A mechanically based approach to non-local beam theories

International Journal of Mechanical Sciences 53 (2011) 676–687

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

A mechanically based approach to non-local beam theories Mario Di Paola a, Giuseppe Failla b, Alba Sofi c, Massimiliano Zingales a,n a b c

Dipartimento di Ingegneria Strutturale, Aerospaziale e Geotecnica (DISAG), Universita di Palermo, Viale delle Scienze, 90128 Palermo, Italy Dipartimento di Meccanica e Materiali (MECMAT), Universita ‘‘Mediterranea’’ di Reggio Calabria, Via Graziella, Localita Feo di Vito, 89124 Reggio Calabria, Italy Dipartimento Patrimonio Architettonico ed Urbanistico (PAU), Universita ‘‘Mediterranea’’ di Reggio Calabria, Via Melissari, Feo di Vito, 89124 Reggio Calabria, Italy

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 March 2010 Received in revised form 4 April 2011 Accepted 11 April 2011 Available online 16 April 2011

A mechanically based non-local beam theory is proposed. The key idea is that the equilibrium of each beam volume element is attained due to contact forces and long-range body forces exerted, respectively, by adjacent and non-adjacent volume elements. The contact forces result in the classical Cauchy stress tensor while the long-range forces are modeled as depending on the product of the interacting volume elements, their relative displacement and a material-dependent distance-decaying function. To derive the beam equilibrium equations and the pertinent mechanical boundary conditions, the total elastic potential energy functional is used based on the Timoshenko beam theory. In this manner, the mechanical boundary conditions are found coincident with the corresponding mechanical boundary conditions of classical elasticity theory. Numerical applications are also reported. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Non-local elasticity Long-range interactions Timoshenko beam theory Total elastic potential energy functional

1. Introduction There exist several physical phenomena, such as dispersion of plane elastic waves of short length [1–4], screw dislocation [5,6], surface tension in fluids [7], fracture [8–10] and softening [11–13] where long-range intermolecular forces and microstructure play a significant role. For this reason, such phenomena cannot be adequately captured by classical local continuum mechanics that is an intrinsically scale-free theory, but require a suitable alternative approach. Although a discrete modeling of the medium may certainly appear as a most natural way to capture microstructural effects, atomic and molecular models can be difficult to formulate accurately and prove very demanding from a computational point of view, even for modern computers. Such a difficulty has motivated, since the pioneering work of Kroner [14], Eringen [15] and Kunin [16] in the late 1960s, a considerable research effort to build enriched continua where the influence of microstructure can be somewhat accounted for, in an average sense, by introducing non-local effects. This is, for instance, the perspective of the integral theory [1,5,6], the gradient elasticity theory [4,17–20], the peridynamic theory [21,22] and, also, the so-called continualization theories, where a discrete medium is translated into a non-local continuous model [23–25]. Recently, a mechanically based non-local theory has been proposed also by some of the authors [26–28]. It has been

n

Corresponding author. E-mail address: [email protected] (M. Zingales).

0020-7403/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2011.04.005

assumed that non-local effects result in long-range body forces mutually exerted by non-adjacent volume elements depending on the product of the volume elements, the relative displacement between the volume elements measured along the line connecting the centroids in the original configuration and, also, on a material-dependent distance-decaying function. Then, equilibrium integro-differential equations have been derived in the displacement field, to which boundary conditions (B.C.), identical to the B.C. of classical elasticity theory, apply. The proposed nonlocal theory has been given a consistent variational framework based on the principle of virtual work [27,28]. Applications for smooth and fractional distance-decaying functions [29] have shown that the proposed non-local theory can be a versatile tool to address non-local problems. Also, since classical B.C. are involved, the drawbacks inherent in the gradient elasticity theory are overcome; 1D applications have also shown some mechanical inconsistencies of the Eringen integral theory [26]. In the context of non-local elasticity, non-local beam theories have been awarded a growing attention in the last decades. Applications of such theories, in fact, have been proposed to address the mechanics of nanobeams (for a review, see Ref. [30]) for which, on one hand, early classical local beam theories [31–33] have soon revealed inadequate due to the small length scales involved [34] while, on the other hand, non-local beam models are of particular interest to avoid computationally demanding atomistic simulations. In general, non-local beam theories have addressed linearly elastic behavior, where non-local effects have been modeled based on the Eringen integral theory; the latter involves a non-local stress–strain relation between the stress at a given

M. Di Paola et al. / International Journal of Mechanical Sciences 53 (2011) 676–687

point and the strain in the whole beam volume. In this context, a considerable number of theoretical studies have been proposed, briefly summarized in the following. First, the elastic response to static loading has been investigated by Peddieson and coworkers [35]. They applied the Euler– Bernoulli beam theory in conjunction with the Eringen integral theory to derive non-local static equilibrium fourth-order differential equations. They concluded that non-local effects are present only in beams subjected to distributed loads. Still based on the Euler–Bernoulli beam theory and the Eringen integral theory, Wang and Shindo [36] investigated the static response of carbon nanotubes. Challamel and Wang [37] analyzed a non-local Euler– Bernoulli cantilever beam under a point load at its tip by applying a gradient elastic model as well as an integral non-local elastic model, where the constitutive relationship is expressed by combining the local and non-local curvatures. Then, in order to account for the small aspect ratios of the carbon nanotubes generally involved in experiments, Wang and Liew [38] used the Timoshenko beam theory in conjunction with the Eringen integral theory. The same approach has been pursued by Wang and coworkers [39,40] who derived the governing equations using the principle of virtual work. Wang and Liew [38], as well as Wang and coworkers [39,40], have considered non-local effects in the normal stress but not in the shear stress. Such an assumption helps deriving the governing equations in a simpler form but appears somewhat not consistent with the general Eringen integral theory, where all the stress tensor components are affected by non-local terms. Also, since the Timoshenko theory is used to account for the shear deformations of the beam, it should be more appropriate to account for non-local effects also in the shear stress constitutive equations. This limitation has been overcome by Reddy [41] and Aydogdu [42], who presented a more general formulation deriving the governing equations for different beam theories including those of Euler–Bernoulli, Timoshenko, Reddy, etc., all endowed with the Eringen integral model, where non-local effects appear also in the shear stress constitutive equations. It is worth mentioning that non-local effects under arbitrary loads have been found by Wang and Shindo [36], Wang and Liew [38], Reddy [41], Challamel and Wang [37] and Aydogdu [42], thus overcoming the paradox encountered by Peddieson et al. [35]. More recently, non-local beam theories have been used to address some specific problems, such as: the influence of temperature change on the vibration characteristics of double-walled carbon nanotubes (DWCNTs) [43], or of single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium [44]; the dynamic effects of a moving nanoparticle on nanotube structures [45]; the vibration analysis of fluid-conveying SWCNTs [46] or DWCNTs [47]; the modeling of electrically actuated nanobeams incorporating surface elasticity [48]. Nonlinear free vibrations of SWCNTs have been also studied by Yang et al. [49] combining von Ka rma n geometric nonlinearity, Timoshenko beam theory and Eringen non-local elasticity theory. This paper aims to present a non-local beam theory relying on the mechanically based non-local theory recently proposed by some of the authors [26–28]. Specifically, upon assuming that each volume element of the beam is acted upon by contact forces and long-range forces, the beam equilibrium equations, along with the pertinent mechanical B.C., are derived by applying the principle of minimum potential energy in conjunction with the Timoshenko beam theory. It will be shown that the proposed approach may offer an interesting alternative to classical nonlocal beam theories: no restrictions hold, as in the previous work by Wang et al. [40], on the stress directions along which non-local effects are significant, since long-range interactions between the volume elements are accounted for in all directions; non-local

677

effects are found for general loading conditions and not necessarily for distributed loads only; also, the B.C. coincide with the B.C. of the classical elasticity theory. The paper develops as follows: a general introduction to the basic concepts of the mechanically based non-local theory is outlined in Section 2; the equilibrium equations of the beam under static loads are derived based on the principle of minimum potential energy in Section 3; numerical applications are given in Section 4; the generalization to dynamic loads is finally presented in Appendix B.

2. Mechanically based approach to non-local elasticity Consider a linearly elastic body of volume V and boundary surface S, S¼Sc[Sf, where Sc and Sf denote the constrained and unconstrained part of S, referred to a Cartesian reference system h iT the O(x,y,z), as shown in Fig. 1a. Denote by uðxÞ ¼ ux uy uz displacement vector, depending on the position vector  T x¼ x y z and be uðxÞ ¼ uðxÞ the displacement vector field prescribed on Sc. Also, denote by pn ðxÞ the external load applied to the unit surface on Sf, of outward normal n; by bðxÞ the external load applied to the unit volume in V and by vn(x) the reaction forces on the unit surface on Sc. In the mechanically based non-local theory proposed by Di Paola et al. [26–28], it is assumed that non-local effects result in long-range internal body forces exchanged between non-adjacent volume elements. Specifically, the long-range force acting on a volume element dV(x), due to a volume element dV(n) at h iT n ¼ x c z , is taken as proportional to the interacting volume elements, that is q(x,n) dV(n) dV(x), where q(x,n) denotes the (specific) long-range force exerted on a unit volume at x by a unit volume at n (see Fig. 1a). The long-range force df(x,n) acting on the unit volume dv(x)¼ 1 and due to a volume element dV(n) is then defined as dfðx, nÞ ¼ qðx, nÞ dVðnÞ: ð1Þ h iT Let rðx, nÞ ¼ rx ry rz be the unit vector associated with the direction x  n, positively oriented from x to n, given by rðx, nÞ ¼

nx nx ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : :nx: 2 ðxxÞ þ ðcyÞ2 þðzzÞ2

ð2Þ

The specific long-range force q(x,n) in Eq. (1) is modeled as a central force having the following form: qðx, nÞ ¼ gðx, nÞ½gT ðx, nÞrðx, nÞ rðx, nÞ,

ð3Þ

where g(x,n) is a material-dependent, symmetric and distancedecaying real-valued scalar function, i.e. g(x,n) ¼g(n,x) and g(x1,n1)og(x2,n2) 8 9x1  n19 49x2  n29; g(x,n) is the vector of the relative displacements between the centroids of the volume elements dV(x) and dV(n):

gðx, nÞ ¼ uðnÞuðxÞ:

ð4Þ

It is worth remarking that, under the assumption of small displacements, the proposed non-local theory is invariant with respect to rigid-body displacements. In fact, the long-range forces (3) vanish for any rigid-body displacement field, including rigid rotations. In the context of the proposed non-local theory, the equilibrium of a volume element dV(x) is attained due to the external body forces bðxÞdVðxÞ, the long-range internal body forces f(x)dV(x) and under contact stresses exerted by the adjacent volume elements, denoted by tðklÞ ðxÞ, where the superscript in parenthesis means local since they are the classical Cauchy stresses (see Fig. 1b) while the subscript indicates that the stress

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M. Di Paola et al. / International Journal of Mechanical Sciences 53 (2011) 676–687

pn (x)

n Sf

u ( x)

q ( x, ξ ) dV ( x ) dV ( ξ )

r

-q ( x, ξ ) dV ( x ) dV ( ξ )

b ( x ) dV ( x )

u (ξ)

x ξ

z , uz

O

y, uy

x, ux Sc u ( x) = u ( x)

t (zl ) ( x ) dAz

t (yl ) ( x ) dAy

z

O

y

q ( x, ξ ) dV ( x ) dV ( ξ )

x

t(l)x ( x ) dAx

b ( x ) dV ( x ) Fig. 1. (a) 3D continuum with long-range interactions and (b) contact and long-range forces on a volume element.

vector acts on the elementary plane at x of outward normal k. Specifically, the long-range internal body force f(x) is the resultant of the long-range forces exerted on the volume element dV(x) by all the volume elements at n, that is Z Z ð5Þ fðxÞ ¼ qðx, nÞ dVðnÞ ¼ gðx, nÞ½gT ðx, nÞrðx, nÞrðx, nÞ dVðnÞ: V

V

The long-range body force f(x) dV(x) is an infinitesimal of the same order as bðxÞ dVðxÞ. For this, in the proposed non-local theory the stress vector tðnlÞ ðxÞ acting on an elementary plane at

x of outward normal n ¼ ½ nx Cauchy stress relation ðlÞ ðlÞ ðlÞ tðlÞ n ðxÞ ¼ tx nx þ ty ny þtz nz :

ny

nz T is still given by the

ð6Þ

Further, the mechanical B.C. coincide with the classical boundary conditions (for a more detailed discussion on the mechanical B.C. for a 1D bar see also Ref. [26]), that is pn ðxÞ ¼ NrðlÞ ðxÞ on Sf ,

ð7Þ

M. Di Paola et al. / International Journal of Mechanical Sciences 53 (2011) 676–687

where r(l)(x) is the vector collecting the Cauchy stress components, i.e. h iT ðlÞ syðlÞ sðlÞ tyz tðlÞ tðlÞ rðlÞ ðxÞ ¼ sðlÞ ð8Þ x xy z xz

formulation. This has been pursued starting from the work identity Z Z Z T T b ðxÞuðxÞ dVðxÞ þ f ðxÞuðxÞ dVðxÞ þ pTn ðxÞuðxÞdSðxÞ V

and N is the boundary equilibrium operator, defined as 2 3 nx 0 0 0 nz ny 6 7 N ¼ 4 0 ny 0 nz 0 nx 5 , 0 0 nz ny nx 0

Sc

nx, ny and nz being the components of the outward normal n to the boundary surface Sf. Then, the equilibrium equation of the elementary volume dV(x) singled out from the solid (Fig. 1b) can be written in the form C T rðlÞ ðxÞ þ bðxÞ þ fðxÞ ¼ 0

in V,

ð10Þ

where f(x) is defined in Eq.(5) and C given by 2@ @ @ 3 0 0 0 @z @x @y 6 @ @ @ 7 7: 0 @y 0 @z 0 @x CT ¼ 6 4 5 @ @ @ 0 0 @z @y @x 0

T

is a differential operator

ð11Þ

eðxÞ ¼ CuðxÞ in V

ð12Þ

to which the kinematic B.C. uðxÞ ¼ uðxÞ, for xASc, apply. In Eq. (12), C is the transpose of the differential operator defined in Eq. (11), while e(x) is the vector listing the strain components, i.e. h iT eðxÞ ¼ ex ey ez gyz gxz gxy : ð13Þ Finally, the constitutive equations relating the local stress field to the strain field must be considered. For a linearly elastic and isotropic material they are given as

rðlÞ ðxÞ ¼ D eðxÞ in V,



6 6 6 6  D ¼6 6 6 6 4

l 

l l

0

0

0

0

0 7 7 7 0 7 7, 0 7 7 7 0 5

l

ð2m þ l Þ

0

0

0

0

0

G

0

0 0





0 0



0 0

3

0

ð2m þ l Þ

G 0

ð15Þ

G

T

r ðxÞeðxÞ dVðxÞ:

ð18Þ

V

V

1 2

Z Z V

qT ðx, nÞgðx, nÞ dVðnÞ dVðxÞ

ð19Þ

V

has been derived, according to which the work identity (18) takes the form Z Z Z T b ðxÞuðxÞ dVðxÞ þ pTn ðxÞuðxÞ dSðxÞ þ vTn ðxÞuðxÞ dSðxÞ V

Sf

Z

Sc

Z Z

1 ¼ rT ðxÞeðxÞ dVðxÞ þ 2 V

V

qT ðx, nÞgðx, nÞ dV ðnÞ dVðxÞ:

ð20Þ

V

It is based on Eq. (20) that the total elastic potential energy functional ð21Þ

can be introduced, where u(x) and e(x) are arbitrary functions satisfying the strain–displacement equations (12) and the kinematic B.C. uðxÞ ¼ uðxÞ on Sc. Specifically, in Eq. (21) F(u,e) denotes the elastic potential energy Z 1 Fðu, eÞ ¼ FðlÞ ðeÞ þ FðnlÞ ðgÞ ¼ eT ðxÞD eðxÞ dVðxÞ 2 V Z Z 1 þ gðx, nÞ½rT ðx, nÞgðx, nÞ2 dVðnÞ dVðxÞ, 4 V V ð22Þ where F(l)(e) and F(nl)(g) denote the local and non-local contributions, respectively, and W(u) is the work done by the external forces, Z

T

b ðxÞuðxÞ dVðxÞ þ

Z Sf

pTn ðxÞuðxÞ dSðxÞ:

ð23Þ

It can be readily seen that the Euler–Lagrange equations and the natural conditions associated to the functional (21) coincide, respectively, with the equilibrium Eq. (16) and the mechanical B.C. (17b) [27,28].

3. Non-local Timoshenko beam theory

where m ¼ b1m and l ¼ b1l, being m and l the Lame elastic constants and being b1 a dimensionless real coefficient, 0 r b1 r1, weighting the amount of local interactions [50]. This is in analogy to the non-local theories where the non-local elastic material is conceived as a two-phase elastic material [51]. Following the stiffness method, the equilibrium equations governing the non-local continuum may be expressed in terms of the displacement field u(x) by replacing Eq. (12) for e(x) into Eq. (14) and then introducing the resulting expression of the vector rðlÞ ðxÞ ¼ D CuðxÞ in the equilibrium Eq. (10), that is n

T

f ðxÞuðxÞ dVðxÞ ¼ 

V

l l 0 0

Z

WðuÞ ¼

with D denoting the elastic matrix



Sf

Z

Further, due to the symmetry of the attenuation function (g(x,n)¼g(n,x)) the relation

ð14Þ

*

ð2m þ l Þ

vTn ðxÞuðxÞ dSðxÞ ¼

Pðu, eÞ ¼ Fðu, eÞWðuÞ

Along with the equilibrium equations (10), the non-local continuum is governed by the strain–displacement equations

2

V

Z þ

ð9Þ

679

n

LuðxÞ þ bðxÞ þ fðxÞ ¼ 0 in V T

ð16Þ



where L ¼ C D C. The associated B.C. take on the following form: uðxÞ ¼ uðxÞ

on Sc ,

pn ðxÞ ¼ ND CuðxÞ

on Sf :

ð17a; bÞ

In a recent paper by some of the authors [27], the above described non-local model has been given a consistent variational

In this section, the equilibrium equations of a Timoshenko beam with long-range interactions are derived by a variational approach based on the energy functional (21). Let us consider an initially straight uniform beam of length L and assume a rectangular Cartesian coordinate system O (x,y,z), where the x-axis coincides with the centroidal axis of the undeformed beam, while the y- and z-axis are the neutral and symmetry axis of the cross-section A, respectively. According to classical beam theories, the displacement components ux, uy, and uz along the x-, y- and z-axis depend only on the coordinates x and z. In the following, arguments will be omitted for conciseness. In the context of the Timoshenko beam theory, the displacement field can be expressed as follows: ux ¼ uðxÞzjðxÞ,

uy ¼ 0,

uz ¼ vðxÞ

ð24a2cÞ

where u(x) and v(x) are the x- and z-components of the displacement of a point (x,0,0) on the centroidal axis; j(x) denotes the rotation of the cross-section about the y-axis, taken as positive if

680

M. Di Paola et al. / International Journal of Mechanical Sciences 53 (2011) 676–687

clockwise in the xz plane. The corresponding nonzero strain components are

ex ¼

@ux ¼ eðxÞ þzwðxÞ; @x

gxz ¼

@ux @uz þ  gðxÞ, @z @x

ð25a; bÞ

where

eðxÞ ¼

ð33Þ

0

@uðxÞ ; @x

wðxÞ ¼ 

@jðxÞ ; @x

gðxÞ ¼

@vðxÞ jðxÞ @x

ð26a2cÞ

denote the axial strain, the bending strain and the transverse shear strain, respectively. Let N(l)(x), T(l)(x) and M(l)(x) be the axial stress resultant, the shear force and the bending moment pertaining to the classical local beam theory, i.e.: Z Z Z ðlÞ ðxÞ dA; M ðlÞ ðxÞ ¼ ysðlÞ NðlÞ ðxÞ ¼ sxðlÞ ðxÞ dA; T ðlÞ ðxÞ ¼ txz x ðxÞ dA: A

A

A

ð27a2cÞ The linearly elastic constitutive relationships between stress resultants and generalized strain components then read NðlÞ ðxÞ ¼ E AeðxÞ;

T ðlÞ ðxÞ ¼ Ks G AgðxÞ;

M ðlÞ ðxÞ ¼ E IwðxÞ

ð28a2cÞ

where En ¼ b1E and Gn ¼ b1G, E being the Young’s modulus and G being the shear modulus, while b1 is the dimensionless coefficient introduced in Section 2; I denotes the second moment of area about the y-axis; finally, Ks is the shear correction factor. Then, as far as the non-local variables are concerned, based on Eqs. (24a–c) the three components of the relative displacement vector (4) between the centroids of the elementary volumes dV(x) and dV(n) are given by

Zx ¼ ½uðxÞuðxÞ½z jðxÞzjðxÞ; Zy ¼ 0;

Zz ¼ vðxÞvðxÞ: ð29a2cÞ

Replacing Eq. (29) for Zx, Zy and Zz in Eq. (3) and performing straightforward manipulations, the following expressions of the components of the specific long-range force q(x,n) are obtained: qx ðx, nÞ ¼ gðx, nÞf½uðxÞuðxÞrx2 ðx, nÞ½zjðxÞzjðxÞrx2 ðx, nÞ þ ½vðxÞvðxÞrx ðx, nÞrz ðx, nÞg; qy ðx, nÞ ¼ gðx, nÞf½uðxÞuðxÞrx ðx, nÞry ðx, nÞ½zjðxÞzjðxÞ rx ðx, nÞry ðx, nÞ þ ½vðxÞvðxÞry ðx, nÞrz ðx, nÞg; qz ðx, nÞ ¼ gðx, nÞf½uðxÞuðxÞrx ðx, nÞrz ðx, nÞ½zjðxÞzjðxÞ rx ðx, nÞrz ðx, nÞ þ½vðxÞvðxÞrz2 ðx, nÞg,

ð30a2cÞ

where rx(x,n), ry(x,n) and rz(x,n) are the x-, y- and z-components of the unit vector r(x,n) associated with the direction x  n (see Eq. (2)). Now, the total elastic potential energy functional (21) can be rewritten in terms of the generalized variables of the Timoshenko beam. Specifically, the elastic potential energy (22) can be recast as Z 1 L  2 Fðe, w, g, Zx , Zz Þ ¼ FðlÞ ðe, w, gÞ þ FðnlÞ ðZx , Zz Þ ¼ ½E Ae þ E Iw2 2 0 Z Z  1 þKs G Ag2 dx þ ½qx Zx þqz Zz  dVðxÞ dVðnÞ 4 V V ð31Þ its first variation is then given by

dF ¼ dFðlÞ þ dFðnlÞ Z

¼

Eq. (32) can be rewritten as Z Z Z LZ LZ Z ½qx dux þqz duz  dVðxÞ dVðnÞ ¼ ½qx dux V V A A 0 0 Z L þ qz duz  dAðxÞ dAðxÞ dx dx ¼ ½R1 du þ R2 dv þ R3 dj dx,

L

½E Ae d e þE Iw d w þ Ks G Ag d g dx Z Z ½qx dux þqz duz  dVðxÞ dVðnÞ ,  0

V

where dA(x)¼dy dz and dA(x)¼dc dz, while the functions R1(x), R2(x) and R3(x) are defined as Z L fI1 ðx, xÞ½uðxÞuðxÞI7 ðx, xÞjðxÞ þ I8 ðx, xÞjðxÞ R1 ðxÞ ¼ 0

þ I9 ðx, xÞ½vðxÞvðxÞgdx, Z L fI9 ðx, xÞ½uðxÞuðxÞI2 ðx, xÞjðxÞ þ I3 ðx, xÞjðxÞ R2 ðxÞ ¼ 0

þI4 ðx, xÞ½vðxÞvðxÞgdx, Z L fI8 ðx, xÞ½uðxÞuðxÞ þI5 ðx, xÞjðxÞI6 ðx, xÞjðxÞ R3 ðxÞ ¼ 0

I3 ðx, xÞ½vðxÞvðxÞgdx,

with Ii(x,x) (i¼1,2,y,9) being integrals defined in Appendix A. In particular, it can be readily verified that, if the cross-section A of the beam has two axes of symmetry, then Ij(x,x)¼0 (j ¼7,8,9). Then, introducing the strain–displacement relationships (26a– c), performing integration by parts on the first integral and taking into account Eq. (33), Eq. (32) can be recast in the following form: ! Z L( @2 u @2 v @j dF ¼  E A 2 du þ Ks G A  dv @x @x2 @x 0 "  # ) @2 j @v j dj dx þ E I 2 þ Ks G A @x @x     x ¼ L @u @v @j j dv þE I dj þ E A du þ Ks G A @x @x @x x¼0 Z L  ½R1 du þR2 dv þR3 dj dx: ð35Þ 0

Next, consider the external work W given by Eq. (23). If Fx(x) and Fy(x) are introduced as generalized measures of the external forces acting on the unit length of the beam, the first variation dW in terms of the generalized variables of the Timoshenko beam can be written as Z L dW ¼ ½Fx du þ Fz dv dxþ ½N0 duð0Þ þ T0 dvð0Þ þ M0 djð0Þ 0

þ NL duðLÞ þ TL dvðLÞ þ ML djðLÞ,

V

where use of Eqs. (5) and (19) has been made. Next, upon replacing Eqs. (30a) and (30c) for qx and qz and carrying out appropriate manipulations, the second integral in the r.h.s. of

ð36Þ

where Ni, Ti and Mi (i¼0,L) are the axial force, transverse force and bending moment at the beam ends, i.e. at x ¼0 and L (where, obviously, du a0, dv a0 and dj a0 only if no corresponding constraint applies). Finally, based on Eqs. (35) and (36) the first variation of the total elastic potential energy functional (21), rewritten in terms of the generalized variables of the Timoshenko beam, takes the form: ! # " # Z L (" @2 u @2 v @j þ R dP ¼  E A 2 þ R1 þ Fx du þ Ks G A  þ F z dv 2 @x @x2 @x 0 " # )   @2 j @v j þR3 dj dx þ E I 2 þ Ks G A @x @x (" # " #  @u  @u NL duðLÞ þ E A N0 duð0Þ þ E A @x x ¼ L @x x ¼ 0 "

ð32Þ

ð34a2cÞ

þ Ks G  A



# " #    @v @v TL dvðLÞ þ Ks G A T0 dvð0Þ j j @x @x x¼L x¼0

) # " # @j  @j þ E I ML djðLÞ þ E I M0 djð0Þ : @x x ¼ L @x x ¼ 0 "



ð37Þ

M. Di Paola et al. / International Journal of Mechanical Sciences 53 (2011) 676–687

Then, accounting for the arbitrariness of du, dv and dj in the interval 0 ox oL and assuming that the beam cross-section has two symmetry axes, i.e. setting Ij(x,x)¼ 0 (j ¼7,8,9) in Eqs. (34a–c), the extremum condition dP ¼0 yields the following Euler– Lagrange equations: Z L @2 uðxÞ þ I1 ðx, xÞ½uðxÞuðxÞdx þ Fx ðxÞ ¼ 0, 2 @x 0 " # Z L @2 vðxÞ @jðxÞ  fI2 ðx, xÞjðxÞ þI3 ðx, xÞjðxÞ þ Ks G A @x @x2 0  

þ I4 ðx, xÞ vðxÞvðxÞ dx þ Fz ðxÞ ¼ 0,   Z L @2 jðxÞ @vðxÞ jðxÞ þ þ K s G A fI5 ðx, xÞjðxÞI6 ðx, xÞjðxÞ E I 2 @x @x 0 E A

I3 ðx, xÞ½vðxÞvðxÞg dx ¼ 0:

ð38a2cÞ

The associated static and kinematic B.C. are: @uðxÞ ¼ N0 or uð0Þ ¼ u0 NðlÞ ð0Þ ¼ E A @x x ¼ 0 @uðxÞ NðlÞ ðLÞ ¼ E A ¼ NL or uðLÞ ¼ uL @x x ¼ L   @vðxÞ jðxÞ T ðlÞ ð0Þ ¼ Ks G A ¼ T0 or vð0Þ ¼ v0 @x x¼0   @vðxÞ jðxÞ T ðlÞ ðLÞ ¼ Ks G A ¼ TL or vðLÞ ¼ vL @x x¼L @jðxÞ M ðlÞ ð0Þ ¼ E I ¼ M0 or jð0Þ ¼ j0 @x x ¼ 0 @jðxÞ M ðlÞ ðLÞ ¼ E I ¼ ML or jðLÞ ¼ jL , @x x ¼ L

ð39Þ

where ui, vi and ji (i¼0,L) are prescribed displacements and rotations at beam ends. The integro-differential Eqs. (38a–c) along with the B.C. (39) govern the equilibrium of a Timoshenko beam with long-range interactions. A notable feature of the proposed non-local beam model is that the mechanical B.C. are the same as those pertaining to the classical Timoshenko beam theory. Indeed, as explicitly stated by Eq. (7), since the long-range interactions are modeled as (internal) body forces, they do not affect the mechanical B.C. Furthermore, it is worth emphasizing that, by virtue of the double symmetry of the cross-section, the axial and transverse responses are uncoupled as it happens when non-local effects are disregarded. Finally, it is noted that suppressing the integral terms on the l.h.s. of Eqs. (38a–c), i.e. neglecting the effects of long-range non-local forces, and setting b1 ¼1 the differential equations governing the classical Timoshenko beam model are recovered. In the literature, several numerical methods are available to build approximate solutions to the integro-differential Eqs. (38a–c). Among these, the finite difference method [26], the Galerkin method [29,52] and the finite element method [53–58], already successfully applied to address non-local problems, can be readily implemented to solve Eqs. (38a–c) as well. Numerical results presented in the next section will be derived based on the finite difference method. However, it is worth mentioning that to deal with more general problems, involving complex geometries and inhomogeneous materials, or to capture discontinuities in the displacement field [57], finite element solutions should be pursued. In this regard, recognize that the non-local stiffness matrices associated to the long-range interactions can be readily derived based on the proposed variational formulation. As a final remark, the above presented non-local beam theory can be straightforwardly generalized to address the beam dynamic response, as outlined in Appendix B. Specifically, the Hamilton’s principle will be applied to derive the equations of

681

motion of a non-local Timoshenko beam under time-varying external loads.

4. Numerical applications The proposed Timoshenko beam theory with long-range interactions has been applied to evaluate the bending response of two non-local beams with different boundary and loading conditions: (i) a cantilever beam subjected to a tip load; (ii) a simply supported beam under a uniformly distributed load. In both cases: a rectangular cross-section with width b and thickness h has been considered (Ks ¼5/6); a material-dependent bi-exponential attenuation function of the form [26]:   :xn: Eð1b1 Þ gðx, nÞ ¼ C exp  ð40Þ , C¼ l0 2A2 l0 has been selected. In the previous equation, l0 denotes the internal length material scale whose value, much smaller than the smallest dimension of the beam, is herein fixed arbitrarily. Notice that the greater the value of the internal length, the wider the so-called influence distance, LI, namely the maximum distance beyond which the attenuation function and therefore the long-range interactions become negligible. In order to investigate the influence of the long-range forces on the beam response, different values of both the parameters l0 and b1, governing the non-local behavior in the context of the proposed model, have been considered. The integro-differential Eqs. (38a–c) ruling the beam equilibrium have been solved under the pertinent B.C. resorting to the standard finite difference method using a uniform grid with mþ1 points and m intervals of amplitude Dx ¼L/m. Furthermore, the integrals Ij(x,x) (j ¼1,y,6) appearing in the long-range terms (see Eqs. (38a–c) and Appendix A) have been computed numerically by means of the trapezoidal rule adopting a uniform grid over both the cross-sections A(x) and A(x), with equal subdivisions along the principal axes, say Dy¼ Dc ¼b/n and Dz ¼ Dz ¼h/s, n and s being appropriate integers. 4.1. Cantilever beam under tip load Consider the cantilever beam depicted in Fig. 2 subjected to a concentrated load P at its tip. This problem is of interest in the field of microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS) since cantilever beams are often used as actuators [35,37,48]. Without loss of generality, a square cross-section with b¼h¼ 1 nm has been considered and the length of the beam, when not taken variable, has been set equal to L¼10 nm. The finite difference solution of the integro-differential Eqs. (38a–c), governing the equilibrium of the non-local beam, has been built considering a grid with m ¼650 intervals. Numerical studies here omitted for brevity indeed have shown that the solution does not significantly change when a more refined grid is adopted. The static response of the cantilever beam is herein characterized in terms of non-dimensional deflection, vðxÞE I=PL3 , and nondimensional rotation, jðxÞE I=PL2 , with En ¼ b1E. The influence of same relevant parameters on the beam response has been investigated in detail through numerical simulations. Fig. 3 displays the non-dimensional static deflection and rotation versus the non-dimensional location x/L obtained by the proposed non-local beam model setting b1 ¼0.5 and considering two different values of the internal length, say l0 ¼0.05 and 0.08 nm. For comparison purposes, the solution pertaining to the classical local Timoshenko beam theory (b1 ¼1.0, l0-0) is also

682

M. Di Paola et al. / International Journal of Mechanical Sciences 53 (2011) 676–687

P

8 h

l0 = 0.05 nm l0 = 0.08 nm

7 b

6

ρv (L)

L Fig. 2. Cantilever beam subjected to a tip load.

5 4 3

0

2 1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

β1 0.2

8

6

0.4 0

0.2

0.4

0.6

l0 = 0.05 nm l0 = 0.08 nm

7

Local Non-local (l0=0.05 nm, β1=0.5) Non-local (l0=0.08 nm, β1=0.5)

0.3

0.8

1

x/L

0.5

ρϕ (L)

v (E*I/PL3)

0.1

5 4 3 2

0.4

ϕ (E*I/PL2)

1 0.1

0.3

0.3

0.4

0.5

0.6

0.7

0.8

0.9

β1 Fig. 4. Ratio of the non-local deflection (a) and non-local rotation (b) to their local counterparts at the free end versus b1 for L¼ 10 nm and two different values of the internal length l0.

0.2 Local Non-local (l0=0.05 nm, β1=0.5) Non-local (l0=0.08 nm, β1=0.5)

0.1

0

0

0.2

0.4

0.6

0.8

1

x/L Fig. 3. Non-dimensional deflection (a) and rotation (b) of the cantilever beam: comparison between the local response (continuous line) and the proposed nonlocal solution for b1 ¼ 0.5 and two different values of the internal length l0 (symbols).

reported. Notice that the long-range forces are modeled in such a way that, for a given value of b1, an increase of the internal length l0 makes the beam stiffer, while, obviously, the beam becomes more flexible as the parameter b1 decreases. In order to give an expressive measure of the effects induced by the long-range forces, the ratio between the non-local response and the corresponding local one (b1 ¼1.0, l0-0), i.e.:

ra ðxÞ ¼

0.2

aðxÞ , aðlÞ ðxÞ

a ¼ v, j

ð41Þ

has been evaluated for both the deflection and rotation along the beam. In Fig. 4, the ratios rv(L) and rj(L) at the free end of the cantilever beam, (x¼L) versus the parameter b1 for two different values of the internal length (l0 ¼0.05 and 0.08 nm) are plotted. As

expected, the ratios rv(L) and rj(L) are decreasing functions of b1 and tend to approach unity as b1-1, whatever the value of the internal length is. Furthermore, it is observed that the greater l0 the smaller the ratio between the non-local and the local response in agreement with the above considerations about the influence of the internal length variation on the beam stiffness. Finally, the comparison between Fig. 4a and b clearly shows that the deflection and rotation at the tip of the cantilever beam undergo almost the same amplification due to the long-range interactions. The influence of the beam length, L, in relation to non-local effects has also been investigated through numerical simulations. Fig. 5 shows the ratios rv(L) and rj(L) at the free end of the cantilever beam (x¼L) versus the length L for two different values of the internal length (l0 ¼0.05 and 0.08 nm). As expected, non-local effects are more significant for shorter beams and decrease with increasing L. Before illustrating the second numerical example, some computational issues concerning the evaluation of the integrals Ij(x,x) (j¼1,y,6) defined in Appendix A need to be discussed. In the context of the finite difference method, once the beam domain [0,L] is subdivided into m intervals of amplitude Dx¼ L/m, each integral must be evaluated for all the couples of abscissas xi ¼(i 1)Dx and xk ¼(k 1)Dx, (i,k ¼ 1,2,. . .,m þ 1; Dx ¼ Dx), that is: Ij(xi, xk), (j¼1,y,6). Thus, for a given abscissa xi ¼ xi , each integral Ij ðxi , xk Þ, (j¼1,y,6) has to be computed for all the points xk (k ¼ 1,2,. . .,m þ 1 ) within the domain, namely mþ1 times. It can be easily inferred

2

0

1.8

0.4 v (102E*I/Fz L4)

ρv (L)

M. Di Paola et al. / International Journal of Mechanical Sciences 53 (2011) 676–687

1.6 1.4

Local Non-local (l0=0.05 nm, β1=0.5) Non-local (l0=0.08 nm, β1=0.5)

0.8

1.2

l0 = 0.05 nm, β1 = 0.5 l0 = 0.08 nm, β1 = 0.5

1.2

1.6

1 5

7.5

10

12.5

15

17.5

0

20

0.2

0.4

0.8

1

0.8

1

6

2

4

ϕ (102E*I/Fz L3)

1.8 1.6 ρϕ (L)

0.6

x/L

L [nm]

1.4 1.2 l0 = 0.08 nm, β1 = 0.5

5

7.5

10

12.5

2 0 -2

Local Non-local (l0=0.05 nm, β1=0.5) Non-local (l0=0.08 nm, β1=0.5)

-4

l0 = 0.05 nm, β1 = 0.5

1 0.8

683

-6

15

20

17.5

L [nm] Fig. 5. Ratio of the non-local deflection (a) and non-local rotation (b) to their local counterparts at the free end versus the beam length L for b1 ¼ 0.5 and two different values of the internal length l0.

0

0.2

0.4

0.6 x/L

Fig. 7. Non-dimensional deflection (a) and rotation (b) of the simply supported beam: comparison between the local response (continuous line) and the proposed non-local solution for b1 ¼ 0.5 and two different values of the internal length l0 (symbols).

therefore the integrals Ij ðxi , xk Þ, (j¼1,y,6), are practically vanishing; (ii) taking into account the double-symmetry of the beam cross-section as well as the symmetry of the attenuation function, it can be easily verified that the following relationships hold true:

Fz

h

b L Fig. 6. Simply supported beam under a uniformly distributed load.

Ij ðxi , xk Þ ¼ Ij ðxk ,xi Þ, j ¼ 1,4,5,6; Ij ðxi , xk Þ ¼ Ij ðxk ,xi Þ, j ¼ 2,3:

ð42a; bÞ

In view of observation (i), for any abscissa xi ¼ xi a number of integrals Ij ðxi , xk Þ, (j¼1,y,6) less than mþ1 has to be computed, namely those pertaining to the abscissas xk falling within the interval ½xi LI ,xi þ LI . A further reduction of the number of integrals to be evaluated can be obviously obtained by exploiting properties (42a,b). 4.2. Simply supported beam under uniformly distributed load

that the computational effort may become prohibitive as m increases since the evaluation of each integral Ij ðxi , xk Þ actually involves a double integration over the cross-sections Aðxi Þ and A(xk). The latter, as already mentioned, can be performed by means of the trapezoidal rule using a sufficiently refined grid of points over each cross-section. Nevertheless, the computational times may be substantially reduced relying on the following observations: (i) for a given abscissa xi ¼ xi , each integral Ij ðxi , xk Þ, (j¼1,y,6) takes on significant values only at points xk belonging to the interval ½xi LI ,xi þ LI  where LI, estimated equal to LI ¼6l0 for the bi-exponential function (40) herein adopted [59], is the influence distance beyond which the attenuation function and

The second example concerns the simply supported beam under a uniformly distributed load Fz shown in Fig. 6. A square crosssection with b¼h¼1 nm has been considered and the length, when not considered variable, has been set equal to L¼10 nm. The integro-differential equilibrium Eqs. (38a–c) have been solved under the pertinent B.C. by means of the finite difference method subdividing the beam domain into m ¼650 intervals. Indeed, no appreciable improvement of the accuracy has been obtained using more refined subdivisions. Numerical results in terms of non-dimensional deflection, vðxÞ102 E I=Fz L4 , and non-dimensional rotation, jðxÞ102 E I=Fz L3 ,

M. Di Paola et al. / International Journal of Mechanical Sciences 53 (2011) 676–687

2 1.9

l0 = 0.05 nm l0 = 0.08 nm

9 8

l0 = 0.05 nm, β1 = 0.5 l0 = 0.08 nm, β1 = 0.5

1.5

5

7.5

10

17.5

20

15

17.5

20

1.9 1.8

l0 = 0.05 nm, β1 = 0.5 l0 = 0.08 nm, β1 = 0.5

5

7.5

10

12.5 L [nm]

5

Fig. 9. Ratio of the non-local midspan deflection (a) and non-local rotation at the left-end (x¼ 0) (b) to their local counterparts versus the beam length L for b1 ¼ 0.5 and two different values of the internal length l0.

4 3 2

5. Concluding remarks

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

β1

10 l0 = 0.05 nm l0 = 0.08 nm

9 8 7 6 5 4 3 2 1

15

2

1.5

6

1

12.5 L [nm]

1.6

7 ρv (L/2)

1.7

1.7

10

ρϕ (0)

1.8

1.6

ρϕ (0)

with E* ¼ b1E, yielded by the proposed approach for b1 ¼0.5 and two different values of the internal length, say l0 ¼0.05 and 0.08 nm, are displayed in Fig. 7 along with the corresponding local solution (b1 ¼1.0, l0-0). It can be seen that, also in this case, for a given value of b1, a larger internal length implies a stiffer behavior of the beam. Fig. 8a displays the ratio rv(L/2) between the non-local midspan deflection and the corresponding local one (see Eq. (41)) versus b1. In Fig. 8b, the dependence of the ratio rj(0) between the non-local and the local rotation (see Eq. (41)) at the left-end support (x¼ 0) on the parameter b1 is shown. In both cases, two different values of the internal length (l0 ¼0.05m and 0.08 nm) have been considered. Notice that the ratios rv(L/2) and rj(0) are decreasing functions of b1 and tend to approach unity as b1-1 for any value of the internal length. Furthermore, as expected, a smaller ratio between the non-local and the local response is obtained as l0 increases since the beam becomes stiffer. Fig. 9a and b show the ratios rv(L/2) and rj(0) versus the length L of the beam for l0 ¼0.05 and 0.08 nm. As already observed in the case of the cantilever beam, the effects of longrange forces on the beam response are remarkable for shorter beams and decrease with increasing L. Finally, by inspection of Figs. 8 and 9, it can be seen that the non-local effects cause almost the same amplification of both beam deflection and rotation.

ρv (L/2)

684

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

β1 Fig. 8. Ratio of the non-local midspan deflection (a) and non-local rotation at the left-end (x¼ 0) (b) to their local counterparts versus b1 for L¼10 nm and two different values of the internal length l0.

A non-local Timoshenko beam model has been presented. Specifically, to derive the equilibrium equations along with the mechanical B.C. a variational approach has been pursued, involving a total elastic potential energy functional built relying on a mechanically based non-local modeling of the 3D continuum, recently proposed in the literature [26–28]. Such a modeling stems from the key assumption that to the equilibrium of each volume element contribute not only contact forces exerted by adjacent volume elements (resulting in the classical Cauchy stress) but also long-range forces exerted by non-adjacent volume elements. The latter are taken as depending on the product of the interacting volume elements, on their relative displacement and on a material-dependent distance-decaying function; also, it is assumed that they act along the line connecting the centroids of the interacting volume elements in the initial configuration. There are some advantages in the proposed non-local Timoshenko beam model: (1) no restrictions are introduced, as in some previous models, on the stress directions along which non-local effects are significant; (2) non-local effects are encountered for arbitrary loading conditions, unlike in some previous studies where non-local effects may disappear depending on the applied load; (2) also, non-local terms are not involved in the mechanical B.C. that, therefore, coincide with the corresponding mechanical B.C. of classical elasticity theory; as a result, the

M. Di Paola et al. / International Journal of Mechanical Sciences 53 (2011) 676–687

governing equations can be readily solved by standard numerical procedures such as, for instance, the finite difference method pursued in the paper. Numerical results have been reported for a variety of non-local theoretical parameters, chosen to enhance non-local effects in the response. The proposed model offers a wide spectrum of parameters (l0 and b1) to be set according to experimental tests on the material under study. As for any of the non-local beam models proposed in the literature to date, the calibration of the non-local parameters is certainly a challenging task, to which further effort is warranted by the authors in the future.

Appendix A The integrals introduced in Eqs. (34a–c) are defined as follows: Z Z I1 ðx, xÞ ¼ gðx, nÞrx2 ðx, nÞ dAðxÞ dAðxÞ, A A

I2 ðx, xÞ ¼ I3 ðx, xÞ ¼

Z Z

ZA ZA

gðx, nÞzrx ðx, nÞrz ðx, nÞ dAðxÞ dAðxÞ,

Upon taking into account Eqs. (24a–c), the kinetic energy of the beam is given by     2 # Z " 1 @ux 2 @uy 2 @uz K¼ r þ þ dVðxÞ 2 V @t @t @t " #  2   Z 1 @u @j 2 @v z r þ dVðxÞ, ðB:2Þ ¼ 2 V @t @t @t where r is the mass density of the beam material which is assumed independent of time t. By applying the rules of calculus of variation and performing integration by parts, the following expression of the first variation of the kinetic energy on the time interval [0,T] is obtained: # Z T Z TZ L" @2 u @2 v @2 j d K dt ¼  m 2 du þ m 2 dv þIr 2 dj dxdt @t @t @t 0 0 0  t ¼ T Z L @u @v @j þ m du þ m dv þ Ir dj dx ðB:3Þ @t @t @t 0 t¼0 where Z m ¼ r dA,

gðx, nÞzrx ðx, nÞrz ðx, nÞ dAðxÞ dAðxÞ,

I4 ðx, xÞ ¼

A A

I5 ðx, xÞ ¼ I6 ðx, xÞ ¼ I7 ðx, xÞ ¼

Z Z

ZA ZA ZA ZA A A

I8 ðx, xÞ ¼ I9 ðx, xÞ ¼

Z Z

ZA ZA

gðx, nÞzzrx2 ðx, nÞ dAðxÞ dAðxÞ, gðx, nÞz2 rx2 ðx, nÞ dAðxÞ dAðxÞ, gðx, nÞðx, nÞzrx2 dAðxÞ dAðxÞ, gðx, nÞzrx2 ðx, nÞ dAðxÞ dAðxÞ,

0

0

rz ðx, nÞ ¼

zz zz ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : :nx: 2 ðxxÞ þðcyÞ2 þ ðzzÞ2

T

Z Z V

0

V V Z T

Z

L

0

0

T Z

0

0

where K and F denote the kinetic and the elastic potential energy of the body, respectively, whereas W is the work done by the external forces.

L

Z Z

½qx dux þ qz duz  dAðxÞ dAðxÞ dx dx dt

A A

0 L

½R1 du þR2 dv þR3 dj dx dt

¼

The dynamic response of non-local beams has been generally investigated based on the same concepts used for static analysis [41,60–63]. In this appendix, the equations of motion of a Timoshenko beam in presence of long-range interactions are derived by applying Hamilton’s principle in conjunction with the above described mechanically based model of non-local elasticity. It is assumed that the beam is subjected to timevarying external body forces per unit length along the x-axis denoted by Fx(x,t) and Fz(x,t). As well-known, Hamilton’s principle states that, among all admissible dynamic paths, the actual motion of a beam with prescribed configurations at the time instants t ¼0 and t¼T, in the interval 0 oxoL, is the one satisfying the following condition: Z T ½KðFW Þ dt ¼ 0 dH ¼ d ðB:1Þ

Z

¼ Z

Appendix B

ðB:5Þ

V

where use of Eqs. (5) and (19) has been made. Upon taking into account Eq. (29a–c) and carrying out appropriate manipulations, the second integral on the r.h.s. of Eq. (B.5) can be rewritten as Z TZ Z ½qx dux þ qz duz  dVðxÞ dVðnÞ dt 0

ðA:2a; bÞ

½qx dux þ qz duz  dVðxÞ dVðnÞ dt



ðA:1a2iÞ

where dA(x)¼dy dz and dA(x)¼ dc dz; g(x,n) is a distance-decaying real-valued attenuation function; rx(x,n) and rz(x,n) are given by

xx xx ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , :nx: 2 ðxxÞ þ ðcyÞ2 þ ðzzÞ2

ðB:4Þ

0

Z

A A

rx ðx, nÞ ¼

rz2 dA: A

Notice that, if it is assumed that the configurations of the beam at the time instants t ¼0 and T are prescribed, i.e. du(x,0) ¼ du(x,T) ¼0, dv(x,0)¼ dv(x,T)¼ 0, dj(x,0)¼ dj(x,T) ¼0 in the interval 0ox oL, then the integrand of the last term on the right-hand side of Eq. (B.3) vanishes. The elastic potential energy F is given by Eq. (31) and its first variation on the time interval [0,T] can be expressed as follows: Z TZ L Z T d F dt ¼ ½E Aede þE Iwdw þKs G Agdg dxdt

gðx, nÞrz2 ðx, nÞ dAðxÞ dAðxÞ,

gðx, nÞrx ðx, nÞrz ðx, nÞ dAðxÞ dAðxÞ,

Z

Ir ¼

A

A A

Z Z

685

ðB:6Þ

0

where dA(x)¼dy dz and dA(x) ¼dc dz, while the functions R1(x,t), R2(x,t) and R3(x,t) are defined as in Eqs. (34a–c). Then, taking into account the strain–displacement relations (26a–c), performing integration by parts on the first integral and introducing Eq. (B.6), Eq. (B.5) can be recast in the following form: ! Z TZ L( Z T @2 u @2 v @j d Fdt ¼  E A 2 du þ Ks G A  dv @x @x2 @x 0 0 0 ) " #   @2 j @v j d j dxdt þ E I 2 þ Ks G A @x @x    x ¼ L Z T @u @v @j j dv þ E I E A du þKs G A dj dt þ @x @x @x 0 x¼0 Z

T

Z

 0

L

  R1 du þR2 dv þR3 dj dx dt:

ðB:7Þ

0

The virtual work done by the external loads Fx(x,t) and Fz(x,t) on the time interval [0,T] is given by Z T Z TZ L d W dt ¼ ½Fx du þ Fz dv dx dt 0

0

0

686

M. Di Paola et al. / International Journal of Mechanical Sciences 53 (2011) 676–687

Z

T

½N0 duð0,tÞ þ T0 dvð0,tÞ þ M0 djð0,tÞ þ NL duðL,tÞ

þ 0

þ TL dvðL,tÞ þML djðL,tÞ dt

ðB:8Þ

where Ni, Ti and Mi (i¼0,L) are the axial force, transverse force and bending moment at the two ends of the beam, i.e. at x¼ 0 and L. Taking into account Eqs. (B.3), (B.7) and (B.8) into Eq. (B.1), in view of the arbitrariness of du, dv and dj in the interval 0ox oL, the statement of Hamilton’s principle yields the following Euler– Lagrange equations: Z L @2 uðx,tÞ @2 uðx,tÞ E A þ I1 ðx, xÞ½uðx,tÞuðx,tÞdx þFx ðx,tÞ ¼ m , 2 @x @t 2 0 " # Z L @2 vðx,tÞ @jðx,tÞ Ks G A  fI2 ðx, xÞjðx,tÞ þ I3 ðx, xÞjðx,tÞ þ 2 @x @x 0  

@2 vðx,tÞ , þ I4 ðx, xÞ vðx,tÞvðx,tÞ dx þ Fz ðx,tÞ ¼ m @t 2   @2 jðx,tÞ @vðx,tÞ jðx,tÞ E I þ Ks G A @x @x2 Z L fI5 ðx, xÞjðx,tÞI6 ðx, xÞjðx,tÞI3 ðx, xÞ½vðx,tÞvðx,tÞg dx þ 0

@2 jðx,tÞ ¼ Ir : @t 2

ðB:9Þ

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