the limits of solid state foaming · 2013. 4. 24. · during powder consolidation. if this is...

Acta mater. 49 (2001) 849–859www.elsevier.com/locate/actamat

THE LIMITS OF SOLID STATE FOAMING

D. M. ELZEY† and H. N. G. WADLEYDepartment of Materials Science and Engineering, School of Engineering and Applied Science, University

of Virginia, Thornton Hall, Charlottesville, VA 22904 VA, USA

( Received 11 October 2000; accepted 27 October 2000 )

Abstract—Ultralight metal foams can be produced by gas expansion in either the molten or solid state bythe release of H2 during the decomposition of TiH2 particles. An alternative approach uses a powder metallur-gical route to deliberately trap a low solubility gas within interparticle spaces during consolidation. This issubsequently used to plastically expand the voided solid during a post-consolidation heat treatment. Porositiesof about 40% have been reported. However this is only about half that of melt-foamed materials and thereis much interest in designing processes that increase it. Micromechanical models for the plastic expansionprocess are developed and used to identify the practical porosity limits in this entrapped gas expansionapproach. It is shown that the porosity is limited by the reduction in pore pressure as voids expand, andultimately by the loss of gas accompanying void coalescence. Increasing the initial pore pressure is shownto also lead to the formation of face sheet delaminations in stiffened, porous core sandwich panels. Itsdependence on the process methodology is considered. Achievable porosities during solid state foaming areshown to be limited to less than 50%; much less than that of metals foamed in the liquid state. A simpleextension of the analysis to semi-solid state expansion shows that much higher porosities could be achievableunder these conditions because void coalescence can be avoided. 2001 Acta Materialia Inc. Published byElsevier Science Ltd. All rights reserved.

Keywords:Powder processing; Titanium; Foams; Plastic deformation

1. INTRODUCTION

Metal and metal alloy foams containing up to 95%porosity are being explored for applications requiringhigh specific strength and stiffness, high mechanicalenergy absorption (e.g. for protection against impactsand explosions), for heat exchangers, flame resistanceand applications motivated by their intrinsic buoy-ancy [1]. They offer many potential benefits forcomponents that operate in loading environments andat temperatures where conventional polymeric foamscannot be used [2, 3: p. 345]. They also promise costadvantages over conventional lightweight honeycombstructures and rib-stiffened panels used in many aero-space and ship structures [4].

Early closed-cell metal foams were created by theeutectic solidification of binary metal–hydrogen sys-tems [5]. More recently, much higher porosity closed-cell metal foams have been created using liquid stateprocesses in which a gas is introduced into the moltenmetal either by direct injection through an orifice orby the decomposition of a dispersed foaming agent,

† To whom all correspondence should be addressed. Tel.:11-804-982-5796; fax:11-804-982-5660

E-mail address:[email protected] (D. M. Elzey)

1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.PII: S1359-6454(00 )00395-5

such as TiH2 particles [6, 7]. The “froth” created atthe top of the melt as gas bubbles rise to the surfacecan be solidified, resulting in a stable, low densitystructure containing closed cells, Fig. 1.

Metals and alloys can also be foamed in the solidstate by entrapping gas within the intraparticle voidsduring powder consolidation. If this is followed byheating, the gas pressure within the voids increases

Fig. 1. Low-density cellular metals produced by liquid statefoaming processes are characterized by thin intra-cellular walls

of relatively uniform wall thickness.

850 ELZEY and WADLEY: THE LIMITS OF SOLID STATE FOAMING

and when it exceeds the resistance to plastic flow, thecomponent expands by either plasticity or creep. TheLow Density Core (LDC) process for Ti-6Al-4V,explored originally by Kearnset al. [8] and morerecently by Martin and Lederich [9: pp. 361–370],Schwartzet al. [10] and Queheillatet al. [11], is anexample of such a solid state foaming process. TheLDC process begins with the hot isostatic pressing(HIP) of a thick-walled Ti-6Al-4V cannister contain-ing Ti-6Al- 4V powder and an inert gas (e.g. Ar) withan initial pressure of typically 3–5 atm [8]. After con-solidation to a relative density greater than 90% tophysically isolate the porosity into discrete, uncon-nected voids, the compact (i.e. the cannister plus par-tially consolidated powder) is rolled into a sheet torefine and homogenize the pore distribution and tocreate a sheet-like structure. These sheets can be cold-pressed to a near-net shape [9], resulting in voidswhich are flat and crack-like. A final heat treatmentis then used to induce the expansion process. Thefinal shaped structure is typically a sandwich panelwith a modestly low relative density core (25–40%porosity) and integrally bonded, fully dense titaniumalloy face sheets derived from the former HIP canwalls, Fig. 2.

It would be desirable to produce structures witheven greater porosity levels in order to exploit thebeneficial property combinations offered by very low-density metallic foams [4]. Experiments have failedto accomplish this and our primary objective here isto develop relatively simple micromechanical modelsdescribing the dependence of porosity on time, initialpore size and spacing and processing conditions, anduse these to explore the limitations on achievableporosity during solid state foaming. Essentially, theattainable porosity is limited by a reduction of gaspressure, which occurs either as a result of the expan-sion itself (an increase in void volume lowers the gaspressure), or because of gas lost through the surfacesof the expanding body when pores become intercon-nected (Fig. 3). This latter effect is a result either offailure to isolate voids properly so that interconnectedpaths for gas flow to the surface exist from the startof the expansion step, or because of void coalescenceand the creation of long-range gas flow paths [5]. Thegeneration of long-range interconnected void pathsvia void coalescence belongs to that class of physicalphenomena known as percolation and can thus beunderstood by the application of concepts germane topercolation theory [12].

We begin by developing a simple model for thecreep expansion of an internally pressurized, voidedsolid. The metallic matrix material is modeledassuming isotropic, power-law creep, the voids areassumed to contain an ideal gas with an initial press-ure, p0, and void interactions are neglected. Themodel can be analyzed to identify the achievableporosity as a function of the temperature–time his-tory, the initial gas pressure and the initial relativedensity for any material for which the creep constants

Fig. 2. The structure and pore morphology of a Ti-6Al-4Vsandwich panel core as it evolves through the solid state pro-cess sequence of HIP’ing, rolling and a thermal expansion heattreatment. (a) Powder compact; (b) after HIP consolidation; (c)

after hot rolling, and (d) after expansion heat treatment.

851ELZEY and WADLEY: THE LIMITS OF SOLID STATE FOAMING

Fig. 3. The mechanisms limiting the solid state foaming of a metal or alloy by the thermal expansion ofentrapped gas are pressure loss due to increase in void volume and due to void coalescence, leading eventually

to loss of gas through the component surfaces.

are known. The model can be modified to account forthe pore flattening that occurs during the rolling step.We then consider the effect of void coalescence usinga two-dimensional idealization in which an array ofrandomly placed, penny-shaped crack-like voidsexpand, coalesce and eventually reach a percolationlimit. The percolation threshold is taken to correspondto the limiting attainable porosity since gas is thenrapidly lost through the component surfaces. Theresult is expressed in terms of achievable porosity asa function of starting density and gas pressure. Thisresult is used to establish the limits of expansion inthe solid state where creep flow dominates. We com-pare this to the behavior encountered in semi-solidflow, which is more appropriately modeled as a vis-cous fluid.

2. SOLID STATE FOAMING

At the start of the expansion process, we considera material containing a relatively uniform distributionof isolated, spherical voids, each containing a giveninitial gas pressure; if the temperature is increased,the gas pressure increases, and if the flow stress isexceeded, it causes the material to expand iso-tropically. We seek to predict the relative density (D),or alternatively, the porosity,w 5 12D, as a functionof time for an arbitrary temperature–time history. Thematrix material is assumed to be nonlinearly viscousin accordance with a power-law formulation

eij 532Bsn21

e sij (1)

where eij are the strain rate components,se, is theequivalent stress,sij are the deviatoric stress compo-nents, andB andn are material parameters. Equation(1) can also be expressed in the form of a strain ratepotential,f

eij 5∂f∂sij

with f 5B

n 1 1sn 1 1

e . (2)

Strain rate potentials have also been developed forpower-law materials containing spherical voids [13].In particular, Duva and Crow [14] considered aspherical shell representation of a solid containing auniform distribution of spherical, non-interacting,monosized voids (Fig. 4). If the shell is assigned aninner radius of unit length, then the outer radius,R,can be chosen such that the volume fraction of voids

Fig. 4. The spherical shell unit cell model for creep of a porous,power-law creeping medium [14] is used to model inelastic

expansion during solid state foaming.


is w 5 1/R3. Duva and Crow showed that the poten-tial for this case can be expressed as

fDC(S) 5B

n 1 1sn 1 1

DC (3)

where S is the macroscopic stress applied to thevoided solid (sij are the components of stress actingon an arbitrary material element within the matrix),andsDC is given by

s2DC 5 aS2

e 1 bS2m. (4)

whereSe andSm are the macroscopic deviatoric andmean stresses, respectively. The coefficientsa andbdepend only on relative density and the material non-linearity and are given by

a(D) 5 F1 123(12D)G · D22n/(n 1 1) (5)

and

b(D) 5 F n(12D)[12(12D)1/n]nG 2

n 1 1· S 3

2nD2

. (6)

The macroscopic strain rate components for thevoided solid are then given by

Eij 5∂fDC

∂Sij

(7)

Given a set of applied stresses,Sij the dilatationalstrain rate from Eq. (7) isEkk 5 (∂fDC)/(∂Sm), andthe rate of expansion/densification is given by

D 5 2DEkk. (8)

Conventional applications of equation (7) include thesimulation of densification during hot isostaticpressing of metallic powders [15] and the analysis ofdilatational effects associated with void growth andcoalescence during tensile straining [16]. In bothcases, the stresses are applied externally to the regionof interest. Here we are concerned with stressesdeveloped internally due to expansion of gas withinthe voids. However, use can be made of the fact thatan internal pressure is equivalent to an externallyapplied hydrostatic tensile stress of the same magni-tude.

If the number of moles of gas in a void remainsconstant, then the pressure decreases as the voidexpands. Assuming ideal gas behavior within a void,the equation of state relating pressure, temperatureand volume is given by

pV 5 nRT (9)

wheren is the quantity of gas in moles,R is the gas

constant (8.315 J/mol-K) andV is the void volume.It is convenient to normalize the state variables inequation (9) by their initial values,p0, V0 and T0,which allows us to rewrite equation (9) in dimen-sionless form as

p 5 V21T (10)

wherep 5 p/p0, T 5 T/T0, and for the spherical shellunit cell, the dimensionless void volume,V, can beexpressed in terms of the relative density by

V 5D0(12D)D(12D0)

(11)

whereD0 is the initial relative density.Substituting equation (11) into (10) and differen-

tiating with respect to time gives an expression forthe rate of change of pressure when the temperatureand relative density vary:

p- 5D

D0(12D)F(12D0)T-

1 pD0

D2DG (12)

where dots are used to indicate time derivatives, andT 5 pV has been used to eliminate the normalizedtemperature. With the rate of change of temperaturegiven as input, the coupled set of differential equa-tions (8) and (12) can be solved numerically to simul-taneously obtain the time-dependent density (orporosity) and internal gas pressure.

Using data for Ti-6Al-4V at 920°C (B 5 6.9310213, n 5 4), Fig. 5 shows the predicted porosityachieved after 1, 15, and 30 h at 920°C as a functionof the initial gas pressure,p0, and starting density,D.The porosity reached is seen to increase with increas-ing initial void pressure and, less strongly, withdecreased starting density. Peak porosities lie in the40–50% range for this particular alloy and heat treat-ment and are limited by the loss of pressure as thevoids expand. Reducing the starting density, increas-

Fig. 5. Predicted achievable porosities in Ti-6Al-4V after 1, 15and 30 h at 920°C as a function of initial pore pressure andrelative density, limited only by loss of pressure due to expan-

sion.


ing the initial pore pressure or the maximum tempera-ture are all beneficial. However, the starting relativedensity of the compacted powder may not bedecreased below 0.9 due to the requirement that voidsbe isolated at the start of the expansion process.While increases in temperature and initial pressurewould be expected to produce still greater porositylevels, other limiting conditions arise here also.Increasing the initial gas pressure leads to rupture ofthe HIP cannister either before or during consoli-dation [10]. Pushing the expansion process further byraising the temperature leads to void coalescence anda consequent loss in gas pressure and degraded mech-anical properties. Void coalescence consists of therupture of ligaments separating adjacent voids,eventually giving rise to the formation of intercon-nected pathways allowing gas within the interior todiffuse to the component’s surfaces where it is eitherlost or, if a dense face sheet exists, to deformation(blistering) of the face sheet. Higher processing tem-peratures may also lead to undesirable microstruc-tures, as in the case of Ti-6Al-4V, in which a Wid-manstaetten structure is formed at temperatures above1000°C. Void coalescence as a limitation on solidstate foaming is discussed next.

3. VOID COALESCENCE

If the relative density following consolidation isgreater than 0.9, then during the initial phase of theexpansion process, the voids are small and well separ-ated, so that each void may be considered indepen-dently of its neighbors. For isolated spherical voidsin an (incompressible) power-law creeping matrixsubjected to a spherically symmetric stress-field, theradial velocity must be given byvr 5 Ar22. Conse-quently, the stresses scale according tos~r23/n. Thestress field decays more rapidly with distance fromthe void asn decreases. However, as two neighboringvoids in an expanding solid metallic foam approacheach other, their stress fields eventually begin to inter-act, accelerating failure of the ligament betweenthem. In addition to this stress concentration effect,premature rupture of the walls separating voids isaided by low ductility (and fracture toughness). Ti-6Al-4V is an excellent candidate for solid state foam-ing due to its superplastic forming (SPF) capabilitywhen processed in the 900–930°C temperature range.In this region, strains of 100% or more can beaccumulated without failure. Kearnset al. [4] wereable to expand Ti-6Al-4V to porosities approaching40% before observing significant void coalescence.

More recently, Dunand and Teisen [17] have notedthe general nature of the void coalescence limitation.They introduce the application of transformationalsuperplasticity (as opposed to micro-structural super-plasticity in which sliding of tine grains is the operat-ive mechanism). This allows the range of attainableporosities to be extended in metals and alloys whichexperience allotropic phase transformations. Dunand

and Teisen showed that transformation superplas-ticity, achieved by cycling the temperature throughthe phase transition temperature during expansion,allowed accelerated expansion and greater porositiesto be attained in commercially pure titanium and Ti-6Al-4V relative to isothermally treated samples. Thepeak porosities reported were approximately 38% forCP-Ti and 27% for Ti-6Al-4V for cycled samples ascompared to 25% and 17% when expanded iso-thermally. The delay in void coalescence due tosuperplastic deformation was found to be responsiblefor the improvement in the attained porosity.

There is a sizable literature on the subject of ductilevoid growth and coalescence, most of it directed atthe tensile straining of initially dense metals andalloys containing hard inclusions which promote earlynucleation of primary voids [e.g. [18–21]]. Failurethen occurs in these materials by void sheeting inwhich a localized plastic instability leads to rapidnucleation, growth and coalescence of voids on shearplanes connecting primary voids [20]. Models forcrack growth under these conditions are often basedon the assumption that crack extension occurs whena critical strain,eF, is reached a distance,xc, ahead ofthe crack tip. The characteristic length,xc, may berelated to microstructural lengths, such as the spacingbetween inclusion particles [21]. Our objective is todevelop a simple model for estimating the limitingrelative density (or porosity) at which long range†paths of interconnected voids develop, and to identifyits dependence on initial void size, spacing and vol-ume fraction. The influence of material parameters,and in particular the strain rate sensitivity as givenby the stress exponent in equation (1), will also beexamined. We make extensive use of concepts whichhave already been developed and described in theliterature on fracture by ductile hole growth.

Use is made of a two-dimensional idealization andthe C*-integral concept [22] to simulate the growthof penny-shaped, crack-like voids in a power-lawcreeping solid. Void interaction effects are taken intoaccount. This approach leads to an estimate of thetime required for two adjacent voids initially separ-ated by a distance,l, to coalesce. Knowing the distri-bution of void spacings, it leads to the density offailed ligaments,NLF, and the relative density, bothas a function of time‡. When plotted parametrically,these two results allow prediction of the extent ofvoid coalescence as a function of density, i.e.NLF(D).Finally, estimating thecritical density of failed liga-ments using percolation theory, theNLF(D)-relationprovides the limiting achievable density (or porosity)due to void coalescence.

† Here, “long range” is taken to mean over distances onthe order of the component’s dimensions.

‡ The expansion model described in Section 2 cannot beused here to computeD(t) since it neglects void coalescence.


3.1. Analysis of void coalescence

The progression of cell wall rupture is modeledusing a two-dimensional idealization in which thevoids are represented as penny-shaped cracks ofradius,a, within a power-law creeping solid (see Fig.6). The voids lie in a single plane and are randomlydistributed, as illustrated schematically in Fig. 6(c).Considering an internally pressurized, representativevoid, the stress field at the void’s periphery (i.e. thecrack tip region) is described by the Hutchinson–Rice–Rosengren (HRR) field [23]

sij 5 S C∗

InBrD1

n 1 1sij (13)

in which B andn are the creep parameters introducedin equation (1),C* is the crack tip parameter charac-terizing the stress intensity,In is a dimensionless fac-tor depending on the crack opening mode and the

Fig. 6. The influence of material nonlinearity and void spacingis explored using a model in which voids are idealized asinternally pressurized, penny-shaped cracks in a power-law

solid.

value of the stress exponent (n), r is the radial coordi-nate specifying the position of a material point rela-tive to the crack tip, andsij is a geometry-dependentfactor which varies withq, the angular coordinate, butnot with r. The crack tip parameter,C* , can beexpressed in terms of the net section stress,snet (theload divided by the area of the uncracked ligament)[24: p. 269]

C∗ 5 aBsn 1 1net g1(2a/l,n) (14)

The factor,g1, which depends only on the ratio ofvoid diameter to spacing andn is [23]

g1 5 S l2a21D h1

(1.455h)n 1 1 (15)

where

h1 5 (0.1220.44(2a/l) 1 1.04(2a/l)2 (16)1 0.46(2a/l)3)/(2a/l)2

h 5 √4s2 1 4s 1 222s21 (17)

s 5 1/((l/2a)21). (18)

Since the internal void pressure,p, is equivalent toan externally applied hydrostatic stress of the samemagnitude, the net section stress is

snet 5 S 112fv

Dp (19)

wherefv is the fraction of area in the plane occupiedby voids, given byfv 5 (2a/l)2. The stress field givenby equations (13–19) leads to an accumulation ofinelastic strain ahead of the crack tip (void periphery),which can be calculated by inserting the stress fieldinto the material law (equation (1)) and integratingover time. The crack growth rate,a, is then determ-ined by requiring the strain at a distance,xc, ahead ofthe crack tip be equal to a critical strain,eF. Riedel[25: p. 274] has presented such a critical strain formu-lation, given by

a 5pse(0)(Bxc)

1n 1 1

eF sin (pa) SC∗

InDaFSa2a0

xcD 1

n 1 1(20)

2G(a)

G(22a)G(2a21)Gin which a 5 n/(n 1 1), a0 is the initial crack length(void radius) andG is the conventional gamma func-tion. Equation (20) can be solved numerically toobtain the void size as a function of time,a(t), for anarbitrary void pressure, initial size and spacing. Fora given initial void size and pressure, application ofequation (20) allows the relationship between thetime required for rupture of the ligament separating


two adjacent voids and their spacing,tR(l), to be plot-ted, as shown in Fig. 7 for several representativevalues of critical strain.

However, the fraction of failed ligaments at anytime depends not only on the time required to rupturea ligament of a given length, but also on the distri-bution of initial void spacings. We assume a GaussianPDF (probability density function) for the distributionof void spacings

j(l) 51

√2pslexpF1

2Sl2lsl D2G (21)

where l and sλ are the mean spacing and standarddeviation. The fraction of all spacings less than or

equal to l9 is then NLF 5 El9

0

j(l)dl. If the time

required to coalesce two voids separated byl9 is t,as given by equation (20), then voids with spacingssmaller thanl9 will also have coalesced att 5 t.Thus the fraction of ligaments ruptured at time,t,

becomesNLF 5 El(t)

0

j(l)dl. Finally, if the number of

void ligaments per unit area (referred to as the liga-ment density) isrL, then the number of ruptured liga-ments at timet is

NLF(t) 5 rL El(t)

0

j(l)dl. (22)

The ligament density is found fromrL 5 lv.rv, wherelv is the average number of ligaments per void (takento be 5) andrv is the number of voids per unit area,given by rv 5 fv/(pa2). Fig. 8 shows the predictedligament failure density,NLF, as a function of time,given a void pressure of 4 MPa and an initial voidfraction of 0.08. The time evolution of the ligamentfailure density is not of so much consequence here,

Fig. 7. The predicted dependence of the time to rupture theligament between adjacent voids on the void spacing. The meaninitial void spacing was assumed to be 0.5 mm with a standard

deviation of 0.1 mm.

Fig. 8. The fraction of failed ligaments (NLF) as a function oftime as predicted by the crack growth model for void coalesc-

ence (using the same void distribution as for Fig. 7).

but rather howNLF correlates with the relative densityor porosity. Therefore, the next step is to compute theporosity, w, as a function of time using the crackgrowth model in equation (20), and to then determineNLF(w), parametrically.

At any time, t, an increment in void area,DA, isassociated with each ligament, i.e. void spacing, from

DA(l,t) 5 p[a(l,t)22a20] (23)

where equation (20) is used to computea(l,t). Then,sincej(l)dl · DA(l,t) is the increase in area associa-ted with spacings betweenl andl 1 dl, the cumu-lative void fraction at time,t, is

fv(t) 5 fv0 1 rL El(t)

0

j(l) · DA(l,t)dl (24)

Equations (22) and (24) can be related parametricallyto obtain the number of failed ligaments per unit areaas a function of the porosity, as shown for severalvalues of the stress exponent in Fig. 9.

Fig. 9. Predicted ligament failure density as a function ofporosity for several values of the creep stress exponent. Theeffect of increasing material non-linearity is to lower the

porosity achieved prior to percolation.


3.2. Percolation of ligament failures

As the number of failed ligaments increases, clus-ters of interconnected voids form. When the lengthof contiguous pathways approaches the size scale ofthe component, expansion ceases because of gas lossthrough the free surfaces of the component. The criti-cal density of interconnections is known as the perco-lation threshold and its dependence on dimensionalityand initial packing configuration is the subject of per-colation theory [26]. Before proceeding, it is neces-sary to distinguish between site percolation and bondpercolation: in the present context, sites are equival-ent to voids and bonds represent the ligamentsbetween voids. Void coalescence, the physicalphenomenon of interest here, corresponds to bondpercolation. The bonds may be blocked (intact) orunblocked (ruptured). Restating the criterion for per-colation in these terms, percolation occurs when thecluster size of unblocked bonds approaches infinity.

While analytical results exist for some two-dimen-sional lattices, e.g. square, the percolation threshold,pc, must be obtained by the Monte Carlo method, orsome similar numerical experiment, for a randompacking regardless of dimensionality. For a two-dimensional, random close packing (RCP), the perco-lation threshold may be estimated from the pseudo-invariant quantity,zpc<2, wherez is the coordinationnumber (i.e. the number of nearest-neighbor sitesassociated with any particular site). Since the coordi-nation number is a distributed quantity in a randomstructure, an average value must be used, say five,which leads topc 5 0.4.

Since the spatial distribution of voids may not cor-respond to a random dense structure, a series ofnumerical experiments were conducted in whichvoids (represented as circles) were placed randomlyin a plane of unit area, such that no overlap of voidsoccurred. Such random loose structures differ fromRCP in their greater variability in site spacing andappear to represent actual void packings better thenRCP. Once a void array was generated, all possiblebonds or ligaments were identified by performing aDelauney triangulation upon the array. Ligamentswere then allowed to rupture (bonds were unblocked)in order of the shortest ligament to the largest and thefraction of unblocked bonds corresponding to perco-lation recorded. Computational investment was keptsmall by considering arrays containing only about 100voids. Although some variability was observed inpc,we will consider the highest value (pc 5 0.48), whichleads to an upper bound estimate of the attainableporosity.

3.3. Limiting expansion due to void coalescence

The limiting porosities imposed by void coalesc-ence may now be obtained by combining the resultsof Sections 3.1 and 3.2. Figure 9 shows the depen-dence of the fraction of failed ligaments on porosityfor several values of the stress exponent,n. Forn 5 4, the upper bound bond fraction, i.e.NLF, corre-

sponding to percolation indicates an achievableporosity of approximately 0.34. For more strain rate-sensitive materials (highern), the achievable porosityat percolation is reduced.

4. SOLID STATE FOAMING LIMITS

The variables affecting achievable porosity duringsolid state foaming include the initial gas pressure,the initial relative density, the size, shape and spatialdistribution of voids, the matrix material selection,and the heat treatment schedule. Although increasingthe initial gas pressure and porosity aid the expansionprocess, both are constrained, as discussed pre-viously. Higher internal gas pressures could be intro-duced while avoiding rupture of the HIP can byincreasing the can’s wall thickness, but greater com-paction stresses would then be needed fordensification. Additionally, the thicker face sheets ofthe final product lessen the impact on specific proper-ties achieved by means of a highly porous core,though this could be mitigated by subsequent milling.

Avoiding premature void coalescence due to frac-ture of intervoid ligaments requires a matrix whichtends to deform uniformly and which has high duc-tility. The selection of an appropriate matrix materialappears to be critical for achieving the highestporosity metals or alloys by solid state foaming.Although the void expansion model (Section 2) indi-cates the loss of gas pressure due to void volumeexpansion can be compensated for by further increasein temperature, void coalescence then becomes thelimiting mechanism. Alloys which exhibit superplas-tic behavior are expected to offer the greatest poten-tial for solid state foaming because of their ability toavoid strain localization while undergoing largedeformations.

However, even if the effects of stress concen-trations and material instabilities leading to prematurevoid coalescence are neglected, percolation would beexpected to occur at a porosity of approximately0.637, corresponding to the percolation threshold fora three-dimensional, random close packing of monos-ized spherical voids. A perfectly close-packed (e.g.fcc or hcp) void lattice, though difficult to achieve inpractice, would allow porosities of approximately 0.7to be achieved. Similarly, a well-controlled size distri-bution in which smaller voids were somehow packedinto the interstices between larger voids might allowporosities in excess of 0.9 to be reached.

Figure 10 summarizes schematically the porositywhich may be achieved during solid state foaming ofa metal or alloy where voids are randomly distributed.The range of porosities achievable by liquid statefoaming (0.85#w#0.95) are shown for comparison.The starting relative density varies from 0.9(determined by the requirement that voids be isolatedat the start of the expansion process) to 1. The gaspressure ranges from zero, in which case no expan-sion can occur, topmax, determined by the rupture


Fig. 10. The porosity levels achievable by solid state foaming are limited by loss of gas pressure due to voidvolume expansion and void coalescence. Even under ideal conditions in which premature rupture of voidligaments is avoided, the achievable porosity is well below that of metal foams produced by liquid state

processes.

strength of the HIP can. At lowp0, the loss of gaspressure due to void expansion is the limiting mech-anism and only small increases in porosity are real-ized. At higher initial gas pressures, premature voidcoalescence limits the attainable porosity to approxi-mately 0.35. For matrix alloys with no capability todeform superplastically, premature void coalescencedue to ligament fracture imposes a fairly severe limi-tation on the foaming potential. Ti-6Al-4V, whichexhibits SPF within a narrow range of temperature atapproximately two thirds of its melting point, is ableto expand substantially before void coalescencebecomes a factor (0.3#w#0.4). Foaming at tempera-tures above the SPF range lowers the resistance tocreep flow, but also lowers the strain to fracture and

results in undesirable microstructures. Superplasticalloys in which the SPF range occurs at still greaterhomologous temperatures (T/Tm>0.6) and for whichthe ratio of void pressure to flow strength is high,might be expected to provide the greatest potentialfor achieving ultralightweight structures by solidstate foaming.

Beyond the practical limitations due to void expan-sion and coalescence, the ideal porosity limit is shownas a surface atw 5 0.67 (Fig. 10). As discussedabove, this porosity might be expected for a “per-fectly superplastic” alloy in which void ligamentscould be deformed with no strain localization or frac-ture. It is clear that even in this “ideal” case, the achi-evable porosities are well below those demonstrated


for liquid state foaming. This ideal case also high-lights the differences in cell structure between met-allic foams produced in the solid state and thoseresulting from liquid state processes. Solid statefoaming leads to thinned cell walls (ligaments separ-ating adjacent voids) with most of the matrix materialconcentrated at cell joints (i.e. at the edges of cellfaces). Foams produced by the solidification of aliquid froth are characterized by a much more uni-form distribution of mass (although some variationin cell wall thickness does occur and results in theformation of plateau borders [3]). These morphologi-cal differences arise from the differing constitutivebehavior of a liquid or semi-solid membrane as it isstretched versus a solid subject to the effects of workhardening and plastic instability.

5. SEMI-SOLID REGIME

Countering the tendency of intervoid ligaments toneck and fracture is the void surface energy, whichprovides a driving force for the elimination of curva-ture. The surface tension force is approximately,Fs5gR, whereg is the surface energy [J/m2] and R isthe average void radius. During solid-state foaming,the forces due to surface tension are much lower thanthose required to deform the ligaments, e.g. by plasticyielding or power-law creep and therefore have a neg-ligible effect on the tendency for neck formation.However, the limiting effects of ligament necking andvoid coalescence during solid state foaming may beovercome by heating above the alloy’s solidus tem-perature. For alloys in which a melting range exists,this leads to a semi-solid whose deformation resist-ance can be characterized by a temperature-dependentviscosity,m(T). As melting occurs, the viscosity willdecrease until surface tension forces are able to over-come the resistance to flow and eliminate the curva-ture present in the inter-void ligaments. With neckingand void coalescence suppressed, much greaterporosities can be achieved [2].

As a measure of the ability of surface tensionforces to eliminate ligament curvature and hencenecking, we define a dimensionless ratio,y, obtainedby computing the ratio of the surface tension force tothe ligament shearing force. For the liquid (or semi-solid), we assume newtonian behavior as a firstapproximation (i.e.t 5 mg, wheret is the shear stressrequired to deform the viscous liquid at a shear strainrate, g), so that the shearing force needed to deformthe ligament isFl 5 2mgR2, leading to

y 5g

4RmS1gD. (25)

For typical values (surface energy around 1–2 J/m2,pore radius of 0.1–1 mm, a viscosity of 0.005–0.05Pa-s, and strain rates of 1028 to 1021 s21), y>106,indicating that necking does not occur during liquidor semi-solid foaming.

If linear viscous (e.g. Nabarro–Herring creep)behavior is assumed during solid state foaming, theviscosity can be expressed asm 5 kTd2/(aDvV),wherek is Boltzmann’s constant,d is the grain size,a is a constant,Dv is the lattice diffusion coefficient,and V is the atomic volume. The necking parameter(i.e. ratio of the surface tension force to the forceneeded to deform the ligament in shear at strain rate,g), becomes

y 5agDvV

4RkTd2S1gD, (26)

where the lattice diffusivity is expressed in Arrheniusform as Dv 5 Dv0 exp (2Qv/RT). Inserting typicalvalues (a 5 10, g 5 2 J/m2, Dv0 5 1.931024 m2/s,Qv 5 284 kJ/mol, V 5 10229 m3), and letting thepore size vary from 0.1 to 1 mm, the grain size from1 to 100µm, and the strain rate from 1028 to 1021

s21, shows thaty is usually much less than one, indi-cating that surface tension forces are unable to coun-ter ligament necking during solid state foaming. Onlyfor small grain size (0.1µm), low strain rate (,1026 s21), and high temperature (T/Tm>0.9), doesyapproach or exceed one. Figure 11 illustrates sche-matically the limiting achievable porosity duringsolid, semi-solid and liquid state foaming. A corre-sponding trend in the necking parameter,y can beseen, which undergoes a rapid transition (from verysmall to large values) during the semi-solid tempera-ture range.

6. CONCLUSIONS

Solid state foaming in which metals or alloys areexpanded in the solid state by means of gas pressuretrapped within small internal voids can be used toachieve moderately lightweight structures. However,the maximum achievable porosities are much lessthan for metals foamed in the liquid state and theresulting cell structures are different. The limitationson the porosity levels achievable by solid state foam-

Fig. 11. The transition to much higher porosity levels duringsemi-solid and liquid state foaming relative to the solid stateis due to the suppression of plastic instabilities (leading tonecking of inter-void ligaments during solid state foaming)under conditions where surface tension forces are on the order

of the forces required for material deformation (y>>1).


ing are determined by loss of gas pressure due toincrease in void volume during expansion or becauseof void coalescence. This latter effect can be reducedby processing under superplastic forming conditionsfor alloys exhibiting SPF capability, but even underideal conditions, limits the attainable porosity to lessthan 65%. Finally, even if the voids can be expandeduniformly, the resulting cell structure is very nonuni-form, with most of the matrix material concentratedin cell junctions. The relatively low porosity levelsachievable by solid state foaming constrain theirpotential applications and prevent them from beingviewed as direct competitors of very low densitymetal foams, such as those produced by semi-solidand liquid state processes.

Acknowledgements—The authors gratefully acknowledge fin-ancial support from the Ultralight Metal Structures MURI Pro-gram (Contract No. N00014-96-1-1028, Steve Wax (DARPA)and Steve Fishman (ONR), program monitors). Helpful dis-cussions with Drs. Howard Stone and Sinisa Mesarovic arealso appreciated.

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