Well-posed nonlocal elasticity model for finite domains and its application to the mechanical behavior of nanorods

Abstract

Eringen’s nonlocal elasticity theory is one of the most attractive approaches to investigate the intrinsic scale effect of nanoscopic structures. Eringen proposed both integral and differential nonlocal models which are equivalent to each other over unbounded continuous domains. Although the Eringen nonlocal models can be used as very useful tools for modeling the mechanical characteristics of nanoscopic structures, however, several researchers have reported some paradoxical results when they used the nonlocal differential model. In this paper, we develop a well-posed nonlocal differential model for finite domains, and its applicability to predict the static and dynamic behavior of a nanorod is investigated. It is shown that the proposed integral and differential nonlocal models are equivalent to each other over bounded continuous domains, and the corresponding elastic problems are well-posed and consistent. In addition, some paradigmatic static problems are solved the and we show that the paradoxical results disappear by using the present model.

Appendix

As discussed in Sect. 5.1, the free end displacements of the rod are independent of the nonlocal parameter. The correctness of this finding can be mathematically demonstrated. To prove the accuracy of the observations, we substitute Eq. (2) into Eq. (1) and obtain the following equation:

$$\begin{aligned} \sigma (x)=\left[ 1-\int \limits _a^b {\alpha (\left| {x-\xi } \right| ,e_{0} l_{i} )\hbox {d}\xi } \right] E\varepsilon (x)+\int \limits _a^b {\alpha (\left| {x-\xi } \right| ,e_{0} l_{i} )E\varepsilon (\xi )\hbox {d}\xi }. \end{aligned}$$

(A.1)

Multiplying both sides of Eq. (A.1) by $\hbox {d}\chi$ and integrating over the domain yields

$$\begin{aligned} \frac{1}{E}\int \limits _a^b {\sigma (\chi )} \hbox {d}\chi =\int \limits _a^b {\varepsilon (\chi )} \hbox {d}\chi +\int \limits _a^b {\left( {\int \limits _a^b {\alpha (\left| {\chi -\xi } \right| ,e_{0} l_{i} )[\varepsilon (\xi )-\varepsilon (\chi )]\hbox {d}\xi } } \right) } \hbox {d}\chi . \end{aligned}$$

(A.2)

Now, let us define a function as follows:

$$\begin{aligned} h(\chi ,\xi )=\alpha (\left| {\chi -\xi } \right| ,e_{0} l_{i} )[\varepsilon (\xi )-\varepsilon (\chi )]. \end{aligned}$$

(A.3)

According to Fubini’s theorem [40], we have

$$\begin{aligned} \int \limits _a^b {\left( {\int \limits _a^b {\alpha (\left| {\chi -\xi } \right| ,e_{0} l_{i} )\varepsilon (\xi )\hbox {d}\xi } } \right) } \hbox {d}\chi =\int \limits _a^b {\left( {\int \limits _a^b {\alpha (\left| {\chi -\xi } \right| ,e_{0} l_{i} )\varepsilon (\chi )\hbox {d}\xi } } \right) } \hbox {d}\chi . \end{aligned}$$

(A.4)

Using Eqs. (A.2) and (A.4), we obtain

$$\begin{aligned} \frac{1}{E}\int \limits _a^b {\sigma (\chi )} \hbox {d}\chi =\int \limits _a^b {\varepsilon (\chi )} \hbox {d}\chi . \end{aligned}$$

(A.5)

Recalling that $\varepsilon (\chi )={\hbox {d}U} / {\hbox {d}\chi }$, Eq. (A.5) becomes

$$\begin{aligned} U(b)=\frac{1}{E}\int \limits _a^b {\sigma (\chi ,t)} \hbox {d}\chi +U(a). \end{aligned}$$

(A.6)

It can be seen from Eq. (A.6) that the free end displacement is independent of the nonlocal kernel function and the nonlocal parameter.