Study of the Fractures in Slowly Driven Dominated Threshold Systems

Fracture mechanics plays an important role in the material science, structure design and industrial production due to the failure of materials and structures are paid high attention in human activities. For this reason, the fracture mechanics can be considered today one of the most important research fields in engineering. The attempts to predict the failure of a material are able to link different disciplines: in this dissertation, a very deep use of the statistical physics will be done in order to try to introduce the disorder of the medium into the breaking and to a give a new point of view to the fracture mechanics. In the following, we will introduce a new kind of model to evaluate the genesis of the crack: the statistical central force model. As we will see, this model tries to compute the genesis of the fracture in a medium by taking into account the presence of defects of the material that are the main cause of the differences between the critical theoretical strength of a material and the real one. This innovation introduced by this model which is difficult to find in other kinds of techniques existing today united to the fact that we try to predict the behaviour of a macro system by knowing exactly the statistical behaviour of the micro-components of the system itself (the trusses) like in complex systems happens, is the main innovation of the statistical central force model. The model consists of a truss structure in which each truss is representative of a little portion of the material. Since this model was already applied in static for a porous medium in literature, we will study it from a mathematical viewpoint and we will apply it to the study of the dynamic of a dry medium before (the applications could be for the study of the fracture in metals and composites with loads changing in time) and of a porous medium later (in order to study the fracking into soils and the fracture of the concrete). Further developments could bring us to develop the same method for the study of the spalling in the concrete because of the application of a thermal load. In the dissertation we will introduce the mathematical tools to understand this model and some simulation on generic media will be realized. This dissertation consists basically of five chapters. In chapter 1 a brief description of the state of the art will be given: we will leave from the birth of the classical elastic fracture mechanics and we will shortly talk about the fracture mechanics in a plastic field. After this we will describe two important techniques used today for the evaluation of the crack: the XFem and the Peridynamics; the first one is a numerical technique allowing the FEM to take into account the possibility to create a breaking into the material. This is done, as we will see, by adding further degrees of freedom to the finite elements. In this way a single finite element will have the possibility to “open” itself and to simulate a discontinuous field of displacements, which is the main problem concerning with FEM in calculating the fracture. The second one is a theory that postulates that each medium can be divided in particles and that each particle interacts with its own neighbours within a given horizon. From which we get the word “peri”. By this assumption it is possible to get some integro-equations that can be defined on the surfaces of the tips and of the cracks as well. In chapter 2 we will talk about the so called Fiber Bundle Model which is the basis of our statistical model. We will talk about the dry FBM that was already studied at the beginning of ’90s from a mathematical viewpoint : it consists of a bundle of fibers clamped at one edge and free to move to the other one. The model is one dimensional and it is probably one of the most naïve models to begin to study the fracture; however, despite to its simplicity, it contains an important tool: the possibility to take into account the defects of the medium by introducing the concept of variable thresholds in stress. As we will see, these thresholds will be picked up by a probability density function. Then we will apply the theory of the statistical ensembles to study one of the extensions of the FBM: the continuous fiber bundle model. This is necessary to have an idea of how the micro-components of our model, the trusses, behave in a truss structure subject to an external load. In chapter 3 we will report briefly the theory of the porous medium according to the mixture theories of De Boer. So an overview about the equations will be given and then we will discretize these equations according to the finite element technique. After this, we will briefly describe in which part of the algorithm the concept of imperfection/threshold in stress enters. We will do this for a dry medium and for a porous medium in dynamics. In chapter 4 we will report the numerical results. Some simulations in dynamics will be done both for a dry medium and for a porous medium. Furthermore we will introduce in the end a new damage law that will have a precise statistical meaning: it will be the average among all the possible realizations of the constitutive laws of our truss structure and for a big number of trusses, it will become the constitutive behaviour of our structure from which to get the damage law. And this result will take into account the disorder of the medium. In chapter 5 we will talk about a controversial argument: the Self Organized Criticality (SOC) that was sticked in previous papers to the statistical central model. We will try to understand what SOC is and if our system with our algorithm to compute the fracture gets the necessary and sufficient conditions to enter into the set of the SOC systems. At the end of our journey we will have hopefully done a first step into the description of a new numerical tool to evaluate the crack into a generic medium without needing an initial discontinuity to develop the crack itself. The next steps will be to validate this technique for existing materials and to compare it to other numerical tools like XFem or Peridynamics. After this, the future will be to extend the technique passing from trusses to 2D elements.