## 2 Dynamics of heterogeneous bodies  
### Governing relations  
Let \(\Omega\) be a heterogeneous three-dimensional body; its smooth outer boundary, with unit outward normal \(\mathbf{m}\), be constituted by the two disjoint sets \(\Gamma_{u}\) and \(\Gamma_{\tau}\), where displacements and tractions are respectively assigned. Without any loss of generality, let us assume that \(\Omega\) is a continuum made of two phases (\(\Omega^{+}\) and \(\Omega^{-}\), with \(\Omega=\Omega^{+}\cup\Omega^{-}\)), tied together in the initial configuration along the flat interface \(\Gamma\), see Figure 1. \(\Gamma\) actually represents the interphase between \(\Omega^{+}\) and \(\Omega^{-}\); since the thickness of this interphase is usually at least one order of magnitude smaller than the characteristic size of \(\Omega\), the interphase itself is modeled as a zero-thickness interface. Damaging processes along \(\Gamma\) may cause opening (mode I) and/or sliding (mode II and mode III) displacement discontinuities in the \(\mathbf{n}\) direction and in the \(\mathbf{s}_{1}-\mathbf{s}_{2}\) plane, respectively.  
The equilibrium of \(\Omega\) at time \(t\) is governed by:  
\[\mathcal{C}^{\mathrm{T}}\boldsymbol{\sigma}+\bar{\mathbf{b}}= \varrho\vec{\mathbf{u}} \text{in }\Omega\backslash\Gamma \tag{1}\] \[\boldsymbol{\mathcal{M}}\boldsymbol{\sigma}= \bar{\boldsymbol{\tau}} \text{on }\Gamma_{\tau}\] (2) \[\boldsymbol{\mathcal{N}}\boldsymbol{\sigma}= -\boldsymbol{\tau} \text{on }\Gamma^{+}\] (3) \[\boldsymbol{\mathcal{N}}\boldsymbol{\sigma}= \boldsymbol{\tau} \text{on }\Gamma^{-} \tag{4}\]  
where \(\Gamma^{+}\) and \(\Gamma^{-}\) are the sides of \(\Gamma\) respectively belonging to \(\Omega^{+}\) and \(\Omega^{-}\), and according to [PERSON]'s notation [10]: \(\boldsymbol{\sigma}\) is the stress vector, which gathers the independent components of the stress tensor; \(\bar{\mathbf{b}}\) and \(\bar{\boldsymbol{\tau}}\) are the assigned loads in the bulk \(\Omega\backslash\Gamma\) and along \(\Gamma_{\tau}\); \(\varrho\) is the mass density of the bulk material; \(\vec{\mathbf{u}}\) is the acceleration in \(\Omega\backslash\Gamma\); \(\boldsymbol{\mathcal{C}}\) is the differential compatibility operator; \(\boldsymbol{\mathcal{M}}\) and \(\boldsymbol{\mathcal{N}}\) are the matrices collecting the components of the unit vectors \(\mathbf{m}\) and \(\mathbf{n}\).  
In the small strain regime, compatibility reads:  
\[\boldsymbol{\varepsilon}= \boldsymbol{\mathcal{C}}\mathbf{u} \text{in }\Omega\backslash\Gamma \tag{5}\] \[\left[\mathbf{u}\right]= \mathbf{u}\bigg{|}_{\Gamma^{+}}-\mathbf{u}\bigg{|}_{\Gamma^{-}} \text{on }\Gamma\] (6) \[\mathbf{u}= \bar{\mathbf{u}} \text{on }\Gamma_{u} \tag{7}\]  
where: \(\boldsymbol{\varepsilon}\) is the strain vector; \(\mathbf{u}\) is the displacement in \(\Omega\); \(\left[\mathbf{u}\right]\) is the displacement discontinuity along \(\Gamma\); \(\bar{\mathbf{u}}\) is the assigned displacement along \(\Gamma_{u}\).  
Figure 1: Geometry of a two-phase body, and notation.  
The body is conceived to be initially at rest, i.e.:  
\[\mathbf{u}_{0}=\mathbf{0},\quad\dot{\mathbf{u}}_{0}=\mathbf{0}\qquad\qquad\text{ in }\Omega \tag{8}\]  
As far as constitutive modeling is concerned, the bulk is assumed to behave elastically according to:  
\[\boldsymbol{\sigma}=\mathbf{E}_{\Omega}\boldsymbol{\varepsilon}\qquad\text{ in }\Omega\backslash\Gamma \tag{9}\]  
where \(\mathbf{E}_{\Omega}\) is the bulk elasticity matrix. Along \(\Gamma\), damage can be incepted once a local strength criterion is satisfied; to simplify matters we assume that the stress waves do not cause any dissipative phenomena. The interface thus behaves elastically too, according to:  
\[\boldsymbol{\tau}=\mathbf{E}_{\Gamma}[\mathbf{u}]\qquad\text{on }\Gamma \tag{10}\]  
\(\mathbf{E}_{\Gamma}\) being the interface elasticity matrix.