Shock Waves (1997) 7: 147 150
An analysis tool for piezoelectric gages
B.B. Lewis
APTEK, Inc., 1257 Lake Plaza Dr., Colorado Springs, Colorado, 80906, USA
Received 6 May 1996 / Accepted 31 October 1996
Abstract. The Piezo-electric Gage Analysis System U.S. has previously been no capability to provide quantitative de-
(pegasus) couples a two-dimensional dynamic structural scription of the enhanced electric fields leading to this con-
finite element code to a two-dimensional electrostatics code duction. Montgomery and coworkers have recently shown
for analysis of piezoelectric gages. The method has a sound the distortions in electric fields with three-dimensional nu-
theoretical basis and is built around two powerful finite ele- merical simulation, but the spatial resolution of their work
ment anlysis codes. The analysis codes provide the solution was more limited than that of the present study (Montgomery
of the time dependent stress state in the gage and the solu- et al., 1995).
tion of the electrostatic equation for each time step. pega- We have recently developed an analytical method for
sus provides the link between the two codes and the steps studying piezoelectric gages in two dimensions. The method
required to carry the analysis through to prediction of gage couples a two-dimensional dynamic finite element analysis
currents. Post-processing of the results allows visual inter- program to a two-dimensional electrostatics code. The goal
pretation of the the electric fields within the gage. Here we of this development was to aid in understanding the differ-
briefly describe the code and show that it can be a valuable ences in response of quartz gages due to gage configuration
tool for understanding the nature of piezoelectric gages. and loading nonuniformity. The analysis method is general
enough to be applied to any type of piezoelectric gage and
Key words: Piezoelectric gages, Dynamic analysis, Electric would be useful for gage development studies, experiment
fields design studies, and data interpretation.
The theoretical basis is described in Sect. 2. In Sect. 3,
we outline the implementation of the analytical method, and
describe a study of shorted and shunted gages in Sect. 4. A
brief summary is presented in Sect. 5.
1 Introduction
Piezoelectric stress gages are widely used to provide nano-
2 Theory
second, time-resolved measurements of materials subjected
to high pressure shock compression loading (Graham et al., The theory of linear piezoelectricity is well known. The use
1965). The active elements in these gages are typically disks of quartz gages for shock compression environments sub-
of X-cut quartz or various crystallographic cuts of lithium jects the sensors to finite strain which results in nonlinear
niobate (Graham, 1993, Graham and Reed, 1978). These materials behaviors (Graham, 1972). These nonlinear ma-
crystal gages are ideally used in a guard-ring configura- terials contributions have been well characterized and are
tion such that the one-dimensional mechanical and electrical relatively easy to incorporate in shock sensor theory. In the
conditions within the inner part of the disk result in approxi- present case in which the objective of the work is to iden-
mately linear relations between measured short-circuited cur- tify and quantify the magnitudes of the electric field en-
rent and stress at the input electrode (Graham et al., 1965). hancements, it is sufficiently accurate to rely principally on
In various engineering applications it has proven useful linear piezoelectric theory, modified for particular conditions
to compromise the one-dimensional condition of a shunted achieved in the experiments used to test the theory.
gage by utilizing a shorted guard ring configuration. How- Piezoelectric polarization can be related to the stress, Ã,
ever the shorted guard ring configuration has limitations due through piezoelectric coefficients, d. The eq. can be written
to the presence of conductors on the radial surfaces of the as:
sensor. These conductors cause enhancement of the electric
Ż ŻŻ
P = dà (1)
fields which results in shock-induced conduction within the
crystal. Experimental studies (Graham, 1975) clearly showed where P is the polarization vector, d is the matrix of piezo-
electrical conduction in shorted guard ring gages, but there electric coefficients, and à is the stress tensor written in
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3 Code development and implementation
We used two public domain (but export controlled) codes
to perform the bulk of the calculations needed to solve a
gage problem. The codes were coupled and the results put
into a useful form by means of additional coding written
specifically for those functions. Together, these comprise the
PiezoElectric Gage Analysis System - U. S. (pegasus).
Fig. 1. Analysis models representing a shorted gage configuration (left) and pegasus was verified by analysis of shunted and shorted
a shunted gage configuration (right) were compared
gages for comparison with expected results.
The first step in implementing the method is the de-
velopment of a suitable finite element mesh. We use pa-
vector form. The overlines indicate that the variable is a
tran (1987) for model development as well as for vec-
vector or matrix. For x-cut quartz gages, the only nonzero
tor plotting of results. Next is analysis using a dynamic
coefficient is that which relates the through-thickness stress
stress analysis program. We use dyna2d (Hallquist, 1988),
to through-thickness polarization.
a two-dimensional explicit hydrodynamic code developed
The electrical eq. is given by:
at Lawrence Livermore National Laboratory. The element
stresses and nodal displacements are output to a plot file at
Å» Å»
D = 
+ P (2)
specified times during the analysis.
The remaining steps are done internally to the pega-
where D is the electric displacement, is the material per-
sus code. First, the element stresses are converted to nodal
mitivity, and E is the electric field. Quartz gages are gen-
polarization vectors by averaging the stresses at the nodes
erally made with electrodes on the top and bottom and with
and multiplying by the piezoelectric constants. In the case
a small resistance between them; essentially a short circuit.
of quartz, only the term relating through-thickness stresses
Since charge can not collect on the surfaces, the following
to through-thickness polarizations is needed. The code is
eq. holds:
written such that piezoelectric materials having additional
piezoelectric coefficients could be correctly handled. Polar-
Å» Å»
"· D= 0 (3)
ization vectors are output for later use in determining electric
displacements and currents. The code performs the operation
where " is the gradient operator.
Å» Å»
"· P needed for Eq. 6. The manipulation is done using nor-
Substitution of equation 2 into Eq. 3 gives
mal finite element techniques involving shape functions and
results in a scalar value for each element. These values are
Å» Å»
"· ( 
+ P) =0. (4)
used as input in the solution of Poisson s eq.
The code used to solve Poisson s eq. is topaz2d (Sha-
The electric field, E, is the spatial gradient of the voltage,
piro, 1986), a two-dimensional heat transfer code devel-
V .
oped at Lawrence Livermore National Laboratory. topaz2d
solves the eq.
Å»

= -"V (5)
Å» Å»
"· k"¸+qg = 0 (8)
By making this substitution, and by using the associative
property of the vector dot product, Eq. 4 becomes
where k is the thermal conductivity, ¸ is the temperature,
and qg is the internal heat generation. By direct replacement
Å» Å» Å» Å»
"· "V - "· P = 0 (6)
of variables, i.e., replace temperature with voltage (¸
V ), conductivity with permitivity (k ), and internal heat
which is Poisson s equation for electrostatics. Although the
generation with the negative of the polarization gradient dot
driving function, i.e., the stress wave, is not static, it trav-
Å» Å»
product (qg -"·P), the heat transfer code can be used to
els through quartz much more slowly than the electrical re-
solve the electrostatics problem. Surfaces that are electroded
sponse. Therefore, it is reasonable to solve the electrical
are given a voltage (temperature) boundary condition of zero
problem statically for any given stress state. Solution of this
for the calculation. Results of the heat transfer code are given
eq. gives the voltage. Back substitution into Eqs. 5 and 2
as temperature and temperature gradients which correspond
gives the electric displacement vector for this stress state.
to voltage and electric field in our analogy. The calculated
The total charge, C, passing through the electrode at any
values for electric field, which are nodal vectors, are saved
of these discrete time steps is the component of the electric
for later use.
displacement, D, normal to the surface integrated over the
The electric displacement is determined from the polar-
electrode surface, S.
ization and electric field vectors previously saved. The code
integrates the electric displacement normal to the electrode
Å»
Ć
C = (D · n)dS (7)
over the electrode area for each time step solved, and out-
S
puts a charge versus time curve. As the final step, current
The current is the derivative of the charge with respect to versus time is obtained by numerically taking the slope of
time. the charge - time curve.
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Fig. 2. Electric field intensities for the axisymmetric right half of a shorted gage configuration for a shock wave location 600 nanoseconds after impact.
The field intensities and directions show distortion due to the presence of the electrical conductor on the edge of the disk
Fig. 3. A close-up view of the electric field vectors for a shorted gage (left) compared to the same area for a shunted gage (right) shows the severe conditions
encountered due to the conductor on the disk of the shorted gage. Of particular interest is the major enhancement in electric field intensity at the edge of
the disk at the location of the shock front. This field enhancement has been termed  the high voltage beast (Graham, 1995); a shorted; b shunted
4 Gage configuration study a guard ring to thickness ratio of 1.5. The guard ring gap
was 0.05 mm (about 0.002 inches). The gage was loaded by
a uniform pressure of 0.90 GPa (9.0 kbar) across the top of
As part of the code verification studies, we analysed a gage
with various boundary conditions and electrode configura- the quartz with a duration longer than the gage read time.
Analysis of the gage was done using shunted and shorted
tions. Of particular interest is the comparison of results for
a shunted gage configuration versus a shorted gage config- electrode configurations as shown in Fig. 1. The finite ele-
ment model used in the analysis contained 25351 nodes and
uration. The gage used in these studies has a thickness of
25000 quadrilateral elements representing an axisymmetric
6.35 mm (0.25 in), a diameter to thickness ratio of 5.0, and
150
section through the gage. The quartz properties used were References
8.68 × 1010 Pa (1.25 × 107 psi) for the elastic modulus, 0.10
1. Graham RA, Neilson FW, Benedick WB (1965) Piezoelectric Current
for Poisson s ratio, and a density of 2.65 g/cc. The piezo-
from Shock-Loaded Quartz  A Submicrosecond Stress Gage, J. Ap-
electric constant used for quartz was 2.30 × 10-12 C/N and
plied Physics, Vol. 36, pp. 1775 1783
the permitivity used was 4.00 × 10-11 F/m.
2. Graham RA (1972) Strain Dependence of Longitudinal Piezoelectric,
Currents and electric fields were predicted and compared
Elastic, and Dielectric Constants of X-Cut Quartz, Physics Review,
for the shorted and shunted configurations. The current pre-
Vol. B6, pp. 4779 4792
dicted for the shorted case is more distorted from ideal than 3. Graham RA (1975) Piezoelectric Current from Shunted and Shorted
Guard-Ring Quartz Gages, Journal of Applied Physics, Vol. 46,
the shunted case, which is a phenomenon seen experimen-
pp. 1901 1909
tally (Graham, 1975). Of particular interest is a plot of the
4. Graham RA, Reed RP (1978) Selected Papers on Piezoelectricity
electric field vectors obtained for the shorted configuration
and Impulsive Pressure measurements, Sandia Report, SAND78 1911,
gage. The electric fields calculated at 600 nanoseconds is
Sandia National Laboratories
shown in Fig. 2 for the symmetric right half of the gage.
5. Graham RA (1993) Solids Under High Pressure Compression: Me-
The area of high gradients at the electroded edge is shown chanics, Physics, and Chemistry, Springer-Verlag
6. Graham RA (1995) Private communication
in greater detail and is compared to the same region of a
7. Hallquist JO (1988) User s Manual for DYNA2D  An Explicit Two-
shunted gage in Fig. 3.
dimensional Hydrodynamic Finite Element Code with Interactive Re-
zoning and Graphical Display, UCID 18756, Lawrence Livermore
Laboratory, Livermore, CA, Rev. 3
5 Conclusion
8. Montgomery ST, Graham RA, Anderson MU (1995) Return to the
Shorted and Shunted Quartz Gage Problem: Analysis With the Subway
Code, in Shock Compression of Condensed Matter, in press
We developed a tool for analyzing piezoelectric gage re-
9. PATRAN Plus User Manual Release 2.4, Publication Number 2191023,
sponse and have used it for studies of quartz gages. Plotting
PDA Engineering, September 1989
of the calculated electric fields allows good visualization and
10. Shapiro AB (1986) TOPAZ2D  A Two-dimensional Finite Element
a better understanding of the gage. Practical applications of
Code for Heat Transfer Analysis, Electrostatic, and Magnetostatic
the pegasus system include investigation of gage configu-
Problems, UCID 20824, Lawrence Livermore Laboratory, Livermore,
rations. One such application was presented and shows the CA
existence of greatly enhanced electric fields in a shorted gage
configuration. These results show that use of shorted gages
without consideration of their limitations can lead to major
errors in measurement.