Finite element method for stress-driven nonlocal beams
Introduction
Small-scale structures have attracted lots of attention due to impressive mechanical, chemical and physical properties [1], [2], [3], [4]. Since atomistic models [5], [6] are computationally expensive for nanostructures of engineering interest, continuum models provide effective tools in the study of small-scale structures. It is well-known that the classical (local) elasticity theory leads to inaccurate results due to the lack of length scale parameters. This fact determined the development of several nonlocal continuum theories such as nonlocal elasticity [7], [8], [9], [10], [11], [12], [13], strain gradient elasticity [14], [15], [16], [17], [18], [19], classical and modified couple stress elasticity [20], [21], [22], [23], stress-driven nonlocal elasticity [24], [25], [26], [27], [28], [29].
Differential elasticity has been widely applied, as compared to the nonlocal integral elasticity due to its simplicity, for the analysis of nanobeams [30], [31], nanoplates [32], [33] and nanoshells [34], [35]. Recent applications of scale-dependent models to nanostructures with multiphysics coupling can be found in [36], [37].
Among methodologies for solving governing differential equations arising in structural analysis of nonlocal elastic structures, Navier’s method [38], [39] and Differential Quadrature Method [40], [41] have been used. It is known that Finite Element Method (FEM) can successfully handle complex geometries, material properties, boundary and loading conditions in contrast to other strategies.
Considerable attention is given in literature on the development of nonlocal finite element formulations. Variational principles in the framework of nonlocal integral elasticity have been adopted in [42], [43]. A two-noded, with six degrees of freedom, nonlocal finite-element is considered in [44] to examine free vibration of FG nanobeams. A five noded, with ten degrees of freedom, nonlocal finite-element model is provided in [45] to examine the free vibration and buckling behaviour of FG nanobeams on the basis of first-order shear deformation theory. A finite element approach based on the third-order shear deformation theory of Reddy has been developed in [46] to analyse the FG plates. Nonlocal nonlinear finite element analysis of laminated composite plates is presented in [47]. Nonlocal free vibration analysis of axial rods embedded in elastic medium is performed in [48] applying the finite element method. Nonlocal damage is modelled by the scaled boundary finite element method in [49]. Scale-effects in orthotropic composite assemblies as micropolar continua is investigated in [50] adopting a finite element nonlocal formulation. A finite element analysis of thin laminated nanoplates is performed in [51]. Nonlocal vibration of nano-hetero-structures in thermal and magnetic fields is investigated in [52] by means of a nonlinear finite element method. The response of a nonlocal beam model when viscoelastic long-range interactions are included, modelled by Caputo’s fractional derivatives, is considered in [53].
The present paper aims at providing a finite element formulation of nonlocal beams based on the stress-driven integral model (SD) which provides well-posed structural problems unlike the strain-driven theory, see e.g. [54], [55], [56], [57]. Long-range interactions have been introduced into the finite element formulation on the basis of the nonlocal elasticity law recently proposed in [25] where the stress-driven model is conceived for structural problems involving discontinuity of bending interaction and elastic bending stiffness fields.
The nonlocal finite element method using the stress-driven approach, formulated in differential terms, will be denoted by SD-FEM. A two-noded element is defined with two degrees of freedom at each node. Nonlocal shape functions are obtained in terms of constitutive boundary conditions (CBCs) and constitutive continuity conditions (CCCs). Also, a procedure to get the nonlocal stiffness matrix and the equivalent nodal forces of a finite element nanobeam is provided.
The global nonlocal stiffness matrix is derived by the assemblage of nonlocal element stiffness matrices involving long-range interactions among the elements.
The benefits of the proposed approach relies on the fact that the SD-FEM provides the exact stress-driven solution using only one two-noded element. The same solution is then obtained by considering any finite element subdivision of the nanobeam. As the small size feature, such structures often exhibit stiffening nature of microstructure-dependent size effects [58], [59], [60]. Thus, the SD-FEM provides a stiffer structural response with respect to the classical (local) behaviour independently of kinematic boundary conditions and applied loads.
In particular, in Section 2, the SD model is formulated in the integral form and, following the approach provided in [25], the equivalent differential form, with constitutive boundary conditions and constitutive interface conditions, is provided. In Section 3 the nonlocal shape functions of the FE nonlocal problem associated with the SD model are provided for the first time and the element-beam stiffness matrix and generalized force vector are derived. Finally, numerical results and discussions are outlined in Section 4.
Section snippets
Stress-driven nonlocal integral model
We examine a straight Bernoulli–Euler nanobeam of length as shown in Fig. 1. The -coordinate is taken along the beam axis, the -coordinate is taken along the thickness and the -coordinate is taken along the width of the beam.
Applied loads and geometry are such that the displacements along the coordinates are independent of the -axis and given by where is the transverse displacement of the cross-section and the symbol denotes the
Nonlocal finite element
The nonlocal SD elastostatic problem can be solved by using the nonlocal finite element approach (SD-FEM) provided in this Section.
The interpolated displacement field pertaining to the two-noded th element of the partition of the domain , see Fig. 2, occupied by the nanobeam is given by adding to the solution of the homogeneous equation associated with Eqs. (11) the complementary integral for the given distributed load with .
The beam element
Case-studies
In this section some numerical examples are presented in order to investigate the results predicted by the proposed nonlocal SD-FEM. The boundary condition of the nanobeam is specified by letter symbols: S, C, F denote simply-supported, clamped and free boundary conditions respectively.
Comparisons between the solution of the SD-FEM and the corresponding closed form solution of nonlocal structural problems are performed to validate the methodology provided in this paper. It is assumed that the
Closing remarks
The outcomes of the present research may be summarized as follows.
- 1.
Small-scale structural systems composed of straight slender beams have been investigated by the well-posed nonlocal stress-driven integral theory of elasticity.
- 2.
A novel computational numerical technique has been conceived in order to obtain technically exact size-dependent elastic responses of nanobeams. A two-noded nonlocal finite element has been formulated by considering long-range interactions involving the whole structure,
Acknowledgement
Financial supports from MIUR in the framework of the Project PRIN 2017—code 2017J4EAYB Multiscale Innovative Materials and Structures (MIMS) - University of Naples Federico II Research Unit and from the research program ReLUIS 2021 are gratefully acknowledged.
References (66)
- et al.
Graphene as biomedical sensing element: state of art review and potential engineering applications
Composites B
(2018) Linear theory of nonlocal elasticity and dispersion of plane waves
Int J Eng Sci
(1972)- et al.
Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory
Comput Mater Sci
(2009) - et al.
A review on the mechanics of functionally graded nanoscale and microscale structures
Int J Eng Sci
(2019) - et al.
Isogeometric analysis of size-dependent isotropic and sandwich functionally graded microplates based on modified strain gradient elasticity theory
Compos Struct
(2018) - et al.
Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: applications to sandwich beams and auxetics
Int J Eng Sci
(2018) - et al.
On the bifurcation buckling and vibration of porous nanobeams
Compos Struct
(2020) - et al.
A microstructure-dependent Timoshenko beam model based on a modified couple stress theory
J Mech Phys Solids
(2008) - et al.
Nonlocal elasticity in nanobeams: the stress-driven integral model
Int J Eng Sci
(2017) - et al.
Exact solutions for bending of Timoshenko curved nanobeams made of functionally graded materials based on stress-driven nonlocal integral model
Compos Struct
(2020)