Journal of Materials Processing Technology, 32 (1992) 119-134 119 Elsevier

Void Growth in Drawing and Extens ion of Duct i l e Meta l s

Paul Dawson, Yong-Shin Lee and Ashish Kumar

Sibley School of Mechanical and Aerospace Engineering Cornell University, Ithaca, New York 14853

A b s t r a c t Two applications of a new void growth model are presented to illustrate the ability

of the model to predict porosity increases in metal forming. The model was developed from analyses of void growth under conditions of macroscopically isotropic and deviatoric stress states. Special attention was given to scaling of the mean stress by a measure of the matrix material strength rather than the effective stress. The advantage of this is evident in the applications in regions where the macroscopic stress state approaches purely hydrostatic.

1. I N T R O D U C T I O N

Hydrostatic tensile stresses can arise in forming operations as a result of the defor- mation zone geometry and the manner in which forces are applied to drive the process. Under hydrostatic tension, voids may nucleate and grow, perhaps to the point of causing failure. Even without failure, increased porosity from void growth can have deleterious effects on mechanical properties. Suppression of void growth in its early stages is de- sirable to assure acceptable quality and reduce manufacturing losses. One well-known example occurs in drawing operations where hydrostatic tension in the center of the workpiece can lead to internal porosity and eventual failure. Forging operations may also induce zones of hydrostatic tension, particularly in axisymmetric geometries where circumferential stresses become tensile.

In this paper two applications of a recently developed void growth model [14,15] are presented. The void growth model stems from analyses of isolated spherical voids embedded in spherical or cylindrical domains. The analyses assumed that the matrix material surrounding a void deformed by dislocation slip and could be modeled well with a rate-dependent, strain hardening, state variable model [10]. The results of many simulations involving a broad range of porosity, strength, mean stress, and triaxiality of the deformation were condensed by scaling to give relations for void growth rate. An important feature of the model is that the mean stress is normalized by the strength when used to compute the driving force for void growth. In forming applications, this is an important distinction from normalization by the effective stress because in regions of little deformation (dead zones) the effective stress may be very small if the material is viscoplastic. The strength remains well-defined even in regions of no deformation, so the problem of scaling by a variable that itself is vanishing is avoided. Strip drawing and

092443136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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extension of notched specimens are presented to illustrate use of the model under quite different forming conditions, but both having zones in which the effective stress vanishes.

2. G O V E R N I N G E Q U A T I O N S

The forming model assumes that under the large deformations imposed by the pro- cesses the inelastic behavior is dominant and that elastic response may be neglected. It is recognized that this limits the utility of the formulation in that aspects of unloading, such as the occurrence of residual stresses, cannot be examined without expanding the formulation to include elasticity. However, it is possible to study the structural alter- ations caused by the large inelastic deformations and compute property changes related to the structural modifications. With a viscoplastic material characterization in mind, the governing equations are summarized and the numerical formulation including the new void growth model is presented.

2.1. Ba lance laws and k inemat ics . Balance of linear momentum, neglecting body forces and inertia, and conservation

of mass provide equations for computing the motion of a workpiece. Balance of energy provides an equation which is used to evaluate the temperature field throughout the workpiece. Mechanical dissipation from the inelastic deformations is computed as the product of the stress and deformation rate. Details of temperature calculations and coupling of the temperature with other portions of the solution can be found elsewhere [6].

The velocity gradient L may be decomposed into the deformation rate, D, and the spin, W:

L = g r a d ( u ) = D + W . (1)

The deformation rate will be further decomposed into deviatoric ( D 1) and volumetric parts (½tr(D)) with the volumetric part being related to the time rate change of porosity through conservation of mass:

D=DI+ tr(D)I and tr(D)-(1-ff~ where qS= 1---.po (2)

po and p are the true and apparent densities, respectively. Because elasticity is neglected and a state variable is used to describe strain hardening, no strain measures are needed in the formulation.

2.2. C o n s t i t u t i v e relat ions. It is assumed that the workpiece material behavior is isotropic and rate dependent.

The plastic response of the fully dense workpiece material is modeled with a state variable representation that consists of three parts. The first part is the yield condition. It requires that the effective stress ale equals the flow stress, rl:

1 71 jr_ Tip o e = = r Iv (3)

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er is the Cauchy stress whose deviatoric and mean components are given by ~r' and am, respectively. The effective stress is proportional to the second invariant of the deviatoric Cauchy stress, r ~ and r ~p are defined in terms of the current temperature, deformation rate, and state variable, a*. ~-~ and r w represent contributions to the flow stress from resistance to dislocation glide from frictional forces acting on slip planes and resistance to dislocation motion by strong barriers, respectively [10].

The hardness, a*, is an isotropic state variable which quantifies the strength of strong barriers in an average sense. Its value increases as a consequence of strain hardening as described by an evolution equation (the second of the three parts):

r n I

k~*/ . ( 4 )

where Die is effective deformation rate (and is proportional to the second invariant of the deviatoric deformation rate). The evolution equation, as proposed by Hart, can accommodate the effects of athermal hardening and dynamic recovery, but neglects static recovery. Rate dependence of the hardening is present in the evolution law through the dependence on T ~p. The final part of the model for the fully dense shearing behavior is the flow law. Because we have assumed isotropy, this equation simply states that the deviatoric deformation rate is proportional to the deviatoric stress:

D ' = 2D'~ oa. (5) 3a'~

Parameters for the model were chosen to represent 1100 aluminum [19] based on stress relaxation and compression tests and are given in Table 1.

The growth of internal microvoids leads to volumetric dilatation [1,2,3,5,7,11,12,13,17]. For scalar void growth models with one state variable, the state variable most often is intended to characterize the internal porosity. In general the evolution equation for such a damage variable depends on the mean stress, the effective stress, the effective deforma- tion rate, the porosity, and the temperature. In all damage models having the form of the above equation, the mean stress has been scaled by the effective stress. Further, the porosity evolution rate often is given an exponential dependence on this ratio of mean stress to effective stress. However, the effective stress is controlled by the external forces, and does not characterize directly the material microstructure. When the effective stress is sufficiently small in comparison to the mean stress that the stress state is close to hydrostatic, the normalization is poor.

A void growth model has been proposed [14] in which the mean stress is scaled by the state variable for plastic flow. The state variable, in this case the hardness, a*, is used because at the scale of individual voids the deformation process is one of isochoric plasticity in the matrix material surrounding the voids and the hardness is a measure of material strength for this process. The model was constructed by computing the growth of voids contained in spheres or cylinders of hardening, viscoplastic matrix materials. The rate dependent, strain hardening behavior of the matrix material was characterized with a simplified (isotropic) version of Hart 's model that retains only the scalar state variable, ~r* [6]. Various conditions of external forces and microstructures of the matrix materials were examined. The resulting model has two basic parts. One part describes the growth of a void in a sphere (shell) of matrix material and is appropriate to the case of a purely hydrostatic macroscopic stress state. The other describes the growth of voids

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in cylinders of matrix material and permits conditions of stress triaxiality. The latter component of the model defines the dominant term in the growth of voids except when the mean stress is large compared to the hardness. The equation appropriate for void growth with macroscopic stress triaxiality is

(~ = ~ P o ~ e X p Cl'-~ D' , where ~Oo = ~ - ( ~ - ) exp c 2 - ROrJ"

In the above equation, cl and c2 are the material constants. For aluminum considered here, cl and c2 are chosen as 1.6 and 24.0, respectively. The reference temperature, 0r, is 373K, the reference deformation rate Der is l s -1 , and the reference hardness, a~ is 64 MPa. fo, m, Ge, Qo and R are parameters in Hart 's model. While no temperature depen- dence appears explicitly, it enters the model implicitly through the effective deformation rate. Note that the dependence of void growth rate on mean stress appears in terms of the ratio of mean stress to hardness. The dependence on porosity exhibits the limits of zero growth rate at zero porosity and infinite growth rate at a porosity of unity (although the validity of the computations used to establish the model doesn't extend beyond a few percent porosity).

At high values of mean stress, the influence of triaxiality is diminished and the ide- alization of a void growing in a spherical shell of matrix material is accurate. Thus, the macroscopic effective deformation rate vanishes and doesn't appear in the growth equa- tion. Based on simulations that spanned a broad range in porosity, hardness, and mean stress, a relationship was constructed that yielded a single master curve for the growth rate of voids. Analytically this curve is given by the combined equations:

l Ln = y ¢ + a 4 Ln ~ - - x ¢ , when L n ( a m / a * ) + X ( ~ I ' ) > X o (7)

Ln (-~.)----Y4~--a4[ x¢-Ln(°'m'~]a6ka* ]J , when Ln(am/cr*)+X(gP)<Xo_ (8)

where

x¢ =Xo-X(~ ) , y¢=Yo-Y (gP) , and D* =fo(~-~e ) exp [ ROrJ' (9)

X ( ¢ ) = al Ln ) - aa Ln( ) , and Y(~) = a3X(~ ). (10)

In the above equations, al, a2, a3, a4, an and a6 are the material constants. For aluminum, the best fit was obtained using al = 21.5, a2 = 0.05, az = -3.0 , a4 = 32.0, as = 0.4, and • a6 = 0.21. Xo = 1.8 and Yo = 28.0. Again, the reference temperature, 0r, is 373K and fo, m, G~, Qo and R are the parameters in Hart 's model.

Since the proposed damage model does not take into account the nucleation of voids, it predicts zero growth rate if the porosity is zero. Therefore, a small amount of internal porosity must exist initially for voids to grow. Most engineering materials do have some small volume fraction of voids at the beginning of deformation processing. However, void nucleation could easily be considered in conjunction with the growth model presented here. Mathur and Dawson [18,19] assumed the damage process to be irreversible based on the work of Murakami and Ohno [20]. This is a consequence of the observation that it is more difficult to close voids than to open them. To preclude the collapse of voids under compressive stresses, it is assumed that in forming applications ¢ = 0 whenever o" m < 0 .

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2.3. B o u n d a r y a n d in i t ia l c o n d i t i o n s . Boundary conditions for the motion consist of either known tractions or velocities

on the surfaces of the workpiece. To accommodate frictional sliding, the traction vector tangential to the tool surface, tf, is taken to be proportional to the difference between tangential velocity components of the workpiece and tool through a coefficient,/3, that may be dependent on temperature and the condition of stress near the surface, tf = /3(Ud -- u~), where ua is the tangential component of the tool velocity and uw is the tangential component of the workpiece velocity. Thermal boundary conditions consist of imposed temperatures or heat fluxes. The chilling effect of the dies is modeled as convection, with high values of the film coefficient.

Initial conditions are specified for the temperature, hardness, and porosity:

0 ( t = t ~ ) = 0 i , a ' ( t = t , ) = 7 and ¢ ( t = t , ) = ~ , . (11)

For a transient Lagrangian simulation, these quantities are specified throughout the do- main at the initial time; for an steady-state Eulerian simulation, they are specified at the inlet of material flow to the domain.

3. F I N I T E E L E M E N T F O R M U L A T I O N

The governing equations discussed above are solved for the velocity, temperature and state variables using finite element approximations. The equations for the motion, tem- perature and state variables are coupled together with a free surface correction algorithm, requiring an iterative scheme to obtain a solution for the complete set of equations as well as constraints. Each of the individual solutions is discussed in the following para- graphs. Of special interest here is the manner in which the void growth equations are implemented in a formulation for viscoplastic flow that permits volumetric deformations.

3.1. V e l o c i t y and pressure . A weighted residual is formed on the equations of balance of momentum using the

weighting functions, w:

f " [V. ~,1 d Y = O. (12) V

The deviatoric response for voided materials may be approximated with a modification of the flow rule for fully dense materials by neglecting the coupling between shear and volume responses at small porosity:

t r ' = 2#~D' where #¢ = (1 - ¢)a'(D'~,o*,O) (13) 3D'

Trial functions are introduced to the velocity and pressure (and introducing matrix notation), {u} = [g]{v} and p = [Np]{P}, where {V} and {P} are nodal values. Following standard finite element procedures, the discretized weak form of the weighted residual becomes:

[Ku] {U} + [G] T { - P } = {F}, where (14)

124

IS,'.] = f[N*l'r[z~][N']dY, [a] = Jtg.l:r{h}T[N']dY, and {F} = fIN] T {t}dS. ( 1 5 )

V V S

[N*] extracts the incompressible portion of the velocity while [D] is a diagonal matrix containing the effective viscosities; [N.'] contains spatial gradients of the trial functions for the velocity; and {h} is a matrix trace operator•

[N*] = [S'] - l{h}{h}w[u'], t r (D) = {h}W{D} = {h}w[g']{U}. (16)

A consistent penalty formulation [8] is used element-by-element to enforce Equation (2) using weighting functions ~:

1i i { -~ ~pdV= ~ t r ( D ) - dV. (17) v~ ve

Here, A is a penalty parameter. The finite element discretization of Equation (17) yields for every element:

[Mp] { - P } = -A[GI {U} + {Fe}, where (18)

=/" /'[Npl w ¢--~dV A E A#¢ IMp] I[NpIT[Np]dV, {re} = A / and = x Wlqp, (19) d d 1 Ve Ve nqp

where w are the standard Gauss weights, nqp indicates the number of quadrature points per element, and A is a large number (on the order of 106). The pressure can be eliminated at an element level by solving Equation (19) for the nodal pressures and then substituting the result into Equation (14), thereby reducing the problem to one involving only nodal point velocities. Thus, we obtain:

[I(, + If,x] {V} = {F} + {F¢}, where [I(~] = A [G] T [Mp]-l[G]. (20)

The submatrix [K,] depends on the velocity field through the nonlinear yield criterion for the plastic flow. Moreover, the force vector {F¢} depends on the velocity field as well as the pressure field because the rate equation governing the accumulation of porosity is nonlinear. This nonlinear system of equations was solved using a direct iteration scheme with the stiffness matrix computed using a secant update.

3 .2 . S t a t e v a r i a b l e e v o l u t i o n a n d free s u r f a c e c o r r e c t i o n . Two state variables have been used in the above formulation: porosity and hardness.

Each is modified by deformation. Correspondingly, the evolution equations that define the rate of change must be integrated for all points within the workpiece. For steady- state Eulerian simulations, the evolution equations are integrated as ordinary differential equations following particle paths defined by the streamlines of the motion. For transient Lagrangian simulations, the evolution equations also may be treated as ordinary differen- tial equations, but because the motion is not steady the integrations follow streampaths of the flow. in either case, the constitutive models provide the evolution equations that are integrated:

125

Table 1: Material Parameters for the Simplified Hart 's Model for Aluminum.

ao fo G a; ( s - ' ) (s -a) (GPa) (MPa)

9.64 x 1052 2.12 x 1019 24. 64.0

6.19 x 10 .9 0.15 5.0

Qo/ R Q'o/ R (K) (K)

1.45×104 1 .45x104

T/, T r / /

4.5 3.5

Free ~ ~tX } ~ z.~ Constant Fie,, S,,rfac¢ r'_. I Velocity

iD,el

I I I i t I I i t I I I i t

~ ,!13Trtttf~,,:', ' ' I l l l l i l i l i l l l : I I I I I i l l l l l l l l l l l ] i I l | l l l l l l l l i l l [ l l l I

Figure 1: Strip Drawing Operation: (a) Schematic, (b) Mesh.

126

where the specific forms for &* and ~ were given previously. A free surface is defined as that part of the surface which has zero traction and across

which no mass flows. In Eulerian simulations the position of the free surface is part of the solution. A weighted residual is formed on the constraint of zero velocity normal to the free surface, from which a surface position can be determined which has the proper slope so that the velocity will be tangent to the surface:

[ t~ [u. n] dS = 0, (22) S

where v~ are the weights. The details of the free surface correction for the steady state problem are reported elsewhere [16].

4. A P P L I C A T I O N S

Two applications of the void growth model are presented to illustrate its use in form- ing applications. The first is steady-state strip drawing, in which the effect of deformation zone geometry on porosity growth is examined (Figure 1). The second involves the exten- sion of notched specimens (Figure 4), either plates or rods, and the comparison of void distributions with computations made using Gurson's model [9,21]. In both applications the material is 1100 aluminum, and the material parameters are those mentioned in the discussion of the constitutive equations.

4 . 1 . S t r i p d r a w i n g . Material that is initially 0.5cm thick is pulled between two flat-faced dies at a constant

1 ~m velocity to reduce its thickness by 20% to 0.4cm. The temperature of material entering the deformation zone is fixed at 373 K and the entering hardness is 64 MPa. The entering porosity is set to 0.0011, based on comparisons by Rogers and Coffin [4] of the initial apparent density of aluminum in their strip drawing experiments. Rogers and Coffin also reported that in their experiments the friction was significant at larger die angles. Consequently, friction was applied between the workpiece and die in the simulations using a value for fl of 10 MPa's. This value is a compromise between frictionless condition appropriate at low die angTe and a higher value that was found to give good agreement between simulation and experiment at high die angle [14].

Three die angles are considered: 10 °, 20 °, and 30 °. By varying the die angle over this range, the deformation becomes increasingly inhomogeneous. The inhomogeneity in deformation can have substantial impact on the degree to which the material strain hardens during drawing, which in turn affects the force needed to conduct the drawing operation. Consequently, the void growth is influenced through higher stresses when the inhomogeneity is greater. The upper half of the strip was discretized with 232 nine-noded Lagrangian elements having a total of 1003 nodal points for velocity and temperature. The mesh used for the 10 ° die angle simulations is shown in Figure lb; meshes for 20 ° and 30 ° are similar, but have smaller deformation zones and larger die angles. The streamline integrations were performed using step sizes of ~ of the length of an element in the direction of flow. Free surface corrections were performed on both the surfaces upstream and downstream of the die.

127

AB 10 ° Die 9c

A B 20 ° Die

A B 30 ° Die

Figure 2(a): Hardness contours: A=75 MPa, Aa* =20 MPa.

A B

ABC

Figure 2(b): Porosity contours: A = l . 3 x l 0 -3, A ~ = 0.3 x 10 -3 .

Figure 3(a): Dis t r ibut ions of a,~lo-* and a,.,,Icr,.

I::L : ; I!L: •

ili~iiiii!i~ii~iiiiii ii!~ii!iii#iiiiiil

/:i?ii:!:f - 1

128

r,D

1.0

0.5

0.0

- 0 . 5

- t .0 2.0

' I I I - -~ ' - O ' m / ~ l l

" ~ O'lm/O'*

f ~

~t.li" 'it \ - /

i * ' J I I [ I [ I i

4.B 6.0 8.0 10.0

Distance, mm

Figure 3(b): Normalized mean stress along the centerline.

t t t °

Table 2: Average Porosity at Exit Plane

Die Angle @~g (%) ~o~./to 10 ° 0.1114 1.013 20 ° 0.1299 1.181 30 ° 0.1425 1.295

A

-l] ,, ,- _.vg

J

Figure 4: Extension of notched plates and rods: (a) Schematic, (b) Mesh.

Table 3: Specimen dimensions.

A [ 12.2870 cm

B I 3.0607 cm C 6.1087 cm D 1.8732 cm

129

The material strain hardens appreciably during the drawing operation, even for the case of low die angle. As is evident from the hardness contours in Figure 2, the hardness increases by more than 40 MPa for the 10 ° die angle case, and in comparison to the other cases is relatively uniform through the strip thickness. As the die angle increases, both the amount of hardening near the surface and the variation in hardness through the thickness increase. The centerline hardness is approximately the same in each case, rising to slightly more than 100 MPa. In the 30 ° die angle case the increase in hardness is more than 80 MPa near the surface of the strip while the through strip variation exceeds 40 MPa. The total force necessary to pull the strip between the dies is correspondingly larger for larger die angles.

The impact of the deformation zone geometry on void growth is illustrated in Fig- ure 2. The porosity distributions show increasing porosity with increasing die angle. The porosity accumulates near the centerline as a consequence of the mean stress being tensile there. Near the surface the reaction forces by the die contribute to hydrostatic compression and suppress the growth of porosity. Further, the increased hardening near the surface will suppress void growth even where there exits hydrostatic tension. Thus, void accumulation is concentrated more strongly in the core of the strip. The magnitudes of the porosity subsequent to drawing are not large and remain well within the limita- tions of small porosity required for validity of the model. The average porosity across the thickness of the strip increases by about 30% for 30 °, although along the centerline it nearly doubles. A summary of the average porosities is given in Table 2.

The scaled mean stress as obtained by normalizing with respect to the hardness is compared to the mean stress scaled by the effective stress in Figure 3. The zones where the effective deformation rate is small correspond to regions of low effective stress. Scaling by a quantity that is approaching zero gives erratic normalized mean stresses, whereas with hardness scaling the normalized mean stress remains smooth.

4 . 2 . E x t e n s i o n o f n o t c h e d p l a t e s a n d r o d s . The void growth model has been used to compute trends in the extension of plates or

rods having a large notch. The geometric constraints of plane strain (plate) versus ax- isymmetric (rod) deformations give rise to qualitatively different responses in terms of the spatial location of the porosity and hardness within the notch. Needleman and Tvergaard [21] previously investigated this application, which provides a basis for comparison.

The specimen is stretched via an applied crosshead velocity that remains constant throughout the entire test at 2.1167 × 10 -~m The initial hardness is 64 MPa and the $ •

initial porosity is 0.0011. The deformations are quite slow, so isothermal analyses were performed with a uniform temperature of 373 K. The specimens were extended to a nominal strain of 1.72 × 10 -2, based on the specimen half length dimension, A. Most of the deformation is concentrated in the notch where the strains are much larger than the nominal value. The upper right quadrant of the specimen was discretized with a total of 90 eight-noded serendipity elements, giving a total of 319 nodal points for the velocity field (Figure 4b). Time increments of 1 s were specified. Integration of the hardness and porosity was performed at quadrature points within the elements.

The evolution of hardness and porosity for the axisymmetric and plane strain geome- tries arc shown in Figures 5 and 6, respectively. Straining produces a gradient in the hardness from the centerline to the notch surface to a greater extent for plane strain than for axisymmetry. The hardness increases overall are higher for axisymmetry for an equivalent level of nominal strain (as given by ~). The porosity growth occurs predomi-

A

A

/ e=6 .00 xlO -3

/ A

A

/

130

A

e = l . 2 0 × 10 -2

J

(b) (a) 6=1.72 x l 0 -2

Figure 5: Hardness contours: (a) Axisymmetric A=72 MPa, Aa" =5 MPa, (b) Plane Strain A=69 MPa, Aa" =5 MPa.

A

A 6=6.00 x 10 -3

A

A

A

131

A

A

e=l.20 x I0 -2

A

)

(a) ¢=1.72 xl0 -2 (b)

Figure 6: Porosity contours: (a) Axisymmetric A = l . 1 3 x l 0 -3, A ¢ = 0.3 x 10 -3, (b) Plane Strain A = l . 1 5 x l 0 -3, A ¢ = 0.1 x 10 -3.

132

J

0.75

r

0.82

W ~ : -0 .31 ~ ' - 0.45 (a) (b)

Figure 7: Contours for the ax i symmet r ic case: e = 1.72 x 10 -2

(a) ~m/~', (b) ~m/~o.

0.15

" ~ 0.10

¢.9

~23 0.05

0.00 -1

~ ~'rn / CT e ........~.-- O'rn / tT *

0 Normalized Mean Stress

(a)

0.151

1 -I

• . . - . . ~ . - . o r r n / O r e

.L O'm / 6r*

";r

t ! ,

0

Normalized Mean Stress (b)

Figure 8: Normal ized mean stress for the ax i symmet r ic case: ¢ = 1.72 x 10 -2

(a) along the centerline, (b) on the specimen surface.

133

nately along the centerline and at the minimum notch radius for axisymmetry, declining substantially outward towards the notch surface. In the case of the plane strain notch, although maximum porosity occurs at the centerline as in the axisymmetric case, poros- ity levels at the notch surface are relatively high. Thus, while failure initiation by the accumulation of porosity would be expected at the center for an axisymmetric specimen, a plane strain specimen may also fail at the notch surface as a consequence of the higher porosity levels there. These observations co~respond qualitatively to those reported by Needleman and Tvergaard [21] and are supported by the results of additional simulations performed at 573K.

The impact of normalizing the mean stress by the hardness instead of by the effective stress is shown in Figures 7 and 8 for the axisymmetric (rod) case. In the deformation zone, the two distributions are similar. However, along the centerline and slightly above the notch the deformation rate is small and the effective stress is low. In this zone the normalization by the effective stress is poor because the effective stress itself is vanishing and the stress is essentially hydrostatic (Figure 8(a), plotted along the centerline). Higher in the specimen, where the stress is relatively unaffected by the notch and approaches the stress state of uniaxial tension, the normalizations are qualitatively similar, although the normalization by the hardness produces a lower numerical value. A similar behavior is seen on the outer surface, Figure 8(b). Thus, in zones where the stress triaxiality is significant, either normalization is well defined. However, where the effective stress vanishes, normalization by the effective stress is ill-posed.

5. S u m m a r y

A new void growth model was developed for conditions of isotropic and deviatoric macroscopic stress states. The matrix behavior was represented by state variable consti- tutive theory appropriate for dislocation slip plasticity so that the evolution of porosity in typical metal forming applications could be studied. Special attention was given to scaling the mean stress by a measure of the material strength related to the barriers to dislocation movement during plastic flow. The model was implemented in a finite element formulation for viscoplastic flow and used to simulate different forming operations. An advantage of the new formulation is that the stress scaling remains well posed in dead zones of the material flow where the effective stress may vanish.

6. Acknowledgements

This work was supported in part by Alcoa, the General Electric Company, and the Allegheny Ludlum Corporation.

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