Formulation ########################## .. _fault_plane-fig: .. figure:: ./Doc_figs/Three_Dimensional_interface.png :alt: Interfac_Plane :align: center A three-dimensional rupture propagation in a plane. Consider a 3D fracture plane as given shown in :numref:`Fig. %s `. The elastodynamic equations at the interface between two dissimilar elastic half-spaces is given as follows (`Kostrov, 1966 `_; `Gupta and Ranjith, 2024 `_): .. math:: A_{ij}(\tau_{i}(x_1,x_3,t)-\tau_{i}^0(x_1,x_3,t))+ \dot{\delta}_i(x_1,x_3,t) = f_i (x_1,x_3,t) where, :math:`x_i` is a Cartesian coordinate system, with :math:`x_1-x_3` plane being the plane of interface and time :math:`t`. :math:`\left[A_{ij} \right]` is a diagonal matrix with :math:`A_{11} = A_{33} = \frac{c_s^+}{\mu^+}`, :math:`A_{22} = \frac{c_d^+}{\lambda^+ + 2 \mu^+}+\frac{c_d^-}{\lambda^- + 2 \mu^-}`. :math:`\tau_{i}`'s are the traction components at the interface with stresses at the interface being :math:`\tau_{i}(x_1,x_3,t)= \sigma_{i2}(x_1,x_2^{\pm},x_3,t)`. :math:`\tau_{i}^0(x_1,x_3,t)` are the initial traction components. :math:`\dot{\delta}_i` is the rate of discontinuity at the interface with :math:`\delta_i` given in terms of displacements :math:`u_i` of the two half-spaces as :math:`\delta_i = u_i^+ - u_i^-`. The superscripts “:math:`\pm`” refer to field quantities corresponding to the top and lower half-space, respectively. :math:`\lambda` and :math:`\mu` are the Lame\'s parameters. The shear and dilatational wave speeds are given by :math:`c_s = \sqrt{\mu/\rho}` and :math:`c_d = \sqrt{(\lambda +2\mu )/\rho}`, respectively, with :math:`\rho` being the density of the elastic half-space. The functionals :math:`f_i`'s involve the spatio-temporal convolutions of traction components at the interface. Spectral BIEM ******************* In spectral BIEM, the field quantities traction and slip (displacement discontinuities) at the interface are taken in spectral form as: .. math:: \begin{Bmatrix} \tau_j(x_1,x_3,t) \\ \delta_j(x_1,x_3,t) \\ \end{Bmatrix} = \begin{Bmatrix} T_j(t;k,m) \\ D_j(t;k,m) \\ \end{Bmatrix} e^{i(kx_1+mx_3)}, where :math:`k` and :math:`m` are wave numbers in :math:`x_1` - and :math:`x_3` - directions. Let, :math:`q = (k , m)` is a two-dimensional wave vector, with a magnitude :math:`q=\sqrt{k^2 + m^2}`. We get the 3D Spectral BIEM relations (see. Eq. 19 in `Gupta and Ranjith, 2024 `_) at the interface of two dissimilar half-spaces as: .. math:: \begin{bmatrix} \frac{c_s^+}{\mu^+} \eta T_1 \left( t; k, m \right) + \frac{\partial D_1\left(t; k, m \right)}{\partial t} \\ \frac{c_s^+}{\mu^+} \xi T_2 \left( t; k, m \right) + \frac{\partial D_2\left(t; k, m \right)}{\partial t} \\ \frac{c_s^+}{\mu^+} \eta T_3 \left( t; k, m \right) + \frac{\partial D_3\left(t; k, m \right)}{\partial t} \end{bmatrix} = \begin{bmatrix} F_1 \left( t; k, m \right) \\ F_2 \left( t; k, m \right) \\ F_3 \left( t; k, m \right) \end{bmatrix} where, :math:`\eta = \left( 1+ \frac{c_s^-}{c_s^+}\frac{\mu^+}{\mu^-}\right)` and :math:`\xi = \left( \frac{c_s^+}{c_d^-} + \frac{c_s^-}{c_s^+}\frac{\mu^+}{\mu^-}\frac{c_s^-}{c_d^-}\right)`, and the convolution terms are given by .. math:: \begin{Bmatrix} F_1 \left( t; k, m \right) \\ F_2 \left( t; k, m \right) \\ F_3 \left( t; k, m \right) \end{Bmatrix} = \int_0^t \begin{bmatrix} C_{11} \left( t'; q \right) & C_{12} \left( t'; q \right) & C_{13} \left( t'; q \right) \\ C_{21} \left( t'; q \right) & C_{22} \left( t'; q \right) & C_{23} \left( t'; q \right) \\ C_{31} \left( t'; q \right) & C_{32} \left( t'; q \right) & C_{33} \left( t'; q \right) \end{bmatrix} \begin{bmatrix} T_1 \left( t - t'; k, m \right) \\ T_2 \left( t - t'; k, m \right) \\ T_3 \left( t - t'; k, m \right) \end{bmatrix} dt' In the above equation, we have five independent convolution kernels which can be expressed as .. math:: C_{11} (t; q) = \frac{1}{q^2} [k^2 (M_{11}^+ + M_{11}^-) + m^2 (M_{33}^+ + M_{33}^-)] \\ C_{22}(t; q) = M_{22}^+ + M_{22}^- \\ C_{33}(t; q) = \frac{1}{q^2} [m^2 (M_{11}^+ + M_{11}^-) + k^2 (M_{33}^+ + M_{33}^-)] \\ C_{12}(t; q) = \frac{k}{m} C_{22}(t; q) = -\frac{m}{k} C_{33}(t; q) = -C_{21}(t; q) = \frac{k}{q^2} [M_{12}^+ - M_{12}^-] \\ C_{13}(t; q) = C_{31}(t; q) = \frac{km}{q^2} [ (M_{11}^+ + M_{11}^-) - (M_{33}^+ + M_{33}^-) ] All above convolution kernels :math:`\left( C_{ij}(t; q) \right)` can be expressed in closed form. The convolution kernels :math:`M_{11}(t; q), M_{12}(t; q), M_{22}(t; q)` and :math:`M_{33}(t; q)` can be written as in the Appendix of `Gupta and Ranjith, 2024 `_): .. math:: \begin{split} M_{11}(t; q) =& c_s \frac{c_s}{\mu} |q| \Bigg\lbrace - \int_0^1 \frac{g(y) - g(y_R)}{y - y_R} y \sin(|q| c_s t) \, dy \\ & + g(y_R) \Bigg[y_R \sin(|q| c_R t) \int_{|q|c_s t - |q|c_R t}^{|q|c_s t + |q|c_R t} \cos \theta \, d\theta - y_R \cos(|q|c_R t) \int_{-|q|c_R t}^{|q|c_R t} \sin\theta \, d\theta \Bigg] \\ & + g(y_R) \frac{\cos(|q|c_s t) - 1}{|q| c_s t} \frac{2}{\pi} \int_1^a \frac{4(1-y^2)\sqrt{1-y^2/a^2}}{(y^2 - y_R^2)(y^2-y_1^2)(y^2-y_2^2)} y \sin(|q|c_s t) \, dy \Bigg\rbrace \end{split} .. math:: \begin{split} M_{22}(t; q) =& c_s \frac{c_s}{\mu} |q| \Bigg\lbrace - \int_0^1 \frac{f(y) - f(y_R)}{y - y_R} y \sin(|q| c_s t) \, dy \\ & + f(y_R) \Bigg[ y_R \sin(|q| c_R t) \int_{|q|c_s t - |q|c_R t}^{|q|c_s t + |q|c_R t} \cos \theta \, d\theta - y_R \cos(|q|c_R t) \int_{-|q|c_R t}^{|q|c_R t} \sin\theta \, d\theta \Bigg] \\ & + f(y_R) \frac{\cos(|q|c_s t) - 1}{|q| c_s t} \frac{2}{\pi} \int_1^a \frac{(2-y^2)\sqrt{1-y^2/a^2}}{(y^2 - y_R^2)(y^2-y_1^2)(y^2-y_2^2)} y \sin(|q|c_s t) \, dy \Bigg\rbrace \end{split} .. math:: \begin{split} M_{12}(t; q) =& -iq c_s \frac{c_s}{\mu} \frac{2}{\pi} \int_1^a \Bigg[ \frac{2y\sqrt{2 - 1 - y^2 / a^2}(2-y^2)}{(y_R^2 - y^2)(y_1^2 - y^2)(y_2^2 - y^2)} \cos(y|q|c_s t) \\ & + \frac{-8y_R^2/a^2 + 8/a^2 - y_R^4 + 4 + 6y_R^2 + 2\sqrt{1-y_R^2/a^2}(2-y_R^2)}{(y_1^2 - y_R^2)(y_2^2 - y_R^2)} \cos(|q|c_R t) \Bigg] \end{split} .. math:: M_{33}(t; q) = c_s \frac{c_s}{\mu} J_1(|q|c_s t) with .. math:: g(y) = \frac{2}{\pi} \frac{4\,(1-y^2)\, \sqrt{1-y^2/a^2} + (2-y^2)^2 \sqrt{1-y^2}}{(y + y_R) \, (y^2 - y_1^2)\, (y^2 - y_2^2)} .. math:: f(y) = \frac{2}{\pi} \frac{4\,(1-y^2/a^2)\, \sqrt{1-y^2} + (2-y^2)^2 \sqrt{1-y^2/a^2}}{(y + y_R) \, (y^2 - y_1^2)\, (y^2 - y_2^2)} where :math:`s = p / \|q\| c_s, a = c_d / c_s`, and :math:`y_R = c_R / c_s`. :math:`c_R` is the Rayleigh wave speed. .. Note that, in the current scheme, all the kernels are obtained as a closed-form solution at the interface. The formulation for these kernels can be found in \cite{Gupta2024Spectral}. Interface Law **************************** We have incorporated the classical linear friction slip-weakening law in the code. The frictional coefficients are vectorized over the fault plane so that one can analyze the effect of local heterogeneities. One can look into the `main.cuf` in the `./src` directory to add more frcitional law as an when necessary. Linear Slip-Weakening law ----------------------------------- The linear slip-weakening friction law is as follows: .. math:: \tau_2^{*}(\delta) = \tau_2 \begin{cases} \mu_s -(\mu_s - \mu_r) \frac{\delta}{\delta_{c}}, & \text{if $\delta < \delta_c$}\\ \mu_r, & \text{otherwise} \end{cases} where, :math:`\tau_2^{*}` is the effective shear strength of fault plane, :math:`\delta=\sqrt{(\delta_1^2 + \delta_3^2)}` is the effective slip, and :math:`\delta_{c}` is the critical slip-weakening distance. :math:`\mu_s` and :math:`\mu_r` are the static and residual frictional coefficients, respectively.