## 1 Introduction  
Components of Nano-Electro-Mechanical-Systems (NEMS), such as sensors and actuators, are usually modeled as nano-rods and nano-beams. It is well known that methods of local continuum mechanics cannot be adopted for such elements. Molecular Dynamics (MD) simulations are time-consuming and micro/nano-scaled experiments are usually difficult to implement. Hence, nonlocal continuum models have been developed for predicting the size-dependent mechanical behavior of nano-structures. In the framework of nonlocal elasticity, [PERSON]'s strain-driven differential model [1] has been widely adopted in the literature (see, e.g., [2; 3; 4; 5; 6]). It is worth noting that recent papers on the strain-driven nonlocal model [7; 8; 9; 10] prove that, if the bending field is expressed as convolution of elastic curvature with an averaging kernel assuming an exponential expression, a solution of this problem exists only if the bending field satisfies constitutive boundary conditions. Accordingly, it is shown in [11] that the nonlocal elastostatic problem is ill-posed in all cases of applicative interest, as acknowledged in the literature (see, e.g., [12; 13; 14; 15; 16; 17; 18]). A modified version of the Eringen integral model is proposed in [19] and has been recently applied to inflected nanobeams in [20]. The gradient elasticity theory [21] assumes that a material at the nano-scale is modeled via gradient terms. Many works investigate the small-scale effects on the static and dynamic behaviors of rods, beams and plates and the effect of stiffness enhancement has been often reported in these strain gradient models (see, e.g., [22; 23; 24; 25; 26]). Recently, the [PERSON]'s integral law [1] has been combined with the strain gradient elasticity in [27] to formulate a higher-order nonlocal theory, thus collecting nonlocal theory and strain gradient theory into a single model. Using such a model, many contributions have been provided to model the size-dependent behavior of nano-rods and beams (see, e.g., [28; 29; 30; 31; 32; 33; 34; 35]) and plates [36; 37]. The procedure consists in considering the integral nonlocal gradient method for structural problems defined on bounded domains equivalent to a differential law of higher-order than the one of the classical local problem. Therefore, additional non-classical suitable boundary conditions must be added to solve the nonlocal strain gradient elastostatic problem. To solve the problem, different choices have been followed in the literature. Two usual choices consist in imposing higher-order boundary conditions pertaining to the strain gradient theory, of kinematic type [38] or static type [39]. It is worth noting that the structural behavior is greatly influenced by such choices. The nonlocal strain gradient theory with higher-order boundary conditions has been recently adopted in [40] to study nano-rods in tension. The closed-form solutions for predicting the axial displacement and the variation of the Young's modulus have been derived for four different nano-rods differing by the choice of the higher-order boundary conditions. In particular, for each nano-rod, the nonlocal parameters have been set to match the variation of the Young's modulus obtained by the MD simulations. The choice of the higher-order boundary conditions is disputed in the literature and is considered an open question (see, e.g., [41]). In the context of nano-beams subjected to flexure, a recent contribution [42] provides a definite solution to this issue. In fact, the non-classical boundary conditions to be imposed to solve the elastostatic problem of nonlocal strain gradient inflected nano-beams are given by constitutive boundary conditions (CBC) that naturally follow from the nonlocal strain gradient integral model. The consistent nonlocal strain gradient strategy has been successfully applied to free vibrations of nano-beams in [43]. In the present paper, the structural behavior of nano-rods in tension is formulated in the framework of the modified nonlocal strain gradient (NSG) theory.