A uniaxial nonlinear viscoelastic constitutive model with damage for M30 gun propellant

A uniaxial nonlinear viscoelastic constitutive model with damage for M30 gun propellant

Mechamcs of Matermls 15 (1993) 323-335 Elsewer 323 A uniaxial nonlinear viscoelastic constitutive model with damage for M30 gun propellant G A Gazon...

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Mechamcs of Matermls 15 (1993) 323-335 Elsewer

323

A uniaxial nonlinear viscoelastic constitutive model with damage for M30 gun propellant G A Gazonas Army Research Laboratory, Propulston and Fhght Dwtston, Aberdeen Prot'mg Ground MD 21005-5066, USA Received 14 August 1992, revised version received 19 March 1993

T h e n o n h n e a r viscoelastic mechanical response of a conventional tank gun propellant, M30, IS modeled using a "modified superposltlOn integral" that incorporates the effects of mlcrostructural fracture damage Specifically, a linear, time-dependent kernel is convolved with the Ilrst-tlme derlvatwe of a power-law function of stress and a damage 'softening" that accounts for damage evolution by a mlcrocrack growth mechanism The mlcrocrack damage function IS a master curve formed from shifted isothermal, compressive, unlaxml constant strain rate (0 01 s - 1 to 420 s - 1 ) data on solid, right-circular cylinders of M30 gun propellant An attractwe feature of the model is its ability to predict work-softening behavior under conditions of monotomcally increasing deformation Time-dependent predictions of stress versus time, failure stress versus failure time, and failure stress versus strain rate, quantitatively agree with experimental results from constant strain rate tests on the propellant Theoretical predictions of time-dependent stresses for Heavlslde and "balhstlc-hke ' strain histories are also provided

I. Introduction

The unlaxaal compressive mechanical response of M30 gun propellant is well documented over the temperature range from - 4 0 to 60°C, and strata rates from quaslstatlc to 104 s - i using drop weight (Lleb, 1989), spht Hopkmson bar (Lleb et al, 1989), and servohydrauhc (Gazonas, 1991, Gazonas and Ford, 1992) test apparatus Unlaxaal compression of nght-orcular cylinders of M30 reduces fracture damage consisting of axial cracks that grow and eventually coalesce to form macroscop~c conjugate shear fractures at large strains Mlcrograph~c evidence reveals that mlcrocracks initially form between the subayaally oriented nltroguamdlne crystalhtes (energetic filler) and the nitrocellulose (binder) The stress-strata response of M30 is slightly nonhnear prior to the maxamum stress level (failure stress), the propellant subsequently work-softens until ultimate failure or rupture occurs (Gazonas and Ford, 1992) Correspondence to G A Gazonas, Army Research Laboratory, Propulsion and Fhght Dwlslon, Aberdeen Ground, M D 21005-5066, U S A

Proving

Desptte the large body of work that documents the mechanical properties of gun propellants, studies related to their constltutwe characterization are scarce A recent study characterizes the viscoelastic response of M30 propellant and shows that the propellant exhibits nonhnear (strata-dependent) power-law relaxation over the time interval 10 -2 to 10 +4 mllhseconds (Gazonas, 1991) A complete consututwe description of the sohd propellant phase ~s crttlcal for accurate description of combustion In numerical models (Gough, 1990) of the interior balhstlc (IB) process because the rate of mass generation of the gaseous phase during combustion ~s proportional to the amount of exposed propellant surface area (see Techmcal Manual, 1955) Early models of combustion assume that t~me-dependent surface area ~s only a function of the differential changes in the initial propellant geometry caused by deflagratlon These models do not account for an increase m surface area due to deformauon and fracture of the propellant Subsequent IB numerical models incorporate the effect of enhanced mass generatlon rate due to fracture by using surface area "multipliers" (Keller and Horst, 1989) It is antic-

0167-6636/93/$06 00 © 1993 - Elsevier Science Pubhshers B V All rights reserved

~24

G d Gazonas / M30 gun propellant

lpated that the constitutive and damage characterization ol slngle-gralns of propellant will provide insight into the physics governing the bulk deformation of granular propellant beds This paper employs a unlaxial specialization of a general three-dimensional constitutive theory for viscoelastic materials with damage (Schapery, 1981) Several features of the constitutive theory make it an attractive candidate for modehng the constitutive behavior of M30 propellant First, the theory can predict the observed work-softening behavior in M30 under monotonically increasing deformation Mlcrocracking materials that •xh i n t work-softening behavior pose special problems for constitutive modelers since it can be argued that "shear" fracture planes that develop in many materials in compression are structural or geometric features that corrupt detection of the true material response of the material as it work-softens Additional problems associated with a loss of hyperbollclty of the wave equation in damaged, work-softening materials have recently been addressed using a nonlocal elasticity approach (Valanls, 1991) Secondly, the constitutive equations can be transformed to those of nonlinear elastic materials through correspondence principles The transformation facilitates the solution of boundary value problems encountered in the theory of nonlinear viscoelasticity Thirdly, mlcrocracklng in M30 is characterized with a damage function that is related to time-dependent surface area evolution in the propellant This relation could be incorporated into IB numerical codes for more accurate prediction of surface area evolution and mass generation rate during propellant combustion Finally, the theory is general enough to successfully describe the nonlinear viscoelastic response of variety of other materials that include marine sediment (Schapery and Rlgglns, 1982), rocket propellant (Schapery, 1982), and ice (Harper, 1986, 1989) Thus, materlal constitution and damage evolution in a variety ol materials can be compared within the framework of a single theory Even though "material" damage is treated herein as a scalar-valued quantity, predictions of stress versus time, failure stress versus failure time, and failure stress versus strain rate quantl-

tatlvely agree with observations from isothermal, uniaxlal, constant strain rate compression tests on the propellant It is shown that mlcrocrackmg m M30 is characterized by an "elhpsoldal" damage function, formed from shifted isothermal constant strain rate test data The form of the "elllpsoldal" damage function is compared with damage functions that have been developed for other materials such as, Ice in compression (Harper, 1986), and oil-shale under dynamic blast conditions (Grady and Kapp, 1980)

2. The constitutive theory The unlaxlal nonlinear viscoelastic constitutive equation for materials that possess a random or regular distribution of mlcrocracks can be written with a so-called "modified superposltlOn integral" (Schapery, 1981) The notation that follows is essentially that given in Schapery (1989) The unlaxlal strain • in a material stress ~ can be written as z

df

• (t) = ErfoD(t - r ) -d-Tdz

(1)

The integral in (1) is also known as an hereditary Integral or as a convolution of the functions D and f Material nonlinearities and damage are usually incorporated in the function f, where )' = f(~r, S~k )

(2)

In (1), Er is an arbitrary constant referred to as the " r e f e r e n c e " modulus with units of stress Elastic behavior with damage is obtained when D=E71 D(t) is the linear viscoelastic creep compliance if all material nonlinearities are incorporated Into f The creep compliance is defined as the strain response normalized to the unit stress input (1 e , D(t)= •(t)/~r o, with o'(t) = % H ( t ) , and H(~') IS the Heavlslde function defined as, H(sr) = 1 for ~ ' > 0 , and H ( ( ) = 0 for ~ ' < 0 ) In (2), the S~a are k, time-dependent damage parameters that influence the time-dependent strain in (1) The o--subscript in (2) refers to damage parameters developed for stress-history inputs The damage parameters S,k are used

G A Gazonas / M30 gun propellant for strain-history inputs In (2), f IS written In product form with a power-law stress function g l and a damage function g2~ as

f = g l ( 0 " ) g 2 ~ ( S ~ ) = (0"/0"2)'e xs~ sgn(0"), where 02, r and A are positive constants damage p a r a m e t e r S~ is written as,

S~ = £10-/0-, Iqf, dt,

(3) The

(4)

where 0-1 and q are positive constants, and I I denotes the absolute value of the quantity The "crack tip material-related coefficient" fl can depend on time and temperature and material aging effects (Schapery, 1981, Schapery, 1989) For example, extension of the theory to account for thermoviscoelastlc effects involves replacing f l in (4) by the factor ( 1 / a t ) , where a r is the shift factor used in time-temperature superposltlon (Schapery, 1989) In the present analysis f l = 1 for isothermal behavior and material aging effects are not considered A damage p a r a m e t e r similar to (4) is derived by Wnuk and Krlz (1985) by integration of the " K a c h a n o v " equation which relates the rate of damage growth to a power-law function of the net-section stress The underlying physical basis for the constitutive theory with distributed damage is a crackgrowth "law" that relates the speed a of a single crack to a power-law function of the mode I stress intensity factor, a = A K °, where A and Q are empirically determined constants Schapery (1975) arrives at this result by generalizing Barenblatt's (1962) cohesive crack-tip model for linear elastic media to linearly viscoelastic media The crack propagation theory has been experimentally verified for a variety of materials Including polymers (Knauss, 1976, Swanson, 1976, Schapery, 1975) and rocks and minerals (Atklnson and Meredith, 1987) The crack propagation theory is later generalized to encompass nonlinear viscoelastic behavior through the use of the so-called "pseudo-variables" (Schapery, 1981, Schapery, 1984) Crack speed is related to a power-law function of a "generalized" J integral, a = f l J ~ , where k is an empirical constant The generahzed J integral is

325

defined to include a region of crack-tip cohesive forces that are excluded In the more familiar path-independent J integral definition used in analyzing crack growth in elastoplastlc media (Rice, 1968) Others have related crack speed to a power law function of the crack-tip energy release rate, a = A ' G q (Kmiock and Young, 1983), where A' and q are empirical constants A complete derivation that extends the singlecrack theory to describe the global response due to distributed mlcrocracking is found in Schapery (1981) (Eqs (122)-(126)) using the generalized J integral relation, or In Schapery (1989) (Eqs (76)-(80)) using a complementary strain-energy release rate relation Briefly, the derivation involves integration of one of the previously defined crack-speed relations for a single-crack The result is then generalized to distributed damage and assumes that growing mlcrocracks do not affect the response of a macroscopic material element until they have reached their final or "arrested" size Growth-arrest of mlcrocracks occurs when the crack propagation path becomes impeded by mlcrostructural fabric elements such as grain boundaries found m ice (Schapery, 1989) or by rigid particles in highly-filled composites (Schapery, 1981) The arrest of mlcrocracks can also occur through crack interaction and coalescence Crack interaction geometries observed in granitic rocks deformed In creep include an "en echelon" and an " e n passant" type (Carter et al, 1981) The latter type of crack arrest mechanism (crack-crack interaction) dominates in M30 propellant since mlcrocrack growth IS unimpeded by the grain boundaries of the relatively slender (3 to 5 Ixm in diameter) nltroguanldlne crystalhtes A further simplification assumes that a single damage p a r a m e t e r (Eq (4)) is sufficient to account for the effects of mlcrocrack orientation on the resulting deformation response Although the general theory provides for multiple damage parameters (Eq (2)), a significant simplification is realized by using a single damage p a r a m e t e r that "smears-out" the effect of crack orientation One might argue that a one-dimensional model does not assist in understanding the three-dimensional mlcrophyslcal fracture processes that occur in a propellant grain Nevertheless, the slmphficatlon

326

G A Gazonas / M30 gun propellant

appears to be vahd for describing the global conStltutlve behaviors of both Ice and M30 subjected to unlaxlal compression Furthermore, if a sufficient number of material tests are conducted using deformation histories similar to those experienced by the gun propellant during firing, one can circumvent (or at least rely less heavily on) the need for the development of more complex mlcrophyslcal models Regardless of the dlmenslonahty of the proposed model, it should ultimately be reducible to a one-dimensional form that can be verified through experimentation The theory can also presumably account for damage accumulation by mechanisms other than by mlcrocracklng (Schapery, 1989) This ts accomplished by replacing the crack speed relation with a similar rate-relation such as the rate of void growth, rate of bond rupture or an other failure mechanism that may be operative in the material Currently however, there does not appear to exist experimental verification of the theory for deformation mechanisms other than by mlcrocrack growth The theory appears general enough, however, to describe the effects of distributed damage (mlcrocracklng) in a variety of materials including marine sediment, composite, polymers, rocket propellant and ice The functions gl and g2,~ and associated constants may be different for characterizing material behavior in compression versus tension or for unloading after significant plasticity The general validity of the theory can be verified if the above constants and functions, gi and g2~, are unique for a variety of stress-history inputs in (1) The slgnum function in (3) IS defined as sgn(cr) = 1 for compression in this study The damage function g2,~(S,~) in (3) reflects material damage due to mlcrocracks The exponential form of g2,~ was originally proposed by Schapery (1981) and was later used to model microstructural damage in ice subjected to unlaxlal compression (Harper, 1986) Later, it is shown that the exponential damage function with characterizes damage in ice does not satisfactorily characterize damage in M30 propellant Instead, an "elhpsoidal" damage function is utilized that is directly determined from the test data From (3) we see that in the absence of damage, S~ = 0 and g2~ = 1, and (1)

then predicts strain in a nonhnear viscoelastic material without damage Linear viscoelasticity is recovered when r = 1 in (3) Material symmetry changes due to damage induced anlsotropy are not addressed in this paper and damage is treated as a scalar-valued quantity More discussion of this topic and higher-order tensorlal descriptions of damage can be found In the papers of Krajclnovlc (1987) and Weltsman (1988) Equations (1)-(4) are suitable for characterizing damage and the strain response of a material if stress is a controlled Input for the test However, if strain is a controlled input for the test, then these equations must be inverted in order to predict stress as a function of strain history

or(t) =O'2e

01/ r

g 2 , ( S , ) sgn(e°),

(5)

where

g 2 , ( S ~ ) = g 2 ~ ( S a ) - l / r , and the pseudostrain e ° is related to strain history by

dE e ° = E r l i f E ( t - ¢) w - d r , Jo

(6)

where E ( t ) is the relaxation modulus Schapery (1982) points out that the utlhty of pseudo-strain, as a strain measure, lies in that fact that stress versus pseudo-strain plots are single-valued or "elastlc-hke" for cyclically-strained materials However, the e ~ e ° transformation in (6) does not produce a single-valued curve for materials that work-soften The damage parameter S, is obtained by substitution of (5) into the time-derivative of (4) Rearrangement and integration of the result leads to S, = f

S,r q/r g2o- ( S a ) dS,~,

(7)

Jo

or q rt

SE = ( 0"2/0"1) / e a0

0 q

r

fl d t

(8)

The lower limit in (7) is zero, which corresponds to a no-damage condition at time t = 0 If an exponential form g2,~(S~)= e As" is used in (7), then the damage function becomes g2¢(S~) = (1 + A q S J r )

-1/q

(9)

G A Gazonas / M30gun propellant

327

A more general damage function is derived herein (AppendLx), tf one assumes a product form for f as given by (3) Substltutton of the first ttme-dertvattve of (4) mto that of (3) results in a nonhnear dlfferenttal equation (Bernoulli equatton), solvable for stress, whtch can be lmeartzed and mtegrated usmg a power-law function of the pseudo-strain as an mtegratmg factor The identtty,

Substitution of (14) into (5) relation 0-1 = 0 - 2 E r1/r ef provides the time-dependent stress

f = •0,

3. E x p e r i m e n t a l

(10)

ts used m the derlvatton and is obtained tf/~/9 = 1, where /~ and /) are the Carson transformed relaxation modulus and creep comphance, respectively A generalized form of the damage functton is then given as

g2,(S~) = (1 + q S J r ) -l/q,

(11)

where the generalized damage p a r a m e t e r is

.

.

Se ~--[0"2/0"1)

. q f t 0q/,dg2~ f ~ d t J0 e dS~ g2cr

(12)

If g2~(S~) = e xs~ m (12), then dg2~/dS~g2~ = A, and the more general (11) reduces to (9)

2 1 Predtcuons for constant stram-rate compression tests

0-(t) =0-2 ~ - n

g2~(S,) s g n ( • ° ) ,

(16)

where sgn(• °) = 1 for monotonically mcreaslng functtons•o

results

Experimental results for tsothermal unlaxlal compression of M30 at four strain rates appear m Fig 1 Each stress-time curve IS a composite curve formed from the average of five tests The compression tests are performed uttlIzing a servohydrauhc test apparatus (MTS 810 High Rate Test System) that ts described m more detail elsewhere (Gazonas, 1991) The maximum piston veloctty is m the order of 12 m / s (39 4 f / s ) (and thts limits the axial compressive stram rate to 500 s -1 In 25 4 m m (1 inch) long specimens) Constant strain-rate tests are performed by computer control of the ptston velocity via feedback from an externally mounted transducer, hnear-variable differential transformer (LVDT, MTS Model 244 11 Force is measured with a 60 kN (13 5 ×

This section provtdes expressions for predicting stress and damage functions for constant stram rate tests For a constant stram rate • mput,

e(t) = etH(t)

(13)

Substitution of (13) into (6) with the power-law relaxation modulus of M30 determined earher (Gazonas, 1991), E ( t ) = Et t-n, (typically 0 < n < 1), and E l = Er, yields the pseudo-stram e ° ( t ) : at O - m / ( 1

-n)

(14)

"d

A

< 0

.~

*

o

n 0



[] OOl l/s

o

• 09 l/s n 89 l/s A 420 l/s

and for the damage parameter, E Se(t)=

1

"re,- ( l - - n )

)q/rta -~-'

-1

(15)

-8

i -6



n -4



n -2



i 0

,

i 2

4

LOgl0 Time (see)

where a = (1 - n ) q / r + 1 The constant 0-1 ts replaced by the reference stram rate, eref, with the

Fig

1 Axial stress versus hme

a t v a r i o u s s t r a t a r a t e s in M 3 0

gun propellant

G A Gazonas / M30 cun propellant

328

10 ~ lb), quartz-piezoelectric force transducer, Klstler Type 9031A, which is mounted on the upper moving piston Specimen displacements are corrected for apparatus distortion which has a measured stiffness of about 91 9 k N / m m (52 4 × l04 lb/ln) Specimen stiffness ranges from 4 5 to 14 0 k N / m m over the strain rates Investigated Tests are conducted at a room temperature of 22 + 1°C Right-circular cylinders of M30 gun propellant (Radford lot#128 B) are prepared by cutting specimens from sLx-inch, sohd, stick propellant using an Isomet, double-bladed diamond saw The inert lubricant, molybdenum disulfide, MoSz, is sparingly applied to the specimen ends to reduce end friction effects and test variability (Gazonas and Ford, 1992) The chemical composition and nominal specimen dimensions of M30 appear in Table 1

4. Determination of model parameters Constitutive response predictions from the nonhnear theory described above are made by first determining constants, °'2, n, r, eref, and q, and the damage function g2, in (14) through (16) The material constants are determined from a limited set of experimental data (constant strainrate tests) for prediction of material response under more general input histories The damage function and constants are obtained by plotting lOgl0g2,(S ,) in (16) versus log11)S, in (15) Experimental o.(t) data are used in (16) Damage curves for a set of arbitrarily

Table 1 Chemical composmon and nominal speomen dtmenstons of M30 gun propellant Component

(%)

nitrocellulose % NC mtratlon nitroglycerin mtroguamdme ethyl centrahte

28 0 12 6 22 0 48 0 20

length (mm) diameter (mm)

25 4 12 3

n= q= r= 02=

01 10 50 800

e4

o ,-a

-2

*"

[]

[]

.

m

m

0 01



0 9 l/s

R

89 1/s

l/s

A 420 1/s

-3 -8

i

i

i

i

i

-6

-4

-2

0

2

Loglo (Se) F~g 2 Damage functmn versus damage parameter as a )unction of strata rate for arbitrary material constants

chosen constants, o-2, r, n, and q, appear in Fig 2 However, if two curves, taken at the strain rate extremes (001 s -1 and 420 s - I ) are shifted, a "master" damage function is formed with an appropriate choice of constants (determined by trial-and-error, see Fig 3a) which are o"2 = 670 MPa (97 2 × l06 p s i ) , r = 1 2, n = 0 1, q = 9, and ere) = 0 01 S- I The master damage function takes an "elllpsoldal" form (solid line in Fig 3b) in logarlthmm coordinates and represents the "softening" effect of the microcracks, iogl0g2E(S~) = - 1 5 ( 1 - vrl - V ) ,

(17)

where = l°glo(S~)/l°glo(S

....

)

and the mlcrocrack "saturation" loglo(S~max) = 10

level

is at

5. Damage function comparison to other forms The damage function developed hereto for M30 propellant is compared to damage functions used for describing mlcrocrack growth in ice in compression (Harper, 1986) and dynamic blast and fragmentation of oil-shale (Grady and Kapp, 1 9 8 0 ) ( F i g 4a) The damage function for ice is

GA Gazonas /M3Ogun propellant

329

2"

01 q= 90 r=12 if2= 670 n=

~0"

~o

o

.=

tkO

~-1

A 420 l/s -3 ~ -12

-8

-4

0 4 Loglo (SE)

8

-3 -12

(a} 12

. -8

.

. -4

. . 0 4 Loglo (SE)

8

12 {b)

Ftg 3 (a) "Master" damage function versus damage parameter formed from shifted constant strain-rate test data, (b) "elhpsoldal" damage funchon

based on an exponential form and IS unsuitable for M30 propellant since the slope of the function g z , ( S , ) in logarithmic coordinates is linear and proportional to - 1 / q (see (9)) whereas experimental data for M30 in these coordinates are nonlinear (Fig 3a) The damage function for dy-

namlc fragmentation of oil-shale, g2~

=

1

--

(18)

D = 1 - a e m t m+ 3,

is microstructurally derived from a two-parameter Welbull crack distribution function (Grady and KJpp, 1980) and has been successfully used to

10

it

08

o

,~ 0~ ~'06

.= O

.-1

o

~

o

o

-I

04

m

o

02

-6

o

od-shalc at 100 l/s

*

i c c a t l l/s

00 -5

-4

-3

-2

-I

LOgl0 Time (sec)

0

1 (a} 2

-6

-5

-4

-3

-2

-1

LOgl0 Time (see)

0

1

(b)

2

Fig 4 (a) Comparison of damage functions versus time for M30 (symbols), (b) comparison of damage versus time for M30

(symbols) (ice ( + ) (Harper, 1986), and od shale (©) (Grady and IQpp, 1987) )

33(1

G A Gazonas / M30 gun propellant

predict fragment size and the fracture stress dependence on strain rate {r(t) = Bet(1 - D )

(19)

Damage is defined here as a scalar quantity, 0 < D < 1, and D = 0 corresponds to a no-damage condition, whereas D = 1 corresponds to complete material failure In (18), a and m are constants, and in (19), • is the strain rate, t is time, and B is the intrinsic elastic modulus Numerical values for the constants used to plot (9) and (18) m Fig 4a can be found in the original references (Harper, 1986, Grady and Klpp, 1980) The ultimate utility of the damage function lies in its ability to accurately model material behavior, regardless of the method used in ItS determination However, models developed from mlcrostructural considerations are intrinsically more appeahng than those developed from empirical data fits

5 1 Stgmficance of parameter D Fracture surface area A number of authors have attributed mlcrostructural significance to D which is assumed to be proportional to (1) the ratio of the mlcrocrack area to total area, Ac/A, (Schapery, 1981, Lemaltre, 1985), (2) the ratio of defect density to a "saturation" defect density, d/d s, (Rousseher, 1981), (3) the ratio of the radius of a single spherical mlcrocrack to the volume of a representative unit cell, a3/V, (Budlansky and O'Connell, 1976), and (4) statistical distributions of the ratio of the number of broken bounds in " b u n d l e " models to the total number of bonds, n/N, (Bolotln, 1969), to name a few For reasons of relative slmphclty, most mlcrostructural models are developed for cracks which grow normal to an apphed tensile stress field For materials in compression, mixed mode crack growth and interaction considerably complicate both the development of mlcrostructural models and the functional relation between crack speed and stress Intensity factor (Costln, 1987) It is not the intention of this study to develop a mlcrostructural model for mlcroeraek growth in M30 in compression However, D could be expressed in terms of

the ratio of mlcrocrack area to the "saturation" mlcrocrack area, Ac/A~, (motivated by the form of g2, in (17)) Figure 4b compares D versus log l0 time for M30, ice, and oil-shale The timerate of change of D increases with strain rate, l e, dD/dt =f(e) Observations indicate that the amount of damage (fracture surface-area) in M30 propellant is relatively insensitive to strain rate from 10 -2 s -1 to 100 s - I Over this strain rate range, fracture surface-area production IS primarily dependent on the amount of axial specimen strain (1 e , D = g(e)) and secondarily dependent on the deformation temperature (Gazonas et a l , 1991) However, theoretical predictions of fragment size in oil-shale subjected to dynamic loading indicate that fragment size decreases as strain rate mcreases (Grady and Klpp, 1980) Furthermore, fragment size in oil-shale decreases from 0 1 to 0 01 m as strain rate increases In the moderate loading rate regime from 10 s-~ to 100 s-1 The damage function and material constants determined in this study may not appropriately model the constitutive response of M30 propellant under more dynamic loading conditions Additional material data obtained at large, dynamic strain rates are needed to accurately predict material behavior over the wide spectrum of strains and loading rates experienced by the propellant in the gun during firing The next section compares data and constitutive predictions of time-dependent stresses in M30 in order to verify the theory and illustrate its general utility

6. Constant strain-rate c o m p r e s s i o n

Time-dependent stresses for constant strain rate deformation of M30 are predicted by solving (17) for g2,(S,) and substituting the result into (16) Stress versus time predictions are plotted in Fig 5 (solid lines) for the strain rate extremes at which the damage function is defined Experimental data (symbols) are also plotted for comparison with the theoretical predictions Superposed curves at Intermediate strain rates (0 9 s - i and 89 s-1) reflect the predictive capability of the theory s i n c e g2~(S E) IS determined from data

G A Gazonas / M30 gun propellant

331

failure time, is determined by solving (15) for tf subject to condition (21), ~'

tf={S.(ef)[e/(1-n)eref]-q/ra}

2

l/~

(22)

The failure stress is obtained by substitution of (22) into (16) to obtain ._~

1

~r(tf) = o.2e f°'/r(tf) g2,(~f)

(23)

.< & A 0

.a

ram

a

II

0 01 l / s

0

0 9 l/s 89 l/s A

-1

-8

4 2 0 l/s

i

i

i

!

i

-6

-4

-2

0

2

LOgl0 Time (sec) Fig 5 Overlay of observed (symbols) and theoretical (sohd) stress versus t~me curves for M30 gun propellant

shifted from the strain rate extremes The theoretical expression in (16) provides a good approximation to actual stress-time data obtained at constant strain rate A cursory inspection of the superimposed stress versus time curves reveals that the theory accurately predicts maximum or "failure" stresses, failure times, and the worksoftening characteristics of M30 propellant A closed-form expression for the failure time (time at maximum stress level) is obtain by setting the stress-rate in (5) equal to zero ( d o . / d t = 0) i e , do-

- -

dt

~

~0 l/r

dt

]~2 + 1

tfe,

(24)

IS independent of strain rate in materials if r = nq (Harper, 1986) This result is deduced by substituting (22) into (24) with r = nq The strain-rate independence of failure strain is also predicted by the theory m materials if r = 1 and n << 1, as in linear elastm materials with damage Both con-

1000 - 092 of = 46 t f

+ g 2 ~ - -

dt

0

(20)

Using the chain rule of differentiation for the first term I n ( 2 0 ) , d ( g E ~ ) / d t = d ( g E ~ ) / d S ~ ( d S ~ / dt), and after some algebraic manipulation, the principal result is that at the maxamum stress, ~f = ~ at failure = constant, 1 e , b~f=

ef =

d(e °1/')

d(g2~ ) - -

The log fadure stress versus log failure t~me is plotted in Fig 6 The predmtlons (solid symbols) compare well with observed failure stresses and observed failure times (open symbols) The failure stress is also observed to monotonically increase with strain rate (open symbols in Fig 7) The monotonic increase m failure stress with strain rate IS also predicted by (23) (solid symbols) and the weak strain rate dependence is given by o.f = 59 e °°93 The fadure stress is insensitive to strain rate ff n << 1 In (22) and (23) The theory predicts that the strain at failure,

0 542,

eL

100

(21)

ar/[lOglo(S, ma~)(1 - n)] Since ~: = loglo(S~)/logm(S, max), the damage p a r a m e t e r at

where fl = 1 5

the failure stress is constant, i e , S,(~:f) = 10542 The time required to reach maximum stress, or

• p~c~d I-1 observed 10

........

0001

I

001

.......

,i

01

........

!

1

........

1

1

Time to Failure (see) Fig 6 Stress at fadure versus t~me to failure

10

G A Gazona~ / M30 gun propellant

3~2

Cr =

59 e

093

1

2"

t~_

I(E)"

/

/

.<

/

/

10

........

01

i

........

l

i

........

i

........

1 10 Strain Rate (l/sec)

i

100

A

0

prechcted [] observed •

II D

........

1000

-1

-8

[]

001

'~

091/s

1/s

I

89 l/s

A

420 l/s

i

i

i

i

i

-6

-4

-2

0

2

4

LOgl0 Time (see)

Fig 7 Stress at f a d u r e v e r s u s s t r a i n r a t e

Fig 8 P r e d i c t e d stress v e r s u s t~me f o r t w o - s t e p i n p u t h i s t o r y c o n s t a n t s t r a i n r a t e f o l l o w e d by c o n s t a n t s t r a i n

stralnt conditions are approximately true for M30 material constants, and this may explain why measured yield strains (approximately equal to failure strains) in M30 are Insensitive to strain rate (Gazonas, 1991, Gazonas and Ford, 1992) In addition to being strain-rate independent, the failure strain is seen to take on a constant value since the slope of failure stress versus failure time curve (Fig 6) and the slope failure stress versus strain rate curve (Fig 7) are equal in magnitude, but

opposite

dependent stress predictions for stress-relaxation of M30 after a period of constant strain rate straining appears in Ftg 8, and (2) a "ballisticlike" input, simulated by a concave-up, two-ramp, input history, with the strain rate in the second ramp and order of magnitude greater than the strain rate in the first ramp (Fig 9) The macroscoplc VlSCOplastlc behavior of the propellant can

in sign

6 1 More compler input histories Closed-form analytical expressions for timedependent stresses are determined if the product of the relaxation modulus and the strain history in (6) is an lntegrable function Stram histories of arbitrary complexity can be approximated using a large number of constant strain-rate ramp functions As an illustration, an input history that consists of two, successive, constant strain-rate ramps is expressed by

E(t) =e,tH(t) + ( e 2 - e l ) t H ( t - t t ) ,

2

m

1

A

II

D []

.< O

,,d

A

0

II

001

(25)

where ej and e: are the constant strain rates of each ramp and t~ IS the time of application of strain rate e 2 Time-dependent stresses are determined by substituting (25) into (6) Two examples that illustrate such ramp inputs include (1) time-

-I

-8

l/s



09

l/s

[]

89 l/s

~.

420 l/s

i

i

|

i

i

-6

-4

-2

0

2

4

LOgl0 Time (see) Fig 9 P r e d i c t e d stress v e r s u s t i m e f o r " b a l h s t l c - h k e " I n p u t history, c o n s t a n t s t r a t a r a t e f o l l o w e d by o r d e r - o f - m a g n i t u d e i n c r e a s e in s t r a i n r a t e

G A Gazonas / M30 gun propellant

be characterized by performing cyclic (sawtooth) compression tests with input histories that consist of a variety of loading and unloading strain rates Information such as the slope of the unloading curve and the amount of permanent (plastic) strain can be used to develop a more general viscoplastic model with damage A viscoplastlc model was not developed because of experimental difficulties associated with cyclic testing of right-circular cylinders of soft propellant Platen separation from the specimen occurs during unloading, ff the unloading rate is more rapid than the specimen relaxation rate Furthermore, the unidirectional h~gh-rate compresswe behavior of the propellant is more relevant to balhstics problems than is the cyclic fatigue behavior of the propellant A comparison of the constitutwe response of M30 under a variety of input histories, such as those gwen in this section, will provide a more general verification of the theory Experimental programs should include input histories that are similar to those experienced by the propellant in the gun durmg firing

7. Concluding remarks A nonlinear theory of viscoelasticity with damage has been shown to accurately model the constttUtlVe response of M30 gun propellant in uniaxial, isothermal compression The exponential damage function e ~s~ originally proposed by Schapery (1981) and used to describe uniaxaal deformation and failure in ice (Harper, 1986) was found unsuitable for describing deformation and failure in M30 propellant Instead, an "elhpsoldal" damage function was determined, directly from the data, which accurately predicted worksoftenIng behavior under monotonically Increasing deformation In ad&tlon, time-dependent predictions of stress versus time, failure stress versus failure time, and failure stress versus strain rate, quantitatively agree with experimental results from constant strum rate tests on the propellant The observed insensitivity of failure strains to strain rate in M30 (Gazonas, 1991) is a result that is also derivable from the theory Fu-

333

ture work ts planned to verify the generality of the constltuttve model under more complex input histories such as those described in the prevtous section However, significant plastic deformation that is observed in the propellant after unloading may pose difficulties for the model An extension of the model to include temperature-dependent behavior with reduced time variables is currently in progress References Atkmson, B K, ed, and P G Meredith (1987), The theory of subcrltlcal crack growth with applications to minerals and rocks, in Fracture Mechamcs of Rock, Academic Press London, pp 111-166 Barenblatt, G I (1962), The mathematical theory of equlhbrlum cracks In brittle fracture, Advances m Apphed Mechamcs, Vol VII, Academic Press, New York, pp 55-129 Bolotm, V V (1969), In J J Brandstatter, ed, Stanstical Methods m Structural Mechanics, Holden-Day, Inc, San Francisco Budlansky, B and R J O'Connell (1976), Elastic moduh of a cracked solid Int J Sohds Struct 12, 81-97 Carter, N L, D A Anderson, F D Hansen and R L Kranz (1981), Creep and creep rupture of granitic rocks, in Geophysical Monograph 24, Mechamcal Behavior of Crystal Rocks, The Handin Volume, AGU, Washington, DC Costln, L S (1987), Time-dependent deformation and failure, in B K Atkmson, ed, Fracture Mechamcs of Rock, Academic Press, London, pp 167-215 Departments of the Army and the Air Force Technical Manual No 9-1910, Mthtary Exploswes, Technical Order No 11A-1-34, US Government Printing Office, Washington, DC, 1955 Gazonas, G A (1991), The mechamcal response of M30, XM39, and JA2 propellants at strain rates from 10 -z to 250 s -1, BRL-TR-3181, US Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD Gazonas, G A , A Juhasz and J C Ford (1991), Strain rate insensitivity of damage-induced surface area in M30 and JA2 gun propellants, BRL-TR-3251, US Army Ballistic Research Laboratory, Aberdeen Proving Ground. MD Gazonas, G A and J C Ford (1992), Unlaxlal compression testing of M30 and JA2 gun propellants using a statistical design strategy, Exp Mech 32, 154-162 Gough, P S (1990), The XNOVAKTC Code, BRL-CR-627, US Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD GradL D E and M E Kapp (1980), Continuum modelling of exploswe fracture in oll shale, Int J Rock Mech Miner Sct Geomech Abstr 17, 147-157 Harper, B D (1986), A unlaxtal nonlinear viscoelastic constitutive relation for ice, J Energy Resources Technol 108, 156-160

334

G A Gazonas / M30 gun propellant

Harper, B D (1989), Some implications of a nonhnear viscoelastic constitutive theory regarding mterrelanonsh~ps between creep and strength behavior of ice at failure, J Offshore Mech Arctic Eng 111, 144-148 Keller, G E and A W Horst (1989), The effects of propellant grain fracture on the interior balhstlcs of guns, BRL-MR3766, US Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD Klnlock, A J and R J Young (1983), Fracture BehaL,tor of Polymers, Applied Science, Barking, England Knauss, W G (1976), Fracture of sohds possessing deformation rate sensmve material properties, The Mechamcs of Fracture, AMD-Vol 19, ASME Krajcmovlc, D (1987), Mlcromechamcal basis of phenomenological models, in eds, D Krajcmovtc and J Lemmtre Continuum Damage Mechamcs Theory and Apphcattons, Springer, pp 195-206 Lemmtre, J (1985), A continuous damage mechamcs model for ductde fracture J Eng Mater Technol 107, 83-89 Lleb R J (1989), The mechanical response of M30 JA2 and XM39 gun propellants to high rate deformation, BRL-TR3023, US Army Ballistic Research Laboratory, Aberdeen Prowng Ground, MD Lleb, R J, T J Fischer and H J Hoffman (1989), High strain rate response of gun propellant using the Hopklnson spht bar, presented at the JANNAFJomt S&MBS/CMCS Meetmg JPL, Pasadena, CA Rice, J R (1968), A path independent mtegral and the approximate analysis of strain concentration by notches and cracks, J Appl Mech 35, 379-386 Rousseher, J (1981), m S Nemat-Nasser, ed, Proc IUTAM Symp on 30 Constttuttt,e Relations and Ductile Fracture, North Holland, pp 331-335 Schapery, R A (1975), A theory of crack mltmtlon and growth In viscoelastic media I, II, and III Int J Fract 11 141-159, 369-388, and 549-562 Schapery, R A (1981), On viscoelastic deformation and fadure behawor of composite materials with distributed flaws, in S S Wang and W L Renton, eds, 1981 AdL'ances m Aerospace Structures and Materials, ASME, AD-01, New York, pp 5-20 Schapery, R A (1982), Models for damage growth and fracture m nonlinear wscoelastlc particulate composites, Proc Ninth US National Congress of Apphed Mechanics ASME, pp 237-245 Schapery, R A and M RIggms (1982), Development of cychc non-hnear viscoelastic constitutive equatmns for marine sediment, in R Dungar, G N Pande and J A Studer, Int Syrup on Numerical Models m Geomechamcs, Zurich, pp 172-182 Schapery, R A (1984), Correspondence principles and a generahzed J integral for large deformanon fracture analysis of wscoelastlc medm, lnt J Fract 25, 195-223 Schapery, R A (1989), Models for the deformation of viscoelastic medm w~th distributed damage and their appllcabdlty to Ice, Proc I U T A M / I A H R Symp on Ice/Structure Interaction, Memorial University of Newfoundland, St John's

Swanson, S R (1976), Apphcatxon of Schapery's theory of viscoelastic fracture to solid propelant, J Spacecraft Rockets 13, 528-533 Valanls, K C (1991), A global damage theory and the hyperbohclty of the wave equation, J Appl Mech 58, 311-316 Weltsman Y (1988), A continuum damage model for viscoelastic materials, J Appl Mech 55, 773-780 Wnuk, M P and R D Knz (1985), CDM model of damage in laminated composites, Int J Fract 28, 121-128

Appendix - Derivation of generalized function and damage parameter

damage

The derwatlon of the generahzed damage f u n c t i o n a n d p a r a m e t e r g w e n by (11) a n d (12) m t h e m a m text b e g i n s by t a k i n g t h e first t i m e derwatwe (dotted quantity) of the function f m (3), (A1)

f = gig2 + gzgl

a n d c o l l e c t i n g t e r m s to o b t a i n , f

g2

gl + -g2 gl

-

f

(A2)

Since gl = g l (O') and g2 = g2(S,,), derwatwes of these quantmes are,

ordinary

d g 1 dcr gl

dcr

dt

(A3)

and d g 2 dS,, g2-

dS,,

dt

(A4)

S u b s t i t u t i o n o f ( A 3 ) a n d ( A 4 ) a n d t h e first t i m e d e r l v a t w e o f t h e d a m a g e p a r m e t e r S,~ g i v e n in (4) f r o m t h e m a i n text i n t o ( A 2 ) y i e l d s t h e f o l l o w i n g n o n h n e a r o r d i n a r y d i f f e r e n t i a l e q u a t i o n in o-, o f Bernoulh form, f of

-

dg2 °q+l

fl

dS,,

g2

+~r, 0 "q

where, fl =fl(t) using, u = 1~/~21 -q

(A5)

A change of variables m (A5)

(A6)

G A Gazonas / M30 gun propellant

335

hence

and _qo--(q+ U

0.2q

1)0 -

(A7)

,

results in the following linear ordinary dlfferentim equation,

u +p(t)u = h(t),

p ( t ) = _+ee<')= +_eClfl ql"

For c = 0 in ( A l l ) , a solution to (A8) is obtained for u given by,

(A8)

r

where

fq p(t) = 77

and

(All)

dS~ g2

dt + c o (A12)

h(t)=q((r2] adgefl r ~,ori

t 0"1 ]

] dScr g2 (A9)

An integrating factor p(t) for (A8) is obtained by letting,

d'=q-lnlfl+c'r (AIO)

Substitution of (A6) into (A12), using the identity (10), f = ~0, from the main text, and letting c o = 1, one arrives at the generalized damage function (11) and damage parameter (12).