Exact solutions for the static bending of Euler-Bernoulli beams using Eringen ' s two-phase local / nonlocal model

Though widely used in modelling nano- and micro- structures, Eringen’s differential model shows some inconsistencies and recent study has demonstrated its differences between the integral model, which then implies the necessity of using the latter model. In this paper, an analytical study is taken to analyze static bending of nonlocal Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. Firstly, a reduction method is proved rigorously, with which the integral equation in consideration can be reduced to a differential equation with mixed boundary value conditions. Then, the static bending problem is formulated and four types of boundary conditions with various loadings are considered. By solving the corresponding differential equations, exact solutions are obtained explicitly in all of the cases, especially for the paradoxical cantilever beam problem. Finally, asymptotic analysis of the exact solutions reveals clearly that, unlike the differential model, the integral model adopted herein h...


I. INTRODUCTION
The nonlocal elasticity advanced by Eringen and his co-workers [1][2][3][4] is based on the hypothesis that the nonlocal stress at a reference point depends not only on the strain at that point but also on all other points of the body.Consequently, in the constitutive equation of nonlocal elasticity Hooke's law (for the classical elasticity) is replaced by an integration.Such integral form constitutive relations account for the forces between atoms and internal length scale, and have been applied to many problems including wave propagation, fracture mechanics, dislocations, etc.
In recent years, with the growing need in designing or analysing size-dependent materials and structures, Eringen's nonlocal elasticity received much attention as reviewed by Arash and Wang. 5 Since it is usually difficult to deal with integro-differential equations, in the literature an approximate differential model once proposed in Eringen 6 for a specific integral kernel is widely used to incorporate nonlocal effects.][9][10][11] To be specific, for all boundary conditions except the cantilever, the model has a softening effect (i.e., large deflections and lower fundamental frequencies) as the nonlocal parameter increases.While for a cantilever beam with concentrated load, there is no nonlocal effect.As far as the authors are aware of, such inconsistences were further clarified most recently in three important pieces of work [12][13][14] which resorted to Eringen's integral models.In the literature, the integral models mainly include the pure nonlocal model 6 and the two-phase local/nonlocal model, 15 and the former (with a specific kernel ) can be viewed as the original integral model of the aforementioned differential model. 6In Khodabakhshi and Reddy, 12 a unified integro-differential nonlocal model, which resembles the two-phase local/nonlocal model, was presented.Numerical study of the nonlocal Euler-Bernoulli beam showed that, other than the simply supported case which showed a stiffening effect, the nonlocal beam has a softening effect for most boundary conditions.More importantly, with this model the paradoxical cantilever beam problem was resolved.Further, it was shown in Fernández-Sáez et al. 13 that, in general, the differential model is not equivalent to its original integral model (i.e., the corresponding pure nonlocal model): the solution of the original integral model actually is obtained by adding those of the differential model and of other two integral equations.The contrasts were illustrated through the static bending analysis of Euler-Bernoulli beam, and the paradox that appears when solving the cantilever beam problem with differential model was also resolved numerically.In Tuna and Kirca, 14 using the pure nonlocal model as in Fernández-Sáez et al., 13 by Laplace transformation exact solutions were claimed to be obtained for both Euler-Bernoulli and Timoshenko beams, and the analytical results showed that the integral model has a consistent softening effect.
Because of the inconsistences in the differential model and some promising results reported by the integral models, especially for the cantilever beam problem which would have many applications in science, there is a great need to investigate the integral models.For structures with Eringen's integral models, numerical methods are usually applied in the literature to deal with the resulting integro-differential equations, 12,13,[16][17][18] which are shown to be time consuming and the numerical errors are sometimes hard to detect for some specific nonlocal kernels. 120][21] Due to the difficulties in dealing with integro-differential equations, analytical solutions of problems in nonlocal elasticity are very few.Most of the analytical solutions were concerned with a nonlocal bar problem firstly proposed in Pisano and Fuschi. 22The closed form solution there were shown to have considerable error, and more accurate analytical solutions were then presented for this problem. 23,24It should be noted that, before any progress can be made in this direction, the aforementioned analytical solutions are only valid for some specific loadings, and thus can not be applied to more complicated cases (say, boundary conditions and loadings).The exact solutions for the pure nonlocal model in Tuna and Kirca 14 as they claimed, however, would incur some unknown errors, and we shall discuss it in Section V. To summarize, there are very few exact solutions neither for the two-phase local/nonlocal model, nor for the pure nonlocal model.So, more comprehensive analytical studies of Eringen's integral models as applied to nano-and micro-structures would be desirable.These have become the motivations of the work.
In this paper, we adopt the two-phase local/nonlocal model 15,25 and aim at searching for the analytical solutions to the bending problem of nonlocal Euler-Bernoulli beams.By reducing the integral equation to a differential equation with mixed boundary conditions, we manage to obtain exact solutions of the beam problem under several typical boundary conditions, especially for the paradoxical cantilever case.Analysis of the solutions shows that, as far as the examples examined, the two-phase local/nonlocal model has a consistent softening effect.Also, with the analytical results obtained, we find that the numerical solutions in Khodabakhshi and Reddy 12 encountered some numerical errors for simply supported beam, the solutions in Tuna and Kirca 14 can be viewed as "limit solutions" of our exact solutions and the possible error in those solutions can be discussed, and as a byproduct, the once controversial nonlocal bar problem 22 is well resolved by the reduction method proved here.The layout of the remaining part is as follows.In Section II, we review briefly Eringen's nonlocal theory and provide the key method in our latter derivations.In Section III, we set up the governing equations for static bending of nonlocal Euler-Bernoulli beam using Eringen's two-phase local/nonlocal model and discuss the applicability of the method.Exact solutions are obtained for various types of boundary conditions and loadings in Section IV.In Section V, some analysis will be taken to the exact solutions obtained, and comparisons will be made with existing analytical and numerical results.In Section VI, we draw conclusions and discuss the future work.

II. ERINGEN'S NONLOCAL ELASTICITY AND METHOD DESCRIPTION
According to Eringen, 25 the long rang force in a linear homogeneous and isotropic elastic material as a consequence of a strain field, is expressed by the following constitutive relation where t(x) is the nonlocal stress tensor, σ k l is the classical stress tensor at x ′ , V is the region occupied by the body.takes the following form where ξ 1 > 0, ξ 2 > 0 and satisfy ξ 1 + ξ 2 = 1.Equation ( 2) can be viewed as a constitutive relation of a two-phase elastic material, in which phase 1 (of volume fraction ξ 1 ) complies with local elasticity, phase 2 (of volume fraction ξ 2 ) complies with nonlocal elasticity.It is usually typed as two-phase local/nonlocal model.As a first attempt, in this paper we adopt the two-phase local/nonlocal model (2) and devote to deriving analytical solutions to bending problems of nonlocal Euler-Bernoulli beam.We consider a beam of length L, with uniform cross section S, Young modulus E and subjected to distributed transverse load q(x) at the top.The coordinate system is introduced as: x coordinate is taken along the length of the beam and z coordinate is taken along the height of the beam, while w denotes the transverse displacement of the middle plane (i.e., z = 0).For a beam problem, we make the assumptions that the nonlocal behavior in the thickness direction is negligible.As commonly used in the literature, by taking the normalized bi-exponential kernel in Eringen, 6 the nonlocal stress along x direction then can be expressed by its local counterpart as where L denotes the length of the beam.
Remark 2.1.In the literature, both of the models (1)(2) have been investigated either numerically 12,13,[16][17][18]21 or analytically for a nonlocal bar problem [22][23][24] and for nonlocal beam problems. 14 t should be noted that for the bi-exponential kernel adopted here, by choosing ξ 1 = 0 constitutive equation (3) then reduces to the corresponding pure nonlocal model It has been demonstrated (for the case of axial behavior of bars 24 and bending of Euler-Bernoulli beams 13 ) that in the literature there is an improper transformation from this pure nonlocal model into the following differential model From a mathematical point of view, models (4)(3) (also generally for (1)(2)) can be typed as Fredholm integral equations of the first kind and the second kind respectively, whose solutions are usually very different.In our latter derivations, it will be further shown that the exact solutions in Tuna and Kirca 14 with model ( 4) can actually be viewed as "limit solution" (i.e., ξ 1 → 0) from our exact solutions.Unfortunately, such solutions may not be the exact solutions to model (4).
To begin with, we present the following proposition, which is the key in our method.
Proposition 2.1.Consider the following linear integral equation of the second kind , twice differentiable) and the parameters λ < 0, A > 0, then the solution y(x) to the integral equation also belongs to C 2 [a, b] and can be uniquely determined by the following differential equation with mixed boundary value conditions The proof is reported in Appendix A. With this "reduction method", for any given initial data that satisfy the assumptions, one can solve the differential equation instead of the integral equation.It should be noted that the method is brought from Polyanin and Manzhirov, 26 and suits for the problem in consideration.Such a procedure was also applied to get the analytical solution of a nonlocal bar problem previously, 24 while without further justifications of its applicability.Regarding the proposition, we make the following comments according to nonlocal elasticity problems.
1.The reduction method requires it seems that the method can not be applied directly, and some related numerical studies have been done. 17As it is reported in the appendix, and one can even deduce that y(x) shares the same regularity with f (x).This is true for the adopted kernel, but it is not always the case in nonlocal elasticity, especially for the pure nonlocal model (1).For instance, the constitutive equation with Gaussian kernel As the integral kernel has the effect of mollification, then for any integrable function σ(x), the nonlocal stress is smooth.Consequently, for any given function t(x) which is smooth, we can not claim that σ(x) is smooth.Or, for the problem in which t(x) is not smooth enough, there would be no solution for σ(x).This issue is also addressed before. 21. The result that y(x) is unique depends on the choice of the parameters.Though it is sufficient for the problem in hand, the condition λ < 0, A > 0 can be made stronger mathematically.As it had been investigated for general theory of well-posedness in nonlocal theory, 19,20 the uniqueness of y(x) here essentially relies on the relations between λ < 0, A > 0 and eigenvalues κ of the corresponding integral operator  b a e λ|x−s| y(s)ds = κ y(x).
3. In some of the cases as considered in the literature numerically, 12,16,17,22 the differential equation can be solved explicitly for given functions f (x).Thus, we can get the analytical solutions, which once is a very difficult task.As to the mixed boundary conditions (8), we can consider it as consistent requirements.While from another point of view, by constructing Green function (Tricomi 27 can be consulted) of the differential operator in equation ( 7) with boundary conditions (8), integral equation ( 6) can be recovered.Thus, the boundary conditions can be viewed as an intrinsic condition in the integral equation, and which, in turn tells us that boundary conditions (8)  are requisites for the unknown solution y(x) in the problem.

III. GOVERNING EQUATIONS
In this section, we adopt constitutive equation ( 3) and set up the governing equation for the nonlocal Euler-Bernoulli beam.Also, we make some discussions about the applicability of the method described in last section.
The strain-displacement relation of Euler-Bernoulli beam can be given as By constitutive equation ( 3), nonlocal stress t x x can be expressed by and the corresponding bending moment is where I =  S z 2 dS.Note that the principle of virtual displacements 28,29 is independent of constitutive models, the equations expressed in terms of stress resultants are valid for local or nonlocal models.So, we can apply it to derive the governing equation and boundary conditions for the unknown transverse displacement w(x).
The internal virtual work due to δw(x) is The external virtual work is The principle of virtual displacements then gives which can be expressed as Integrating the first term twice by parts, we arrive at By the Fundamental Lemma of Variational Calculus, the Euler equation is or, in terms of displacement (cf.( 13)) 085114-6 Wang, Zhu, and Dai AIP Advances 6, 085114 (2016) The boundary conditions involve specifying one element of each of the following two pairs at x = 0 and x = L: For given external load q(x), after integrating both sides of governing equation (20), we can get the following integral equation for y(x)( y(x) = w ′′ (x)) as where C 1 , C 2 are integration constants to be determined.Once y(x) is obtained, the unknown displacement w(x) can be obtained as where C 3 ,C 4 are actually the values for w(0), w ′ (0) respectively.Now, we examine the applicability of the reduction method to our problem.Comparing equation (23) with integral equation ( 6), we have in our case λ = − 1 τ < 0, A = ξ 2 2τξ 1 > 0. Thus, it only requires the right hand side function to be in C 2 [0, L], and hence q(x) should belong to C[0, L] (i.e., continuous), which can be generally satisfied.So, if q(x) is continuous, then according to proposition 2.1 equation ( 23) can be uniquely solved by the corresponding differential equation with the mixed boundary value conditions where . The general solution to equation ( 26) can be expressed as with C 5 ,C 6 being uniquely determined from boundary value conditions (27).That is, if C 1 ,C 2 are known from the boundary conditions, then we can get a unique pair C 5 ,C 6 , or vise versa.So, we actually have four unknowns C 1 ,C 2 ,C 3 ,C 4 as usual, and all of which can be determined from the prescribed boundary conditions.
Remark 3.1.Note that in nonlocal beam theory we usually have integral (nonlocal) boundary conditions (e.g., for clamped-clamped and clamped-pinned ends), however, as the solution to the integral equation can be expressed explicitly, the unknown coefficients can be uniquely determined by solving a linear system as before.The lengthy calculations can be performed with the help of mathematical tools, such as Mathematica. 30

IV. BENDING SOLUTIONS
In this section, we shall derive analytical solutions for static bending problem of nonlocal Euler-Bernoulli beam with various boundary conditions.In all of the examples, the external loads q(x) ∈ C[0, L].
(1). (S-S) Simply supported beam with uniformly distributed load q 0 For a beam simply supported at x = 0 and x = L, the boundary conditions are Using the boundary conditions on the moment, we firstly obtain On substitution of the values for C 1 ,C 2 into equation ( 25) and using equations ( 26) (27), we can get C 5 ,C 6 as ) With the obtained solution for y(x) = w ′′ (x) and the boundary conditions on the displacement, we can get ) .
By equation (24), the exact solution for transverse displacement w(x) can be expressed as By differentiating with respect to x (or symmetry of the solution), we can find that the maximum deflection occurs at x = L/2, and it is given by In order to reveal the nonlocal effect more clearly, we can take some asymptotic analysis to the exact solutions for w(x), w max .Since L/τ is relatively large and 0 < ξ 1 < 1, the O(e −L/( √ ξ 1 τ) ) terms can be neglected.Asymptotic analysis then gives Wang, Zhu, and Dai AIP Advances 6, 085114 (2016) Hereafter, "E.S.T." denotes the O(e −L/( √ ξ 1 τ) ) terms.It can be easily checked by setting ξ 1 = 1 in the exact solution that the classical solution can be fully recovered.
(2). (C-C) Clamped beam with uniformly distributed load q 0 For a beam with clamped ends at x = 0 and x = L, the boundary conditions are The determination of the unknown coefficients are omitted for brevity, expressions for these coefficients will not be presented hereafter and we just give the solutions.The exact solution for transverse displacement w(x) is expressed as The maximum deflection occurs at x = L/2, which can be expressed as After taking some asymptotic analysis to w(x) and w max , we get Wang, Zhu, and Dai AIP Advances 6, 085114 (2016) (3). (C-P) Clamped-Pinned beam with uniformly distributed load q 0 For this case, the boundary conditions are The lengthy expression for the exact solution of w(x) is reported in Appendix B, and the asymptotic solution can be expressed as The deflection of w m at x = (15 − √ 33)L/16 (which is the maximum position in local theory) can be expressed by
(4).(C-F) Cantilever beam with concentrated load or distributed load Case A (CFP): For a cantilever beam with concentrated load P at the free end, the boundary conditions are The exact solution for transverse displacement is ) The maximum deflection occurs at x = L, which can be expressed as The asymptotic solutions for w(x) and w max are To the best of our knowledge, it is the first time that exact solutions to a cantilever beam problem is obtained with Eringen's integral model.

Case B (CFT):
A cantilever beam with triangularly distributed load (1 − x/L)q 0 .The load is obviously nonuniform, 12 but its second derivative is zero (for the purpose of comparing with Case C).The boundary conditions are The exact solution for the transverse displacement is ) ) Hereafter, c denotes L 3 + 3L 2 τ + 6Lξ 1 τ 2 + 6ξ 1 τ 3 .The maximum deflection occurs at x = L, which can be expressed as ) ) After taking some asymptotic analysis to w(x) and w max , we get Wang, Zhu, and Dai AIP Advances 6, 085114 (2016) Case C (CFE): A cantilever beam with exponentially distributed load q 0 e x/L .The load is nonuniform, and its second derivative is nonzero. 11The boundary conditions are the same with those in Case B.
The exact solution for transverse displacement can be obtained and the lengthy expressions are listed in Appendix B. The asymptotic expressions can be expressed as and the maximum deflection occurs at x = L, which can be expressed as

V. ANALYSIS AND COMPARISONS OF SOLUTIONS
In this section, we shall examine the exact solutions obtained in last section more closely, which mainly includes nonlocal effect of the model, the effect of the boundary conditions, the relations with existing numerical and analytical solutions.The normalized deflections w (with respect to the maximum deflection of the local beam) of the beam for τ/L = 0.01, 0.03, 0.05 with ξ 1 = 0.1 in each case are shown in Figures 1,2,3,4,5,6, respectively (denoted by Group I).The normalized maximum deflections w max (with respect to the maximum deflection of the local beam) for different values of τ/L and ξ 1 in each case are shown in Figures 7,8,9,10,11,12, respectively (denoted by Group II).Note that the maximum deflection of the local beam in each case is as follows [10][11][12] S-S: w max = 5q 0 L 4 384EI ; C-C: w max = q 0 L 4 384EI ; C-P: w max = q 0 L 4 65536EI ( 39 + 55 1. nonlocal effects and boundary conditions.From the figures in Group I, it is clear that in all of the cases the deflection of the nonlocal beam is larger than that of the local one, and an increase in τ/L would increase the deflection of the beam, i.e., a softening effect.This can be also found from the solutions, especially the asymptotic one: the additional terms incorporated with AIP Advances 6, 085114 ( 2016 parameter τ/L contribute to nonlocal effect, especially the positive leading order term concerned with O((τ/L) 2 ) in the maximum deflection strongly shows the softening effect.The corresponding curve for τ/L = 0.01 for S-S case is very close to the local one, and we provide a partial enlarged view to show the behavior near the mid-span.It should be pointed out that the softening effect predicted by the analytical solutions in this case is opposite to that in Khodabakhshi and Reddy. 12In that paper, a FEM analysis is taken and the nonlocal beam is found to be slightly stiffer than the local one.This may due to the numerical error (some other comparisons can also be made 13,14 ).A comparison among figures in Group I then tells that nonlocal effect for each boundary condition from largest to least generally would be: C-C, C-P, C-F, S-S.From the figures in Group II, it is clear that a decrease in ξ 1 would lead to an increase in the deflection of the beam.Moreover, the dependence of the maximum deflection on ξ 1 for S-S is nearly linear, while for the other cases it is nonlinear.This can also be seen from the leading order terms in the maximum deflections.Thus, in general, higher τ/L and lower ξ 1 both denote stronger softening effect in the nonlocal beam for all of the boundary conditions.2. Cantilever beam: effect of loadings.Of interest here we have considered the cantilever beam with three types of loadings: concentrated load at the free end, nonuniformly distributed load whose second derivative is either zero or nonzero.In all of the cases, the model shows a consistent softening effect.It should be mentioned that, though the results are not presented here, for a cantilever beam with uniformly distributed load there is also a softening effect.The nonlocal effects in the cases CFP and CFT are consistent with the numerical results in Khodabakhshi and Reddy, 12 where a similar model is adopted.The nonlocal effect for CFE is opposite to that in Cheng et al. 11 (latter discussed by Preethi et al. 31 combined with surface stress effects), where the differential model ( 5) is used and a stiffening effect is found.It may be partially due to the integral kernel chosen here.
3. Limit solutions when ξ 1 = 1, ξ 1 → 0. By setting ξ 1 = 1 in the nonlocal solutions obtained in last section, it can be easily checked that local solutions can be recovered in all of the cases.By letting ξ 1 → 0 in the solutions, we can get a set of "limit solutions".For example, for cases S-S and CFP, if we let ξ 1 → 0 in the solutions (cf.equations (35)(48)), then we obtain The above "limit solutions" coincide with the solutions in Tuna and Kirca 14 (equations ( 29) and (34) in that paper).For S-S case, equation (58a) also coincides with the solution by Reddy and Pang 10 using differential model (5).In order to check whether such "limit solutions" w 0 (x) are exact ones for pure nonlocal model (4), we can simply calculate the resulting nonlocal bending moment.That is, we need to check whether the corresponding version for pure nonlocal model (4) of equation ( 23) (i.e., by setting ξ 1 = 0) is satisfied by y(x) = w ′′ 0 (x).We can take the S-S case as an example.Note that, from equation ( 29), the right handside of (23)(59) should be the same, i.e., the bending moment calculated directly from the boundary conditions are both M(x) = q 0 (Lx − x 2 )/2.After substituting y(x) = w ′′ 0 (x) (i.e., equation (58a)) into the left handside of equation (59), the resulting nonlocal bending moment M 0 (x) can be expressed as Obviously, M 0 (x) M(x), especially at the boundaries x = 0, L, and the non-dimensional error is O(τ/L).That is, the boundary conditions are not exactly satisfied, hence the exact solutions for pure nonlocal model (4) can not be recovered by the "limit solutions".As w ′′ 0 (x) = lim ξ 1 → 0 w ′′ (x, ξ 1 ) and w ′′ (x, ξ 1 ) satisfies equation (23) for any ξ 1 > 0, using equations (60) and ( 23 which implies the underlying reason is that the bi-exponential integral kernel here does not allow an exchange between the limit (of ξ 1 ) and the integration.From an integral equation point of view, it means that the limit of the solution to the second kind integral equation ( 23) is not the solution to its first kind version (59), a phenomenon usually encountered in the field of integral equations.It is the same situation for CFP case (probably for all of the other cases).
As the analytical solutions by Tuna and Kirca 14 share the same expressions with the "limit solutions", they are not exact solutions to pure nonlocal model (4), neither.For example, for S-S case in that paper, although the expression in equation (25) for the bending moment meets the boundary conditions, however, if one substitutes equation (29) into equation (7) to compute the resulting bending moment at either x = 0 or x = L, the values are then nonzero and the error is O(τ/L) (κ in that paper has the same meaning as τ).Hence the boundary conditions are not exactly satisfied.Actually, as the numerical solution in Fernández-Sáez et al. 13 showed that, for S-S case, the relative error of such analytical solution at x = L/4 is about 0.1 percent.Moreover, we think some exponential-like terms, which would affect the behavior near the boundaries, may be neglected in such analytical solutions.Thus, the relative error near the boundaries would be even larger.To summarize, we conclude that the "limit solutions" (or the solutions in Tuna and Kirca 14 ) are not exact solutions for pure nonlocal model (4).However, for relatively small τ/L, they may be good approximations to the inner part of the beam, and it has been shown in Tuna and Kirca. 14

VI. CONCLUSIONS AND FUTURE WORK
In this work, a reduction method in the literature is proved rigorously and applied to study bending problem of nonlocal Euler-Bernoulli beam with various boundary conditions.Eringen's two-phase local/nonlocal model is adopted.As far the examples examined, the following conclusions can be made: 1. Exact solutions are obtained for nonlocal beams with the integral model, especially for the paradoxical cantilever beam problem.
2. For the local/nonlocal model adopted here, through examining several typical boundary conditions and loadings, there is a consistent softening effect.
The analytical solutions are validated through comparisons with existing analytical and numerical solutions.It appears that the integral model considered here has some advantages as compared with differential model (5).That is, it has a consistent softening effect for bending, and there is no paradox when solving a cantilever beam problem.So, it would be interesting to investigate the buckling and vibration response of the model.Also, the reduction method proved here can be applied to related analytical studies of this model.
FIG. 1. w of a simply supported beam with uniformly distributed load for different values of τ/L.

)FIG. 5 .
FIG. 5. w of a cantilever beam with triangularly distributed load for different values of τ/L.
FIG. 9. w max of a clamped-pinned beam with uniformly distributed load for different values of ξ 1 and τ/L.
) we can finally express "M 0 (x) M(x)" as  The scalar function α(|x-x ′ |, τ) is called kernel function or influence function, which is positive and decays rapidly with the increase of |x-x ′ |.One additional constraint on the kernel function is that for τ → 0 we obtain Dirac delta function.Due to this property, nonlocal elasticity reverts to classical elasticity in the limit τ → 0. The former constitutive equation is usually referred to as pure nonlocal model.Another constitutive model in the literature ) )