PSI - Issue 14

Rahul Saini et al. / Procedia Structural Integrity 14 (2019) 362–374 R. Saini, S. Saini, R. Lal, and I. V Singh / Struct ral Integrity Procedia 00 (2018) 000–000

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�� � � , � � � � , �� � � , � � � � , �� � �� � �� � 0 . Here, the consideration of non-linear terms in strain-displacement relations doesn’t mean that non-linear vibrations are being considered. The equation of motion has been obtained using Hamilton’s principle, given by � � � ��� � � � � � � � ��� � 2 � � � � � � ��� � ��� � � � ��� � � �� 1 � � � � � � � ��� � ��� � � � ��� � � �� 1 � � � ℎ � � � 0, �10� together with the conditions at the periphery: � � Clamped �C� ∶ � � ��� � 0, � � ��� � 0 , � � Simply supported �S� ∶ � � ��� � 0 , �� � � ��� � � � � � � ��� � 0, where, � ��� , ��� � � � � � ��� , ��� � , ��� ���� � ��� , ��� � � � � ��� , ��� � ��� ���� . Assuming � � � � ��� (for harmonic solution), the non-dimensional form of equation (10) and periphery conditions lead to, where, � � � � Ω � , � � 1 � � 1 ∗ � ��� � ��� � � � ��� � � � � � ∗ � , � � 1 � � 1 ∗ � ��� � ��� � � � ��� � � � � � ∗ � , � � 2 � � � ��� ∗ �, � � � � ��� ∗ , � � � � , Ω � � � � ℎ ∗ , � � ℎ 1 � � , � ��� ���� , value 0 is with respect to the ceramic constituent and ω is the radian frequency. Equations (7) and (11) are the fourth-order differential equations with variable coefficients whose exact solutions are not possible. Then approximate solutions with appropriate boundary conditions have been obtained by using GDQR. � � � ��� � � � 0, �11� � � ��� � � � ��� � 0 ; � � ��� � � � � � � �∗ � ��� � 0 , �12��1��