## 2 Forward problem: layered composite dynamics
Since both opening/closing (along \(\mathbf{n}\)) and sliding (in the \(\mathbf{s}_{1}^{j}-\mathbf{s}_{2}^{j}\) plane) discontinuities are expected to take place under general loading conditions, interface kinematics is effectively represented by the displacement discontinuity \([u]\) (in what follows, for ease of notation, we shall drop subscript \(j\)), defined as [22; 23; 24]:  
\[[u]=\sqrt{[u]_{n}^{2}+\kappa^{2}[u]_{s}^{2}} \tag{9}\]  
where: \([u]_{n}=[\mathbf{u}]^{\mathsf{T}}\mathbf{n}\) and \([u]_{s}=|[\mathbf{u}]-[u]_{n}\mathbf{n}|\) are the opening and sliding displacement discontinuities, respectively; \(\kappa\) is a coupling coefficient, which phenomenologically accounts for the local ongoing  
\[\mathbf{\mathcal{N}}\mathbf{\sigma}=-\mathbf{\tau}_{j}\qquad\text{on }\Gamma_{j}^{+}, \mathbf{\mathcal{N}}\mathbf{\sigma}=\mathbf{\tau}_{j}\qquad\text{on }\Gamma_{j}^{-}, j=1,...,n_{\Gamma} \tag{2}\]0 Piecewise linear (PWL) and linear-exponential (L-E) effective traction-displacement discontinuity relationships. mixed-mode damaging in the inter-laminar phase. The effective traction \(\tau\), work-conjugate to \([u]\), reads (see [22, 23]):  
\[\mathbf{\mathcal{N}}\mathbf{\sigma}=-\mathbf{\tau}_{j}\qquad\text{on }\Gamma_{j}^{+}, \mathbf{\mathcal{N}}\mathbf{\sigma}=\mathbf{\tau}_{j}\qquad\text{on }\Gamma_{j}^{-}, j=1,...,n_{\Gamma} \tag{2}\]1  
where \(\tau_{n}\) and \(\tau_{s}\) are the normal and shear traction components. Since quasi-brittle materials (like inter-phase resins) show a high ratio between compressive and tensile strengths, inter-laminar damage in compression due to low velocity impacts can be disregarded. On the other hand, under tensile stress pulses the response of inter-phase materials is characterized by softening beyond the attainment of a peak traction. Two simple effective descriptions of this behavior are here envisaged: a piecewise linear (PWL) law  
\[ \begin{cases}\tau=K[u]&\text{if }[u]\leq[u]_{e}\\ \tau=\tau_{M}+Q\left([u]-[u]_{e}\right)&\text{if }[u]_{e}<[u]\leq[u]_{U}\\ \tau=0&\text{if }[u]>[u]_{U}\end{cases}  \tag{11}\]  
and a linear-exponential (L-E) law  
\[ \begin{cases}\tau=K[u]&\text{if }[u]\leq[u]_{e}\\ \tau=\tau_{M}\exp\left(-\varsigma([u]-[u]_{e})\right)&\text{if }[u]>[u]_{e} \end{cases}  \tag{12}\]  
In these cohesive-like envelopes: \([u]_{e}\) is the effective displacement discontinuity at the attainment of the peak traction \(\tau_{M}=K[u]_{e}\) (elastic limit); \(K\) is the stiffness in the elastic regime; \(Q\) is the (negative) slope of the softening branch in the PWL case; \([u]_{U}=\left(1-\frac{K}{Q}\right)[u]_{e}\) is the effective displacement discontinuity beyond which the interaction of opened crack faces is annihilated in the PWL case; \(\varsigma\) defines the slope of the softening branch in the L-E case. The two effective models (11) and (12) are compared, for \([u]>0\), in Figure 2 at assigned effective fracture toughness, or work of separation \(G=\int\limits_{0}^{\infty}\tau\,d[u]\). It is worth noting that, since the L-E law assumes the interaction between crack faces to continue up to \([u]\rightarrow\infty\), delamination growth is modeled via a breakdown threshold; according to [25, 26], we assume this interaction to suddenly vanish as soon as the current traction \(\tau\) becomes smaller than a pre-defined fraction (say 5-10 %) of the peak value \(\tau_{M}\). To allow for irreversibility of the micro-mechanical damage processes, unloading from the tensile envelope follows a radial path to the origin of the \(\tau-[u]\) plane. In rate form, constitutive modeling for interface \(\Gamma_{j}\) can thus be written:  
\[\dot{\mathbf{\tau}}_{j}=\mathbf{E}_{\Gamma}[\mathbf{\dot{u}}]_{j} \tag{13}\]  
where \(\mathbf{E}_{\Gamma}\) is the interface tangent stiffness matrix. For additional details, readers are referred to [23, 24].