Finite element solutions for nonhomogeneous nonlocal elastic problems

Mechanics Research Communications 36 (2009) 755–761

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Finite element solutions for nonhomogeneous nonlocal elastic problems Aurora A. Pisano *, Alba Sofi, Paolo Fuschi Dipartimento DASTEC, Università Mediterranea di Reggio Calabria, via Melissari, 89124 Reggio Calabria, Italy

a r t i c l e

i n f o

Article history: Received 21 January 2009 Received in revised form 13 May 2009 Available online 24 June 2009 Dedicated to Prof. Castrenze Polizzotto on the occasion of his 85th birthday. Keywords: Nonhomogeneous nonlocal elasticity Nonlocal finite elements 2D mechanical problems

a b s t r a c t A finite element procedure for analysing nonhomogeneous nonlocal elastic 2D problems is presented and discussed. The procedure grounds on a variationally consistent approach known, in the relevant literature, as Nonlocal Finite Element Method. The latter is recast making use of a recently theorized phenomenological strain-difference-based nonhomogeneous nonlocal elastic model. The peculiarities of the numerical procedure together with the pertinent nonlocal operators are expounded and discussed. Two simple numerical 2D examples close the paper. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The key idea of the nonlocal elastic approaches resides on the concept that within a nonlocal elastic medium the particles influence one another not simply by contact forces and heat diffusion (as it occurs in local materials), but also by long range cohesive forces which imply some long distance energy interchanges (see e.g. Kröner, 1967; Edelen, 1976). Such assumption allows one to overcome within the context of continuum theories the inability of the local theory to describe physical phenomena affected by events arising at microstructure or, even, at atomic level. The singular stress field predicted at a sharp crack-tip in a continuum fracture mechanics problem is a typical example. A very extensive list of contributions is traceable in the relevant literature, see e.g. Eringen and Kim (1974) or Zhou and Wang (2005) just to quote one of the early contribution and one of the more recent one. An effective nonlocal continuum approach for solving problems involving (spontaneous) formation of discontinuities, so including fracture mechanics problems, is the one known as peridynamic model proposed by Silling (2000); see also the recent contributions of Silling et al. (2003) and Emmrich and Weckner (2007). The application of nonlocal elastic continuum approaches to nanomaterials, where size effects often become prominent and have to be accounted for, is another current example. The list of contributions is also in this case quite extensive, starting with the work of Peddieson et al. (2003) who developed a nonlocal Euler–Bernulli beam model, among others, some of the more recent papers are those of Zhang et al. (2005); Ece and Aydogdu (2007); Heireche et al. (2008) addressing problems of vibration, buckling and wave propagation in carbon nanotubes on the base of a nonlocal Timoshenko beam theory. Variational principles for multi-walled carbon nanotubes have recently been presented by Adali (2008) who assumed a continuum modelling which takes into account small scale effects via the nonlocal theory of elasticity. Till the very recent and remarkable contribution of Aifantis (2009), related to gradient elasticity, but showing that continuum elasticity can indeed describe a variety of problems at micro/nano regime if long range or nonlocal material point interactions and surface effects are taken into account. The list of the above quoted references is far to be exhaustive; to this concern reference can be made to the

* Corresponding author. Tel.: +39 965 3223141; fax: +39 965 21158. E-mail address: [email protected] (A.A. Pisano). 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.06.003

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remarkable contributions of Kunin (1982) and Rogula (1982). A recent study on the relations between nonlocal theories and atomistic models can be found in Chen et al. (2004). The nonlocal elastic material here considered is, by hypothesis, able to transmit information to neighbouring points within a certain distance beyond which the diffusion processes vanish. This distance, herein named influence distance, is strictly related to an internal length material scale; the latter enters the constitutive material model in different ways, i.e. by considering body couples as in polar elasticity; or by gradient operators or, moreover, by integral operators (see e.g. Rogula, 1982; Eringen, 2001; Aifantis, 2003; Polizzotto, 2003; Lazar et al., 2006). In the following attention will be focused on nonlocal integral elasticity (see e.g. Eringen and Kim, 1974, Eringen, 2001). The latter is characterised by a stress–strain constitutive law of convolutive-type with a kernel expressed by a positive, symmetric, scalar, attenuation function aimed to capture the diffusion processes of the nonlocality effects. In the context of nonlocal integral elasticity a variationally consistent formulation of the finite element method has been presented in Polizzotto (2001). The method, therein named NL-FEM—where NL stands for ‘‘NonLocal”—, is based on a nonlocal total potential energy principle conceived as an extension of the analogous principle of classical (local) elasticity theory. An attractive peculiarity of the NL-FEM lies on the circumstance that each element, besides its standard (local) element stiffness matrix, is endowed with a set of nonlocal stiffness element’s matrices devised to interpret the postulated nonlocal material behaviour. The global (nonlocal) stiffness matrix resulting from the contributions of all the local and nonlocal elements’ matrices turns out to be symmetric, positive semi-definite and banded with a bandwidth larger than in the standard FEM. The NL-FEM has been very recently implemented by the authors for solving simple 2D nonlocal elastic problems (Sofi et al., 2008). The results obtained, although confirming the validity of the method in a 2D context, are often distressed by some numerical oscillations or incoherencies related to the assumed Eringen-type model and arising at sharp material discontinuities like domain boundaries. They are due, in practice, to the circumstance that the support of the attenuation function exceeds the integration domain, the weighting process resulting so altered. A known consequence is, for example, the one detectable in an Eringen-type nonlocal bar under uniform tension (see e.g. Pisano and Fuschi, 2003) in which the strain profile exhibits rising queues close to the bar ends. In a recent study, Polizzotto et al. (2006), the above drawbacks have been overcome by proposing a new phenomenological nonlocal elastic model. This model, based on a firm thermodynamic formulation, is a two components local/nonlocal model for (macroscopically) nonhomogeneous linear elastic materials in isothermal conditions. In particular, the stress is expressed as the sum of two contributions: one is the standard local stress and the other is of nonlocal nature. A distinctive feature of the formulation concerns the nonlocal part of the stress which is expressed in terms of an averaged strain difference field. Such an assumption, besides its proven variational consistency, seems able to assure numerical stability avoiding also the above cited incoherencies on the strain distribution. The nonlocal constitutive model behaves as a local one in the presence of uniform strain field both in terms of stress and energy while preserves the symmetry of the nonlocal operators. It is worth noting that the solution uniqueness of the pertinent continuum boundary-value problem has already been proven and the related total potential energy principle for NL-FEM discretizations is available. Nevertheless, the potentialities of the strain-difference-based nonlocal model have been so far explored within a simplified context of a 1D bar structure postponing further investigations oriented to validate the model in a 2D context. The main goal of the present study is indeed to rephrase the NL-FEM for dealing with nonhomogeneous nonlocal elastic problems in 2D making use of the quoted strain-difference-based constitutive model. The latter, which also in 2D appears able to ensure numerical stability, is utilised assuming: nonhomogeneous elastic moduli, constant internal length and an attenuation function depending on the Euclidean distance. The paper points out the novelties concerning the numerical implementation of the NL-FEM and tries to show its applicability to problems of engineering interest. The results obtained for two 2D academic examples do not exhibit macroscopic bugs or incoherencies but, of course, a validation either via experimental tests or analysis of more complex problems appears to be necessary. The plan of the paper is the following. After this introductory section, in Section 2 the constitutive assumptions are briefly summarised. Section 3 is devoted to the NL-FEM description also with the aid of a flow-chart type scheme. In Section 4, which closes the paper, two simple numerical examples are presented and discussed drawing also a few concluding remarks. Notation: A compact notation is used throughout, with bold-face letters for vectors and tensors. The ‘‘dot” and ‘‘colon” products between vectors and tensors   denote simple and double index contraction operations, respectively, for instance u  v ¼ ui v i , r : e ¼ rij eij , r  n ¼ rij nj , D : e ¼ Dijhk ehk . The symbol :¼ means equality by definition. Other symbols will be defined in the text where they appear for the first time.

2. Constitutive assumptions: a nonhomogeneous nonlocal elastic material Let a 3D Euclidean domain, V, be occupied by a nonlocal elastic linear nonhomogeneous material in its undeformed state. By hypothesis the considered material obeys to the following stress–strain constitutive relation (Polizzotto et al., 2006):

rðxÞ ¼ DðxÞ : eðxÞ  a

Z

Jðx; x0 Þ : ½eðx0 Þ  eðxÞdx0

8ðx; x0 Þ 2 V:

ð1Þ

V

Eq. (1) states that the stress response, rðxÞ, to a given strain field, eðxÞ, is the sum of two contributions. The former is the local contribution, governed by the standard symmetric and positive definite elastic moduli tensor (variable in space), DðxÞ, and

A.A. Pisano et al. / Mechanics Research Communications 36 (2009) 755–761

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related to the strain at point x; the latter is the nonlocal contribution, governed by the symmetric nonlocal tensor Jðx; x0 Þ and depending on the strain difference field ½eðx0 Þ  eðxÞ; a is a positive scalar coefficient. Moreover, the nonlocal tensor Jðx; x0 Þ is defined as:

Jðx; x0 Þ :¼ ½cðxÞDðxÞ þ cðx0 ÞDðx0 Þgðx; x0 Þ  qðx; x0 Þ;

ð2aÞ

where

Z

gðx; x0 Þdx0 ; Z gðx; zÞgðx0 ; zÞDzdz: qðx; x0 Þ :¼

cðxÞ :¼

ð2bÞ

V

ð2cÞ

V

In the above operators gðx; x0 Þ is a positive, symmetric, scalar attenuation function devoted to model the nonlocality effects diffusion processes. In practice gðx; x0 Þ assigns a ‘‘weight” to the (nonlocal) effects induced at the field point x by a phenomenon acting at the source point x0 ; it contains the internal length material scale, say l, and rapidly decreases with increasing (Euclidean) distance between points x and x0 in V. Plots explaining how the defined kernels operators are given in the above quoted paper and are here omitted for lack of space. By inspection of Eq. (1) it is clear that a uniform stress field is provided whenever the strain field is uniform, just like in the case of local elasticity and consistently with what can be deduced from some experimental findings on thin wires in tension executed by Fleck et al. (1994) dealing with strain gradient plasticity. Moreover, it is worth noting that the averaged strain difference, which vanishes identically for any uniform strain field, can be considered the nonlocal counterpart of the strain gradient. A notable alternative form of Eq. (1) is the following:

rðxÞ ¼ DðxÞ : eðxÞ þ a

Z

Sðx; x0 Þ : eðx0 Þdx0

8ðx; x0 Þ 2 V;

ð3Þ

V

in which the (singular) tensor Sðx; x0 Þ has the dimension of a stress per unit volume and it can be interpreted as nonlocal stiffness tensor. Eq. (3) explicitly shows the additional nonlocal stress contribution at x due to the strain at x0 . Sðx; x0 Þ is defined as:

Sðx; x0 Þ :¼

 1 2 c ðxÞDðxÞ þ c2 ðx0 ÞDðx0 Þ dðx0  xÞ  Jðx; x0 Þ 8ðx; x0 Þ 2 V 2

ð4Þ

with dðx0  xÞ being the Dirac delta function. Eq. (3) is derivable from Eq. (1) by simple substituting the expression of Jðx; x0 Þ obtainable from Eq. (4) and taking into account that the following relation holds true:

Z

Jðx; x0 Þdx0 ¼ c2 ðxÞDðxÞ 8x 2 V:

ð5Þ

V

Finally, by inspection of Eqs. (1) and (2a) or Eqs. (3) and (4) it can be observed that the material nonhomogeneity affects both the local and the nonlocal part of the stress distribution and this through the elastic moduli tensor D(x). 3. Nonlocal finite elements: variational basis; nonlocal operators; numerical procedure The continuum boundary-value problem for a body made of nonhomogeneous nonlocal elastic material as the one assumed in the previous section, with the further hypothesis of infinitesimal displacements, is governed by the standard equilibrium and compatibility equations, besides the stress–strain law (1). The pertinent total potential energy functional can be written as, (see e.g. Polizzotto, 2001; Polizzotto et al., 2006):

p½uðxÞ :¼

1 2

Z

ruðxÞ : DðxÞ : ruðxÞdx þ

V

: ruðx0 Þdx0 dx 

Z

bðxÞ  uðxÞdx 

Z Z

a 2 Z

V

V

ruðxÞ : Sðx; x0 Þ V

tðxÞ  uðxÞdx;

ð6Þ

St

where V is the body volume within which assigned body forces bðxÞ act; St is the portion of the boundary surface of V, say S, where boundary tractions tðxÞ are given; uðxÞ is the unknown displacement field required to satisfy the boundary conditions  ðxÞ on Su ¼ S  St . By substituting Eq. (4), Eq. (6) can be written as: uðxÞ ¼ u

p½uðxÞ :¼

1 2

Z

ruðxÞ : DðxÞ : ruðxÞdx þ

V

: ruðx0 Þdx0 dx 

Z V

bðxÞ  uðxÞdx 

Z

a 2 Z

ruðxÞ : c2 ðxÞDðxÞ : ruðxÞdx 

V

tðxÞ  uðxÞdx:

a 2

Z Z V

ruðxÞ : Jðx; x0 Þ

V

ð7Þ

St

The functional (7) can then be used for a consistent NL-FEM formulation, to this aim the standard FEM notation is used; precisely within the nth element, say of volume V n , the displacement and the strain fields are expressed by interpolating the element node displacements vector, say dn , as:

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uðxÞ ¼ N n ðxÞdn ;

eðxÞ ¼ Bn ðxÞdn ; 8x 2 V n ;

ð8Þ

where N n (x) and Bn (x) denote the matrices of the element shape functions and their derivatives, respectively. By substituting Eq. (8) in (7), a discretized form of the total (nonlocal) potential energy functional can be obtained, i.e.:

p½dn  ¼

" # Ne Ne Ne Ne X X 1X aX T loc T nonloc T nonloc T dn kn dn þ dn k n dn  dn knm dm  dn f n ; 2 n¼1 2 n¼1 m¼1 n¼1

ð9Þ

where the following positions have been set up: loc

Z

BTn ðxÞDðxÞBn ðxÞdx; Z nonloc :¼ BTn ðxÞc2 ðxÞDðxÞBn ðxÞdx; kn Vn Z Z nonloc BTn ðxÞJðx; x0 ÞBm ðx0 Þdx0 dx; knm :¼ Vn Vm Z Z N Tn ðxÞbðxÞdx þ N Tn ðxÞtðxÞdx: f n :¼ kn :¼

ð10aÞ

Vn

Vn

ð10bÞ ð10cÞ ð10dÞ

StðnÞ

loc

nonloc

nonloc

In the above positions kn is the standard element (local) stiffness matrix, kn and knm are the element nonlocal stiffness matrices and f n is the standard element equivalent nodal forces vector. nonloc nonloc and knm is witnessed by the nonlocal operators cðxÞ and Jðx; x0 Þ, as given by Eqs. (2a–c), The nonlocal nature of kn nonloc entering the expressions (10b, c), respectively. Moreover, looking at the structure of Eq. (10c), it is plain that knm represents the ‘‘nonlocality effects” of the mth element on the current nth one and is so called cross- stiffness nonlocal element matrix. nonloc It is worth noting that knm vanishes when the elements #n and #m are far from each other with respect to the influence distance, LR , which coincides with the maximum distance beyond which the value of the attenuation function, gðx; x0 Þ, and consequently the nonlocality effects are almost negligible. A second remark concerns the evaluation of the nonlocal operators cðxÞ and qðx; x0 Þ as well as the computation of the nonloc cross-stiffness matrices knm . By inspection of Eqs. (2b, c), it can be inferred that both cðxÞ and qðx; x0 Þ require integrations over the whole domain. The latter, in a standard FE treatment, can be carried out by a Gauss–Legendre quadrature involving nonloc all the FEs (and the related Gauss points) in the mesh. On the other hand, the evaluation of knm , given by Eq. (10c), implies cross integrations between elements which can be performed by standard quadrature considering just the Gauss points belonging to elements #n and #m. Nevertheless, such integrations might be computationally prohibitive even when dealing with simple structures. To reduce the computational efforts, the nonlocal operators cðxÞ and qðx; x0 Þ can be evaluated taking into account only the Gauss points falling within an influence region determined by the influence distance LR . In a similar nonloc way, for the generic element #n, the computation of the cross-stiffness matrices knm can be confined to the elements #m falling within the influence region. A third remark concerns the numerical integrations involved in Eqs. (10a–d). Different standard procedures, available in many commercial codes (see e.g. Mathematica5, 2003), have been used for sake of comparison but, for the run cases, it appears that a standard Gauss quadrature provides reliable approximations of the nonlocal operators and cross-stiffness matrices. An advantage of the adopted integration rule is also that it allows the implementation of the presented nonlocal FE procedure just by enriching standard (local elastic) FE codes with apposite subroutines. The main steps of the expounded strain-difference-based NL-FEM are given in Table 1 in a flow-chart style and this for a twofold reason: to give a concise summary of the computational scheme; to highlight the peculiarities of the proposed method with respect to the standard FEM. 4. Numerical examples and concluding remarks The expounded NL-FEM has been applied for analysing two simple nonhomogeneous nonlocal boundary-value problems under plane stress conditions. In both cases a piecewise constant Young modulus has been considered. Precisely, a bar-like structure under tension has been chosen in order to validate the implemented numerical procedure by comparison with the solution given, in a 1D context, by Polizzotto et al. (2006); a nonhomogeneous square plate under tension has then been analysed to show some of the potentialities of the proposed NL-FEM. The material behaviour is governed by the strain-difference-based nonlocal constitutive model (1); the attenuation func0 2 tion of the form gðx; x0 Þ ¼ k0 ejxx j=l (with k0 ¼ 1=ð2pl tÞ denoting the normalization factor, t the thickness of the structure and LR ¼ 6lÞ has been adopted; 8-node serendipity elements with 3  3 Gauss points per element have been used. The following material properties have finally been assumed: Young moduli E1 ¼ 0:4E0 and E2 ¼ E0 with E0 ¼ 2:1  106 da N=cm2 , Poisson ratio m ¼ 0:2, internal length l ¼ 0:1 cm and a ¼ 50 (the positiveness of the latter material parameter being required to avoid numerical instabilities, refer to the above quoted paper for details). Example 1. The nonhomogeneous bar-like structure shown in Fig. 1a with: length L ¼ 50 cm; wideness h ¼ 1 cm; thickness  x ¼ 0, and it is subjected to a t ¼ 0:1 cm has been considered. The bar is constrained at the cross-section x ¼ 0, where u  x ¼ 0:1 cm at the free end section x ¼ L. A uniform mesh of 800 FEs with 200 subdivisions uniform prescribed displacement u

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A.A. Pisano et al. / Mechanics Research Communications 36 (2009) 755–761 Table 1 Main steps of the strain-difference-based NL-FEM.

 input: geometry; boundary and loading conditions; material data; FE mesh (say of a number of elements equal to N e Þ  start element loop #1, say for n ¼ 1; . . . ; Ne r locate the elements neighbours at the current element #n, i.e. the ones falling within the pertinent influence region, say of a number equal to M e  start Gauss points loop on element #n, say on GPs(n)  at the current Gauss point GP(n), say at x, evaluate: r the matrix Bn loc

r the local element stiffness matrix kn (Eq. (10a)) r the nodal force element vector f n (Eq. (10d)) R r the nonlocal operator cðxÞ :¼ V gðx; x0 Þdx0 with x0 ranging over the GPs of the Me neighbouring elements nonloc

r the nonlocal element stiffness matrix kn  end Gauss points loop on element #n

(Eq. (10b))

loc nonloc ^ r assemble kn and kn in the global matrix K h i ^ :¼ PNe C T kloc C n þ aPNe C T knonloc C n  PNe C T knonloc C m K n n n¼1 n n n¼1 m¼1 n nm P e T r assemble f n in the global force vector F :¼ N n¼1 C n f n  start element loop #2, say for m ¼ 1; . . . ; Me  start Gauss points loop on element #n, say on GPs(n)  start Gauss points loop on element #m, say on GPs (m)  at the current Gauss points GP(n) and GP(m), say at x and x0 , evaluate: r the matrices Bn and Bm r the attenuation function gðx; x0 Þ R r the nonlocal operator qðx; x0 Þ ¼ V gðx; zÞgðx0 ; zÞDðzÞdz, with z ranging over the GPs of the M e neighbouring elements R 0 r the nonlocal operator cðx Þ ¼ V gðx0 ; xÞdx with x ranging over the GPs of the Me neighbouring elements r the nonlocal operator Jðx; x0 Þ :¼ ½cðxÞDðxÞ þ cðx0 ÞDðx0 Þgðx; x0 Þ  qðx; x0 Þ nonloc

r the nonlocal self- and cross-stiffness element matrices knm  end Gauss points loop on element #m  end Gauss points loop on element #n nonloc ^ in the global matrix K r assemble k

(Eq. (10c))

nm

 end element loop #2  end element loop #1  solve global equation system  output: displacements, stresses, strains, etc.

(a)

(b)

y

0.0032 0.0028

u x = 0.1cm

h

E2

L/2

L/2

εx (x,y)

0.0024

E1

x

0.002 0.0016

Local NL-FEM Polizzotto et al. (2006)

0.0012 0.0008 0

10

20

30

40

50

x[cm] Fig. 1. (a) Bar-like structure under tension with piecewise constant Young modulus and (b) strain distribution line), nonlocal (solid line) and bench mark (thin solid line) solutions.

ex versus x at y ¼ 0:472 cm: local (dashed

 ¼ 0:472 cm provided by the presented NLalong x and 4 along y has been adopted. In Fig. 1b the strain profile ex versus x at y FEM is plotted against the 1D solution assumed as bench mark solution. As clearly shown in the enlargement of Fig. 1b, the NL-FEM solution appears to be in very good agreement with the bench mark one thus testifying the accuracy and effectiveness of the implemented numerical procedure. Also with the present 2D NL-FEM formulation around the section x ¼ L=2, where the abrupt change of the elastic modulus occurs, the strain profile is smoother than the one obtainable with a local elastic analysis (dashed line in Fig. 1b). The sharpness of the slope of the strain curve within this middle zone of the nonhomogeneous bar depending on the selected values of l and a driving the ‘‘degree” of material nonlocality. Outside the above zone and also close to the bar end sections the strain profile is constant and coincident with the local solution. The present 2D NL-FEM solution is then confirming the capability of the strain-difference-based constitutive model to overcome the drawbacks inherent in an Eringen-type nonlocal model, i.e. the alteration of the weighting process of the nonlocal effects close to the domain boundaries. Finally, it is worth mentioning that assuming a constant elastic modulus, i.e. E1 ¼ E2 ¼ E0 for 0 6 x 6 L, the strain distribution recovered by applying the NL-FEM is uniform and exactly coincident with ex ¼ ux =L ¼ 0:002Þ whatever l and a are, as it has to be. the local elastic solution (

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A.A. Pisano et al. / Mechanics Research Communications 36 (2009) 755–761

Example 2. The nonhomogeneous square plate sketched in Fig. 2a with: a ¼ 1cm and thickness t ¼ 0:5cm has been  y ¼ 0, and it is subjected to a uniform displacement x ¼ u analysed. The plate is fixed along the edge at x ¼ 0, where u  x ¼ 0:001 cm at the opposite edge x ¼ 5a. The Young modulus is equal to E2 ¼ E0 over the whole structure except in a u square region located at the centre of the plate where it is fixed to a smaller value E1 ¼ 0:4E0 . The analysis has been carried out assuming a uniform mesh of 400 FEs with 20 subdivisions both along x and y. A 3D Plot of the NL-FEM solution, in terms of strain distribution ex ðx; yÞ over the plate is displayed in Fig. 2b.  ¼ 2:528 cm, respectively. For comFigs. 3a and b show plane sections of the 3D plot given in Fig. 2b at  x ¼ 2:528 cm and y parison purposes, the corresponding local FEM solutions are also reported. By inspection of these plots, it is observed that the assumed nonlocal behaviour of the material has a notable influence on the strain distribution arising around the transition sections between E1 and E2 and this both along the x and y directions. In analogy with the bar-like example, Fig. 3a shows that around the sections at x ¼ 2a and x ¼ 3a, where the value of the elastic modulus abruptly changes, the strain profile  x (refer also to Fig. 2a), is smoother than the one given by the local apalong the direction of the prescribed displacement u proach. Near the boundaries, the nonlocal solution is very close to the local one thus confirming that, also in a more general 2D context, the strain-difference-based constitutive model allows to eliminate undesired boundary effects. Finally, it is worth mentioning that for both examples, several FE meshes have been considered and the obtained results, not reported for brevity, do not exhibit mesh dependence. Nevertheless, a rigorous proof to this concern is still missing and further investigations would certainly be necessary. The results obtained, even if confined to simple 2D nonlocal nonhomogeneous elastic problems, seem to confirm the ability of the adopted strain-difference-based constitutive nonlocal model to overcome some incoherencies of the nonlocal integral approach arising at the domain boundaries as well as its ability to avoid numerical instabilities. The NL-FEM in this enhanced version appears then an effective tool for the numerical analysis of problems the nonlocal (integral) continuum elasticity theory is oriented to. A variety of engineering problems can be envisaged in the current literature, from the ones related to fracture mechanics in composite materials to the ones encountered in nanotechnology applications, but a lot of work has still to be done to apply the NL-FEM in these contexts. On some of these themes is mainly focused the authors’ present research activity.

(a)

y

(b)

u x = 0.001 cm

E2

2a

a

E1

2a x

a

2a

2a

Fig. 2. (a) A nonhomogeneous plate under tension with piecewise constant Young modulus and (b) 3D plot of the strain distribution ex ðx; yÞ over the plate.

(a) 0.0004

(b) Local-FEM

0.00035

Local-FEM

0.00036

NL-FEM

0.0003

NL-FEM

0.00032

εx (x,y)

εx (x,y)

0.0004

0.00025

0.00028

0.0002

0.00024

0.00015

0.0002

0.0001

0.00016 0

1

2

3

4

5

x[cm] Fig. 3. Strain profiles of the local (dashed lines) and nonlocal (solid lines) solutions: a)

0

1

2

3

4

5

y[cm]

ex versus x at y ¼ 2:528 cm; b) ex versus y at x ¼ 2:528 cm.

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