# Damage Behaviour of Quasi-Brittle Composites: Mathematical and Computational Aspects
## 4 Possible Generalizations
### Structured Deformation
This makes it possible to introduce the space of structured deformation \[SD(\Omega)=\{(\mathfrak{G},G):\,\mathfrak{G}\in SBV(\Omega,\mathcal{R}^{N}),\,G\in L ^{1}(\Omega,\mathcal{R}^{N\times N})\}\,; \tag{34}\]  
\(\
abla\mathfrak{G}-G\) derived from (34) refers to the disarrangement tensor. The following convergence result is available. For any \((\mathfrak{G},G)\in SD(\Omega)\), there exists such \(\mathfrak{U}_{n}\in SBV(\Omega,\mathcal{R}^{N})\) that  
\[\mathfrak{U}_{n}\to\mathfrak{G}\text{ in }L^{1}(\Omega,\mathcal{R}^{N})\,, \qquad\
abla\mathfrak{U}_{n}\overset{\simeq}{\rightharpoonup}G\text{ in }\mathcal{M}(\Omega,\mathcal{R}^{N\times N})\,, \tag{35}\]  
using the standard notion of strong- and weak-star convergence. The original proof of this result by [99] combines the Alberti theorem by [100] with certain approximations of functions with bounded variations by piecewise constant ones; an alternative proof using a weaker approximation (avoiding [100]) can be found in [94]. The jumps in \(\mathfrak{F}_{n}\) contribute to the interfacial part of an initial energy response \(E(\mathfrak{F}_{n})\). They diffuse throughout portions of the body and generate, in the limit, all bulk and interfacial parts of a relaxed energy response \(I(\mathfrak{G},G)\). Introducing bulk densities \(W\) and interfacial densities \(\Psi\), a \((N-1)\)-rectifiable jump set \(S(u)\) with its unit normals \(\
u(\mathfrak{U})\) for any admissible function \(u\) with non-zero jumps \([\mathfrak{U}]\) there, together with the functional  
\[E(\mathfrak{U})=\int_{\Omega}W(\
abla\mathfrak{U}(x))\,\mathrm{d}x+\int_{S( \mathfrak{U})}\Psi\big{(}[\mathfrak{U}(x)],\
u(\mathfrak{U})\big{)}\,\mathrm{ d}s(x)\,. \tag{36}\]  
The aim is to come from (36) to  
\[I(\mathfrak{G},G)=\inf\Bigl{(}\liminf_{n\to\infty}E(\mathfrak{U}_{n})\Bigr{)} \tag{37}\]  
with \(\mathfrak{U}_{n}\in SBV(\Omega,\mathcal{R}^{N})\) satisfying (35). For a rather wide class of energy functionals with carefully introduced \(W\) and \(\Psi\), ref. [99] performs such relaxation utilising the so-called blow-up method by [101]. For more details on explicit formulae for interfacial energies, optimal design, and dimension reduction, see [93]. Now, we are able to come back to the considerations of Section 4.3, with \(\theta\) referring to the positive Kelvin temperature. Let us use the symbol \(\partial_{z}\) for partial derivatives with respect to a variable \(z\) here, as well as upper dots for partial time derivatives. Taking, instead of \(I(\mathfrak{G},G)\) by (37), some \(\psi(F,G,\mathcal{Q},\theta)\), interpretable as the Helmholtz free energy, containing the deformation gradient \(F=\
abla g\), admitting needed multiplicative decompositions, one can formulate the internal energy  
\[\omega(F,G,\mathcal{Q},\theta,\mathfrak{H})=\psi(F,G,\mathcal{Q},\theta)+ \mathfrak{H}\theta\,, \tag{38}\]  
whose setting  
\[\mathfrak{H}=-\partial_{\theta}\psi(F,G,\mathcal{Q},\theta) \tag{39}\]  
for the specific entropy \(\mathfrak{H}\) is well known as the Gibbs relation; \(\mathcal{Q}\) denotes a vector of appropriate internal parameters, and \(\zeta(\hat{F},\hat{G},\hat{Q})\) will refer to certain potential of dissipative forces. Differentiating (38) in time, thanks to the obvious consequence of (39) \(\partial_{\theta}\psi\dot{\theta}+\dot{\theta}\mathfrak{H}=0\), we obtain  
\[\dot{\omega}=\partial_{F}\psi:\hat{F}+\partial_{G}\psi:\hat{G}+\partial_{ \mathcal{Q}}\psi\cdot\dot{\mathcal{Q}}+\theta\dot{\mathfrak{H}}\,. \tag{40}\]  
The derivation of the system of partial differential equations of evolution, coming from (40), comparable with (31), is rather difficult, with expectable complicated results, even in special cases significant in engineering practice; for more details, see [48].