PSI - Issue 14

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

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��� � � � � , � � � � � � � � � � � � , ��� � � � � , � � � � � � � 1 � � � � � , ��� � ��� � ��� � � ,

⎪⎬ ⎭ ⎪⎫

���

� , � � � , � �1 � � � , � � �� , � ∆ � � �1 � � , ∆ � � � � � � � , where, � being the initial temperature and subscript is used for the deformations due to thermal environment. Introducing the non-dimensional variables � � � / , � / and � / , the thermo-elastic equilibrium equation of the plate has been obtained using Hamilton’s principle, given by � � � ��� � � � � �, ��� together with the conditions at the periphery: � � Clamped �C� ∶ � � � ��� � � � � ��� � � , � � Simply supported �S� ∶ � � � ��� � � � � � � � �∗ � � ∗� � ��� � �, ⎪⎬ ⎪ ⎭ ⎫ ���, ��� where , � � 1 � � � � � ∗ �, � � � 1 � � � � � ∗ �, � � 2 � , � � �∗ � � ∗ , �∗ � � ∗ ∗ � � � ∗ , ∗� � � ∗ , � � � � . � . � , �� �/� ��/� , ∗ � � ℎ � 12�1 � � � , � value � is with respect to the ceramic constituent � and � � , � � � � � � , ��1, � �/� ��/� are the flexural rigidities of the plate. 2.2. Vibration analysis For axisymmetric motion of the plate, the displacement components are given by � , , � � � , � , , � � �, � , , � � � , �. To analyse the effect of thermal environment on the vibration characteristics of the plate, the total displacement components of point � , , � becomes � , , � � � � , � , � , , � � � � , � and � , , � � � � , � , respectively. Using the strain-displacement relations (Malekzadeh et al., 2011) �� � � 1 2 � � � � � � � , �� � � � 2 � , �� � �� � �� � � , the stress-displacement relations for small amplitude vibrations around the thermal equilibrium position are given by