XXI ICTAM, 15β21 August 2004, Warsaw, Poland
A GEOMETRICALLY NON-LINEAR FINITE SHELL ELEMENT WITH PIEZOELECTRIC
LAYERS
Sven Lentzen, RΓΌdiger Schmidt
IAM, Institute for General Mechanics,
Technical University Aachen, D-52062 Aachen, Germany
EXTENDED SUMMARY
In recent years the potential
of integrating piezoelectric materials into smart structures has aroused the interest of many
researchers.
Theoretical models have been developed in order to fully understand the behaviour of smart structures
with
integrated piezoelectric materials. These models range from relatively simple linear beam theories to
complex geometrically
as well as physically non-linear models.
Many models are developed in order to investigate
the mentioned smart structures. These models are based on either a
beam or a plate/shell theory. Piefort
[1] has pointed to the fact that the application of a beam theory to model thin smart
structures will
not produce satisfying results when collocated systems are considered. A plate or shell model is therefore
to
be preferred.
Considerably less work can be found based in the area of geometrically non-linear shell
theories. This is quite surprising
considering the fact that many authors refer to a particular benchmark
problem also described in [1]. In this benchmark
problem a cantilever piezoelectric bimorph beam
acting as a sensor is deflected reasonably far away from the geometrically
linear regime.
In this work
a moderate rotation theory is deployed in the finite element formulation of a composite shell with integrated
piezoelectric
0_20_40_60_80_100
layers. The strain-displacement relations are valid for small strains but moderate
rotations (see Schmidt and
Reddy [2]).
A total Langrangian formulation is applied to define the internal
virtual work. This requires not only the application of
the 2nd Piola-Kirchhof stress and the Green-Lagrange
strain tensor, but also the introduction of electrical field quantities
defined in material curvilinear
coordinates. The covariant elements of the electric field vector 0Ei and the contravariant
elements
of the electric displacement vector 0Di in material curvilinear coordinates can be written as:
0Ei
= τ
@
@i and 0Di = J tDi; (1)
where J denotes the determinant of the deformation gradient, i
denotes the surface parameter in direction i, is the
electric potential and the lower left indices 0
and t refer to quantities defined in material respectively spatial coordinates.
It can be easily shown
that both quantities 0Ei and 0Di are energetically conjugated.
These new definitions together with the
well-known linear direct and converse piezoelectric effect and the moderate rotation
theory is then assembled
into a finite shell element. An additional assumption is made in allowing only an electric
field
in transverse direction which is uniform between the positive and negative pole of an electrode pair.
Considering
the benchmark problem discussed in [1] it can be shown that not only considerable differences
0_20_40_60_80_100
exist between
the analytical beam solution and the finite shell element approximation, but also between
the geometrically linear and
non-linear finite element results. These differences are even larger than
the primarily mentioned ones. In the benchmark
problem a cantilever piezoelectric bimorph beam is simulated
(100 5 1 mm), E = 2:0 GPa (Youngβs modulus),
d31 = 2:2 10τ11 Cb/N (piezoelectric constant)
and 33 = 1:062 10τ10 F/m (dielectric constant). In order to investigate
the actuator and
sensor properties of the beam it is either imposed with an electrical voltage or with a transverse tip
force.
In the latter case the beam is covered with five equal electrode pairs.
To make the differences
between the results of beam analysis, the geometrically linear and non-linear finite element approximations
more
apparent, the boundary conditions have been changed from clamped-free to clamped-hinged in the
actuator
case and to clamped-clamped in the sensor case. The results are displayed in figure 1.
Actuator
In
figure 1a the piezoelectric bimorph beam with the mentioned boundary conditions is imposed with
200 V. The finite
element analysis is performed twice. One case is calculated with the normal boundary
conditions as mentioned above,
and the other case is calculated with adapted boundary conditions to invoke
beam-like behaviour. Then both results are
compared with the analytical result:
3d31
4h2L
τLx2
τ
0_20_40_60_80_100
x3 ; (2)
where h is the total thickness of the beam and L its length. From this comparison one can
conclude that the difference
between the geometrically non-linear finite shell element approximation
and the analytical result is explained by the
boundary conditions: Due to the clamping conditions in finite
element analysis (MRT shell) high stress concentrations are
induced in these areas. Once these effects
are cancelled (MRT beam) the non-linear results agree well with the analytical
results. It can then
be concluded that the effects of the geometrical non-linearity is not profound in the actuator application.
Sensor
In
figure 1b the bimorph beam is used as a sensor. Five equally large electrode pairs are
attached to the beam and in the
middle a transverse force acts downwards to obtain a mid-point deflection
of 2 mm. Comparing the analytical results to
the linear finite shell element approximation, the influence
of the clamping conditions can be noticed in these areas.
Due to the induced membrane stresses,
which are considered in the geometrically non-linear case, the required force to
obtain a deflection of
2 mm (1.1843 N), is much greater than in the linear analysis (0.32133 N). For comparison purposes
figure
1b additionally displays the results in case the force remains equal (deflection: 1.09 mm). Now one
can conclude
that the effect of the geometrically non-linearity is more profound than the clamping effect,
in the sensor application.
Another phenomenon which appears in this case is the fact that the relative
0_20_40_60_80_100
voltage distribution changes when the geometrically
non-linear case is considered.
0 0.025 0.05 0.075
0.1 0 0.025 0.05 0.075 0.1
0 β500
1 β300
β100
2
100
3
300
4
5
x [m]
Deflection [mm]
Theory
MRT
beam
MRT shell
Sensor Position [m]
Sensor Voltage [V]
Theory
Linear [2mm/0.32133N]
MRT [2mm/1.1843N]
MRT
[1.09mm/0.32133N]
200 V
F
a b
ττττττ
Figure 1. a:
bimorph beam acting as an actuator with 200V. b: bimorph beam acting as sensor with a mid-point displacement
of 2mm.
The effects shown in this benchmark problem, where a beam is clamped on both sides, are
representative for many cases
in which plates or shells as 2- or 3-dimensional objects are clamped along
the sides. Not only the absolute sensor voltage,
but also the relative distribution is influenced by
the non-linearity. This could influence the effectivity of, for instance,
modal damping algorithms used
in linear dynamic systems.
Further examples, concerning beams/plates and shells with integrated piezoelectric
layers, will be presented to validate
the finite shell element formulation and to demonstrate
the effect of geometrical non-linearity on actuator and sensor
applications.
References
[1] Piefort, V.:
Finite Element Modelling of Piezoelectric Active Structures. PhD Thesis, UniversitΓ© Libre de Bruxelles,
2001.
[2] Schmidt R., Reddy J.: A Refined Small Strain and Moderate Rotation Theory of Elastic Anistropic
Shells. J. Appl Mech 55:611-617, 1988.0_20_40_60_80_100