Identifying Second-Gradient Continuum Models in Particle-Based Materials with Pairwise Interactions Using Acoustic Tensor Methodology

Abstract

This paper discusses wave propagation in unbounded particle-based materials described by a second-gradient continuum model, recently introduced by the authors, to provide an identification technique. The term particle-based materials denotes materials modeled as assemblies of particles, disregarding typical granular material properties such as contact topology, granulometry, grain sizes, and shapes. This work introduces a center-symmetric second-gradient continuum resulting from pairwise interactions. The corresponding Euler-Lagrange equations (equilibrium equations) are derived using the least action principle. This approach unveils non-classical interactions within subdomains. A novel, symmetric, and positive-definite acoustic tensor is constructed, allowing for an exploration of wave propagation through perturbation techniques. The properties of this acoustic tensor enable the extension of an identification procedure from Cauchy (classical) elasticity to the proposed second-gradient continuum model. Potential applications concern polymers, composite materials, and liquid crystals.

Appendices

Appendix A

Hereafter, we undertake algebraic computations to derive Eq. (11). Starting with Eq. (4), we have

$$ \begin{aligned} &[\mathbf{f}(\overline{\mathbf{x}},\mathbf{x},t)]_{ip}[ \mathbf{f}(\overline{\mathbf{x}},\mathbf{x},t)]_{iq} \\ &\quad =[\mathbf{f}^{(1)}( \mathbf{x},t)]_{ip}[\mathbf{f}^{(1)}(\mathbf{x},t)]_{iq} \\ &\qquad {}+\frac{1}{2}\left ( \frac{\partial [\mathbf{f}^{(1)}(\mathbf{x},t)]_{ip}}{\partial x_{k}}[ \mathbf{f}^{(1)}(\mathbf{x},t)]_{iq}+[\mathbf{f}^{(1)}(\mathbf{x},t)]_{ip} \frac{\partial [\mathbf{f}^{(1)}(\mathbf{x},t)]_{iq}}{\partial x_{k}} \right )(\overline{x}_{k}-x_{k}) \\ &\qquad {}+\frac{1}{4} \frac{\partial [\mathbf{f}^{(1)}(\mathbf{x},t)]_{ip}}{\partial x_{k}} \frac{\partial [\mathbf{f}^{(1)}(\mathbf{x},t)]_{iq}}{\partial x_{s}}( \overline{x}_{k}-x_{k})(\overline{x}_{s}-x_{s})\,. \end{aligned} $$

(44)

Considering Eqs. (5) and (6) yields

$$\begin{aligned} &[\mathbf{f}(\overline{\mathbf{x}},\mathbf{x},t)]_{ip}[\mathbf{f}( \overline{\mathbf{x}},\mathbf{x},t)]_{iq} \\ &\quad =[\mathbf{c}^{(1)}( \mathbf{x},t)]_{pq}+\frac{1}{2}\mathbf{c}_{pqk}^{(12)}(\mathbf{x},t) \, (\overline{x}_{k}-x_{k}) +\frac{1}{4}\mathbf{c}_{pqks}^{(2)}(\mathbf{x},t)\, (\overline{x}_{k}-x_{k})( \overline{x}_{s}-x_{s})\,. \end{aligned}$$

(45)

Considering tensor $[\mathbf{e}(\overline{\mathbf{x}},\mathbf{x},t)]$, defined by Eq. (10), results in

$$\begin{aligned}{} [\mathbf{e}(\overline{\mathbf{x}},\mathbf{x},t)]_{pq}={}&\frac{1}{2} \left ([\mathbf{c}^{(1)}(\mathbf{x},t)]_{pq}-[\,\mathrm{I}\,]_{pq} \right )+\frac{1}{4}\mathbf{c}_{pqk}^{(12)}(\mathbf{x},t)\, ( \overline{x}_{k}-x_{k}) \\ &{} +\frac{1}{8}\mathbf{c}_{pqks}^{(2)}(\mathbf{x},t)\,(\overline{x}_{k}-x_{k})( \overline{x}_{s}-x_{s}), \end{aligned}$$

(46)

Finally, substituting the tensors $[\mathbf{e}^{(1)}(\mathbf{x},t)]$, $[\mathbf{e}^{(12)}(\overline{\mathbf{x}},\mathbf{x},t)]$, and $[\mathbf{e}^{(2)}(\overline{\mathbf{x}},\mathbf{x},t)]$, defined by Eqs. (7), (8), and (9), we obtain Eq. (11).

Appendix B

In this appendix, we show that ${\mathbb{m}}^{(0)}(\mathbf{x})=1$ in Proposition 3. Since $\Omega$ is $\mathbb{R}^{3}$, ${\mathbb{m}}^{(0)}(\mathbf{x})$ is given by

$${\mathbb{m}}^{(0)}(\mathbf{x})={\int }_{{\mathbb{R}}^3}\alpha (\bar{\mathbf{x}},\mathbf{x})d\bar{\mathbf{x}},$$

(47)

where $\alpha (\overline{\mathbf{x}},\mathbf{x})$ is defined in Eq. (18). It can be seen that $\overline{\mathbf{x}}\mapsto \alpha (\overline{\mathbf{x}}, \mathbf{x})$ is the probability density function of a Gaussian $\mathbb{R}^{3}$-valued random variable with mean vector $\mathbf{x}$ and covariance matrix $\xi ^{2}\, [\,\mathrm{I}\,]$. Consequently, we have ${\mathbb{m}}^{(0)}(\mathbf{x})=1$.

Appendix C

Proof

Consider $\pi ^{\mathrm{kin}}$ defined by Eq. (19). The first variation $\delta \pi ^{\mathrm{kin}}$ is expressed as

$$ \delta \pi ^{\mathrm{kin}}(\mathbf{u};\delta \mathbf{u})=\int _{t_{0}}^{t_{1}} \int _{\Omega}\rho (\mathbf{x}) \frac{\partial u_{k}(\mathbf{x},t)}{\partial t} \frac{\partial \delta u_{k}(\mathbf{x},t)}{\partial t}\,d\mathbf{x}\,dt \,. $$

(48)

Subsequently, integrating by parts and since, for all $\mathbf{x}$, $\delta \mathbf{u}(\mathbf{x},t_{0})=0$ and $\delta \mathbf{u}(\mathbf{x},t_{1})=0$, Eq. (48) can be transformed into

$$ \delta \pi ^{\mathrm{kin}}(\mathbf{u};\delta \mathbf{u}) =-\int _{t_{0}}^{t_{1}} \int _{\Omega}\rho (\mathbf{x}) \frac{\partial ^{2}u_{k}(\mathbf{x},t)}{\partial t^{2}}\delta u_{k}( \mathbf{x},t)\,d\mathbf{x}\,dt\,. $$

(49)

Considering $\pi ^{\mathrm{def}}(\mathbf{u})$, $[\boldsymbol{\sigma}(\mathbf{u},\mathbf{x},t)]$, and $\mathbb{h}(\mathbf{u},\mathbf{x},t)$ defined in Eqs. (15), (23), and (24), the first variation $-\delta \pi ^{\mathrm{def}}(\mathbf{u};\delta \mathbf{u})$ is expressed as

$$\begin{array}{rl}-\delta {\pi }^{\mathrm{def}}(\mathbf{u};\delta \mathbf{u})=& -{\int }_{t_0}^{t_1}{\int }_{\mathrm{\Omega }}{[\mathit{\sigma }(\mathbf{u},\mathbf{x},t)]}_{ij}\frac{\partial \delta u_i(\mathbf{x},t)}{\partial x_j}d\mathbf{x}dt\\ & -{\int }_{t_0}^{t_1}{\int }_{\mathrm{\Omega }}{\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t)\frac{\partial \delta u_i(\mathbf{x},t)}{\partial x_j\partial x_q}d\mathbf{x}dt.\end{array}$$

(50)

Using an integration by parts, the first integral in the right-hand side of Eq. (50) can be written as

$$ \begin{aligned} -\int _{t_{0}}^{t_{1}}\int _{\Omega}[\boldsymbol{\sigma}( \mathbf{u},\mathbf{x},t)]_{ij} \frac{\partial \delta u_{i}(\mathbf{x},t)}{\partial x_{j}}\,d \mathbf{x}\,dt=-\int _{t_{0}}^{t_{1}}\int _{\partial \Omega}[ \boldsymbol{\sigma}(\mathbf{u},\mathbf{x},t)]_{ij}\,n_{j}( \mathbf{\mathbf{x}})\, \delta u_{i}(\mathbf{x},t)\,d\mathbf{x}\,dt \\ +\int _{t_{0}}^{t_{1}}\int _{\Omega} \frac{\partial [\boldsymbol{\sigma}(\mathbf{u},\mathbf{x},t)]_{ij}}{\partial x_{j}} \delta u_{i}(\mathbf{x},t)\,d\mathbf{x}\,dt\,, \end{aligned} $$

(51)

and the second integral in the right-hand side of Eq. (50) can be expressed as

$$\begin{array}{rl}& -{\int }_{t_0}^{t_1}{\int }_{\mathrm{\Omega }}{\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t)\frac{\partial \delta u_i(\mathbf{x},t)}{\partial x_j\partial x_q}d\mathbf{x}dt\\ & =-{\int }_{t_0}^{t_1}{\int }_{\partial \mathrm{\Omega }}{\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t)\frac{\partial \delta u_i(\mathbf{x},t)}{\partial x_j}{n}_q(\mathbf{x})dsdt\\ & +{\int }_{t_0}^{t_1}{\int }_{\partial \mathrm{\Omega }}\frac{\partial }{\partial x_q}({\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t))n_j(\mathbf{x})\delta u_i(\mathbf{x},t)dsdt\\ & -{\int }_{t_0}^{t_1}{\int }_{\mathrm{\Omega }}\frac{\partial }{\partial x_q\partial x_j}({\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t))\delta u_i(\mathbf{x},t)dsdt.\end{array}$$

(52)

Let $[\mathbf{q}(\mathbf{x})]$ and $[\mathbf{p}(\mathbf{x})]$ denote the orthogonal and parallel projection operators, respectively. The Kronecker delta $[\boldsymbol{\delta}]_{rj}$ is equal to the sum of $[\mathbf{q}(\mathbf{x})]_{rj}$ and $[\mathbf{p}(\mathbf{x})]_{rj}$. Given that $[\mathbf{q}(\mathbf{x})]_{rj}=n_{r}\,n_{j}$ and $[\mathbf{p}(\mathbf{x})]_{rj}=[\mathbf{p}(\mathbf{x})]_{sj}[ \mathbf{p}(\mathbf{x})]_{rs}$, the first integral in the right-hand side of Eq. (52) can be transformed into

$$\begin{array}{rl}& -{\int }_{t_0}^{t_1}{\int }_{\partial \mathrm{\Omega }}{\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t)n_q(\mathbf{x})\frac{\partial \delta u_i(\mathbf{x},t)}{\partial x_j}dsdt\\ & =-{\int }_{t_0}^{t_1}{\int }_{\partial \mathrm{\Omega }}{\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t)n_j(\mathbf{x})n_q(\mathbf{x})\frac{\partial \delta u_i(\mathbf{x},t)}{\partial \mathbf{n}}dsdt\\ & -{\int }_{t_0}^{t_1}{\int }_{\partial \mathrm{\Omega }}{\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t)\frac{\partial \delta u_i(\mathbf{x},t)}{\partial x_r}\left({[\mathbf{p}(\mathbf{x})]}_{sj}{[\mathbf{p}(\mathbf{x})]}_{rs}\right)n_q(\mathbf{x})dsdt\end{array}$$

(53)

The second integral in the right-hand side of Eq. (53) can be rewritten as

$$\begin{array}{r}-{\int }_{t_0}^{t_1}{\int }_{\partial \mathrm{\Omega }}{\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t)\frac{\partial \delta u_i(\mathbf{x},t)}{\partial x_r}\left({[\mathbf{p}(\mathbf{x})]}_{sj}{[\mathbf{p}(\mathbf{x})]}_{rs}\right)n_q(\mathbf{x})dsdt\\ =-{\int }_{t_0}^{t_1}{\int }_{\partial \mathrm{\Omega }}\frac{\partial }{\partial x_r}({\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t)n_q(\mathbf{x}){[\mathbf{p}(\mathbf{x})]}_{sj}\delta u_i(\mathbf{x},t)){[\mathbf{p}(\mathbf{x})]}_{rs}dsdt\\ +{\int }_{t_0}^{t_1}{\int }_{\partial \mathrm{\Omega }}\frac{\partial }{\partial x_r}({\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t)n_q(\mathbf{x}){[\mathbf{p}(\mathbf{x})]}_{sj}){[\mathbf{p}(\mathbf{x})]}_{rs}\delta u_i(\mathbf{x},t)dsdt\end{array}$$

(54)

Using the Gauss divergence formula for bounded surfaces and since $[\mathbf{p}(\mathbf{x})]_{sj}\,\nu _{s}(\mathbf{x})=\nu _{j}( \mathbf{x})$, the first integral in the right-hand side of Eq. (54) becomes

$$\begin{array}{rl}& -{\int }_{t_0}^{t_1}{\int }_{\partial \mathrm{\Omega }}\frac{\partial }{\partial x_r}({\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t)n_q(\mathbf{x}){[\mathbf{p}(\mathbf{x})]}_{sj}\delta u_i(\mathbf{x},t)){[\mathbf{p}(\mathbf{x})]}_{rs}dsdt\\ & =-{\int }_{t_0}^{t_1}{\int }_{\partial \partial \mathrm{\Omega }}{\mathbb{h}}_{ijq}(\mathbf{u},\mathbf{x},t)n_q(\mathbf{x}){\nu }_j(\mathbf{x})\delta u_i(\mathbf{x},t)d\ell dt.\end{array}$$

(55)

Sequentially substituting Eq. (55) into Eq. (54), Eq. (54) into Eq. (53), Eq. (53) into Eq. (52), Eqs. (52) and Eq. (51) into Eq. (50), and finally substituting Eqs. (50), (49), and (20) into Eq. (21), we obtain the Euler-Lagrange equations as presented in Proposition (2). □