peek composites

NlJClIAN

NA'rNH ELSEVIER

Mechanics of Materials 19 (1994) 1-14

Constitutive modeling of high strain-rate deformation and spall fracture of graphite/peek composites J.A. N e m e s a, P.W. Randles b a Mechanical Engineering Department, McGill University, Montreal, Quebec, Canada H3A 21(6 b Field Command, Defense Nuclear Agency / FCTTS Kirtland AFB, NM 87117-5669, USA

Received 19 October 1993,accepted 27 January 1994

Abstract

A constitutive model that describes the response of graphite/peek composite materials to high rate uniaxial deformation with resulting damage and spall fracture is presented. A nonlinear, rate-dependent description is used in conjunction with a continuum damage description of evolving delamination damage. Results of flyer plate impact experiments performed on 25 mm thick, transversely isotropic laminates are given. Comparison of measured rear-face particle velocity histories to those predicted by the theory show good agreement over a wide range of impact velocities. Computed damage profiles are compared to visual and ultrasonic observation of the post-impacted specimens.

1. Introduction

The propagation of stress waves in materials arising from impact, explosions, or intense pulsed radiation has been of considerable interest for several decades, but its study has principally been confined to homogeneous materials, mainly metals (Davison and Graham, 1979). An important class of materials that has received much less attention is the laminated fiber-reinforced polymer composites. Increased use of these materials, however, particularly in aerospace and naval applications, requires better understanding of their dynamic behavior. In considering these composite materials, difficulties arise in modelling the stress wave response not only due to the material nonlinearities of the constituents, but also in treating the effects of the geometric heterogeneities. One-dimensional stress waves propagating

through the periodic laminates has been treated by several authors (Ting, 1980) using homogenization techniques. For fiber-reinforced composites, however, there are several levels of heterogeneities to consider: heterogeneity between layers, between the fiber and matrix, and also heterogeneities arising from processing the composites in the form of voids. This fact combined with the nonlinearity of the constituents makes "exact" treatments of the wave propagation behavior through the composite unfeasible. The alternate approach is the development of empirical constitutive equations for the composite that describes the response of the material, including dispersion and attenuation of the wave, but allowing the composite to be treated as an equivalent homogeneous material. Several different models have been applied to the stress wave response of fiber-reinforced com-

0167-6636/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0167-6636(94)00024-B

2

J.A. Nemes, P.W. Randles/Mechanics Of Materials 19 (1994) 1-14

posites with varying degrees of success in capturing the important features of dispersion and attenuation. Barker (1971) used a Maxwell viscoelatic model to simulate the wave attenuation effects in laminated composites. Cohen and Berkowitz (1972) treated Quartz-phenolic composites using an elastic-plastic, strain-rate insensitive, work-hardening constitutive model to predict fracture. The same material was considered by Boade (1970) using used a porosity or "P-a" model (Herrrnan, 1969a). Tokheim et al. (1990) used a constitutive model of Curran et al. (1974) with a nonlinear pressure-volume relationship and viscoelastic-plastic behavior for the deviatoric response for modelling Kevlar/epoxy. The effects of geometric dispersion were simulated using "artificial viscosity". In all of these studies the experimental data was too limited to fully assess the ability of the constitutive equations to model the stress wave propagation. An area of concurrent interest with stress wave propagation is the dynamic fracture or spallation of materials. Spallation occurs as the result of superposition of direct shock and release waves, which produces a net tensile stress in the direction of wave propagation. Complete separation occurs if the stress is of sufficient magnitude and duration, whereas for lower stress levels a n d / o r shorter duration some form of material damage may occur, which partially degrades the material. As in the case of stress wave propagation, spallation studies have been carried out for decades, with primary emphasis on metals (Curran et al., 1987). The limited number of spallation studies conducted on fiber-reinforced composites (Cohen and Berkowitz, 1972; Roylance, 1980; Tokheim et al., 1990) have included little quantitative experimental data, instead being limited to qualitative observations on determining conditions under which spallation or damage occur. Several models to predict spallation have been considered, most of which are based on early models for predicting spallation in metals. Some of these models are based on the impulse associated with the tensile stress pulse, thus including the effect of stress level and time at stress, originally proposed by Tuler and Butcher (1968) for metals. Cohen and Berkowitz (1972) and Roy-

lance (1980) have used such models to predict spall in composites. These models are limited in that they predict only "spall-no spall" conditions and thus are not useful in predicting material damage that results when conditions are such that complete separation does not occur. Tokheim et al. used a version of the nucleation-and-growth models developed by Seamen et al. (1976), which considers size and distribution of cracks throughout the material, to model spall in Kevlar/epoxy with some success. In this paper we consider the stress wave propagation and resulting damage and spallation be-havior of thick (25.4 mm) graphite/peek laminated composites, resulting from plate-impact~ Experiments are conducted to study the behavior of the material under moderate amplitude (< 1000 MPa) short pulse (~ 1-4 Ixs) duration shock loading. The plate impact experiments produce a deformation field that can be considered as uniaxial, with strain rates of 103--105 s -I encountered. Experimental monitoring of the rearsurface particle velocity of the impacted composite plates permits a quantitative comparison between the experiments and the computed results, which has been absent from previous studies of fiber-reinforced composite behavior subjected to shock wave loading, and which shows distinct evidence of spallation. Study of the damage and spallation of the composite, which is the emphasis of this paper, requires that the preceeding compressive stress pulse also be modelled accurately since details of this pulse determine the shape of the tensile pulse which has been shown to greatly affect damage and spallation (Roylance, 1980). Therefore we consider the compressive response of the material using a nonlinear, rate-dependent constitutive model treating the composite as a homogeneous solid. Dispersion of the compressive waves arises from the relaxation in the unloading behavior of the material. The model for material damage and spallation is guided by a model previously developed by the authors for thick composites (Randles and Nemes, 1992). Emphasis is given here to developing a model requiring as few material parameters as possible while still providing reasonable results.

L4. Nemes, P. IV.. Randles / Mechanics of Materials 19 (1994) 1-14

2. Experiments Experiments were conducted for the U.S. Naval Research Laboratory on the Phillips Laboratory Impact Facility 102 mm gas-gun, using standard plate-impact techniques. The sample and impactor were aligned so that tilt was less than 1.0 mrad, thus producing a near planar impact and resulting in the uniaxial strain field. Shorting pins placed adjacent to the sample were used for precise measurement of the projectile planarity and velocity just prior to impact. Measurement of the rear-surface particle velocity of the composite was accomplished using a VISAR technique (Barker and Hollenbach, 1972) Two types of flyer-plate impact experiments were conducted (Smith, 1991). The first of which was transmitted wave experiments depicted in Fig. 1. In this configuration a plate of polymethylmethacrylate (PMMA) impacts the composite target, which is contained in a phenolic holder. The composite is backed by a PMMA buffer containing the reflecting surface that permits the particle velocity measurements to be made. The transmitted wave experiments permit accurate timing of the arrival of the compressive wave and measurement of the complete compressive wave profile. Since the mechanical impedance of the PMMA is

3

similar to the graphite/peek, there is little reflection at the interface. These experiments were used primarily to develop an understanding of the compressive response of the composite. The second configuration is the spallation experiment, shown in Fig. 2. Here it can be seen that there is no PMMA buffer plate, so that the rear surface of the composite is free. In this way the compressive wave reflects off the surface as a release wave, which, when superimposed with the initial compressive wave, leads to a net tensile stress in the material. Particle velocity in these experiments was recorded directly at the rear surface. The composite here is loosely supported at the edge such that after impact the sample travels to a soft recovery tank for subsequent examination. Table 1 summarizes the geometry parameters for the experiments discussed later in the paper. 2.1. Material The material specimens were manufactured by McDonnell Douglass Aircraft Company of Berkely, MO. Unidirectional tape containing fibers of AS4 graphite impregnated with a matrix of APC-2 polyetheretherketone (PEEK) was used to fabricate the quasi-isotropic laminates. The

J PMMA IMPACTOR

/

/

GRAPHI11E/PEEK TARGET

J

X___ Fig. 1. Configuration for transmitted wave experiments.

VISAR

4

J.A, Nemes, P.W. Randles/Mechanics of Materials 19 (1994) 1-14

Table 1 Experimental geometry(Smith, 1991) Experiment Type Flyer Sample Buffer Impact No. thick, thick, thick, velocity (mm) (mm) (mm) (m/s) 3349 3350 3360 3357 3362 3358 3368 3375

TW TW Spall Spall Spall Spall spall spall

5.17 5.85 3.49 3.45 1.37 3.50 3.52 3.53

26.17 25.97 25.67 25.83 25.82 25.91 25.79 25.55

1.54 1.50 n/a n/a n/a n/a n/a n/a

3. Modelling

3.1. Balance laws

101 505 149 93 92 57 36 20

One of the most attractive aspects of the flyer-plate experiments is that the resulting deformation can be considered to be uniaxial. Thus the balance of mass and momentum equations are simplified, reducing to the following P0

0x3

p

(1)

and 00"33 0x 3

plies were 0.13 mm thick and contained 62% fiber volume. The layup for the nominally 25 mm thick sample was ([+_452/02/902]s)12 for a total of 192 plies. Density of the material was found to be 1.57 g / c m 3. Ultrasonic measurements gave a longitudinal (thru-thickness) velocity of 2900 m / s and a shear wave velocity of 1580 m / s .

(

J

du 3 P dt '

(2)

where here the x 3 axis is taken to be the direction of wave propagation. In the expressions P0 and p are the initial and current material densities, X 3 and x 3 are the Lagrangian and spatial

, IMA 13AC1"(

t~Hr I~Lrr

Fig. 2. Configuration for spall experiments.

J~,t. Nemes, P.W. Randles / Mechanics of Materials 19 (1994) 1-14

5

material coordinates, 033 represents the normal stress component and u 3 is the particle velocity. In the subsequent development the use of subscripts will be dropped. Solution of (1) and (2) was performed numerically with the computer code WONDY (Kipp and Lawrence, 1982), which uses the corresponding finite difference expressions and explicit integration.

taneous modulus, and G is the relaxation function, taken respectively as

3.2. PMMA

tre(•) =(ke +le• +meeZ +nee3)•.

Since the development of the material model for graphite/peek is performed iteratively by comparing the computed particle velocities to those recorded for the corresponding impact experiment, it is necessary to specify the constitutive behavior of the PMMA flyer and backing plates a priori. PMMA is commonly used as a material in impact experiments and its response has been extensively studied. As a result, PMMA behavior is well characterized. In order to minimize uncertainty in the model development of the graphite/peek, the PMMA is modelled as accurately as possible. Here we have utilized a nonlinear, viscoelastic model developed by Schuler and Nunziato, (1974). The constitutive equation is written in rate form as d-= E ( • ) ~ + G(tr, •),

= k I + l i e + m l e2 + n l e 3 ,

G ( g , e) =

•

(4)

Note that the definition in (4) results in strains that are positive in compression, which is in accordance with convention used in shock wave physics. In (3), () denotes a()/Ot, E is the instan-

(6)

•)

In (6) 0-E is the equilibrium stress-strain relationship given by (7)

It should be noted that the relaxation time z is taken as a function of stress and strain rather than a constant as is often done in viscoelastic models. This was found to be necessary by Shuler and Nunziato to model the hysteresis upon unloading. The form for ~- employed in their model is r = r0 e x p [ - ( - ~ - ~ ) ] ,

0-- trE> 0,

The parameters determined by Schuler and Nunziato for the PMMA and employed here are given in Table 2.

3.3. Graphite/peek In developing a model that describes the behavior of the graphite/peek the following observations and conclusions were made from the transmitted wave experiments: (1) The arrival times of the compressive wave indicated that the wave speed increased for higher impact velocity, as is common for most materials. The increase was small, but not insignificant. It was therefore necessary to include nonlinear stress-strain behavior.

Table 2 M a t e r i a l c o n s t a n t s for P M M A ( S c h u l e r a n d N u n z i a t o , 1974) k I = 9.031 x 103 M P a l I = 1.414 × 105 M P a rn I = - 6 . 7 7 9 x 105 M P a n x = 4.16 x 106 M P a

(5)

(3)

where in uniaxial deformation the strain • is given by P • = 1 - --. P0

E(e)

k E = 8.979 x 103 M P a l E ~ 7.00 × 104 M P a

7 o = 0.25 Ixs k = 800 M P a

m E = - 5 . 8 6 9 X 105 M P a

P0 = 1.185 g / c m a

n E = 1.9652 x 106 M P a

6

J.A. Nemes, P.W. Randles/M e c hani c s o f Materiab 19 (1994) 1-14

(2) The initial wavefront was essentially a single steep shock. There was no evidence of a two-wave structure as is found in metals with separate elastic and plastic waves (Hermann, 1969b). Therefore no yield phenomena would be included as had been done in the work of Cohen and Berkowitz (1972) and Tokheim et al. (1992). (3) The transition from the shock front to the plateau is very abrupt, indicating an absence of rate-dependence on loading. (4) The arrival of the unloading portion of the wave also indicated that the wave speed increased at higher impact velocities, but these wave speeds are higher than those obtained for the loading, as can be seen from the wave speeds shown in Table 3. Therefore, the material is unloading on a path different than the path taken during loading. Unlike the loading, however, the unloading of the material appears to be rate-dependent, with significant relaxation occurring. In view of these observations, the material model developed for the compressive behavior of the material is described by the following equations: ,

I~ > 0, ~ __>0,

(9a)

d ' = C u ( e ) ~ + G ( t r , e),

e > 0, ~ < 0.

(9b)

Or =

CL(~S)E

Since the nonlinearity of the material on loading in the range of impact considered is not strong, a single higher order term is considered CL(e ) = k,(1 + k 2 e )

(10)

where k I is the stiffness in the through-thickness direction, which can be determined from the ultra-sonic longitudinal wave speed (11)

k l __ 2 • -- P oU L

Table 3 L o a d i n g a n d u n l o a d i n g wave velocities (Smith, 1991) Experiment

I m p a c t vel. s(m/s)

L o a d i n g wave vel. ( m / s )

U n l o a d i n g wave vel. ( m / s )

3363 3349 3348 3365 3350

100 101 240 339 505

2990 3010 3010 3080 3110

3100 3120 3200 3320 3410

Table 4 M a t e r i a l p r o p e r t i e s for g r a p h i t e / p e e k Po = 1.579 g / c m 3 cl = 1.42× 104 M P a r t = 13.0 # s

k I = 1.38 × 104 M P a c2 = 5.37 cr(~o = 35 M P a

k 2 = t.56 r=2.5#s n = 2

k 2 is then determined by a fit to the loading wave speeds given in Table 3. Cu(e) is also taken to have one higher order term C u ( e ) = cl( 1 + c2e ) .

(12)

Here both c t and c 2 are taken to fit the unloading wave velocities. The relaxation function in (9b) is given the form G(or, e) =

ORE(')

(13)

T

For the g r a p h i t e / p e e k the equilibrium stressstrain curve will be taken to be the same as the loading so that Ore(e) = CL(e)e,

(14)

in effect saying that the material relaxes to the stiffness found upon loading. Finally, ~- in (13) is a time constant which must be found by fitting the computation to the compressive wave profiles. The parameters appearing in (9)-(13) are given in Table 4. The stress-strain compressive response for loading, which is rate-independent, and for unloading at three different rates is shown schematically in Fig. 3. In actuality, the nonlinearity and difference in unloading paths is not nearly as pronounced as shown in the schematic.

3.4. Comparison to transmitted wave experiments The material model described in the previous section was used with the computer code W O N D Y to simulate experiments 3350 and 3349. These two were selected because they represented the highest and lowest impact velocities of the transmitted wave experiments. Comparison of the recorded particle velocity in the PMMA buffer to that computed at the corresponding finite-difference location in the model is shown in Fig~ 4. As seen in the figures, there is excellent agreement on the wave arrival time and on the slope of

J.A. Nemes, P.W. Randles / Mechanics of Materials 19 (1994) 1-14

0

> e2> e~

~E3

7

relaxation function cannot be taken to be the same as the stress-strain curve on loading. However, for either of these the possible increased accuracy of the results comes at the expense of increased complexity of the model with additional parameters, which was felt to be unjustified. Thus, the material description given by (9)-(14) was taken to adequately describe the important features of the compressive wave response. This permits the compressive pulse to be accurately computed, which is necessary for study of the ensuing material damage.

4. Damage 1 Fig. 3. Schematic r e p r e s e n t a t i o n of loading and unloading behavior at three rates.

the wave. The level of the plateau is slightly overshot in experiment 3350, most likely due to the fit of (10) with a single nonlinear term. The unloading arrival (decrease in particle velocity) also agrees well. The slope of the release wave is good in 3350 and good initially in 3349, but at later times, the computed slope is too steep. Thus, it appears that the relaxation time is also nonlinear, as found by Schuler and Nunziato (1974) for PMMA which would require a functional description similar to that in (8) or the

4.1. Model development As a result of the uniaxial deformation produced in the flyer plate experiment, a tensile stress state results in the through-thickness direction with the absence of shear stresses. Thus the ensuing damage is in the form of delamination type cracks (Fig. 5). If damage is expressed as a vector quantity (Talreja, 1985; Randles and Nemes, 1992) the magnitude of the vector represents the degree of cracking and the direction of the vector is normal to the cracks, in this case the x 3 direction. In these experiments a particular damage mode (delamination cracking) can be related to a single stress component.

250

(b)

50-

20O ~vE150

\

>w

<

"!

60-

(a)

100- - -

_i 30: i~ 20-:

~

10.:

50 - 0 .I -10. 8

10 12 14 TIME(microsec)

16

18

Fig. 4. C o m p a r i s o n of c o m p u t e d and m e a s u r e d particle velocity(- - - - 3349.

6

computed, - -

10 12 14 TIME (microsec)

16

18

experiment): (a) shot 3350; (b) shot

8

J.A. Nemes, P. W. Randles/Mechanics (if'Materials 19 (1994) 1-14

The model that will be used here is based on one previously given by Randles and Nemes, simplified to the case of uniaxial deformation. Since the tensile stresses that can develop are relatively low, stress can be linearly related to strain, but nonlinearity enters due to the presence of damage. Thus cr = C ( V 3 ) e ,

e < 0,

(15)

where V3 is the component of the damage vector with normal in the x 3 direction, corresponding to the delamination cracks. Again since this is the only damage component present, the subscript will subsequently be dropped. Eq. (15) is written in rate form as d C ( V ) l)'e.

d----Y--

(16)

In this form it can be clearly seen that the change in stress results not only from increments of strain, but also from the evolving damage, which is responsible for the stress softening that occurs. Dependence of the material stiffness on damage is taken to be C ( V ) = k t ( 1 - V2).

(17)

The range of the damage parameter is from 0, for virgin material, to 1, for the completely cracked or spalled condition. Damage evolution is taken to be a function of the current state of damage and some overstress above a threshold, d = - ( o r + ~rG) (recalling that tension is considered negative). The form postulated for the damage evolution is dV dt

(d/~rG0)" r/(1 - V2) '

where n is a positive power term tor d/cr6o, ,, is a time constant which governs the damage rate magnitude and o-c0 is a material constant that will be discussed shortly. Note that the (1 -- V ~) term in the denominator leads to sudden increase in the rate of damage as V--* 1. This term appears to be appropriate for modelling the brittle fracture associated with complete delamination or spall. As stated previously d is the stress above a threshold, where the threshold itself is a function of damage. This assumption is in accordance with the observation that the onset of damage is at a lower stress level in a material that has previous damage than in the virgin material. The form for the threshold stress is crc; = ,re;,,( 1 - V 2)

(19)

From (19) it can be seen that for V = 0, ~ra0 can be interpreted as the virgin material threshold stress. The evolution of delamination damage due to thru-thickness tensile stress given by (15)-(19) requires determination of three material parameters: n, 7, and ~ra0. o'c0 can be found from the experiments as the stress below which no damage is found, taken here to be 35 MPa, which also approximates the thru-thickness tensile strength at quasi-static rates. The parameter n is taken as 2, which also has been used by Dechaene et al. (1992) in a similar damage evolution law applied to the study of impact in fiber glass epoxy composites. The remaining parameter ~7 was determined from iteration between the computed and measured rear-surface particle velocity histories. 4.2. Comparison to spall experiments

(18)

Fig. 5. Delamination cracking in laminated composites.

The model discussed in the previous section, using the material parameters shown in Table 4 was used in numerical simulation of 5 spall flyer plate impact experiments. These experiments represent a wide variety of material response, ranging from impact velocities where no visible damage was detected to ones in which multiple spall planes were observed. In addition, the thickness of the flyer was varied in the experiment by a factor of 2.5, which results in a similar variation in duration of the tensile pulse. With this range of experiments, reasonable simulation of the re-

J.A. Nemes, P.W. Randles / Mechanics of Materials 19 (1994) 1-14

9

pulse. Particularly important are the local minimum in particle velocity, the subsequent local peak and the timing of the following oscillations. Figure 6a shows the comparison for shot 3360, which is for the highest impact velocity considered. There is very good agreement in all of the features noted above. Agreement for shot 3357,

sults was considered to be a good test for the model's ability to describe the material behavior. The computed and measured rear-surface velocities for the experiments are shown in Fig. 6. The aspects of the history that are directly related to damage or spallation, and of primary interest here, occur on the release portion of the 140

(a~2o

i

lOO

70-:

g

802--

w

~ 6o > ~

f/i fl -

..i 50: "~.

402

40.:

~ 30.:

~ 2o

20 -:

o

10-:

-2o

i

z

8

6

10 12 14 16 TIME (microsecs)

18

20

0-: 6

18

20

6

10 12 14 16 TIME (microsecs)

18

20

10

18

20

70 :

~ 60 5so ~40 ,< o.

30:

I

20

lo 0

r

6

,

,

8

10 12 14 16 TIME (microsecs)

12 14 16 TIME (microsecs)

8

(e)3350~_

t

lO]E s=I-

::21.,

. . . . . .

8

i , , ' i

/ t

.

.

.

.

.

10 12 14 16 TIME (microsecs)

.

.

.

18

20

Fig. 6. Comparison of computed and measured particle velocity in spall experiments ( . . . . . . shot 3360; (b) shot 3357; (c) shot 3362; (d) shot 3358; (e) shot 3368.

computed, - -

experiment): (a)

10

ZA. Nemes, P. W, Randles / Mechanics c~" Materials 19 (1994) 1- i4

¸¸i

Fig. 7. Photographs of post-impact specimens: (a) shot 3360 - showing two complete spall planes; (b) shot 3357 - ortc complet~ spall plane with edge cracks; (c) shot 3362 edge crack in rear (lower) surface.

J.A. Nemes, P.W. Randles / Mechanics of Materials 19 (1994) 1-14

shown in Fig. 6b, is reasonable, although the dwell time at the local minimum particle velocity is too short, resulting in mistiming of the subsequent peaks. Even with this mistiming, the slope of the computed particle velocity curve is very close to the measured value, indicating the degradation of material stiffness is being approximated. Shot 3362 is for an almost identical impact velocity as 3357 but with a much thinner flyer, which produces a shorter duration pulse as previously noted. As a result, the local minimum particle velocity before rebound is lower for shot 3362 in both the computed and experimental results. Comparison of the timing of subsequent reverberations is also very good. Figure 6d is the comparison for shot 3358, which clearly is not as good as the others. The reason for this may be due to the fact that this experiment produces damage that might be described as incipient spall, i.e. just reaching complete separation. Therefore it would be expected that there would be a wide variation in material behavior over a small range of impact velocities. The comparison for shot 3368, shown in Fig. 6e, is for an impact velocity

11

producing some material damage but not sufficient to cause a rebound in particle velocity, indicated by the flattening of the computed velocity and the slightly negative measured values.

4.3. Obserued damage Post impact examination of the graphite/peek specimens was conducted both visually and ultrasonically. Fig. 7 shows photographs of three of the specimens. Shot 3360, impacted at the highest velocity shows two clear spall planes near the rear surface of the specimen. Shot 3357, impacted at 93 m/s, shows one complete separation and one plane where edge cracking is evident. Shot 3362, which was impacted at an almost identical velocity and therefore experiences the same peak stress, shows damage that is much less severe than 3357. There is only a single incomplete spall plane near the rear surface. The contrast between Figs. 7b and 7c gives clear evidence of the time-dependent nature of the evolving damage. Examination of the specimen from shot 3358 showed slight cracks around the edge indicating

35OO

3500

3000

3000

2500

2500

I 2000

'J,, 2000

1500

1500

1000

1000

500

5OO

0 0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

DIMENSIONLESS THICKNESS

DIMENSIONLESS THICKNESS

a.

b.

0.8

Fig. 8. Histogram of crack location with depth from surface for shot 3358: (a) front surface scan; (b) rear surface scan.

J.A. Nemes, P. [~ Randle.s / Mechanics of MateriaLs 19 (1994) /- ]4

12

in a specimen showing no visible damage and one in which the ultrasonic pulse could be transmib ted, indicating little or no internal damage. Thus, the results from this experiment were used to establish the damage threshold parameter, '-"~;o. Further ultrasonic examination was performed to obtain information on the distribution of dam-

incipient development of a spall plane. The specimen from shot 3368 showed no apparent damage under visual examination, but subsequent ultrasonic inspection found that a pulse could not be transmitted through the specimen, indicating substantial internal damage. Shot 3375, which was conducted at the lowest impact velocity resulted (a)

1

(b) 0 g~

0.9:

[

0,8 ~

[

,

07 - ~ - - i ~ ,

0.7-

i ]

i

E

1--t

,

!

! .... ~t-q

uJ 0.6-

0.5~ 0.4

0.4~ 0.30.2-

-

0.1 o,

0-

0

0 0.1 0.2 0.3 0.4 0,5 0.6 0.7 0,13 0.9 1 NORMALIZED THICKNESS

--i--4-

-4--

~ii 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

NORMALIZED THICKNESS

(c) l i

(d)

0.9 =

~

i

i'

'

'

i

i

i

0.8 0.7

,

+

uJ 0.6-:

!

0.5 . . . . .

I

i

<

//

~

]

0,4-

0.3.--.---t-~-4---q---

0.3- - - - -

~ ~-

0.2-! 0.1-

[

,

i o

0 .i 0.1 0.2 0.3 0,4 0.5 0,6 0.7 0.8 0.9 1 NORMALIZED THICKNESS

4,1J,

i1,,

~. . . . r ,

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 NORMALIZED THICKNESS

(e) 1 0.9-

:_

0.7 o

0.~:

i~

'

!

. . . .

<0.40.3 0,2 I ~ L . .

~1 "

0.1 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 NORMALIZED THICKNESS Fig. 9. C o m p u t e d damage profile through the specimen thickness (normalized): (a) shot 3360 b) shot 3357 c) shot 3362 d) shot 3358 e) shot 3368.

J.A. Nemes, P.W. Randles / Mechanics of Materials 19 (1994) 1-14

age through the thickness of the material, particularly for specimens not resulting in complete separation. The front and rear surfaces of the specimens were scanned in a pulse echo mode over a large number of sampling points or pixels ( ~ 26 000). For each pixel the time of arrival of the reflected pulse above a preset threshold was measured and the depth of the reflecting surface (crack or delamination) computed from the wave velocity of the material. A histogram was then constructed by calculating the pixel count, in successive depth intervals from the scan surface, and over a large sample surface area of the scan surface. This resulted in a measure of the density distribution of reflecting area seen by the ultra sonic waves versus sample depth. The histograms for shot 3358 are shown in Fig. 8, to illustrate the variation of damage through the thickness. It should be noted that the presence of cracks near the surface effectively shields the central portion of the sample from examination, preventing complete quantitative examination. The technique, however, still provides valuable information as to the extent of material damage through the thickness.

13

tion. The computed location of both of the planes is in good agreement with the observation. Similar results are found for shot 3357. Experiment 3362 has a predicted single spall plane at 23.98 mm from the front surface, where the specimen shows the plane to be at 23.50 mm. The damage profile computed for shot 3358 shows a narrow plane reaching the maximum damage, which may be indicative of the incipient spall observed in the specimen. The damage profile for shot 3368 shows some damage distributed throughout the thickness of the specimen, which would result in a degradation and variation of stiffness of the material. Whether this decrease would account for a lack of a transmitted ultrasonic pulse requires further study. Comparison of the co~'aputed damage profiles to the histogram in Fig. 8 shows good qualitative agreement concerning the extent of damage toward the front and rear surfaces. It is also interesting to note that although the impact velocity for this shot is lower than that of 3360 or 3357, the level of damage is higher than those two except near the spall planes. Thus, it appears that the higher impact velocities produce well defined spall planes, but lower impact velocities produce more distributed damage.

4.4. Computed damage Since damage in the theory has been treated as a field variable it can be examined spatially to compare the computed results to those observed visually and ultrasonically. Figure 9 shows the computed damage, which ranges from 0 for virgin material to 1 for complete delamination, as a function of the normalized thickness of the specimen (front surface is zero; rear surface is one). Temporally, the distribution is shown at the final computed time increment, which is of sufficient duration that little further damage would occur. Shots 3360, 3357, 3362, and 3358 all show a plane where damage has reached its maximum value, which is in agreement with the observation that an ultrasonic pulse could not pass through the material. Comparison of the computed locations of the spall planes to those observed in the specimens shows several interesting results. As seen in Fig. 7c, shot 3360 resulted in two distinct spall planes, which is also captured in the computa-

5. Conclusion

A material model for predicting delamination type damage and spallation under one-dimensional high rate loading has been presented. The damage model itself requires specification of three parameters, one of which was obtained from finding the threshold from experiments, one of which was set somewhat arbitrarily, and the final one was established iteratively by comparison to the particle velocity profiles measured in the experiment. The resulting model provides a good description of evolving damage under the uniaxial conditions considered, which is a first step toward development of an overall damage model for high rate loading. Comparison of measured particle velocity histories to those predicted by the model show very good agreement over a range of impact velocities and conditions, establishing credibility for the model's capability. In

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J.A. Nemes, P. W. Randles /Mechanics o(Materials 19 (1994) 1-14

addition, the calculation of damage through the thickness agrees very well in a qualitative sense with that observed both visually and ultrasonically. Continuing efforts to quantify the damage and correlate its effects on material behavior are in progress and will be reported at a later date.

Acknowledgements The authors wish to acknowledge the support of the DARPA Naval Technology Office, in particular Mr. Jim Kelly and to acknowledge Dr. Robert Badaliance of the Naval Research Laboratory for support of the experimental work. Gratitude is also expressed to Mr. Larry Lee and Mr. Eric Smith of Ktech Corporation for performing the experiments and providing insight into their interpretation and to Mr. Kirth Simmonds of the Naval Research Laboratory for performing the ultrasonic examination.

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