Finite element method for stress-driven nonlocal beams

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Abstract

The bending behaviour of systems of straight elastic beams at different scales is investigated by the well-posed stress-driven nonlocal continuum mechanics. An effective computational methodology, based on nonlocal two-noded finite elements, is developed in order to take accurately into account long-range interactions present in the whole structural domain. The idea consists in partitioning the beam in subdomains and in observing that the nonlocal stress-driven convolution integral, equipped with the Helmholtz averaging kernel, can be equivalently formulated by expressing nonlocal bending interaction fields in terms of zero-th and second-order derivatives of elastic curvature fields which have to fulfil appropriate non-classical constitutive boundary and interface conditions. Relevant mesh-dependent shape functions governing the FEM technique are analytically detected. Each element is characterized by shape functions whose number is equal to four times the number of elements of the considered mesh. A simple analytical strategy to obtain nonlocal stiffness matrices and equivalent nodal forces of a finite element is exposed. The global nonlocal stiffness matrix is got by assembling the nonlocal element stiffness matrices accounting for long-range interactions among the elements. The proposed numerical approach is examined by exactly solving exemplar nonlocal case-problems of current interest in nano-engineering. The presented nonlocal strategy extends previous contributions on the matter and offers designers a consistent computational tool.

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 L as shown in Fig. 1. The x-coordinate is taken along the beam axis, the y-coordinate is taken along the thickness and the z-coordinate is taken along the width of the beam.

Applied loads and geometry are such that the displacements sx,sy,sz along the coordinates x,y,z are independent of the z-axis and given by sxx,y,z=xvxy,syx,y,z=vx,szx,y,z=0where v is the transverse displacement of the cross-section and the symbol xn(x) denotes the n

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 vihx pertaining to the two-noded ith element Ωi of the partition of the domain Ω, see Fig. 2, occupied by the nanobeam is given by adding to the solution v̄ix of the homogeneous equation associated with Eqs. (11) the complementary integral vˆix for the given distributed load qyi(x) vihx=v̄ix+vˆixwith i1,,N.

The beam element Ωi=xi

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.

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