Constitutive relations for copper under shock wave loading: Twinning activation

Mechanics of Materials 38 (2006) 173–185 www.elsevier.com/locate/mechmat

Constitutive relations for copper under shock wave loading: Twinning activation J. Petit *, J.L. Dequiedt Centre d’Etudes de Gramat, 46500 Gramat, France Received 18 May 2004; received in revised form 8 March 2005

Abstract The simulation of shock-loading sequences in numerical hydrocodes includes the representation of shock fronts as continuous transitions in which the equations of mechanics are solved just as in the rest of the structure. Nonetheless, the elasto-plastic transformations are specific and, for some materials, the elasto-plastic models usually used in dynamic problems do not apply within these fronts. A constitutive law is developed in this study for copper, for dynamic problems to be used both inside and outside shock fronts. The Zerilli and Armstrong formulation is modified to take into account twinning activation under such conditions: twinning and dislocation glide are treated as two competing plastic deformation mechanisms, each having its own flow stress. The mechanical characterization of samples recovered after various shock tests and the numerical simulation of these tests are used to identify coefficients for this plasticity model.
2005 Elsevier Ltd. All rights reserved. Keywords: Shock-loading; Constitutive equations; Twinning; Plate impact tests; Explosive loading tests; Copper

1. Introduction In the case of the explosive forming of metals, the large deformation phase, which produces the final shape of the device, is preceded by a shockloading phase during which the device is crossed

*

Corresponding author. Tel.: +33 5 65 10 53 54; fax: +33 5 65 10 54 09. E-mail address: [email protected] (J. Petit).

by a succession of shock and release waves. Shock phases can modify the microstructure of the material and thus the subsequent behavior. They start with the propagation of a shock front in which extremely high strain rates are reached and specific microstructural transformations can be activated. The microstructure of shock-loaded metals and the consequences of shock-loading on mechanical characteristics have been studied by many authors. Smith (1958) was one of the first to perform impact tests with plates projected by an explosive pad; he observed the resulting effects on small copper and

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2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2005.06.005

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iron samples. Later various authors, Murr and Grace (1969) and Gray (1991) for example, did the same on a large variety of materials: the tests used both explosively projected impactors and gas guns and the experimental set-ups were progressively improved. They observed the evolution of the substructure (dislocation cells, twinning, phase transformations) and of the mechanical characteristics (hardness, yield stress, stored energy) with the shock pressure and pulse duration. Different approaches have been proposed to model a shock phase and its effects. Smith (1958), Hornbogen (1962) and Meyers (1994) chose to model the shock front at the scale of the microstructure: they represent it as a dislocations front moving in the crystal lattice. In other studies, like those of Wallace (1981) and Clifton (1971), the front is represented as a continuous transition ruled by thermo-mechanical relations. Conversely, some other researchers did not study the shock front itself but modeled the effect of the whole shock phase on the post-shock behavior: Gazeaud et al. (1989) built an empirical relation between post-shock yield stress and shock pressure. Grace (1969) links the yield stress and stored energy to the mechanical work done in the shock-release cycle. The aim of this work was to develop an elastoplastic model to be used in numerical hydrocodes, for shock-loading sequences. In these cases, the simulation includes both the internal structure of the shock front—represented as a continuous transition of thermo-mechanical characteristics but in which specific elasto-plastic transformations may be activated—and the evolution outside this front. The constitutive relations must be chosen appropriately. A model like this was elaborated for copper from the analysis of different shock-loading tests: plate impact experiments and plane explosive loading tests. In plane shock tests, due to the very shallow shock front, in situ measurements are not possible. Nevertheless, the mechanical characterization of both as-received and shocked samples allows return to a constitutive model applicable inside and outside the shock front, with the help of numerical simulation. In the first paragraph, an elasto-plastic model is set up for copper under classical quasi-static

and dynamic loading conditions. The hypothesis used is one of isotropic strain-hardening and Zerilli and ArmstrongÕs model is chosen: in such models, the internal state variable is the equivalent plastic strain. This model is quite suitable but a few terms had to be improved. The model coefficients are identified with the help of quasi-static and dynamic compression tests on the as-received material. Its validity field is then extended with the help of a Taylor test. In the second paragraph, the different shock tests and the characterization of shocked samples are presented: the activation of twinning under shock-loading is observed and its effect on the subsequent behavior is quantified. In the third paragraph, a constitutive model is built for copper to be used under shock-loading conditions, both inside and outside the shock front: due to the activation of twinning, the previous Zerilli and Armstrong model in its initial formulation does not apply. Therefore, it is improved in the following way: twinning is considered as a second plastic deformation mechanism competing with dislocation glide, a twinning yield stress is defined and the associated equivalent plastic strain is taken as a second internal state variable for the model. The new parameters are fitted with the help of the characterization of the shocked samples and of the numerical simulation of the shock tests.

2. Constitutive relations for annealed material Before studying the elasto-plastic behavior under shock conditions and modeling the thermomechanical evolution inside a shock front, a constitutive model was fitted for classical quasi-static and dynamic conditions. The Zerilli and Armstrong formulation was selected for the flow stress and the Preston and Wallace formulation for the shear modulus. In this study, all the experiments were carried out on OFHC copper. It was cross-rolled and annealed, the final average grain size D was about 18 lm leaving out the annealing twin boundaries and its hardness was 45 ± 0.5 HV 1.

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2.1. Shear modulus

with

Shear wave velocity measurements were performed at the Commissariat a` Energie Atomique (Valduc Center) to elaborate a constitutive relation for the shear modulus. The Preston and Wallace formulation (1992) was
chosen:  T Gðp; T Þ ¼ ðG0 þ Gp pÞ 1  GT ð1Þ T m ðqÞ where Tm(q) is the melting temperature given by Lindemann relation. 2.2. Elaboration of an elasto-plastic model Quasi-static compression tests and dynamic Hopkinson bar tests were performed at the Centre Technique dÕArcueil (C.T.A.) on cylindrical specimens 6 mm in diameter and 7 mm in thickness. For the elasto-plastic model, the Zerilli and Armstrong formulation was selected. In the initial formulation given by the authors (Zerilli and Armstrong, 1987), the flow stress is: pffiffiffiffi ry ðep ; e_ p ; T Þ ¼ C 0 þ ðC 1 þ C 2 ep Þ  exp½ðC 3 þ C 4 Ln_ep ÞT þ C 5 ðep Þn ð2Þ in which the internal state variable which characterizes strain hardening is the equivalent plastic strain ep. This model links the flow stress to the thermal activation of the dislocation glide. The thermally activated part of this stress is linked to short range obstacles; the athermal part of the flow stress is linked to long range obstacles. With this initial form, we did not manage to fit properly both quasi-static and dynamic stress– strain curves; so, we introduced a few modifications some of them being suggested by the authors themselves. The resulting ‘‘modified Zerilli–Armstrong model’’ (the corresponding flow stress and its components are written with index 1 since another yield stress will be introduced later for the activation of twinning) was: ry1 ðep ; e_ p ; T Þ ¼

Gðp; T Þ Gðp ¼ 0; T ¼ 300 KÞ  ðr01 þ r11 ð_ep ; T Þ þ r21 ðep Þ þ r31 ðe; e_ p ; T ÞÞ

175

ð3Þ

r11 ð_ep ; T Þ ¼ B eðb0 þb1 Ln_ep ÞT r21 ðep Þ ¼ C 5 ðera ð1  e 1 r31 ðep ; e_ p ; T Þ ¼ rTh 2

ep =era

ð4Þ n

ÞÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! C d T e_ p 1þ 1þ4 rTh

rTh ðep ; e_ p ; T Þ ¼ B0 emr ð1  eep =er Þp eða0 þa1 Ln_ep ÞT

ð5Þ ð6Þ ð7Þ

with er ð_ep ; T Þ ¼



er0

eða0 þa1 Ln_ep ÞT

1 m1

ð8Þ

In particular, the dependence on the shear modulus, integrated by Goldthorpe et al. (1994) to the athermal part of the flow stress, is extended here to all the terms. In the case of fcc metals, the principal short range obstacles to dislocation glide are the ‘‘dislocation trees’’; the thermally activated part of the flow stress thus depends on the amount of stored dislocations, and consequently of ep. r11 ð_ep ; T Þ, associated with short range obstacles independent of strain hardening, is left out of account. Thus, we consider: B ¼ b0 ¼ b1 ¼ 0

ð9Þ

The strain hardening saturation with plastic strain is taken into account in both the athermal and thermally activated part of the flow stress; for the latter, the saturation strain er introduced by Zerilli and Armstrong (1997) is expressed here as a function of temperature and strain rate. The viscous drag effect, which leads to high increase of the flow stress at high strain rates, is introduced in the form suggested by Armstrong and Zerilli (1988) with parameter Cd. With this new model, the compression graphs fitted quite satisfactorily (Fig. 1); the associated set of coefficients is given in Table 1. The optimization process showed that the viscous drag effect could be neglected in the field of compression tests; this meant that the optimum value for coefficient Cd necessarily satisfies: Cd < 10 Pa s/K. Incidentally, it is noteworthy that Cd = 10 Pa s/K is the value proposed by Armstrong and Zerilli (1988). To extend the model validity field and establish a suitable value for Cd, dynamic tests are needed in

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CTA experiments

stress σy1 (MPa)

500

. . -1 εp = 1.5 × 103 s -1 T0 = 293 K .εp = 0.15 s-3 , -1 εp = 2 × 10 s , T0 = 293 K T 0 = 293 K . ε = 0.15 s-1, T0 = 373 K T 0 = 398 K .p -3 -1 εp = 2 × 10 s , T0 = 373 K T 0 = 523 K . ε = 0.15 s-1, T0 = 573 K .p εp = 2 × 10-3 s-1, T0 = 573 K

400 300 200 100 0 0.0

fitted curves

0.1

0.2 0.3 0.4 plastic strain εp

0.5

0.6

0.0

0.1 0.2 0.3 plastic strain εp

Fig. 1. Compression tests and modeling with the modified Zerilli–Armstrong model (relations (3)–(8)).

Table 1 Modified Zerilli–Armstrong model—coefficients for copper r0

C5

era

n

B0

a0

a1

er0

m

p

Cd

G0

Gp

GT

59.24 MPa

163.45 MPa

0.62

0.779

1232.1 MPa

3.2 · 105 K1

1.66 · 106 K1

1.01

0.987

0.695

<10 Pa s/K

50.0 GPa

1.36

0.4559

which larger strains and higher strain rates are reached. To do this, a Taylor test was carried out as presented in the next section. 2.3. Validation test The Taylor test consists in performing an impact experiment on a cylinder of the material to be tested. The constitutive model is validated by comparing the final shape of the cylinder to the one obtained by numerical simulation. In this study, the test was performed in the symmetrical configuration, which avoids friction problems at the impact surface; two identical cylinders (7 mm in diameter and 50 mm in length) of as-received copper were machined then impacted at a relative velocity of 305.6 m/s. The experimental set-up was suggested by Erlich et al. (1982) and used later in CEG by Chartagnac et al. (1988). The target cylinder was recovered and its final shape determined with a three-dimensional control machine: the shape had no significant anisotropy. A micrograph of a longitudinal section revealed ductile damage; the damage area was however localized in the center of the sample and its effect has been left out of account in the following.

Numerical simulation of the Taylor test was performed with Ouranos code. It proved that strains up to 100% and strain rates up to 5 · 104 s1 are obtained during this test but only in a very small volume near the impact surface. Two simulations were achieved using the modified Zerilli–Armstrong model with two values for the viscous drag parameter Cd: Cd = 0 and Cd = 10 Pa s/K. The experimental and simulated final shapes were compared (Fig. 2). While the model aligns quite well with Cd = 0, the numerical shape is significantly underdeformed with Cd = 10 Pa s/K. More precisely, further computations proved that the optimum value satisfies: C d < 1 Pa s=K

ð10Þ

Even if the Taylor test still remains a quite complex structure problem, this tends to prove that the modified Zerilli–Armstrong model is validated in a loading field up to 100% of strain and strain rates up to a few 104 s1. Moreover, the viscous drag effect is such that it becomes effective at higher strain rates than those of the Taylor test, which has already been referred to in a previous study by Petit et al. (1999).

J. Petit, J.L. Dequiedt / Mechanics of Materials 38 (2006) 173–185

6 radius (mm)

l0 = 50 mm, r0 = 3.5 mm, impact velocity = 305.6 m/s

t = 150 µs

detail

177

5 experiment 4 3

0 2 length (mm)

4

0

10 20 length (mm)

30

40

numerical results, with modified ZA model with Cd = 0 with Cd = 10 Pa s/K

Fig. 2. Taylor test—experimental and simulated final shapes.

3. Shock tests and post-shock characterization Once we had elaborated a constitutive model for our copper in a wide range of quasi-static and dynamic conditions, we performed shock tests on this material to evaluate the effect of a shockloading phase. After each of these tests, the shocked sample was recovered, optical micrographs were made and the post-shock mechanical characteristics were identified by compression tests.

guard ring

plane wave sample explosive pad generator comp. B or 96 % HMX Fig. 3. Plane explosive loading set-up.

3.1. Presentation of shock experiments Two kinds of plane shock tests have been used at the Centre dÕEtudes de Gramat (CEG) to produce shocked samples: the plane explosive loading test and the plate impact test. In the plane explosive loading test, developed in CEG by Petit and Picard (1987), a copper disk is projected by an explosive pad in which a plane detonation wave is generated thanks to an appropriate set-up (Fig. 3). The sample is surrounded by a momentum ring in such a way that it remains in a state of single-axis deformation during the shock phase. The projected sample is recovered gently through several meters of gradually densified foam. The last stage of the recovery system is a water tank used to quench the samples and avoid post-shock annealing and re-crystallization which were sometimes observed under inappropriate experimental conditions (as mentioned in Zukas and Fowler (1960) and could be suspected in Aeberli and Pratt (1985)). This experiment is quite simple to perform. Moreover, the sample keeps its plane geometry

but it is crossed by a shock-loading phase that is similar to the one of an explosive shaping sequence. Nonetheless, the range of shock pressures that can be reached is quite limited. Besides, the crossing of Taylor release waves from the explosive and the successive reflected waves at the free surfaces generates cyclic loading and spalling, which makes the simulation of an experiment like this somewhat difficult. Two explosive loading tests were performed (Table 2) with two different explosives (V401 which is 96% HMX and comp. B) generating shock pressures of about 50 Pa. The plate impact test generates a loading sequence that is simpler to analyze: a shock front followed by a pressure plateau and a release wave. The experimental set-up which was proposed by Gray (1991) is given in Fig. 4; the back up plates in four pieces, developed by Ponsonnaille and He´reil (1997), trap spalling and guard rings, maintain single-axis deformation in the sample during the shock phase. The calculation of the dimensions of all these devices is given in Gray (1991).

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Table 2 Plane explosive loading tests Test

Explosive pad

No. 3 No. 4

Sample

Nature

Length (mm)

1st shock pressure (GPa)

Initial thickness (mm)

Post-shock hardness HV 1

Thickness reduction (%)

Comp. B V 401

120 50

44.6 ± 3 51.3 ± 3

7 7

109 ± 1 107 ± 1.5

7.8 0.1

First shock pressure is derived from numerical computation.

guard rings

• the last test where the impactor was driven by an explosive pad at a velocity of 2310 m/s.

3.2. Characteristics of recovered samples

impactor v = 524 to 2310 m/s

sample

back up plates (4 angular sectors)

Fig. 4. Plate impact set-up.

In the case of plate impact tests, the difficulty comes from controlling the tilt angle between the two impact surfaces and the possible lateral release waves due to the clearance between the different parts of the experimental set-up. Nevertheless, the shock pressure depends on the impactor velocity and thus a larger shock pressures field can be reached than with the explosive loading test. The pressure pulse duration can be modified by changing the impactor thickness, but its influence was not evaluated in this study. Three impact tests were performed as summarized in Table 3: • the two first tests where the impactor was launched with Demeter gas gun at velocities of 524 and 885 m/s,

The post-shock hardness was measured on the recovered samples (Tables 2 and 3). The thickness reduction of a shock-loaded sample is indicative of the quality of the experimental device and process as was illustrated in Gray et al. (1989). For our shock tests, the associated values are also reported in Tables 2 and 3. Optical micrographs were made (Fig. 5) and revealed a high twin density for the highest shock pressures (tests nos. 3, 4 and 5). For tests nos. 1 and 2, it was impossible to be sure that twins due to shock-loading exist among annealing twins. It must be borne in mind that copper does not twin under classical loading conditions except at very low temperatures. On the other hand, twinning was observed in shocked copper many times since the first studies by Smith (1958). TEM micrographs confirmed the existence of twins in copper after 15 GPa loading when the grain size is 200 lm (as reported in Murr (1981a)); Gray et al. (1989) exhibited some twins with a grain size of 40 lm and after a 20 GPa shock test.

Table 3 Plate impact tests Test

No. 1 No. 2 No. 5

Impactor

Sample, initial thickness 7 mm

Thickness (mm)

Velocity (m/s)

1st shock pressure (GPa)

Post-shock hardness HV 1

Thickness reduction (%)

2.35 2.35 6

524 885 2310

10.1 ± 0.5 18.4 ± 0.5 60.5 ± 2

86 ± 1 100 ± 1 120.5 ± 1.5

1.1 2.9 9

First shock pressure is derived from numerical computation.

J. Petit, J.L. Dequiedt / Mechanics of Materials 38 (2006) 173–185

179

Fig. 5. Optical micrographs of recovered samples after impact tests.

3.3. Post-shock elasto-plastic behavior Specimens were machined in the samples and compression tests were performed at C.T.A. The tests were realized under the same loading conditions as for the annealed material except at the highest temperatures to avoid the re-crystallizing of the shocked specimens, which have gone through strong work-hardening. The re-loading stress–strain curves revealed that shock-loading in copper resulted in significant work-hardening which increases with shock pressure: a temporary ‘‘memory effect’’ is observed at the beginning of these curves, vanishing as the strain increases, but a residual strain hardening effect remains. Then, we chose to concentrate on the second effect and leave out of account the very beginning of the reloading curves. Before modeling the behavior under shockloading conditions, we first tried to quantify the

600

after plate impact no. 5 2310 m/s - 60.5 GPa

fitted curves with relation (12) and εp0 = 0.294, ∆σ 0 = 45.2 MPa

þ r31 ððep0 þ ep Þ; e_ p ; T ÞÞ

400 300 200 100 0 0.0

0.1

ð11Þ

This first approach did not give good results since we did not manage to fit simultaneously quasi-static and dynamic reloading graphs. In other words, the effect of shock-loading on the mechanical characteristics is not equivalent to one of pre-strain which would have been performed under conventional conditions, not surprising due to the activation of twinning. Moreover, it is usually admitted that twinning is equivalent to grain size reduction increasing the

. εp = 1.5 × 10 3 s-1, T0 =233 K . K εp = 1.5 × 10 3 s-1, T0 =398 . εp = 2 × 10-3 s-1, T0 = 293 K . εp = 2 × 10-3 s-1, T0 = 373 K

500 stress σy1 (MPa)

CTA experiments

effect of the shock on the post-shock mechanical behavior. The first step consisted in trying, for each shock test, to getting the reloading curves to fit by adding a pre-strain ep0 to the modified Zerilli–Armstrong model with the initial coefficients: Gðp; T Þ ðr01 þ r21 ðep0 þ ep Þ ry1 ¼ Gð0; 300Þ

0.2 0.3 0.4 plastic strain εp

0.5

0.6

Fig. 6. Fit of post-shock compression tests—impact no. 5.

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Table 4 Optimized values for Dr0 and ep0 for the different shock tests

4.1. Presentation of the model

Test

Just as was suggested by Meyers et al. (1995), we supposed that dislocation glide and twinning were two competing plastic deformation mechanisms, each having its own flow stress (ry1 and ry2 respectively). This approach is slightly different from that of Zerilli and Armstrong (1997) who only bring in the effect of twinning as a shift of the stress–strain curves. The total plastic strain increment is the sum of the two corresponding strain increments and the two equivalent strains, ep1 and ep2 respectively, are the two internal state variables of the model: ð13Þ e_ p ¼ e_ p1 þ e_ p2

No. No. No. No. No.

1 2 3 4 5

Dr0 (MPa)

ep0

7.4 ± 3.0 16.7 ± 2.0 35.0 ± 4.0 38.0 ± 2.0 45.2 ± 5.0

0.101 ± 0.025 0.146 ± 0.025 0.262 ± 0.025 0.237 ± 0.025 0.294 ± 0.025

athermal part of the flow stress. Thus, in a second step we tried to fit the reloading curves by adding to the initial model both a pre-strain ep0 and an additive athermal stress Dr0: ry1 ¼

Gðp; T Þ ðr01 þ Dr0 þ r21 ðep0 þ ep Þ Gð0; 300Þ þ r31 ððep0 þ ep Þ; e_p ; T ÞÞ

¼

ð12Þ

This gave far better results than with a prestrain alone (Fig. 6). The optimized ep0 and Dr0 values are given in Table 4 for each shock test. The need to include a shift Dr0 in the yield stress of the annealed material to describe the effect of twins in the post-shock behavior has already been reported by Meyers et al. (1995) after 50 GPa shock-loading and by Rohagti et al. (2001) after 35 GPa shock-loading.

¼

¼

In each case, the activation of one or two mechanism depends on the relative level of the flow stresses. These stresses are supposed to have the following form: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gðp; T Þ 1 0 r01 þ k D1 þ k m ep2 ry1 ¼ Gð0; 300Þ D ! þr21 ðep1 Þ þ r31 ðep1 ; e_ p1 ; T Þ and Gðp; T Þ ry2 ¼ r0 þ k D2 Gð0; 300Þ 02

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 þ k m ep2 D

ð14Þ

ð15Þ

4. Constitutive behavior during a shock phase In the former paragraph, the mechanical and microstructural characteristics of the samples recovered after the different shock tests have been identified. From these experimental data, we then tried to elaborate an elasto-plastic model which applies both inside and outside a shock front. This can be achieved by the simulation of the shock tests, provided that a few hypotheses are made. There is no reason why the Zerilli–Armstrong law exhibited in paragraph 2 should be totally abandoned. Nonetheless, this model is based on the activation of dislocation glide as the microstructural mechanism associated with plastic deformation. So, the activation of twinning which appears in shock conditions and its effect on strain hardening had to be brought into the model.

The dislocation glide flow stress is the one given by the initial modified Zerilli–Armstrong model with the coefficients previously identified. The viscous drag coefficient Cd had to be determined. The new terms brought into the two flow stresses are the following: • the influence of deformation twins was brought into both flow stresses. Twinning was supposed to have the same effect as grain size reduction, each twin dividing one grain into two apparent sub-grains. The athermal part of the flow stresses was thus written as a function of the resulting ‘‘effective grain size’’, with two Hall–Petch coefficients kD1 and kD2. Coefficient km links the effective grain size to the twinning equivalent strain:

J. Petit, J.L. Dequiedt / Mechanics of Materials 38 (2006) 173–185

ð16Þ

(The demonstration of this formula and an expression for km are given in Appendix A). • the athermal yield stress r01 identified in paragraph 2 is the one without twins; in other words: rffiffiffiffi 1 0 r01 ¼ r01 þ k D1 ð17Þ D • the twinning strain was supposed to be independent of strain rate and temperature since it is generally admitted that it is unaffected by thermal activation. • without any precise information on this subject, we assumed that the twinning stress was independent of the number of stored dislocations and thus of ep1.

yield strength σy

1 1 ¼ þ k m ep2 Deff D

181

σy1 with Cd > 0 σy2

σy1 with Cd = 0 strain rate Fig. 7. Relative position of the two flow stresses as a function of strain rate.

km and Cd are the free parameters which had to be optimized with the help of the shock phase simulation. These simulations were achieved with the CEG unidimensional code UNIDIM. 4.2. Identification of parameters

Among the new coefficients, r002 , kD1, and kD2 were taken from previous studies (Table 5). These studies exhibited, for instance, that the twinning stress ry2 was more sensitive to the effective grain size than the dislocation glide stress ry1: thus, kD2 is higher than kD1. An estimated value of km is given in Appendix A but it is simply an order of magnitude for this parameter. We might note that Cd controls the high increase of ry1 at very high strain rates and the characteristic strain rate for this effect. A strictly positive value is needed in order to activate twinning as schematized in Fig. 7. In this case, the plastic strain increment is the sum of the two terms ep1 and ep2 and the equivalent stress is simultaneously equal to both ry1 and ry2. Otherwise (if Cd = 0), we always have ry1 < ry2 and only dislocation glide is activated. Nonetheless, the non-activation of the viscous drag effect under Taylor test loading conditions already gave the upper limit: 0 < Cd < 1 Pa s/K. Table 5 Two mechanism model coefficients r 0 02 kD1 kD2

370 MPa (in Vo¨hringer, 1976) 5.3 MPa mm1/2 (in Zerilli and Armstrong, 1987) 20.25 MPa mm1/2 (Vo¨hringer, 1976)

Coefficients km and Cd then had to be worked in. The optimization process was achieved by comparing, for each test, the post-shock mechanical characteristics identified experimentally with the results obtained by numerical simulation. The first simulations of shock-loading sequences showed that for any choice of km and Cd < 1 Pa s/K (with the other coefficients previously identified), twinning was only activated inside the first shock front. In the other parts of the loading sequence, the dislocation glide stress always remains below the twinning stress. This can be considered as a first validation of the model since twinning was not observed after usual compression tests and a Taylor test. Moreover, for the different shock tests, the computation of the whole loading sequence appeared to be too unwieldy, especially for explosive loading tests. As we established that the activation of twinning was limited to the first shock front, it was possible to limit ourselves to the computation of this first shock front, in order to evaluate the final twinning strain. So, for each shock test, we carried out a simulation of the symmetrical impact of two thin plates at an impact velocity such that the shock pressure is conserved. As we will see below, we conducted these simulations in such a way that

J. Petit, J.L. Dequiedt / Mechanics of Materials 38 (2006) 173–185

the resulting shock front is as representative as possible of the one of the associated shock tests. We then expected that when simulation was complete, the final equivalent twinning strain, ep2f, would be consistent with the experimental additive athermal stress identified in paragraph 3 in other words, we expected that: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 þ k m ep2f Dr01 ðep2f Þ ¼ r001 þ k D1 D rffiffiffiffi! 1 0  r01 þ k D1 D ¼ Dr0

ð18Þ

This would constitute a partial validation of the two-mechanism model under shock-loading conditions. To be more precise, the identification of the latter parameters was carried out as follows: inside the shock front, the sharing of the plastic strain between the two mechanisms is highly dependent on the ‘‘loading history’’ and especially the strain rate that is reached. The strain rate inside the front depends on the viscosity which controls the shock front thickness. This viscosity does not appear as a physical parameter in constitutive laws but is indirectly introduced in numerical simulation through the term of pseudo-viscosity which is a function of the mesh size. We do not have experimental data on the shock front thickness. Thus, we selected a mesh size of 2500 meshes per mm, such that this thickness was consistent with the order of magnitude given by Chhabildas and Asay (1979). We searched for a pair of values (km, Cd) so that Dr01(ep2f)  Dr0, for all the shock tests (Dr0 is recalled in Table 6). The optimization process was

Table 6 Experimental additive athermal stress and theoretical twin density Test No. No. No. No. No.

1 2 3 4 5

Dr0 (MPa)

Ntwins

7.4 ± 3.0 16.7 ± 2.0 35.0 ± 4.0 38.0 ± 2.0 45.2 ± 5.0

0.8 ± 0.3 2.0 ± 0.2 5.1 ± 0.7 5.7 ± 0.4 6.9 ± 1.0

experimental data

numerical results

50 ∆σ0, ∆σ01 (MPa)

182

k m = 1500 mm-1 Cd = 0.6 Pa s/K

40 30 20

k m = 2830 mm-1 Cd = 0.02Pa s/K

10 0

0

10

20 30 40 50 shock stress (GPa)

60

70

Fig. 8. Additive athermal stress after the shock tests—experimental and numerical results with initial and optimized km and Cd values.

carried out in the following way. An initial value of km was tested; we took km = 2830 mm1 given in Appendix A. For this km, we identified a value for Cd in such a way that Dr01(ep2 f)  Dr0 for impact test no. 5 (for km = 2830 mm1 we found Cd = 0.02 Pa s/K). This pair of values however, underestimated the additive athermal stress for the other shock tests. We then repeated the operation for other values of km until Dr0 was reasonably approached for all the tests (km = 1500 mm1, Cd = 0.6 Pa s/K) (Fig. 8). With this set of parameters, the constitutive law developed in the former paragraph is able to model twinning activation within a shock front: the additive athermal yield stress which characterizes the post-shock behavior of copper is reproduced satisfactorily for shock levels of about 10–60 GPa.

5. Conclusion In this work, we tried to build a constitutive model for copper to be used in the simulation of shock-loading sequences that is inside and outside the shock fronts. To do this, we used the mechanical characterization of samples recovered after different shock tests in a plane configuration and a partial numerical simulation of these tests. We established that a usual elasto-plastic model, validated in classical dynamic conditions, did not apply under shock conditions. We used the hypothesis that this was mainly due to the

J. Petit, J.L. Dequiedt / Mechanics of Materials 38 (2006) 173–185

activation of twinning inside the shock front, which is consistent with the post-shock athermal stress displayed on the reloading graphs. We then produced a model including two internal state variables associated with two competing plastic deformation mechanisms: dislocation glide and twinning. Each mechanism has its own flow stress and the total plastic strain is written as the sum of two strains. In the case of copper, the different terms of the model will be refined thanks to more precise microstructural analysis; for example, TEM micrographs would help quantify more precisely the number of twins in the samples. Nonetheless, at this stage, the model is able to describe accurately the additive athermal stress after a plane shock-loading phase. We will then have to add certain elements to describe the complete thermo-mechanical evolution during a whole shock phase and under any type of loading conditions and geometry: effect of cycling loading in the case of explosive loading, effect of pulse duration (see for instance Murr, 1981b), anisotropy of the internal state behind the shock front, evolution of dislocation substructure with strain rate. Some of these effects would need further experimental developments. The latter was observed by certain authors when reloading samples after pre-straining at various strain rates and temperature; it was taken into account in the models developed by Klepaczko (1975) and Follansbee and Kocks (1988). One important problem remains in the simulation of shock phases: the use of sophisticated models in the shock front requires the exact description of the loading path in it and especially of the strain rate level that is actually attained. This excludes the artificial spreading of the front with the introduction of a pseudo-viscosity. Consequently, very fine mesh sizes shall be required which are very costly in numerical codes, especially in threedimensional computations.

Acknowledgments This work was supported by DGA/DSA/ SPNuc/SDAN of the French Department of Defense. The authors would like to thank Alain

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Halgand, Yves Sadou and co-workers who performed the shock tests and Jacques Clisson of C.T.A. who performed the compression tests on both annealed and shocked copper specimens.

Appendix A. Approximate relation between twinning strain and ‘‘effective grain size’’ The purpose of this Appendix A is to demonstrate the relation between the twinning strain ep2 and the ‘‘effective grain size’’ Deff and to work out a relation for coefficient km. This relation links these two quantities to the average number of twins per grain Ntwins. Each twin is supposed to divide one grain into apparent sub-grains. The effective grain size is then: Deff ¼

D 00

M N twins þ 1

ð19Þ

M00 is the mean orientation factor between dislocation glide and twinning planes. On the other hand, the twinning strain is linked to the average number of twins per grain in the following way, suggested by Armstrong and Worthington (1973). The resolved shear strain inside a twin is a constant for FCC metals: ctwin = 0.707 (given in Franc¸ois et al., 1991). If all the twins were parallel in an idealized cubic grain, the resolved shear strain in the grain would be: cgrain ¼ N twins

etwin c D twin

ð20Þ

with etwin, the average twin thickness. The same relation is written for the macroscopic equivalent twinning strain with a second orientation factor which takes into account the twin orientation distribution and the relation between the resolved shear strain in one grain and the macroscopic strain. Further, it is assumed that all the grains are affected by twinning and that they have the same twin density. etwin ep2 ¼ M 0 cgrain ¼ M 0 N twins c ð21Þ D twin We then deduce a relation between the equivalent twinning strain and the effective grain size:

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1 1 M 00 1 ep2 ¼ þ 0 Deff D M etwin ctwin

ð22Þ

Thus km ¼

M 00 1 0 M etwin ctwin

ð23Þ

An estimation of the different parameters is made: • in the case of single-axis deformation, the twin orientations are accepted as being distributed with maximum density in the two directions +30 and 30 from the direction of loading; this was observed by Smith (1958) on shocked copper. In this case, it could be deduced that M 0 < 0.5. Here we choose to keep the value M 0 = 0.5. • in the same case of single-axis deformation, the dislocations are admitted to glide in planes at 45 from the direction of loading; an average value for M00 is then close to 0.5. We choose to take M00 = 0.5. • etwin is evaluated on the micrographs of shocked samples: etwin  0.5 lm. Thus an estimated value for km is: km = 2830 mm1 which gives a relation between ep2 and Deff. It is interesting, thanks to these relations, to evaluate a ‘‘theoretical’’ average number of twins per grain associated with the experimental additive athermal stress Dr0 for the different shock tests: ! pffiffiffiffi
2 1 Dr0 D N twins ¼ 00 þ1 1 ð24Þ M k D1 This theoretical twin density is given in Table 6 for the different shock tests. Although it is difficult to evaluate Ntwins precisely on optical micrographs, these theoretical values seem reasonable.

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