Wave Propagation in Heterogeneous Media with Local and Nonlocal Material Behavior

Abstract

Wave propagation in heterogeneous solids has been an interest of researchers due to industrial applications. Some of the heterogeneous materials can exhibit power law scaling in material behavior which can be characterized by the fractal dimension of the microstructure. In this study, wave propagation in heterogeneous media with self-similar structure is investigated via fractional calculus along with space-time discontinuous Galerkin method. One and two dimensional problems are studied to demonstrate the capability of the proposed model in modeling heterogeneous media. The results show that the proposed model is a good candidate for modeling the mechanical behavior of disordered materials.

Appendices

Appendix A: Derivation of Equilibrium Equation for Fractal Media

Let us consider a body occupying a space Ω ^α embedded in Euclid space such that Ω ^α⊂Ω and $\alpha\in\mathbb{R}^{+}$ is the Hausdorff dimension or box counting dimension of the quasi-fractal space Ω ^α. S ^α denotes the closed boundary of Ω ^α and n is the unit outward normal vector to S ^α. The conservation of linear momentum can be written as

$$\begin{aligned} \begin{array}{lll} \int\rho\ddot{\mathbf{u}}d\varOmega^\alpha=\int\boldsymbol {\sigma}\cdot\mathbf{n}dS^\alpha+\int\mathbf{b}d\varOmega^\alpha. \end{array} \end{aligned}$$

(28)

The first term on the right side of Eq. (28) can be written as follows by using the Gauss divergence theorem for fractional vector operators given in [3]

$$\begin{aligned} \int\boldsymbol{\sigma}\cdot\mathbf{n}dS^\alpha=\int \nabla ^{\alpha}\cdot\boldsymbol{\sigma}d\varOmega^\alpha. \end{aligned}$$

(29)

Assuming that dependent variables are continuous in Ω ^α and shrinking Ω ^α to a point, similar to what is done in classical continuum mechanics [19], one can write the differential form of the equilibrium equation for fractal media as

$$\begin{aligned} \begin{array}{lll} \rho\ddot{\mathbf{u}}=\nabla^{\alpha}\cdot\boldsymbol{\sigma }+\mathbf{b}. \end{array} \end{aligned}$$

(30)

Appendix B: Derivation of Strain Tensor

Here, the strain tensor is derived for small deformations. We assume that the deformation is 2 dimensional for simplicity; extending it to 3 dimensions is straightforward.

2.1 B.1 Extensional Strain

Let u(x) be the displacement field, where u={u ₁,u ₂} and x={x ₁,x ₂}. $\mathbf{x^{\alpha}}=\{x_{1}^{\alpha},x_{2}^{\alpha}\}$ are the coordinates in fractal space. Let us consider the differential element occupied by the volume Ω ^α.

Using Lemma 3.1 in [25], the strain in the x ₁ direction can be written as

$$\begin{aligned} \varepsilon_{11}=\frac {\partial u_1}{\partial x_1}=\lim _{\Delta x_1\rightarrow0}\frac{\Delta u_1}{\Delta x_1}=\lim_{\Delta x_1\rightarrow0} \frac{\Delta^\alpha u_1}{(\Delta x_1)^\alpha} \frac{(\Delta x_1)^{\alpha-1}}{\varGamma(1+\alpha)}=C(L,\alpha) \frac{\partial u_1}{\partial x^\alpha_1}. \end{aligned}$$

(31)

In Eq. (31) coefficient C(L,α) is the constant which relates the fractional derivative to the fractal media [43].

2.2 B.2 Shear Strain

The shear strain is the sum of the angles between the edges of the differential element and the coordinate axis. For small deformations the relation between the fractional derivatives and the shear strain can be written as

$$\begin{aligned} \varepsilon_{12}=\frac{\partial u_1}{\partial x_2}+ \frac{\partial u_2}{\partial x_1}=C(L,\alpha) \biggl(\frac{\partial^\alpha u_1}{\partial x^\alpha_2}+\frac{\partial^\alpha u_2}{\partial x^\alpha_1} \biggr). \end{aligned}$$

(32)

Appendix C: Bilinear and Linear Operators for the Discontinuous Galerkin Method

Linear operator and bilinear operators for the discontinuous Galerkin method with local and nonlocal fractional derivatives can be written, respectively, as

$$\begin{aligned} L(w) =&\sum_{\varOmega^{\alpha}_e\in\mathcal{P}_h}\int_{\varOmega^{\alpha}_e} w \rho\mathbf{b}d\varOmega^{\alpha}_e+\int_{\varGamma^{\alpha}_D}\nabla^{\alpha} w\cdot\mathbf{n}~\bar{ \mathbf{u}}d\varGamma^{\alpha}_D+\int_{\varGamma_N}w \bar{\mathbf{t}}d\varGamma_{N}, \end{aligned}$$

(33)

$$\begin{aligned} A_L(\mathbf{u},w ) =&\sum_{\varOmega^{\alpha}_e\in\mathcal{P}_h} \biggl\{ \int_{\varOmega^{\alpha}_e} \nabla^{\alpha} w \cdot\boldsymbol{\sigma}d\varOmega^{\alpha}_e \biggr\} -\int_{\varGamma^{\alpha}_{\mathrm{int}}}c_3(\alpha,\mathbf{x})[w]\langle \boldsymbol{\sigma}\rangle\cdot\mathbf{n}~d\varGamma^{\alpha}_{\mathrm{int}} \\ &{}+\int_{\varGamma^{\alpha}_{\mathrm{int}}}\bigl\langle \boldsymbol{\sigma} \bigl( \nabla^{\alpha} w \bigr)\bigr\rangle \cdot\mathbf{n}[\mathbf{u}]d \varGamma^{\alpha}_{\mathrm{int}} -\int_{\varGamma^{\alpha}_{D}}w\boldsymbol{\sigma}\cdot\mathbf{n}~d \varGamma^{\alpha}_{D} +\int_{\varGamma^{\alpha}_{D}}\boldsymbol{\sigma} \bigl(\nabla^{\alpha} w \bigr) \cdot\mathbf{n}~\mathbf{u}d\varGamma^{\alpha}_{D} \\ &{}+\mu\frac{\gamma_\mu}{h}\int_{\varGamma^{\alpha}_{\mathrm{int}}}[w]\cdot[\mathbf{u}]d \varGamma^{\alpha}_{\mathrm{int}} +\int_{\varGamma^{\alpha}_{\mathrm{int}}} \biggl[\lambda\frac{\gamma^{\alpha}_\lambda}{h} w \biggr] \cdot\mathbf{n}[\mathbf{u}]\cdot\mathbf{n}d\varGamma^{\alpha}_{\mathrm{int}} \\ &{}+\int_{\varGamma^{\alpha}_{D}}\mu\frac{\gamma_\mu}{h}w\cdot\mathbf{u}d \varGamma^{\alpha}_{D} +\int_{\varGamma^{\alpha}_{D}}\lambda\frac{\gamma_\lambda}{h}w \cdot\mathbf{n}~ \mathbf{u}\cdot\mathbf{n}d\varGamma^{\alpha}_{D}, \end{aligned}$$

(34)

$$\begin{aligned} A_{NL}(\mathbf{u},w ) =&\sum_{\varOmega_e\in\mathcal{P}_h} \biggl\{ \int_{\varOmega^{\alpha}_e} \nabla^{\alpha} w \cdot\boldsymbol{\sigma}d\varOmega^{\alpha}_e\biggr\} -\int_{\varGamma^{\alpha}_{\mathrm{int}}}\varGamma^2(\alpha+1)[w]\langle \boldsymbol{\sigma}\rangle\cdot\mathbf{n}~d\varGamma^{\alpha}_{\mathrm{int}} \\ &{}+\int_{\varGamma^{\alpha}_{\mathrm{int}}}\bigl\langle \boldsymbol{\sigma} \bigl( \nabla^{\alpha} w \bigr)\bigr\rangle \cdot\mathbf{n}[\mathbf{u}]d \varGamma^{\alpha}_{\mathrm{int}} -\int_{\varGamma^{\alpha}_{D}}w\boldsymbol{\sigma}\cdot\mathbf{n}~d \varGamma^{\alpha}_{D} +\int_{\varGamma^{\alpha}_{D}}\boldsymbol{\sigma} \bigl(\nabla^{\alpha} w \bigr) \cdot\mathbf{n}~\mathbf{u}d\varGamma^{\alpha}_{D} \\ &{}+\mu\frac{\gamma_\mu}{h}\int_{\varGamma^{\alpha}_{\mathrm{int}}}[w]\cdot[\mathbf{u}]d \varGamma^{\alpha}_{\mathrm{int}} +\int_{\varGamma^{\alpha}_{\mathrm{int}}} \biggl[\lambda\frac{\gamma^{\alpha}_\lambda}{h} w \biggr] \cdot\mathbf{n}[\mathbf{u}]\cdot\mathbf{n}d\varGamma^{\alpha}_{\mathrm{int}} \\ &{}+\int_{\varGamma^{\alpha}_{D}}\mu\frac{\gamma_\mu}{h}w\cdot\mathbf{u}d \varGamma^{\alpha}_{D} +\int_{\varGamma^{\alpha}_{D}}\lambda\frac{\gamma_\lambda}{h}w \cdot\mathbf{n}~ \mathbf{u}\cdot\mathbf{n}d\varGamma^{\alpha}_{D}. \end{aligned}$$

(35)

Note that these last two equations define bilinear operators for the models based on local and nonlocal fractional derivatives, respectively. h is a parameter given by

$$\begin{aligned} h= \begin{cases} 2 (\frac{\mathit{length}(\varGamma)}{\mathit{area}(\varOmega_e)}+\frac{\mathit{length}(\varGamma)}{\mathit{area}(\varOmega_{nb})} )^{-1} & \mbox{for }\varGamma\subset\varGamma_e\cap\varGamma_{nb}\\ \frac{\mathit{area}(\varOmega_e)}{\mathit{length}(\varGamma_{e})} & \mbox{for }\varGamma\subset\varGamma_e\cap\partial\varOmega. \end{cases} \end{aligned}$$

(36)

In Eqs. (34) and (35), γ _μ and γ _λ are penalty parameters and w is a weight function. Difference and averaging operators are defined as follows

$$\begin{aligned}{} [\phi] =& (\phi_{nb}-\phi_{e}), \end{aligned}$$

(37)

$$\begin{aligned} \langle\phi\rangle =& (\phi_{nb}+\phi_{e})/2. \end{aligned}$$

(38)

Appendix D: Numerical Implementation

Numerical implementation of Eq. (27) is done by first discretizing the momentum equation in space by using the bilinear and linear operators which are given in Eqs. (33), (34) and (35). Then the resulting systems of ordinary differential equations are solved by using the time discontinuous Galerkin method.

4.1 D.3 Space Discretization

Let u ^h(t,x) be the approximate solution for u(t,X), where $\mathbf{u}^{h}(t,\mathbf{x})=\sum_{i} \mathbf{u}^{h}_{i}(t) \mathbf{N}_{i}$ and summation is carried out over the nodes of elements and N _i is the base function. The bilinear operator can be written as follows after evaluating the integrals

$$\begin{aligned} A \bigl(\mathbf{u}^h(t,\mathbf{x}),\mathbf{w} \bigr)=\sum _{e=1}^N \bigl(\mathbf {K_e} \mathbf{u}^h_e-\mathbf{F_e} \bigl( \mathbf{u}^h_e \bigr)-\mathbf {F_{nb}} \bigl( \mathbf{u}^h_{nb} \bigr) \bigr). \end{aligned}$$

(39)

In the above equation, $\mathbf{K_{e}}$ is the element stiffness matrix. $\mathbf{F_{e}}$ and $\mathbf{F_{nb}}$ arises from the surface integrals along the element boundaries and can be regarded as force vectors acting on the surface of the element due to the internal and external displacements. For notational convenience, the superscript h is dropped for the following equations. The fractional derivatives of the base functions are evaluated analytically.

Body forces and boundary integrals subject to boundary conditions can be written as

$$\begin{aligned} \sum_{e=1}^N\mathbf{F_{b}} =&\sum_{\varOmega_e}\int_{\varOmega^{\alpha}_e}w \rho\mathbf{b}d\varOmega^{\alpha}_e +\lambda\sum_{\varGamma_e\in\varGamma_D}\int_{\varGamma_e}\nabla^{\alpha}w\cdot\mathbf{n}\hspace{1mm}\bar{\mathbf{u}}d\varGamma^{\alpha}_e +\sum_{\varGamma_e\in\varGamma_{N}}\int_{\varGamma_e}w\bar{\mathbf{t}}d\varGamma^\alpha_{e}. \end{aligned}$$

(40)

Then one can write the semi-discrete balance equation as follows

$$\begin{aligned} \sum_{e=1}^N \bigl( \mathbf{M_e}\ddot{\mathbf{u}}_e+\mathbf{K_e} \mathbf {u}_e-\mathbf{F_e}(\mathbf{u}_e)- \mathbf{F_{nb}}(\mathbf {u}_{nb})-\mathbf{F_{ext}} \bigr)=0. \end{aligned}$$

(41)

In the above equations $\mathbf{M_{e}}$ is the element mass matrix and $\mathbf{F_{ext}}=\mathbf{F_{b}}+\mathbf{F_{nb}}$. The semi-discrete balance equation can be written in the more compact form

$$\begin{aligned} \sum_{e=1}^N( \mathbf{M_e}\ddot{\mathbf{u}}_e+\tilde{\mathbf{K}}_{\mathbf{e}}\mathbf{u}_e-\mathbf{F_{ext}})=0, \end{aligned}$$

(42)

where $\tilde{\mathbf{K}}_{\mathbf{e}}$ is the modified element stiffness matrix which is defined as $\tilde{\mathbf{K}}_{\mathbf{e}}=\mathbf{K}_{\mathbf{e}}+\frac{\partial\mathbf{F}_{\mathbf{e}}}{\partial\mathbf{u}_{e}}$.

Equation (42) can be solved element by element by using the block Gauss-Seidel method along with the time integration method.

4.2 D.4 Time Integration

The time discontinuous Galerkin method (TDG) is used for the time integration. In the application of TDG, a double field formulation is used. The double field formulation consist combining the velocity and equation of motion in the form

$$\begin{aligned} &\int_I \mathbf{w}_t\left (\left [ \begin{array}{c@{\quad}c} \tilde{\mathbf{K}}&0\\ 0&\mathbf{M} \end{array} \right ] \left [ \begin{array}{c} \dot{\mathbf{u}}\\ \dot{\mathbf{v}} \end{array} \right ]+\left [ \begin{array}{c@{\quad}c} 0&-\tilde{\mathbf{K}}\\ \tilde{\mathbf{K}}&0 \end{array} \right ] \left [ \begin{array}{c} \mathbf{u}\\ \mathbf{v} \end{array} \right ] \right )dt +\mathbf{w}_t \bigl(t^+_n \bigr) \left [ \begin{array}{c@{\quad}c} \tilde{\mathbf{K}}&0\\ 0&\mathbf{M} \end{array} \right ] \left [ \begin{array}{c} \mathbf{u}(t^+_n)\\ \mathbf{v}(t^+_n) \end{array} \right ] \\ &\qquad {}-\mathbf{w}_t \bigl(t^+_n \bigr) \left [ \begin{array}{c@{\quad}c} \tilde{\mathbf{K}}&0\\ 0&\mathbf{M} \end{array} \right ]\left [ \begin{array}{c} \mathbf{u}(t^-_n)\\ \mathbf{v}(t^-_n) \end{array} \right ] \\ &\quad {}=\int_I \mathbf{w}_t\left [ \begin{array}{c} 0\\ \mathbf{F} \end{array} \right ] dt. \end{aligned}$$

(43)

In the above equations the subscripts of the vectors and matrices are dropped for the sake of simplicity. Thus, in the following section no subscript will be used to show the element-wise values. In Eq. (43) w _t is the weight function in time. We use 1st order Lagrange polynomials a base function. After evaluating the integrals in Eq. (43), we reach the following linear equation system

$$\begin{aligned} \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \frac{1}{2}\tilde{\mathbf{K}}&\frac{1}{2}\tilde{\mathbf{K}}&\frac{-1}{3}\mbox{dt} \tilde{\mathbf{K}}&\frac{1}{6}\mbox{dt} \tilde{\mathbf{K}}\\ \frac{-1}{2}\tilde{\mathbf{K}}&\frac{1}{2}\tilde{\mathbf{K}}&\frac{-1}{6}\mbox{dt} \tilde{\mathbf{K}}&\frac{-1}{3}\mbox{dt} \tilde{\mathbf{K}}\\ \frac{1}{3}\mbox{dt}\tilde{\mathbf{K}}&\frac{1}{6}\mbox{dt}\tilde{\mathbf{K}}&\frac{1}{2} \mathbf{M}&\frac{1}{2}\mathbf{M}\\ \frac{1}{6}\mbox{dt}\tilde{\mathbf{K}}&\frac{1}{3}\mbox{dt}\tilde{\mathbf{K}}&\frac{-1}{2}\mathbf{M}&\frac{1}{2}\mathbf{M} \end{array} \right ] \left [ \begin{array}{c} \mathbf{u}^+_{n}\\ \mathbf{u}^-_{n+1}\\ \mathbf{v}^+_{n}\\ \mathbf{v}^-_{n+1}\\ \end{array} \right ] =\left [ \begin{array}{c} \tilde{\mathbf{K}} \mathbf{u}_{n}^-\\ 0\\ \mathbf{F_{n}}+ \mathbf{M} \mathbf{v}_{n}^-\\ \mathbf{F_{n+1}}\\ \end{array} \right ]. \end{aligned}$$

(44)

Equation (44) is solved for each element sequentially as is done in the block Gauss-Seidel method in order to decrease the computational cost.