High strain-rate plastic deformation of molybdenum: Experimental investigation, constitutive modeling and validation using impact tests

International Journal of Impact Engineering 96 (2016) 116–128

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International Journal of Impact Engineering j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / i j i m p e n g

High strain-rate plastic deformation of molybdenum: Experimental investigation, constitutive modeling and validation using impact tests Geremy Kleiser a,b, Benoit Revil-Baudard a,*, Crystal L. Pasiliao b a b

Department of Mechanical and Aerospace Engineering, University of Florida, REEF, 1350 N. Poquito Rd., Shalimar, FL 32579, USA Air Force Research Laboratory, Eglin, FL 32542, USA

A R T I C L E

I N F O

Article history: Received 29 June 2015 Received in revised form 21 January 2016 Accepted 22 May 2016 Available online 2 June 2016 Keywords: Polycrystalline molybdenum Constitutive modeling Taylor impact tests Finite element simulation

A B S T R A C T

In this paper, an experimental study on the quasi-static behavior and dynamic behavior of a polycrystalline molybdenum material is presented. Due to the material’s limited tensile ductility, successfully acquiring data for impact conditions is very challenging. For the first time, Taylor impact tests were successfully conducted on this material for impact velocities in the range of 140–165 m/s. For impact velocities beyond this range, the very high tensile pressures generated in the specimen immediately after impact lead to failure. A constitutive model accounting for the key features of the Mo plastic behavior, i.e. its tension–compression asymmetry and plastic anisotropy was developed. An implicit solver was used to simulate the impact deformation. A good agreement was obtained between predictions and experimental outlines of the specimens. Furthermore, it was shown that the model can be used to gain understanding of the dynamic deformation process in terms of time evolution of the pressure, the extent of the plastically deformed zone, distribution of the local plastic strain rates, and when the transition to quasistable deformation occurs. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The high-temperature stability and creep resistance of refractory materials, and in particular of molybdenum (Mo), make them very attractive for high-temperature applications (e.g. turbines). However, of major concern is the limited ductility of Mo at roomtemperature. Recently, experimental studies devoted to the investigation of the room-temperature response in tension and the forming properties of polycrystalline Mo were reported (e.g. Walde [1], Oertel et al. [2]). However, the tension–compression asymmetry in yield and flow stresses and its hardening directionality were not documented. Of particular importance is the investigation of the mechanical response of Mo under impact where the strain-rates experienced are very high. Such strain rates (of the order of 104–105 s−1, Nicholas [3]; Wilkins and Guinan [4]; Cerreta et al. [5]) can be achieved in a Taylor impact test, which consists of launching a solid cylindrical specimen at high velocity of the order of 100–300 m/s against a stationary rigid anvil (see Taylor [6]; Cristescu [7]; Wilkins and Guinan [4]). When the specimen impacts the rigid stationary anvil, an elastic compressive wave that is generated at the impact interface, travels through the specimen in the axial direction with a speed

* Corresponding author. Department of Mechanical and Aerospace Engineering, University of Florida, REEF, 1350 N. Poquito Rd., Shalimar, FL 32579, USA. Tel.: +1(850) 833 9350; Fax +1(850) 833 9366. E-mail address: revil@ufl.edu (B. Revil-Baudard). http://dx.doi.org/10.1016/j.ijimpeng.2016.05.019 0734-743X/© 2016 Elsevier Ltd. All rights reserved.

equal to the sound speed in the respective material. When this compressive wave reaches the other end of the specimen, it is reflected as a tensile wave. For a sufficiently high impact velocity, for which the magnitude of the compressive wave reaches the yield stress of the material, the impacted end undergoes plastic deformation. The plastic front with maximum stress magnitude equal to this yield/flow stress starts propagating from the impact interface at a much lower speed than the elastic wave. The travelling plastic front interacts with the reflected precursor elastic wave at some intermediate point along the length of the specimen. The elastic wave then gets reflected at the elastic–plastic interface and travels toward the free end of the specimen. This back and forth movement of the elastic wave results in deceleration of the specimen. For more details about wave propagation during an impact event, the reader is referred to the seminal books of Rakhmatulin and Demyanov [8] and Cristescu [7]. Taylor [6] developed a one-dimensional wave propagation analysis to estimate from this test the dynamic yield strength of a given material. Later on, semi-analytical models have been proposed to determine the yield strength as a function of density, impact velocity, initial and final length, and length of the un-deformed region of the recovered specimen (see for example, Gilmore et al. [9]; Lu et al. [10]; Wang et al. [11]). Currently, the Taylor test is used more as a means of validating plasticity models and codes for the simulation of dynamic deformation (see Holt et al. [12]; Zerilli and Armstrong [13]; Maudlin et al. [14]; etc.). It is also important to note that Taylor impact test data have been reported only for materials that exhibit large tensile ductility at

G. Kleiser et al. / International Journal of Impact Engineering 96 (2016) 116–128

room-temperature (e.g. for iron, see Zerilli and Armstrong [13]; for tantalum, see Zerilli and Armstrong [15]; Maudlin et al. [14]; for titanium, see Revil-Baudard et al. [16]). Recently, Taylor impact tests on copper and aluminum alloys have been performed and numerical studies on damage evolution during such tests have been conducted (e.g. for aluminum alloys, see Rakvåg et al. [17]; Moc´ko et al. [18]; for copper alloys, see Wei et al. [19]; Borodin and Mayer [20]). Concerning the very high strain rate deformation of refractory metals, the literature is very limited. For spall tests on such materials, the reader is referred to Chhabildas et al. [21], while for shock testing, see Kleiser et al. [22]. For a polycrystalline high-purity molybdenum material, which shows no ductility for quasi-static strain rates of order of 10−2/s, even the feasibility of a Taylor test is questionable. This is because the very large tensile stresses generated at impact would result in immediate fracture with disintegration of the specimen. To the best of the authors’ knowledge, Taylor impact test results on molybdenum have not been reported. In this paper, we present an experimental investigation and a new three-dimensional model that was developed in order to account simultaneously for the anisotropy, tension–compression asymmetry and strain-rate sensitivity of the plastic deformation of a high purity polycrystalline molybdenum. Furthermore, Taylor impact tests conducted on the same material are reported. The results of these tests will serve to assess the predictive capabilities of the model. The outline of the paper is as follows. Section 2 presents the polycrystalline material, and the experimental tests that were conducted. Section 3 presents the elastic/viscoplastic modeling framework adopted. Finite-element (FE) simulations of the dynamic Taylor impact tests using this model and a dynamic implicit solver are presented in Section 4. The model predictions are compared with the measurements of the deformation of the impacted specimens. Discussion of the capabilities of the model and its potential to be used for “virtual testing” for high-rate application is also given. The summary of the main findings and conclusions are given in Section 5. 2. Experimental characterization 2.1. Material and mechanical tests The material used in this work was a high purity (99.98%) polycrystalline molybdenum rolled plate. The chemical composition, reported in Table 1, was determined using a LECO combustion technique for carbon (C), oxygen (O), and nitrogen (N) content, and glow discharge mass spectrometry for the remaining elements. The maximum quantity for a non-reported element was 0.1 parts per million (ppm) and any element with ppm less than 4 were summed and listed “as other”. In order to quantify the influence of the loading direction, and thereby the texture, on the mechanical response room-temperature quasi-static (nominal strain rate of 10−5 s−1) uniaxial compression tests and uniaxial tension tests were conducted. Many factors influence the ductility of molybdenum based materials in uniaxial tension at room temperature. These include alloy composition and impurity content, strain rate, surface defects in the sample, and thermo-mechanical processing (see Lement and Kreder [23]). For the commercially pure polycrystalline Mo material studied, first we

Table 1 Chemical composition of the polycrystalline molybdenum plate investigated. C

O

N

Si

K

Cr

Fe

Ni

W

Other

5

16

5

4.7

15

13

35

6.2

80

27.96

117

conducted a uniaxial tensile test along the rolling direction (RD) at a strain rate of 10−2/s. At this strain-rate, the elongation at failure is less than 1% (for more details about the quasi-static deformation of Mo, the reader is referred to Kleiser et al. [24]). Given the strain-rate sensitivity of the brittle-to-ductile transition of refractory metals, uniaxial tension tests at a strain rate of 10−5/s (i.e. three order of magnitude lower) were also conducted. Note that for this strain rate the specimen exhibited ductile behavior, the failure strain being of 22%. To investigate the effect of the loading orientation on the mechanical response a systematic investigation was conducted. All the quasi-static tests were conducted at 10−5/s. The tensile specimens are of rectangular cross-section (3.175 mm by 1.588 mm), the gauge length being of 25.4 mm. Tests were conducted on specimens taken along RD and six other in-plane orientations i.e. θ = 15°, 30°, 45°, 60°, 75° and 90° (TD) to RD. On the basis of all tests, it can be concluded that the material displays anisotropy in tensile yield stresses, the largest yield stress corresponding to θ = 60° ( σ T θ =60 = 349.6 MPa ), while the lowest yield stress is along RD ( σ T θ =0 = 308.6 MPa). For θ < 45° the yield stress is increasing with the angle θ ( σ T θ =45 = 347.6 MPa ) and then remains almost constant for 45° < θ < 90° ( σ T θ =90 = 347.5 MPa ). Lankford coefficients, defined as the ratio of the width to thickness plastic strain increments, were measured in the uniaxial tension tests along RD, 45°, and TD, respectively. It was found that the r-value at 45° to the RD is the lowest one ( r θ =45 = 0.665 ), the r-value along RD being 0.75 while the r-value along TD is almost 1 ( r θ =90 = 0.93 ). The compression specimens were right circular cylinders (5.23 mm in diameter by 9.83 mm long) that were machined such that the axes of the cylinders were along RD and two other inplane directions at 45°, and 90° (i.e. TD) with respect to RD. In addition, compression tests were also conducted on specimens with the axis along ND, the through-thickness direction of the plate. In uniaxial compression, the polycrystalline Mo exhibits anisotropy in yield stresses, the largest yield stress being for the 45° orientation ( σ C θ =45 = 384 MPa ) and the lowest yield stress being along RD ( σ C θ =0 = 318 MPa ), while along TD direction, σ C θ =90 = 372 MPa. The yield stress in compression along ND is of 341 MPa. Due to the material’s anisotropy, any specimen cross-section, which was initially circular, becomes slightly elliptical, the ellipticity ratio (ratio of the minor over major axes) varying between 0.94 for the compression specimen taken at 45° to the RD and 0.99 for the TD specimen (for more details concerning the material’s anisotropy revealed by the quasi-static tests performed for various intermediate orientations and specimen geometry, the reader is referred to Kleiser et al. [24]). As an example, a comparison between the experimental stress– strain curves in uniaxial tension and compression along RD and along TD directions, respectively, are shown in Fig. 1. Note that the material is harder in compression than in tension. Furthermore, the tension–compression asymmetry ratio depends on the loading orientation, being the lowest along RD and the largest for the specimen loaded at an angle of 45° to RD. High-strain rate compression tests were conducted using a split-Hopkinson pressure bar (SHPB) pressure bar technique to characterize the strain-rate sensitivity of the material. The SHPB system used has nickel-based alloy Inconel pressure bars with diameters of 15.9 mm and the lengths of the striker, incident, and transmitted rods are 1.37 m, 6.53 m, and 3.27 m, respectively. The specimens used for the SHPB compression tests were cylinders 5.08 mm in diameter by 5.08 mm long (0.2 inches). A small film of MoSi2 industrial lubricant was used between ends to minimize frictional effects during radial expansion. The striker bar was accelerated using a torsion spring. The loading durations for the SHPB experiments were nominally 560 μs. The average strain rates for compression experiments were around 400 s−1. Strain gages were placed 1.6 m from each specimen interface and sampled every 2 μs using a Win600 digital oscilloscope. Compressive SHPB tests were conducted for specimens

118

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600 TD Compression

RD Compression 500 RD Tension

400

Stress (MPa)

Stress (MPa)

500

300 200 100

400

TD Tension

300 200 100

0

0 0

0.05

0.1 True strain

0.15

0.2

0

0.05

(a)

0.1 True strain

0.15

0.2

(b)

Fig. 1. Experimental stress–strain response in quasi-static uniaxial tension (10−5/s) and compression for a polycrystalline Mo: (a) rolling direction (RD), (b) transverse direction (TD), respectively.

(see Table 3). It is to be noted that the major diameter corresponded to ND. Thus, in compression, the ellipticity ratio does not depend on the strain-rate. For the RD specimen, the average hardening response in compression at the quasi-static strain rate (10−5/s) and at the high-rate of 400/s is shown in Fig. 4. With increasing strain rate there is an increase in the flow stress; however, the strainhardening rate is almost the same.

1200 45 1000

TD RD

Stress (MPa)

with the long axis along the following in-plane orientations: θ = 0 (RD), 45°, 90° (TD), and the through-thickness plate direction (ND). High-rate tension tests using the SHPB configuration were not conducted since Mo has no ductility in tension beyond a strain-rate of 10−2/s. To check the repeatability of the SPBH experimental data, at least three tests have been performed for each orientation. As an example, in Fig. 2 are plotted the stress–strain curves obtained in the three different tests along RD. It is to be noted that the dispersion of the elastic wave in the rod is accounted in the data analysis. The method developed by Malvern et al. [25] was used to extract the stress– strain curve from the raw signals. As seen previously, in quasi-static uniaxial compression, the polycrystalline Mo exhibits anisotropy in yield stresses. Therefore, dynamic tests were also conducted for different specimen orientations. In Fig. 3 are plotted the experimental split-Hopkinson bar compression tests results on specimens loaded along the rolling (RD), transverse (TD), at 45° to the RD directions and normal direction (ND), respectively. Note that the largest yield stress is obtained for the 45°orientation (889 MPa) and the lowest yield stress being along RD (795 MPa) (see Table 2 and Fig. 3). Recovered specimens were measured to quantify the change in cross-section. The ellipticity of the recovered specimen cross-section is similar to that of the posttest specimens subjected to the quasi-static strain-rate of 10−5/s

800

600

RD 45 to RD TD ND

400

200

0 0 1200

0.05

0.1

0.15

Plastic strain Fig. 3. Experimental split-Hopkinson bar compression tests results on specimens loaded along the rolling (RD), transverse (TD), at 45° to the RD directions and normal direction (ND), respectively. Strain rate of 400/s.

1000

Stress (MPa)

800 Table 2 Anisotropy of yield stresses in compression for a strain-rate of 400/s.

600 RD-2 RD-1 RD-3

400

Loading direction

Yield stress (MPa)

Range of variation (MPa)

795 889 842 821

± 10 ± 15 ± 20 ± 19

RD (θ = 0°) In plane (θ = 45° to the RD) TD (θ = 90°) Through-thickness (ND)

200

0 0

0.05

0.1

0.15

Plastic strain Fig. 2. Experimental stress–strain response at high-rate (400/s) in uniaxial compression along the rolling direction (RD) obtained using the split-Hopkinson bar technique for a polycrystalline Mo showing the good repeatibility of the tests.

Table 3 Ellipticity ratio (ratio of the minor over major axes) of deformed SPHB specimens loaded in compression along RD, 45°, and TD directions, respectively. In-plane loading direction θ [degrees]

0

45

90

Ellipticity ratio

0.97

0.95

0.99

G. Kleiser et al. / International Journal of Impact Engineering 96 (2016) 116–128

119

1400 High rate (400 s-1)

1200

Stress (MPa)

1000 800 600

Quasi-static (10-5 s-1)

400 200 0 0

0.05

0.1

0.15

Plastic strain Fig. 4. Experimental stress–strain response in uniaxial compression along the rolling direction of polycrystalline molybdenum under quasi-static (10−5 s−1) and highrate (400 s−1) loadings. Symbols represent the experimental data, while lines shows the predictions using the Johnson–Cook [26] law.

Fig. 6. Photograph depicting the configuration of the Taylor impact test: close-up view of the anvil and the barrel reflection.

directions of material symmetry, and ultimately evaluate the effect of the anisotropy of the material on its dynamic response. The specimens were launched toward an anvil as shown in Figs. 5 and 6 using a smooth bore barrel and propellant. The barrel had a bore diameter of 5.36 mm. An anvil of 190.5 mm in diameter and 149.2 mm in length, made of steel 4340 alloy with a hardness of 58 HRC, was used. It should be noted that the dimensions of the anvil are very large compared to the ones of the specimen (almost three times larger in length, and fifteen times larger in diameter. During any given experiment, the specimen was placed inside the barrel near the breech, and a cartridge containing the propellant was placed in the breech behind the specimen. The propellant was ignited using a percussion primer. A small polymer obturator was

2.2. Taylor cylinder impact testing of a polycrystalline molybdenum All the Taylor cylinder impact tests have been conducted at the Air Force Research Laboratory, Eglin AFB, FL. For these tests, cylindrical specimens, having an initial diameter of 5.33 mm and an initial length of 53.3 mm (i.e. length-to-diameter ratio of 10) were used (Fig. 5.a). The cylinders were cut from the plane of the plate using an EDM process and followed by a turning operation with a lathe. Specimens were machined with the long axis either along RD, 45°, and 90° to RD, respectively. For each specimen, the throughthickness direction of the plate (ND) was marked by an arrow on the cross-section. This allowed the material axes to be tracked and thus document the deformation of the specimens in the different

(a)

(b)

(c) Fig. 5. (a) Dimensions of the cylindrical specimens (mm) used for the Taylor impact tests. (b) Cross-sectional view of the Taylor impact test set-up showing the specimen, the barrel, and the target (anvil). (c) Finite-element mesh of a quarter of the specimen with zoom of the impact surface.

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(a)

(b)

(c)

Fig. 7. Photographs of a post-test Taylor specimen launched at a velocity of 180 m/s: (a) Deformed region near the impact surface showing separation cracks due to large tensile loading; (b) Deformed region near the impact surface showing excessive radial cracking. (c) Impacted surface.

placed between the specimen and the cartridge to minimize gas slipping around the specimen during propellant by-product expansion and acceleration through the barrel. Both impacting surfaces (i.e. anvil and specimen) were polished to minimize friction (see Fig. 6). Special care was taken such as to ensure that the axis of the barrel bore is aligned at 90° to the impact surface of the anvil, so normal impact of the specimen is achieved. Velocity measurements were acquired using a set of pressure transducers and an optical light detector connected to a high-speed camera. The pressure transducers mounted in series at the exit end of the barrel were used to detect the leading edge of the pressure change due to the expansion of the propellant gas. A white light source was mounted beneath the table and illuminated a detector above the flight path of the specimen. As the specimen moved toward the anvil it blocked the light, which triggered the high-speed camera. While both techniques were used to determine the specimen velocity, the measurements of the velocity using the high-speed camera images were deemed to be the most accurate (for more details, see Kleiser [27]). The geometry of the recovered specimens were measured using a Deltronic DH214 optical comparator, which was accurate to within ± 0.001 mm. Due to the anisotropy of the material, the initial circular cross-sections became elliptical. The major and minor axes of the cross-section of the deformed specimen were recorded along its length, the respective faces of the deformed specimens being labeled major and minor profiles, respectively. As mentioned, due to the material’s limited tensile ductility conducting Taylor tests and obtaining information on the dynamic deformation of Mo is particularly challenging. For an impact velocity greater than 170 m/s, substantial failure of the specimen is likely to occur. As an example, the photographs of the recovered specimens for velocities of approximately 180 m/s are shown in Fig. 7, while that of a recovered specimen for a velocity of 151 m/s is shown in Fig. 8(a).

Irrespective of the impact velocity, special care needs to be taken such as to prevent failure due to a slight misalignment between the barrel and the anvil. Typically, to ensure normal impact, the barrel is brought into contact with the anvil, and then the barrel is retracted and fixed in position. For more ductile materials than Mo, this procedure would be sufficient (for e.g. see Taylor impact test results on Ti reported in Revil-Baudard et al. [16]). However, in the case of Mo, more rigorous assessments of the perpendicularity (e.g. using levels) had to be conducted. As an example, in Fig. 8 are shown the photographs of deformed specimens following impact at approximately 150 m/s. In the case shown in Fig. 8(b), there was catastrophic failure due to a misalignment by an angle of less than 2° between the barrel and the anvil.

(a)

(b)

Fig. 8. Photographs of post-test Taylor specimens launched at 151 m/s showing the consequences of a slight misalignment between the barrel and the anvil: (a) an intact specimen with impact velocity of 151 m/s is recovered, if the perpendicularity between the barrel and the anvil is properly ensured. (b) catastrophic failure due to a very slight misalignment (less than 2°).

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121

and minor diameters of the impacted cross-section were of 3.635 mm and 3.489 mm, respectively. Plastic deformation extended to approximately 25 mm from the impact surface. On the other hand, both the RD and the TD specimen exhibited more axial deformation than the 45° specimen, and more radial expansion. As a general conclusion, irrespective of the specimen orientation, there is more deformation (change in length and radial expansion) with increasing impact velocity (see also Fig. 11). Fig. 9. Schematic representation of the specimen after impact.

3. Constitutive model for the polycrystalline molybdenum In summary, very high-strain data for refractory metals such as Mo can be acquired using the Taylor impact technique if special care is taken. A total of twelve Taylor impact tests at impact velocities in the range 140–165 m/s were successfully conducted. Upon impact, stress wave propagation is complex particularly during the first few microseconds of deformation and the material at the impact surface is subjected to the highest magnitude of stress and strain-rate during the initial transient stage of deformation due to the shock at impact [7,28]. Additionally, the impact surface is subjected to loading for duration greater than any other cross-section along the axial length of the specimen; therefore it should undergo the most radial expansion. Indeed, the attenuation of the strain-rate and stress along the length of the cylinder results in a reduction of radial deformation with increasing distance from the impact surface. Nevertheless, due to the anisotropy of the material, any cross-section becomes elliptical (for example, see the schematic representation of the post-test specimen shown in Fig. 9). Irrespective of the specimen orientation (i.e. RD, 45 or TD), the major axis of the cross-section of the recovered specimen is along the ND direction of the plate. This indicates that the material’s anisotropy remains the same irrespective of the strain-rate experienced by the material. On the basis of all the tests conducted i.e. quasistatic tests, high strain-rate SHPB tests, and Taylor tests, it can be concluded that the hardest to deform direction for this material is the ND direction. For each specimen orientation, the measurements of the geometry of the recovered specimens following the Taylor tests at about the same impact velocity (~151 m/s) is provided in Fig. 10. Note that for the 45° specimen, the final length was of 49.23 mm, the major

3.1. Governing equations To describe the combined effects of anisotropy and tension– compression asymmetry observed experimentally as well as the strain-rate influence on the plastic deformation of the polycrystalline Mo material studied, a macroscopic/structural level model will be developed within the framework of the mathematical theory of plasticity. The total rate of deformation D (the symmetric part of  −1 , where F is the deformation gradient) is considered to be the FF sum of an elastic part and a plastic part Dp . The elastic response is described as:

σ = Ce : (D − Dp )

where σ is an objective rate (Green–Naghdi derivative, see Green and Naghdi [29], ABAQUS, [30]) of the Cauchy stress tensor σ , Ce is the fourth-order stiffness tensor while “:” denotes the doubled contracted product between the two tensors. Assuming small elastic strains and linear isotropic elasticity, with respect to any coordinate system

2 ⎞ ⎛ e Cijkl = 2 G δ ikδ jl − ⎜ K − G⎟ δ ijδ kl, ⎝ 3 ⎠

(2)

with i, j, k, l = 1…3, δij being the Kronecker unit delta tensor while G and K are the shear and bulk modulus, respectively. Key in the formulation is the use of a yield function that accounts for both anisotropy and strength differential effects proposed by Cazacu et al. [31]. The equivalent stress, σ , according to this criterion is:

4.2

4.2 Initial Major Minor

3.8 3.4

Radius (mm)

Radius (mm)

(1)

3

Initial Major Minor

3.8 3.4 3 2.6

2.6 5

10 15 20 25 Axial position x (mm) (a) RD – 151m/s 4.2 Radius (mm)

0

30

0

5

10 15 20 25 Axial position x (mm) (b) 45° – 149m/s

30

Initial Major Minor

3.8 3.4 3 2.6 0

5

10 15 20 25 Axial position x (mm) (c) TD – 151m/s

30

Fig. 10. Measured major and minor outlines of post-test Taylor impact specimens, i.e. radius vs. the distance from the impacted end (x = 0) for : (a) RD specimen launched at a velocity of 151 m/s; (b) 45° specimen launched at an impact velocity of 149 m/s; (c) TD specimen launched at an impact velocity of 151 m/s.

G. Kleiser et al. / International Journal of Impact Engineering 96 (2016) 116–128

4.2

Major Radius (mm)

Major Radius (mm)

122

Initial 140m/s 151m/s 161m/s

3.8 3.4 3 2.6

4.2 Initial 148m/s 149m/s 154m/s

3.8 3.4 3 2.6

5 10 Axial position x (mm) (a) RD 4.2 Major Radius (mm)

0

15

0

5 10 Axial position x (mm) (b) 45° to the RD

15

Initial 141m/s 151m/s 161m/s

3.8 3.4 3 2.6 0

5 10 Axial position x (mm) (c) TD

15

Fig. 11. Measured major outlines of post-tests Taylor specimens for several impact velocities, i.e. radius vs. the distance from the impact surface from the impacted end (x = 0): (a) RD specimen; (b) 45° specimen; (c) TD specimen.

σ = m ( Σ1 − k ⋅ Σ1 ) + ( Σ 2 − k ⋅ Σ 2 ) + ( Σ 3 − k ⋅ Σ 3 ) , 2

2

2

(3)

where Σ1, Σ 2, Σ 3 are the principal values of the transformed stress tensor Σ defined as:

Σ = L :σ ′

(4)

where σ ‘ is the deviator of the Cauchy stress tensor, i.e. σ ′ = σ − σ m I , with I being the second-order identity tensor. In Eq. (4), L is a fourthorder symmetric tensor that describes the plastic anisotropy of the material. Modeling the anisotropy by means of a 4th order symmetric and orthotropic tensor ensures that the material’s response is invariant under any orthogonal transformation belonging to the symmetry group of the material (i.e. rotations along the symmetry axes). Furthermore, the minimum number of independent anisotropy parameters that is needed such as to satisfy these symmetry requirements is introduced. For example, for an orthotropic material, in the coordinate system associated with the material symmetry axes (x, y, z) and in Voigt notations the tensor L is represented by a 6 × 6 matrix given by:

⎡ L11 ⎢L ⎢ 12 ⎢ L13 L=⎢ ⎢0 ⎢0 ⎢ ⎣0

L12 L22 L23 0 0 0

L13 L23 L33 0 0 0

0 0 0 L44 0 0

0 0 0 0 L55 0

0⎤ 0⎥ ⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ L66 ⎦

(5)

The equivalent stress σ given by Eq. (3) is homogeneous of degree one in its arguments. Thus, if we replace Lij by ωLij, ω being any positive number, the expression for the effective stress remains unchanged. Hence, the anisotropy coefficients can be scaled by L11, or equivalently, set L11 = 1. Thus, for orthotropic materials the criterion involves six anisotropy coefficients. Note that for isotropic materials, L = I4, where I4 is the fourth-order symmetric identity tensor. In the expression of the effective stress given by Eq. (3), k is a material parameter, while m is a constant defined such that the equivalent stress, σ , reduces to the compressive stress along x (or RD) direction. Thus, m is expressed in terms of the anisotropy coefficients Lij, with i, j = 1…3 and the material parameter k as follows:

m =1

( Φ1 − kΦ1 )2 + ( Φ 2 − kΦ 2 )2 + ( Φ 3 − kΦ 3 )2

(6)

where Φ1 = − (2L 11 − L 12 − L 13 ) 3 , Φ 2 = − (2L12 − L 22 − L 23 ) 3;

Φ 3 = − (2L13 − L 23 − L 33 ) 3 .

It is assumed that isotropic hardening is governed by ε p , the equivalent plastic strain, associated to σ given by Eq. (3) using the work-equivalence principle [32]. To account for the combined effects of strain hardening and strain-rate, the Johnson–Cook [26] law is considered, i.e.

σ = Y (ε p, ε *)

(7)

where

Y (ε p, ε *) = (C1 + C 2ε pn ) (1 + C 3 ln (ε *)) .

(8)

The expression in the first bracket of the Johnson–Cook law (Eq. 8) represents the strain-hardening contribution to the flow stress, the constant C2 being the hardening coefficient while n is a strainhardening exponent. The expression in the second bracket models the influence of the strain-rate, with the constant C3 denoting the strain-rate sensitivity coefficient and ε * being a dimensionless plastic strain rate, defined as the ratio of the plastic strain rate ε to a reference strain rate ε0 (i.e. ε * = ε ε0 ). Assuming associated flow rule, the plastic part of the rate of deformation is:

∂σ Dp = λ ∂σ

(9)

where λ is the plastic multiplier and the equivalent stress, σ , is given by Eq. (3). The test results in unixial tension and compression for different orientations θ were used to identify the material parameters involved in the model, namely the anisotropy coefficients as well as the parameter k (see Eq. (3)–(6)). The theoretical expressions necessary for writing the cost function to be used in the minimization are given in Cazacu et al. [31] paper. The numerical values of these parameters at yielding are given in Table 4.

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Table 4 Model parameters for polycrystalline Mo. L11

L22

L33

L12

L13

L23

L44

L55

L66

k

1.0

0.997

1.0218

0.9285

0.933

0.9389

0.0641

0.06411

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Represented in Fig. 12 is the projection in the plane (σRD, σTD) of the theoretical yield surface of the material according to Eq. (3) in comparison with the experimental yield stresses (symbols). Note that the yield criterion accounts for the anisotropy and the tension– compression asymmetry of the material. The material parameters involved in the hardening law of Eq. (8) were identified based on uniaxial compression data at the strain rates of 10−5 s−1 (quasistatic) and 400 s−1 (SHPB). The numerical values are: C1 = 300 MPa; C2 = 320 MPa; C3 = 0.075; n = 0.3; ε0 = 10−5 s−1. Fig. 4 shows a comparison between the experimental stress–strain curves in RD compression at these strain-rates and the model. A user material subroutine (UMAT) was developed for the constitutive model described by Eqs. (1)–(9) and implemented in the commercial implicit FE solver ABAQUS Standard [30]. A fully implicit integration algorithm was used for solving the governing equations. 4. Model validation: simulation of the plastic deformation under impact loading FE simulations using the constitutive model described in Section 3 in conjunction with the dynamic implicit solver were performed for three impact velocities for both the RD and TD specimens. Given that a dynamic solver was used the equilibrium equations were solved for each increment. The time step was kept extremely small (between 10−9 and 10−7 s) in order to capture with highfidelity the propagation of the elastic and plastic waves. Due to the symmetry of the problem, only a quarter of the specimen was meshed with 6732 hexahedral linear elements (ABAQUS C3D8R) with reduced integration. The mesh is refined in the impact zone (see zoom shown in Fig. 5(c)) to accurately capture the deformation of the impacted surface of the specimen. The mesh size in the impact zone is of 0.1 mm. A mesh refinement study was carried to ensure that the numerical results are mesh insensitive. Furthermore, to ensure accurate F.E. calculations, the time step and mesh sized have been defined such that the elastic wave cannot pass through one element in less than two time increments. More precisely, for any

500

simulation, the maximal time step is of 10−7 s and considering that the wave speed in Mo is of 5400 m/s, the size of the mesh should be smaller than 5.4 mm. In the mesh used in this study, the largest element is located at the free-end of the specimen and is of length of 2.5 mm to ensure that the elastic wave propagation is accurately captured. The simulations were performed in two phases, associated to the specimen launching and impact, respectively. In the first phase, the impact velocity was applied to all of the nodes such as to represent the free flight or launch of the specimen. In the second phase, the impact of the specimen with the rigid anvil was reproduced by imposing a null velocity only to the nodes belonging to the impact surface while the other nodes were not constrained anymore and driven only by inertia. The end of the impact event is considered to occur when the axial velocity at the end of the specimen becomes null. In this manner, the experimental conditions are reproduced with fidelity. The elastic parameters values used in all simulations are: E = 330 GPa, ν = 0.3, where E is the Young modulus and ν is the Poisson coefficient (typical values taken from the literature). The density value was ρ = 10200 kg/m3. In the following, we present a detailed analysis of the deformation predicted by the model and quantitative comparison with the experimental measurements. As discussed in Section 2, irrespective of the orientation of the specimen, the geometry of the post-test specimen changes along its length or with the distance from the impact surface (see also data presented in Fig. 10). Figs. 13–15 show the profiles of the RD posttest specimen after impact at a velocity of 140, 151 and 165 m/s. Irrespective of the impact velocity, the major outline is along the ND (y) direction while the minor outline is along the TD (z) direction. In these figures, the predicted outlines are represented by solid lines while the experimental measurements taken at several distances from the impacted end (impacted end corresponds to x = 0) are shown as symbols. Note the good overall agreement between the numerical predictions and the experimental data for all impact velocities. Due to the orthotropy of the material, the deformed cylinder cross-sections are slightly elliptical, with the maximum radius being along the ND direction and the minor radius along the 90° or TD plate direction (z direction). It is also very worthy to note that the very little ellipticity of the specimen’ cross-section is well captured by the model. Furthermore, the fact that there is very little difference in the ellipticity of

0

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Fig. 12. Projection of the theoretical yield surface in the (σRD, σTD) plane (i.e. RD rolling and TD transverse directions) according to Cazacu et al. [31] orthotropic criterion in comparison with experimental yield stresses (symbols) for polycrystalline Mo.

5 4 3 2 1 0 -1 0 -2 -3 -4 -5

Major outline measured along ND

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Fig. 13. Comparison of the simulated and measured minor and major outlines (i.e. radius vs. the distance x from the impacted end) of a RD specimen launched at an impact velocity of 140 m/s Data are represented by symbols.

5 4 3 2 1 0 -1 0 -2 -3 -4 -5

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Major outline measured along ND

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Minor outline measured along RD Axial position x (mm)

the cross-section all along the specimen reinforces the observation made based on SHPB data, namely that the anisotropy of this material is very little influenced by the strain-rate (i.e. the coefficients of anisotropy could be considered independent of the strain rate). Comparison between the simulated outlines of the post-test TD specimens and the experimental measurements for all the three impact velocities tested are shown in Figs. 16–18. Note the very good agreement, also evidenced by comparing the measured and predicted axial deformation for each specimen (see Table 5). Next, the model will be used to get insights into the deformation process. Analysis of the local state fields at different instances in the deformation process is provided for the RD specimen and the impact velocity v = 151 m/s. Specifically, we will discuss the predicted evolution in time of state variables at various locations along the specimen axis (see Fig. 19). Each location of interest is defined

by the distance x from the impacted end in the initial configuration. The evolution of the hydrostatic pressure ( p = −1 3 tr (σ ) ) for the total duration of the test is plotted in Fig. 20 (a). Note that the model predicts a large amplitude oscillation in pressure (transient loading), which attenuates into a more stable uniform distribution

5 4 3 2 1 0 -1 0 -2 -3 -4 -5

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Minor outline measured along RD Axial position x (mm)

Fig. 18. Comparison of the simulated and measured minor and major outlines (i.e. radius vs. the distance x from the impacted end) of a RD specimen launched at an impact velocity of 161 m/s. Data are represented by symbols.

50

Table 5 Measured and predicted change in axial dimension of the specimens after impact. Specimen

Minor outline measured along TD Axial position x (mm)

Fig. 15. Comparison of the simulated and measured minor and major outlines (i.e. radius vs. the distance x from the impacted end) of a RD specimen launched at an impact velocity of 165 m/s Data are represented by symbols.

5 4 3 2 1 0 -1 0 -2 -3 -4 -5

Radius (mm)

Fig. 17. Comparison of the simulated and measured minor and major outlines (i.e. radius vs. the distance x from the impacted end) of a TD specimen launched at an impact velocity of 151 m/s. Data are represented by symbols.

Radius (mm)

Fig. 14. Comparison of the simulated and measured minor and major outlines (i.e. radius vs. the distance x from the impacted end) of a RD specimen launched at an impact velocity of 151 m/s Data are represented by symbols.

RD RD RD 90° 90° 90°

Impact velocity

Experimental LFinal/LInitial

Predicted LFinal/LInitial

140 151 165 141 151 161

0.923 0.915 0.903 0.930 0.916 0.906

0.915 0.905 0.890 0.920 0.911 0.902

4

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Fig. 16. Comparison of the simulated and measured minor and major outlines (i.e. radius vs. the distance x from the impacted end) of a TD specimen launched at an impact velocity of 141 m/s. Data are represented by symbols.

2

Distance from Impact Surface 1: x= 0.42 mm 2: x= 3.20 mm 3: x=19.0 mm 4: x= 49.0 mm

Fig. 19. Schematic view showing the locations of interest along the centerline of the specimen. Each location is defined by the distance from the impact surface.

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Fig. 21. Predicted axial velocity at various locations along the centerline for a RD specimen launched at an impact velocity of 151 m/s.

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within 15 μs (quasi-steady loading). The largest oscillation in pressure is predicted at distances closest to the impacted end. For example, at x = 0.42 mm from the impacted end it is predicted a peak positive pressure of 7.5 GPa and a peak negative pressure of −2.2 GPa (recall that according to the sign convention, compressive pressures are positive). The development of such high tensile pressures is consistent with experimental results and the explanation of the lateral fracture that may occur/potential pulling apart of the specimen is in agreement with published literature values [21].Oscillation in pressure is also predicted at x = 3.2 mm from the impacted end, but the magnitude is significantly reduced, the peaks being of 4 GPa and −1.5 GPa, respectively. It also interesting to note that within 15 μs, at locations beyond 3.2 mm from the impacted end, pressure oscillations are less than ±0.5 GPa (see Fig. 20(b)). It is also important to note that within 3 μs of the impact, the pressure reaches about 35% of the length of the specimen, yet at the respective location (x = 19 mm from impacted end) its amplitude is significantly smaller than closer to the impacted end and it is almost exclusively compressive (Fig. 20(b)). At t = 20 μs from the impact, the pressure at the location x = 19 mm is higher than that predicted at 0.42 mm, 3.2 mm, and 49 mm and ranges between 225 and 375 MPa. On the other hand, at x = 49 mm, the pressure exhibits rapid oscillations between -100 MPa and 350 MPa, with an average of approximately 125 MPa. In summary, the model predicts large amplitude oscillation in pressure which attenuates dramatically within 15 μs. It is also worthwhile analyzing the predicted attenuation of the velocity and the

distribution of the plastic strain-rate in the specimen, and relate to the predicted pressure distribution. In Fig. 21 is shown the evolution of the axial velocity as a function of time for the same locations along the specimen centerline. Note that at the location near the impacted end (x = 0.42 mm), an extremely rapid deceleration occurs followed by a slight increase in velocity and a more gradual decrease toward zero velocity, which is reached at t = 15 μs. At the location x = 3.2 mm, the oscillation in velocity occurs during the first 5 μs after impact, and there is a steady decrease in velocity thereafter. Nevertheless, after 15 μs, at both x = 0.42 mm and x = 3.2 mm, the axial velocity is zero. At location x = 19 mm and x = 49 mm the decrease in velocity is more gradual, and similar, which is consistent with these material points experiencing quasi-steady loading. The time at which the velocity becomes zero at those locations is of 46 μs and 55 μs, respectively. The same trends in velocity attenuation are predicted irrespective of the impact velocity, but with different magnitude of the oscillations and time at which the velocity becomes zero (see Fig. 22 for the predicted evolution of the axial velocity with time for several impact velocities v = 140 m/s, v = 151 m/s, and v = 165 m/s). The free end of the specimen oscillates due to the propagation of the elastic wave. For example, this is seen in Fig. 21 (see the predicted velocity evolution at location 4). However, the most important in Taylor experiment is the accurate prediction of the plastic wave propagation, since it determines the final shape of specimen. The predicted axial velocity evolution along the centerline indicates that the specimen underwent a very rapid deceleration. For the impact velocity, v = 151 m/s in Fig. 23 are shown the isocontours of the equivalent plastic strain rate ε p within the specimen at different times after the impact. First, let us note that at t = 5.3 μs very

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Fig. 22. Predicted evolution of the axial velocity with time at the locations x = 3.2 mm (solid lines) and x = 19 mm (interrupted lines) along the centerline for an RD specimen launched at various impact velocities: v = 140 m/s (blue), v = 151 m/s (red), and v = 165 m/s (black). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 23. Predicted isocontours of plastic strain rate at different times from the impact of an RD specimen launched at 151 m/s.

high plastic strain rates of the order of 104 s−1are predicted. The largest strain rates, of the order of 5.6.104 s−1, are near the impacted end and along the centerline while at the outer radius the plastic strain rate is only 1.6.104 s−1. Furthermore, the largest strain rates were experienced by the material during the initial, transient phase of the deformation when non-zero plastic strain rates occur only in the vicinity of the impacted end. At 8.4 μs, this zone of peak plastic strain-rate has grown, but peak strain rates of the same order also appear along the outer radius of the specimen. By 10.3 μs from impact, these regions along the outer radius with high plastic strainrate values have propagated toward the centerline and coalesced forming a circular region experiencing high plastic strain-rate that is moving away from the impacted end. On the other hand, the plastic strain-rate of the material at the impact surface is rapidly diminishing. By 22.5 μs from impact, the material at the impacted end is no longer experiencing high strain-rates, the specimen geometry undergoing expansion much further away from the impact surface. The predicted isocontours of the plastic strain rate at different

times from impact and the sequence of events taking place are consistent with the interpretation of the deformation of the specimen due to Taylor [6]. To further illustrate the difference in magnitude in plastic strainrate experienced at different locations along the centerline in Fig. 24 is shown the plastic strain-rate evolution with time at x = 0.42 mm, 3.2 mm, and 19 mm, respectively. The plastic strain-rate at x = 0.42 mm has a peak value of 15.104 s−1 whereas at x = 3.2 mm the peak value is of 6.5.104 s−1. At x = 19 mm the peak value is of only 0.8.104 s−1 at approximately 45 μs. At x = 49 mm, plastic deformation does not occur. This is consistent with the pressure evolution predicted at these locations (see Fig. 20). The evolution in time of the equivalent plastic strain at the same center-line locations is shown in Fig. 25. As expected the strain near the impacted end (i.e. at x = 0.42 mm) is the largest, reaching a peak of 0.57 at t = 14 μs. An interesting observation concerns the predicted final peak value of the plastic strain at x = 3.2 and x = 19 mm respectively. Although

G. Kleiser et al. / International Journal of Impact Engineering 96 (2016) 116–128

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Plastic Strain Rate (1/s)

1

1: x= 0.42 mm 2: x= 3.20 mm 3: x=19.0 mm

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50000

127

Fig. 26 shows the distribution of the equivalent plastic strain for the RD specimen corresponding to the end of the test. Note that a zone of small equivalent plastic strain is predicted very close to the impacted end. Furthermore, there are locations with the same final equivalent plastic strain (e.g. the two highlighted locations), which is consistent with the predicted transition from transient to quasistable wave propagation.

2 3

5. Conclusions

0 0

5

10 15 20 25 30 35 40 45 50 55 60 Time (μs)

Fig. 24. Predicted evolution of the plastic strain-rate with time at various locations along the centerline for an RD specimen launched with an impact velocity of 151 m/s.

this final value is the same, the deformation experienced by these two material points during the test is completely different. For example, at the location x = 3.2 mm, a plastic strain of ~ 0.14 is experienced after t = 15 μs from impact but at the location x = 19 mm, the same level of plastic strain is reached only after t = 46 μs . This is consistent with the evolution in axial velocity and pressure predicted at these locations, and the time from the impact at which quasi-stable deformation occurs.

Equivalent Plastic Strain

0.6 1

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1: x= 0.42 mm 2: x= 3.20 mm

0.3

3: x=19.0 mm

0.2

2

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0 0

5 10 15 20 25 30 35 40 45 50 55 60 Time (μs)

Fig. 25. Predicted evolution of the equivalent plastic strain with time at various locations along the centerline for an RD specimen launched with an impact velocity of 151 m/s.

In this paper, an experimental study on the quasi-static and highrate plastic deformation of a polycrystalline molybdenum was presented. Quasi-static uniaxial tensile and compression tests were performed to characterize the plastic behavior of the material. Based on the quasi-static data, it can be concluded that the material exhibits tension–compression asymmetry (harder in compression than in tension) and plastic anisotropy. The Lankford coefficients vary from 0.66 to 0.93 depending on the relative orientation of the considered specimen. Evaluation of the ellipticity of the deformed compression specimens allowed uncovering that although the material exhibits strong strain anisotropy in tension, it has a weak strain anisotropy in uniaxial compression. Compression tests were conducted at high-strain rates (400/s) using the split–Hopkinson pressure bar technique to evaluate the strain-rate sensitivity of the material. A dynamic increase factor of three in flow stress magnitude was observed, while the hardening rate remained unchanged for all orientations. Due to the Mo’s limited tensile ductility, characterization of the behavior under impact is particularly challenging. To the best of our knowledge, for the first time, Taylor impact tests were successfully conducted on this material for impact velocities in the range of 140–165 m/s for specimens taken along the rolling direction, the transverse direction and 45° to the RD. For impact velocities beyond this range, very high negative pressures (tensile loading) are generated in the specimen immediately after impact leading to failure. A constitutive model was developed to describe the observed behavior. Key in the formulation was the use of a yield function proposed by Cazacu et al. [31] that accounts for both anisotropy and tension/compression asymmetry. The anisotropy coefficients and strength differential parameter were identified from the quasistatic test data. The observed strain-rate sensitivity was modeled with a Johnson–Cook law that was identified based on the SHPB data in compression. The Taylor impact tests, which involve very high strain rates (to the order of 105 s−1) served for only for validation.

Fig. 26. Distribution of the equivalent plastic strain within the RD specimen after impact. Note the two highlighted regions have similar strain values.

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The model was implemented in the FE code ABAQUS. For this purpose, an UMAT was developed. A fully-implicit algorithm was used for solving the governing equations. The predictive capabilities of the model were assessed by comparing the measured outlines of the Taylor impact specimens to the F.E. predictions. Given that the F.E. simulations were carried out using an implicit dynamic solver, the equilibrium equations are solved for every time increment. This in turn ensured high-fidelity in the prediction of the profiles of the deformed specimens. Furthermore, the model was used to gain understanding of the dynamic deformation process in terms of time evolution of the pressure, the extent of the plastically deformed zone, distribution of the local plastic strain rates, and when the transition to quasi-stable deformation occurs. It was thus shown that this model has the potential to be used for virtual testing of complex systems.

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