Magnetically applied pressure-shear (MAPS) is a new experimental technique that provides a platform for direct measurement of material strength at extreme pressures. The technique employs an imposed quasi-static magnetic field and a pulsed power generator that produces an intense current on a planar driver panel, which in turn generates high amplitude magnetically induced longitudinal compression and transverse shear waves into a planar sample mounted on the drive panel. In order to apply sufficiently high shear traction to the test sample, a high strength material must be used for the drive panel. Molybdenum is a potential driver material for the MAPS experiment because of its high yield strength and sufficient electrical conductivity. To properly interpret the results and gain useful information from the experiments, it is critical to have a good understanding and a predictive capability of the mechanical response of the driver. In this work, the inelastic behavior of molybdenum under uniaxial compression and biaxial compression-shear ramp loading conditions is experimentally characterized. It is observed that an imposed uniaxial magnetic field ramped to approximately 10 T through a period of approximately 2500 $\mu s$ and held near the peak for about 250 $\mu s$ before being tested appears to anneal the molybdenum panel. In order to provide a physical basis for model development, a general theoretical framework that incorporates electromagnetic loading and the coupling between the imposed field and the inelasticity of molybdenum was developed. Based on this framework, a multi-axial continuum model for molybdenum under electromagnetic loading is presented. The model reasonably captures all of the material characteristics displayed by the experimental data obtained from various experimental configurations. In addition, data generated from shear loading provide invaluable information not only for validating but also for guiding the development of the material model for multiaxial loadings.

## I. INTRODUCTION

To characterize material behavior under large amplitude, high rate loading, shock wave compression has been commonly used as a loading method. For weak shocks at relatively low pressures, the rise times of the associated shock waves are long enough to be resolved by diagnostic instruments. The structure of the wave can then be used to deduce material behavior for the loading regime associated with the applied shocks. For strong shocks, the wave profile is overdriven; the structure is not discernible and only the equilibrium state is measured. This equilibrium state constitutes one point of the principal Hugoniot, but contains little information on the functional dependence of the material properties on the loading path or history.

To generate large amplitude structured waves in the multi-megabar pressure regime, ramp wave loading over sufficiently long time scales has been an essential experimental approach. In the past two decades, significant progress has been made on the development of a pulsed-power-based technique to initiate ramp wave loading.^{1–4} The Z machine at Sandia National Laboratories is a fast pulsed power generator which is typically used for producing ramp waves to multi-megabar pressures.^{1–3} Veloce, a smaller version of this generator, has also been developed for studying material behavior in the 10–20 GPa range under ramp loading.^{4}

A typical experimental configuration for the Veloce pulser is shown in Figure 1 and consists of a pair of load panels for carrying the current produced by the machine. The bottom panel includes only a laser window for interferometry studies, while the top panel has a material sample between the panel and the window. Upon firing, a nearly planar stress wave is produced in the central region of the panel, sample, and window for times of interest. The longitudinal stress response of the sample is thus tested for uniaxial strain conditions. In this configuration, the measurement obtained from the bottom panel is used as a reference for characterizing the load applied to the sample. The planar ramp wave is initiated along the longitudinal direction of the sample by the Lorentz force, $J\xaf\xd7B\xaf$, resulting from an applied current pulse density ($J\xaf$), which is the applied uniform current divided by the width of the panel, and the associated magnetic field ($B\xaf$) formed between the panels due to this current flowing in a loop. On Veloce, the peak load current is ramped to about 2.5 MA over about 400 ns, resulting in a ramped stress input to the sample assembly.

One important material property associated with the extreme loading regime is material strength, which represents a material's ability to resist deformation. Different techniques have been proposed to determine material strength under extreme loading.^{5} To extract material strength from wave profiles obtained from uniaxial-strain compression experiments, a self-consistent method was developed.^{6,7} This technique has been applied to shock^{8} and ramp wave loading.^{9,10} To characterize the strength at the peak stress state, additional release and reshock waves are generated after initial shock loading^{8} by either unloading or reshocking the sample material from the shocked state. For ramp compression, unloading can be achieved by reduction of current after its peak. In both cases, Lagrangian analysis methods were used to extract material strength^{9,10} from the unloading or reloading wave propagation. A priori for this method is measurement of a pair of *in-situ* particle velocity histories at two different sample locations, ideally at the input and output sides of a planar sample. Usually, these measurements are made with interferometer techniques through a transparent window bonded to the sample (see Figure 1). To determine the *in-situ* velocities from measured velocities, either at the sample/window interface or the free surface of a sample, several methods have been developed, including the incremental impedance matching method,^{8–10} backwards integration method,^{11,12} a technique referred to as the backward characteristics analysis method,^{13–15} and more recently, the transfer function mapping method.^{16} All of these methods require considerable assumptions and analysis to infer strength properties at high pressure.

A more recent development for direct strength measurement is to use the Veloce pulser to generate combined compression and shear ramp waves.^{17} The configuration for this technique, referred as MAPS (Magnetically Applied Pressure-Shear), is shown in Figure 2, where an additional shear traction, $J\xaf\xd7B\xaf0$, is generated simultaneously with the applied longitudinal traction through the interaction between the drive current (J) and an externally imposed magnetic field (B_{0}) oriented along the longitudinal direction. The external field is applied for a total duration of about 5 $ms$(field strength is approximately Gaussian with rise and fall times approximately 2.5 $ms$ each) and is within 1% of peak for about 250 $\mu s$ before the main current pulse (roughly 2 $\mu s$ in duration) is applied to the drive panel. This gives sufficient time for the imposed field to uniformly diffuse into the sample assembly, including the panel, which is necessary to generate the shear traction. The combined waves that are produced consist of a planar longitudinal wave with a peak stress of about 10–15 GPa that propagates through the panel and into the sample, followed by a planar shear wave with a maximum amplitude limited by the shear strength of the drive panel. The transmitted shear wave can then be used to determine the strength of the sample, provided that its strength is less than that of the driver.^{17} The detailed interaction between the longitudinal and shear wave loadings will be illustrated in Section VII. Compared to the indirect strength measurements based on uniaxial compression experiments, the MAPS technique provides a potential new platform for direct measurement of material strength, which has been a goal of the shock wave community for several decades.

For longitudinal compression experiments alone, as illustrated in Figure 1, annealed 1100 aluminum^{4} with a minimum of purity of 99% and an initial compressive strength of about 0.1 GPa has typically been used as the driver material. Since this type of experiment is driven by a longitudinal wave, typically referred to as a pressure wave, materials with negligible strength, such as pure aluminum, are preferred so the driver strength will minimally affect the experimental observation and analysis of the waves produced in the sample. However, for MAPS experiments, the load panel must have sufficient strength in order to transmit shear traction to the test sample that exceeds its shear strength. This was the main reason for selecting molybdenum as the driver material for the MAPS experiments. Molybdenum yield strength has been reported to be 1.3,^{18} 1.4,^{19} and 1.6 GPa.^{20} The corresponding shear yield strength can be estimated as half of the above values using Tresca yield criterion, i.e., 0.6–0.8 GPa. The Hugoniot elastic limit (HEL) was reported to be around 2.15–24 GPa^{18} and 2.3–2.8 GPa.^{19} As mentioned earlier, the shear yield strength limits the amplitude of the shear wave that can be applied in MAPS.

The finite strength associated with the load panel brings up some new issues and challenges for the MAPS technique. In order to properly design the experiment and interpret the results, it is critical to have a thorough understanding and a predictive capability for the driver's mechanical response to the applied electromagnetic loading. For driver materials whose strength can be ignored, such as pure aluminum, an equation of state that describes the mean stress part of the stress tensor should, in general, be sufficient for predicting its behavior under imposed ramp loading. However, for materials with finite strength, an appropriate strength model to describe the deviatoric component of the stress tensor must be an integral part of the predictive capability. This issue is further complicated by the mixed mode loading, i.e., combined compression and shear, experienced by the driver and specimen in MAPS experiments.

Another issue is related to the analysis of the experiments. As mentioned earlier, backward analyses^{11–15} have been used to determine the *in-situ* material response and reconstruct the stress or velocity inputs for the ramp wave experiments.^{9} The underlying assumption for the backward analyses is that the driver material has negligible strength. This assumption is reasonable for pure aluminum, but obviously not for molybdenum. Even with a reliable strength model, the backward approach for a load panel with significant strength is still questionable as the material strength in general has strong path-dependence. Furthermore, all the techniques developed so far for retrieving *in-situ* material response have been limited to uniaxial strain loading. Extraction of the *in-situ* material response under mixed mode, or biaxial loading is an area that has not been explored. Another related issue is the numerical simulation of the ramp wave experiments. As indicated above, backward analysis has been used to extract the equivalent stress or velocity inputs for the ramp wave experiments. Thus even though the loading is through a current pulse, the simulation can often be carried out as purely mechanical greatly simplifying the analysis. Again, this approach does not work for molybdenum panel owing to the aforementioned issues associated with the backward analyses.

In summary, for analyzing MAPS experiments, because of the finite strength of the molybdenum panels and mixed mode loading, a forward simulation which is based directly on the imposed current input is necessary. The simulation needs to incorporate the complex interaction between the electromagnetic field and the resultant mechanical loading. A key component for this approach is a molybdenum material model that is applicable to electromagnetically applied compression-shear loading. This brings up the main objective of the current work.

In this study, the mechanical behavior of molybdenum under ramp wave loading with and without an imposed magnetic field is experimentally characterized. Based on the insights gained from the experiments, a continuum model for molybdenum under applied magnetic field and mixed compression-shear mechanical loadings is proposed and validated against the experiments. It is also demonstrated that data generated from shear loading provide some unique insights on the mechanical behavior of molybdenum, which are vital not only for validation but also for guiding the model development.

## II. EXPERIMENTAL DATA FOR MOLYBDENUM

The experiments performed in this work are summarized in Table I. The configurations with aluminum panels were typically used for baseline material characterization under uniaxial compression,^{9} and those with molybdenum panels were for the MAPS experiments. Likewise, experiments with LiF windows were focused on the compressive response, while MAPS experiments required the use of zirconia anvils to support shear waves. Both types of experiments were used in this study. The variety of the experiments also provides valuable information for validating the material model for different experimental configurations and loading conditions.

. | Driver . | Sample . | Window/Anvil . | . | ||
---|---|---|---|---|---|---|

Experiment . | Material . | Thickness (mm) . | Material . | Thickness (mm) . | Material . | B_{0} (T)
. |

EX1A-T | Al | 1.504 | Mo | 1.397 | LiF | 0 |

EX1A-B | Al | 1.504 | LiF | 0 | ||

EX1B-T | Al | 1.506 | Mo | 1.397 | LiF | 0 |

EX1B-B | Al | 1.506 | LiF | 0 | ||

EX2A-T | Al | 1.503 | Mo | 1.419 | LiF | 10.6 |

EX2A-B | Al | 1.503 | LiF | 10.6 | ||

EX2B-T | Al | 1.500 | Mo | 1.412 | LiF | 7.9 |

EX2B-B | Al | 1.500 | LiF | 7.9 | ||

EX3-T | Mo | 1.164 | Mo | 1.408 | LiF | 0 |

EX3-B | Mo | 1.164 | LiF | 0 | ||

EX4-B | Mo | 1.158 | ZrO_{2} (2.471 mm) | 0 | ||

EX5-B | Mo | 1.122 | ZrO_{2} (1.981 mm) | 9.3 |

. | Driver . | Sample . | Window/Anvil . | . | ||
---|---|---|---|---|---|---|

Experiment . | Material . | Thickness (mm) . | Material . | Thickness (mm) . | Material . | B_{0} (T)
. |

EX1A-T | Al | 1.504 | Mo | 1.397 | LiF | 0 |

EX1A-B | Al | 1.504 | LiF | 0 | ||

EX1B-T | Al | 1.506 | Mo | 1.397 | LiF | 0 |

EX1B-B | Al | 1.506 | LiF | 0 | ||

EX2A-T | Al | 1.503 | Mo | 1.419 | LiF | 10.6 |

EX2A-B | Al | 1.503 | LiF | 10.6 | ||

EX2B-T | Al | 1.500 | Mo | 1.412 | LiF | 7.9 |

EX2B-B | Al | 1.500 | LiF | 7.9 | ||

EX3-T | Mo | 1.164 | Mo | 1.408 | LiF | 0 |

EX3-B | Mo | 1.164 | LiF | 0 | ||

EX4-B | Mo | 1.158 | ZrO_{2} (2.471 mm) | 0 | ||

EX5-B | Mo | 1.122 | ZrO_{2} (1.981 mm) | 9.3 |

EX1A and EX1B are nominally identical experiments. Both are included here to illustrate the typical experimental variations as shown in Figure 3. The configuration is shown in Figure 1. On the top panel, a molybdenum sample is sandwiched between an aluminum driver and a LiF window. On the bottom panel, the LiF window is attached directly to the aluminum driver. The measurement obtained from the bottom panel is typically used as a reference for retrieving and/or characterizing the load applied to the sample. The molybdenum used in these tests was 99.95% pure commercially available molybdenum (ASTM B386 type 361). Samples were cut from plate stock and had an average density of 10.20 g/cm^{3}.

Figure 4 shows the effect of an externally applied magnetic field on the mechanical behavior of molybdenum under ramp loading. Experiment EX2A is similar to EX1A/B in Figure 3, but with an external magnetic field (B_{0}) of 10.6 T imposed normal to the plane of current flow. The imposed field was ramped up to peak intensity through a period of approximately 2500 $\mu s$ and held within 1% of the peak for about 250 $\mu s$ before the ramp compression was applied to the sample. The imposed field is approximately Gaussian and remains within 1% of the peak value for another 250 $\mu s$ following the ramp compression testing. The total active ramp compression and subsequent release lasted for about 2 $\mu s$ with the compressive part of the pulse of interest in this work lasting for about 1 $\mu s$. Hence, the applied field can be considered constant during these experiments. As shown by Figure 4, the elastic-plastic transition point for EX2A is lower than that observed in EX1A, indicating a decreased strength with application of a peak applied magnetic field of 10.6 T for a few millisecond applied duration. The lowered strength suggests that an imposed magnetic field, even for a short time period, appears to have induced a discernible softening effect. Magnetic annealing has been reported in the literature, but mostly on a time scale longer than that for the experiment reported here and for lower applied magnetic fields. In the study by Pavlov, Pereturina, and Pecherkina,^{21} it was found that imposition of a magnetic field of 0.25 T had reduced the strength of molybdenum during low strain rate, longer time-scale, experiments. In addition to molybdenum, enhanced thermal annealing by the addition of imposed magnetic fields was also reported for other paramagnetic materials, such as cold-rolled aluminum alloy^{22,23} and cold-rolled pure titanium.^{24,25} In these studies,^{22–25} the magnetic fields were around 17–20 T and the soaking periods ranged from minutes to hours. The magnetic-field induced annealing effect is generally attributed to the enhanced dislocation mobility, or so-called magneto-plasticity, due to the unlocking of dislocations that are pinned by paramagnetic defects through spin-dependent interactions.^{26} EX2B, which is similar to EX2A but with a reduced applied field of 7.9 T, is shown in Figure 5 and indicates a similar annealing effect.

Before discussing the constitutive model proposed to describe the observed mechanical response, the remaining experiments are briefly summarized. Each will be discussed further in relation to relevant simulation results in Sections VI and VII.

The experiments discussed so far were all based on aluminum drive panels. Shown in Figure 6 are the experimental results (EX3) obtained with the configuration shown in Figure 1, but with molybdenum drive panels. On the bottom panel, EX3-B, the Mo thickness was 1.164 mm. On the top panel, EX3-T, a molybdenum sample was placed between the panel and a LiF window, effectively resulting in a thicker drive panel so that the effect of wave propagation in the molybdenum for a combined thickness of 2.572 mm could be studied. The data shown in Figure 6 are the measured particle velocities at the molybdenum-LiF interface. As shown by this figure, the data exhibit a stiff response for the thinner sample, but the elastic-plastic precursor becomes more pronounced as the thickness increases.

Shown in Figure 7 is the result from experiment EX4-B, which has the same configuration as EX3-B, but with the LiF window replaced by zirconia. The velocity was measured at the zirconia free surface. It is noteworthy that the data also exhibited a very stiff response, i.e., no discernible elastic-plastic transition, in agreement with EX3-B shown in Figure 6. Compared to LiF whose acoustic impedance is around 22 MRayl, zirconia (Z ∼ 44 MRayl) has better impedance match with molybdenum (Z ∼ 65 MRayl). Although wave interaction is inevitable and contributes to part of the wave profile, the fact that both experiments exhibited similar behavior indicates that the stiff response is not a sole result of wave interaction. As will be shown later, it is mainly attributed to the rate sensitivity of the mechanical behavior of molybdenum.

Shown in Figure 8(a) are the longitudinal velocity data obtained from experiment, EX5-B, which is similar to EX4-B, but with an externally imposed magnetic field (B_{0} = 9.26 T). The data obtained with the imposed field also showed a very stiff response similar to EX3-B that did not have an external field. Due to the imposed B_{0}, a shear traction was simultaneously induced in the sample, transmitted into the ZrO_{2} anvil, and the corresponding transverse velocity was measured on the free surface of the anvil, which is shown in Figure 8(b). As discussed earlier, these measurements allow a direct measure of the shear response of molybdenum under dynamic longitudinal stress loading.

## III. CONSTITUTIVE FRAMEWORK FOR MECHANICAL ANALYSIS

### A. General framework

To provide a basis for electromagnetic modeling and simulation, the previously developed constitutive model framework for mechanical analysis^{27} is summarized first.

The rate of deformation or logarithmic strain rate tensor ($\epsilon \u0307ij$) is the sum of elastic, $\epsilon \u0307ije$, and plastic strain rate tensor, $\epsilon \u0307ijp$, and the total stress ($\sigma ij$) is the sum of $\sigma ije$ and $\sigma ijv$, where $\sigma ije$ is the equilibrium stress and $\sigma ijv$ is the viscous stress. In this work, the polycrystalline material studied is assumed to be isotropic. Thus, $\sigma ije=P\delta ij+\sigma ije\u2032$, where *P* is the equilibrium pressure, $\sigma ije\u2032$ is the equilibrium deviatoric stress, and $\delta ij$ is the Kronecker delta. Furthermore, $\sigma ijv$ is assumed to be a contribution of the bulk or mean viscous stress ($Q$) only and the deviatoric viscous stress is included in the visco-plastic constitutive model that will be discussed later. In other words, $\sigma ijv=Q\delta ij$, where $Q=\sigma kkv/3$ is the mean viscous stress.

The general form of the constitutive model is in the following form:

where $\theta $ is the absolute temperature, $s$ the entropy, $Bijkl$ the adiabatic stress strain coefficient, $\rho $ the current material density, $\Gamma ij$ the Grüneisen parameter, $Cv$ the specific heat at constant elastic configuration, and $qi$ the heat flux vector.

With the aforementioned isotropy assumption, the expressions for $Bijkl$ and $\Gamma ij$ can be simplified significantly. Specifically, $\Gamma ij=\Gamma \delta ij$, and the equilibrium pressure is determined by the Mie-Grüneisen equation of state expressed in the following form:

where $PH$ is Hugoniot pressure, $\mu =(v0/v)\u22121$, with $v0$ and $v$ being the initial and current specific volumes, respectively, and $e$ is the internal energy per unit mass. The Grüneisen parameter is given as $\Gamma =\Gamma 0vv0$ with $\Gamma 0$ being the initial value. $PH$ is expressed as a third degree polynomial, i.e.,

where $K1$, $K2$, and $K3$ are material constants.

The viscous mean stress is a function of volumetric strain rate, i.e.,

with functional form given by

where b is arbitrarily set to $1\xd710\u22125\u2009GPa\u22c5s1/2$, which corresponds to a very small bulk viscosity. This term is used mainly to enhance numerical stability.^{28}

The equilibrium deviatoric stress is determined by the relation

where G is the shear modulus. The plasticity model for $\epsilon \u0307ijp$ varies with materials and will be described later.

For the thermal response, a linear Fourier heat conduction law was used, i.e.,

where $k$ is the thermal conductivity.

### B. Mechanical viscoplastic model for molybdenum

The mechanical model for molybdenum is based on an over-stress viscoplasticity formulation in which the plastic strain rate is given by

with

where $\sigma \xaf=(32\sigma ije\u2032\sigma ije\u2032)1/2$ is the effective stress; $\epsilon \xaf\u0307p=(23\epsilon \u0307ijp\epsilon \u0307ijp)1/2$ is the effective plastic strain rate; Y, with the initial value Y_{0}, is a threshold strength below which the material behaves elastically; and *A* is a constant. The evolution of the shear modulus (G) and the threshold strength (Y) essentially follows the Steinberg, Cochran, and Guinan (SCG) model,^{29} but with some modification of the temperature dependence. The SCG model does not specifically deal with melting, the temperature at which material strength is completely lost. As will be discussed later, in the electromagnetic simulations, Joule heating, which is not considered in the mechanical simulation, can result in the melting of the driver material. Hence, the temperature dependence term, $(G\theta \u2032G0)(\theta \u2212\theta 0)$, in the SCG model is replaced by that used in the Johnson-Cook model,^{30} i.e., $(1\u2212\theta *m)$, where $\theta *=(\theta \u2212\theta room)/(\theta melt\u2212\theta room)$ with $\theta room$ and $\theta melt$ being the room and melting temperatures, respectively. In other words, in the modified SCG model

subject to the limitation that

where $G0$ and $Y0$ are the initial shear modulus and yield strength, respectively, $\eta =v0/v=1+\mu $, $\beta $ and $n$ are strain hardening parameters, $\epsilon \xafp$ was defined earlier, $\epsilon \xafip$ is the initial value of $\epsilon \xafp$, and $Ymax$ represents the maximum attainable strength.

The temperature dependences of thermal conductivity (*k*) and the specific heat ($Cv$) are assumed to follow the empirical formulae given in Ref. 31, i.e.,

and

## IV. INCORPORATING MAGNETIC EFFECTS IN THE ELECTROMAGNETIC MODEL

### A. General framework

Before introducing an empirical electromagnetic model for molybdenum, a general framework is proposed first. The framework provides a basis for the model and also serves as a tool to systematically study and understand the experiments.

The framework is based on the dislocation density. The plastic strain rate is related to dislocation motion as follows:^{32}

where *b* is the Burgers vector, $\rho D$ the dislocation density, and $v\xaf$ the average dislocation velocity. The dislocation density could be further decomposed into mobile and immobile dislocations, and strictly speaking, $\rho D$ in Eq. (17) should refer to mobile dislocation density. However, these details are not considered here. The dislocation velocity $v\xaf$ depends on stress ($\sigma \xaf$), temperature ($\theta $), magnetic field (B), and the material's resistance to dislocation motion. The mobility of dislocations could be enhanced through thermal or magnetic activation, or impeded by phonon or electron damping. As observed by Kravchenko,^{33} the magnetic field can also increase the electron component of viscous drag, which would tend to increase the flow stress.

To complete the formulation, an evolution rule for the dislocation density and its correlation with material strength is needed. The evolution can be written as

where $h(\rho D,\theta ,\epsilon \xaf\u0307p)$ is a hardening or dislocation generation function and $d(\rho D,\epsilon \xaf\u0307p,\theta ,B)$ is a dynamic recovery function. Eqs. (17) and (18) also relate the dislocation density to plastic strain. To include the static recovery due to temperature or magnetic field, a static recovery term, $r(\rho D,\theta ,B)$, can be included in Eq. (18) as

The possible pressure dependence of dislocation density evolution, such as that used in the Steinberg-Cochran-Guinan model mentioned earlier, can also be incorporated in Eq. (19) by including pressure as an independent variable. The threshold strength Y can be related to dislocation density (immobile dislocation density, strictly speaking), for example, through Taylor's hardening law,^{34} i.e.,

### B. Empirical electromagnetic model for molybdenum

Characterization of the framework described above would be a very challenging and time-consuming task. Since the general understanding of the effects of magnetic field on mechanical behavior is quite limited and the available experimental data are very scarce, it is unfeasible to attempt to develop functional expressions for the framework discussed above. Instead, this framework is used as a physical basis to modify the viscoplastic model described in Section III B to incorporate an external magnetic effect. The correlation between the viscoplasticity model and the dislocation-density based frame work can be illustrated by transforming Eq. (17) into a form similar to Eq. (11).

Assume that the stress dependence of $v\xaf$ is given by

where *A* in Eq. (11) is now equal to $(3/2)b\rho DD$, i.e., proportional to the dislocation density.

Based on the framework, magnetic annealing is essentially related to the static recovery term in Eq. (19), $r(\rho D,\theta ,B)$. As mentioned earlier, the magnetic-field induced annealing effect is generally attributed to the enhanced dislocation mobility due to the unlocking of dislocations that are pinned by paramagnetic defects through the spin-dependent interaction. This would lead to a decrease of immobile dislocations in Eq. (20) and an increase of mobile dislocation density in Eq. (17), both of which could contribute to the decrease of initial strength as observed in Figure 4. Due to the very short time scale involved here, the degree of dislocation annihilation is anticipated to be very limited and the overall dislocation density is not expected to change much. Based on the correlation discussed above, instead of finding an explicit expression for the static recovery function, *r*, which is unfeasible with the very limited amount of information available at this point, the effect of magnetic annealing is modeled by simply changing the initial value of the threshold strength, Y_{0}, from 1.2 GPa to 1.1 GPa. The change of mobile dislocation density could also have some effect on the coefficient *A*, which is related to material's rate sensitivity. However, since the current experimental data suggested no evidence that the rate sensitivity was affected by magnetic annealing, it was left unchanged. As mentioned earlier, the magnetic field could also increase the electron component of viscous drag, which would increase the plastic flow resistance and potentially offset the effect of increased mobile dislocations. It should be noted that the Y_{0} used here, i.e., 1.2 GPa, is smaller than the yield strengths reported in Refs. 18–20. This is because the proposed model incorporates rate effect which tends to increase the apparent strength as the loading rate gets higher. Accordingly, the initial value which represents the intrinsic material strength should be lower than those from the measurements or models which ignore the rate effects.

The relevant material constants used in the above formulation are summarized in Table II. Many of them were collected from Refs. 20 and 35. Others were added or modified according to the model described above and the available experimental data.

$\rho 0$ (kg/m^{3}) (initial density) | 10200 | ||

K_{1} (GPa) | 276.1 | ||

K_{2} (GPa) | 353.2 | ||

K_{3} (GPa) | 317.0 | ||

$\Gamma $_{0} | 1.59 | ||

G_{0} (GPa) | 100.00 | ||

Y_{0} (GPa) | 1.2 (without B_{0}) | ||

Y_{0} (GPa) | 1.1 (with B_{0}) | ||

Y_{max} (GPa) | 2.8 | ||

n | 2.0 | ||

A | $4.0\xd7105$ | ||

$\beta $ | 20 | ||

q | 0.15 | ||

$Gp\u2032/G0\u2009(TPa\u22121)$ | 11.4 | ||

$\u2212GT\u2032/G0\u2009(kK\u22121)$ | 0.152 | ||

$\theta melt$ (K) | 2890 | ||

$m$ | 0.4 | ||

$\eta 0$ ($\Omega -m$) | $5.34\xd710\u22128$ | ||

$\alpha $ | 0.0046 |

$\rho 0$ (kg/m^{3}) (initial density) | 10200 | ||

K_{1} (GPa) | 276.1 | ||

K_{2} (GPa) | 353.2 | ||

K_{3} (GPa) | 317.0 | ||

$\Gamma $_{0} | 1.59 | ||

G_{0} (GPa) | 100.00 | ||

Y_{0} (GPa) | 1.2 (without B_{0}) | ||

Y_{0} (GPa) | 1.1 (with B_{0}) | ||

Y_{max} (GPa) | 2.8 | ||

n | 2.0 | ||

A | $4.0\xd7105$ | ||

$\beta $ | 20 | ||

q | 0.15 | ||

$Gp\u2032/G0\u2009(TPa\u22121)$ | 11.4 | ||

$\u2212GT\u2032/G0\u2009(kK\u22121)$ | 0.152 | ||

$\theta melt$ (K) | 2890 | ||

$m$ | 0.4 | ||

$\eta 0$ ($\Omega -m$) | $5.34\xd710\u22128$ | ||

$\alpha $ | 0.0046 |

## V. ELECTROMAGNETIC SIMULATION

As discussed earlier, for experiments with molybdenum as the driver material, a direct simulation using the imposed current as the load input is necessary. For consistency, the same type of simulation was also applied to the experiments with an aluminum driver. To carry out simulations with the current input, Maxwell equations need to be solved. The relevant basic formulation can be found in Ref. 36. Besides the Maxwell equations, the Lorentz force $(J\xaf\xd7B\xaf)$ and Joule heating need to be incorporated in the momentum and energy conservation equations, respectively, i.e.,

where $vi$ is the particle velocity and $\eta $ is the electrical resistivity.

In addition to the aforementioned governing equations, an electrical conductivity model is also needed. In this study, a simple linear relation was used to describe the change of resistivity as a function of temperature,^{37} i.e.,

where $\eta 0$ is the initial resistivity and $\alpha $, the temperature coefficient of resistance, is constant. Following Ref. 37, the $\alpha $ values used for aluminum and molybdenum in this study were 0.0043/K and 0.0046/K, respectively. These values are consistent with the data reported in Ref. 38 up to the melting temperature. In other words, Eq. (25) is expected to be reasonably valid over the temperature range to melting. The initial magnetic field (B) induced by the imposed current pulse ($I$) is calculated by

where $\mu 0$ is the magnetic permeability, *w* is the panel width, and $kE$ is an experimental coefficient to accommodate the simplification of the actual 2D experimental configuration with a 1D approximation.^{4} As described in Ref. 4, in an ideal case where the current flows uniformly and entirely on the inner surface of the panel $kE$ would be unity. However, in reality, the current diffuses into the panel and is nonuniform at the panel edges. As a result, $kE$ is less than 1. In this study, this factor was used to scale down the input current amplitude to match the peak of the measured particle velocity. A similar but slightly different form of Eq. (26), i.e., $B=\mu 0I(t)/S$ where *S* is a scale factor, was used in Ref. 39.

A more precise way to determine the input current or magnetic field is the unfold technique developed by Lemke *et al*.^{40} which is conceptually similar to the backward analysis mentioned earlier, but applied to electromagnetic loading. The technique has been applied previously to aluminum panels.^{41} Due to much more complicated material behavior compared to that of aluminum, further development work, as will be discussed later in Section VIII, will be needed to apply this technique to molybdenum panels. Accordingly, the simulations conducted in this study were based directly on the experimentally measured input current.

In the simulation, zirconia was modeled as an elastic, perfectly plastic material with a constant yield strength (Y_{0}) and shear modulus (G_{0}). The relevant material properties are listed in Table III.^{42,43} The models and relevant material constants for aluminum and LiF were given in Refs. 44 and 45.

$\rho 0$ (kg/m^{3}) (initial density) | 6070 | ||

K_{1} (GPa) | 190.0 | ||

K_{2} (GPa) | 329.1 | ||

K_{3} (GPa) | 263.1 | ||

$\Gamma 0$ | 1.403 | ||

C_{v} (J/Kg-K) | 400 | ||

k (W/m-K) | 2.0 | ||

Y_{0} (GPa) | 12.0 | ||

G_{0} (GPa) | 102.3 |

$\rho 0$ (kg/m^{3}) (initial density) | 6070 | ||

K_{1} (GPa) | 190.0 | ||

K_{2} (GPa) | 329.1 | ||

K_{3} (GPa) | 263.1 | ||

$\Gamma 0$ | 1.403 | ||

C_{v} (J/Kg-K) | 400 | ||

k (W/m-K) | 2.0 | ||

Y_{0} (GPa) | 12.0 | ||

G_{0} (GPa) | 102.3 |

## VI. COMPARISON OF SIMULATIONS WITH LONGITUDINAL EXPERIMENTAL DATA

The simulated longitudinal responses for each experiment are shown in Figures 3, 5–7, and 8(a). As demonstrated by these figures, the simulations in general match well with the experimental data obtained from various configurations. Some deviations between the experiments and simulations could be attributed to uncertainties associated with current measurements (∼10%).^{36,46} For example, in Figure 3(b) (the bottom panel of experiment EX1A), the simulated particle velocity profile appears to have slightly larger rise time than the data. The same trend was also observed in Figure 3(a) (the top panel of experiment EX1A). Similar results were also observed in Figures 5(a) and 5(b). All these could likely be due to a systematic error in drive current measurement which would be reflected in both top and bottom panels. In Figure 7, the simulated velocity profile showed slightly more pronounced elastic-plastic transition than the experimental data. Besides the experimental uncertainties mentioned above, other factors related to electromagnetic loading such as 1D simplification of the 3D experimental configuration and uncertainties associated with the coupling between electromagnetic field and mechanical response could also contribute to the observed deviation.

## VII. SIMULATION OF THE TRANSVERSE RESPONSE OF THE MAPS DATA AND ITS IMPLICATIONS

The simulated result for the transverse part of experiment EX5-B is shown in Figure 8(b). Unfortunately, only one experiment produced good transverse particle velocity measurements in this study, so the present results are tentative. However, the simulation matches well over most of the loading portion of the data, but shows a higher peak value. This overshoot was determined to be likely due to wave interactions present in the experimental configuration and correlated to the leveling-off of the later part of the longitudinal velocity data shown in Figure 8(a) and others. By incorporating a more complicated model for zirconia, including features such as damage which might be initiated at the later stage of the experiments, the overshoot could be eliminated and good agreement obtained for the later part of the shear data. However, these issues are beyond the scope of this paper which focuses on the strength of molybdenum and the details of these issues will not be discussed here. Additional experiments are in progress to resolve these issues.

To gain insight on the loading part of the transverse response displayed by Figure 8(b), the various stress histories experienced by molybdenum at the molybdenum/zirconia interface were examined in detail. For a better clarification, molybdenum was modeled as a rate-independent, perfectly plastic material in these simulations. In this case, i.e., rate effects were ignored, Y_{0} was set to 1.4 GPa which was consistent with that reported in Refs. 18 and 19. Other more complicated material features, such as rate sensitivity and hardening and/or softening, described earlier, were neglected to avoid unnecessary and somewhat irrelevant complexities. The deviatoric stress experienced by an isotropic material under combined compression shear loading can be expressed as follows:

where $\sigma x$, $\sigma y$ are the normal longitudinal and lateral stress components, $\tau xy$ the shear component of the Cauchy stress tensor ($\sigma ij$), and $Sx$ the axial component of the deviatoric stress tensor ($\sigma \u0303\u2032$). The resultant histories for $\sigma x$, $Sx$, $\tau xy$, $\sigma \xaf$($\sigma effective$), and Y at the interface of Mo/ZrO_{2} are shown in Figure 9(a). To gain further insights on the actual material behavior at the interface, another simulation to recover the *in-situ* material response was also performed. In this simulation, zirconia was replaced by molybdenum. The corresponding results are shown in Figure 9(b). As shown by both figures, initially the increase of $\sigma \xaf$ is contributed by $Sx$ since the longitudinal stress travels faster. When the shear wave arrives, $\tau xy$ starts to play a role in driving the inelastic deformation. However, its contribution to the effective stress and evolution are dominated by the faster-moving longitudinal stress, i.e., the increase of $\tau xy$ is limited by both the current value and the rate of change of $Sx$. As also shown by Figure 9, at yielding, $\tau xy$ essentially stays constant, which corresponds to the plateau region mentioned above. Furthermore, Figure 9 also indicates that the strength or effective stress in this plateau region is contributed by both $Sx$ and $\tau xy$, but the former is the major contributor again because the longitudinal stress arrives earlier.

## VIII. DISCUSSION

The experimental data shown in Figure 8(b) provided a critical piece of information for validating the model's capability to predict molybdenum's response to mixed mode compression shear loading. However, there is another very important contribution from this piece of information, that is, to guide the model development. The specific issue to be discussed here is the stiff response observed in Figures 6–8. Based on the model presented above, this stiff response is essentially another manifestation of the material's strong rate sensitivity. In the early stage of the model development, it was conjectured that the stiff response of the molybdenum panel could be due to the interaction between material strength and magnetic field. During the ramp wave experiments, the magnetic field diffuses into the panel.^{4,47} As mentioned earlier, the coupling between the magnetic field and material behavior could manifest itself in two opposing material responses:^{26,33} enhancement of the dislocation mobility and resistance to dislocation motion by electron damping. The relative dominance is expected to depend on the loading conditions, e.g., strain rate, and the magnitude and perhaps even the orientation of magnetic field. In the early stage of study, the possibility of the strengthening effects induced by the diffused magnetic field was once conjectured to be responsible for the stiff response. As the panel thickness increases, the diffused field and the strengthening effect faded away (see Figure 6). A formulation based on this conjecture worked well for modeling the longitudinal part of the mechanical behavior. However, the increase in the strength was obviously not supported by the shear data shown in Figure 8(b). Without this part of the data, the stiff longitudinal response could have been misinterpreted.

Besides the MAPS technique, compression-shear experiments have also been conducted with gas guns.^{48–50} In these experiments, the impactor and anvil remained elastic and the longitudinal stress (or pressure) was maintained at the peak shock stress, i.e., constant. These restrictions significantly simplify the experiments and subsequent analysis of the data, but also limit the applicable stress range because of the velocity range of suitable gas guns^{5,16} and the limited elastic range of anvil materials. Compared to the gas-gun experiments, MAPS, as of now, is more complicated partially due to the varying longitudinal stress. In the future, it may be possible to tailor the current pulse with MAPS to achieve a more constant final stress state. However, using electromagnetic loading, MAPS can be applied to much higher stresses,^{1–3} an area that has not been explored with other complex loading techniques. Obviously, extending the current model to higher stress ranges will require considerable additional research and development. For example, the linear resistivity model^{37} as used in the current study would certainly not be applicable. A full-scale resistivity model such as the Lee-More-Desjarlais^{51,52} (LMD) model and its coupling with the mechanical behavior of materials will need to be part of the predictive capability.

Finally, it should be emphasized that the magnetic effects reported in this paper, such as annealing, are based on a very limited number of experiments. Further experiments are needed to both confirm and systematically quantify the interaction between molybdenum and the magnetic field. However, the framework proposed in this study is expected to be able to accommodate further experimental findings as they become available.

## IX. CONCLUSION

The MAPS technique provides a potential new platform for direct measurement of material strength, which has been a goal of the shock wave community for several decades. In order to properly design the experiment and interpret the results, it is critical to have a thorough understanding and a predictive capability for the driver's mechanical response to the applied electromagnetic loading. In this study, the inelastic behavior of a potential driver material, namely, molybdenum, under ramp wave uniaxial compression and mixed compression-shear loading was experimentally characterized. Based on the insights gained from the experiments, a continuum model for molybdenum under applied magnetic field and mixed compression-shear mechanical loading was proposed and validated against the experiments. In addition, this study also demonstrated that shear data generated from MAPS provided some unique insights on the mechanical behavior of molybdenum, which were vital not only for validation but also for guiding the model development.

## ACKNOWLEDGMENTS

Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Company, for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.

## References

_{1}, K

_{2}, and K

_{3}in Table II should be corrected to 70, 107.4, and 128.8, respectively; and those in Tables III and IV should be 196.8, 259.8, and 256.6, respectively. The denominator Y in Eqs. (17), (22), and (23) should be corrected to Y

_{0}, and the unit for B in Table IV should be cm/s).

_{1}, K

_{2}, and K

_{3}in Table I should be corrected to 196.8, 259.8, and 256.6, respectively, and the unit for B in the same table should be cm/s).