PSI - Issue 47

794 6

Rosa Penna et al. / Procedia Structural Integrity 47 (2023) 789–799 Author name / Structural Integrity Procedia 00 (2019) 000–000

� � � � � � � � � , � ��� � � � � � �∗ � � �

� � (14b)

where M 1 and M 2 are the bending moments at the left and right of abscissa d , respectively. As it is well-known, by choosing a special function kernel � � equal to � � � , � �� 2 1 � exp �� | | � � the convolutions in Eq.14 are equivalent to following second-order differential equations according to Penna (2023) � 1 � �� � � � � �� � � � �∗ 0 � � (16a) � 1 � �� � � � � �� � � � �∗ � � (16b) if and only if the conventional constitutive boundary conditions (CBCs) of the stress-driven nonlocal theory ( Romano & Barretta, 2017 ) and constitutive continuity boundary conditions (CCBCs) ( Caporale et al., 2020 ) are satisfied � � 0 � � 1 � � � 0 �� 0 (17a) � � � � 1 � � � �� 2 � � , � � � (17b) � � � � 1 � � � ��� 2 � � , � � � (17c) � � � � 1 � � � �� 0 (17d) where � , � � � � � � � � � , � ��� � � � � � �∗ � � � (18a) � , � � � � � � � � � , � ��� � � � � � �∗ � � � (18b) By manipulating Eq.16, the expression of the stress-driven nonlocal resultant moments incorporating surface effects can be obtained as � �� �∗ � � � �∗ �� � � � � 0 � � (19a) � �� �∗ � � � �∗ �� � � � � � � (19b) Finally, by substituting Eq. 19 into Eq. 6, the governing equations incorporating surface energy effect in terms of transverse displacement for the first and the second part of an inflected FG nanobeam are obtained as (15)