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Interface crack between isotropic Kirchhoff plates

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Abstract

In this work Kirchhoff plate theory is used to calculate the energy release rate function in delaminated isotropic plates. The approximation is based on the consideration of the equilibrium equations and the displacement continuity between the interface plane of a double-plate model. It is shown that the interface shear stresses are governed by a fourth order partial differential equation system. As an example, a simply supported delaminated plate subjected to a point force is analyzed adopting Lévy plate formulation and the mode-II and mode-III energy release rate distributions along the crack front were calculated by the J-integral. To confirm the analytical results the 3D finite element model of the delaminated plate was created, the energy release rates were calculated by the virtual crack-closure technique and the J-integral. The results indicate a good agreement between analysis and numerical computation.

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Acknowledgements

This work was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and the National Science and Research Fund (OTKA) under Grant No. T34040 (69096). This work is connected to the scientific program of the “Development of quality-oriented and harmonized R+D+I strategy and functional model at BME” project. This project is supported by the New Hungary Development Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR-2010-0002).

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  1. Department of Applied Mechanics, Budapest University of Technology and Economics, Műegyetem rkp. 5, 1111, Budapest, Hungary

    András Szekrényes

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Correspondence to András Szekrényes.

Appendix

Appendix

To derive the governing equations for the in-plane forces N x and N y we express first the moments from Eqs. (3), (9) and (10):

$$ \everymath{\displaystyle}\begin{array}{@{}l} M_x = - \frac{{2I_1 }}{ {t^2 }}N_x + \frac{{2I_1 E_1 k_{sh} }}{ t} \\[13pt] \hphantom{M_x =}{}\times\biggl( {\frac{{\partial^2 N_x }}{ {\partial x^2 }} - (1 + \nu )\frac{{\partial^2 N_{xy} }}{ {\partial x\partial y}} + \nu \frac{{\partial^2 N_y }}{ {\partial y^2 }}} \biggr) \\[13pt] M_y = - \frac{{2I_1 }}{ {t^2 }}N_y + \frac{{2I_1 E_1 k_{sh} }}{ t} \\[13pt] \hphantom{M_y =}{}\times \biggl( {\frac{{\partial^2 N_y }}{ {\partial y^2 }} - (1 + \nu )\frac{{\partial^2 N_{xy} }}{ {\partial x\partial y}} + \nu \frac{{\partial^2 N_x }}{ {\partial x^2 }}} \biggr) \end{array} $$
(73)
(74)

Incorporating the equilibrium equations (Eqs. (1), (2)) and the compatibility equation (Eq. (14)) one can obtain the following coupled PDE system:

(75)
(76)

where the constants are:

$$ \everymath{\displaystyle}\begin{array}{@{}l} A_1 = \frac{1}{ 9}\frac{{t^3 (4 + \nu - \nu^2 )}}{ {1 - \nu^2 }} \\[13pt] A_2 = \frac{1}{ 9}\frac{{t^3 (5 - \nu^2 )}}{ {1 - \nu^2 }} \\[13pt] A_3 = - \frac{1}{ {12}} \frac{{t(2 + \nu )}}{ {1 + \nu }}, \qquad A_4 = \frac{1}{ 9}\frac{{t^3 }}{ {1 + \nu }} \\[13pt] A_5 = - \frac{1}{ {12}}\frac{t}{ {1 + \nu }}, \qquad A_6 = \frac{1}{ 9} \frac{{t^3 (3 + \nu )}}{ {1 - \nu^2 }} \\[13pt] A_7 = \frac{1}{9}\frac{{t^3 (3 + \nu^2 )}}{ {1 - \nu^2 }}, \qquad A_8 = A_5 \\[13pt] A_9 = - \nu A_4 ,\qquad A_{10} = - \nu A_8 \end{array} $$
(77)

The above PDE system can be solved by using the Fourier series solutions for N x and N y .

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Szekrényes, A. Interface crack between isotropic Kirchhoff plates. Meccanica 48, 507–526 (2013). https://doi.org/10.1007/s11012-012-9611-9

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