Generalized Plane Strain
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.
The above assumption have the following 3D solution for the displacement field:
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 out-of-plane shear strains are zero, εxz = εyz = 0, and under the assumption of small displacements and rotations the normal strain reads
(3-8)
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.
The generalized plane strain approximation is good when the structure is free to expand in the out-of-plane direction, and the possible bending curvature is small with respect to the extents of the structure in the xy-plane. In the case where there is no out-of-plane bending, the out-of-plane strain component simplifies to εz = c.
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), u/∂Y = ∂u/∂Z = 0, and εxy = εy = 0. 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 εy = b and εz = c.
For the 1D axisymmetric representation, only the radial component of the displacement field is considered, and only gradients with respect to this direction are computed, this is, u = u(R), and u/∂Φ = ∂u/∂Z = 0. It is possible to apply the generalized plane strain assumption to the rϕ-plane, so the strain is augmented to εz = c.
Geometric Nonlinearity
In case of geometric nonlinearity, the strains are represented by the Green–Lagrange strain tensor:
Consider the following displacement field expressed in terms of the material coordinates:
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
We obtain the strain components by dropping quadratic and higher order terms in the coefficients a, b, and c:
Thus, in the leading order approximation, the strains become independent of the out-of-plane coordinate Z.
Out-of-Plane Waves
When a 2D plane strain model represents a cross-section of the structure that has a significant uniform extension in the out-of-plane Z direction, the following 3D solution can be sought in form of the amplitude expansion:
The first term, u0, represents a static in-plane prestress deformation:
This can be obtained by a standard static analysis using a 2D geometry for the cross section with the corresponding boundary conditions.
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.