One possible extension of the Plane Strain formulation is to assume that the displacement field depends on the out-of-plane coordinate
Z, but in-plane strains are independent of it.
here, u0(
X,
Y) and
v0(
X,
Y) are the in-plane displacement components; and
a,
b, and
c are constants independent of the
X,
Y, and
Z coordinates. The gradient of the displacement field then reads:
At the cross section Z = 0, the in-plane deformation is fully characterized by the in-plane displacement components
u0(
X,
Y) and
v0(
X,
Y). The displacement gradient then simplifies to
The above conditions differ from the Plane Strain formulation only by the fact that the out-of-plane strain component
εz can vary linearly throughout the cross section.
In COMSOL Multiphysics, the coefficients a,
b, and
c in
Equation 3-8 are modeled as extra degrees of freedom that are constant throughout the model (global variables).
For the 1D representation, only the x-component of the displacement field is considered, and only gradients with respect to this direction are computed, this is,
u =
u(
X),
, and
. It is possible to apply the generalized plane strain assumption to either the
xy-plane, the
xz-plane, or to both planes, in which case the strain components are augmented to
and
.
Coefficients a,
b, and
c are assumed to be small. Then, using the above displacement field in the displacement gradient evaluated in the plane
Z = 0 simplifies to
The first term, u0, represents a static in-plane prestress deformation:
The second part of the solution, u1, presents a time-harmonic linear perturbation with an amplitude that can be a function of the in-plane coordinates
X and
Y. Such a perturbation can be seen as an out-of-plane wave, with a small amplitude that propagates in the
Z direction, a wavelength
L, and phase velocity
c:
Note that in contrast to the prestress solution u0, the perturbation amplitude
u1 can have nonzero values in all the displacement components:
There are two alternative approaches. The wavelength L, and thus the wave number
kZ, can be considered as a parameter. Then,
ω can be computed by an eigenfrequency analysis for the 2D cross section with all three displacement components taken as dependent variables. As a result, one obtains
Thus, the wave speed for the out-of-plane wave is computed as a function of the wavelength L and possible prestress
u0 in the material. The dependence of the wave speed on the wavelength is often called dispersion.
Alternatively, the frequency f (and thus
ω) can be taken as a parameter. Then, the solution can be computed via eigenvalue analysis with respect to the wave number
kZ using the 2D cross section geometry. Hence,
which determine the wavelength L and phase velocity
c for the wave that propagates out-of-plane for a given frequency
f under given in-plane prestress deformation
u0. Such interpretation of the perturbation solution is sometimes called a signaling problem.