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## Free Edge Approximation for the Angle-Ply Laminate

A tool which extends the Pipes-Pagano solution to include mechanical and thermal effects in the angle-ply laminate

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#### Abstract

A closed-form, analytic solution for prediction of the displacement, stress and strain fields within the m-layer, symmetric and balanced angle-ply composite laminate of finite-width and subjected to uniform axial extension was developed earlier [4]. In the present paper, the authors have extended that solution to treat thermal stresses and deformations induced by a uniform change in
laminate temperature. The results have revealed not only the complex fields within the laminate, but also inter-relationships between the lamina axial and shearing coefficients of thermal expansion and the effective laminate coefficients of thermal expansion. Further, the solution is shown to recover
laminated plate theory predictions for thermally induced fields at interior regions of the laminate, thereby confirming the boundary layer nature of the interlaminar phenomena for the thermoelastic case. Finally, the results exhibit the anticipated response in congruence with the mechanical solution of
reference [4] and the thermoelastic results satisfy the conditions of self-equilibration necessary for the finite-width laminate subjected to free thermal deformation. Integration of the stress σx over the laminate cross-section in the y-z plane is shown to converge to zero as the number of Fourier terms is
increased. Consistent with the solution for mechanical loading, an apparent singularity in the interlaminar shearing strain is seen to occur at the intersection of the free edge and planes between lamina of lamina of +Θ and -Θ orientation. The analytical results show extraordinary agreement with the finite-element predictions for the same boundary-value problem.

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#### References

1. Pipes, R. B., and Pagano, N. J., "Interlaminar Stresses in Composite Laminates under Uniform Axial Extensions," Journal of Composite Materials, Vol. 4, (1970), 538-548.
2. Pagano, N. J., and Pipes, R. B., "The Influence of Stacking Sequence on Laminate Strength," Journal of Composite Materials, Vol. 5, (1971), pp. 50-57.
3. Pagano, N. J., and Pipes, R. B., "Some Observations on the Interlaminar Strength of Composite Laminates," International Journal of Mechanical Sciences, Vol. 15, (1973), pp. 679-688.
4. Pipes, R. B., and Pagano, N. J., "Interlaminar Stresses in Composite Laminates - An Approximate Elasticity Solution," Journal of Applied Mechanics, Vol. 41, No. 3, (1974), pp. 668-672.
5. Mittelstedt, C. and Becker, W., “The Pipes-Pagano-problem revisited: Elastic fields in boundary layers of plane Laminated specimens under combined thermomechanical load,” Composite Structures, Vol. 80, (2007), pp. 373-395.
6. Mittelstedt, C. and Becker, W., “Free Edge Effects in Composite Laminates,” Applied Mechanics Reviews, Vol. 60, (2007), pp. 217-244.

#### Publications

Pipes, R. B.; Goodsell, J.; Ritchey, A.; Dustin, J. and Gosse, J. H. "Interlaminar Stresses in Composite Laminates: Thermoelastic Deformation." Submitted Composites Science and Technology

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