deformation processing of metal … processing of metal powders: part i--cold isostatic pressing r....

Pergamon Int. J. Mech. S6. Vol. 36, No. 4, pp. 343-357. 1994 1994 Elsevier Scienoe Ltd

Printed in Great Britain. All fights reserved 0020-7403/94 S6.00 + 0.00

DEFORMATION PROCESSING OF METAL POWDERS: PART I--COLD ISOSTATIC PRESSING

R. M. GOVINDARAJAN* and N. ARAVAS Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania,

297 Towne Building, 220 S. 33rd Street, Philadelphia, PA 19104, U.S.A.

{Received 9 March 1993)

Abstract--Analytical and numerical techniques are used to analyze in detail the stage I Cold Isostatic Pressing (ClPing) of metal powders. Plastic yielding is considered to be the only densification mechanism and the constitutive model developed recently by Fleck, Kuhn and McMeeking [(d. Mech. Phys. SolMs 40, 1139 (1992)] is used to describe the elastoplastic behavior of the metal powder. It is shown that, for the case of powder consolidation in a long cylindrical tube, a change in length is not possible, unless it is accompanied by some distortion ("shape change") of the CIPed specimen. The numerical implementation of the constitutive elastoplastic equations in a finite element program is discussed. The finite element method is used to analyze the C1Ping of titanium powder. The predictions of the finite element solution agree well with available experimental data.

1. INTRODUCTION

In many industrial applications, metal powders are processed using "isostatic pressing" to manufacture near net shape structural components or advanced structural materials such as toughened ceramics, metal matrix composites and intermetallics. In isostatic pressing, the powder is packed into a metallic container of the desired shape; the can is evacuated, sealed and subjected to high pressure until the powder consolidates. Depending upon whether consolidation occurs at an ambient or higher temperature, the process is called Cold- or Hot-lsostatic-Pressing (CIPing or HIPing). The pressure transmitting medium is usually liquid in CIPing and gas in HIPing [ l] .

Ashby and co-workers [2-8] studied in detail the micromechanisms of powder consolidation and developed HIP diagrams for a number of materials. They conclude that densification during isostatic pressing occurs due to the plastic yielding and creep of the material composing the powder as well as due to boundary and surface diffusion. Recently, McMeeking and co-workers [9-11] have developed constitutive models for metal powders that densify due to the three aforementioned mechanisms.

The first part of the densification process, in which the relative density D is less than 0.90, is known as stage I [12, 13]. During stage I, the pores in the powder are interconnected and the porosity decreases as necks grow at the contact points between the particles. When the relative density level is about 0.90 or higher (stage II), the individual pores are sealed off, and, from the modelling point of view, the material can be thought of as a solid containing isolated voids [13].

In CIPing, consolidation occurs due to plastic flow of the metal particles in contact. Typical HIP cycles, however, involve temperatures higher than 60% of the melting temperature of the metal composing the powder. At those temperatures, most materials exhibit time-dependent behavior, and creep and diffusion become additional consolidation mechanisms.

A common defect in CIP and HIP products is the distortion ("shape change") of the manufactured component. McMeeking [14] and Xu and McMeeking [15] studied recently the effects of the canister on shape change and concluded that the final shape of CIPed components depends strongly on the can design. In HIPing, shape change is known to occur in schedules where heat is applied rapidly enough to cause substantial temperature gradients within the HIPed specimen [7].

* Current address: Hibbitt, Karlsson and Sorensen, Inc., 1080 Main St, Pawtucket, RI 02860, U.S.A.

343

344 R M. (iOVI'xII)~.R.'O,XN and N AR,xv.xx

A detailed analysis of stage I CI Ping of metal powders is presented in this paper, i'last~,. flow of the metal particles is assumed to be the only densification mechanism. The effect, of creep and diffusion, which become important at higher temperatures, are addressed i~ Part 11 of this work, where a detailed analysis of stage I HIPing is presented. In section 2 ~,1 this paper, the constitutive equations used to describe the elastic-plasuc beh;.r~ioi ~,f Ihc powder compact are discussed. A detailed analysis of powder consolidation in a Itmg cylindrical tube is presented in section 3, where it is shown that a change m length is m,t possible in perfectly straight cylinders, i.e. a change in length is alwa)s accompanied b~ shape change. The numerical implementation of the elastic plastic constitutive equations m a general purpose finite element program is discussed in section 4. Finite clement method i~ used to analyze the C1Ping of titanium powder in a long cylindrical tube. and the predictions of the finite element solution agree well with available experimental data.

Standard notation is used throughout. Boldface symbols denote tcnsors the order (ff which are indicated by the context. All tensor components are written with respect to a tixed Cartesian coordinate system, and the summation convention is used for repeated indices. A superscript T indicates the transpose of a second-order tensor, a superposed dot the material time derivative, and the subscripts s and a the symmetric and anti-symmetric parts of a second-order tensor. If A and B are the second-order tensors, and C a fourth-order tensor, then the following products are used in the text (A. B),, = A~kB~, . IAB) i j k~ . . . . l:iBk,, and (C :A)~j = CijMAkt.

2. ( 'ONSTITtJTIVE EQUATIONS FOR METAl. POWI)I:R~;

The deformation rate D is defined as the symmetric part of the spatial velocity gradient and is written as the sum of an elastic and a plastic part, i.e.

D = D e + D p. ~11

The constitutive equations used to describe the mechanical behavior of the powder compact are described in the following.

2.1. E l a s t i c i t y

The elastic part of the constitutive equations is written as:

';' = C e D e. o " ~ 2 )

where a is the Cauchy (or true) stress tensor, C e is the fourth-order elasticity tensor, v

= # + o ' W - W . o is the Jaumann or co-rotational derivative of o, and W is the continuum spin defined as the anti-symmetric part of the spatial velocity gradient.

The elastic response is assumed to be isotropic and the tensor of the elastic moduli is written as:

C[jk, = G(6 ik6S l + ~u~sk ) + ( K - ~ G ) 6 u f k t , (3)

where G is the shear modulus, K is the bulk modulus and 6 u is the Kronecker delta. The elastic moduli K and G depend on the relative density D of the metal powder, and are assumed to vary according to the following two expressions that have been suggested by Budiansky [16]:

1 - D D . . . . . . . . + . . . . . . . . . . . . . . . . . I , 14) 1 -- a 1 - a + a ( K , , / K )

1 .... D D . . . . . . . . . . . + . . . . . . . 1 , ~5) I --- b 1 - h + b ( G ~ . , " G )

where, , ( , 2

'~ = J \ i - - - ~ , / h = 1 5 \ i - , , / ~ o )

and v is the Poisson's ratio of the composite given by:

3K - 2G v - - - (7)

6K + 2G"

In the above, K,, and G,, are the bulk and shear moduli of the metal composing the powder.

Deformation processing of metal powders: Part I 345

2.2. Plasticity Fleck et al. I'9] developed a constitutive law for the plastic yielding of a random aggregate

of spherical metal particles. The particles are assumed to be bonded perfectly by isolated contacts and deformation occurs by plastic yielding of material at and near these contacts. Using several kinematically admissible velocity fields and optimizing the corresponding upper-bound solutions, they propose the following macroscopic yield function for the metal powder compact:

0(~, e~, D) = 5 + -9 2am-(e~) + 4B - 4B2(D) = 0, (8)

where,

9D2(D - Do) (9) B(D)=2 ( l - D 0 )

In the above equations, q = (1.5 a[~a'u) ~n is the macroscopic equivalent Mises stress, a ¢ is the deviatoric part of the Cauchy stress tensor a, p = - a~,/3 is the hydrostatic stress, Do is the initial relative density of the powder compact, and a,, is the uniaxial yield stress of the powder material. The microscopic yield stress a,, depends on the hardening characteristics of the metal and is a function of the local (microscopic) plastic strain e~.

Under pure hydrostatic loading the predicted yield pressure Pr is:

3D2(D - Do) ~r,,, (10)

Pr = (1 - Do)

which is consistent with the findings of Ashby and co-workers [4, 61. The yield pressure Pr vanishes when D = Do, because then the particles touch only at points. The above yield condition is considered valid for aggregates with relative densities in the range of 0.60-0.90 of the theoretical fully dense level. In this range of relative densities the pores are interconnected and the porosity decreases as the necks grow at the contact points between particles. The yield condition, Eqn (8), is not meant to apply above D = 0.90, so that the predicted finite value of the yield pressure Pr when D = 1 is meaningless.

The direction of the plastic deformation rate is outwardly normal to the yield surface. A sketch of the yield surface on the p-q plane is shown in Fig. 1. A notable feature of the proposed yield surface is the presence of a vertex along the p-axis. The direction of the plastic strain rate is not unique at the vertex, and in the presence of pure pressure the plastic flow can have a volumetric as well as a deviatoric part.

At points where the normal to the yield surface is uniquely defined, the plastic deformation rate D p is written as:

D p = ,,if (1) (I l) t311'

where ~l is a non-negative plastic multiplier.

MS 36:4-E

q 1.5 P,

0.5

0.0 . . . . . 0.0 0.5 1.0 1.5

F_ P,

FIG. 1. Macroscopic yield surface for metal powders.

346 R.M. GOVINDARAJAN and N. AR.WAS

The rate of change of relative density is derived from the conservation ol mass a~

D = - D DfA. ~', 2,

The equivalent plastic strain in the powder material ~,~ is assumed to vary according tc the equivalent plastic work expression:

l) a,,i:~ = ai iD~ j. i l31

3. POWDER CONSOLIDATION IN A LONG CYLINDRICAl, TUBE AND THE "'SHAPE CHANGE"

Consider a circular tube so long that end effects can be neglected. The powder is packed into the tube at an initial relative density Do and an external hydrostatic pressure p is applied to the tube. A solution to this problem has been developed by McMeeking i-14] for the special case of thin-walled cylinder. The analysis that follows is quite general and places no restriction on the thickness of the tube.

The tube is modelled as a rigid perfectly plastic material obeying the von Mises yield criterion with associated flow rule. The metal composing the powder is also assumed to be rigid perfectly plastic, i.e. am = ag = constant. The effects of hardening in the can and the powder are discussed briefly in section 3.4. The powder motion confirms itself to the container deformation and is assumed to bond completely to the can. The stress state in the powder is assumed to be a uniform hydrostatic pressure. In view of the vertex on the yield surface along the pressure axis, the deformation in the powder can vary from nearly uniaxial through purely isostatic to nearly axisymmetric [9, 14]. In fact, it is the kinematic freedom at the vertex point that drives the stress state to the vertex on the yield surface. The assumption that the powder is subject to a pure hydrostatic stress is validated from the results of detailed finite element calculations of a cylindrical CIP specimen that are presented in section 3, where it is shown that, away from the end points, the stress state in the powder is indeed a pure hydrostatic stress.

With the exception of regions near the ends of the tube, the deformation of the can and the powder can be described using a generalized plane strain model. The solution is developed with respect to a cylindrical coordinate system (r, 0, z), in which the -,-axis is along the axis of symmetry of the tube. In view of the axial symmetry of the problem, the only non-zero components of the deformation rate tensor are the radial and angular (D, , Do), which are functions of r, and the longitudinal D:, which is constant and equal to i:. The corresponding non-zero velocities and stresses are v,(r), Vz(Z) = i:z, a , ( r ) , ao(r) and ,r:(r).

The superscripts c and p are used in the following to denote quantities ill the can and the powder, respectively.

Let r~ and ro be the inner and outer radii of the canister. The traction boundary condition on the outer surface of the canister is:

cr~,(r,,) = -- Po. ~14J

where Po is the applied pressure. The continuity conditions along the canister powder interface at r = r, are:

t'f/r =t'~l,, and al l , =a~[ , ,. ~15)

Finally, overall equilibrium in the z-direction requires that:

por, 2, = 2 a f r d r - 2 a ~ r d r . 116)

3.1. The canis ter The equilibrium equations and the plastic flow rule for the canister can be written as:

+ -- 0, q17) dr r

Deformation processing of metal powders: Part 1 347

and c

_ _ = ' , c' _ = 2 c a ~ ' , k - _ , ~ c ~ ' , (18) dv'~ 2 a , , v, • dv~ dr r dz

where )5 is the non-negative plastic multiplier. The yon Mises yield condition for the canister becomes:

¢ '2 C' C' ' 2 tr, + tr, o'o + a~ 2 = ~ , (19)

where a~ is the constant yield stress of the canister material. The flow rule Eqns (18) imply that:

dv, ~ v, c - - + - + ~ = O, (20) dr r

which expresses the incompressibility of the canister. Eqns (17-20) can be readily solved for the radial velocity and the deviatoric stress

components (see also Hill [17], p. 267):

( 1 ) c' l + ~ R 2 try' 1 - ~ R 2 a, = (21)

v,~=c~ 0tR-- , tr~Tx/~= x/1 + 3~2Ra, a~/x/~ x/1 + 3°t2R* ,

where ~ is an integration constant, ct = - krd(2O) is proportional to the longitudinal strain k, and R = r/r~ is the dimensionless radial coordinate.

The pressure p* = - a~kd3 and the axial stress a, , in the canister are computed by using the equilibrium Eqn (17) together with Eqns (21) and the traction boundary condition at r = ro [Eqn (14)]:

pC = Po S(~t, R) x/1 + 3~t2R* J , (22)

a~= --po + ac°,--FS(ct, R) 1 + 3ctR 2 ] x/3 L x/1 + 3 ~ ' _ 1 ' (23)

where,

2 4 S(~t, R) = c o t h - l x / l + 3c~2R 4 - c o t h - t x / i + 3ct Ro,

S(0, R ) = 2In ( ~ ) ,

and Ro = ro/r~.

for ct # 0, (24)

(25)

3.2. The powder The stress state in the powder is assumed to be a pure hydrostatic stress, i.e. tr~ = - p~6 o.

The equilibrium equations a~j. ~ = 0 imply then that/9, is uniform within the powder. The yield condition for the powder reduces to:

pP 3D2(D - Do) = 0. (26)

ag 1 -- Do

Let ~, be the volumetric part of the deformation in the powder. Noting that the axial strain rate ~ is the same in both the powder and the canister, we can write:

-d-~-r + r d V " p v~+ ~ = _ kv. (27)

The above equation is solved for v, p to obtain:

= - ½(~ + ~v)r = (otk - ½kvr,)R. (28)

The rate of densification (12) is given by:

/) = D~ or D = Doe *U. (29)

348 R.M. GOVINDARAJAN and N. ARAVAS

3.3. Matchiny o f solutions Using the continuity conditions (15), we find that:

1 ! d = 5i.,ri and pP = p, ~a~oS(a, 1), [30}

X//3

which also show that ~ = - ~/'i:... Using Eqn (23) for a~ and taking into account that a~= - p P , we can write the

equilibrium equation in the z-direction (16) as:

F(~) = S(~, 1) + 2 S ( ~ , R ) RdR = 0 . t311 \//I + 3ct 2 R 4

Using the definition of S(~, R) given by Eqns (24) and (25), one can readily show that:

F ( ~ ) = ~ ( ~ / l + 3 a z - v / ~ + 3 ~ 2 R o 4) for ~ # 0 , t32~

F(0) = 0. (33~

The last two equations show that ~ = - U?,o = 0 is the only solution to Eqn (31). Thus, there is no axial straining in a perfectly straight cylinder, i.e. the tube wall thickens and the diameter diminishes at constant length. This conclusion is consistent with the results of detailed finite element calculations that are presented in section 6. McMeeking [14] reached the same conclusion for thin-walled specimens. In most practical situations, a uniform shrinkage of the specimen, including a reduction in length, is desired. The results of the analysis presented in this section suggest that a change in length is not possible in perfectly straight cylinders--put in other words, a change in length is always accompanied by some distortion (shape change) of the specimen.

Returning to the analysis of the problem we can write the solution for the canister and the powder as:

and:

a~," = - a5 - a~° pC a" ro ,c i f , = P o - 2 I n - - - 1 v ,= - -" - - . (34) ~/3 ,v/3\ r ' 2C"r

2 a b l n • = 3D2(D - Do) D PP=P° V3 I - Do cry, ~:.,=lnD00. (35)

For a final value of the external pressure po, the solution can be obtained incrementally using D as the driving variable. The relative density is incremented from D = Do, and the values of po, pP and ev are found using Eqn (35). At the end of each increment the values of the inner and outer radii of the tube are updated using Eqn (34c), i.e.

Ll l \ r~ ) (ri)n~l = (r,)n-- ;Ae.,(r,)n and (ro)n+ I = (ro). - ~ A e ~ , [ - . . 136)

3.4. The plane strain solution for hardening materials The results of the finite element calculations presented in section 6 take into account the

hardening capacity of the canister and the metal composing the powder, and show that the axial strain e, vanishes in that case as well, i.e. the deformation occurs under conditions of plane strain.

In the following, we discuss briefly the plane strain solution for hardening materials. The yield stress of the metal composing the powder depends now on the microscopic plastic strain, i.e. a,. = ~r,,(e~). The flow stress of the canister material o ~ is also a function of the equivalent plastic strain ~P, which is defined as ~P = ~[(2/3)/~D~j] 1/2 dr, where D c is the deformation rate in the canister and t the time.


The solution for the powder region is readily found to be:

and:

I pP 3D2(D - Do) D = - ~ r , ~ ( ~ ) 1 - Do = 0, e~ = ln~o, (37)

l.SD~ . + D D _

The derivation of the last equation for ~ is presented in detail in section 5 below. The solution in the can is found to be:

(38)

where:

- - - - - - , - I - - d r , ( 3 9 ) ' f ' = = - ,fi, =po+ ,/3J, r

~ o~(~ ~) and ~P 1 . r, = -- . / ~ v ~ . (40)

The traction continuity condition at the interface r = r~ requires that:

~f'°~dr pn = Po x /~ J,, r " (41)

The solution is developed incrementally using the relative density D as the driving variable. For a given value of D, the quantities eL, ev and pP are found using Eqns (38) and (37). The solution for ~P, o x and Po is found using Eqns (41) and (40). At the end of each increment, the position of each material point is updated using Eqn (39a), i.e.

r.+1 = r. - ~A~ (42) n"

The results of the plane strain solution are compared with those of detailed finite element calculations of a cylindrical CIP specimen in section 6.

4. NUMERICAL IMPLEMENTATION OF THE CONSTITUTIVE EQUATIONS

The constitutive model described in section 2 is implemented in a finite element program. In a fni te element environment, the solution of the elastoplastic problem is developed incrementally and the constitutive equations are integrated at the element Gauss points; in a displacement based finite element formulation the solution is deformation driven. At a material point, the solution (F. , ~., (en.)., D.) at time t. as well as the deformation gradient F.+I at time t.+l = t. + At are known and one has to determine the solution (~.+ 1, (e~).+ 1, D.+ 1).

The time variation of the deformation gradient F during the time increment I t . , t. + 1] can be written as:

F(t) = AF( t ) -F . = R ( t ) . U ( t ) . F . , t. ~< t ~< t .+l , (43)

where R(t) and U(t) are the rotation and stretch tensors associated with AF(t), with:

A F ( t . ) = I and A F ( t . + I ) = F .+ I 'F ~ ~ (44)

The deformation rate and spin tensors D and W are defined as:

D(t) = [It'(t) • F - 1 (t)]~ = R(t)- [ ~ ( t ) . U - ' (t)]s" Rr( t ) (45)

W ( t ) = [ . l '¢ ( t ) . F - l ( t ) ] o = R(t).RT(t)+ R(t).[fg(t).U-l(t)]..RT(t). ( 4 6 )

It is assumed that the logarithmic strain rate is constant over the increment ['18, 19], i.e.

Ig = constant or E ( t ) - lnU(t ) = ( t - t.)lg Yt ~ [ t . , t . + l ] , (47)

350 R . M . GOVlNDARAJAN and N. AR,~V,.xs

where E(t) is the logarithmic strain relative to the start of the increment, i.e. E(t.) .... 0 Integration of the above equation using the initial condition U(t.) = i. and the transforma- tion r = t - t. yields:

so that:

| [ ~, " 3 U(t) = exp(rE) = ! + rl~ + 5-! r2~2 + 3~.r E" 4 - . , !4X)

1 T2 ~ 3 U(t) = I~ + zl~. 2 + ~ + " " = E ' e x p ( r l ~ ) = !~. U(t), (49)

which implies that:

U ( t ) - U act) = I~ = symmetric. (50)

Therefore, the expressions for D and W [Eqns (45) and (46)] reduce to [18]:

D ( t ) = R ( t ) - I ~ ' R r ( t ) and W ( t ) = R ( t ) - R r ( t ) . 151)

We consider next the co-rotational stress and deformation rates defined as [18, 20]:

6(t)=Rr(t) '~(t) 'R(t) and l ) ( t ) = R T ( t ) ' D ( t ) . R ( t ) = I~. (52)

Using Eqns (51b) and (52a), we can readily show that: v o = R ' b - R T and D = R ' i~ 'R r. (53)

The elasticity Eqns (2) can now be written as:

R . b . R r = C e : ( R - ~ e . R r) or b = C ~ : ~ L (54)

in view of the isotropy of C ~. Similarly, since the yield function • in Eqn (8) is an isotropic function of its arguments,

the plastic flow rule can be written in terms of the co-rotational quantities as:

£p = ) . ~ . (,i, D, ,:~). (55)

In the finite element computations. 1~ is defined from the known incremental stretches at the end of the increment, i.e.

= ~ t l n U.+ ~ = constant Vt E [ t . . t .+~] . (56)

The governing elastoplastic equations can now be written for the time period [t.. t.+ t ] as follows:

1 E~ Ep, = A;ln L ' . . , = + (57)

b = c , ' . ~7. (58)

I~P: 2 .w. (59)

• (6, D, c~) = 0. (60)

~.~ _ ~'J E~'~ (61) D a m "

t~ = - D E~'~. (62)

The above set of constitutive equations is integrated using the backward Euler method described by Aravas [21].

In the numerical calculations, the vertex on the yield surface is replaced by a small spherical cap as described in the Appendix. The normal to the yield surface is now uniquely


defined at all points and the flow rule (55) can be used everywhere on the yield surface. It should be noted that the conditions for strain localization depend critically on subtle features of the elastoplastic model such as vertex-like yielding effects and plastic compress- ibility [22-24]. Issues concerning flow localization are not addressed in our work; therefore, the "rounding off" of the corner on the yield surface has only a minimal effect on the obtained results.

The constitutive equations discussed above are implemented in the ABAQUS general purpose finite element program [25]. This code provides a general interface so that a specific constitutive model can be introduced as a "used subroutine".

5. PURE PRESSURE DENSIFICATION

In order to check the consistency of the finite element formulation and the numerical implementation of the algorithm described in the previous section we carry out a single element test subject to pure hydrostatic loading. We ignore the effects of elasticity in developing the exact solution.

The yield condition (8) for pure pressure loading can be written as:

p = ~ am(e~)B(D). (63)

The plastic part of the deformation rate is written as D~ = D~ - (1/3)kv6~. Using the evolution Eqn (12) for D, we can readily show that:

D = Doe ~. (64)

The equivalent plastic work expression (13) can be written as:

D trm g~ = p~v. (65)

Using Eqns (9), (63) and (64) to substitute for B, p and D, we can integrate the last equation to find:

1.5Oo e~ = [1 + e~(e ~ - 2)].

1 - Do

In view of Eqn (64), the above equation can be written as:

e ~ = [ - ~ - ° 1 + - 2 .

(66)

(67)

Finally, substituting the last equation into Eqn (63), we determine the pressure p in terms of the relative density D.

P In Fig. 2 the pressure p and the microscopic plastic strain e m are plotted versus D; open symbols in this figure indicate the results of the finite element calculations and solid lines the

P ( k s i )

40

30

20

10

0 - - 0.62

-- FEM /

0.64 0.66 0.68 0.70 0.72 D

(a)

¢Pm 0.045

0.036

0.027

0.018

0.009

0 - 0.62

-- FEM /

0.64 0.66 0.68 0.70 0.72 D

(b)

FIG. 2. Pure pressure densification: (a) pressure vs density, (b) microscopic equivalent plastic strain vs density (FEM and analytical results).

352 R.M. GOVINDARAJAN and N. AR^VAS

predictions of the exact solution. In the calculations, the initial relative density Do is I).02 and a= is a linear function of e~ of the form a,, = ao + he~, where ao = 62 x 103 psi and h = 688 x 103 psi. The results of the finite element calculations agree well with the exact solution.

6. C O L D I S O S F A T I ( P R E S S I N G ( C I P I N G I O F I I T A N I U I M P ( ) W D F : R

In the CIPing experiment of Fields [26], single phase titanium powder was packed into a copper can and pressurized to 40 ksi. A schematic representation of the CIP specimen is shown in Fig. 3. The flow stress of the copper material used as the canister was found to vary according to the formula a '~ " = ao + hOlY. The plastic properties of the titanium composing the powder were obtained by H|Ping single phase titanium to thll density and then carrying out a uniaxial test on the fully dense specimen; the hardening of titanium was

P , P found to be of the form am = ag + h *:,~ [26]. The elastic properties of fully dense copper and fully dense titanium are reported in Fig. 3. where the values of ag, h p, ao and h" are also given. The powder inside the canister is loosely packed and has a random packing initial relative density of about 62%. or in other words Do = 0.62.

The specimen shown in Fig. 3 is analyzed using the finite element method. The method described in section 4 for the numerical integration of the elastoplastic constitutive equations is implemented in the ABAQUS general purpose finite element program, which is used for the computations. Four-node isoparametric axisymmetric elements with 2 × 2 Gauss integration points are used to model both the powder and the canister. The powder is assumed to be completely bonded to the can. Because of symmetry, only the upper right quarter of the specimen is analyzed. The analysis is carried out incrementally and the applied external pressure Po is gradually increased to its final value• The results presented it~ the following correspond to the final value of Po = 40 ksi.

t 0.5" l

Id

I Col)per canister

T i tan ium powder

9"

U canister I powder I v 0.33 0.327

E (psi) 17 x l0 s 16.1 x l0 s ao (psi) 25 x 103 62 x 103 h (psi) 125 x 103 688 x 103

0.625"

0.064"--~ b--

FIG. 3. Schematic representation of the CIP specimen; the table shows the material properties.


FIG. 4. Deformed finite element mesh (solid lines) of the top 1 inch of the CIP specimen against the original mesh (broken lines); the powder region is shaded for clarity.

• 4

J PR~SURE (Lsi) DENSrrY

I (a) (b)

FIG. 5. Pressure and density contours of approximately the top I/2 inch of the powder region in the CIP specimen.

354 r . M . (JOVINI)ARAJAN a n d N.. , '~r,xx .xh

Figure 4 shows the deformed finite element mesh (solid lines) superpused t,, tn~ .nu~- formed mesh (broken lines) near the top 1/2 inch of the specimen: the shaded area , , ~i.u figure corresponds to the powder. Figure 5 shows contours of the hvdtoshttiu .qr~.-.., p = - %,.:3 and relative density D in the powder: the region sho,.vn .3 I ~g 5 corrc-q-,ond, t,. the shaded area of Fig. 4. Figures 4 and 5 show that the end effects arc h n3il,..'d '.~, ,| ic~..m L~i less than 1/2 inch near the ends of the specimen: the rest of the spcci,uc,t experience, a uniform hydrostatic stress and relative density.

Figure 6 shows the radial variation of the stress and logarithmic strain componenl?, ,tt the middle of the specimen. In that figure, the results of the finite clcmenl calculations lopen symbols) are shown together with the predictions of the plane .,,tram ~,olut~on discussed in section 3.4. The results of the finite element solution agree well with the pFcdictiot!> ,ff the plane strain solution of section 3.4. The results shown in Fig. 6 make it clear that the ,,tre,,s state in the powder is close to being purely hydrostatic and that the long,tudinal sir,tin m the specimen is practically zero. It is interesting to note that the hydrostatic ,tres< in vi~e powder is about 10 ksi less than the applied external pressure po. The radiai x armtton ,~f" It',: relative density D of the powder is shown in Fig. 7. The results of the tinitc element ,~olut IO~! agree well with those of the analytical solution and they both predict ~+ uniform rehtlixc density in the powder.

The predictions of the finite element solution are compared with the experimental data of Fields [26] in Fig. 8, where the variation of the relative density D and the outer diameter of the specimen d are plotted against the applied pressure p,,. The measured value of D corresponds to the average relative density of the powder which is determined from the change of the total volume of the specimen, and the value of D corresponding to the linite element solution is obtained at the middle cross-section of the tube: the outer diameter d is the value

a -20 I- /

; k,i I | . . . I

e e o .e

t

-dO t . . . . Analy t ica l "~- / ' . ~ , , FEM

,. t:3 ,"oo I ' T M

I ',2: rr:: ] ' I ' ; M .(-5 ~ ! 6) - .

-('X) ('>

t

O.O8

0 .04

1) .....

-0 .04 +

- .\re,l>',, a i "4.

L ; , . , I " I ' ; M -...~_

n , . . I " E M

. ~ , 1." I'~ M

' - - ¢ (J ::7 !-: i:. ,-.,

J ~T.~L.7,...~L I ~i'7 F ~ ,~ ~ 1 ~ , ~/

-80 1 . . . . , .......... ~ -O.Og .... ' .......... ~ ............... 0 O . l 0.2 0.3 0 O. I ,q 2 ( ) .

l" ( i : l l I :11~

U t j

FI(;. 6. R a d i a l v a r i a t i o n s o f ( a ~ s t r e s s a n d ib) s t r a i n c o m p o n e n t s ( F E M a n d a t ' , a l> twal rc,,uh~,l

0.76

0.74 r

0.72

0.70

0.6g

0.66 1) 0.06

: I I,M .-. ~,~ ~ ill,l i. HI,Ill

4-..3. •. q,,- ---.- -@---

, i , ~ . , i ,

O I 2 I).18 <).24 O.31) ,. : t , .

Fpu. 7. R a d i a l v a r i a t i o n s o f d e n s i t y ( F E M a n d a n a l y t i c a l r e s u h s k

Deformation processing of metal powders: Part ! 355

D 0.72

0.70

0.68

0.66

0.64

0.62 q

0.042 0 Experiment O

0.O36

0.030

0.024

0.018

0.012

0.006

0 = 10 20 30 40 0

p (ksi) (~)

®

C) Experiment /

I0 20 30 40 p (ksi)

(b)

FIG. 8. History of (a) density and (b) diametral strain (FEM and experimental results).

measured at the middle cross-section of the specimen for both the methods. The predictions of the finite element solution agree well with experimental data. The "kink" on both curves shown in Fig. 8 (po ~- 7.5 ksi), corresponds to the point where the whole cross-section of the canister deforms plastically for the first time.

Acknowledgements--This work was carried out while the authors were supported by NADC (Contract No. N62269-91-C-0247, BDM Subcontract No. 06S10052) and by the NSF MRL program at the University of Pennsylvania under Grant No. DMR-9120668 The ABAQUS finite element code was made available under academic license from Hibbitt, Karlsson and Sorensen, Inc., Providence, RI, U.S.A.

R E F E R E N C E S

1. P. J. JAMES (Editor), Isostatic Pressing Technology. Applied Science Publishers, Barking (1983). 2. M. F. ASHBY, A first report on sintefing diagrams. Acta Metall. 22, 275 (1974). 3. F. B. SWINKELS and M. F. ASHBY, A second report on sintering diagrams. Acta Metall. 29, 259 (1981). 4. E. ARZT, M. F. ASHBY and K. E. EASTERLING, Practical application of hot-isostatic pressing diagrams: four

case studies. Metall. Trans. 14A, 211 (1983). 5. F. B. SWlNKELS, D. S. WILKINSON, E. ARZT and M. F. ASHBY, Mechanism of hot-isostatic pressing, Acta

Metall. 31, 1829 (1983). 6. A. S. HELLE, K. E. EASTERLING and M. F. ASHBY, Hot-isostatic pressing diagrams: new developments. Acta

Metall. 33, 2163 (1985). 7. W.-B. LI, M. F. ASHBY and K. E. EASTERL1NG, On densification and shape change during hot isostatic pressing.

Acta Metall. 35, 2831 0987). 8. H. N. G. WADLEY, R. J. SCHAEFER, A. H. KAHN, M. F. ASHBY, R. B. CLOUGH, Y. GEFFEN and J. J. WLASSICH,

Sensing and modeling of the hot isostatic pressing of copper pressing. Acta Metall. 39, 979 (1991). 9. N. A. FLECK, L. T. KUHN and R. M. McMEEKING, Yielding of metal powders bonded by isolated contacts.

J. Mech. Phys. Solids 40, 1139 (1992). 10. L. T. KUnN and R. M. MCMEEKING, Power law creep of metal powder bonded by isolated contacts. Int. J.

Mech. Sci. 34, 563 (1992). I 1. R. M. MCMEEKING and L. T. KUNN, Diffusional creep law for powder compacts. Acta Metall. 40, 961 (1992). 12. M. F. ASNBY, HIP 6.0, Background Reading, The modeling of hot-isostatic pressing. Cambridge University

Report, England (1990). 13. M. F. ASHBY, Powder compaction under non-hydrostatic stress: an initial survey. Cambridge University

Report, England (1990). 14. R. M. MCMEEKING, The analysis of shape change during isostatic pressing. Int. J. Mech. Sci. 34, 53 (1992). 15. J. Xu and R. M. MCMEEKING, An analysis of the can effect in an isostatic pressing of copper powder. Int. J.

Mech. Sci. 34, 167 (1992). 16. B. BUDIANSKY, Thermal and thermoelastic properties of isotropic composites. J. Comp. Marls. 4, 286 (1970). 17. R. HILL, The Mathematical Theory of Plasticity. Oxford University Press, Oxford (1950). 18. J. C. NAGTEGAAL, On the implementation of inelastic constitutive equations with special reference to large

deformation problems. Comp. Meth. Appl. Mech. Engng 33, 469 (1982). 19. J. C. NAGTEGAAL and F. E. VELDPAUS, On the implementation of finite strain plasticity equations

in a numerical model, in Numerical Analysis of Forming Processes (Edited by J. F. T. PP.~'rMAN, O. C. ZIENKIEWICZ, R. D. WOOD and J. M. ALEXANDER), p. 351. John Wiley, New York (1984).

356 R, M. GOVINDARAJAN and N. ARAVAS

20. G. G. WEBER, A. M. LUSH, A. ZAVAL1ANGOS and L. ANAND, An objective time-integration procedure lot isotropic rate-independent and rate-dependent elastic-plastic constitutive equations. Int. J. Plast 6. 71~I (1990).

21. N. ARAVAS, On the numerical integration of a class of pressure-dependent plasticity models. Int J. Num ~e th Engng 24, 1395 (1987).

22. J. R. RICE, The localization of plastic deformation. Proceedings of the 14th I U T A M Congress (Edited by W I KorrERL p. 207. North-Holland, Amsterdam (1976).

23. J. W. RUDNICKI and J. R. RICE, Conditions for the localization of deformation in pressure-sensitive dilatanI materials. J. Mech. Phys. Solids 23, 371 (1975).

24. K.-T. CHAU and J. W. RUDNICKI, Bifurcations of compressible pressure-sensitive materials in plane strata tension and compression. J. Mech. Phys. Solids 3g, 875 (1990)

25. H. D. HmBrrT, ABAQUS/EPGEN--A general purpose finite element code with emphasis on nonlinear applications. Nucl. Engno. Des. 77, 271 (1984).

26. R. FIELDS, Private communication (1990). 27. J. C. SIMO, J.-W. Ju, K. S. PISTER and R. L. TAYLOR, Assessment of cap model: consistent return algorithms and

rate-dependent extension. ASCE J. Engng Mech. !14, 191 (1988).

A P P E N D I X

A. Modified yield surface The yield surface (8) proposed by Fleck et al. [9] has a vertex in the p--q plane (Fig. 1) along the p-axis. The

normal to the yield surface is not uniquely defined at the vertex point, and the plastic deformation rate has the freedom to lie anywhere between the limiting normals to the adjacent segments, of the yield surface. A detailed description of the difficulties associated with the numerical integration of elastoplastic equations at such singular points on the yield surface has been given by Simo et al. [27]. In our calculations, we replace the vertex on the yield surface by a small spherical cap, so that the algorithm described in section 4 and Ref. [21] can be used.

The original yield condition is written as:

where:

3 + (15~ + ~ - 1 = 0 , (681

p q 3D2(D - Do) x = - - , y = - - , and p,. a=. (69/

P~ Pr 1 - Do

The vertex on the x-axis is rounded off by an arc of radius:

R = x / ( x , - x,} 2 + y,2 I70)

where (x,, y,) is the intersection point of the quadratic yield surface and the circular cap of radius R, and x, is the center of the circle, as shown in Fig. AI.

& qv

/ \ \

/ / \ \

e \ • (xo,0)

P,

FIG. AI. Modified yield surface (solid lines); the sketch is not to scale.


The parameters xo, y, and xc are chosen in such a way that the yield function and its normal vary continuously at the intersection point (x., 3,.), i.e.

Ya -- ~ X~

In our calculations, we set x. = 0.95 and determine y., x~ and R from eqns (71) and (70). Summarizing, we mention that the yield condition used in the calculations is of the form:

+ ~ + ~ ) - ~ -- 0, for x.< x., (72)

y2"+(x_x,) 2 -R 2~0 for x > I x . . (73)

deformation processing of metal … processing of metal powders: part i--cold isostatic pressing r....

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