Impact induced depolarization of ferroelectric materialswebhome.auburn.edu/~vza0013/files/Agrawal_Bhattacharya... · 2018. 4. 3. · commonly used ferroelectric ceramics are lead

Journal of the Mechanics and Physics of Solids 115 (2018) 142–166

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Journal of the Mechanics and Physics of Solids

journal homepage: www.elsevier.com/locate/jmps

Impact induced depolarization of ferroelectric materials

Vinamra Agrawal a , Kaushik Bhattacharya b , ∗

a Department of Aerospace Engineering, Auburn University, Auburn AL 36849, USA b Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA

a r t i c l e i n f o

Article history:

Received 19 July 2017

Revised 13 March 2018

Accepted 13 March 2018

Available online 17 March 2018

a b s t r a c t

We study the large deformation dynamic behavior and the associated nonlinear electro-

thermo-mechanical coupling exhibited by ferroelectric materials in adiabatic environments.

This is motivated by a ferroelectric generator which involves pulsed power generation by

loading the ferroelectric material with a shock, either by impact or a blast. Upon impact, a

shock wave travels through the material inducing a ferroelectric to nonpolar phase transi-

tion giving rise to a large voltage difference in an open circuit situation or a large current

in a closed circuit situation. In the first part of this paper, we provide a general contin-

uum mechanical treatment of the situation assuming a sharp phase boundary that is pos-

sibly charged. We derive the governing laws, as well as the driving force acting on the

phase boundary. In the second part, we use the derived equations and a particular con-

stitutive relation that describes the ferroelectric to nonpolar phase transition to study a

uniaxial plate impact problem. We develop a numerical method where the phase bound-

ary is tracked but other discontinuities are captured using a finite volume method. We

compare our results with experimental observations to find good agreement. Specifically,

our model reproduces the observed exponential rise of charge as well as the resistance

dependent Hugoniot. We conclude with a parameter study that provides detailed insight

into various aspects of the problem.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

Ferroelectric materials exhibit spontaneous polarization below a certain Curie temperature. A typical polarization vs. elec-

tric field curve of a ferroelectric material is shown in Fig. 1 a; note that the material has non-zero polarization even at zero

electric field indicating spontaneous polarization. Ferroelectric materials, commonly those with a perovskite crystal struc-

ture and often in polycrystalline form, have found a wide range of applications including as sensors and actuators. Two most

commonly used ferroelectric ceramics are lead zirconate titanate (PZT, a solid solution of lead zirconate and lead titanate)

and barium titanate (BaTiO 3 ). PZT is most commonly used for transducers and actuators, while barium titanate is used for

capacitors and electro-optic modulators Damjanovic, 1998; Jona and Shirane, 1993; Xu, 1991 ). Despite the highly nonlinear

electro-thermo-mechanical coupling, most of the applications work in the linear range of the ferroelectric response, to avoid

hysteresis and cyclic degragation.

One application of ferroelectric materials is in pulsed power generation as ferroelectric generators (FEGs). FEGs generate

a large pulse of power for a short duration of time by loading the ferroceramic with a shock, either through a plate impact

or a blast. Depending on the electrical boundary condition, a large current (short circuit case) or a large voltage (open

∗ Corresponding author. E-mail address: [email protected] (K. Bhattacharya).

https://doi.org/10.1016/j.jmps.2018.03.011

0022-5096/© 2018 Elsevier Ltd. All rights reserved.

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V. Agrawal, K. Bhattacharya / Journal of the Mechanics and Physics of Solids 115 (2018) 142–166 143

Fig. 1. (a) Characteristic polarization vs. electric field behavior of a ferroelectric material showing non-zero polarization at zero electric field ( Altgilbers

et al., 2010 ). (b) Pressure temperature diagram for PZT 95/5 ( Fritz and Keck, 1978 ). Solid lines indicate forward phase transitions while broken lines

indicate backward transitions. (c) The electric displacement D vs. electric field E behavior of PZT 95/5 at various values of the imposed hydrostatic stress

σ h ( Valadez et al., 2013 ). The loop closes at aroung 400 MPa, representing stress induced ferroelectric to nonpolar phase transition.

circuit case) output is obtained. Under shock loading, the material is subjected to a large stress and this results in a loss

of spontaneous polarization through a ferroelectric to nonpolar (antiferroelectric or depoled) phase transformation. This in

turn results in a large voltage spike or current depending on the electrical boundary condition.

PZT 95/5 (95:5 being the ratio of Zirconium and Titanium) doped with 2% Niobium is the most commonly used fer-

roceramic for FEG applications. Fig. 1 b shows the temperature-pressure phase diagram of PZT 95/5 due to Fritz and Keck

(1978) where the phases indicated by the letter F are ferroelectric and those indicated by the letter A are antiferroelectrics.

Fig. 1 c shows the electric displacement vs. electric field curves measured by Valadez et al. (2013) for various values of

applied hydrostatic stress. Notice that the material loses polarization and becomes non-polar above 400 MPa.

The first electromechanical study on ferroelectric ceramics was conducted by Berlincourt and Krueger (1959) on PZT

and BaTiO 3 . Uniaxial compressive stresses were applied slowly and rapidly on axially and normally poled samples in short

and open circuit configurations. It was observed that the charges developed on the electrodes in open circuit configuration

prevented domain reorientation. Burcsu et al. (2004) studied the large electromechanical coupling in BaTiO 3 single crystals.

Halpin (1966, 1968) conducted plate impact experiments on short-circuited axially (but opposite to the direction of rem-

nant polarization) poled samples of normally sintered PZT 95/5, hot pressed PZT 95/5 and PSZT 68/7. It was observed that

peak currents (pulse duration) increased (decreased) with impact speeds, but decreased (increased) beyond a certain point.

Cutchen (1966) investigated the effect of polarity in the axially oriented short-circuited PZT 65/35 samples, and reported

increased sensitivity to stress fluctuations when polarization was opposite to the direction of shock. Lysne (1973) studied

the dielectric breakdown of PZT 65/35 and found that the stress of 10kbar was enough to cause dielectric breakdown. In

144 V. Agrawal, K. Bhattacharya / Journal of the Mechanics and Physics of Solids 115 (2018) 142–166

Fig. 2. (a) Schematic of a typical plate impact experiment. The electrodes are placed to the left and right of the ferroelectric sample. For short circuit

conditions, the resistance load is kept extremely small ( ≈ 1 �) in ( Furnish et al., 20 0 0 ). (b) Charge profiles obtained in ( Furnish et al., 20 0 0 ) for different impact speeds. The highest charge output corresponds to 65 m/s impact speed. The charge output decreased because of dielectric breakdown.

another experiment ( Lysne and Percival, 1975 ), shock experiments were conducted on doped PZT 95/5, and it was observed

that electric field ahead of the shock depended on the properties of unshocked material, which influenced the shock wave

itself. Experiments by Fritz (1978) demonstrated that shock induced depoling was caused by domain reorientation in cer-

tain stress ranges, beyond which phase transition dominated. Impact experiments were conducted on normally poled and

unpoled PZT 95/5 samples by Dick and Vorthman (1978) in short circuit and finite resistance cases. It was observed that

the point of phase transformation and the kinetics did not depend on the poling state of the material. The Hugoniot for

an unpoled doped PZT 95/5 sample was measured by Setchell (20 02, 20 03, 20 05) . It was observed that the Hugoniot curve

changes corresponding to ferroelectric and antiferroelectric phases, and onset of pore collapse in the ceramic at higher

speeds. Distinct profiles were obtained for unpoled, axially poled and normally poled samples.

Furnish et al. (20 0 0) conducted systematic plate impact experiments on axially poled, short-circuited PZT 95/5. The ex-

perimental configuration is shown schematically in Fig. 2 a. The stresses varied from 0.9 GPa to 4.6 GPa, corresponding to

the impact speeds from 65m/s to 344 m/s. The measured charge profiles obtained by integrating the current are reproduced

in Fig. 2 b. From these it was concluded that complete depoling happened at around 0.9 GPa corresponding to the impact

speed of 65 m/s. Beyond that, the current output decreased suggesting dielectric breakdown in the medium.

In this paper, we develop a model of impact induced depolarization of ferroelectrics in the framework of ( Abeyaratne

and Knowles, 1990; 1991; 1994 ). In the first part of the paper, we provide a general continuum mechanical treatment of

the situation assuming a sharp phase boundary that is possibly charged. We derive the governing laws, as well as the

driving force acting on the phase boundary. In the second part, we use the derived equations and a particular constitutive

relation that describes the ferroelectric to nonpolar phase transition to study a uniaxial plate impact problem focussing on

the experiments of Furnish et al. (20 0 0) .

Devonshire (1949, 1951, 1954) provided the first theoretical treatment of ferroelectricity (see also Fatuzzo and Merz,

1967 for a detailed treatment) based on the framework of Landau and Ginzburg. This theory is the basis of many “phase

field” models of ferroelectric materials and electromechanical coupling (see for example Nambu and Sagala, 1994; Wang

et al., 2004; Zhang and Bhattacharya, 2005a; Zhang and Bhattacharya, 2005b and the references therein). However, these are

either static or quasistatic, and generally consider small strains. Toupin (1956) provided a systematic continuum treatment

of polarized media in finite deformations, but did not consider phase transitions or domain wall motion. McMeeking and

Landis ( McMeeking and Landis, 2005 ) presented a formulation for deformable dielectric materials using principle of virtual

work in an isothermal environment. Later, Su and Landis (2007) formulated a thermodynamic theory for the motion of

ferroelectric domain walls. More recently, calculations were performed for composites under finite deformations with sharp

interfaces in isothermal environments ( Tevet-Deree, 2008 ).

In Section 2 , we consider a ferroelectric material with a propagating boundary across which particle velocity, deformation

gradients and polarization can suffer a jump subject to dynamic electrical and mechanical loading. We allow the boundary

to be charged since charge buildup at the boundaries has been suggested as one of the mechanisms that ultimately lead

to dielectric breakdown. We consider inertia, but treat electromagnetics quasistatically since the depolarization and acoustic

waves are slow compared to the speed of light. We use the balance laws, the entropy inequality and the arguments of Cole-

man and Noll to derive the governing equations. A key result is the identification of the driving force across the propagating

boundary. We postulate a kinetic relation relating the normal velocity of the boundary to the driving force. Our analysis

uses ideas in James (2002) who studied magnetoelastic bodies and Xiao and Bhattacharya (2008) who studied ferroelectrics

with space charges.


C

C

W

We used a simplified version of the model in one space dimension to study the plate impact experiments in Section 3 .

We develop a numerical method where the phase boundary is tracked but other discontinuities are captured using a finite

volume method following ( Purohit and Bhattacharya, 2003; Zhong et al., 1996 ). We validate the model using the experimen-

tal observations of Furnish et al. (20 0 0) . We then conduct a parameter study. We point out the critical role of the kinetic

relation which may be viewed as a generalization of the Lax entropy condition commonly used in the study of shocks. We

also show how the Hugoniot can change with electrical boundary condition.

2. Continuum analysis

2.1. Notation

We present the list of notations used in Section 2 .

� Domain of the ferroelectric region in reference configuration

∂� Boundary of � v Electrode kept at constant potential ˆ φ Voltage prescribed on C v q Electrode kept at constant charge

B Arbitrary subdomain in the reference configuration

y (B ) Corresponding domain in the deformed configuration

y Deformation mapping, point in the deformed configuration

x Point in the reference configuration

F Deformation gradient, ∇ x y J Determinant of F

p 0 Polarization per unit reference volume

p Polarization per unit deformed volume

S 0 Phase boundary in the reference configuration

S Phase boundary in the deformed configuration

κ Curvature of Ss 0 n Normal velocity of S 0 s n Normal velocity of S

ˆ n 0 Normal to S 0 ˆ n Normal to S

v Particle velocity

σ Surface charge density on St Traction on a surface in deformed configuration

t 0 Traction on a surface in reference configuration

f Self-force experienced by S

f 0 Pull-back of f to reference configuration

φ Electric potential E Electric field ( E = −∇ y φ) E t Tangential component of E

E 0 Pull back of E to reference configuration

D Electric displacement

D 0 Pull back of D to reference configuration

ε 0 Vacuum permittivity ˆ t Tangent to S

� ·� Jump across an interface 〈·〉 Average across an interface ψ A smooth function defined over R 3

T M Maxwell stress tensor

T 0 M

Pull-back of T M to reference configuration

ρ Density of the material in deformed configuration ρ0 Density of the material in reference configuration S Stress tensor in deformed configuration

S 0 Stress tensor in reference configuration

E Total energy of an arbitrary part F Rate of work done by external forces on the part Q Heat input to the part

E 0 Stored internal energy in B

H 0 Helmholtz energy of B

σ Interfacial energy density per unit deformed area


V Velocity of a point on boundary of S ∩ y (�) ˆ w Unit vector tangential to S ∩ y (�) but normal to ∂(S ∩ y (�)) κ∂y (B ) Curvature of the boundary of y (B ) ˙ r Volumetric rate of heat generation

q Heat flux in deformed frame

q 0 Heat flux in reference frame

η0 Entropy per unit reference volume θ Temperature d Driving force acting on the phase boundary

2.2. Preliminaries

We consider a ferroelectric medium occupying a region � ⊂ R 3 in the Lagrangian frame as shown in Fig. 3 . The systemis subjected to a deformation y : � → R 3 under the action of traction t bringing it in contact with electrodes C v ⊂ R 3 andC q ⊂ R 3 . The electrode C v has a fixed potential ˆ φ while the electrode C q has a fixed charge Q . The deformation gradientis F = ∇ x y , where ∇ x denotes derivative with respect to the reference position x . Further, y is assumed to be invertibleand orientation preserving with J = det F > 0 . The polarization of the ferroelectric material per unit volume is given by p :y (�) → R 3 in deformed configuration. Since the deformation is assumed invertible, we can define polarization in referenceconfiguration by p 0 : � → R 3 through

p 0 (x ) = J p (y (x )) . (1) Further, it is assumed that the conductor C q is fixed in space while C v deforms with negligible elastic energy. In practice this

assumption is reasonable because the electrodes are usually very thin compared to the ferroelectric medium.

A phase boundary denoted by S 0 in reference configuration ( S in deformed configuration) propagates in the ferroelectric

with a normal velocity s 0 n ( s n in deformed configuration). The deformation is continuous across the phase boundary, but the

deformation gradient and the polarization may suffer jumps across it. Consequently, we have the Hadamard jump condition

� F � = a � ˆ n 0 , � v � = −s 0 n � F � ̂ n 0 (2)where v is the particle velocity, F the deformation gradient, and ˆ n 0 is the normal to S 0 in reference configuration. Further,

the phase boundary may be charged with a charge density σ , and also experiences a force f per unit deformed area.

2.3. Electrostatics

We denote the electric potential at any point by φ : R 3 → R and the electric field as the (spatial gradient) of the electricpotential E = −∇ y φ. We denote the electric displacement D : R 3 → R 3 and note that

D (y ) = −ε 0 ∇ y φ(y ) + p χ(y (�)) . (3)where ε0 is the vacuum permittivity and χ ( B ) is the standard characteristic function of a set B ⊂ R 3 :

χ = {

1 on B, 0 else .

(4)

We assume that all processes of interest are slow compared to the speed of light, and thus we assume electrostatics at

each time. Then the Maxwell’s equation reduces to Gauss’ charge balance: {∇ y · (−ε 0 ∇ y φ + p χ(y (�)) = 0 in R 3 \ (C v ∪ C q ) , ∇ y φ = 0 on C v ∪ C q (5)

Fig. 3. Schematic of the continuum formulation for a ferroelectric material under large deformation dynamic loading with generalized electromechanical

loading. A phase boundary S propagates through the material in deformed configuration ( S 0 in reference configuration).


subject to ⎧ ⎪ ⎨ ⎪ ⎩

� −ε 0 ∇ y φ + p ) � · ˆ n = σ on S, ∫ ∂C q

−ε 0 ∇ y φ · ˆ m dS y = Q , φ = ˆ φ on C v , φ → 0 as | y | → ∞

(6)

where ˆ m denotes the outward normal to the conductor C q , C v . Above we have used the notation � a � = a + − a − for defininga jump in a across the interface.

Alternately, we can rewrite this in weak form as follows

−∫

D \ (C v ∪ C q ) (−ε 0 ∇ y φ + p χ(y (�))) · ∇ y ψdy +

∫ ∂D \ (C v ∪ C q )

(−ε 0 ∇ y φ + p χ(y (�)) · ˆ l dS y

−∫

D ∩ S σψdS y −

∫ D ∩ ∂C v

ˆ φ(−ε 0 ∇ y φ) · ˆ m dS y −∫

D ∩ ∂C q ψ(−ε 0 ∇ y φ) · ˆ m dS y = 0 , (7)

∫ ∂C q

−ε 0 ∇ y φ · ˆ m dS y = Q (8)for every smooth function ψ and every domain D ⊂ R 3 . Specifically for a part B ⊂�, we have

−∫

y (B ) ∇ y ψ · (−ε 0 ∇ y φ + p ) dy +

∫ ∂(y (B ))

ψ(−ε 0 ∇ y φ + p ) · ˆ n d S y −∫

S∩ y (B ) σψd S y = 0 (9)

for every smooth function ψ . Note that the time 1 and spatial derivatives of the potential φ can suffer jumps across S . However, since the potential is

continuous, the jumps in the derivative can not be arbitrary but have to satisfy the Hadamard jump conditions (for scalars)

� ∇ y φ� · ˆ t = 0 , � φ,t � + s n � ∇ y φ · ˆ n � = 0 (10)where s n is the interface speed in the deformed configuration. Combining these equations with the first condition of (6) , we

conclude

� ∇ y φ� = − 1 ε 0

(σ − � p · ˆ n � ) ̂ n , � φ,t � = 1 ε 0

(σ − � p · ˆ n � ) s n (11)For future use, the Maxwell stress tensor T M is defined in terms of electric field E = −∇ y φ and electric displacement D as

T M = E � D − ε 0 2

E · EI (12)

2.4. Conservation of linear momentum

For an arbitrary part B ⊂� of the body, the conservation of linear momentum requires d

dt

∫ y (B )

ρ ˙ y dy = ∫ ∂y (B )

t dS y + ∫

S∩ y (B ) f dS y (13)

where t is the traction and f is the force per unit area acting on S . Note that we ignore body forces, but the treatment is

modified easily to account for it. We can equivalently write this in the reference configuration as

d

dt

∫ B

ρ0 ̇ y dx = ∫ ∂B

t 0 dS x + ∫

S 0 ∩ B f 0 dS x (14)

where f 0 (respectively t 0 ) is the pullback of f (respectively t ) to the reference frame

f 0 = (

ˆ n · JF −T ˆ n 0 )f , t 0 =

(ˆ n · JF −T ˆ n 0

)t . (15)

Taking B to be tetrahedron away from S and localizing leads to the existence of stress tensors S and S 0 that satisfy

t = S ̂ n , t 0 = S 0 ̂ n 0 , S 0 = JSF −T . (16)Finally localizing (14) away from S and at S (after using a transport theorem) gives

ρ0 ̈y = ∇ x · S 0 , (17)

� ρ0 ̇ y � s 0 n + � S 0 ̂ n 0 � = f 0 (18)where s 0 n is the normal speed of the phase boundary in the reference frame.

1 Since the deformation and the phase boundary are time dependent, the electrostatic potential is also time dependent; however, it satisfies the equations

above at each time with time as a parameter.


2.5. Conservation of angular momentum

For an arbitrary part B ⊂�, the conservation of angular momentum requires d

dt

∫ y (B )

y × ρ ˙ y dy = ∫ ∂y (B )

y × t dS y + ∫

S∩ y (B ) y × f dS y

⇔ d dt

∫ B

y × ρ0 ̇ y dx = ∫ ∂y (B )

y × S ̂ n dS y + ∫

S∩ y (B ) y × f dS y .

Using the transport theorem and expressing this in indicial notation (corresponding to a fixed rectangular Cartesian

frame), we obtain ∫ B

S 0 JK y i,K iJL dx = 0 , ⇔ S , S 0 F T symmetric . (19)

2.6. First law of thermodynamics or conservation of energy

For any part B ⊂�, the conservation of energy in its rate form can be written as dE dt

= F + dQ dt

(20)

where E is the total energy of the part, F is the rate of work done by forces external to the part and dQ dt

is the rate of heat

input to the part.

The total energy E of B comprises of four parts: energy stored within the ferroelectric medium, interfacial energy atthe phase boundary surface, the energy from the electrostatic field and the kinetic energy of the system (see Shu and

Bhattacharya, 2001 for static case):

E = ∫

B

E 0 dx + ∫

S∩ y (B ) W σ dS y + ε 0

2

∫ y (B )

|∇ y φ| 2 dy + ∫

B

ρ0 2

| ̇ y | 2 dx (21)where E 0 is the stored internal energy per unit reference volume and W σ is the interfacial energy density per unit deformed

area of the phase boundary surface, with W σ (σ = 0) = 0 . The rate of work done F on B is given by a combination of rate of work done by external traction and the rate of work

done by external fields on the boundary of the subdomain. This can be written as

F = ∫ ∂y (B )

t · v dS y −∫ ∂y (B )

φd

dt

(D · ˆ n

)dS y −

∫ ∂y (B )

φ(D · ˆ n

)κ∂y (B ) (v · ˆ n ) dS y

+ ∫ ∂(S∩ y (B ))

(W σ + φσ )(V · ˆ w ) dl (22)

where V is the velocity of a point on ∂( S ∩ y ( B )), w is the unit vector on the boundary of S ∩ y ( B ) tangential to the surface butnormal to the curve ∂( S ∩ y ( B )) and κ∂y ( S ) is the curvature of the boundary. The terms in the rate of working related to theelectrostatics can be inferred by examining the rate of change of electrostatic field energy (see for example the calculation

in Appendix A.1.2 .) Physically, the presence of a tangential electric field and the (bound or D · ˆ n ) charges contributes anadditional energy flux during motion. The second integral is related to the rate of change of (bound) charges, and the third

is related to the convection of (bound) charges through the boundary.

Finally the rate of heat input is given by a combination of volumetric heat generation and heat flux going out through

the boundary.

dQ

dt =

∫ B

˙ r dx −∫ ∂y (B )

q · ˆ n dS y

= ∫

B

˙ r dx −∫

B

∇ x · q 0 dx + ∫

S 0 ∩ B � q 0 · ˆ n 0 � dS x (23)

where q 0 = JF −1 q . We then calculate the rate of change of energy and further manipulate the first law as described in Appendix A . We

conclude that the first law (20) can be rewritten as ∫ B

[˙ E 0 + ∇ y φ · ˙ p 0 − ˙ r + ∇ x · q 0 − (S 0 − T 0 M ) · ˙ F + (ρ0 ̈y − ∇ x · S 0 ) · v

]dx

+ ∫

S 0 ∩ B

(−� E 0 + ∇ y φ · p 0 + ( ̂ n · T M ̂ n ) J − (〈 S 0 ̂ n 0 〉 − 〈 D 0 · ˆ n 0 〉 E t ) · (F ̂ n 0 ) � )s 0 n dS x

−∫

S ∩ B

[(W σ + φσ ) κ〈 J〉 s 0 n − � q 0 · ˆ n 0 �

]dS x

0


∫

+ S∩ y (B )

(W̊ σ + φσ̊

)dS y

−∫

S∩ y (B )

(f −E t σ + (W σ + φσ ) κ ˆ n

)· 〈 v 〉 dS y

= 0 , (24)where κ is the total curvature of S and å denotes the normal time derivative of a following the surface S . Further, D 0 = JF −1 Dis the pull-back of the electric displacement D to the reference frame.

2.7. Second law of thermodynamics

The second law of thermodynamics on the arbitrary part B ⊂� requires d

dt

∫ B

η0 dx ≥∫

B

˙ r

θdx −

∫ ∂B

q 0 · ˆ n 0 θ

dS x (25)

where η0 is the entropy per unit reference volume. 2 Using the transport and divergence theorems, we obtain

∫ B

(˙ η0 −

˙ r

θ+ ∇ x ·

(q 0 θ

))dx +

∫ S 0 ∩ B

(−� η0 � s 0 n −

� q 0 · ˆ n 0 θ

� )dS x ≥ 0 (26)

Localizing away from and on the interface, we obtain

˙ η0 −˙ r

θ+ ∇ x ·

(q 0 θ

)≥ 0 in �\ S, (27)

−� η0 � s 0 n −� q 0 · ˆ n 0

θ

� ≥ 0 on S. (28)

It is natural to introduce the Helmholtz free energy density as

H 0 = E 0 − θη0 . (29)Plugging (29) in (24) and using (17), (27) and (28) gives, ∫

B

[˙ H 0 + η ˙ θ + ∇ y φ · ˙ p 0 − ˙ r + ∇ x · q 0 − (S 0 − T 0 M ) · ˙ F

]dx +

∫ S 0 ∩ B

(ds 0 n − � q 0 · ˆ n 0 �

)dS x +

∫ S∩ y (B )

(W̊ σ + φσ̊

)dS y

−∫

S∩ y (B )

(f − E t σ + (W σ + φσ ) κ ˆ n

)· 〈 v 〉 dS y ≥ 0 . (30)

where

d := � H 0 + 〈 η〉 θ + ∇ y φ · p 0 − F ̂ n 0 · (〈 S 0 ̂ n 0 〉 − 〈 D 0 · ˆ n 0 〉 E t ) + ( ˆ n · T M ̂ n ) J� + (W σ + φσ ) κ〈 J〉 . (31)2.8. Constitutive equations

We specialize to adiabatic process where q 0 = 0 . We make the following constitutive assumptions H 0 = H 0 (F , p 0 , θ ) , η0 = η0 (F , p 0 , θ ) , T 0 = T 0 (F , p 0 , θ ) , W σ = W σ (σ ) . (32)

We now follow arguments similar to Coleman and Noll (1963) to simplify the constitutive relations. First, considering

only smooth processes, we conclude from (30) that

˙ H 0 + η ˙ θ + ∇ y φ · ˙ p 0 − (S 0 − T 0 M ) · ˙ F ≥ 0 in �, (33)Exploiting the constitutive relations (32) , we conclude

∂H 0 ∂p 0

+ F −T ∇ x φ = 0 , (34)

∂H 0 ∂θ

+ η0 = 0 , (35)

∂H 0 ∂F

− (S 0 − T 0 M ) = 0 (36)

2 Note that we have ignored interfacial entropy density. We could introduce it, but this would require us to introduce an interfacial temperature distinct

from the limiting temperatures from the two sides and heat flux along the interface following for example the approach in ( Javili et al., 2013 ). However,

this becomes much more involved, and is not necessary for the current study of shocks where the interface is adiabatic.


in �. Now, considering processes with no motion but only changes in electrostatics, we conclude

dW σ

dσ+ φ = 0 (37)

on S . Then, considering processes involving boundaries stationary in the reference configuration, we conclude that

f = E t σ − (W σ + φσ ) κ ˆ n (38) The force f is an extra self-force that has two components. The first component appears because of the tangential component

of electric field across the charged surface. The second component is due to the curvature of the charged surface. The

absence of surface charges also means f = 0 . Finally, we conclude that

d s 0 n ≥ 0 (39) where d is the driving force on the phase boundary given by (31) . Following Abeyaratne and Knowles (1991) , we make the

final constitutive assumption as a kinetic relation that relates the driving force to the normal velocity of the boundary

s 0 n = K(d) (40) where the function K satisfies K ( d ) d ≥ 0.

Before we proceed, it is useful to substitute these constitutive equations into the energy balance or first law (24) . We

obtain ∫ B

(θ ˙ η − ˙ r ) dx −∫

S 0 ∩ B (d + 〈 θ〉 � η� ) s 0 n dS x = 0 (41)

2.9. Driving force

We now discuss the expression (31) for the driving force which is the main finding of this part of the paper. We begin

by specializing to a series of special cases.

Isothermal, purely mechanical setting. In this case, the expression reduces to

d = � H 0 − F ̂ n 0 · 〈 S 0 ̂ n 0 〉 � = ˆ n 0 · � H 0 I − F T 〈 S 0 〉 � ̂ n 0 (42)or the normal component of the jump in Eshelby’s energy momentum tensor in agreement with previous works (e.g.

Abeyaratne and Knowles, 1990 ).

Isothermal rigid body. Here F = I and J = 1 , and the referential and current quantities are the same. The expression for thedriving force reduces after some calculations to

d = � H 0 − p · 〈 E 〉 � + 〈 E · ˆ n 〉 σ + (W σ + φσ ) κ. (43)Note that the first term is the electrical analog of the jump in Eshelby’s energy momentum tensor. The second term

describes the force acting on charged interface subjected to a normal electric field. The final term is the result of curvature.

The term (W σ + φσ ) represents an interfacial free energy this results in a driving force (see for example Cermelli et al.,2005 ).

Isothermal, small displacement setting. We assume that the displacement and displacement gradients are small. Specifically,

we assume that the displacement gradient is very small, i.e., | ∇u | < < 1 where y = x + u . So we linearize the relation(31) around F = I to obtain

d = � H 0 − ((∇u ) ̂ n 0 ) · 〈 S 0 ̂ n 0 〉 − p · 〈 E 〉 � + 〈 E · ˆ n 〉 σ + (W σ + φσ ) κ , (44)The first term which agrees with the results of Mueller et al. (2006) and Su and Landis (2007) is a combination of the

purely mechanical and the purely electrical situations described above. The final two terms are a result of interfacial charges

and interfacial free energy as described above.

Isothermal, no interfacial charges. The expression (31) now becomes

d := � H 0 + ∇ y φ · p 0 − F ̂ n 0 · (〈 S 0 ̂ n 0 〉 − 〈 D 0 · ˆ n 0 〉 E t ) + ( ˆ n · T M ̂ n ) J� . (45) James (2002) derived a similar expression in the context of ferromagnetism. However, our expression differs from his

through the term � F ̂ n 0 · 〈 D 0 · ˆ n 0 〉 E t � . James uses a condition (Eq. (35) in James, 2002 ) which holds on the boundary butnot in the interior.

The expression for driving force (31) contains terms corresponding to the electro- thermomechanical coupling of the

material. This expression is complicated by the fact that the mechanical terms are natural in the reference configuration

(see the purely mechanical setting above) while the electrical term are natural in the current configuration (see the rigid

setting above).


Fig. 4. Schematic of the one dimensional impact problem. A thermoelastic flyer traveling at speed v imp hits a ferroelectric material. The phase boundary x = s separates the two phases. Phase 1 denotes the ferroelectric phase while Phase 2 is the anti-ferroelectric phase. The phase boundary is nucleated very close to x = 0 , and it travels through the material upon impact. Current flowing through the resistor R is monitored.

3. Shock wave studies in ferroelectrics

In this section, we specialize the equations above to study the canonical uniaxial impact experiment shown in Fig. 4

motivated by the work of Furnish et al. (20 0 0) . We work in an one-dimensional setting. A thermoelastic flyer x ∈ (−L f , 0)traveling at speed v imp hits a ferroelectric target x ∈ (0, L ) with electrodes at the two ends connected together through aresistor R . The ferroelectric material of the target has two phases – a ferroelectric (FE) Phase 1 and a non-polar or anti-

ferroelectric (AFE) Phase 2 . The target is initially entirely in Phase 1. Upon impact, a phase boundary nucleates at the point

of impact and propagates through the body. The position of the phase boundary is given as x = s (t) . As it propagates, thecurrent I flowing through the resistor is tracked. The left end of the flyer and the right end of the target have traction free

boundary conditions. To simplify the analysis, we assume small strain consistent with ferroelectric ceramics, and we neglect

any charges on the phase boundary.

3.1. Notation

We present the list of notations used in Section 3 .

L f Length of the flyer

L Length of the target

v imp Speed of the flyer R External resistor

I Current flowing through R

V Voltage across R

s Location of the phase boundary

˙ s Speed of the phase boundary

p 0 Polarization

p 1 Remnant polarization of target in FE phase

ε Strain θ Temperature v Particle velocity H 0 Helmholtz energy of ferroelectric target

H p Polarization component of H 0 H ε Thermo-mechanical component of H 0 ρ Density of the target αp Electric coefficient in shocked AFE phase α′ p Electric coefficient in unshocked FE phase E Elastic modulus of target in unshocked phase

E ′ Elastic modulus of target in shocked phase α Thermal expansion coefficient of target c 1 Heat capacity per unit volume of target

εT Transformation strain of target θ T Transformation temperature of target M Material constant connected to latent heat

σ c Cauchy stress of the target E p Electric field within target

D Electric displacement within target

η0 Entropy per unit volume of the target c , c ′ Wave speeds of target in unshocked and shocked phases d Driving force acting on the phase boundary

E Total stored energy of the target (or a sub-domain) F Rate of external work done on target (or sub-domain) H

f 0

Helmholtz energy of the flyer

ρ f Density of the flyer


E f Elastic modulus of the flyer

αf Thermal expansion coefficient of the flyer θ Tf Reference temperature of the flyer c 1 f Heat capacity per unit volume of the flyer

c f Wave speed of the flyer

˜ · Non-dimensional quantity N Number of discretized intervals (equal length) for target

h Length of discretized interval in target

N f Number of discretized intervals (equal length) for flyer

h f Length of discretized interval in flyer

�˜ t Non-dimensional time step

˜ p L , ˜ p R Non-dimensional p 0 to the left and right of the phase boundary ˜ E L , ˜ E R Non-dimensional E to the left and right of the phase boundary

C 0 , S Hugoniot parameters for target

d s , d c Parameters for kinetic relation

3.2. Model description

3.2.1. Ferroelectric material

We chose the Helmholtz energy of the ferroelectric material as follows

H 0 (p 0 , ε, θ ) = H p (p 0 ) + H ε (ε, θ ) where ε denotes the strain. As such the Helmholtz energy is a combination of two energies: one dependent purely onpolarization, while other being a phase transforming thermoelastic material. This is consistent with the fact that in typical

ferroelectric material with FE-AFE phase transition, strain is considered the primary order parameter ( Fatuzzo and Merz,

1967 ). So we take H ε to be a two phase material with two wells while we take H p to have phase dependent spontaneous

polarization. We also ignore ordinary electrostriction because the contributions are an order of magnitude smaller. The

choice of H ε follows the one described in ( Abeyaratne and Knowles, 1994 ) with some modifications. Further, the AFE phase

formed after shock loading is referred as a shocked phase, while the FE phase is denoted by unshocked phase. Specifically,

H p (p 0 ) =

⎧ ⎨ ⎩

αp 2

p 2 0 in shocked phase

α′ p 2

(p 0 − p 1 ) 2 in unshocked phase and (46)

H ε (ε, θ ) =

⎧ ⎨ ⎩

E

2 ε 2 − αEε(θ − θT ) − c 1 θ log ( θ/θT ) in unshocked phase

E ′ 2

((ε − ε T ) 2 + ε T M(θ − θT )

)− αE ′ ε (θ − θT ) − c 1 θ log ( θ/θT ) in shocked phase

. (47)

where E , E ′ , α, c 1 , εT and θ T represent the elastic modulus (unshocked and shocked phase), thermal expansion coefficient,heat capacity per unit reference volume, transformation strain and transformation temperature respectively. M is a material

constant that relates to the latent heat of the material. The exact values of all the parameters are discussed later. Following

(36) , we have the expressions for Cauchy stress as,

σc = {

E ( ε − α(θ − θT ) ) in unshocked phase E ′ ( ε − ε T − α(θ − θT ) ) in shocked phase . (48)

The schematic forms of H ε and σ c are represented in Fig. 5 a and b. The strains ε1 ( θ ) and ε2 ( θ ) are given by, ε 1 (θ ) =α(θ − θT ) and ε 1 (θ ) = ε T + α(θ − θT ) , where εT is independent of temperature. Also, the expressions for electric field andentropy are given by,

E p = {αp p 0 in shocked phase α′ p (p 0 − p 1 ) in unshocked phase (49)

η0 = {αEε + c 1 ( 1 + log ( θ/θT ) ) in unshocked phase αE ′ ε + c 1 ( 1 + log ( θ/θT ) ) − E ′ ε T M in shocked phase . (50)

3.2.2. Flyer material

We choose a thermoelastic material for the flyer with

H f 0

= E f 2

ε 2 − α f E f ε(θ − θT f ) − c 1 f θ log (θ/θT f ) (51) where E f , αf , c 1 f and θ Tf denote elastic modulus, thermal expansion coefficient, specific heat capacity per unit referencevolume and reference temperature for thermal expansion for the flyer.


Fig. 5. (a) Schematic diagram of double energy wells of H ε( ε, θ ). The transformation strain εT is fixed with temperature, but the points ε 1 ( θ ), ε 2 ( θ ) and

the height of energy wells are changing with temperature. (b) Cauchy stress in the two phases.

3.2.3. Non-dimensionalization

The choice of the energy in Section 3.2 allows us to define characteristic speeds c = √

E ρ , c

′ = √

E ′ ρ and c f =

√ E f ρ f

for the

target (unshocked and shocked phases) and the flyer respectively. Here ρ and ρ f are the densities of the target and the flyerrespectively. We can now non-dimensionalize various quantities as follows.

˜ x = x L , ˜ t = tc

L , ˜ v = v

c , ˜ p 0 = p 0

p 1 , ˜ θ = θ

θT , ˜ c ′ = c

′ c

, ˜ c f = c f

c , ˜ ρ f =

ρ f ρ

From here on, non-dimensional quantities are represented with a tilde accent. We can further non-dimensionalize the

Helmholtz energy as follows.

˜ H 0 = H 0 E

=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

1 2 ε 2 − ˜ αε( ̃ θ − 1) − ˜ c 1 ̃ θ log ˜ θ + ˜ α

′ p

2 ( ̃ p 0 − 1) 2 in unshocked phase

1

2 ˜ E ′ ((ε − ε T ) 2 + ε T ˜ M ( ̃ θ − ˜ θT )

)− ˜ α′ ε( ̃ θ − 1)

− ˜ c 1 ̃ θ log ˜ θ + ˜ αp 2

˜ p 2 0 in shocked phase

(52)

˜ H f 0

= H f

0

E =

˜ E f

2 ε 2 − ˜ α f ε( ̃ θ − ˜ θT f ) − ˜ c 1 f ˜ θ log

(˜ θ

˜ θT f

). (53)

Here ˜ E ′ = E ′ /E, ˜ α = αθT , ˜ α′ = αθT ˜ E ′ , ˜ c 1 = c 1 θT /E, ˜ αp = αp p 2 1 /E, ˜ α′ p = α′ p p 2 1 /E and ˜ M = MθT for the ferroelectric material,while ˜ E f = E f /E, ˜ α f = α f E f θt /E, ˜ θT f = θT f /θT and ˜ c 1 f = c 1 f θT /E for the flyer material. We can now obtain non-dimensionalelectric field and entropy for the ferroelectric material as follows.

˜ E p = ∂ ˜ H 0

∂ ̃ p 0 =

{˜ α′ p ( ̃ p 0 − 1) in unshocked phase ˜ αp ̃ p 0 in shocked phase

(54)

˜ η0 = −∂ ˜ H 0

∂ ̃ θ=

{˜ αε + ˜ c 1 (1 + log ( ̃ θ )) in unshocked phase −ε T ˜ M + ˜ α′ ε + ˜ c 1 (1 + log ( ̃ θ )) in shocked phase

(55)

We can now consistently define the non-dimensional electric displacement as ˜ D p = D p /p 1 = ˜ p 0 + ˜ ε 0 ̃ E where ˜ ε 0 = Eε 0 /p 2 1 .This also gives us a consistent form of non-dimensional Maxwell stress ˜ σM = ˜ E p ̃ D p − ˜ ε 0 2 ˜ E 2 p =

σM E Further, we can get the

non-dimensional Cauchy stress ˜ σc = ∂ ̃ H 0 ∂ε = σc E .

3.3. Governing equations

The governing equations, given a trajectory s ( t ) of the phase, boundary are the conservation of linear momentum (13) ,

and conservation of energy (20) . Our numerical method uses both the integral forms (simplified to the one-dimensional

adiabatic setting) as well as the solutions to Riemann problems. We need the jump conditions across the interface for the

latter.

� ̃ σ � + ˙ s � ̃ v � = 0 , (56)˜ d + 〈 ̃ θ〉 � ̃ η� = 0 . (57)


See (41) for the latter. We use strain and velocity (instead of displacement) as our kinematic variables, and thus we need

a kinematic compatibility condition:

d

d ̃ t

∫ ˜ b ˜ a

εd ̃ x = ˜ v | ˜ b ˜ a , � ̃ v � + ˙ s � ε� = 0 . (58) Since we assume that there are no interfacial charges, we have the jump condition

� ̃ D p � = 0 (59) To close the system, we need to determine the trajectory s ( t ). We do so using a kinetic relation (40) . Finally, Ohm’s law is

enforced across the external resistance

˜ I = −d ̃ D p

d ̃ t =

˜ V

˜ R (60)

where ˜ R = p 2 1

cR

EL 2 and ˜ V = ∫ 1 0 ˜ E p d ̃ x are the non-dimensional resistance and voltage respectively.

3.4. Method

In this section, we describe the computational method used to solve the impact problem. The computational method

employed is a variation of Gudonov scheme for phase transforming materials used in ( Purohit and Bhattacharya, 2003 ) and

( Zhong et al., 1996 ). The main idea is to capture elastic and other minor waves, but track the major discontinuities like the

phase boundary. We work in the Lagrangian frame of reference.

3.4.1. Discretization

The flyer and target are spatially discretized into N and N f intervals respectively of equal lengths h f and h t . We add a

nodal point at the phase boundary to avoid averaging over the two different phases and getting values in unstable region of

stress-strain curve. In order to do that, the mesh needs to be updated at every time step to keep track of the moving phase

boundary. We assume ε, p 0 and v to be piecewise constant in each interval. Further, we define θ at every node point. Wediscretize time with time steps of �˜ t .

3.4.2. Initial condition and nucleation

We take time ˜ t = 0 to be the time of impact. The flyer plate is unstressed and travelling at a uniform impact velocity˜ v imp . The target plate is at rest and unstressed in the ferroelectric state. A phase boundary is nucleated on impact at theflyer-target interface ˜ x = 0 . We model this by introducing a phase boundary at ˜ x = ̃ h t = h t /L, with ferroelectric phase lying tothe right ˜ x ∈ [ ̃ h t , 1] and the anti-ferroelectric phase lying on the left ˜ x ∈ [0 , ̃ h t ] . Initial polarization across the phase boundaryis determined by maintaining continuity of electrical displacement and ensuring zero voltage across the resistor R .

˜ p L + ˜ ε 0 ̃ αp ̃ p L = ˜ p R + ˜ ε 0 ˜ α′ p ( ̃ p R − 1) ˜ V = ˜ h t ̃ E L + (1 − ˜ h t ) ̃ E R = ˜ h t ̃ αp ̃ p L + (1 − ˜ h t ) ̃ α′ p ( ̃ p R − 1) = 0

where ˜ p L and ˜ p R denote the non-dimensional polarization to the left and the right of the phase boundary. These equations

can be solved to give the initial polarization across the nucleated phase boundary.

3.4.3. Evolution

Given the information about states at time ˜ t , we seek to calculate the states at time ˜ t + �˜ t . Since we are in the smallstrain setting and our constitutive relation does not have any electrostriction, the polarization and the electric field is un-

affected by sound waves. Instead, they are only affected by the phase boundary. Further, we have a single phase boundary.

So the polarization remains piecewise constant jumping across the single phase boundary. We continue to use ˜ p L and ˜ p R to

denote these values.

At any node ˜ x i in the target at time ˜ t , we look at the domain ω t i

= (a, b) × ( ̃ t , ̃ t + �˜ t ) where a = 1 2 ( ̃ x i + ̃ x i −1 ) and b =1 2 ( ̃ x i + ̃ x i +1 ) . We follow a staggered approach.

• We first update ε, ˜ p 0 and ˜ v in the domain ω t i by keeping the temperature ˜ θ constant. Notice that the data at time˜ t is piecewise constant and so we obtain an isothermal Riemann problem. We use this structure, the conservation of

momentum as well as the kinetic relation to find ε, ˜ p 0 and ˜ v at time ˜ t + �˜ t . • We then update the temperature ˜ θ while holding ε, ˜ p 0 and ˜ v constant using the conservation of energy. • Finally the states are updated for the next time step by averaging over the new solution. We average the temperature

over the entire interval (a,b) while we average ε, ˜ p 0 and ˜ v over (a, ̃ x i ) and ( ̃ x i , b) .

The various problems are shown in Fig. 6 and explained in detail in the following sections.


Fig. 6. The various problems involved in the evolution. (a,b) Node in the ferroelectric target away from the phase boundary, (c,d) Node in the ferroelectric

target at the phase boundary, (e,f) Node in the target-flyer plate interface, (a,c,e) The conservation of momentum to update ε, ˜ p 0 and ˜ v . (b,d,e). The conservation of energy to update ˜ θ .

Ferroelectric target. Fig. 6 a shows the conservation of momentum problem to be solved at a point ˜ x i which is not at the

phase boundary or at the two ends of the material. We solve the conservation of momentum and compatibilty

[∫ b a

˜ v dx ]˜ t +�˜ t

˜ t

−∫ ˜ t +�˜ t

˜ t

˜ σ | b a d ̃ t = 0 (61)

∫ ˜ t +�˜ t ˜ t

˜ v | b a + ∫ b

a

ε| ˜ t +�˜ t ˜ t

= 0 (62)

to update ε and ˜ v . Fig. 6 c shows the Riemann problem at the phase boundary. In this case, ˜ p L and ˜ p R also get updated at time ˜ t + �˜ t . Aside

from (61) and (62) , we also have following equations across the phase boundary at time ˜ t + �˜ t . ˜ σ (ε ˜ t +�˜ t x iL ,

˜ θ ˜ t , ˜ p ˜ t +�˜ t L ) − ˜ σ (ε ˜ t +�˜ t x iR , ˜ θ˜ t , ˜ p ˜ t +�˜ t R ) + ˜ ˙ s ( ̃ v ˜ t +�˜ t x iL − ˜ v

˜ t +�˜ t x iR

) = 0 ˜ v ˜ t +�˜ t x iL − ˜ v

˜ t +�˜ t x iR

+ ˜ ˙ s ( ̃ ε ˜ t +�˜ t x iL − ˜ ε ˜ t +�˜ t x iR

) = 0


˜ D p ( ̃ p ˜ t +�˜ t L ) − ˜ D p ( ̃ p ˜ t +�˜ t R ) = 0

˜ d ( ̃ ε ˜ t +�˜ t x iL , ˜ ε ˜ t +�˜ t x iR

, ˜ p ˜ t +�˜ t L , ˜ p ˜ t +�˜ t R ,

˜ θ ˜ t ) = f ( ̃ ˙ s )

˜ V = ˜ s ˜ t +�˜ t ˜ E p ( ̃ p ˜ t +�˜ t L ) + (1 − ˜ s ˜ t +�˜ t ) ̃ E p ( ̃ p ˜ t +�˜ t R ) = − ˜ R d ̃ D p ( ̃ p

˜ t +�˜ t R

)

d ̃ t

The first two equations are jump conditions across the phase boundary. The third equation is the continuity of electrical

displacement across the phase boundary with no surface charges. In the fourth equation, ˜ d is the non-dimensional driving

force acting on the phase boundary given by ˜ d = d/E where d = � H 0 − ε〈 σ 〉 + p〈 E〉 � . (63)

The final equation is the non-dimensionalized version of the Ohm’s Law, where ˜ s = ˜ x i is the position of the phase boundary.Using these equations, we obtain ε ˜ t +�˜ t x iL , ε

˜ t +�˜ t x iR

, ˜ v ˜ t +�˜ t x iL , ˜ v ˜ t +�˜ t x iR

, ˜ p ˜ t +�˜ t L

, ˜ p ˜ t +�˜ t R

and ˜ ˙ s .

The energy balance problems corresponding to Fig. 6 a and c are represented in Fig. 6 b and d respectively. We update the

temperature using the conservation of energy, [∫ b a

˜ E dx ]˜ t +�˜ t

˜ t

= ∫ ˜ t +�˜ t

˜ t

˜ F ∣∣b

a d ̃ t (64)

where

˜ E = ˜ E 0 + 1 2

˜ v 2 + 1 2

˜ ε 0 ̃ E 2 p

˜ F = ˜ σ ˜ v + ˜ φ d ̃ D

d ̃ t

where ˜ E 0 (ε, ˜ θ, ˜ p 0 ) is the internal energy per unit reference volume of the material. In the problem in Fig. 6 b, the electricalcomponents in ˜ E and ˜ F do not play a role because the polarization remains constant.

Target-flyer interface : The Riemann problem at the target-flyer interface is shown in Fig. 6 e. The state of polarization

during this calculation is kept constant at p ˜ t L . The equations we solve are still (61) and (62) , but stresses for ˜ x = a < 0 are

now calculated using the Flyer model. The conservation of energy problem is shown in Fig. 6 f and the temperature at the

interface is updated by solving (64) . Since we are working in the Lagrangian frame, the interface does not move. Hence,

there is no dissipation term in (64) for this problem.

Flyer : Since there is no phase boundary in the flyer, there is only one kind of Riemann problem represented in Fig. 6 a

with ˜ c = ˜ c f . We solve (61) and (62) for the Riemann problem and (64) for updating the temperature.

3.4.4. Mesh update and averaging

Since the phase boundary has moved in the time �˜ t , as depicted in Fig. 6 c, we need to update the mesh in order to

prevent averaging over it. This will make some intervals longer and some shorter. We need to ensure that the minimum

length of any interval in the ferroelectric material should be larger than 2�˜ t . This is because in the domain of computation

ω ˜ t i , maximum wave speed is ˜ c = 1 . If the size of the domain is less than 2�˜ t , information will leak out of the domain

and then averaging will lead to instability of the solution. The procedure for the mesh update is explained in ( Purohit and

Bhattacharya, 2003 ). In order to obtain the final state of strains and particle velocities in the new time step, we implement

an averaging scheme over the interval i . Since nodes are added and/or removed as the phase boundary moves, the averaging

scheme needs to be updated.

3.5. Results: comparison with experiments

We use the results by Furnish et al. (20 0 0) presented in Fig. 2 b to compare the numerical simulations. Since complete

depolarization was achieved at an impact speed of ≈ 65 ms −1 , numerical computations are performed for 65 ms −1 . Weneed a number of material and experimental parameters. Some of these are given in ( Furnish et al., 20 0 0 ), and these are

listed in Table 1 . We fit the rest to these experiments and these are given in Table 2 . We note that all of these values are

consistent with generally accepted experimental ranges ( Fritz and Keck, 1978; Furnish et al., 20 0 0; Valadez et al., 2013 ).

We employ a stick-exponential slip kinetic relation as presented in Fig. 7 a. In it’s exact form, the relation is given as

˙ s = {

0 when | d| ≤ d c c (1 − exp

( | d|−d c τ

))sign (d c ) when | d| > dc, (65)

where the parameters d c and τ are chosen as 10 6 Pa and 8 × 10 6 Pa respectively. The charge profile is computed and com-pared with that obtained by Furnish as shown in Fig. 7 b. The profile obtained from the simulation matches closely with ex-

periments. The current profiles obtained are exponential and achieve a steady value as the phase boundary moves into the

material. The peak current achieved in the process is 37 A/cm 2 which matches with the peak current achieved in ( Furnish

et al., 20 0 0 ), as evident by the matching slopes in Fig. 7 b.


Table 1

Setup and material parameters in

experiments by Furnish ( Furnish

et al., 20 0 0 ).

Parameters Values

L 4 mm

L f 6 mm

ρ 7300 kg/m 3

c 4163 ms −1

p 1 ≈ 0.30 μC/cm 2 v impact 65 ms −1 R 1 ohm

Table 2

Choice of material and setup param-

eters for simulations ( Fritz and Keck,

1978; Furnish et al., 20 0 0; Valadez

et al., 2013 ).

Parameters Values

αp 6 × 10 7 Pa C −2 m 4 α′ p 2 × 10 9 Pa C −2 m 4 p 1 0.28 C/m

2

c , c ′ 4163 ms −1 ρ 7300 kg m −3

α 10 −6 K −1

θ T 198 K

c 1 10 7 JK −1 /m 3

εT −0 . 008 M 8 × 10 −4 K −1 ρ f 7800 kg m

−3

c f 3200 ms −1

αf 10 −6 K −1

θ Tf 298 K

c 1 f 10 6 JK −1 / m 3

L 5 mm

L f 6 mm

v impact 65 ms −1 R 1 ohm

Fig. 7. (a) Stick-exponential slip kinetic relation (b) Comparison of charge profiles from experiments and simulations.

Finally, a current of 37 A/cm 2 across a 1 ohm resistance leads to a electric field of the order of 30 kV/m. This is well

below the dielectric breakdown limit of PZT of the order of 10 MV/m. This is consistent with the observations and justifies

our insulating assumption.

3.6. Results: Hugoniot

It is a common practice to specify the Hugoniot of the material under consideration, in terms of shock speed and particle

velocity. Experiments conducted over a wide range of materials show a linear dependence of shock speed and particle


Fig. 8. Shock speed - Particle speed Hugoniot for different R in the external circuit. For high R , the material experiences an open-circuit boundary condition.

Table 3

Variation of Hugoniot parameters of

the ferroelectric material for different

values of R in the external circuit. The

Hugoniot, and hence the material be-

havior, shows strong dependence on

the external circuit which is consis-

tent with experimental observations.

R C 0 S

0.001 � 3056 . 18 ms −1 1.28 0.01 � 3057 . 88 ms −1 1.31 0.1 � 3054 . 85 ms −1 1.50 1 � 3002 . 07 ms −1 2.59 10 � 2838 . 44 ms −1 5.25 100 � 2731 . 17 ms −1 6.69 10 0 0 � 2711 . 34 ms −1 7.00

velocity ( Marsh, 1980 ). This is expressed as

U s = C 0 + Sv (66)

where U s is the shock speed, C 0 is characteristic speed of the material (typically the bulk sound speed), S is a material

constant derived from experiments and v is the particle speed. Fig. 8 shows the shock speed – particle speed Hugoniot dataof the ferroelectric material, obtained through our simulations for seven different values of R in the external circuit. Most

of the material parameters used to recreate the Hugoniot were taken from Table 1 , except for α′ p and p 0 , which were takento be 10 9 Pa C −2 m 4 and 0.33 C/m 2 to have better variation over the range of impact speeds. To generate every Hugoniot, sixdifferent impact speeds were taken from 55 ms −1 to 105 ms −1 with intervals of 10 ms −1 . The speed of the phase boundarywas averaged to remove any numerical oscillations caused by mesh update and averaging.

Fig. 8 shows that our model indeed gives rise to a linear U s − v Hugoniot of the material. The slope and the interceptschanges depending on R . The values of C 0 and S as computed are reported in Table 3 . Both parameters show a strong

dependence on the external circuit or electrical boundary conditions. This is because as R increases, the boundary condi-

tions transition from short-circuit configuration to resemble an open-circuit configuration which significantly influences the

material behavior. This is consistent with experimental observations.

3.7. Results: parameter study

We now explore the phenomena through a parameter study. Many of the material parameters are measured reliably in

independent experiments ( Fritz and Keck, 1978; Furnish et al., 20 0 0; Setchell, 20 02; 20 03; Valadez et al., 2013 ). We take

them as fixed. Some are known only upto order of magnitude in shock conditions, and we choose consistent values. There

are listed in Tables 4 and 5 . Much less is known about the kinetic relations. So we study two classes, linear and stick-slip

linear, in detail. We also vary experimental configurations starting from base numbers in Table 6 .


Table 4

Choice of ferroelectric material parameters.

Parameters Values

αp , α′ p 10 7 Pa C −2 m 4

p 1 0.33 C/m 2

E , E ′ 100 GPa ρ 50 0 0 kg m −3

α 10 −6 K −1

θ T 258 K

c 1 10 6 JK −1 / m 3

εT −0 . 005 M −4 × 10 −5 K −1

Table 5

Choice of flyer material parameters.

Parameters Values

ρ f 7800 kg m −3

c f 3200 ms −1

αf 10 −6 K −1

θ Tf 298 K

c 1 f 10 6 JK −1 / m 3

Table 6

Choice of parameters in setting up the

problem.

Parameters Values

L 1 mm

L f 1 mm

N 100

N f 150

�t 10 −9 s v impact 50 − 125 ms −1 R 10 −4 ohm

Fig. 9. (a) Linear kinetic relation. (b) X − t diagram of the impact problem. The colors correspond to the strain in the flyer and the ferroelectric target. The speed of the impact is 75 m/s. We see a phase boundary propagating in the ferroelectric target marked by a large strain change. The phase boundary is

preceded by an elastic precursor which reflects off the free edge of the target and interacts with the phase boundary. The strain state changes to tensile

after this reflection. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3.7.1. Linear kinetics

We start our calculations with a linear kinetic relation d = d s ̇ s as shown in the Fig. 9 a. The choice of d s is taken to bed s = 10 3 kg m −2 s −1 . Fig. 9 b shows the strain map on the X − t diagram for the impact problem. Upon impact, and elasticprecursor propagates into the ferroelectric target followed by a phase boundary. Note that the elastic precursor is not sharp

since we capture it numerically. On the other hand, the phase boundary is sharp because it is numerically tracked. The

elastic precursor reflects off the free edge of the target and interacts with the phase boundary on its way back. The state of

strain changes to tensile during this point, as evident by the strain map.


Fig. 10. (a) Current profile obtained flowing through the resistor R has an exponential profile. (b) X − t diagram of the impact problem. The colors correspond to the temperature in the flyer and the ferroelectric target. (For interpretation of the references to color in this figure legend, the reader is referred

to the web version of this article.)

Fig. 11. (a) Variation of current profile with increasing impact speeds. (b) Variation of current profiles with changing values of d s for impact speed of 75

m/s.

Fig. 12. (a) Variation of current profile with target length. (b) Variation of current profiles with flyer length.

Fig. 10 a shows the current output flowing through the resistor R . We obtain an exponential profile of the current, which

is consistent with the experiments. The current obtained is consistent in magnitude with the results obtained in ( Furnish

et al., 20 0 0 ). The decrease in the current output corresponds to the phase boundary slowing down upon the interaction of

reflected elastic wave with the phase boundary. Fig. 10 b shows the temperature map on the X − t diagram of the impactproblem. Initially the material is at room temperature θ = 298 K . As the phase boundary goes through, we observe a tem-perature rise of ≈ 5 − 10 K in the ferroelectric material. This is largely due to the latent heat. The temperature rise acrossthe elastic waves due to thermal expansion is very small.

Now we change the impact speeds and plot the current through the resistor R . From Fig. 11 a, we observe that as the

impact speed increases, the current output also increases. Fig. 11 b shows the variation of current output for different values

of d s . The current increases with decreasing d s . This is because the speed of the phase boundary decreases with increasing

d s and hence the steady state current magnitude also decreases.

Next, we change the target length and study the current profile upon impact. From Fig. 12 a, we observe that for a shorter

target length, the magnitude of the current output is larger while the duration of the pulse is shorter. We also change the

flyer length to study the effect of the release wave from the free edge of the flyer. Fig. 12 b shows the variation in the current

profile as the flyer length changes. We observe the change in pulse length with the change in flyer length. This is because


Fig. 13. Variation of current profiles with R in the external circuit. As R , decreases, the current output declines. The current profiles achieve steady value

very quickly for very low R .

Fig. 14. (a) Combination of Stick-slip and Linear kinetic relation. (b) X − t diagram of the impact problem. The colors correspond to the strain in the flyer and the ferroelectric target. The speed of the impact is 75 m/s. (For interpretation of the references to color in this figure legend, the reader is referred to

the web version of this article.)

Fig. 15. (a) Comparison of current profiles for Linear and Stick-Slip Linear Kinetic relation. (b) X − t diagram of the impact problem. The colors correspond to the temperature in the flyer and the ferroelectric target. (For interpretation of the references to color in this figure legend, the reader is referred to the

web version of this article.)

the only difference flyer length makes is to control the time when the release wave hits the phase boundary. Hence the

current magnitude remains the same until the release wave hits the phase boundary and the current output goes down.

Profiles for L f = 0 . 4 mm , 0.7 mm and 1 mm overlap because the elastic precursor reflects off the free edge of the target andinteracts with the phase boundary before the release wave can interact with the phase boundary.

Finally, the current profiles are tracked for different values of R in the external circuit. Fig. 13 shows the variation of cur-

rent profiles with R . The current profiles are significantly different for different R values. For very low R , the current profile

achieves the steady value very quickly. The profile for R = 10 −5 � shows oscillations due to numerical errors associated withmesh updating and averaging.


3.7.2. Stick-slip linear kinetics

Next, we work with a combination of stick slip and linear kinetic relations as shown in Fig. 14 a. The strain map on

the X − t diagram is shown in Fig. 14 b for the impact speed v impact = 75 m/s and d c = 10 6 Pa and d s = 10 3 kg m −2 s −1 . Justas in Fig. 9 b, in this case, the phase boundary is clearly visible by the large change in strains. The phase boundary is

again preceded by elastic precursor which reflects off the free edge of the target and interacts with the propagating phase

boundary. The profile of the current is presented in Fig. 15 a along with temperature variations in Fig. 15 b. The comparison

of the two profiles show very similar qualitative behavior in linear and stick-slip linear case. The magnitude differs due to

the fact that an additional minimum driving force d c is required to move the phase boundary, which makes it slow. The

temperature rise of ≈ 5–10 K is again observed in this case.

4. Conclusions

In this paper, we studied impact induced phase transformation in ferroelectric ceramics. In the first part of the paper,

we developed a continuum model for a propagating discontinuity in a ferroelectric material subjected to large deformation

dynamic loading under adiabatic conditions. The developed model also allows for surface charges on the phase boundary,

which corresponds to the possibility of dielectric breakdown in the system. Using conservation laws and second law of

thermodynamics, we derive the governing equations and the expression for driving force acting on the phase boundary.

This generalizes the many special cases studied in the literature.

In the second part of the paper, we analyze plate impact on a ferroelectric material connected with a resistance across

the two ends. We develop a novel numerical method that captures sound waves, but tracks phase boundaries. We then

study in detail the experiments conducted by Furnish et al. (20 0 0) . These experiments measure the current and charge

across the resistor. Our results match the observations very well ( Fig. 7 b). Further we also show that our model reproduces

the experimentally observed linear Hugoniot relation between impact and shock velocity, and the fact that the Hugoniot

depends on the resistance. Finally, we conduct a detailed parameter study that gives new insight to various aspects of the

problem.

We conclude with a comment regarding the charge on the interface. Ferroelectrics are wide-band-gap semiconductors

and as such can sustain a space charge density. Xiao et al. (2005) developed a model of ferroelectrics that incorporates space

charges and showed that boundaries can cause localized charge density. However that work did not consider deformation

or dynamics. A combination of these analysis remains a topic for future investigation.

Acknowledgments

We are grateful to Chris Lynch for numerous discussions and many insightful comments during the course of this work.

This work was made possible by the financial support of the US Air Force Office of Scientific Research through the Center

of Excellence in High Rate Deformation Physics of Heterogeneous Materials (Grant : FA 9550-12-1-0091 )

Appendix A. Details of the first law of thermodynamics

A1. Rate of change of total energy

We begin by calculating the rate of change of total energy,

dE dt

= d dt

∫ B

E 0 d x + d d t

∫ S∩ y (B )

W σ (σ ) d S y + d d t

∫ y (B )

ε 0 2

|∇ y φ| 2 d y + d d t

∫ B

ρ0 2

| ̇ y | 2 d x. (67)

A1.1. Rate of change of stored energy

Looking at the first term in (67) and using the transport theorem,

d

dt

∫ B

E 0 dx = ∫

B

˙ E 0 dx −∫

S 0 ∩ B � E 0 � s 0 n dS x . (68)

A1.2. Rate of change of electrostatic energy

The calculation of the rate of change of electrostatic energy, the second term in (67) , is rather involved. The steps are

similar to ( James, 2002 ) and ( Xiao and Bhattacharya, 2008 ) but with modifications and generalizations. The calculation is

split over four parts listed below.

Rate of change of electrostatic energy: part 1

Setting ψ = φ in the weak form of Maxwell’s equation (9) we obtain ∫ y (B )

ε 0 |∇ y φ| 2 dy = ∫

y (B ) ∇ y φ · p dy +

∫ S∩ y (B )

φσdS y −∫ ∂y (B )

φ(D · ˆ n ) dS y . (69)

We note using the transport theorem that

d

dt

∫ y (B )

∇ y φ · p dy = d dt

∫ B

∇ y φ · p 0 dx

https://doi.org/10.13039/100000181


= ∫

B

(d

dt (∇ y φ) · p 0 + ∇ y φ · ˙ p 0

)dx −

∫ S 0 ∩ B

� (∇ y φ) · p 0 � s 0 n dS x (70)

= ∫

B ( (∇ y φ,t + v (∇ y ∇ y φ)) · p 0 + ∇ y φ · ˙ p 0 ) dx −

∫ S 0 ∩ B

� (∇ y φ) · p 0 � s 0 n dS x . We recall the transport theorem for surfaces in ( Cermelli et al., 2005 ) and we apply it to S ∩ y ( B ):

d

dt

∫ S∩ y (B )

φσ dy = ∫

S∩ y (B )

(˚(φσ ) − φσ s n κ

)dS y +

∫ ∂(S∩ y (B ))

φσ (V · w ) dl (71)

where V is the velocity of a point on ∂( S ∩ y ( �)), κ is the total curvature of S , w is the unit vector on the boundary of Stangential to the surface but normal to the curve ∂S ∩ y ( B ), and å denotes the time derivative of a following a trajectory thatis always normal to S ( Cermelli et al., 2005 ). Using (70) and (71) , the time derivative of (69) can be written as,

d

dt

(∫ y (B )

ε 0 |∇φ| 2 dy )

= ∫

B ( (∇ y φt + v (∇ y ∇ y φ)) · p 0 + ∇ y φ · ˙ p 0 ) dx −

∫ S 0 ∩ B

� (∇ y φ) · p 0 � s 0 n dS x

+ ∫ ∂(S∩ y (B ))

φσ (V · w ) dl + ∫

S∩ y (B ) ( φ̊σ + φσ̊ − φσ s n κ) d S y − d

d t

∫ ∂y (B )

φD · ˆ n dS y .

(72)


We use the Reynolds’ transport theorem to get

d

dt

[∫ y (B )

ε 0 2

|∇ y φ| 2 dy ]

= ∫

y (B ) ε 0 ∇ y φ · ∇ y φ,t dy +

∫ ∂y (B )

ε 0 2

|∇ y φ| 2 v · ˆ n dS y −∫

S∩ y (B )

� ε 0 2

|∇ y φ| 2 �

s n dS y (73)

Next, we apply the weak form of Maxwell equation (9) to y ( B ± ) with ψ = φ,t to obtain ∫ y (B )

ε 0 ∇ y φ · ∇ y φ,t dy = ∫

B

∇ y φ,t · p 0 d x −∫ ∂y (B )

φ,t D · ˆ n d S y + ∫

S∩ y (B ) � φ,t D · ˆ n � d S y (74)

Plugging (74) in (73) gives us

d

dt

[1

2

∫ y (B )

ε 0 |∇ y φ| 2 dy ]

= ∫

B

∇ y φ,t · p 0 dx −∫ ∂y (B )

φ,t (−ε 0 ∇ y φ + p ) · ˆ n dS y + ∫ ∂y (B )

ε 0 2

|∇ y φ| 2 v · ˆ n dS y

+ ∫

S∩ y (B ) � φ,t D · ˆ n � d S y −

∫ S∩ y (B )

� ε 0 2

|∇ y φ| 2 �

s n d S y (75)

Now consider the last two terms above: ∫ S∩ y (B )

� φ,t (D · ˆ n ) � dS y −∫

S∩ y (B )

� ε 0 2

|∇ y φ| 2 �

s n dS y

= ∫

S∩ y (B )

(� φ,t � 〈 D · ˆ n 〉 + 〈 φ,t 〉 � D · ˆ n � − ε 0

2

� |∇ y φ| 2

� s n

)dS y

= ∫

S∩ y (B )

((−� ∇ y φ� 〈 D · ˆ n 〉 − 〈∇ y φ · ˆ n 〉 � D · ˆ n � − ε 0

2

� |∇ y φ| 2

� )s n − φ̊σ

)dS y

= ∫

S∩ y (B ) � ̂ n · T M ̂ n � s n dS y +

∫ S∩ y (B )

φ̊σ dS y . (76)

where we have used the jump conditions in the second equality and the definition of T M in the third.

Next, we deal with ∂y ( B ) terms in (75) . We can use dφdt

= φ,t + ∇ y φ · v on ∂y ( B ) to conclude ∫ ∂y (B )

φ,t (D · ˆ n ) − ε 0 2

|∇ y φ| 2 v · ˆ n dS y = ∫ ∂y (B )

dφ

dt D · ˆ n dS y +

∫ ∂y (B )

T M ̂ n · v dS y . (77)

Plugging (76) and (77) in (75) , we get

d

dt

[1

2

∫ y (B )

ε 0 |∇ y φ| 2 dy ]

= ∫

B

∇ y φ,t · p 0 dx −∫ ∂y (B )

dφ

dt D · ˆ n dS y +

∫ S∩ y (B )

φ̊σ dS y −∫ ∂y (B )

T M ̂ n · v dS y

+ ∫

S∩ y (B ) � ̂ n · T M ̂ n � s n dS y (78)



Finally we subtract (78) from (72) to get the final expression.

d

dt

∫ y (B )

ε 0 2

|∇ y φ| 2 dy = ∫

y (B ) (∇ y ∇ y φ) v · p dy +

∫ B

∇ y φ · ˙ p 0 dx −∫

S 0 ∩ B � ∇ y φ · p 0 � s 0 n dS x +

∫ ∂y (B )

T M ̂ n · v dS y

+ ∫ ∂y (B )

dφ

dt D · ˆ n d S y − d

d t

∫ ∂y (B )

φ D · ˆ n dS y

−∫

S∩ y (B ) � ̂ n · T M ̂ n � s n dS y +

∫ S∩ y (B )

(φσ̊ − κσφs n ) dS y + ∫ ∂S∩ y (B )

φσ (V · w ) dl. (79)

A1.3. Rate of change of kinetic energy

Using the typical Reynolds’ transport theorem, we obtain

d

dt

∫ B

ρ0 2

| ̇ y | 2 dx = ∫

B

ρ0 ̈y · ˙ y dx −∫

S 0 ∩ B

� ρ0 2

˙ y · ˙ y �

s 0 n dS x . (80)

A1.4. Rate of change of interfacial energy

The rate of change of the interfacial energy can be written as (following Cermelli et al., 2005 ),

d

dt

∫ S∩ y (B )

W σ dS y = ∫

S∩ y (B ) ( W̊ σ − W σ s n κ) dS y +

∫ ∂S∩ y (B )

W σ (V · w ) dl (81)

A1.5. First law of thermodynamics

Finally putting all the expressions together, (20) becomes ∫ B

(˙ E 0 + ∇ y φ · ˙ p 0 − ˙ r + ∇ x · q 0 + ρ0 ̈y · v

)dx

+ ∫

y (B ) ( ∇ y ∇ y φ) v · p dy +

∫ ∂y (B )

(T M ̂ n · v − t · v

)dS y

+ ∫

S 0 ∩ B

[ −� E 0 + ∇ y φ · p 0 � s 0 n −

� ρ0 2

˙ y · ˙ y �

s 0 n − � q 0 · ˆ n 0 � ]

dS x

+ ∫

S∩ y (B )

[−� ̂ n · T M ̂ n � s n − (W σ + σφ) κs n + (W̊ σ + φσ̊ )]dS y

+ ∫ ∂y (B )

[dφ

dt D · ˆ n + φ d

dt

(D · ˆ n

)+ φ(D · ˆ n )(v · ˆ n )

]d S y − d

d t

∫ ∂y (B )

φ D · ˆ n dS y = 0 (82)

Next, we rearrange and simplify certain terms in (82) . First,

(∇ y ∇ y φ) v · p = −φ,i j p j v i = φ,i j (D j + ε 0 φ, j ) v i = −

(E i D j −

ε 0 2

E k E k δi j

), j

v i = −( ∇ y · T M ) · v (83) where the Maxwell equation (5) is used in the third step. So, ∫

y (B ) ( ∇ y ∇ y φ) v · p dy +

∫ ∂y (B )

T M ̂ n · v dS y = −∫

y (B )

((T M i j v i

), j

− T M i j v i, j )

dy + ∫ ∂y (B )

T M ̂ n · v dS y

= ∫

S∩ y (B ) � T M i j v i � n j dS y +

∫ y (B )

T M i j v i, j dy

= ∫

S 0 ∩ B

� T 0 M ̂ n 0 · v

� dS x +

∫ B

T 0 M · ˙ F dx (84)

where T 0 M

= JT M F −T is the pullback of the Maxwell stress tensor into the reference frame. Next, we focus on the tractionterm as follows. ∫

∂y (B ) t · v dS y =

∫ ∂y (B )

S ̂ n · v dS y = ∫ ∂B

S 0 ̂ n 0 · v dS x

= ∫

B

∇ x ·(S T 0 v

)dx +

∫ S 0 ∩ B

� S 0 ̂ n 0 · v � dS x =

∫ B

[( ∇ x · S 0 ) · v + S 0 · ˙ F

]dx +

∫ S 0 ∩ B

� S 0 ̂ n 0 · v � dS x (85) We use the following equality (see James, 2002 ) to manipulate the terms associated with s n in (82) , ∫

S

gs n dS y = ∫

S

g v ± · ˆ n dS y + ∫

S

g J ±s 0 n dS x , (86)

0


∫ S

� g� s n dS y = ∫

S

� gv · ˆ n � dS y + ∫

S 0

� g J� s 0 n dS x . (87)Starting with the term associated with Maxwell stress and s n in (82) , ∫

S∩ y (B ) � ̂ n · T M ̂ n � s n dS y =

∫ S∩ y (B )

� ( ̂ n · T M ̂ n )(v · ˆ n ) � dS y + ∫

S 0 ∩ B � ( ̂ n · T M ̂ n ) J� s 0 n dS x

= ∫

S∩ y (B ) � v · T M ̂ n � dS y −

∫ S∩ y (B )

� (E t · v )(D · ˆ n ) � dS y +

∫ S 0 ∩ B

� ( ̂ n · T M ̂ n ) J� s 0 n dS x where E has been decomposed into tangential and normal components in the second step. The jump term on E t can further

be simplified as ∫ S∩ y (B )

� ( E t · v ) (D · ˆ n )� dS y = ∫

S∩ y (B ) 〈 E t · v 〉 � D · ˆ n � + � E t · v � 〈 D · ˆ n 〉 dS y

= ∫

S∩ y (B ) E t · 〈 v 〉 σ + E t · � v � 〈 D · ˆ n 〉 dS y

= ∫

S∩ y (B ) E t · 〈 v 〉 σ dS y −

∫ S 0 ∩ B

E t · � F ̂ n 0 � 〈 D 0 · ˆ n 0 〉 s 0 n dS x , (88)where the Hadamard compatibility equation � v � + � F ̂ n 0 � s 0 n = 0 is used and D 0 = JF −1 D is the pullback of D in the referenceframe. Putting this back we get, ∫

S∩ y (B ) � ̂ n · T M ̂ n � s n dS y =

∫ S∩ y (B )

� v · T M ̂ n � dS y + ∫

S 0 ∩ B � ( ̂ n · T M ̂ n ) J� s 0 n dS x

−∫

S∩ y (B ) E t · 〈 v 〉 σ dS y +

∫ S 0 ∩ B

E t · � F ̂ n 0 � 〈 D 0 · ˆ n 0 〉 dS x . (89)Looking at κ term associated with s n in (82) ∫

S∩ y (B ) (W σ + φσ ) κs n dS y =

∫ S∩ y (B )

v ± · (W σ + φσ ) κ ˆ n dS y + ∫

S 0 ∩ B (W σ + φσ ) κs 0 n J ± dS x

= ∫

S∩ y (B ) 〈 v 〉 · (W σ + φσ ) κ ˆ n dS y +

∫ S 0 ∩ B

(W σ + φσ ) κs 0 n 〈 J〉 s 0 n dS x (90)Finally, looking at the term corresponding to the jump in kinetic energy ∫

S 0 ∩ B

� ρ0 2

˙ y · ˙ y �

s 0 n dS x = ∫

S 0 ∩ B s 0 n � ρ0 ̇ y � · 〈 v 〉 dS x

= ∫

S 0 ∩ B

(−� S 0 ̂ n 0 � + f 0 ) · 〈 v 〉 dS x (91)

The last two terms in (82) cancel out. Finally, putting (84), (85), (89), (90) and (91) in (82) , we obtain (24) .

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