## 2 Forward problem: layered composite dynamics  
Let us consider a layered body \(\Omega\), possessing a smooth outer boundary with unit outward normal \(\mathbf{m}\) (see Figure 1). \(\Omega\) be crossed by \(n_{\Gamma}\) non-intersecting surfaces, or interfaces \(\Gamma_{j}\), \(j=1,...,n_{\Gamma}\). Even though the local orientation \(\mathbf{n}_{j}\) of these interfaces may vary in \(\Omega\), for ease of notation we shall assume it to be constant and equal to \(\mathbf{n}\) along every \(\Gamma_{j}\): interfaces are thus flat and parallel to each other. The equilibrium of \(\Omega\) at time \(t\) is governed by:  
\[\mathbf{\mathcal{C}}^{\mathrm{T}}\mathbf{\sigma}+\bar{\mathbf{b}}=\varrho\vec{\mathbf{u}} \qquad\text{in }\Omega\backslash\Gamma \tag{1}\]  
\[\mathbf{\mathcal{N}}\mathbf{\sigma}=-\mathbf{\tau}_{j}\qquad\text{on }\Gamma_{j}^{+}, \mathbf{\mathcal{N}}\mathbf{\sigma}=\mathbf{\tau}_{j}\qquad\text{on }\Gamma_{j}^{-}, j=1,...,n_{\Gamma} \tag{2}\]  
\[\mathbf{\mathcal{M}}\mathbf{\sigma}=\bar{\mathbf{\tau}}\qquad\text{on }\Gamma_{\tau} \tag{3}\]  
where: \(\Gamma=\cup_{j=1}^{n_{\Gamma}}\Gamma_{j}\); \(\Gamma_{\tau}\) is the portion of the outer boundary of \(\Omega\) where tractions \(\bar{\mathbf{\tau}}\) are assigned. Treating each \(\Gamma_{j}\) as a locus where the displacement field \(\mathbf{u}\) may suffer jumps (alike a crack), \(\Gamma_{j}^{+}\) and \(\Gamma_{j}^{-}\) are the two sides of \(\Gamma_{j}\); these two sides can not be distinguished in the initial configuration (at \(t=0\)) but, as soon as delamination occurs, they experience a relative movement. In the above equations, according to Voigt notation: \(\mathbf{\sigma}\) is the stress vector; \(\mathbf{\tau}_{j}\) is the traction vector acting along surface \(\Gamma_{j}\); \(\bar{\mathbf{b}}\) is the assigned bulk load in \(\Omega\backslash\Gamma\); \(\varrho\) is the bulk mass density; \(\vec{\mathbf{u}}\) is the acceleration field in \(\Omega\backslash\Gamma\); \(\mathbf{\mathcal{C}}\) is the differential compatibility operator; \(\mathbf{\mathcal{M}}\) and \(\mathbf{\mathcal{N}}\) are the matrices collecting the components of unit vectors \(\mathbf{m}\) and \(\mathbf{n}\), respectively. Assuming linearized kinematics, compatibility reads:  
\[\mathbf{\varepsilon}=\mathbf{\mathcal{C}}\mathbf{u}\qquad\text{in }\Omega\backslash\Gamma \tag{4}\]  
\[[\mathbf{u}]_{j}=\mathbf{u}\bigg{|}_{\Gamma_{j}^{+}}-\mathbf{u}\bigg{|}_{ \Gamma_{j}^{-}}\qquad\qquad j=1,...,n_{\Gamma} \tag{5}\]  
Figure 1: Notation adopted for layered bodies (here \(n_{\Gamma}=2\)). \[\mathbf{u}=\bar{\mathbf{u}}\qquad\text{on }\Gamma_{u} \tag{6}\]  
where: \(\boldsymbol{\varepsilon}\) is the strain vector; \(\bar{\mathbf{u}}\) is the assigned displacement along portion \(\Gamma_{u}\) of the outer boundary of \(\Omega\); \([\mathbf{u}]_{j}\) is the displacement discontinuity along surface \(\Gamma_{j}\). The body is assumed to be initially at rest, in an undeformed and unstressed state:  
\[\mathbf{u}_{0}=\mathbf{0},\qquad\dot{\mathbf{u}}_{0}=\mathbf{0}\qquad\qquad \text{in }\Omega \tag{7}\]  
In case of low velocity impacts, dissipation due to intra-laminar damage is assumed to be negligible when compared to that due to delamination. Hence, laminae are thought to behave elastically according to:  
\[\boldsymbol{\sigma}=\mathbf{E}_{\Omega}\boldsymbol{\varepsilon}\qquad\text{ in }\Omega\backslash\Gamma \tag{8}\]  
where \(\mathbf{E}_{\Omega}\) is the elasticity matrix of the bulk. As for inter-laminar phases, to be treated as lumped surfaces along which debonding is the result of complex micro-scale damaging processes, a phenomenological coarse-grained modeling is adopted. Strength reduction preceding delamination is governed by softening constitutive laws [3; 16; 17; 18; 19; 20; 21], which locally link tractions \(\boldsymbol{\tau}_{j}\) (see Eq. 2) to the displacement jump \([\mathbf{u}]_{j}\) (see Eq. 5).