Static bending of granular beam: exact discrete and nonlocal solutions

Abstract

This study is an attempt towards a better understanding of the length scale effects on the bending response of the granular beams. To this aim, a unidimensional discrete granular chain composed of a finite number of rigid grains is studied. It is assumed that shear and rotational interactions exist at the rigid grain interfaces. This granular model can be classified also as a discrete Cosserat chain with two independent degrees of freedom (DOF) for each grain (the deflection and the rotation). Subsequently, such a discrete model permits to introduce the size effect (grain dimension) in the bending formulation of a microstructured granular beam. It is shown that the bending deformation solutions of this chain asymptotically converge towards the continuum beam model of Bresse–Timoshenko (neglecting the length scale). The exact solutions of this granular model subjected to a uniform distributed loading, are investigated for various boundary conditions which are defined at the grain level. Accordingly, a twin numerical problem is studied to compare the exact analytical results with the numerical ones simulated by discrete element method (DEM). Eventually, through the continualization of the coupled difference equations system governing the discrete beam, a nonlocal elasticity Cosserat continuum model is obtained. The process of continualization consists in approaching the difference equations by differential equations applied either by the polynomial or the rational development in which a length scale appears. It is shown that both the granular model and the nonlocal beam model give very close and eventually coincident results.

Appendices

Appendix A: The general solution of the discrete granular beam

Basically, the general solutions of Eq. (12) could be considered as

$$\begin{aligned} W_{i} = & W_{i}^{h} + W_{i}^{p} ; \\ \theta_{i} = & \theta_{i}^{h} + \theta_{i}^{p} \\ \end{aligned}$$

(63)

Note that $W_{i}^{h}$ and $\theta_{i}^{h}$ are the homogenous solutions of the associated homogenous equations of Eq. (12) and $W_{i}^{p}$ and $\theta_{i}^{p}$ are the particular solutions depending on the loading type. The homogenous parts admit the cubic polynomial solution:

$$\begin{aligned} W_{i}^{h} = & A_{1} + B_{1} \left( {ai} \right) + C_{1} \left( {\frac{{a^{2} }}{2}i^{2} } \right) + D_{1} \left( {\frac{{a^{3} }}{6}i^{3} } \right); \\ \theta_{i}^{h} = & A_{2} + B_{2} \left( {ai} \right) + \overline{C}\left( {\frac{{a^{2} }}{2}i^{2} } \right) + D_{2} \left( {\frac{{a^{3} }}{6}i^{3} } \right) \\ \end{aligned}$$

(64)

where $A_{i} , B_{i} , C_{i}$ and $D_{i}$ are constants. Equation (64) could be simplified as follows by substituting in the homogenous difference equation system of Eq. (5).

$$\begin{aligned} W_{i}^{h} = & W_{0} + \left( {a\theta_{0} + \left( {\frac{a}{6} - \frac{{2k_{r} }}{{k_{s} a}}} \right)\beta } \right)i + \left( {\frac{a}{2}\alpha } \right)i^{2} + \left( {\frac{a}{3}\beta } \right)i^{3} ; \\ \theta_{i}^{h} = & \theta_{0} + \alpha i + \beta i^{2} \\ \end{aligned}$$

(65)

where $W_{0} ,\theta_{0}$, $\alpha$ and $\beta$ are constants that are obtained through the boundary conditions. A particular solution of Eq. (11) for a uniform loading can be found as:

$$\begin{aligned} W_{i}^{p} = & \left( {\frac{{a^{2} Q}}{{24k_{r} }} - \frac{Q}{{2k_{s} }}} \right)i^{2} + \frac{{a^{2} Q}}{{24k_{r} }}i^{4} ; \\ \theta_{i}^{p} = & \frac{aQ}{{6k_{r} }}i^{3} \\ \end{aligned}$$

(66)

Appendix B: Exact solutions of the granular beam for C–S, C–C and C–F boundary conditions

• 1. Clamped–Simply (C–S) Supported Granular Beam

Considering the clamped condition located at the left and the simply support boundary at the right end. Thus, the boundary conditions for such a beam are given by

$$\begin{aligned} & W_{0} = 0,\,\,\theta_{0} = 0;\,\,W_{n} = 0, \\ & M_{n - 1/2} - \left( \frac{a}{2} \right)V_{n - 1/2} = 0 \to - ak_{s} W_{n - 1} - \frac{{a^{2} }}{2}k_{s} \left( {\theta_{n - 1} + \theta_{n} } \right) + 2k_{r} \left( {\theta_{n - 1} - \theta_{n} } \right) = 0 \\ \end{aligned}$$

(67)

By replacing the general solutions of the discrete beam (Eq. (13)) into the aforementioned set of exact boundary conditions, the deflection and rotation can be obtained readily by:

$$\begin{aligned} W_{i} = & \left( {\left( {\frac{{2k_{r} }}{{k_{s} a^{2} }} - \frac{1}{6}} \right)\frac{{a^{2} nQ\left( {5k_{s} L^{2} - 2k_{s} a^{2} + 12k_{r} } \right)}}{{4k_{r} \left( {4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} } \right)}}} \right)i \\ & \quad + \left( {\frac{{6k_{s} a^{2} n^{4} - 2k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} }}{{24k_{r} \left( {4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} } \right)}}a^{2} Q} \right)i^{2} \\ & \quad - \left( {\frac{{a^{2} nQ\left( {5k_{s} L^{2} - 2k_{s} a^{2} + 12k_{r} } \right)}}{{12k_{r} \left( {4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} } \right)}}} \right)i^{3} + \left( {\frac{{a^{2} Q}}{{24k_{r} }}} \right)i^{4} + \left( {\frac{{a^{2} Q}}{{24k_{r} }} - \frac{Q}{{2k_{s} }}} \right)i^{2} ; \\ \theta_{i} = & \frac{{6k_{s} a^{2} n^{4} - 2k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} }}{{12k_{r} \left( {4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} } \right)}}aQi - \frac{{anQ\left( {5k_{s} a^{2} n^{2} - 2k_{s} a^{2} + 12k_{r} } \right)}}{{4k_{r} \left( {4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} } \right)}}i^{2} + \frac{aQ}{{6k_{r} }}i^{3} \\ \end{aligned}$$

(68)

For an infinite number of grains, the aforementioned discrete solutions could be compared well by the ones of Wang et al. [44] as follows

$$\begin{aligned} W\left( x \right) = & \left( {\frac{{Lq\left( {5{\mathcal{K}}GAL^{2} + 12EI} \right)}}{{2{\mathcal{K}}GA\left( {4{\mathcal{K}}GAL^{2} + 12EI} \right)}}} \right)x + \left( {\frac{{6{\mathcal{K}}GAL^{4} }}{{24EI\left( {4{\mathcal{K}}GAL^{2} + 12EI} \right)}}q} \right)x^{2} \\ & \quad - \left( {\frac{{Lq\left( {5{\mathcal{K}}GAL^{2} + 12EI} \right)}}{{12EI\left( {4{\mathcal{K}}GAL^{2} + 12EI} \right)}}} \right)x^{3} + \left( \frac{q}{24EI} \right)x^{4} + \left( { - \frac{q}{{2{\mathcal{K}}GA}}} \right)x^{2} ; \\ \theta \left( x \right) = & \frac{{6{\mathcal{K}}GAL^{4} }}{{12EI\left( {4{\mathcal{K}}GAL^{2} + 12EI} \right)}}qx - \frac{{L\left( {5{\mathcal{K}}GAL^{2} + 12EI} \right)}}{{4EI\left( {4{\mathcal{K}}GAL^{2} + 12EI} \right)}}x^{2} + \frac{q}{6EI}x^{3} \\ \end{aligned}$$

(69)

Once the solutions of the discrete beam have been obtained, the shear and bending interactions of the boundary grains could be expressed readily by using Eq. (68) in Eq. (7) as follows

$$\begin{aligned} V_{1/2} = & \frac{Q}{2}\left( {\frac{{5k_{s} a^{2} n^{2} - 2k_{s} a^{2} + 12k_{r} }}{{4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} }}n - 1} \right); \\ M_{1/2} = & \frac{aQ}{4}\left( {\frac{{n\left( { k_{s} a^{2} n^{2} \left( {2n - 5} \right) - 2k_{s} a^{2} \left( {n - 1} \right) - 12k_{r} } \right)}}{{4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} }} + 1} \right); \\ V_{n - 1/2} = & - \frac{Q}{2}\left( {\frac{{3n\left( {k_{s} a^{2} n^{2} + 4kr} \right)}}{{4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12kr}} - 1} \right); \\ M_{n - 1/2} = & - \frac{aQ}{4}\left( {\frac{{3n\left( {k_{s} a^{2} n^{2} + 4kr} \right)}}{{4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12kr}} - 1} \right) \\ \end{aligned}$$

(70)

The reaction forces could be obtained through the equilibrium conditions of the granular beam as follows

$$\begin{aligned} & F_{ry}^{1} - \frac{Q}{2} - V_{1/2} = 0,\,\,M_{rz}^{1} - \left( \frac{a}{2} \right)V_{1/2} - M_{1/2} = 0; \\ & F_{ry}^{2} - \frac{Q}{2} + V_{n - 1/2} = 0,\,\,M_{n - 1/2} - \left( \frac{a}{2} \right)V_{n - 1/2} = 0 \\ \end{aligned}$$

(71)

which leads to

$$\begin{aligned} F_{ry}^{1} = & \frac{{5k_{s} a^{2} n^{2} - 2k_{s} a^{2} + 12k_{r} }}{{4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} }}\left( \frac{nQ}{2} \right),\,\,M_{rz}^{1} = \frac{{an^{2} Q\left( { k_{s} a^{2} n^{2} - k_{s} a^{2} } \right)}}{{8k_{s} a^{2} n^{2} - 2k_{s} a^{2} + 24k_{r} }}; \\ V_{n - 1/2} = & \frac{{k_{s} a^{2} n^{2} + 4kr}}{{4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12kr}}\left( \frac{3nQ}{2} \right) \\ \end{aligned}$$

(72)

Thus, the bending moment and shear equations are given by

$$\begin{aligned} V_{i + 1/2} = & - Q\left( {i + \frac{1}{2} - \frac{{5k_{s} a^{2} n^{2} - 2k_{s} a^{2} + 12k_{r} }}{{8k_{s} a^{2} n^{2} - 2k_{s} a^{2} + 24k_{r} }}n} \right); \\ M_{i + 1/2} = & \frac{aQ}{2}\left( {\left( {i + \frac{1}{2}} \right)^{2} - \frac{{5k_{s} a^{2} n^{2} - 2k_{s} a^{2} + 12k_{r} }}{{4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} }}n\left( {i + \frac{1}{2}} \right) + \frac{{n^{2} \left( { k_{s} a^{2} n^{2} - k_{s} a^{2} } \right)}}{{4k_{s} a^{2} n^{2} - k_{s} a^{2} + 12k_{r} }} + \frac{1}{4}} \right) \\ \end{aligned}$$

(73)

Replacing the continuum terms and neglecting the length scale leads to following continuum local equations

$$\begin{aligned} V\left( x \right) = & - q\left( {x - \frac{{5{\mathcal{K}}GAL^{2} + 12EI}}{{8{\mathcal{K}}GAL^{2} + 24EI}}L} \right); \\ M\left( x \right) = & \frac{q}{2}\left( {x^{2} - \frac{{5{\mathcal{K}}GAL^{2} + 12EI}}{{4{\mathcal{K}}GAL^{2} + 12EI}}Lx + \frac{{L^{2} \left( {{\mathcal{K}}GAL^{2} } \right)}}{{4{\mathcal{K}}GAL^{2} + 12EI}}} \right) \\ \end{aligned}$$

(74)

These results converge to the ones of the C–S Timoshenko beam [44].

• 2. Clamped–Clamped (C–C) Granular Beam

  The exact conditions of the clamped ends beam can be considered with

  $$\begin{aligned} & W_{0} = 0,\theta _{0} = 0; \\ & W_{n} = 0,\theta _{n} = 0 \\ \end{aligned}$$

  (75)

  Replacing the general for deflection and rotation of Eq. (13) in boundary conditions (75) leads to:

  $$\begin{aligned} W_{i} = & \left( {\left( {\frac{{2k_{r} }}{{k_{s} a}} - \frac{a}{6}} \right)\frac{{anQ}}{{4k_{r} }}} \right)i + \left( {\frac{{a^{2} n^{2} Q}}{{24k_{r} }}} \right)i^{2} - \left( {\frac{{a^{2} nQ}}{{12k_{r} }}} \right)i^{3} + \left( {\frac{{a^{2} Q}}{{24k_{r} }}} \right)i^{4} + \left( {\frac{{a^{2} Q}}{{24k_{r} }} - \frac{Q}{{2k_{s} }}} \right)i^{2} ; \\ \theta _{i} = & \frac{{an^{2} Q}}{{12k_{r} }}i - \frac{{anQ}}{{4k_{r} }}i^{2} + \frac{{aQ}}{{6k_{r} }}i^{3} \\ \end{aligned}$$

  (76)

  For an infinite number of grains, the solutions of the Bresse–Timoshenko beam on clamped ends might be considered as follows

  $$\begin{aligned} W\left( x \right) = & \left( {\frac{Lq}{{2{\mathcal{K}}GA}}} \right)x + \left( {\frac{{L^{2} q}}{24EI}} \right)x^{2} - \left( {\frac{Lq}{{12EI}}} \right)x^{3} + \left( \frac{q}{24EI} \right)x^{4} - \left( {\frac{q}{{2{\mathcal{K}}GA}}} \right)x^{2} ; \\ \theta \left( x \right) = & \frac{{L^{2} q}}{12EI}x - \frac{Lq}{{4EI}}x^{2} + \frac{q}{6EI}x^{3} \\ \end{aligned}$$

  (77)

  The recent results could be compared well with the ones proposed by Wang et al. [44]. The maximum values are given by

  $$\begin{gathered} W_{max} = \left( {\frac{{qL^{4} }}{24EI}} \right)\left[ {\frac{1}{8} - \frac{{a^{2} }}{{4L^{2} }} + \frac{3EI}{{{\mathcal{K}}GAL^{2} }}} \right] < f_{CC}^{\infty } ; \hfill \\ \theta_{max} = \theta \left( {\frac{3 \pm \sqrt 3 }{6}L} \right) = \left( {\frac{3 \pm \sqrt 3 }{{36}}\frac{{qL^{3} }}{EI}} \right)\left[ {\frac{1}{2} - \frac{3 \pm \sqrt 3 }{4} + \left( {\frac{3 \pm \sqrt 3 }{6}} \right)^{2} } \right] \hfill \\ \end{gathered}$$

  (78)

  where $f_{CC}^{\infty }$ represents the maximum displacement of the continuum beam which was obtained by Timoshenko [43] as follows

  $$f_{CC}^{\infty } = \left( {\frac{{qL^{4} }}{24EI}} \right)\left[ {\frac{1}{8} + \frac{3EI}{{{\mathcal{K}}GAL^{2} }}} \right]$$

  (79)

  These two equations Eq. (78) demonstrate that the length scale influence only the maximum values of beam deflection. This predicts that the granular beam behaves more rigidly than the equivalent local continuum one. The interaction shear and bending moment could be obtained for the boundary grains by using Eq. (76) in the definitions of Eq. (7)

  $$\begin{aligned} V_{1/2} = & \frac{Q}{2}\left( {n - 1} \right),\,\,M_{1/2} = - \frac{aQ}{4}\left( { - \frac{{n^{2} }}{3} + n - \frac{2}{3}} \right) \\ V_{n - 1/2} = & - \frac{Q}{2}\left( {n - 1} \right),\,\,M_{n - 1/2} = - \frac{aQ}{4}\left( { - \frac{{n^{2} }}{3} + n - \frac{2}{3}} \right) \\ \end{aligned}$$

  (80)

  Also, the reaction forces of the boundary would be obtained by using the equilibrium conditions for the boundary grains as follows

  $$\begin{aligned} & F_{ry}^{1} - \frac{Q}{2} - V_{1/2} = 0,\,\,M_{rz}^{1} - \left( \frac{a}{2} \right)V_{1/2} - M_{1/2} = 0; \\ & F_{ry}^{2} - \frac{Q}{2} + V_{{n - \frac{1}{2}}} = 0,\,\, - M_{rz}^{2} + M_{n - 1/2} - \left( \frac{a}{2} \right)V_{n - 1/2} = 0 \\ \end{aligned}$$

  (81)

  Note that $F_{ry}^{1}$ and $M_{rz}^{1}$ are the vertical force and moment reaction of the left clamped end and $F_{ry}^{2}$ and $M_{rz}^{2}$ are the reactions of the right clamped boundary which are obtained by

  $$F_{ry}^{1} = F_{ry}^{2} = \frac{nQ}{2},\,\,M_{rz}^{1} = M_{rz}^{2} = \frac{aQ}{{12}}\left( {n^{2} - 1} \right);$$

  (82)

  The distribution of bending moment and shear forces for clamped ends beam could be found eventually by applying the conditions of Eq. (80) the discrete general solutions of Eq. (18),

  $$V_{i + 1/2} = - Q\left( {i + \frac{1}{2} - \frac{n}{2}} \right),\,\,M_{i + 1/2} = \frac{aQ}{2}\left( {\left( {i + \frac{1}{2}} \right)^{2} - n\left( {i + \frac{1}{2}} \right) + \frac{1}{12} + \frac{{n^{2} }}{6}} \right)$$

  (83)

  It could be concluded that, for an infinite number of grains, the distribution of bending moment and shear forces converge to the ones that refer to the local continuum model of Bresse–Timoshenko as follows [44].

  $$V\left( x \right) = - q\left( {x - \frac{L}{2}} \right), \,\,M\left( x \right) = \frac{q}{2}\left( {x^{2} - Lx + \frac{{L^{2} }}{6}} \right)$$

  (84)

  These equations are almost the same for the Euler–Bernoulli beam subjected to uniformly distributed load.

  • 3. Clamped–Free (C–F) Granular Beam

We assumed here that the 2 DOF of the clamped boundary (for instant left side) are blocked while for the free side, there is no constraint. The reaction forces could be found through the application of the equilibrium equations of the whole system by

$$F_{ry}^{1} - nQ = 0,\,\,M_{rz}^{1} - \left( \frac{an}{2} \right)\left( {nQ} \right) = 0$$

(85)

$F_{ry}^{1}$ and $F_{ry}^{1}$ are respectively the vertical reaction and the bending reaction of the clamped end. Applying the equilibrium conditions to the individual boundary grains

$$\begin{aligned} & F_{ry}^{1} - \frac{Q}{2} - V_{1/2} = 0,\,\,M_{rz}^{1} - \left( \frac{a}{2} \right)V_{1/2} - M_{1/2} = 0; \\ & V_{n - 1/2} - \frac{Q}{2} = 0,\,\,M_{n - 1/2} - \left( \frac{a}{2} \right)V_{n - 1/2} = 0 \\ \end{aligned}$$

(86)

In view of Eq. (85) and Eq. (86), the shear and bending interactions of the boundaries might be obtained by

$$\begin{gathered} V_{1/2} = \frac{Q}{2}\left( {2n - 1} \right),\,\,M_{1/2} = \frac{aQ}{4}\left( {2n^{2} - 2n + 1} \right) \hfill \\ V_{n - 1/2} = \frac{Q}{2}, \,\,M_{n - 1/2} = \left( \frac{a}{2} \right)V_{n - 1/2} = \frac{aQ}{4} \hfill \\ \end{gathered}$$

(87)

Applying these conditions in the discrete general form solutions of the shear and bending moment distribution (Eq. (18)) leads to

$$V_{i + 1/2} = - Q\left( {i + \frac{1}{2} - n} \right),\,\,M_{i + 1/2} = \frac{aQ}{2}\left( {\left( {i + \frac{1}{2}} \right)^{2} - 2n\left( {i + \frac{1}{2}} \right) + \frac{1}{4} + n^{2} } \right)$$

(88)

Ignoring the length scale for an infinite number of grains refers to the Bresse–Timoshenko beam which has the same moment and shear distributions as the Euler–Bernoulli for the C–F conditions. Furthermore, the bending moment and shear equations could be found for the local continuum model as

$$V\left( x \right) = - q\left( {x - L} \right),\,\,M\left( x \right) = \frac{q}{2}\left( {x^{2} - 2Lx + L^{2} } \right)$$

(89)

substituting the shear and bending moment interactions through the kinematics terms (Eq. (7)) into Eq. (86) leads to

$$\begin{aligned} & W_{0} = 0,\,\,\theta_{0} = 0; \\ & k_{r} \left( {\Theta_{n} - \Theta_{n - 1} } \right) - \frac{a}{4}Q = 0,\,\,k_{s} \left( {W_{n} - W_{n - 1} - \frac{a}{2}\left( {\Theta_{n} + \Theta_{n - 1} } \right)} \right) - \frac{1}{2}Q = 0 \\ \end{aligned}$$

(90)

The solutions could be found by replacing the general solutions of Eq. (13) in the aforementioned boundary conditions. Thus, the deflection and micro rotations of the system are given by

$$\begin{aligned} W_{i} = & \left( {\left( {\frac{{2k_{r} }}{{k_{s} a}} - \frac{a}{6}} \right)\frac{ anQ}{{2k_{r} }}} \right)i + \left( {\frac{{6a^{2} n^{2} Q + a^{2} Q}}{{24k_{r} }}} \right)i^{2} - \left( {\frac{{a^{2} nQ}}{{6k_{r} }}} \right)i^{3} + \left( {\frac{{a^{2} Q}}{{24k_{r} }}} \right)i^{4} + \left( {\frac{{a^{2} Q}}{{24k_{r} }} - \frac{Q}{{2k_{s} }}} \right)i^{2} ; \\ \theta_{i} = & \frac{{6aQn^{2} + aQ}}{{12k_{r} }}i - \frac{anQ}{{2k_{r} }}i^{2} + \frac{aQ}{{6k_{r} }}i^{3} \\ \end{aligned}$$

(91)

These converge asymptotically to the ones obtained by Bresse–Timoshenko [43] and Wang et al. [44] for continuum beam assuming an infinite number of grains as follows

$$\begin{aligned} W\left( x \right) = & \left( {\frac{Lq}{{{\mathcal{K}}GA}}} \right)x + \left( {\frac{{L^{2} q}}{4EI}} \right)x^{2} - \left( {\frac{Lq}{{6EI}}} \right)x^{3} + \left( \frac{q}{24EI} \right)x^{4} - \left( {\frac{q}{{2{\mathcal{K}}GA}}} \right)x^{2} ; \\ \theta \left( x \right) = & \frac{{6qL^{2} }}{12EI}x - \frac{Lq}{{2EI}}x^{2} + \frac{q}{6EI}x^{3} \\ \end{aligned}$$

(92)

The maximum deflection and micro angle occur at $i = na$ and obtained as follows

$$\begin{aligned} W_{max} = & \frac{{qL^{4} }}{8EI}\left[ {1 + \frac{4EI}{{{\mathcal{K}}GAL^{2} }}} \right] = f_{CF}^{\infty } ; \\ \theta_{max} = & \frac{{qL^{3} }}{6EI}\left[ {1 + \frac{{a^{2} }}{{2L^{2} }}} \right] \\ \end{aligned}$$

(93)

$f_{CF}^{\infty }$ refers to the maximum displacement of the C–F continuum beam which was obtained by Timoshenko [43]. It could be concluded that the length scale only affects $\theta_{max}$ for clamped–free boundary conditions while $W_{max}$ is independent of the grain dimension. The maximum values of micro-rotation are estimated bigger than the local continuum ones for this case.

Appendix C: Investigating the differential equations of the nonlocal beam

The development of the difference operators is done neglecting the higher-order terms in $a^{4}$ for deflection and rotation field as follows:

$${\mathcal{K}}GA\left( {1 + \frac{{a^{2} D_{x}^{2} }}{12}} \right)D_{x}^{2} W\left( x \right) - {\mathcal{K}}GA\left( {1 + \frac{{a^{2} D_{x}^{2} }}{6}} \right)D_{x} \Theta \left( x \right) = - q$$

(94)

$${\mathcal{K}}GA\left( {1 + \frac{{a^{2} D_{x}^{2} }}{6}} \right)D_{x} W\left( x \right) + \left( {EI\left( {1 + \frac{{a^{2} D_{x}^{2} }}{12}} \right)D_{x}^{2} - {\mathcal{K}}GA\left( {1 + \frac{{a^{2} D_{x}^{2} }}{4}} \right)} \right)\Theta \left( x \right) = 0$$

(95)

Here, a is the characteristic length of the nonlocal model which can be computed from the microstructure cell size (grain diameter for instance). Multiplying Eq. (95) by the term $- \left( {1 - \frac{{a^{2} D_{x}^{2} }}{12}} \right)D_{x}$ with neglecting the higher order terms in $a^{4}$ leads to

$$- {\mathcal{K}}GA\left( {1 + \frac{{a^{2} D_{x}^{2} }}{12}} \right)D_{x}^{2} W\left( x \right) - \left( {EID_{x}^{3} - {\mathcal{K}}GA\left( {1 + \frac{{a^{2} D_{x}^{2} }}{6}} \right)D_{x} } \right)\Theta \left( x \right) = 0$$

(96)

Summing the previous equation with Eq. (94) leads to

$$EI\Theta ^{\prime\prime\prime} = q$$

(97)

On the other hand, the following auxiliary equation could be obtained with the multiplication of Eq. (94) by the term $\left( {1 - \frac{{a^{2} D_{x}^{2} }}{12}} \right)D_{x}^{2}$ and ignoring the higher-order terms in $a^{4}$

$${\mathcal{K}}GAD_{x}^{4} W\left( x \right) - {\mathcal{K}}GA\left( {1 - \frac{{a^{2} D_{x}^{2} }}{12}} \right)D_{x}^{3} \Theta \left( x \right) = - q\left( {1 - \frac{{a^{2} D_{x}^{2} }}{12}} \right)D_{x}^{2}$$

(98)

Thus, the deflection differential equations of the nonlocal system for the displacement could be obtained for a uniform constant distributed loading through the application of relation Eq. (97) as follows

$$EIW^{\prime\prime\prime\prime} = q$$

(99)

Appendix D: Alternative methods for studying nonlocal S–S beam

• 1. Continualization of Discrete Bending Moment

  The nonlocal bending moment and shear distribution of the S–S continuum beam could be obtained from the continualization of Eq. (22) by substituting $x = ai, L = an$ and $Q = qa$ as follows

  $$V\left( x \right) = - q\left( {x - \frac{L}{2}} \right),\,\,M\left( x \right) = \frac{q}{2}\left( {x^{2} - Lx + \frac{{a^{2} }}{4}} \right)$$

  (100)

  According to Eq. (52) and Eq. (94) by considering the corresponding bending moments on the boundaries, the nonlocal conditions could be obtained the same as Eq. (55).

  • 2. Continualization of the Cinematic Boundary Conditions

    This method is based on the continualization of the cinematic boundary conditions presented for the discrete system by Eq. (24). This could be expressed for the nonlocal beam as

    $$\begin{aligned} W\left( 0 \right) = & 0,\quad a{\mathcal{K}}GAW\left( a \right) - \frac{{a^{2} }}{2}{\mathcal{K}}GA\left( {\theta \left( a \right) + \theta \left( 0 \right)} \right) \\ & \quad + 2EI\left( {\theta \left( a \right) - \theta \left( 0 \right)} \right) = 0; \\ W\left( L \right) = & 0,\quad - a{\mathcal{K}}GAW\left( {L - a} \right) - \frac{{a^{2} }}{2}{\mathcal{K}}GA\left( {\theta \left( {L - a} \right) + \theta \left( L \right)} \right) \\ & \quad + 2EI\left( {\theta \left( {L - a} \right) - \theta \left( L \right)} \right) = 0 \\ \end{aligned}$$

    (101)

    Using the general nonlocal continuum solutions of Eq. (49) in the abovementioned conditions reflects the same nonlocal solutions that have been obtained by Eq. (56).

    • 3. Continualization of the Cinematic Boundary Conditions Using Taylor Series

Another approach to continualize the cinematic conditions could be done by applying the polynomial expansions. Developing the difference terms using the Taylor series up to the quartic order $a^{4}$ for displacement and cubic order $a^{3}$ for rotation in Eq. (101) leads to

$$\begin{aligned} W\left( 0 \right) = & 0, \quad a{\mathcal{K}}GA\left( {aW^{\prime}\left( 0 \right) + \frac{{a^{2} }}{2}W^{\prime\prime}\left( 0 \right) + \frac{{a^{3} }}{6}W^{\prime\prime\prime}\left( 0 \right) + \frac{{a^{4} }}{24}W^{\prime\prime\prime\prime}\left( 0 \right)} \right) \\ & \quad - \frac{{a^{2} }}{2}{\mathcal{K}}GA\left( {2\theta \left( 0 \right) + a\theta^{\prime}\left( 0 \right) + \frac{{a^{2} }}{2}\theta^{\prime\prime}\left( 0 \right) + \frac{{a^{3} }}{6}\theta^{\prime\prime\prime}\left( 0 \right)} \right) \\ & \quad + 2EI\left( {a\theta^{\prime}\left( 0 \right) + \frac{{a^{2} }}{2}\theta^{\prime\prime}\left( 0 \right) + \frac{{a^{3} }}{6}\theta^{\prime\prime\prime}\left( 0 \right)} \right) = 0; \\ W\left( L \right) = & 0,\quad - a{\mathcal{K}}GA\left( {aW^{\prime}\left( L \right) + \frac{{a^{2} }}{2}W^{\prime\prime}\left( L \right) + \frac{{a^{3} }}{6}W^{\prime\prime\prime}\left( L \right) + \frac{{a^{4} }}{24}W^{\prime\prime\prime\prime}\left( L \right)} \right) \\ & \quad - \frac{{a^{2} }}{2}{\mathcal{K}}GA\left( {2\theta \left( L \right) - a\theta^{\prime}\left( L \right) + \frac{{a^{2} }}{2}\theta^{\prime\prime}\left( L \right) - \frac{{a^{3} }}{6}\theta^{\prime\prime\prime}\left( L \right)} \right) \\ & \quad - 2EI\left( {a\theta^{\prime}\left( L \right) - \frac{{a^{2} }}{2}\theta^{\prime\prime}\left( L \right) + \frac{{a^{3} }}{6}\theta^{\prime\prime\prime}\left( L \right)} \right) = 0 \\ \end{aligned}$$

(102)

The aforementioned developed conditions again lead to the solutions of Eq. (56).

• 4. Continualization of the Static Boundary Conditions with Cinematic Variables

The equilibrium of the bending moment for the boundaries of the nonlocal beam could be considered by

$$M\left( \frac{a}{2} \right) + \frac{a}{2}V\left( \frac{a}{2} \right) = 0,\,\,M\left( {L - \frac{a}{2}} \right) - \frac{a}{2}V\left( {L - \frac{a}{2}} \right) = 0$$

(103)

Substituting Eq. (50) and Eq. (51) in these equations leads to

$$\begin{aligned} & \left( {EI\Theta^{\prime}\left( \frac{a}{2} \right) + \frac{{a^{2} }}{24}q} \right) + \frac{{a{\mathcal{K}}GA}}{2}\left( {W^{\prime}\left( \frac{a}{2} \right) - \Theta \left( \frac{a}{2} \right) - \frac{{a^{2} }}{12}W^{\prime\prime\prime}\left( \frac{a}{2} \right)} \right) = 0; \\ & \left( {EI\Theta^{\prime}\left( {L - \frac{a}{2}} \right) + \frac{{a^{2} }}{24}q} \right) - \frac{{a{\mathcal{K}}GA}}{2}\left( {W^{\prime}\left( {L - \frac{a}{2}} \right) - \Theta \left( {L - \frac{a}{2}} \right) - \frac{{a^{2} }}{12}W^{\prime\prime\prime}\left( {L - \frac{a}{2}} \right)} \right) = 0 \\ \end{aligned}$$

(104)

The constants of the general solutions of Eq. (49) could be obtained the same as the ones that have been expressed by Eq. (56).

Appendix E: Exact solutions of the nonlocal beam for C–S, C–C and C–F boundary conditions

• 1. Clamped–Simply Nonlocal Model

For C–S boundary conditions, ones could be obtained for the bending moment and shear distribution, through the continualization of the corresponding discrete equations. These are given respectively as follows

$$\begin{aligned} V\left( x \right) = & - q\left( {x - \frac{{5{\mathcal{K}}GAL^{2} - 2{\mathcal{K}}GAa^{2} + 12EI}}{{8{\mathcal{K}}GAL^{2} - 2{\mathcal{K}}GAa^{2} + 24EI}}L} \right), \\ M\left( x \right) = & \frac{q}{2}\left( {x^{2} - \frac{{5{\mathcal{K}}GAL^{2} - 2{\mathcal{K}}GAa^{2} + 12EI}}{{4{\mathcal{K}}GAL^{2} - { \mathcal{K}}GAa^{2} + 12EI}}Lx + \frac{{L^{2} \left( {{\mathcal{K}}GAL^{2} - { \mathcal{K}}GAa^{2} } \right)}}{{4{\mathcal{K}}GAL^{2} - { \mathcal{K}}GAa^{2} + 12EI}} + \frac{{a^{2} }}{4}} \right) \\ \end{aligned}$$

(105)

The nonlocal C–S boundary conditions could be considered in view of Eq. (52) and Eq. (105), by:

$$W\left( 0 \right) = 0;\,\,\Theta \left( 0 \right) = 0;\,\,W\left( L \right) = 0;\,\,\Theta^{\prime}\left( L \right) = \frac{{a^{2} q}}{12EI}$$

(106)

Thus, ones could be obtained for the deflection and rotation of the nonlocal beam by replacing the nonlocal general solutions (Eq. (49))

$$\begin{aligned} W\left( x \right) = & \left( {\left( {\frac{2EI}{{{\mathcal{K}}GA}} - \frac{{a^{2} }}{6}} \right)\frac{{Lq\left( {5{\mathcal{K}}GAL^{2} - 2{\mathcal{K}}GAa^{2} + 12EI} \right)}}{{4EI\left( {4{\mathcal{K}}GAL^{2} - { \mathcal{K}}GAa^{2} + 12EI} \right)}}} \right)x \\ & \quad + \left( {\frac{{6{\mathcal{K}}GAL^{4} - 2{\mathcal{K}}GAa^{2} L^{2} - {\mathcal{K}}GAa^{4} + 12EIa^{2} }}{{24EI\left( {4{\mathcal{K}}GAL^{2} - {\mathcal{K}}GAa^{2} + 12EI} \right)}}q} \right)x^{2} \\ & \quad - \left( {\frac{{Lq\left( {5{\mathcal{K}}GAL^{2} - 2{\mathcal{K}}GAa^{2} + 12EI} \right)}}{{12EI\left( {4{\mathcal{K}}GAL^{2} - {\mathcal{K}}GAa^{2} + 12EI} \right)}}} \right)x^{3} + \left( \frac{q}{24EI} \right)x^{4} \\ & \quad + \left( {\frac{{qa^{2} }}{24EI} - \frac{q}{{2{\mathcal{K}}GA}}} \right)x^{2} ; \\ \theta \left( x \right) = & \frac{{6{\mathcal{K}}GAL^{4} - 2{\mathcal{K}}GAa^{2} L^{2} - {\mathcal{K}}GAa^{4} + 12EIa^{2} }}{{12EI\left( {4{\mathcal{K}}GAL^{2} - {\mathcal{K}}GAa^{2} + 12EI} \right)}}qx \\ & \quad - \frac{{L\left( {5{\mathcal{K}}GAL^{2} - 2{\mathcal{K}}GAa^{2} + 12EI} \right)}}{{4EI\left( {4{\mathcal{K}}GAL^{2} - {\mathcal{K}}GAa^{2} + 12EI} \right)}}x^{2} + \frac{q}{6EI}x^{3} \\ \end{aligned}$$

(107)

These could be also obtained from the continualization of the corresponding discrete solutions (Eq. (68)).

• 2. Clamped–Clamped Nonlocal Model

  The shear and bending distribution of the nonlocal continuum beam could be obtained by continualizing the ones which were found for the discrete system. This could be done by considering the continuum terms $x = ai, L = an$ and $Q = qa$. Accordingly, Eq. (80) leads to

  $$V\left( x \right) = - q\left( {x - \frac{L}{2}} \right),\,\,M\left( x \right) = \frac{q}{2}\left( {x^{2} - Lx + \frac{{a^{2} }}{12} + \frac{{L^{2} }}{6}} \right)$$

  (108)

  For this case, the boundary conditions might be defined the same as the discrete or local ones by

  $$W\left( 0 \right) = 0;\,\,\Theta \left( 0 \right) = 0;\,\,W\left( L \right) = 0;\,\,\Theta \left( L \right) = 0$$

  (109)

  Applying the aforementioned set of boundary conditions in the general solutions of the nonlocal beam (Eq. (49)) leads to:

  $$\begin{aligned} W\left( x \right) = & \left( {\frac{Lq}{{2{\mathcal{K}}GA}}} \right)x + \left( {\frac{{L^{2} q}}{24EI} - \frac{q}{{2{\mathcal{K}}GA}}} \right)x^{2} - \left( {\frac{Lq}{{12EI}}} \right)x^{3} + \left( \frac{q}{24EI} \right)x^{4} + \left( {\frac{{a^{2} q}}{24EI}} \right)\left( {x - L} \right)x; \\ \theta \left( x \right) = & \frac{{L^{2} q}}{12EI}x - \frac{Lq}{{4EI}}x^{2} + \frac{q}{6EI}x^{3} \\ \end{aligned}$$

  (110)

  Similarly, an alternative method to obtain these results is through the continualization of the corresponding discrete solutions of Eq. (76). The maximum displacement occurs at the middle of the beam and is given by

  $$W_{max} = W\left( {L/2} \right) = \left( {\frac{{qL^{4} }}{24EI}} \right)\left[ {\frac{1}{8} - \frac{{a^{2} }}{{4L^{2} }} + \frac{3EI}{{{\mathcal{K}}GAL^{2} }}} \right]$$

  (111)

  It is noteworthy to state that the maximum value is equal to Eq. (78). Although the boundary conditions and the governing differential equations of the local and nonlocal beam are the same, the scale effect appears (only in the displacement equation) in the results. This stems from the nonlocal gradient coupled differential equations system expressed by Eq. (94) and Eq. (95).

  • 3. Clamped–Free Nonlocal Model

Replacing $x = ai, L = an$ and $Q = qa$ in Eq. (88) leads to the moment and shear equations of the nonlocal continuum beam as follows

$$V\left( x \right) = - q\left( {x - L} \right),\,\,M\left( x \right) = \frac{q}{2}\left( {x^{2} - 2Lx + \frac{{a^{2} }}{4} + L^{2} } \right)$$

(112)

For the free boundary, we have $V\left( L \right) = 0$ and $M\left( L \right) = \frac{{a^{2} q}}{8}$. Applying Eq. (50) and Eq. (51) leads to the following nonlocal variational boundary conditions

$$\begin{aligned} W\left( 0 \right) = & 0;\,\,\Theta \left( 0 \right) = 0; \\ \Theta^{\prime}\left( L \right) = & \frac{{a^{2} q}}{12EI} ; W^{\prime}\left( L \right) - \Theta \left( L \right) - \frac{{a^{2} }}{12}W^{\prime\prime\prime}\left( L \right) = 0 \\ \end{aligned}$$

(113)

Also, regarding Eq. (52) and knowing $M^{\prime}\left( L \right) = - V\left( L \right) = 0$, an equivalent boundary conditions could be assumed

$$\begin{aligned} W\left( 0 \right) = & 0;\,\,\Theta \left( 0 \right) = 0; \\ \Theta^{\prime}\left( L \right) = & \frac{{a^{2} q}}{12EI};\,\,\Theta^{\prime\prime}\left( L \right) = 0 \\ \end{aligned}$$

(114)

On the other hand, defining the bending moment and shear force of the free end through the cinematic parameters leads to an alternative set of boundary conditions for C–F nonlocal beam

$$\begin{aligned} & W\left( 0 \right) = 0;\,\,\Theta \left( 0 \right) = 0; \\ & \Theta \left( L \right) - \Theta \left( {L - a} \right) = \frac{{a^{3} q}}{4EI};\,\,W\left( L \right) - W\left( {L - a} \right) - a\frac{{\Theta \left( L \right) + \Theta \left( {L - a} \right)}}{2} = \frac{{a^{2} q}}{{2{\mathcal{K}}GA}} \\ \end{aligned}$$

(115)

Applying the nonlocal beam solutions of Eq. (49) in one of the aforementioned boundary conditions (e.g. Equation (113)) leads to:

$$\begin{aligned} W\left( x \right) = & \left( {\frac{Lq}{{{\mathcal{K}}GA}}} \right)x + \left( {\frac{{L^{2} q}}{4EI} - \frac{q}{{2{\mathcal{K}}GA}}} \right)x^{2} - \left( {\frac{Lq}{{6EI}}} \right)x^{3} + \left( \frac{q}{24EI} \right)x^{4} + \left( {\frac{{a^{2} q}}{12EI}} \right)\left( {x - L} \right)x; \\ \theta \left( x \right) = & \frac{{6L^{2} q + a^{2} q}}{12EI}x - \frac{Lq}{{2EI}}x^{2} + \frac{q}{6EI}x^{3} \\ \end{aligned}$$

(116)

These results coincide with the ones that could be found from the continualization of the discrete solutions which have been presented in Eq. (91). The maximum deflection happens at the free side of the beam and is obtained as follows

$$W_{max} = W\left( L \right) = \frac{{qL^{4} }}{8EI}\left[ {1 + \frac{4EI}{{{\mathcal{K}}GAL^{2} }}} \right]$$

(117)

This equation reflects also the same values as Eq. (93).