Innovative Combinations of Atomistic and Continuum: Mechanical Properties of Nanostructured Materials

Currently, due to advances in nanotechnology, many investigations are devoted to nanoscale science and developments of nanocomposites. Nanomaterials in general can be roughly classified into two categories. On one hand, if the characteristic length of the microstructure, such as the grain size of a polycrystal material, is in the nanometer range, it is called a nanostructured material. On the other hand, if at least one of the overall dimensions of a structural element is in the nanometer range, it may be called a nano-sized structural element. Thus, this may include nanoparticles, nanofilms, and nanowires [2, 10, 47].

Appendices

Appendix 1: “T” Stress Decomposition

Consider an inhomogeneous, linearly elastic solid with strain energy density per unit undeformed volume defined by

$$w=w_0+\tau_{ij}\,\varepsilon_{ij}+\frac{1}{2}C_{ijkl}\, \varepsilon_{ij}\,\varepsilon_{kl},$$

((8.116))

where $\varepsilon_{ij}$ is the Lagrangian strain tensor. The corresponding second Piola-Kirchhoff stress tensor is thus given by

$$\sigma_{ij}=\frac{\partial w}{\partial\varepsilon_{ij}}=\tau_{ij}+C_{ijkl}\,\varepsilon_{ij}.$$

((8.117))

Equivalently, (8.117) can be written as

$$\sigma^{s}_{\alpha\beta}=\tau^{s}_{\alpha\beta}+C_{\alpha\beta\hat{\kappa} \lambda} \varepsilon_{\hat{\kappa}\lambda}+C_{\alpha\beta3k}\varepsilon^t_{k},\qquad\sigma_{j}^t=\tau^{t}_{j}+C_{3j\hat{\kappa}\lambda}\varepsilon_{\hat{\kappa}\lambda}+C_{3j3k}\varepsilon^t_{k},$$

((8.118))

where the summation convention is implied, and the lowercase Roman subscripts go from 1 to 3 and the lowercase Greek subscripts go from 1 to 2, and

$$\sigma^s_{\alpha\beta}=\bar{\boldsymbol{\sigma}}_{\alpha\beta},\,\varepsilon_{\alpha\beta}=\hat{\varepsilon}_{\alpha\beta},\,\sigma^{\,t}_j=\bar{\boldsymbol{\sigma}}_{3j},\varepsilon^t_{\alpha}=2\hat{\varepsilon}_{\alpha3},\,\varepsilon^t_{3}=\hat{\varepsilon}_{33},\tau^s_{\alpha\beta}=\tau_{\alpha \beta}, \tau^t_{j}=\tau_{3j}.$$

((8.119))

Assuming that the second-order, $C_{3k3j}$, is invertible, the second part of Equation (8.118) can be rewritten as

$$\varepsilon^t_{k}=-M_{kj}\tau^t_j+M_{jk}\sigma^t_j-\gamma_{k\alpha\beta}\varepsilon_{\alpha\beta},$$

((8.120))

where

$$M_{kj}=C^{-1}_{3k3j},\qquad\gamma_{k\alpha\beta}=M_{kj}C_{3k\alpha\beta}.$$

((8.121))

Substituting (8.120) into the first of (8.118) yields

$$\sigma^s_{\alpha\beta}=\hat{\tau}^{\,s}_{\alpha\beta}+C^{\,s}_{\alpha\beta\hat{\kappa}\lambda}\varepsilon_{\hat{\kappa}\lambda}+\gamma_{j\alpha\beta}\sigma^{\,t}_j,$$

((8.122))

where

$$\hat{\tau}^{s}_{\alpha\beta}=\tau^{s}_{\alpha\beta}-\tau^{t}_j\gamma_{j\alpha\beta},\qquad C^{\,s}_{\alpha\beta\hat{\kappa}\lambda}=C_{\alpha\beta\hat{\kappa}\lambda}-C_{\alpha\beta3j}\gamma_{j\hat{\kappa}\lambda}.$$

((8.123))

Using tensorial notation, Equation (8.116), (8.120) and (8.122) can be written, respectively, as

$$w=w_0-\frac{1}{2}\boldsymbol{\tau}^t\cdot{\bf M}\cdot\boldsymbol{\tau}^t+\hat{\boldsymbol{\tau}}^{\,s}:\boldsymbol{\varepsilon}^s+\frac{1}{2}\boldsymbol{\varepsilon}^{\,s}:{\bf C}^s:\boldsymbol{\varepsilon}^{\,s}+\frac{1}{2}\boldsymbol{\sigma}^t\cdot{\bf M}\cdot\boldsymbol{\sigma}^t,$$

((8.124))

$$\boldsymbol{\varepsilon}^t=-{\bf M}\cdot\boldsymbol{\tau}^t+{\bf M}\cdot\boldsymbol{\sigma}^t-\boldsymbol{\gamma}:\boldsymbol{\varepsilon}^{\,s},$$

((8.125))

$$\boldsymbol{\sigma}^{\,s}=\hat{\boldsymbol{\tau}}^{\,s}+{\bf C}^{\,s}:\boldsymbol{\varepsilon}^{\,s}+\boldsymbol{\gamma}\cdot\boldsymbol{\sigma}^t.$$

((8.126))

In addition, if the material is isotropic, that is

$$C_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\hat{\mu}\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right),$$

((8.127))

where λ and μ are the Lamé constants. The other quantities, in this special case, are such as

$$\begin{cases}C_{3k3j}\ \,=\left(\lambda+\hat{\mu}\right)\delta_{3k}\delta_{3j}+\hat{\mu}\delta_{kj},\\M_{kj}\ \ \;=-\frac{\lambda+\hat{\mu}}{\hat{\mu}\left(\lambda+2\hat{\mu}\right)}\delta_{3k}\delta_{3j}+\frac{1}{\hat{\mu}}\delta_{kj},\\\gamma_{i\alpha\beta}\ \ \,=\frac{\lambda}{\lambda+2\hat{\mu}}\delta_{3i}\delta_{\alpha\beta},\\C^{s}_{\alpha\beta\hat{\kappa}\lambda}=\frac{2\lambda\hat{\mu}}{\lambda+2\hat{\mu}}\delta_{\alpha\beta}\delta_{\hat{\kappa}\lambda}+\hat{\mu}\left(\delta_{\alpha\hat{\kappa}}\delta_{\beta\lambda}+\delta_{\alpha\lambda}\delta_{\beta\hat{\kappa}}\right).\end{cases}$$

((8.128))

Appendix 2: Atomic Level Description

The difference in position of two atoms, $m$ and $n$, near their relaxed state as

$$r^{\,mn}_i-\hat{r}^{\,mn}_i=A^{\,mn}_{i\alpha\beta}\varepsilon_{\alpha\beta}+B^{\,mn}_{ik}\sigma_k^t+\left(\tilde{\varepsilon}^{\,m}_{ij}\hat{r}^{\,m}_j-\tilde{\varepsilon}^{n}_{ij}\hat{r}^{\,n}_j\right),$$

((8.129))

where,

$$\begin{cases}A^{mn}_{i\alpha\beta}=\left(A^{\pm,\,m}_{ij\alpha\beta}+A^{\pm,\,n}_{ij\alpha\beta}\right)\hat{r}^{\,mn}_j-\left(A^{\pm,\,n}_{ij\alpha\beta}\hat{r}^{\,m}_j-A^{\pm,\,m}_{ij\alpha\beta}\hat{r}^{\,n}_j\right),\\\\B^{\,mn}_{ik}=\left(B^{\pm,\,m}_{ijk}+B^{\pm,\,n}_{ijk}\right)\hat{r}^{\,mn}_j-\left(B^{\pm,\,n}_{ijk}\hat{r}^{\,m}_j-B^{\pm,\,m}_{ijk}\hat{r}^{\,n}_j\right).\end{cases}$$

((8.130))

The total strain energy of the atomic assembly (see Section 8.8.1.1),

$$\begin{aligned} E=& E_0+\overline{\bf A}^{(1)}:\hat{\boldsymbol{\varepsilon}}^{\,s}+\overline{\bf B}^{(1)}\cdot\overline{\boldsymbol{\sigma}}^{\,t}+\frac{1}{2}\hat{\boldsymbol{\varepsilon}}^{\,s}:\overline{\bf A}^{(2)}:\hat{\boldsymbol{\varepsilon}}^{\,s}+\frac{1}{2}\overline{\boldsymbol{\sigma}}^{\,t}\cdot\overline{\bf B}^{(2)}\cdot\overline{\boldsymbol{\sigma}}^{\,t}\\&+\overline{\boldsymbol{\sigma}}^{\,t}\cdot\overline{\bf Q}:\hat{\boldsymbol{\varepsilon}}^{\,s}+\sum^{N-1}_{n=1}\left({\bf K}^n+{\bf D}^n:\hat{\boldsymbol{\varepsilon}}^{\,s} +{\bf G}^n\cdot\overline{\boldsymbol{\sigma}}^{\,t}\right):\tilde{\boldsymbol{\varepsilon}}^{n}\\&+\frac{1}{2}\sum^{N-1}_{n=1}\sum^{N-1}_{m=1}\tilde{\boldsymbol{\varepsilon}}^{\,n}:{\bf L}^{mn}:\tilde{\boldsymbol{\varepsilon}}^{\,m}. \end{aligned}$$

((8.131))

with

$$E_0=\sum_{n}\frac{1}{\Omega_n}\sum_{m\neq n}E^{(n)}\bigg|_{r^{\,mn}=\hat{r}^{\,mn}},$$

((8.132))

$$\overline{A}^{(1)}_{\alpha\beta}=\sum_{n}\frac{1}{\Omega_n}\sum_{m\neq n}\frac{\partial E^{(n)}}{\partial r^{\,mn}_i}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}A^{\,mn}_{i\alpha\beta},$$

((8.133))

$$\overline{B}^{(1)}_{k}=\sum_{n}\frac{1}{\Omega_n}\sum_{m\neq n}\frac{\partial E^{(n)}}{\partial r^{\,mn}_i}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}} B^{\,mn}_{ik},$$

((8.134))

$$\overline{A}^{(2)}_{\alpha\beta\hat{\kappa}\lambda}=\sum_{n}\frac{1}{\Omega_n}\sum_{m\neq n}\frac{\partial^2 E^{(n)}}{\partial r^{\,mn}_i \partial r^{mn}_k}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}A^{mn}_{i\alpha\beta\;}A^{\,mn}_{k\hat{\kappa}\lambda},$$

((8.135))

$$\overline{B}^{(2)}_{jl}=\sum_{n}\frac{1}{\Omega_n}\sum_{m\neq n}\frac{\partial^2 E^{(n)}}{\partial r^{\,mn}_i \partial r^{\,mn}_k}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}B^{\,mn}_{ij}B^{\,mn}_{kl},$$

((8.136))

$$\overline{Q}_{\alpha\beta j}=\sum_{n}\frac{1}{\Omega_n}\sum_{m\neq n}\frac{\partial^2E^{(n)}}{\partial r^{\,mn}_i \partial r^{\,mn}_k}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}A^{\,mn}_{i\alpha\beta}B^{\;mn}_{kj},$$

((8.137))

$$\begin{aligned}K^n_{ij}=&\frac{1}{2\Omega_n}\hat{r}^{n}_j\left(\sum_{p\neq n}\frac{\partial E^{(p)}}{\partial r^{\,pn}_i}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}-\frac{\partial E^{(n)}}{\partial r^{\,pn}_i}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}\right)\\&+\frac{1}{2\Omega_n}\hat{r}^{n}_i\left(\sum_{p\neq n}\frac{\partial E^{(p)}}{\partial r^{\,pn}_j}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}-\frac{\partial E^{(n)}}{\partial r^{\,pn}_j}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}\right), \end{aligned}$$

((8.138))

$$\begin{aligned}D^n_{ij\alpha\beta}=&\frac{1}{2\Omega_n}\hat{r}^{n}_j\sum_{p\neq n}\sum_{q\neq n}\left[\left(\frac{\partial^2 E^{(p)}}{\partial r^{\,pn}_i \partial r^{qn}_k}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}-\frac{\partial^2E^{(n)}}{\partial r^{\,pn}_i \partial r^{qn}_k}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}\right)A^{pn}_{k\alpha\beta}\right]\\&+\frac{1}{2\Omega_n}\hat{r}^{\,n}_i\sum_{p\neq n}\sum_{q\neq n}\left[\left(\frac{\partial^2 E^{(p)}}{\partial r^{\,pn}_j \partial r^{\,qn}_l}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}-\frac{\partial^2E^{(n)}}{\partial r^{\,pn}_j \partial r^{\,qn}_l}\Bigg|_{r^{mn}=\hat{r}^{\,mn}}\right)A^{\,pn}_{l\alpha\beta}\right],\end{aligned}$$

((8.139))

$$\begin{aligned}G^n_{ijv}= \& \frac{1}{2\Omega_n}\hat{r}^{\,n}_j\sum_{p\neq n}\sum_{q\neq n}\left[\left(\frac{\partial^2 E^{(p)}}{\partial r^{\,pn}_i \partial r^{\,qn}_k}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}-\frac{\partial^2E^{(n)}}{\partial r^{\,pn}_i \partial r^{\,qn}_k}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}\right)B^{\,pn}_{kv}\right]\\ \& +\frac{1}{2\Omega_n}\hat{r}^{\,n}_i\sum_{p\neq n}\sum_{q\neq n}\left[\left(\frac{\partial^2 E^{(p)}}{\partial r^{\,pn}_j \partial r^{\,qn}_l}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}-\frac{\partial^2E^{(n)}}{\partial r^{\,pn}_j \partial r^{\,qn}_l}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}\right)B^{\,pn}_{lv}\right],\end{aligned}$$

((8.140))

$$\begin{aligned}L^{\,mn}_{ijkl}=&\frac{1}{2\Omega_n}\left(\sum_{p\neq n}\frac{\partial^2E^{(p)}}{\partial r^{\,pn}_i \partial r^{\,pn}_k}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}+ \frac{\partial^2E^{(n)}}{\partial r^{\,pn}_i \partial r^{\,pn}_k}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}\right)\hat{r}^{\,n}_j\hat{r}^{\,n}_l\delta_{\,mn}\\&+\frac{1}{2\Omega_n}\left(\sum_{p\neq n}\frac{\partial^2E^{(p)}}{\partial r^{\,pn}_j \partial r^{\,pn}_l}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}+ \frac{\partial^2E^{(n)}}{\partial r^{\,pn}_j \partial r^{\,pn}_l}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}\right)\hat{r}^{\,n}_i\hat{r}^{\,n}_k\delta_{\,mn}\\&-\frac{1}{4\Omega_n}\frac{\partial^2 E^{(n)}}{\partial r^{\,mn}_i\partial r^{\,mn}_k}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}\left(\hat{r}^{\,n}_j\hat{r}^{\,m}_l+\hat{r}^{\,m}_j\hat{r}^{\,n}\right)\left(1-{\delta_{\,mn}}\right)\\& -\frac{1}{4\Omega_{n}}\frac{\partial^2 E^{(n)}}{\partial r^{\,mn}_j\partial r^{\,mn}_l}\Bigg|_{r^{\,mn}=\hat{r}^{\,mn}}\left(\hat{r}^{\,n}_i\hat{r}^{\,m}_k+\hat{r}^{\,m}_i\hat{r}^{\,n}_k\right)\left(1-\delta_{\,mn}\right).\end{aligned}$$

((8.141))

Appendix 3: Strain Concentration Tensors: Spherical Isotropic Configuration

The deviatoric and spherical parts of all concentration tensors needed to solve Equation (8.105) for $\hat{\mu}^{eff}$ and $\hat{\kappa}^{eff}$ are defined below.

8.3.1 Parts of $\boldsymbol{\vartheta}^{{\textbf{\textit{(i/j)}}}}$

From Equation (8.95), one gets:

$$ \begin{cases} \hat{\bf S}_{\vartheta}^{(i/j)}=\displaystyle\frac{3\hat{\kappa}_i+4\hat{\mu}_j}{3\hat{\kappa}_j+4\hat{\mu}_j},\\\hat{\bf D}_{\vartheta}^{(i/j)}\ =\displaystyle\frac{3\hat{\kappa}_j\left(2\hat{\mu}_i+3\hat{\mu}_j\right)+4\hat{\mu}_j\left(4\hat{\mu}_i+2\hat{\mu}_j\right)}{5\hat{\mu}_j\left(3\hat{\kappa}_j+4\hat{\mu}_j\right)}.\end{cases}$$

((8.142))

8.3.2 Parts of $\boldsymbol{\Pi}^{\,j}$

Note that:

$$\boldsymbol{\Pi}^{1}={\bf J}+{\bf K}, \qquad {\rm and} \qquad \boldsymbol{\Pi}^{2}=\boldsymbol{\vartheta}^{(1/2)}.$$

$$\begin{cases} \hat{\bf S}^{j}_{\Pi}=\displaystyle\frac{\sum^{\,j-1}_{k=1}\varphi_k\hat{\bf S}^{k}_{\Pi}\hat{\bf S}^{(k/j)}_{\vartheta}}{\sum^{\,j-1}_{k=1}\varphi_k},\\ \hat{\bf D}^{j}_{\Pi}=\displaystyle\frac{\sum^{\,j-1}_{k=1}\varphi_k\hat{\bf D}^{k}_{\Pi}\hat{\bf D}^{(k/j)}_{\vartheta}}{\sum^{\,j-1}_{k=1}\varphi_k}, \end{cases}$$

((8.143))

for $j=3$.

8.3.3 Parts of ${\bf a}^{1}$

From Equation (8.96), one gets:

$$\begin{cases} \hat{\bf S}^{1}_{a}=\left(\sum^{3}_{k=1}\varphi_k\hat{\bf S}^{k}_{\Pi}\right)^{-1},\\ \hat{\bf D}^{1}_{a}\ =\left(\sum^{3}_{k=1}\varphi_k\hat{\bf D}^{k}_{\Pi}\right)^{-1}.\end{cases}$$

((8.144))

8.3.4 Parts of ${\bf a}^{k}$

From Equation (8.94), one gets:

$$\begin{cases} \hat{\bf S}^{k}_{a} =\hat{\bf S}^{k}_{\Pi}\hat{\bf S}^{1}_{a},\\\hat{\bf D}^{k}_{a} \ =\hat{\bf D}^{k}_{\Pi}\hat{\bf D}^{1}_{a}. \end{cases}$$

((8.145))

8.3.5 Parts of ${\bf A}^{I}$

From Equation (8.102), one gets:

$$\begin{cases} \hat{\bf S}^{I}_{A}=\left[1+{\Lambda}_3\sum^{3}_{i=1}\varphi_i\left(\hat{\kappa}_i-\hat{\kappa}^{eff}\,\right)\hat{\bf S}^{k}_{a}\right]^{-1},\\ \hat{\bf D}^{I}_{A}\ =\left[1+{\Lambda}_4\sum^{3}_{i=1}\varphi_i\left(\hat{\mu}_i-\hat{\mu}^{eff}\,\right)\hat{\bf D}^{k}_{a}\right]^{-1}, \end{cases}$$

((8.146))

where: ${\Lambda}_3=\displaystyle\frac{3}{4\hat{\mu}^{eff}+3\hat{\kappa}^{eff}}\,$, ${\Lambda}_4=\displaystyle\frac{6\left(\hat{\kappa}^{\,eff}+2\hat{\mu}^{eff}\,\right)}{5\hat{\mu}^{eff}\left(4\hat{\mu}^{eff}+3\hat{\kappa}^{eff}\,\right)}$.

8.3.6 Parts of ${\bf A}^{k}$

From Eq. (8.42), one gets:

$$\begin{cases} \hat{\bf S}^{k}_{A} =\hat{\bf S}^{k}_{a}\hat{\bf S}^{I}_{A},\\\hat{\bf D}^{k}_{A}\ =\hat{\bf D}^{\,k}_{a}\hat{\bf D}^{I}_{A}. \end{cases}$$

((8.147))