly plastic (N tending to infinity) metal deformation laws [12]. In Eq. (14), on the other hand, there is only one constant. The reason is that, for consistency in the approach, compatible constants must appear in expressions giving the modulus and the yield stress of a foam made of an elastic perfectly plastic metal (Eq. (8) with n = 0). If we adapt the two constants C4 and C5 for consisp tency in this p regard, then we must have C4 = 2/ A, and C5 = 1/(2 A) (Eq. (8) with n = 0 and Eq. (11) with a = 2). Eq. (15) then becomes r0f ¼ r0 ðN þ 2ÞðN Þ 1
1þN N N1 ð1þ3N Þ 2 N Að 2N Þ V m 2N ð2N þ 1Þ
ð16Þ
now showing the same dependence as Eq. (14) on both A and Vm. The ratio R between the two predictions (Eq. (16) divided by Eq. (14)) is then simply R ¼ ðN þ 2ÞðN Þ 1
N N 1 2N ð2N þ 1Þ
ð17Þ
This is plotted in Figure 1. As seen, R remains close to one for all N values between 1 and 20, deviating by at most 17% from unity (and by less at high N values, where the rate of variation of strain-rate with stress is highest). The two approaches are therefore consistent provided the yield stress and the modulus are evaluated consistently with one another. Another simple test of Eq. (10) can be obtained by considering low-density foams made of linked straight struts. For such foams, the engineering beam analysis of Andrews, Gibson and Ashby is expected to apply relatively well, meaning that the elastic modulus and creep rate of these materials should rank roughly as the resistance to elastic and non-linear bending, respectively, of individual beams subjected to bending (only roughly: even foams made of straight struts also see strut torsion in addition to bending; e.g. Ref. [14]). Now, significant variations in the stiffness and flow stress of such foams can be produced by hollowing out the foam arms, keeping the relative density constant. Predictions of Eq. (10)
R. Mueller et al. / Scripta Materialia 57 (2007) 33–36
Figure 1. The variation with N of the ratio, R, between the value of r0f for a foam where the Young’s modulus is related to that of the metal by a simple power-law determined by the current analysis and that determined by the engineering beam analysis of Andrews et al. [12] (Eq. (16) divided by Eq. (14)).
can thus be compared with estimations of engineering beam theory, from which Eq. (15) was derived. Equations for the deformation of hollow beams are given in Refs. [15,16]. Focusing on struts with a square cross-section, engineering beam analysis leads to the conclusion that a hollow in the beam, the width of which is a fraction f of the total beam width, produces (all else being constant) at any given Vm an increase in Ef/Em by a factor KE ¼
1 þ f2 1 f2
ð18Þ
and in C/c by a factor Kr ¼
1 f 3þn ð1 f 2 Þ
3þn 2
Eq. (8), on the other hand, states that 1þn 1þn 1 þ f2 2 2 Kr KE ¼ 1 f2
35
Figure 2. Variation in the factor describing the increase in C/c in foams with hollow struts, Kr (plotted on a logarithmic scale), with the width, f, of the hollow in the beam. Equations derived by the current approach and engineering beam analysis are plotted for various values of N.
ined; these are N = 4, Q = 176 kJ mol1. The value of K = 5.06 · 1020 s1 may be estimated using the data they report and the value of r0 (=G) = 22.02 GPa for pure aluminium given in Ref. [17]. Values for Young’s modulus of these foams can be estimated by fitting experimental data given in the literature for such foams, [18–21]. yielding F ¼ 2:39 V 2:37 m Figure 3 compares the experimental data found by Andrews et al. for the steady-state creep rate of Duocel foam at 275 C under a stress of 0.42 MPa (their Fig. 11) with (i) the predictions of the model proposed by Andrews et al. (Eq. 8 of Ref. [12]) and (ii) Eq. (10). The ordinate axis on Figure 3 normalizes for temperature
ð19Þ
ð20Þ
The two predictions are compared in Figure 2 for various values of N = 1/n; as seen, the agreement between flow stress values predicted by the two models for a given strain or strain-rate is again within around 10%. Comparison of predictions of Eq. (10) with experimental data on the steady-state creep rate of porous or microcellular metals requires also knowing the elastic modulus of the same porous material (because F must be known, Eq. (1)). This can be done for the data of Andrews et al. on the open-cell aluminium foam produced by ERG and known as Duocel foam, using which steady-state strain-rates were measured for a variety of applied stresses and temperatures, for foams with three different Vm, namely 0.06, 0.09 and 0.14 [12]. Creep constants of the alloy from which this foam was made (6101-T6 aluminium alloy) were also measured by Andrews et al. in the temperature range exam-
Figure 3. Comparison of experimental ([12]) data for the steady-state creep rate for open-celled aluminium foam (Duocel) of various densities at 275 C under a stress of 0.42 MPa, with the predictions of the model proposed by Andrews et al. [12] and the predictions of Eq. (10) in this work. The ordinate axis compensates for the effect of varying temperature and applied stress.
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and stress. It is seen that the results of the calculations using Eq. (10) (based on the elastic properties of the foam) compare well with the actual creep data recorded for this material, the prediction being somewhat closer to the data than that from the model of Andrews et al. In short, the variational estimates of Ponte Castan˜eda, Suquet et al. can be used to derive simple expressions for the non-linear plastic or creep deformation of highly porous metals knowing their linear elastic modulus; however, it must be kept in mind that there are limitations to equations presented here. (i) In the formulation of the equations, a small deformation is assumed; foams deform with large local strains, which cause even foams of linear elastic metal to behave non-linearly at higher strain [14,22–24], cause the relative density to evolve during deformation, and finally cause the foam to become anisotropic after a significant cell shape change. So in all rigour equations given here apply only for the early stages of deformation. (ii) The fact that isotropic foam properties were assumed from the onset also means that the analysis does not apply to anisotropic foams, including the cubic or hexagonal symmetry structures most often assumed in numerical simulations of foam mechanical behaviour. (iii) Foams often show behaviour governed by local instability events, such as buckling of beams or membranes (see Ref. [25] for an illustration). In particular, in stretching-dominated foams of high beam coordination, ‘‘yield’’ is indeed often dominated by buckling rather than stable plastic flow [26]. For such cases what precedes is also clearly invalid. (iv) Errors may be introduced due to the nature of creep behaviour. Steady-state creep is assumed: primary creep may dominate initial (low strain) foam creep data. Also, in creep, one must keep in mind that a single steady-state deformation law is assumed. Local peak stresses outside its range of validity may cause shifts in behaviour, for example by power-law breakdown at high strain-rates, as proposed by Wanner et al. [27,28], or at low applied stress (and high temperature) by diffusion creep, such as reported by Goussery [29]. Finally, one must also keep in mind that the predictive precision in terms of strain-rate at a given stress of any power-law creep model is limited when the creep exponent N is high. For instance, a 10% error in the average stress will result in errors in the creep rate by factors of 1.6 and 2.6 with N = 5 and 10, respectively. In summary, simplifying Ponte Castan˜eda and Suquet’s variational estimates for non-linear composite deformation yields a simple expression for the uniaxial steady-state flow rate of power-law creeping metal foams knowing the foam elastic modulus as a fraction of that of the metal. The estimate is shown to be consistent with the data of Andrews et al. for open-celled aluminium foam, and with the conclusions of other models in the literature for the steady-state power-law creep of isotropic microcellular materials.
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