Soutenance mémoire
Why a regularisation?
The study of the solutions of an evolution problem, in the case of imposed displacements as well as imposed forces, has shown that these latter are all unstable [52], [18]. Moreover, local damage models exhibit strong dependency of the numerical solutions to the mesh that can lead to infinite localisation of the damage. Since the width of the damage band is controlled by the mesh element size, if the mesh is innitely rened, the damage zone surface tends to be null, and the dissipated energy tends to zero. We see clearly that breaking a structure without any cost of energy is not physically acceptable, as is the fact that the localisation of damage be a process driven by the mesh size. To remedy these problems, Francfort and Marigo introduced a regularisation of the damage solution. The idea consists in penalising the localisation of the damage so that if localises on a very thin band, it has to cost it a lot of energy. Practically speaking, a term containing a spatial gradient of damage is introduced in the energy density, so that the damage at one point of the structure depends also of the damage on the points surrounding it, and the damage model becomes a non local damage model. The previous approach where we developed the construction of the local (or non regularised) models will now reveal itself very powerful for the construction of non local models. Indeed, the evolution problem and the stability of the states will be formulated in the same way, the only dierence lying in the expression of the energy. We will use the three previously established principles of irreversibility, stability and energy balance to obtain the new evolution problem.
Project background and organization of the work
In the past years, gradient damage models or phase field models, have been developed in many frameworks, and coupled with many phenomenons. They were coupled with temperature, which lead to the simulations of thermal shocks seen in [11], or with plasticity to model ductile fracture in [1] and more recently in [58]. Usually studied with a quasi-static loading, they are now also adapted to dynamic fragmentation, by calculating the displacement with the elastodynamics equation, see for example [41]. Recent work of [2] show that they are also relevant to model fatigue, provided that a dissipation potential which explicitly depends on the strain history be introduced, that enables the dissipated energy to decrease when the accumulated strain increases. Although gradient damage models have been widely studied in many contexts in small deformation, their study has never been extended to large deformation. Indeed, the modelling of crack nucleation and propagation for rubbery material is not a subject that has been widely investigated in the past years. Yet, it is essential to understand these mechanisms because many materials of such a nature undergo loadings and solicitations that can lead to fracture. This is the case for the industry of tires that would greatly benefit from a better understanding of these phenomena. Using the J-integral method, industrials are able to predict for what load a pre-existing crack will propagate, but for the most part, they are still unable to predict the initiation of damage in a sound structure, i.e. where and when the damage will begin. This work has therefore emerged from such a context, and aims at having a better understanding of the initiation of rupture in large deformation, while adapting the gradient damage models to the framework of large deformation. In that perspective, the results are decomposed into four parts. In Chapter II, analytical solutions for three problems are constructed. Because of the viscous eects in polymers, two rheological models are investigated, in small and large deformation. This leads to the study of damaging viscoelastic materials, of hyperelastic materials and hyper-viscoelastic materials. Chapter III exposes the implementation of the gradient damage models in large deformation in an academic nite element code [16]: a recall is made of the method used to solve the displacement problem in finite strain, and numerical results are shown. The damage laws used in this chapter are such that the hydrostatic pressure does not play any role in the initiation of damage for an incompressible material. Therefore, Chapter IV presents a new damage law that overcomes this weakness. Both analytical and numerical results are exposed. Finally, Chapter V introduces the cavitation phenomenon, a famous damage process in polymers. It is first investigated as a purely hyperelastic bifurcation, then the competition that takes place between the cavitation and the damage is studied.
Rheological models and energy density
The connection between stress and strain for a viscoelastic material can be set through the use of rheological models. In the following, we will focus on the Maxwell and the PoyntingThomson model, shown respectively on Figures II.12a and II.10b. The Maxwell model is the association in series of an elastic spring and a viscous dashpot, which, in the Poynting model, is replaced by a Kelvin-Voigt element. Note that in small deformation, Zener and Poynting-Thomson have similar responses, and thus are equivalent.
Rupture energy
The rupture energy is the integral over the damage band of the dissipated energy when a crack has appeared. For the Maxwell model, the damage width band 2d is the same as for linear elastic material. Moreover, when a crack has developed in the bar, there remains only the dissipated energy, and the viscous energy as well as the elastic energy do not play any role. Consequently, the rupture energy Gc is the same for a Maxwell material or a linear elastic material. For the Poynting-Thomson model, we have seen previously that the damage width band takes values that are always superior to the values of the Maxwell model. Since the rupture energy is the integral over the damaged region of the damage dissipated energy, it can only be greater or equal to the dissipated energy obtained with the Maxwell model.
Convergence in the alternate minimisation
One of the difficulties that may arise during the computation of a test, is that the convergence in the alternate minimisation is too fast, i.e. that the damage grows and reaches its maximum value too quickly. This triggers convergence problems of the displacement problem, because it is as if a substantial loading was imposed all of a sudden on the structure. In this situation, the non linear solver has troubles converging to the solution. To prevent a growing of the damage that is too fast, we used mainly two techniques. The first consists in calculating the difference in two successive maximum values of α in the alternate minimisation: if this dierence is larger than a given threshold, the damage vector is replaced by a combination of its previous value weighted by a chosen parameter. The second way to reduce the convergence of the damage problem is to multiply the Hessian matrix (which is a constant) by a scalar so that the convergence is slowed down. An example is given in Figure III.6 where a comparison is made on two plates submitted to traction with the same material and numerical parameters, except that in one case, the Hessian matrix is multiplied by 2. Figure III.6a shows that if the Hessian matrix is untouched, the damage reaches a value close to 1 very quickly, and the calculation (depending on the tolerance) can be made in 20 iterations. If the Hessian matrix is multiplied by 2, the convergence is smoother, and it takes more iterations to converge on α, which is exactly the aim of this method. Figure III.6b illustrates these explanations: at a given time step of the alternate minimisation (here, the 20th), the damage prole found with an untouched Hessian matrix is much more advanced (in terms of final result) than the one found if the Hessian is multiplied by 2. In the end of the alternate minimisation, the damage proles are quasi identical. We could argue that since more iterations are required to converge in the alternate minimisation when the Hessian is multiplied, this is in fact a loss of time. Yet, since the displacement problem converges much more easily when the damage does not grow too quickly, there is no loss of time, and it even accelerates the convergence. In this particular case, the calculations with the natural Hessian took 168 seconds while the calculations with a doubled Hessian took 164 seconds.
Test inducing bi-axiality
Geometry After validation of the mixed implementation with damage, tests that would enhance the role of the pressure had to be performed. We had to nd simple geometries such that the loading would trigger pressure development, as well as deviatoric stresses, but not in the same zones of the sample. We chose to work on the geometry displayed on Figure IV.16: it is made of a rectangle of length L and height H, minus two half circles of diameter D = H. On the green boundary, a null displacement on the two components is imposed, while a growing x displacement is imposed on the red opposite boundary. The top and bottom edges are free of stress. Dirichlet boundary conditions of null damage are imposed on the green and red boundaries. With such a geometry, zones of high pressure and high deviatoric stresses appear in different zones, as can be observed on Figure IV.17. The pressure is very high near the poles of the half-circles and has a conic shape, while the deviatoric part of the stress is high in the middle and on four borders, which could be seen as an x-shape. The boundary conditions on α are such that there is no damage on the boundaries, and consequently we can expect that when the damage should initiate in zones of high deviatoric stresses, it would then develops in the middle of the sample, and not and the edges. This choice is based on the fact that we want to observe damage initiation and rupture in the bulk, and not on the boundaries, that would rather correspond to delamination.
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Table des matières
I Introduction
I.1 History of damage models
I.2 Construction of damage gradient models
I.2.1 Non regularised models
I.2.1.a Modelling ingredients
I.2.1.b Evolution law of α
I.2.1.c Softening and hardening behaviours
I.2.1.d Energy construction and stability
I.2.1.e Variational formulation of the evolution problem
I.2.2 Regularised model
I.2.2.a Why a regularisation?
I.2.2.b Irreversibility, stability, energy balance
I.2.2.c Localisation on a 1D bar
I.2.2.d Strength of the model
I.2.3 Numerical implementation
I.3 Notations
I.4 Project background and organization of the work
II Unidimensional study of gradient damage models for viscoelastic materials
II.1 Damage and viscoelasticity
II.1.1 Energy density, first order stability and energy balance in 3D
II.1.1.a Rheological models and energy density
II.1.1.b First order stability
II.1.1.c Energy balance
II.1.2 Homogeneous damage
II.1.2.a Viscoelastic phase
II.1.2.b Damaging phase
II.1.3 Damage localisation
II.1.3.a Damage profiles
II.1.3.b Rupture energy
II.1.4 Numerical implementation
II.2 Damage and hyperelasticity
II.2.1 Unidimensional hyperelastic potential
II.2.2 Homogeneous damage evolution in a 1D hyperelastic bar
II.2.3 Localised damage evolution in a 1D hyperelastic bar
II.3 Damage and hyper-viscoelasticity
II.3.1 Maxwell model
II.3.1.a Energy
II.3.1.b First order stability
II.3.1.c Hyper-viscoelastic phase
II.3.1.d Homogeneous damage
II.3.1.e Damage localisation
II.3.1.f Numerical implementation
II.3.2 Zener model
II.3.3 Conclusion
III Numerical study of damage gradient models for large deformation
III.1 Numerical implementation
III.1.1 Recall: displacement problem in large deformation
III.1.1.a Choice of the hyperelastic law
III.1.1.b Variational formulations
III.1.2 Damage problem
III.1.3 Remark: implementation with the FEniCS library
III.2 2D simulations
III.2.1 General considerations
III.2.2 Particular case of a 2D plate under uni-axial traction
III.3 3D simulations
III.4 Conclusion
IV Damage initiation in zones of high hydrostatic pressure
IV.1 Motivations
IV.1.1 Take micro defects into account
IV.1.2 Properties of the desired model
IV.2 Inextensible 1D bar
IV.2.1 Analytical study
IV.2.1.a Homogeneous response
IV.2.1.b Localised response
IV.2.1.c Mixed response
IV.2.1.d Energies
IV.2.2 Numerical implementation
IV.3 Extension to higher dimensions
IV.3.1 Numerical implementation
IV.3.2 2D plate under uni-axial traction
IV.3.2.a Analytical developments
IV.3.2.b Numerical application
IV.3.3 Test inducing bi-axiality
IV.3.3.a Geometry
IV.3.3.b Results
IV.4 Comparison with experiments
IV.5 Large deformation
IV.6 Conclusion
V Cavitation phenomenon
V.1 Motivations and state of the art
V.2 Hyperelastic bifurcation
V.2.1 Equilibrium equation
V.2.2 Stability of the equilibrium equation
V.2.2.a Second derivative of the total energy
V.2.2.b Stability of the solution f0 for Ri = 0
V.3 Damage and cavitation
V.3.1 Comparison of critical loads
V.3.2 2D numerical tests
V.4 Conclusion
Conclusion
Bibliography
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