Void growth and coalescence during high velocity impact

MECHANICS

NP4AIlIE ELSEVIER

Mechanics of Materials 19 (1995) 293-309

Void growth and coalescence duringhigh velocity impact M.J. W o r s w i c k a, R.J. Pick h a Carleton University, Ottawa, Ontario, K1S 5B6, Canada b University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

Received 12 January 1994, revised version received 28 February 1994

Abstract

The extent of void growth and cracking due to ductile fracture occurring during symmetric Taylor cylinder impact tests on leaded brass has been determined experimentally. Void growth occurs within these predominantly compressively-loaded specimens through the development of large tensile hydrostatic stresses along the specimen axis near the impact face during expansion of the cylinder, termed "mushrooming". The measured porosities have been compared to predictions using a constitutive model based on the Gurson ( 1975, Ph.D. Thesis, Brown University) yield function, implemented within the DYNA2D finite element code. The initiation of void coalescence and subsequent crack development was also predicted using the approach of Tvergaard and Needleman ( 1984, Acta Metall. 32, 157) based on a critical porosity criterion. The calculations were able to qualitatively predict the development of the porous zone and void coalescence within the impact specimens; however, the predicted void growth exceeded that observed experimentally and the predicted extent of void coalescence was too large. It is suggested that the primary source of error lies in excessively high predicted void growth rates using the Gurson yield function at high stress triaxiality levels. Keywords: Ductile fracture; High strain rate; Impact

1. I n t r o d u c t i o n

Ductile fracture occurs within plastically deforming metals through the nucleation, growth and coalescence of small voids to form a crack. This mechanism is operative under high strain rate conditions in problems involving impact (Curran et al., 1987) and explosive forming (Worswick et al., 1990). This paper examines the application of the Gurson (1975) constitutive model, as modified by Tvergaard and Needleman (1984) to incorporate void coalescence, to predict the onset of ductile fracture under high strain rate deformation. This model was originally formulated by Gurson (1975) based on upper bound solutions for the flow within a quasi-statically

loaded, rigid plastic sphere containing a central sphericai void. Numerous investigations have shown that this model provides reasonable predictions of porosity evolution and constitutive softening due to porosity under quasi-static loading (Becker et al., 1988; Worswick and Pick, 1990). The addition of a coalescence criterion based on critical void volume fraction has also been shown to capture the fracture behaviour of uniaxial and notched tensile specimens under axisymmetric and plane strain conditions (Needleman and Tvergaard, 1984). The application of this model to inertial conditions has received only limited attention (Johnson and Adessio, 1988; Fyfe and Rajendran, 1982). Experimental work by Dumont and Levalllant (1990)

0167-6636/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDIOI67-6636(94)00041-E

294

M.J. Worswick, R.J. Pick~Mechanics of Materials 19 (1995) 293-309

suggests that the rate of increase in void volume fraction with applied plastic strain is not substantially altered by strain rate at rates up to 104 s -1. The effect of strain rate and inertia on void coalescence is uncertain; however, it is conceivable that material strain rate hardening and inertia could combine to increase the level of deformation required to cause void coalescence. In the current work, ductile fracture occurs in an experiment known as the symmetric Taylor cylinder test during the impact of two identical cylinders, one stationary and one travelling at velocities up to 300 m/s. During the impact, plastic "mushrooming" of the cylinders occurs at the impact face causing the development of large tensile triaxial stresses along the specimen axis which can result in ductile cracking through void nucleation, growth and coalescence. Deformation is extremely rapid, with strain rates well in excess of 104 s -1 . By varying the impact velocity, the extent of damage occurring within the specimens can be altered to obtain: (i) void growth at low velocity; (ii) limited void coalescence at intermediate velocities, or (iii) widespread coalescence and crack propagation at high impact velocities. It is the ability to control the level of damage development that makes this experiment interesting since it is very difficult to arrest void coalescence and crack growth during quasi-static tensile tests. In the cylinders, the loading is inertial and the damaged region is surrounded by compressively loaded material which limits the final size of the crack. Experiments and corresponding numerical simulations were performed at impact velocities in the range 175-300 m/s. Measured porosities within the specimens impacted at lower speeds were used to assess the predicted void growth rates. The specimens impacted at higher velocities were used to assess a critical void volume fraction to coalescence criterion. Several critical void volume fraction levels were considered to examine their influence on the predicted coalescence zone size or crack size. Mesh convergence studies were undertaken in conjunction with the numerical studies to examine the sensitivity of the predictions to element size.

2. Material Symmetric Taylor cylinder impact experiments were performed on UNS C36000 free cutting brass. This material is predominantly an alpha brass containing 2.5% dispersed lead phase, added to improve machinability, and small regions of beta brass. The as-received material was annealed at 850°C for two hours in order to promote spheroidization of the lead phase as seen in Fig. 1. This heat treatment resulted in an average particle size of roughly 3 - 4 / z m and aspect ratio near unity. For the purposes of the current study, these lead particles serve as void nucleation sites since the low strength lead phase readily bursts or tears during tensile plastic deformation to form voids (Fig. 2). Using standard quantitative metallographic techniques, the volume fraction of lead particles was determined to be 0.025 with a standard deviation of 0.008. The material grain size was determined to be 55 /zm and the "nearest neighbour" lead particle spacing was approximately 15/zm. The material stress-strain behaviour was described using a polynomial fit to quasi-static uniaxial tensile data by Worswick and Pick (1992). The flow stress obtained using this fit was then scaled using a logstrain rate relationship: 6- = (94.4 + 1594@p - 1693(~P) 2 + 645(~p)3) ( 1 ÷ 0.0009 ln(e)) (MPa),

(1)

in which 6- is the material flow stress, gP is the effective plastic strain and ~ is the effective strain rate. The logarithmic term was introduced by Wong (1990) to account for the effect of strain rate in modelling Taylor cylinders using the EPIC-2 (Johnson and Stryk, 1986) finite element code. The constant 0.0009, preceding the logarithmic term, represents the strain rate sensitivity of the material and was shown to provide a good match between predicted and measured impacted cylinder profiles (Wong, 1990). The use of a quasi-static stress-strain curve, corrected for strain rate in this manner, was viewed as being somewhat crude since thermal softening effects have been neglected and the magnitude of the correction for strain rate appears quite small. Even at strain rates of 105 s-1, the magnitude of the strain rate term will only result in a 1% change in flow stress. It is also known that the Taylor cylinder test is not a particularly accurate

M.J. Worswick, R.J. Pick/Mechanics of Materials 19 (1995) 293-309

295

@

•

o 0 R

D

0

0~

Q

° @

I

•

~

I 30/~m

Fig. 1. Optical micrograph of leaded brass showing the spheroidized lead phase after annealing at 850°C for 2 hours (polished only).

e @e I

I

@

,0 @ 9

b

•

"

@

0

@ O

@ @

1

|

Q

0

_

• -~

30

~m

Fig. 2. Optical micrograph showing the central tear or burst of lead particles after tensile plastic deformation (polished only).

296

M.J. Worswick, R.J. Pick/Mechanics of Materials 19 (1995) 293-309

method for determining constitutive behaviour. However, brass is not particularly strain rate-sensitive at room temperature, although the sensitivity will likely increase at elevated temperatures. Better constitutive data was not available for this material, so Eq. (1) was adopted in the current study. Direct impact Hopkinson bar tests are planned and will be used to obtain an improved constitutive model for future work 1.

3. Experiment In the symmetric Taylor cylinder test, a 34.7 mm long, 9 nun diameter cylinder impacts an identical stationary cylinder at a velocity in the range 175-300 m/s. The advantage in using a symmetric impact lies in eliminating sliding and friction at the impact face. During the impact, noticeable mushrooming and barreling of the cylinders occurs, causing the development of a zone of tensile hydrostatic stress along the specimen axis, just beyond the impact face. If the impact velocity is sufficiently high, ductile cracking can occur within this zone along the axis (Fig. 3). In order to determine the effect of impact velocity on the level of damage occurring within the specimens, tests were made with velocities in the range 175-300 m/s. The recovered specimens were sectioned longitudinally and examined metallographically to determine the extent of void growth, coalescence and/or cracking. Table 1 summarizes the impact velocities, final specimen geometries and metallographic observations of the sectioned specimens. Extensive void coalescence did not occur for impact velocities below 210 m/s. At 230 m/s, a small region with coalesced voids was observed on the specimen axis, as shown in Fig. 3b. Large regions of void coalescence and cracking were observed at impact speeds of 267 and 300

Table 1 Summary of final specimen geometry and observed damage for the range of impact velocities considered. Impact velocity

Measured

Predicted

Lf/L i

Rf/Ri

Lf/Li

Rf/Ri

Observed damage

175 210 230

-0.84 0.81

1.12 1.17 1.21

0.88 0.84 0.83

1.18 1.24 1.27

267

0.78

1.24

0.79

1.34

300

0.74

1.35

0.76

1.41

(m/s) Void growth only Void growth only Coalescence on axis Coalescence and shear bands Coalescence and large crack

each location were then averaged over all of the cylinders to obtain an averaged distribution. This averaging process was intended to account for the statistical variation in initial porosity within the as-received material.

4. Model The specimen geometry and the impact were assumed to be axi-symmetric, permitting a twodimensional analysis using the DYNA2D explicit dynamic finite element code due to Whirley and Hallquist (1992). This code employs explicit time integration with inherent small time steps suitable for tracking the stress wave propagation occurring within the impacting cylinders.

4.1. Constitutive model

m/s.

In order to facilitate comparison between the predicted and measured void growth, seven Taylor cylinder experiments were performed with recorded impact velocities equal to 230 + 1 m/s. This velocity resulted in significant levels of void growth with void coalescence confined to a small region (Fig. 3b). Measurements of void volume fraction were taken at locations along the axis of each cylinder; the measurements at t R.J. Pick and M.J. Worswick, work in progress.

The elastic response of the material was modelled using Hooke's law for a linear elastic isotropic material. The macroscopic plastic response was modelled using the Gurson (1975) constitutive model for porous continuum plastic materials as modified by Tvergaard and Needleman (1984). Central to this model is the use of the Gurson yield function to determine the macroscopic stresses, "~ij, required to initiate or sustain plastic flow within a plastically dilating porous solid:

M.J. Worswick, R.J. Pick~Mechanics of Materials 19 (1995) 293-309

297

(a)

Fig. 3. Optical micrographs showing the extent of damage near the impact face of cylinders impacted at velocities between 210 and 300 m / s . The lines overlaid on the micrographs represent the predicted specimen and coalescence zone outlines at 100/zs. The solid lines correspond to f¢ = 0.15 and f f = 0.25 and the dashed lines to f¢ = 0.25 and f f = 0.30. (a) Impact velocity = 210 m / s . (b) Impact velocity = 230 m / s . (c) Impact velocity = 267 m / s . (d) Impact velocity = 300 m / s .

298

M.J. Worswick, R.J, Pick~Mechanics of Materials 19 (1995) 293-309

(d) ,I mm .

Fig. 3. (Continued)

M.J. Worswick,R.J. Pick~Mechanicsof Materials 19 (1995) 293-309 ~b =

(_~)2

(

3~hyd'~

+ 2f*ql cosh k q 2 - " ~ - ]

poe

- 1 - q3f *x

= 0,

(2)

in which 2eq is the equivalent stress, defined by ~ 2 ~t yt with ~ijt being the deviatoric components = 32"ij-ij, of 2ij and 2?hydis the hydrostatic component of stress, given by 2hyd = 1-ykk. The coefficients ql, q2 and q3 are "calibration" coefficients introduced by Tvergaard ( 1981 ) to better represent the effects of porosity in plastically deforming materials. The values adopted were ql = 1.25, q2 = 0.95 and q3 = q2, as used by Worswick and Pick (1990). f* is initially equal to the void volume fraction f but is modified to account for the onset of void coalescence according to the following function proposed by Tvergaard and Needleman (1984): f f* =

if f_< fc

3'u - f c f e + ~ffZ ~c ( f -

fc)

if f > ft.

(3)

Following Tvergaard and Needleman (1984), fc is the critical value of porosity at which void coalescence commences. To simulate the effect of void coalescence and the resultant loss of material strength, void growth is accelerated once the porosity exceeds this level according to Eq. (3)• Coalescence is assumed to be complete once f reaches a final critical value f f and the material strength vanishes according to Eqs. (2) and (3) with f* = 1/ql. Using Eq. (2) as a flow potential and applying the normality condition (Gurson, 1975), one obtains:

1 ( 3 ~ j + O[~ij) (3.,~tkl

~poo = -n k 20"

~ 20" "+ Ol(~kl) ~kl,

(4)

where f

3 27hyd"~

oc= ½fqlq2 sinh kq2 ~ ;

(5)

and

h [(~_~_~)2 -30a(1-

3~hydl2

J

(

f)[qlc°shkq2

3 ~Y'hyd'~

299

5" .j - q 3 f ] .

(6)

The term h is the slope of flow stress versus effective plastic strain curve (Eq. (1) ) and 8ij is the Kronecker

delta. The components of plastic strain ~:ij refer to the macroscopic strains rather than the detailed distribution near individual voids• Using work equivalence, the effective plastic strain rate in the matrix, kP, is given by

:p

~ij ~poe =

(7)

(1 - f)5-"

The rate of increase in porosity will be due to the growth of existing voids and the nucleation of new voids,

f ----"fgrowth+ ?nucleation.

(8)

The growth of existing voids is given by •

fgrowth = (1 -

.poo

f)ekk ,

(9)

since the matrix material is assumed to be incompress.poe ible and aki is due purely to void expansion. Void nucleation is modelled as plastic straincontrolled. It is assumed that voids nucleate at second phase particles and that there will be a statistical variation in the nucleation strain for individual particles. Assuming that the nucleation strain for the total population of particles follows a normal distribution, one can use the following equation adopted by Gurson (1975): inucleation =

Ah~ p,

(10)

where

1 fN e x p [ _ l ( g P - - e N ) A - hSNv'~-~ ff~

2] .

(11)

The term fN represents the volume fraction of void nucleating particles while eN and SN are the average and standard deviation of the strains at which the particles nucleate voids• In free cutting brass, the globular lead phase would constitute the void nucleating particles. Eqs. (2) through ( 11 ) combine to describe a constitutive model for the evolution of porosity during plastic deformation and the reduction in material loadcarrying capacity, referred to as constitutive softening• This set of equations was implemented as a Gurson constitutive routine in DYNA2D. A subincrement, elastic predictor-normal corrector scheme was used to integrate Eqs. (2) through ( 11 ).

300

M.J. Worswick,R.J. Pick~Mechanics of Materials 19 (1995) 293-309

This scheme, described briefly here and in more detail by Worswick (1994), was adopted to allow for the severe pressures, high strain rates and large incremental strains occurring during impact loading. The method used to integrate a single strain increment is referred to as an elastic predictor-normal corrector. At the beginning of a constitutive integration step, trial or predictor stress components X/~ are calculated by assuming the entire increment to be elastic. The yield function, Eq. (2), is evaluated using the trial stresses and if ~b( ~;/~, O-, f ) < 0, the increment is elastic and the integration is complete. If ~b(2~, O-, f ) > 0, then the increment is plastic and the stresses must be returned to the yield surface while enforcing the normality condition. The yield surface normal is estimated using O(~/tg,~ij evaluated at the trial stress, £/~. A Newton-Raphson iterative scheme is then used to return the trial stress to the yield surface along Oc~/OZij to satisfy the condition ~b = 0 to within a small tolerance. This elastic predictor-normal corrector method enforces the normality condition reasonably well provided that the trial stress components remain "close" to the yield surface so that the change in yield surface normal is small over the increment. This condition is satisfied as long as the strain increment is small and the level of triaxiality is relatively low. The limiting stable time step size used to integrate the equations of motion within an explicit finite element code, such as DYNA2D, is given by the Courant criterion and is calculated as the sound transit time across the smallest element within the model. The strain increment associated with this time step is usually small except in problems involving high rates of strain as in the current work. Furthermore, high triaxial stress states can occur in impact problems as will be described below. This combination of large strain increment and high triaxiality can lead to numerical difficulty and inaccuracies in applying the elastic predictor-normal corrector integration. To alleviate these problems, the total strain increment is divided into a number of subincrements n given by n=NINT

(,)

(12)

d~(XO'°"f) I~max

'

in which NINT refers to the nearest non-zero integer, N/~ is the trial elastic stress based on the total strain increment and ~maxis a user-specified param-

eter set equal to unity in the current work. Thus the number of subincrements is based on the size of the "excursion" from the yield surface as calculated using ~ ( Z/~, O', f ) . The value for n is often equal to unity for time steps in which the magnitude of the triaxiality and the strain rate are low. As the loading becomes more severe, the use ofEq. (12) has the effect of automatically adjusting the subincrement size to keep the trial stress state near the yield surface. Other subincrement criteria and integration schemes are possible and will be studied in future work. The initial void volume fraction was modelled as zero and the volume fraction of void nucleating particles fN was set equal to the volume fraction of lead particles, i.e. fN = 0.025. Based on observations by Worswick and Pick (1992), the average strain and standard deviation to nucleate voids, EN and SN, were set equal to 0.13 and 0.05, respectively. The parameters fc and ff, governing the onset and completion of void coalescence were varied systematically as part of this research. The initial values considered were those suggested by Tvergaard and Needleman (1984), fc = 0.15 and f f = 0.25. Using these values, an individual finite element will fail or lose all strength once f = ft. Once failure occurs, the element is eliminated from the calculations. A number of other values for fc and f f were considered to determine their effect on the size of the predicted coalescence zone. 4.2. Finite element model

A symmetric impact can be modelled as a single cylinder impacting a rigid, frictionless wall at one-half of the velocity of the symmetric impact. A total of six finite element meshes were considered in which the element size was successively refined to examine the mesh convergence. Table 2 summarizes the mesh designations and element sizes in the region of the cylinder where void growth occurs. The first three meshes are termed regular, indicating that the element size was constant throughout. The latter three are graded in order obtain a refined mesh near the impact face while reducing the overall problem size. Fig. 4 shows the refined portion of mesh 5 near the impact face and serves to illustrate the simple grading techniques used. As seen in Table 2, the smallest element in mesh 4 was very close to the element size in mesh 3 in order to assess the effect of grading the mesh. Note that the el-

M.J. Worswick. R.J. Pick~Mechanics of Materials 19 (1995) 293-309

301

z

9.9 mm

IL

Fig. 4. Refined portion of finite element mesh 5. The impact face is located at z = o.

ement sizes considered are quite fine when compared to the material grain size of 55/zm. Calculations were run for a period of 100/~s after which time the strain rates dropped to low levels and deformation was essentially complete.

5. Results Table 1 summarizes the ratio of final cylinder length to initial length (Lf/Li) and radius ratio ( R f / R i ) for each impact velocity considered (the 175 m/s specimen was damaged during metallographic preparation and its final length was not determined.) The level of deformation increases with velocity as reflected by a decrease in the final length and an increase in the impact face width due to greater mushrooming of the specimen. Figs. 3a-d are optical micrographs taken Table 2 Summary of finite element meshes considered. Mesh

Number of nodes

Number of elements

Element size Ar × Az (mm)

Discretization

1 2 3 4 5 6

585 2193 8481 2534 9851 21211

512 2048 8192 2392 9568 20800

0.563 0.281 0.141 0.139 0.069 0.035

regular 8 × 64 regular 16 × 128 regular 32 × 256 graded graded graded

× × × x × ×

0.642 0.271 0.136 0.139 0.069 0.030

near the impact face of cylinders impacted at velocities between 210 and 300 m/s. Micrographs from specimens impacted at 175 m/s or less appear similar to Fig. 3a which shows no large scale void coalescence. Some isolated void coalescence was observed in the specimens impacted at 230 m/s, widespread coalescence was observed at 267 m/s and a large crack formed at 300 m/s. Void coalescence occurred through localized shearing between voids and was transgranular (Fig. 5). Fig. 3c also shows a network of shear bands running between clusters of coalesced voids which is the mechanism leading to the formation of larger cracks as seen in Fig. 3d. The measured distribution of porosity along the axis of the cylinders impacted at 230 m/s is shown in Fig. 6. This distribution is an average measure from seven cylinders, all impacted at 230+1 m/s. Despite the large scatter in the data, it is observed that a large increase in porosity occurs just behind the impact face with peak mean porosity levels of over 0.15. This value represents a six-fold increase over the initial porosity which can be assumed to be equal to the lead volume fraction of 0.025.

5.1. Finite element predictions The predictions of final specimen geometry are summarized in Table 1 and agree quite well with experiment. The predicted final lengths were roughly 0-3% longer than the recovered specimen lengths

302

M.J. Worswick,R.J. Pick~Mechanicsof Materials 19 (1995) 293-309

Fig. 5. Optical micrograph showing a cluster of adjacent voids that have coalesced through localized shearing between voids. while the predicted radii were within 8% of the actual values. This level of error is quite acceptable, particularly when one considers that some misalignment of the cylinders occurred during impact, as evidenced by slight asymmetries in the recovered specimen profiles.

0.35 0.30-

INITIAL VOID FRAETION fo =0.0255 ,S.D. = 0.00B5

0.25f

MEAN VOID FRACTION_+ONE STANDARD DEVIATION (S.D.) FINITE ELEMENTS : ELEMENTS ON AXIS ELEMENTS WITHIN 280,urn

l

0.20" 0.15

.....

0.10 O.O5 0

0

5

10

15

20

25

z (mm)

Fig. 6. Measured porosity distribution along the specimen axis. Impact velocity= 230 m/s. The scatter bands indicate one standard deviation from measurements on seven specimens. The solid curve shows the predicted porosity distribution from the row of elements on the specimen axis. The dashed curve is the average porosity from elements within 280/zm of the axis.

5.1.1. Stress state and void growth history

Before examining mesh convergence and comparing predictions with experiment, it is useful to examine the loading history and stress state experienced during the impact. Fig. 7 plots the predicted stress triaxiality ~nyd/0", void volume fraction f and strain rate ~ versus time history for a number of elements positioned along the axis of the cylinder impacted at 230 m/s. The elements chosen are located on the impact face, 1.5 m m and 4.5 mm away from the impact face, 4.5 m m being equal to the cylinder radius. The element 1.5 mm away from the impact face (solid curves) is located at the centre of the porous zone during an impact of 230 m / s using mesh 5 and experiences the greatest amount of void growth. At 4.5 mm away from the impact face (dashed curves) the extent of void growth is somewhat smaller while the void growth at the impact face (dashed-dotted curves) is quite limited. These data are taken from calculations in which void coalescence had been disabled. Calculations considering void coalescence are discussed below. During the initial loading in the period up to 1.5/xs, the triaxiality is compressive and reaches a value of approximately - 2 6 . 0 . Strain rates during this period are extremely high and vary considerably (Fig. 7c). No void expansion takes place since the stresses are

M.J. Worswick, R.J. Pick~Mechanics of Materials 19 (1995) 293-309

303

(a) 4.0

-----

I

-----

IMPACT FACE 1.5 mm 4,5 mm

2.0

i i [ I

-4.0

510

0.0

1;.0

15.0

TIME (/~s)

(b)

0.40

0.30 --------

IMPACT FACE 1.5 mm 4.5 mm

0.20 /* // 0.10 ..........

-

0.00

|

i 10.0

5.0

0.0

15.0

TIME (ps) 6.0

(c) L

!

'1°f"rA~ ~

4.0

3.0

llL!f

2"00.0

i

5'.0

10.0

TIME (pa)

15.0

Fig. 7. Time histories of porosity f , triaxiality Zhyd/0" and strain rate lOgl0 ~ for the elements on the cylinder axis. (a) Triaxiality versus time. (b) Porosity versus time. (c) Strain rate versus time.

304

M.J. Worswick, R.J. Pick~Mechanics of Materials 19 (1995) 293-309

----

MESH 3 MESH 5

S

a._

IIIIIIIIIIIIIIIIIIII

_ iiiiiiiiiiiiliiiii

III

ii iii

III]111

iii

i ..............................

ii ii i I ........

Iii

........

ill

.......

ii

i iii

ii

.........

iiiii

i

i

Fig. 9. Contours of void volume fraction, 100 bts after impact at 230 m/s, predicted using meshes 3 and 5. Calculations performed with the coalescence model disabled.

-- IMPACT FACE

Fig. 8. Plot of deformed finite element mesh in the vicinity of the impact face, 4 Ns after impact at 230 m/s.

compressive. As mushrooming or spreading of the impact face begins, radial unloading waves propagate inwards from the specimen surface, arriving at the dements in order of their proximity to the impact face as seen by the sharp rise in triaxialities in Fig. 7a and the sudden increase in porosity seen in Fig. 7b. Once the unloading waves converge on the specimen axis, a tensile triaxial stress state is established, with peak triaxiality levels reaching 4.5. This triaxiality is extremely high, particularly when one considers that the triaxiality ahead of a blunting crack tip is roughly 4.0 (McMeeking, 1977) and that the rate of void growth is known to be an exponential function of triaxiality (Rice and Tracey, 1969). During this period, extremely rapid void growth occurs due to the combination of high stress triaxiality and strain rates up to 105 s -1 . The porosity decreases during the period 812/xs due to subsequent compressive loading of the porous zone. One consequence of the high stress triaxiality during the period 1.5-7.0/.ts is the predicted development of a gap between the impact face and the cylinder near the cylinder axis. This gap is shown in Fig. 8 which plots the deformed finite element mesh 4/xs after impact at 230 m/s. The gap exists during the period of high triaxiality and closes once the material near the axis is re-loaded in compression. Physical evidence of

the formation of this gap is observed in the recovered specimens which exhibit different surface roughnesses at the center of the impact face versus the outer periphery. Close examination of the deformed mesh also reveals radial expansion of the elements located on the axis. This radial component of deformation causes large tensile radial and hoop strains which result in net volumetric expansion of the dements even though the axial strains are compressive. 5.1.2. Mesh convergence In order to determine the sensitivity of the model to the mesh design and element size, the 230 m / s impact was modelled using each of the six meshes summarized in Table 2. The first set of calculations were performed with the void coalescence model (Eq. ( 3 ) ) disabled. Comparison of the results of these calculations without coalescence simplifies the assessment of mesh convergence. It was determined that the deformed shapes from each mesh were virtually identical, indicating that even the most coarse mesh was capable of capturing the bulk deformation. However, there were large differences in the predicted porosity levels at the specimen axis. Fig. 9 shows contours of void volume fraction f at 100/zs after impact from calculations using meshes 3 and 5. The distributions are similar with each mesh predicting a tear-drop shaped porous region centered on the specimen axis, near the impact face. The pre-

305

M.J. Worswick, R.J. Pick~Mechanics of Materials 19 (1995) 293-309

Table 3 Summaryof coalescencezone sizes from mesh convergencestudy. , (a)

Mesh

z

MESH: 2 (b)

J

Zone length (ram)

1

--

--

2 3 4 5 6

0.470 0.494 0.483 0.401 0.355

1.330 2.313 2.270 2.774 3.130

Computational time (hrs.) 0.2 0.7 2.8 3.7 8.1 37.7

43

r

)

Zone radius (mm)

z

MESH: 4

S

6

Fig. 10. Outline of the region of coalesced elements, 100/zs after impact at 230 m/s, predicted using meshes 2 to 6 (fc = 0.15, ff = 0.25.) (a) Results from meshes 2-4. (b) Results from meshes 4-6. dicted peak porosity was higher using the fine mesh since smaller elements are better suited to capture the steep porosity gradient developed near the specimen axis. Fig. 10 plots the predicted outlines of the coalescence zones using meshes 2 through 6. No coalescence zone formed using mesh 1. Table 3 summarizes the coalescence zone "sizes", the length referring to the extent along the specimen axis and the radius being the maximum radius of the zone. The results demonstrate that the zone radius decreases and the length increases as the element size is reduced. The results for the regular and graded mesh with similar sized elements in the coalescence zone (meshes 3 and 4) are very close, as seen in Fig. 10a, demonstrating that graded meshes, with their lower computing costs, are acceptable provided that the coalescence zone or crack does not grow out of the finely meshed region. Initiation and growth of the coalescence zone occurs through progressive failure of elements and proceeds in a step-wise manner. In view of the steep gradient in porosity, the ability of the calculation to accurately re-

solve the coalescence zone will depend strongly upon the element size. From Table 3, the expected radius of the zone is about 0.4 mm. The element size in mesh 1 (Table 2) is greater than this value resulting in no predicted coalescence using this mesh. The results for the finer meshes demonstrate reasonable convergence, as reflected in the predicted outlines in Fig. 10b. The computational times for each analysis, using a 30 MIP workstation, are also reported Table 3 and increase substantially as the mesh is refined. Accordingly, mesh 5 was chosen for the balance of the calculations as a compromise between accuracy and computation cost.

5.1.3. Predicted void growth Predicted porosity distributions along the specimen axis for an impact velocity of 230 m / s are compared with experiment in Fig. 6. These curves were obtained from an analysis which neglected void coalescence. The upper curve is the distribution obtained from the row of elements located closest to the axis. The lower curve represents the average of values from elements which lay within 280/zm of the axis. These elements lie within the region of material that was examined during the metallographic measurements to obtain the experimental values. Thus the lower curve should be in better agreement with the experimental results. The predicted porosity levels are considerably higher than the measured average values and lie just above the edge of the scatter band of the measurements suggesting that the predicted void growth rates are too high. The model does capture the distribution in porosity within the specimen quite well, however, as seen in Fig. 6. The effect of impact velocity on porosity evolution is shown in Fig. 11. The curves correspond to final predicted porosity distributions for impact velocities of

306

M.J. Worswick, R.J. Pick~Mechanics of Materials 19 (1995) 293-309 0.6

VELOCITY ----.--

175 m/s 230 mls 300 mls

/

~4 >

0.2

CD

175m/s 0 0

0',5

1'.0

115 2'.0

2'.5

3.0

z COORDINATE

Fig. 11. Predicted porosity distribution along cylinder axis for impact velocities of 175, 230 and 300 m/s. 175,230 and 300 m/s, from calculations in which void coalescence was disabled. There is a marked increase in porosity within the porous zone with increased impact velocity. The predicted peak porosities are all in excess of the levels for void coalescence according to the criterion of Tvergaard and Needleman (1984), even though void coalescence only occurred for impact velocities in excess of 210 m/s. 5.1.4.

C o a l e s c e n c e z o n e size

Calculations were performed using the coalescence criterion of Eq. (3) for the five impact velocities considered. There was little difference in the deformed shape of the cylinders predicted using models run with and without coalescence, probably due to the small volume of material experiencing void growth and coalescence. The predicted coalescence zone sizes for each impact velocity are compared in Fig. 12 and are summarized in Table 4. As expected, the length and width of the coalescence zone increases with impact velocity. For the two higher velocities, the coalescence Table 4 Summary of coalescencezone sizes for each impact velocity. Impact velocity (m/s)

Zone radius (mm)

Zone length (ram)

175 210 230 267 300

0.106 0.270 0.401 0.595 0.702

1.143 2.467 2.779 3.197 3.365

210 230 267 300mls

Fig. 12. Outline of the region of coalesced elements 100/zs after impact (fc = 0.15, ff = 0.25.) zone extends to the impact face. The predicted extent of coalescence is compared with experiment by overlaying the outline of each predicted zone (solid lines) onto the micrographs in Figs. 3a-d. From Fig. 3a it is evident that the model does predict the formation of a zone of coalescence at an impact velocity below which coalescence is observed experimentally. A smaller coalescence zone was predicted at an impact velocity of 175 m/s (Fig. 12) but was also not observed experimentally. At higher velocities, the agreement between experiment and model is considerably better, with the heavily damaged regions falling largely within the predicted zone (Figs. 3c and d). A value of fc = 0.07 was used by Becker et al. (1988) to model coalescence during quasi-static loading of sintered iron with an initial porosity of 0.026. This porosity is very close to the initial volume fraction of lead in the brass material considered here. However, use of this value of fc in the current analyses would lead to coalescence at impact velocities well below 210 m/s and would result in excessively large predicted coalescence zones. A number of calculations were undertaken in which the values for fc and ff were varied in an exercise to determine if the agreement between the actual and predicted coalescence zones could be improved. The coalescence zone predictions from one set of calculations using fc = 0.25 and ff = 0.3 have been plotted as dashed lines in Fig. 3. The zones predicted using these values lie inside those using the values suggested by Tvergaard and Needleman (1984) and appear to provide qualitatively better agreement with the observed coalescence zones in the micrographs. Critical values of 0.25-0.30 void volume fraction to cause coales-

M.J. Worswick, R.J. Pick~Mechanics of Materials 19 (1995) 293-309

cence are viewed as unrealistically high. The improved agreement between experimentally observed and predicted coalescence zones using these values is more likely due to errors in the predicted void growth rates which appear to be too high.

/

[ [--HARINI /efal.(1985)

2J

]

The results of this study have shown that while the Gurson-based constitutive model does qualitatively capture the dynamic fracture behaviour of the cylinders, it is apparent that a number of problems exist. Comparison between measured and predicted porosity levels has shown that the model over-predicts the void growth. As a consequence, void coalescence is predicted to occur at impact velocities well below those required to cause coalescence in the experiments. In addition, the size of the predicted coalescence zone is larger than that observed in the micrographs. A number of reasons for the excessively high void growth rates can be suggested that are not considered in Gurson's (1975) model. It is possible that material strain rate sensitivity could retard void growth as seen in analytical studies by Budiansky et al. (1982). The bulk strain rates in the porous zone are in excess of 104 S--l and one would expect the local strain rates in the vicinity of individual voids to be higher still. Increased resistance to deformation with strain rate could reduce the void growth rate; however, brass does have a comparatively low strain rate sensitivity. Another possible source of error lies in the chosen material constitutive model which neglects thermal softening; however, this effect would tend to increase void growth rates by further reducing material strength. Inertial effects could also retard void growth in a manner not considered by the Gurson-based model considered here. It is also likely that void coalescence could be delayed by inertial stabilization of the plastic flow associated with void growth (Cortes, 1992). One potentially large source of error in the predicted void growth rates could lie in the accuracy of the Gurson yield function at very high stress triaxiality levels. Very few experimental studies have attempted to determine void growth rates at the triaxiality levels experienced by the cylinders, largely because under quasi-static conditions, triaxiality levels of 4-5 normally occur only ahead of a crack tip. The majority

n=0.0 n-01 ~/ ,,./'n~.0.2

Ill ~-o/t,~" ¢~"1/,~.,-" I/1 /~,0_'~.@Z,/~"_/'~-RICEANDTRACEY

tn(0 / I ~ T Z / / / 14 )q~,',~,~" - - |

6. Discussion

/H~/~'/ ] /~,~p~ 01 ~ / ;,El

[~ - 1 ]/~ -2

307

" Q .,L I

d969) tn=O)

BUaANSKY,~

ot (m2)

BARNBYe, °1. (198/,1 SPITZI5 et tal. (1988) WORSWICKAND PICK (1992)

LBEREMIN (1961) "~-BOURCIER et at. (1966)

1 ~-hyd /

Fig. 13. Void growth rate D as a function of stress ~axiality.

of experimental data is concerned with triaxiality levels below 2 as seen in Fig. 13, which compares void growth rates from a number of experimental studies and analytical predictions by Rice and Tracey (1969) and Budiansky et al. (1982). All of the studies shown in the figure considered quasi-static conditions. The figure plots void growth rate as characterized by

D= f ( 1 - / f)~poo,

(13)

as a function of stress triaxiality. The parameter D, introduced by Rice and Tracey (1969), is thought to be an exponential function of stress triaxiality. Concern lies in the fact that little verification of predictions using the Gurson (1975) or Rice and Tracey (1969) models of void growth has been undertaken at high levels of triaxiality. The one experimental study by Barnby et al. (1984) does suggest that the void growth rates ahead of a crack tip with a triaxiality level Xhyd/O" = 4.1 are much lower than predicted using these models. The Taylor test was of interest in the current study since damage propagation is arrested within the compressively-loaded bulk of the impact specimen, allowing the extent of damage to be controlled. In quasi-static tensile tests, the onset of void coalescence often coincides with a global instability, resulting in compete fracture of the specimen. A major drawback of the Taylor test, however, is the wide variation in stress state and strain rate both as functions of time and position within the cylinder, as shown in Fig. 7. Additional uncertainty is introduced through the use

308

M.J. Worswick, R.J. Pick~Mechanics of Materials 19 (1995)293-309

of the matrix constitutive model described by Eq. (1) which is based on quasi-static behaviour with a correction for strain rate effects. Direct impact Hopkinson bar experiments are planned to obtain constitutive data for this material valid in the strain rate regime 104_105 s -1" It is evident that further experimentation is necessary to ascertain whether models such as the Gurson yield function can be used under high strain rate and high triaxial loading conditions, particularly since these two conditions often exist together during impact loading. More controlled plate impact (Worswick et al., 1993a) and tensile split Hopkinson bar (Worswick et al., 1993b) experiments using the leaded brass material studied here are underway in an attempt to isolate the various effects of strain rate and stress triaxiality. The numerical calculations have shown the coalescence zone size to be highly dependent on impact velocity and on the critical values of porosity to cause void coalescence, as one would expect. There also exists a strong sensitivity to the element size used in the calculations. Examination of Tables 3 and 4 reveals that the variation in the length of the coalescence zone size due to changes in the element size was almost equal to the variation due to the range of impact velocities considered in the experiments. Thus in continuum damage studies involving progressive failure of elements, it becomes particularly important to establish mesh convergence in order to verify calculations. Also, a significant level of subjectiveness is introduced since the critical values for fc and ff derived from one type of specimen or geometry may not be suitable for another geometry unless the element sizes are kept the same. In Fig. 7, the triaxiality versus time history reveals that the porous region of the cylinder experiences compressive loading which leads to closure of the voids and a drop in porosity. Although the time histories plotted were taken from calculations in which the void coalescence criterion had been disabled, it is expected that the material in which void coalescence occurred would also experience this compressive loading and it is conjectured that this material would support compressive loads. The model, however, reduces the stresses supported by elements to zero once their porosity level satisfies the coalescence criterion so that the compressive loads at later times would not be supported. Neglecting these loads could cause differences

in the final shape of the coalescence zone which does change during the later stages of the calculation due to deformation of the surrounding material.

7. Summary The results of this study have shown that Gursonbased models of porous plastic solids are capable of providing qualitative predictions of the ductile fracture process occurring at high rates of strain. Uncertainty exists in the accuracy of the predictions of void growth rate, with the model over-predicting the rate of growth of voids. The determination of critical values of porosity to cause void coalescence is thus hindered by this uncertainty in the void growth rates. Further work is planned using more controlled experiments in order to attempt to isolate the separate effects of high strain rate and high stress triaxiality.

Acknowledgement Financial support for this work was provided to R.J. Pick and M.J. Worswick from the Natural Sciences and Engineering Research Council of Canada and to R.J. Pick from the Defence Research Establishment Suffield.

References Barnby, J.T., Y.W. Shi and A.S. Nadkarni (1984), On the void growth in C-Mn Steel during plastic deformation,Int. J. Frac. 25, 273. Becker,R., A. Needleman,O. Richmond,and V.Tvergaard (1988), Void growth and failure in notchedbars, J. Mech. Phys. Solids 36, 317. Beremin (1981), EM., Study of criteria for ductile rupture of A508 steel, in: Advances in Fracture Research, ICF 5, Vol. 2, p. 809. Boucier, R.L, R.J. Koss, R.E. Smelser and O. Richmond (1986), The influence of porosity on the deformation and fracture of alloys, Acta Metall. 34, 2443. Budiansky, B., J.W. Hutchinson and S. Slutsky (1982), Void growth and collapse in viscous solids, in: H.G. Hopkins and M.J. Sewell, eds., Mechanics of Solids, The Rodney Hill 60th Anniversary Volume, PergamonPress, Oxford, p. 13. Cortes, R. (1992), Dynamicgrowthof microvoidsundercombined hydrostatic and deviatoric stresses, Int. J. Solids Struct. 29, 1637.

M.J. Worswick, R.J. Pick~Mechanics of Materials 19 (1995) 293-309 Curran, D.R., L. Seaman and D.A. Shockey (1987), Dynamic failure of solids, Phys. Rep. 147, 253. Dumont, C. and C. Levaillant (1990), Ductile fracture of Cu1% Pb at High Strain Rates, in: M.A. Meyers, L.E. Murr and K.P. Staudhammer, eds., Shock Wave and High-Strain-Rate Phenomena in Materials, Dekker, New York p. 181. Fyfe, I.M. and A.M. Rajendran (1982), Dynamic pre-straln and inertia effects on the fracture of metals, J. Appl. Mech. 49, 31. Gurson, A.L. (1975), Plastic Flow and Fracture Behaviour of Ductile Materials Incorporating Void Nucleation, Growth and Interaction Ph.D. Thesis, Brown University. Johnson, J.N. and EL. Adessio (1988), Tensile plasticity and ductile fracture, J. Appl. Phys. 64, 6699. Johnson, G.R. and R.A. Stryk (1986), User Instructions for the EPIC-2 Code, Honeywell Inc., Edina, MN, Contract Report AFATL-TR-86-51, September, 1986. Marini, B., E Mudry and A. Pineau (1985), Experimental study of cavity growth in ductile rupture, Eng. Frac. Mech. 22, 989. McMeeking, R.M. (1977), Finite deformation analysis of a cracktip opening in elastic-plastic materials and implications for fracture, J. Mech. Phys. Solids 33, 25. Needleman, A. and V. Tvergaard (1984), An analysis of dimpled rupture in notched bars, J. Mech. Phys. Solids 32, 461. Rice, R.J. and D.M. Tracey (1969), On the ductile enlargement of voids in triaxial stress fields, J. Mech. Phys. Solids 17, 201. Tvergaard, V. ( 1981 ), Influence of voids on shear band instabilities under plane strain conditions, Int. J. Frac. 17, 389.

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Tvergaard, V, and A. Needleman (1984), Analysis of the cup-cone fracture in a round tensile bar, Acta Metall. 32, 157. Whirley, R.G. and J.O. Hallquist (1992) DYNA2D Users Manual, Lawrence Livermore National Laboratory, UCRL-MA-110630. Wong, B.J. (1990), Void Growth Modelling in a Dynamically Loaded Material, Ph.D. Thesis, Department of Mechanical Engineering, University of Waterloo, Canada. Worswick, M.J. (1994), Int. J. Numer. Meth. Eng., to be submitted Worswick, M.J. and R.J. Pick (1990), Void growth and constitutive softening in a periodically-voided solid, J. Mech. Phys. Solids 38, 601. Worswick, MJ. and R.J. Pick (1991), Void growth and fracture in plastically deformed free-cutting brass, J. Appl. Mech. 58, 631. Worswick, M.J., N. Qiang, P. Niessen and R.J. Pick (1990), Microstructure and fracture during high-rate forming of iron and tantalum, in: M.A. Meyers, L.E. Murr and K.P. Staudhammer, eds., Shock Wave and High-Strain-Rate Phenomena in Materials, Dekker, p. 87. Worswick, M.J., H. Nahme, J. Clarke, and J. Fowler (1993a), Ductile Fracture During Tensile Hopkinson Bar and Plate Impact Testing, Proc. 8th DYMAT Technical Conference, CRC Joint Research Centre, ISPRA, Italy, October 12-13. Worswick, M.J., J. Clarke and R.J. Pick (1993b), Dynamic fracture under impact and high strain rate loading, Can. J. Physics, submitted for publication.