PSI - Issue 47

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

� � � � � � � � � � � � � � � � �

(8c)

(8d) being � , � , � , � , � ��� and � ��� the bending moments, the shear stress and the transverse distributed loads associated to the first and second part of the FG nanobeam, respectively, and where C and F represent a concentrated

couple and a concentrated force applied at the abscissa d . As introduced in (Penna 2023), the quantity � is defined as � � � � ��

(9)

3. Constitutive law In this section, the novel approach proposed by Penna 2023, based on the coupling of the stress-driven model ( Romano & Barretta, 2017 ) and of the surface elasticity theory [ Gurtin & Murdoch, 1975 , 1978 ] and used for the bending analysis of FG nanobeams under constant distributed transverse load, is extended to analyze their response in the case of internal load discontinuities. 3.1 Stress-Driven Model incorporating surface energy effects (SSDM) The bending curvature at a point of FG nanobeam at the abscissa x in absence of concentrated loads is defined as the following integral convolution according to Penna (2023) � � � � � � , � � �� � � � � �∗ � � � (10) where M is the bending moment, � � represents an averaging kernel depending on the small-scale parameter L c , and �∗ is the equivalent bending stiffness as �∗ ��� � �� � � � � � � � �� (11) being � 2 ℎ �� ℎ � � � 3 4 �� �� � �� � � 2 1 � �� � �� � (12) and � � � � �� (13) In presence of a loading discontinuity and/or concentrated loads (force and couple) at an interior point of abscissa d , Eq.10 can be rewritten as � � � � � � � � � , � ��� � � � � � �∗ � � � 0 � � (14a)