Equivalent Single Layer (ESL) Theory
Composite laminates are formed by stacking layers of different materials and/or having different fiber orientations. In general their planar dimensions are two orders of magnitude larger than their thickness. Often laminated structures are used in applications requiring high membrane and bending strengths. Therefore, in many cases composite laminates can be modeled using a shell element based on an equivalent single layer theory.
This is a classical way of modeling composite laminates. This theory treats a heterogeneous laminated composite as a statically equivalent single layer. This theory reduces a 3D continuum problem to an equivalent 2D problem, thus reducing the size and computational time of the problem.
In addition to simplicity and low computational cost, this theory provides sufficiently accurate description of the global response for a thin to moderately thick laminates such as gross deflections, critical buckling loads, and eigenfrequencies with corresponding mode shapes.
Classification
The ESL theories can be classified into various groups based on the description of the transverse shear stresses.
Classical Laminate Plate Theory (CLPT)
The classical laminate plate theory is an extension of Kirchhoff or classical plate theory used for single layer thin shells. In this theory, transverse shear stresses are neglected and the deformation is entirely due to the bending and in-plane stretching.
First Order Shear Deformation Theory (FSDT)
The first order shear deformation theory is similar to the Mindlin-Reissner shell theory used for single layer thick shells. This theory extends the kinematics of CLPT by including the gross transverse shear deformation. The transverse shear strain is assumed to be constant with respect to the thickness coordinate. As the transverse shear strain has a constant value, this theory requires a shear correction factor.
Higher Order Shear Deformation Theory
This theory is the extension of FSDT and here the displacement field is approximated in such a way that the transverse shear strain varies quadratically with respect to the thickness coordinate. It makes the transverse shear stresses zero at the top and bottom surfaces of the laminate and thus eliminates the need of a shear correction factor.
First Order Shear Deformation Theory (FSDT)
In COMSOL Multiphysics, first order shear deformation (ESL-FSDT) theory is one of the options for analyzing composite laminates. This theory is implemented in the Layered Linear Elastic Material model in Shell interface. It has an MITC (mixed interpolation of tensorial components) formulation. As this theory accounts for the transverse shear deformation, it can be used for rather thick shells.
Key features
•
Degrees of freedom (3 displacements, 3 rotations) are defined only at the midplane or reference plane of the composite laminate.
•
Suitable for modeling thin to moderately thick laminates.
•
Suitable for finding global response of the laminate e.g. gross deflections, eigenfrequencies, critical buckling load etc.
•
Not very computationally expensive, and thus suitable for analysis of laminates having large number of layers.
•
Requires shear correction factor for thicker laminates where transverse shear stresses are not negligible
In the Structural Mechanics Module User’s Guide: Theory for FSDT Laminated Shell