## Final Activity Report Summary - NONLOCCOMPHOM (Nonlocal computational homogenisation)

The failure processes of real-world materials are usually explained by the propagation of pre-existing micro-defects, resulting in a progressive loss of material integrity up to the point of macroscopic collapse. Currently, two mature theories are available for the damage description. The macro-scale phenomena can be efficiently addressed in the framework of continuum damage mechanics, which models the microstructural changes using a set of internal variables and a heuristically postulated evolution law. On the other hand, the effects of distributed heterogeneities can be rather accurately quantified using the tools of theory of heterogeneous materials.

Connecting these two theories into one coherent framework is, however, far from being trivial, mainly due to the fact that the continuum damage model needs to be non-local to provide an objective description of localised failure patterns. This renders the majority of composite theories inadequate as they are based on the separation-of-scales hypothesis, which is an intrinsically local argument. Therefore, the project focused on the development of a consistent theoretical framework allowing a rigorous derivation of damage theories from a microstructural model. Due to the problem complexity in a general setting, the stochastic discrete lattice models were selected as the point of departure.

Lattice models of fracture represent a heterogeneous material as an assembly of discrete units interacting via brittle elements. In the traditional setting, the effective response of such systems is assessed using the "crude" Monte-Carlo simulations. By randomising the individual links' properties and performing repeated analyses, the models are able to simulate very complex phenomena using few parameters. Such direct simulations, however, do not provide any insight into the governing equations of the collective behaviour, such as the non-local interaction between fractured elements.

In the adopted global energetic philosophy, based on recent advances in the analysis of rate independent processes developed by Mielke and co-workers, the evolution of a discrete system is understood as a competition between a globally stored elastic energy and a dissipation distance, quantifying the energy lost in the irreversible changes in the material's structure. The first quantity is assumed to depend on a vector of kinematically admissible displacements and a vector of internal variables, while the dissipation distance depends solely on the internal variables.

For the deterministic case, all quantities can be defined on a local basis. In particular, individual links are assumed to behave in an elastic-ideally brittle manner and the fracture distribution is described in a binary fashion. The change of an internal variable occurs when the energy stored in a unit exceeds a certain threshold value. Uncertainty is incorporated into the model by randomising the thresholds or, equivalently, by the randomisation of discrete fracture states. The generalised internal variables are then interpreted as the probability of simultaneous failure of two elements.

Next, individual components of the model need to be generalised as well. In particular, a non-local expression for the generalised stored energy can be derived by slightly improving the recently developed methods of theory of random composite materials. The derivation of the generalised dissipation distance follows from the classical procedures of the theory of reliability.

After completing the last step, the global energetic formalism leads to an incremental convex programming problem, solvable by tools of mathematical programming. In overall, the outcome of the project provides a unique framework for understanding the multi-scale origins of damage and fracture mechanisms, a problem which remains on the forefront of engineering science.

Connecting these two theories into one coherent framework is, however, far from being trivial, mainly due to the fact that the continuum damage model needs to be non-local to provide an objective description of localised failure patterns. This renders the majority of composite theories inadequate as they are based on the separation-of-scales hypothesis, which is an intrinsically local argument. Therefore, the project focused on the development of a consistent theoretical framework allowing a rigorous derivation of damage theories from a microstructural model. Due to the problem complexity in a general setting, the stochastic discrete lattice models were selected as the point of departure.

Lattice models of fracture represent a heterogeneous material as an assembly of discrete units interacting via brittle elements. In the traditional setting, the effective response of such systems is assessed using the "crude" Monte-Carlo simulations. By randomising the individual links' properties and performing repeated analyses, the models are able to simulate very complex phenomena using few parameters. Such direct simulations, however, do not provide any insight into the governing equations of the collective behaviour, such as the non-local interaction between fractured elements.

In the adopted global energetic philosophy, based on recent advances in the analysis of rate independent processes developed by Mielke and co-workers, the evolution of a discrete system is understood as a competition between a globally stored elastic energy and a dissipation distance, quantifying the energy lost in the irreversible changes in the material's structure. The first quantity is assumed to depend on a vector of kinematically admissible displacements and a vector of internal variables, while the dissipation distance depends solely on the internal variables.

For the deterministic case, all quantities can be defined on a local basis. In particular, individual links are assumed to behave in an elastic-ideally brittle manner and the fracture distribution is described in a binary fashion. The change of an internal variable occurs when the energy stored in a unit exceeds a certain threshold value. Uncertainty is incorporated into the model by randomising the thresholds or, equivalently, by the randomisation of discrete fracture states. The generalised internal variables are then interpreted as the probability of simultaneous failure of two elements.

Next, individual components of the model need to be generalised as well. In particular, a non-local expression for the generalised stored energy can be derived by slightly improving the recently developed methods of theory of random composite materials. The derivation of the generalised dissipation distance follows from the classical procedures of the theory of reliability.

After completing the last step, the global energetic formalism leads to an incremental convex programming problem, solvable by tools of mathematical programming. In overall, the outcome of the project provides a unique framework for understanding the multi-scale origins of damage and fracture mechanisms, a problem which remains on the forefront of engineering science.