Approximate closed-form solutions for vibration of nano-beams of local/non-local mixture

Abstract

This paper presents an approach to natural vibration of nano-beams by a linear elastic constitutive law based on a mixture of local and non-local contributions, the latter based on Eringen’s model. A perturbation in terms of an evolution parameter lets incremental field equations be derived; another perturbation in terms of the non-local volume fraction yields the variation of the natural angular frequencies and modes with the ‘small’ amount of non-locality. The latter perturbation does not need to comply with the so-called constitutive boundary conditions, the physical interpretation of which is still debated. The possibility to find closed-form solutions is highlighted following a thorough discussion on the compatibility conditions needed to solve the steps of the perturbation hierarchy; some paradigmatic examples are presented and duly commented.

Appendix: Details and passages

In the following subsections we will present with more detail some of the passages omitted in the text for sake of space and readability.

1.1 Perturbation expansion

The governing equations for free vibration of Euler-Bernoulli nanobeams composed of the above described mixture of linear local and nonlocal elastic materials in terms of nondimensional quantities are obtained by replacing Eq. (9) into Eq. (8):

$$\begin{aligned} \begin{array}{*{20}{c}} \dfrac{dv}{dz}\!=\!-\Omega ,&{}\left( 1-\xi \right) \dfrac{d\Omega }{dz}\!+\!\xi K*\dfrac{d\Omega }{dz}\!=\!M,\\ \dfrac{dT}{dz}=-\lambda v,&{}\dfrac{dM}{dz}=T, \end{array} \end{aligned}$$

(1)

Inserting the formal power series expansions of the field functions and of the eigenvalue up to the second order in the nonlocal fraction $\xi$ we have

$$\begin{aligned} \begin{array}{*{20}{l}} \dfrac{dv_0}{dz}+\xi \dfrac{dv_1}{dz}+\dfrac{\xi ^2}{2}\dfrac{dv_2}{dz}=\\ \qquad =-\Omega _0-\xi \Omega _1-\dfrac{\xi ^2}{2}\Omega _2,\\ \dfrac{d\Omega _0}{dz}+\xi \left( -\dfrac{d\Omega _0}{dz}+\dfrac{d\Omega _1}{dz}+K*\dfrac{d\Omega _0}{dz}\right) +\\ \qquad +\xi ^2\left( -\dfrac{d\Omega _1}{dz}+\dfrac{1}{2}\dfrac{d\Omega _2}{dz}+K*\dfrac{d\Omega _1}{dz} \right) =\\ \qquad =M_0+\xi M_1+\dfrac{\xi ^2}{2}M_2,\\ \dfrac{dT_0}{dz}+\xi \dfrac{dT_1}{dz}+\dfrac{\xi ^2}{2}\dfrac{dT_2}{dz}=-\lambda _0 v_0-\xi \left( \lambda _0v_1+\right. \\ \qquad \left. +\lambda _1v_0 \right) -\dfrac{\xi ^2}{2}\left( \lambda _0v_2+\lambda _1v_1+\lambda _2v_0 \right) ,\\ \dfrac{dM_0}{dz}+\xi \dfrac{dM_1}{dz}+\dfrac{\xi ^2}{2}\dfrac{dM_2}{dz}=T_0+\xi T_1+\dfrac{\xi ^2}{2}T_2, \end{array} \end{aligned}$$

(2)

By collecting like powers of $\xi$ we obtain the expressions reported in Eqs. (13)–(15).

1.2 Fundamental matrix

For $z_0=0$, the components of the fundamental matrix $\mathbf{{Y}}(z,0)$ are listed below.

$$\begin{aligned} \begin{array}{l} {Y_{11}} = \dfrac{1}{2}\left( {\cos \lambda _0^{1/4}z + \cosh \lambda _0^{1/4}z} \right) \\ {Y_{12}} = - \dfrac{{\sin \lambda _0^{1/4}z + \sinh \lambda _0^{1/4}z}}{{2\lambda _0^{1/4}}}\\ {Y_{13}} = \dfrac{{\sin \lambda _0^{1/4}z - \sinh \lambda _0^{1/4}z}}{{2\lambda _0^{3/4}}}\\ {Y_{14}} = \dfrac{{\cos \lambda _0^{1/4}z - \cosh \lambda _0^{1/4}z}}{{2\lambda _0^{1/2}}},\quad Y_{21}=\lambda _0Y_{13},\\ Y_{22}=Y_{11},\quad Y_{23}=-Y_{14},\quad Y_{24}=-Y_{12}\\ Y_{31}=\lambda _0Y_{12},\quad Y_{32}=-\lambda _0Y_{14},\quad Y_{33}=Y_{11}\\ Y_{34}=-\lambda _0Y_{13},\quad Y_{41}=\lambda _0Y_{14},\quad Y_{42}=-\lambda _0Y_{13}\\ Y_{43}=-Y_{12},\quad Y_{44}=Y_{11}. \end{array} \end{aligned}$$

(3)

1.3 Orthogonality of the eigenmodes

By using the inner product defined in Eq. (20), the condition of orthogonality for the family of eigensolutions $\{M_0^j\}$ associated with the j-th eigenvalue $\lambda _0^j$ is

$$\begin{aligned} \left\langle DM_0^l,M_0^m\right\rangle \!=\!\!\int _{0}^{1}\!\!\left[ (M_0^l)^{IV}\!M_0^m\!-\!\lambda _0^lM_0^lM_0^m\right] \!dz\!=\!0 \end{aligned}$$

(4)

This may be written, by the aid of variational principles,as

$$\begin{aligned} \begin{array}{l} \!\left\langle DM_0^l,M_0^m\right\rangle \!=\!\!\int _{0}^{1}\!\!\left[ (M_0^l)^{\prime \prime }(M_0^m)^{\prime \prime }\!-\!\lambda _0^lM_0^l M_0^m \right] \!dz\\ \quad +\left[ (M_0^l)^{\prime \prime \prime }M_0^m-(M_0^l)^{\prime \prime }(M_0^m)^\prime ) \right] _0^1 \end{array} \end{aligned}$$

(5)

In the previous equation, the boundary terms vanish due to Eq. (27); then, easy calculations show that

$$\begin{aligned} \left\langle M_0^l,DM_0^m\right\rangle =\int _{0}^{1}\left[ (M_0^l)^{\prime \prime }(M_0^m)^{\prime \prime }-\lambda _0^mM_0^l M_0^m \right] \end{aligned}$$

(6)

Then, by subtraction of the previous two equations,

$$\begin{aligned} \begin{array}{l} \left\langle DM_0^l,M_0^m\right\rangle -\left\langle M_0^l,DM_0^m\right\rangle =\\ \quad =(\lambda _0^m-\lambda _0^l)\int _{0}^{1}M_0^lM_0^mdz=0 \end{array} \end{aligned}$$

(7)

By recalling Eq. (20) and accounting for the inequality $\lambda _0^m\ne \lambda _0^l\ \forall \,l\ne m$, the last equality of the previous equation leads to the searched orthogonality condition

$$\begin{aligned} \left\langle M_0^l,M_0^m\right\rangle =0 \end{aligned}$$

(8)

1.4 Fredholm Compatibility

Inserting the eigensolution expansion into the first-order equation for the $l^{th}$ mode gives

$$\begin{aligned} \begin{array}{l} (M_0^l)^{IV}+\sum \limits _{l\ne j}b_{lj}(M_0^j)^{IV}-\lambda _0^lM_0^l-\lambda _0^l\sum \limits _{l\ne j}b_{lj}M_0^j=\\ =(M_0^l)^{IV}-K*(M_0^l)^{IV}+\lambda _1 M_0^l \end{array} \end{aligned}$$

(9)

Inserting in Eq. (9) the zeroth-order field equation, Eq. (22)-1, we obtain

$$\begin{aligned} \begin{array}{l} \sum \limits _{l\ne j}b_{lj}\lambda _0^jM_0^j\!-\!\lambda _0^l\sum \limits _{l\ne j}b_{lj}M_0^j\!=\!\\ \quad =\lambda _0^lM_0^l\!-\!K\!*\!(\lambda _0^lM_0^l)\!+\!\lambda _1^lM_0^l \end{array} \end{aligned}$$

(10)

Multiplying both sides by $M_0^m$ gives

$$\begin{aligned} \begin{array}{l} \sum \limits _{l\ne j}b_{lj}\lambda _0^jM_0^jM_0^m\!-\!\lambda _0^l\sum \limits _{l\ne j}b_{lj}M_0^jM_0^m\!=\\ \quad =\!\lambda _0^lM_0^lM_0^m\!-\!\lambda _0^l(K\!*\!M_0^l)M_0^m\!+\!\lambda _1^lM_0^lM_0^m \end{array} \end{aligned}$$

(11)

and integrating both sides over the domain provides, in terms of the inner product defined in Eq. (20),

$$\begin{aligned} \begin{array}{l} \sum \limits _{l\ne j}b_{lj}\lambda _0^j\left\langle M_0^j,M_0^m\right\rangle -\lambda _0^l\sum \limits _{l\ne j}b_{lj}\left\langle M_0^j,M_0^m\right\rangle =\\ \quad =\lambda _0^l\left\langle M_0^l,M_0^m\right\rangle -\lambda _0^l\left\langle (K\!*\!M_0^l),M_0^m\right\rangle \\ \quad +\lambda _1^l\left\langle M_0^l, M_0^m\right\rangle \end{array} \end{aligned}$$

(12)

When $l=m$ we have

$$\begin{aligned} \begin{array}{l} \sum \limits _{m\ne j}b_{mj}\lambda _0^j\left\langle M_0^j,M_0^m\right\rangle -\lambda _0^m\sum \limits _{m\ne j}b_{mj}\left\langle M_0^j,M_0^m\right\rangle =\\ \quad =\lambda _0^m\left\langle M_0^m,M_0^m\right\rangle -\lambda _0^m\left\langle (K\!*\!M_0^m),M_0^m\right\rangle +\\ \quad +\lambda _1^m\left\langle M_0^m,M_0^m\right\rangle \end{array} \end{aligned}$$

(13)

Exploiting the orthogonality of eigensolutions proved above, $\forall j\ne m, \left\langle M_0^m,M_0^j \right\rangle =0$, Eq. (13) turns out to be

$$\begin{aligned} \begin{array}{l} \lambda _0^m\left\langle M_0^m,M_0^m\right\rangle -\lambda _0^m\left\langle (K\!*\!M_0^m),M_0^m\right\rangle +\\ \quad +\lambda _1^m\left\langle M_0^m,M_0^m\right\rangle =0 \end{array} \end{aligned}$$

(14)

which provides Eq. (37).