H.-C. Lin
1 Components Technology Division, Argonne National Laboratory, Argonne, ill.H.-C. Wu
Assistant Professor, Division of Materials Engineering, The University of Iowa, Iowa City, IowaStrain-Rate Effect in the
Endochronic Theory of
Viscoplasticity
An explicit function of material parameter 0i is proposed to accommodate the rate effect in the endochronic theory of viscoplasticity developed by Valanis. The strain-rate effects are treated in a manner that includes strain-strain-rate history dependence. The theoretical stress-strain curves at constant strain rates are presented and compared with the existing experimental data. Based on this constitutive equation, the solution of one-dimensional plastic wave propagation in thin rods is obtained for annealed alumi-num and copper. The theoretical strain-time profiles are in qualitative and quantitative agreement with the experimental results. It has been shown theoretically that the con-sideration of the strain-rate effect in the endochronic theory lowers the final constant strain states in the strain-time profiles.
Introduction
Experiments on plastic wave propagation in thin rods or tubes (Bell [1-4] ,2 Kolsky and Douch [5], Dillon [6], Efron and Malvern [7], Yew and Richardson [8], Hsu and Clifton [9]),3 have indicated the effect of strain rate. A main conclusion from those experiments is that the mechanical behavior of strain-rate sensitive materials cannot be described by a single stress-strain curve, but may be ex-plained by a set of curves each corresponding to a specific strain rate. During the past two decades, various constitutive equations have been proposed to accommodate the strain-rate effect. Most of them, based on the observed phenomenon, have assumed that the plastic rate of strain is a function of the dynamic overstress (Mal-vern [11], Sokolovskii [12], Perzyna [13], etc.). Cristescu [10, 14] has subsequently generalized the idea into a full quasi-linear con-stitutive equation as follows:
— = * ( o , e ) — + * ( < r , e)
at at (1)
1 Formerly, Division of Materials Engineering, The University of Iowa, Iowa City, Iowa.
2 Numbers in brackets designate References at. end of paper.
3 An extensive literature survey on the experimental evidence of the strain-rate effect may be found in the book by Cristescu [10].
Contributed by the Applied Mechanics Division for publication in the JOURNAL OP APPLIED MECHANICS.
Discussion on this paper should be addressed to the Editorial Depart-ment, ASME, United Engineering Center, 345 East 47th Street, New York, N. Y. 10017, and will be accepted until June 1,1976. Readers who need more time to prepare a Discussion should request an extension of the deadline from the Editorial Department. Manuscript received by ASME Applied Me-chanics Division, November, 1974; final revision, July, 1975. Paper No. 76-APM-G.
where the functions * and <6 represent, respectively, the nonins-tantaneous and the insnonins-tantaneous response of the material to the increment of stress a; t is the small strain; and t is the time. How-ever, this equation does not account for the dependence of the ma-terial behavior on the strain-rate history. Recently, Klepaczko [15, 16] has shown that the strain-rate history effect can play a very important role in the wave propagation problems.
All the aforementioned works are based on the flow theory of plasticity. This theory inherits the uncertainties from the defini-tion of a yield surface and also from the proposidefini-tion of a strain-hardening law. In the previous papers [17, 18], the authors applied a theory of viscoplasticity—the endochronic theory developed by Valanis [19]—to obtain simple wave solutions of a thin-walled tube subjected to a combined longitudinal and torsional step loading, without considering the strain-rate effect. The constant state gion between slow and fast waves which would be expected as a re-sult of the flow theory,4 was not apparent in the authors' theoreti-cal analysis. This result was in good agreement with the experi-ment [21]. Furthermore, in [18], the authors proposed a method of extending the domain of simple wave solutions into the fourth quadrant of the stress space, which was totally different from the concept based on the flow theory. There was, however, a slight quantitative difference between theoretical and experimental re-sults, i.e., the predicted final strain states were consistently higher than the experimental values. It is believed that the aforemen-tioned discrepancy may be accommodated by taking into account the influence of strain rate in the endochronic theory of viscoplast-icity.
In this paper, an explicit function of material parameter /?i
'' Banerjee and Malvern [20] have considered the strain-rate effect using the flow theory. However, the constant state region still clearly appeared.
which appears in the endochronic theory, is proposed based on the experimental observation. This proposal makes the constitutive equation depend not only on the deformation history but also on the strain-rate history. The theoretical stress-strain relations for various constant strain rates (in the range of 10~4 to 103 per sec) are then presented and compared with the experimental data of Karnes and Ripperger [2] for annealed aluminum. Based on this constitutive equation, the solution of plastic wave propagation in thin rods is obtained. Several numerical examples for annealed aluminum and copper are presented together with their compari-sons with the experimental results of Bell [2-4].
Time-Dependency in the Endochronic Theory
In the endochronic theory of viscoplasticity [19], it is proposed that the time scale is not the absolute time scale measured by a clock, but is a material property itself. According to this theory, the current stress is determined by the memory of a material with respect to the endochronic time. Based on the fact that materials are, in general, history as well as strain-rate dependent, a time scale df is now proposed and given as5
d f = K i M ) H
(2)where K\ is a general function of e and e to be determined. For sim-plicity, it is assumed that K\ is a function of e only, i.e.,
df = M e ) H
(3)Then, this time scale has the same form as that of the case of the strain-rate independent uniaxial stress, except that k\ is now a function of strain rate.
From the constitutive equation and the associated relations given by Valanis [19]
-s:
E(z -z') — dz' dz' • l o g ( l + j8f) 2 0 0 150 100 5 0 1 1 1 1 1 1 1° EXP. DATA FROM KARNES AND RIPPERGER V THEORETICAL CURVE: - X ^ /9, = 2 l 6 - 2 5 . 2 5 LOG (£/<=,,)
\ .
1~
-Fig. 1 /3, as a function of strain rate
EXP. CURVES FROM KARNES AND RIPPERGER THEORETICAL
(4)
(5) Fig. 2 Constant strain-rate stress-strain curves for annealed aluminum
and
E(z) = Eoe-"' (6) it is easily obtained, under the condition of constant strain rate,
Eo where and •(1 + Uf) 1
-ft« I (i + m
n n = 1 +-m) = pkdi)
ai(e) = ctki(k)The material parameters have the following relations:
EQ n = — (7) (8) (9) (10) (11)
5 The endochronic time scale was originally defined by Valanis as dp = kHt2 + g2dt2
for the problem of uniaxial stress, where k and g are material parameters, k may be a function of e, and t is the real time. Since the foregoing equation can be written as
dr = K|d6|;K = V k'i + ~
and since for each internal variable there may exist a corresponding endo-chronic time scale, equation (2) is a generalization of the previous expres-sion. The authors are grateful to Prof. K. C. Valanis for his discussion about this point [23],
01 Et (12)
where a and (1 are constants; Eo is the initial slope of the quasi-static stress-strain curve; Et is the tangent modulus of the
asymp-totic straight line of the static stress-strain curve at large e; and OQ is the intercept of this asymptotic line on the stress axis. Equation (4) together with equations (5) and (6) may be combined and writ-ten in the differential form as follows:
-+
da E0 =-af 1 + W «f
which then, using equation (3), reduces to
V l + flf/
(13)
(14) Thus, for each strain rate, there corresponds a stress-strain curve. The experimental data of Karnes and Ripperger [22] for annealed aluminum show that the stress-strain relations at different strain rates are almost parallel6 to each other at least in the range of e < 2 percent for this specific material, i.e., Et remains constant for all strain rates. Based on these experimental curves and equation (12), Fig. 1 has been constructed and the data points are nicely fit-ted by the expression
= da ~ ft, log ( - ) \eo'
(15)
6 For the nonparallel case, reported by Hauser, et al. [24], this theory is still vaiid with the modification that Et may depend on the strain rate, so
Q C
0 x
Fig. 3 Characteristic diagram and Integration scheme
10
x 6
/ / -
BELL'S QUASI-STATIC 0" = 3.32 x I 04 €3 / 8 ENDOCHRONIC QUASI-STATICBELL'S DYNAMIC O" = 5.6 x I 04 €l / z
J_
0.5 1.0 1.5 2.0 2.5
e (%)
Fig. 4 Stress-strain curves for annealed aluminum 6 0
4 0
2 0
3.0
where ft, and fib are material parameters, and to is the strain rate under quasi-static condition (taken as 1.5 X 1 0- 4 per second in the present case). The result of the theoretical curves using equation (7) and the experimental data from [22] are shown in Fig. 2. The comparison is obviously good. Note t h a t for /3t, —- 0, fti —• ft, = constant. T h e rate-independent material parameter may thus be recovered. Furthermore, fti —• fia when e —- ko- Therefore, ft, can be
found from the quasi-static stress-strain curve, and ft; can be in-terpreted as a parameter of the rate-sensitivity. In view of equa-tions (14) and (15), it is seen that the stress is dependent not only on the strain history but also on the strain-rate history.
Viscoplastic Wave Propagation in a Thin Rod
To show the influence of strain rate on the dynamic material re-sponse, the problem of longitudinal wave propagation in thin rods will be considered as an example. The constitutive equation for the uniaxial stress with strain rate effect as derived in the previous section can be written as
[
« i ( c ) f f lE0~—-L = 0
ftf-i
T h e equation of motion and the compatibility condition are ax + pUt = 0
(16)
(17) (18) where x is the distance from the impact end of the rod to the cross section under consideration, a and e are the longitudinal stress and strain, respectively, and U is the longitudinal particle velocity. Compressive stress, compressive strain, and velocity in the positive x -direction are taken to be positive. Subscripts x and t denote the partial differentiation with respect to the corresponding variable.
Equations (16)-(18) constitute a system of equations for five functions: a, e, V', ki, and f. This system can also be exhibited as a system of first-order quasi-linear partial differential equations if one introduces the strain rate as a sixth function. Numerical solu-tions were obtained by means of the predictor-corrector difference method using a rectangular mesh. During the calculation, the strain rate was considered constant for each small time increment, and the following characteristics were used to obtain the numerical solution for the system of equations (16)-(18):
dx . I\ i a\a \ dt V p \ 1 + ftf/ and (19) (20) dx = 0
The corresponding differential equations along the characteristics for this numerical procedure are
dx
and
da — pc2dt = 0; on dx = 0 (22)
The upper and lower signs in equations (21) correspond to each other.
The solution at a mesh point at time t has been obtained by in-tegration of incremental relations (21) and (22) along characteris-tics based on the solution at neighboring mesh points at time t — At. Fig. 3 shows an element of a finite size network in the x-t plane. In the predictor step, all values were elevated at time t — At. Points P and Q were found by linear interpolation between points B and A, and A and C. Once the position of P and Q have been determined, the values of a, U, and c at these points could be found from interpolations. Thereafter, the conventional procedure applied to find the solution at point R.
The correction step was performed based on the mean of the value at R computed from the predictor step and the solution at time step t — At. Thus the corrector position of P and Q was de-termined. The same procedures as in the predictor step applied to complete the calculation except that the value of c in equations (21) and (22) was replaced by the mean of its value at point R from the predictor step and the corresponding value at point P, Q, or A. For more accurate solution, the corrector procedure was repeated. Note that the finite grid size was chosen such that the Courant, et al., stability criterion |c| At/Ax < 1 was satisfied.
The velocity boundary condition was assumed to be of a smooth transition consisting of two parabolic segments during the period of rise-time similar to that used by Hsu and Clifton [9], i.e.,
1/(0, t)
= 0.
5[/
0[l
+( ^ - l ) ( 2 - | l - l | ) ]
(23)da ± pcdU = 0; on — = ±c
dt (21)
where C/o is the prescribed constant value of U after the rise-time, and th is one half of the rise-time. Here, the rise-time was taken to be 10 ixsec.
The present theoretical predictions are compared with the ex-perimental data obtained by Bell [2-4] for annealed aluminum and copper. Fig. 4 shows the stress-strain relations of annealed alumi-num given by Bell [2, 14] and the theoretical quasi-static stress-strain curve used in this paper. From this curve all material pa-rameters, except the strain-rate sensitivity parameter ft,, have been determined (ft, = 68.75). The value of fib must be determined from a set of dynamic stress-strain curves which have, however, not been provided by Bell's papers. In the present case, it is as-sumed that fti = 3.8. The strain-time profiles of annealed alumi-num for impact velocity t/o equal to 800 ips (20.32 m/s) is shown in Fig. 5. Results based on the rate-dependent and rate-independent theories are presented together with the experimental data given by Bell [4], The comparison between the rate-dependent theory and experiment is apparently better than that involving the rate-independent theory. The rate effect on the strain-time profiles can
2.5 BELL'S EXP. RATEDEP. -RATE-INDEP. 100 150 TIME (ilsec)
Fig. 5 Strain-time profiles for annealed aluminum (U0 = 800 ips, or 20.32
m/s)
100 150 TIME i/lsec)
Fig. 6 Stress-time profiles for annealed aluminum (U0 = 800 ips, or 20.32 m/s)
100 150 TIME (/isec)
Fig. 7 Velocity-time profiles for annealed aluminum (UB = 800 ips, or
20.32 m/s) 12 10 £ 6 fc 4
1
"
—
X
X
1 / ^ 'V -
1 r i 1' > "
y/
y ^
/ ' i y / -•'y/ --/
<r J * * j ^ S j S END0CHR0NIC DYNAMIC, £b = 5.0 -ENDOCHR0NIC QUASI-STATIC BELL'S PARABOLA cr = 6.48 x i o4el / 2 1 1 1 80 60 40 | 20 0.5 1.0 1.5 2.0 6 (%)Fig. 8 Stress-strain curves for annealed copper 2.5
easily be detected, i.e., the final strain levels are always lower in the rate-dependent theory than in the rate-independent theory. It is to be noted, however, that the experimental first arrival time at x = 3.937 in. (10 cm) in Fig. 5 is different from the value calculated from the conventional "elastic" modulus E. As indicated by Bell [4], the elastic moduli may not have the same values for different batches of materials. Consequently, the first arrival time may vary from experiment to experiment. Figs. 6 and 7 show the corre-sponding stress and velocity-time profiles at several locations. In the illustrations, the solid lines represent the results of rate-depen-dent theory while the dotted lines those of the rate-indepenrate-depen-dent theory. The final stress states are higher in the rate-dependent so-lution than in the rate-independent theory as one may expect from the results of the strain-time profile. Furthermore, Figs. 6 and 7 show that the stress and the velocity decay with the distance x. This is also expected since the material considered is dissipative.
Fig. 8 shows the stress-strain curves from the annealed copper (with /3a = 88.125). The corresponding strain-time profiles for Ua = 472.44 ips (12 m/s) at x = 2 in. (5.08 cm) are shown in Fig. 9. The "endochronic dynamic" curve shown in Fig. 8 is the resulting stress-strain relation arising from this impact. The strain rate dur-ing the impact varies although it remains about constant shortly after the beginning of the impact. The same features for these two fee metals may be concluded as far as the wave phenomenon is concerned. 2.5 2 . 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 - ANNEALED COPPER, 99.9 u0 = I200cm/sec ~ x = 5.08cm , - o
-BELL'S EXP. / - R A T E - D E P . / - RATE-INDEP. / / Ji 7' Jl ••• 7 ' J i li y ' J i 7 1 j i 7 ' P > / t t p ^ ' i | I i r % ,-'" / / no Q-1 ^ 1 1 1 1 1 1-"
-1 1.0 0.5 O 5 0 IOO 150 TIME ( t i s e c )Fig. 9 Strain-time profiles for annealed copper (U0 =
m/s)
2 0 0
I t is i n t e r e s t i n g t o n o t e t h a t t h e a s y m p t o t i c a p p r o a c h of t h e s t r a i n - t i m e profiles t o t h e final c o n s t a n t s t a t e s is p r e d i c t e d b y b o t h t h e r a t e - d e p e n d e n t a n d t h e r a t e - i n d e p e n d e n t t h e o r i e s . I n t h e a n a l y s i s of a t h i n - w a l l e d t u b e s u b j e c t e d t o c o m b i n e d l o n g i t u d i n a l a n d t o r s i o n a l l o a d i n g a s r e p o r t e d i n [17, 18], s h a r p c h a n g e s of s l o p e w e r e o b t a i n e d for t h e s t r a i n t i m e profiles close t o t h e final cons t a n t cons t a t e region. T h e d i cons c r e p a n c y icons p r i m a r i l y d u e t o t h e r i cons e -t i m e effec-t. W h i l e i n -t h i s p a p e r , a fini-te r i s e - -t i m e is a s s u m e d w i -t h velocity b o u n d a r y c o n d i t i o n satisfying e q u a t i o n (23), in t h e p r e v i -ous a n a l y s i s [17,18] t h e r i s e - t i m e w a s n e g l e c t e d .
Concluding Remarks
F r o m t h e r e s u l t s p r e v i o u s l y p r e s e n t e d , i t is seen t h a t b y c o n s i d e r i n g t h e s t r a i n r a t e effect in t h e e n d o c h r o n i c t h e o r y of v i s c o p l a s t icity, it is p o s s i b l e t o d e s c r i b e t h e s t r e s s s t r a i n r e l a t i o n s u n d e r d y -n a m i c l o a d i -n g c o -n d i t i o -n s w i t h i -n c e r t a i -n a c c u r a c y . T h e r e s u l t i -n g s t r a i n t i m e profiles of t h e p l a s t i c w a v e p r o p a g a t i o n a r e in q u a l i t a -t i v e a n d q u a n -t i -t a -t i v e a g r e e m e n -t w i -t h -t h e e x p e r i m e n -t a l r e s u l -t s . T h e lowering of t h e final s t r a i n s t a t e s is a t t r i b u t e d t o t h e i n f l u e n c e of s t r a i n r a t e . I t is w o r t h w h i l e t o n o t e t h a t t h e g o v e r n i n g e q u a t i o n s in t h e s t r a i n - r a t e - d e p e n d e n t t h e o r y a r e e x a c t l y t h e s a m e a s t h o s e u s e d in t h e s t r a i n - r a t e - i n d e p e n d e n t t h e o r y [17, 18] of w a v e p r o p a g a t i o n , w i t h t h e e x c e p t i o n t h a t t h e m a t e r i a l p a r a m e t e r s i n v o l v e d i n t h e r a t e - d e p e n d e n t t h e o r y a r e d e p e n d e n t o n t h e s t r a i n r a t e . T h u s t h e r a t e - i n d e p e n d e n t s o l u t i o n of w a v e p r o p a g a t i o n p r o b l e m s c a n b e o b t a i n e d f r o m t h e r a t e d e p e n d e n t t h e o r y b y s i m p l y s e t t i n g m a t e r i al p a r a m e t e r /fc = 0. T h e r e f o r e t h e t h e o r y of p l a s t i c w a v e p r o p a g a t i o n in t h e e n d o c h r o n i c t h e o r y of v i s c o p l a s t i c i t y is u n i f i e d i r r e s p e c -t i v e of -t h e s -t r a i n r a -t e . F i n a l l y , it is t o b e r e m a r k e d t h a t in t h i s o n e - d i m e n s i o n a l c a s e of p l a s t i c w a v e p r o p a g a t i o n , t h e p r e s e n t r a t e - i n d e p e n d e n t s o l u t i o n is t h e s a m e a s t h a t o b t a i n e d b y t h e K a r m a n - T a y l o r t h e o r y . A c k n o w l e d g m e n t T h e a u t h o r s w i s h t o e x p r e s s t h e i r a p p r e c i a t i o n t o Prof. K . C. V a -lanis for v a l u a b l e d i s c u s s i o n s . T h i s r e s e a r c h was s u p p o r t e d b y A r m y R e s e a r c h Office, D u r h a m , N . C , t h r o u g h A r m y W e a p o n s C o m m a n d , R o c k I s l a n d A r s e n a l , R o c k I s l a n d , 111.R e f e r e n c e s
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15 Klepaczko, J., "Strain-Rate History Effects for Polycrystalline Alu-minum and Theory of Intersections," Journal of the Mechanics and Physics of Solids, Vol. 16,1968, pp. 255-266.
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20 Banerjee, A. K., and Malvern, L. E., "Plastic Waves of Combined Stress From Longitudinal Impact of a Pretorqued Tube, Analysis by Rate-Dependent Theory," Research Report, Department of Engineering Science, Mechanics and Aerospace Engineering, University of Florida, 1973.
21 Lipkin, J., and Clifton, R. J., "Plastic Waves of Combined Stresses Due to Longitudinal Impact of a Pretorqued Tube, P a r t I: Experimental Re-sults, Part II: Comparison of Theory With Experiment," JOURNAL OF AP-PLIED MECHANICS, Vol. 37, TRANS. ASME, Vol. 92, Series E, 1970, pp. 1107-1120.
22 Karnes, C. H., and Ripperger, E. A., "Strain-Rate Effects in Cold Worked High-Purity Aluminum," Journal of the Mechanics and Physics of Solids, Vol. 14,1966, pp. 75-88.
23 Vaianis, K. C , "Some Recent Applications of the Endochronic Theo-ry: Temperature and Strain Rate Effects in Aluminum," Report G378-ChME-74-003, Department of Chemical and Materials Engineering, Univer-sity of Iowa, 1974.
24 Hauser, F. E., Simmons, J. A., and Dorn, J. E., "Strain-Rate Effects in Plastic Wave Propagation," Response of Metals to High Velocity Defor-mation, eds., Shewmon and Zackay, Interscience, N. Y., 1961.
