## 2 Forward problem: layered composite dynamics
Upon space discretization via displacement-based finite elements, the weak form of the equilibrium equations for the layered body reads:  
\[\mathbf{M}\mathbf{\dot{u}}^{h}+\mathbf{K}_{\Omega}\mathbf{u}^{h}+\sum\limits_{ j=1}^{n_{\Gamma}}\mathbf{R}^{j}=\mathbf{F} \tag{14}\]where \(\mathbf{u}^{h}\) is the vector of nodal displacements. In (14) the mass matrix \(\mathbf{M}\), the bulk stiffness matrix \(\mathbf{K}_{\Omega}\), the internal force vectors \(\mathbf{R}^{j}\), \(j=1,...,n_{\Gamma}\), and the external load vector \(\mathbf{F}\) are respectively given by:  
\[\mathbf{M}= \int\limits_{\Omega\setminus\Gamma}\varrho\,\mathbf{\Phi}^{ \mathrm{T}}\mathbf{\Phi}\,d\Omega \tag{15}\] \[\mathbf{K}_{\Omega}= \int\limits_{\Omega\setminus\Gamma}\mathbf{B}_{\Omega}^{ \mathrm{T}}\mathbf{E}_{\Omega}\mathbf{B}_{\Omega}\,d\Omega\] \[\mathbf{R}^{j}= \int\limits_{\Gamma_{j}}\mathbf{B}_{\Gamma_{j}}^{\mathrm{T}} \boldsymbol{\tau}_{j}\,d\Gamma_{j}\] \[\mathbf{F}= \int\limits_{\Omega\setminus\Gamma}\mathbf{\Phi}^{\mathrm{T}} \bar{\mathbf{b}}\,d\Omega+\int\limits_{\Gamma_{\tau}}\mathbf{\Phi}^{\mathrm{ T}}\bar{\tau}\,d\Gamma_{\tau}\]  
Here: \(\mathbf{\Phi}\) is the matrix of nodal shape functions; \(\mathbf{B}_{\Omega}\) and \(\mathbf{B}_{\Gamma_{j}}\) are, respectively, the compatibility matrices for the bulk \(\Omega\backslash\Gamma\) and surface \(\Gamma_{j}\)[7, 27]. In the space-discretized formulation here above we have implicitly assumed that the displacement field \(\mathbf{u}^{h}\) may be discontinuous only along inter-element boundaries. Smarter formulations [23, 28, 29, 30, 31] allow also for intra-element discontinuities by exploiting the partition of unity property of standard nodal shape functions [32]; since delamination in layered continua occurs along the a-priori known inter-laminar surfaces, the description of the discontinuity topology is simple and the aforementioned feature of the smart (usually termed extended) finite element formulation is of no help. We now need to adopt a time stepping scheme for (14), able to reduce to a minimum or even avoid spurious high frequency oscillations of the velocity field in the fully-discretized solution. In [33] it was shown that local corrective flux terms can be adopted to accurately track discontinuities in the velocity field due to the propagation of stress wave fronts. Since this time stepping scheme is computationally expensive while our target is a low-cost scheme, a less expensive (and, indeed, less accurate) explicit \(\alpha-\)method is adopted [34, 35]. Having partitioned the time interval of interest according to \([t_{0}\quad t_{N}]=\cup_{i=0}^{N-1}[t_{i}\quad t_{i+1}]\), the solution of (14) at time \(t_{i+1}\  
* correction: \[\mathbf{u}_{i+1}^{h}= \tilde{\mathbf{u}}_{i+1}^{h}+\Delta t^{2}\beta\,\tilde{\mathbf{u}} _{i+1}^{h}\] (19) \[\dot{\mathbf{u}}_{i+1}^{h}= \tilde{\mathbf{u}}_{i+1}^{h}+\Delta t\gamma\,\tilde{\mathbf{u}} _{i+1}^{h}\] (20)  
To get oscillations of small amplitude in the velocity field, the following values of the algorithmic parameters \(\alpha\), \(\beta\) and \(\gamma\) have been adopted: \(\alpha=-0.3\); \(\beta\) and \(\gamma\) finely tuned around values allowing second-order accuracy in linear elastodynamics [36]. The time step size \(\Delta t\) has been set so as to fulfill the Courant condition in the bulk. In case of growing delamination, interface elements in the process zone (where damage is locally reducing the inter-laminar strength and stiffness) need to be sized so as to accurately resolve the traction field; this requirement is satisfied if the characteristic size of interface elements is a fraction (say \(\frac{1}{3}-\frac{1}{5}\)) of the process zone length, estimated e.g. through [PERSON]'s law [37]. Since the surrounding FEs in the bulk are associated to interface elements via matching grids, it turns out that inter-laminar strength and toughness, which affect the process zone length, actually affect the time step size \(\Delta t\) too; for details, readers are referred to [38; 39].