# Summary of Results on Laminated Plates

By using the Eshelby-Stroh formalism, analytical solutions in the form of infinite series are obtained for linear elastic anisotropic laminated plates and cylindrical bending of laminated panels under different (clamped, simply supported, traction free) boundary conditions. For a laminated plate, edges of a ply may have boundary conditions different from those of the ply either above or beneath it. Furthermore, the four edges of a ply may have different boundary conditions. Thus one can realistically simulate boundary conditions for each individual ply. The approach has also been extended to hybrid laminates with one or more plies made of a piezoelectric material, and to thermo-mechanical deformations of a laminated plate.

Some of the salient results are summarized below.

The through-the-thickness distribution of the transverse shear stress near an edge strongly depends upon the boundary conditions at the edge, is not generally parabolic as is commonly assumed in plate theories, the slope of the curve is discontinuous at the interface between two plies, and depends upon the lamination scheme.

For a monolithic plate, the through-thickness variation of the transverse shear stress is nearly parabolic at the mid-span and almost uniform at the clamped edges.

For a [0/90°] laminate, the transverse normal stress near clamped and traction free edges exhibits severe gradients (or boundary layers) at interfacial points near the edges. Boundary layers do not exist for simply supported edges.

For a [0/90°] square laminate of aspect ratio 5, the maximum change in the plate thickness near the traction free edges equals 0.6qH/E where q is the amplitude of the load that varies sinusoidally over the plate surface, H is the thickness of the undeformed plate, and E equals Young’s modulus in the fiber direction.

For the [0 PVDF/90° PVDF] square laminate of aspect ratio 5 and a mechanical load applied on the top surface, the through-the-thickness distribution of the electric potential in each layer is parabolic and the magnitude depends upon the boundary conditions. For an electric potential applied to the top and the bottom surfaces, the change in the plate thickness depends upon the boundary conditions, and the induced electric displacement in the plate thickness direction and the transverse shear stress exhibit boundary layers near an edge of the plate except when it is simply supported and electrically grounded.

For a homogeneous square plate of aspect ratio 5 with two opposite edges simply supported and of the other two edges one clamped and the other traction free with a temperature field applied on the top surface, the slopes at the clamped edge of the deflection curve predicted by the first-order shear deformation theory (FSDT) and the 3-dimensional linear thermo-elasticity theory are nonzero; however, the two theories give different values of the slope. Due to the presence of boundary layers in the distributions of the transverse shear stress near a clamped and a traction free edge, the differences between solutions from the FSDT and the 3-D elasticity theories are large near these edges. Boundary layers are not predicted by the FSDT.

References:

- S. Vel and R. C. Batra, Analytical Solutions for Rectangular Thick Laminated Plates Subjected to Arbitrary Boundary Conditions, AIAA J., 37, 1464-1473, 1999.
- S. S. Vel and R. C. Batra, The Generalized Plane Strain Deformations of Thick Anisotropic Composite Laminated Plates, Int. J. Solids Structures, 37, 715-733, 2000.
- S. S. Vel and R. C. Batra, Closure to The Generalized Plane Strain Deformations of thick Anisotropic Composite Laminated Plates, Int. J. Solids Structures, 38, 483-489, 2000.
- S. S. Vel and R. C. Batra, Three-dimensional Analytical Solutions for Hybrid Multilayered Piezoelectric Plates, J. Appl. Mechs., 67, 558-567, 2000.
- S. S. Vel and R. C. Batra, Generalized Plane Strain Thermoelastic Deformation of Laminated Anisotropic Thick Plates, Int. J. Solids Structures, 38, 1395-1414, 2001.
- S. S. Vel, R. Mewer and R. C. Batra, Analytical Solution for the Cylindrical Bending Vibration of Piezoelectric Composite Plates, Int. J. Solids & Structures, 41, 1625-1643, 2004.

### Romesh C. Batra, PhD

**University Distinguished Professor, Clifton C. Garvin Professor
Fellow - ASME, ASEE, AAM, SES, USACM **

Department of Biomedical Engineering and Mechanics (0219)

Norris Hall, Room 333-E, Virginia Tech

495 Old Turner Street

Blacksburg, VA 24061

Research Activities in the Computational Mechanics Lab (poster)