## 2 Dynamics of heterogeneous bodies
### Space-time discretized solution  
The weak form of the equilibrium equations (1)-(4), allowing for bulk and interface constitutive laws (9)-(10), reads [11; 12; 13]:  
\[ \begin{split}\text{find}\;\mathbf{u}\in\mathcal{U}:\quad\int \limits_{\Omega\backslash\Gamma}\varrho\,\mathbf{v}^{\text{T}}\ddot{\mathbf{u} }\,d\Omega\,+\int\limits_{\Omega\backslash\Gamma}\boldsymbol{\varepsilon}_{ \boldsymbol{v}}{}^{\text{T}}\mathbf{E}_{\Omega}\boldsymbol{\varepsilon}\,d \Omega+&\int\limits_{\Gamma}[\mathbf{v}]^{\text{T}}\mathbf{E}_{ \Gamma}[\mathbf{u}]\,d\Gamma\\ &=\int\limits_{\Omega\backslash\Gamma}\mathbf{v}^{\text{T}} \bar{\mathbf{b}}\,d\Omega+\int\limits_{\Gamma_{\tau}}\mathbf{v}^{\text{T}} \bar{\boldsymbol{\tau}}\,d\Gamma_{\tau}\quad\forall\mathbf{v}\in\mathcal{U}_{ 0}\end{split}  \tag{11}\]  
where: \(\mathbf{v}\) is the test function, and \(\boldsymbol{\varepsilon}_{\boldsymbol{v}}=\boldsymbol{\mathcal{C}}\mathbf{v}\); \(\mathcal{U}\) is the trial solution space, featuring displacement fields \(\mathbf{u}\) continuous in \(\Omega\backslash\Gamma\), possibly discontinuous along \(\Gamma\), and fulfilling the boundary condition (7) on \(\Gamma_{u}\); \(\mathcal{U}_{0}\) is the relevant variation space. In view of the assumed linearized kinematics, \(\Gamma\equiv\Gamma^{+}\equiv\Gamma^{-}\) holds for the interface. Now, let the finite element approximation of the displacement, displacement jump and deformation fields be:  
\[\mathbf{u}\cong \boldsymbol{\Phi}\mathbf{u}^{h} \tag{12}\] \[\left[\mathbf{u}\right]\cong \mathbf{B}_{\Gamma}\mathbf{u}^{h}\] (13) \[\boldsymbol{\varepsilon}\cong \mathcal{C}\boldsymbol{\Phi}\mathbf{u}^{h}=\mathbf{B}_{\Omega} \mathbf{u}^{h} \tag{14}\]  
where: \(\mathbf{u}^{h}\) is the nodal displacement vector; \(\boldsymbol{\Phi}\) is the matrix of nodal shape functions; \(\mathbf{B}_{\Gamma}\) and \(\mathbf{B}_{\Omega}\) are, respectively, the interface and bulk compatibility matrices. The semi-discretized equations of motion of the solid turn out to be:  
\[\text{M}\ddot{\mathbf{u}}^{h}+\left(\mathbf{K}_{\Omega}+\mathbf{K}_{\Gamma} \right)\mathbf{u}^{h}=\mathbf{F} \tag{15}\]  
where the mass matrix \(\mathbf{M}\), the bulk and interface stiffness matrices \(\mathbf{K}_{\Omega}\) and \(\mathbf{K}_{\Gamma}\), and the external load vector \(\mathbf{F}\) are:  
\[\mathbf{M} = \int\limits_{\Omega\backslash\Gamma}\varrho\,\mathbf{\Phi}^{\mathrm{ T}}\mathbf{\Phi}\,d\Omega \tag{16}\] \[\mathbf{K}_{\Omega} = \int\limits_{\Omega\backslash\Gamma}\mathbf{B}_{\Omega}^{\mathrm{ T}}\mathbf{E}_{\Omega}\mathbf{B}_{\Omega}\,d\Omega\] (17) \[\mathbf{K}_{\Gamma} = \int\limits_{\Gamma}\mathbf{B}_{\Gamma}^{\mathrm{ T}}\mathbf{E}_{\Gamma}\mathbf{B}_{\Gamma}\,d\Gamma\] (18) \[\mathbf{F} = \int\limits_{\Omega\backslash\Gamma}\mathbf{\Phi}^{\mathrm{ T}}\bar{\mathbf{b}}\,d\Omega+\int\limits_{\Gamma_{\tau}}\mathbf{\Phi}^{ \mathrm{ T}}\bar{\boldsymbol{\tau}}\,d\Gamma_{\tau} \tag{19}\]  
To advance the solution of (15) in time we employ an explicit Newmark method. Having partitioned the time interval of interest according to \([t_{0}\quad t_{N}]=\cup_{i=0}^{N_{t}-1}[t_{i}\quad t_{i+1}]\), within the time step \([t_{i}\quad t_{i+1}]\) a predictor-integrator-corrector splitting is followed according to:  
* predictor: \[\tilde{\mathbf{u}}_{i+1}= \mathbf{u}_{i}+\Delta t\,\dot{\mathbf{u}}_{i}+\Delta t^{2}\left( \frac{1}{2}-\beta\right)\ddot{\mathbf{u}}_{i}\] (20) \[\dot{\mathbf{u}}_{i+1}= \dot{\mathbf{u}}_{i}+\Delta t(1-\gamma)\ddot{\mathbf{u}}_{i}\] (21) where \(\Delta t=t_{i+1}-t_{i}\);
* explicit integrator: \[\left[\mathbf{u}\right]\cong \mathbf{B}_{\Gamma}\mathbf{u}^{h}\]0 (22)
* corrector: \[\left[\mathbf{u}\right]\cong \mathbf{B}_{\Gamma}\mathbf{u}^{h}\]1 (23) \[\left[\mathbf{u}\right]\cong \mathbf{B}_{\Gamma}\mathbf{u}^{h}\]2 (24)  
To resemble the average acceleration scheme, which is unconditionally stable and second-order accurate, \(\beta=0.25\) and \(\gamma=0.5\) are adopted in the preceding equations.