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

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1. Introduction In recent years (Bassiouny S. et al. 2020), functionally graded composite materials and nanotechnologies are being more and more combined in order to develop hi-tech devices for applications in emerging engineering fields where sensors, actuators, biomaterials and multifunctional materials are increasingly used. As we all know, compared to macrostructures, ultrasmall structures exhibit a size-dependent behavior, due the nano-size dimensions and they are also influenced by surface energy effects, due the large surface area to bulk volume ratio which results in the non-negligible energy associated to atoms near the free surface. However, classical continuum mechanics models are unable to capture any of these effects above, and they also use the properties of the bulk as the material overall properties. Over the years, several non-classical models have been proposed in order to capture the size dependent behavior of small structure such as the Strain Gradient Mindlin’s Elasticity Theory (1968), the Strain-Driven Eringen’s Nonlocal Model (1972, 1983), the Stress-Driven Romano and Barretta’s Model (2016) and other coupled theories, including the Nonlocal Strain Gradient Theory developed by Lim (2015), as well as the Two-Phase Local/Nonlocal Stress and Strain gradient Model proposed by Barretta et al. (2019). As widely argued in Barretta et al. (2017), the Eringen’s model may lead to mathematically inconsistencies due to the incompatibility between the requirements of equilibrium and the higher order constitutive boundary conditions while the stress-driven Romano& Barretta’s model is a consistent approach to analyzed the mechanical behavior of structures at nanoscale (Penna et al. 2021). On the other hand, to take into account surface energy effects, Gurtin and Murdoch (1975,1978) proposed a surface elasticity theory in which the surface layer is considered as a zero-thickness membrane perfectly adhered to the bulk continuum with different material properties and constitutive laws than the bulk. In the present study, the surface stress-driven model (Penna 2023), in which the stress-driven model and the surface elasticity theory are coupled, is enriched for the bending analysis of Bernoulli-Euler nanobeams with loading discontinuities. Following the procedure proposed by Caporale et al. (2020), continuity boundary conditions are needed at nanobeam internal point where loading discontinuity occurs. The work is structured as follows. The problem formulation is summarized in Section 2. In Section 3 is reported the constitutive law and the nonlocal governing equations of the elastostatic bending problem in the presence of discontinuities. Results of the parametric analysis are discussed in Section 4. Some closing remarks are given in Section 5. 2. Problem formulation Figure 1 shows a functionally graded (FG) nanobeam with length L and a rectangular cross-section Σ ���ℎ� made of a bulk volume ( B ), composed by of a mixture of metal ( m ) and ceramic ( c ), and a thin surface layer ( S ) perfectly adhered to the bulk continuum. According to the power law distributions proposed in Shahab (et al. (2017), the bulk elastic modulus of elasticity, � � � � � , the surface modulus of elasticity, � � � � � and the residual surface stress, � � � � � , continuously vary along transverse direction, z, as � � �� � �� � � � �� 1 2 � ℎ � � (1) � � �� �� �� � � � �� �� 1 2 � ℎ � � (2) � � �� �� �� �� � �� �� 1 2 � ℎ � � (3) where denotes the gradient index of the FG material � � 0 � . The Poisson’s ratio is here assumed to be constant ( � � � � ).