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New ideas for the Variational Approach to Brittle and Cohesive Fracture

Periodic Reporting for period 1 - BriCoFra (New ideas for the Variational Approach to Brittle and Cohesive Fracture)

Reporting period: 2018-10-01 to 2020-09-30

During World War I the aeronautical engineer A.A. Griffith formulated a theory to explain failure of materials, based on the idea that crack growth is the result of the competition between surface energy spent to produce fracture and energy stored in the uncracked region.
This found a rigorous mathematical setting in the 'variational formulation for quasistatic evolutions (QSE)' by Francfort-Marigo. They devised, in 1998, a general procedure trough an implicit time-sheme based on energy minimisation, which calls for new techniques in Calculus of Variations.
Remarkably, it provided a general framework for also post-Griffith theories, in particular the one due to Barenblatt.
Griffith and Barenblatt theories differ in the surface energy dissipated in the fracture: the first is proportional to the measure of the fracture set (surface in 3d, length in 2d), the latter depends also on the amplitude of the opening between the two sides of the crack. Microscopically, in the first case (brittle fracture) any material point is either broken or sound, in the latter (cohesive fracture) restorative forces, depending on the opening, are present between the lips of the crack set. In fact, in formation of cohesive cracks the role of plastic deformations is determinant: these deformations are dissipative (and irreversible), so that the material is not forced to dissipate energy only along cracks and the fracture process is slowed down; in contrast, in brittle fracture the solid is subject only to elastic deformations (reversible) in the uncracked zone, which causes sudden cracks and fast propagation.

The application in Numerical Analysis relies on the approximation of fracture energies in terms of so-called damage energies. The minimisation of fracture energies is a classical 'free discontinuity problem', namely it contains a set variable (the crack) as unknown, which makes the numerical simulation hard. In damage models the role of fracture is played by a function variable, supported on a thin 3d-neighborhood of the crack surface, and the resulting energy is more regular in the damage variable.
Brittle fracture is approximated by coupling of damage and elasticity, while coupling of damage and elasto-plasticity approximates cohesive fracture (cf. the mechanical interpretation above), in terms of Gamma-convergence, guaranteeing convergence of minima and minimisers of the associated (static) problem. Such approximations, called 'à la Ambrosio-Tortorelli', are nowadays employed in thousands of papers modelling fracture. (In figure, approximation of discontinuity through a smooth profile, restricted to 1d)

The overall objective of the project is to progress the mathematical analysis of brittle/cohesive fracture, in synergy with that of the associated damage models.
The fellow (VC below) studied with A.Chambolle the minimisation of the Griffith energy for brittle fracture. This passes trough a 'weak formulation' of the problem, following a general approach to free discontinuity problems devised by De Giorgi: a suitable energy space of vector fields with surface discontinuities is introduced, consisting of fields whose discontinuity set (corresponding to the crack) is a countable union of surfaces with continuous normal vector and which are 'regular enough' outside the discontinuity set. In fact, the symmetric part of the gradient is a square-integrable function outside the crack set. Assuming the crack set regular enough, the vector field would be Sobolev in the uncracked zone, which results by Korn's inequality controlling the full gradient in terms of its symmetric part in Lipschitz domains. However, the discontinuity set may be very irregular, not even topologically closed.

In some sense this completed a long path of works that progressively reduced the 'a priori' assumptions on the problem, first on the regularity of the discontinuity set and then on the integrability of the vector field. A compactness and semicontinuity theorem allowing to solve the Dirichlet minimum problem for the Griffith energy in full generality is obtained, albeit only in the weak formulation. Such result also completes the Ambrosio-Tortorelli type approximation for the Griffith energy, proved with Chambolle too.
Then the 'strong formulation' has been addressed, consisting to minimise the Griffith energy in the class of fields with continuous gradient outside a finite union of surfaces with continuous normal vector (called 'good discontinuous functions' in the following). The strategy is to prove regularity for 'weak minimisers'.

The techniques developed have been exploited in many directions.
First, considering fracture energies with a brittle behaviour in opening and a cohesive one in the propagation of the crack. VC proved a density result of 'good discontinuous functions' with respect to the considered mixed brittle/cohesive energy. This is a careful refinement of our work on the approximation for the Griffith energy. Then the approximation for these mixed energies was studied, employing also the density result.

Moreover, with Friedrich the works on brittle fracture have been refined in a different way, in order to prove existence of equilibria for thin films subject to 'epitaxial growth'. Such films are crystalline materials deposited on a rigid substrate crystal with a slight mismatch in the crystalline structure. The process is usual for instance in additive manufacturing for electronic devices.
The mismatch in the two atomic structures causes an elastic strain in the film. The corresponding mathematical model is then based on the competition between the elastic energy of the film and the surface tension localised in its free surfaces with respect to both substrate and air.

Further, with Solombrino and collaborators, VC developed a discrete-to-continuous approximation for the Griffith energy, and an 'integral representation' for functionals bounded in terms of the Griffith energy, guaranteeing these are themselves of fracture type.

Eventually, VC generalised with Chambolle the compactness result to include non homogeneous brittle fracture energy, characterised by elastic properties and fracture toughness varying pointwise.

As for cohesive fracture, the analysis of free discontinuity problems turned out to be too hard. Nevertheless, the understanding of elasto-plastic damage models (or damage models with fatigue) was progressed in three works, together with R.Rossi Orlando, and Alessi-Orlando.

The Action gave the chance to present these results in several workshop and conferences, among which those at BIRS Station of Banff (Canada), at RAM3 Meeting in Sapienza University in Rome, at COMPLAS in Barcelona, with the participation/organisation of engineers, and invited talks at important mathematical meetings such as the Italian National Congress of Calculus of Variations.
VC could fruitfully collaborate with many people in all Europe, which has been crucial to extend his network, obtain results, and improve his experience, knowledge, visibility.
In conclusion, the fellow believes that the work in BriCoFra contributed to enrich the big picture in the variational approach to fracture (tried to be summarised at a glance in a diagram). The tools developed look promising in order to continue the general analysis.
Many engineers have been made aware of the progresses and future collaborations seem possible, hoping this could bring to concrete applications in society.
Cartoon of Ambrosio-Tortorelli approximation
Summarizing diagram