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Quasi-static analysis on void growth
The behavior of porous materials under quasi-static loading has been analyzed by several authors, since early works of McClintock , Rice and Tracey , Gurson , Budiansky et al. , Tvergaard  and Needleman and Tvergaard .
McClintock  investigated the growth of cylindrical and elliptical pores in an infinite viscous material. The author developed an approximate closed-form analytical expression for the void radius evolution and fracture criterion. Rice and Tracey  investigated the stage of void growth for mostly spherical void in an infinite matrix under general stress conditions. A variational principle has been proposed where the matrix is rigid-plastic (perfectly plastic or strain hardening). Results of McClintock  were retrieved for cylindrical voids when adopting the variational principle. Both analyses of McClintock  and Rice and Tracey  have revealed the exponential amplification of void growth by stress triaxiality. Budiansky et al.  have considered a viscous power law constitutive relationship for the matrix behavior, to identify the influence of the stress-triaxiality on the void radius growth. McClintock , Rice and Tracey , Budiansky et al.  were all considering dilute voids. In an engineering material, as observed in Fig. 2.1 to Fig. 2.6, voids are not isolated in the matrix phase, so the volume fraction of voids has to be integrated in the modelling. For that purpose, using a micro-mechanical approach, Gurson  derived an approximate yield criterion based on a trial velocity for porous materials with spherical and cylindrical voids under static conditions.
The GTN ( Tvergaard  and Needleman and Tvergaard ) model is extending the Gurson model, which has been widely used for quasi-static investigation of fracture process in finite element calculations. Gologanu et al.  proposed a yield surface for porous material containing prolate voids, known as GLD model. The GLD yield surface can be viewed as an extension of the Gurson  model. A brief explanation of the quasi-static investigation from some of the previous works are pointed out in next pages.
McClintock  developed a criterion for the fracture of cylindrical pores under prescribed stress at infinity. The criterion was developed for perfectly plastic, linear viscous materials and also extended to strain hardening material by extrapolation of the analysis. McClintock  observed that fracture takes place for very small porosity for both plastic and viscous flow with very strong inverse dependence of fracture strain on hydrostatic tension. McClintock  considered a porous medium subjected to stress loading at infinity. Calculations are done Figure 2.7: Cylindrical unit cell with inner radius R subjected to axial strain rate _~ »z and radial strain rate _~ »r. The resulting strain rates are due to imposed stress at infinity, McClintock . in the principal direction frames. McClintock  investigated the case of generalized plane strain with a hole in an infinite medium. The coalescence stage (and the associated fracture) is triggered when the radius of the hole reaches the mean spacing between voids. The problem is solved adopting cylindrical coordinates r; ; z (Fig. 2.7). From the conservation of linear momentum in quasi-static conditions one obtains: @r @r+ r r = 0 (2.1).
Gurson  developed an approximate plastic constitutive theory for porous materials. An approximate velocity field is adopted, similar to Rice and Tracey  in such a way that the velocity field satisfies the kinematic boundary conditions on the external surface of the unit cell. Homogenous kinematic boundary conditions are considered: v = D x (2.8).
where D is the macroscopic strain rate tensor. The material is assumed incompressible and the volume change of the RVE is only due to void growth. The approximate yield criterion is initially proposed for rigid perfectly plastic materials. Gurson studied the case of a matrix material containing both long circular cylindrical voids (Fig. 2.8) and spherical voids (Fig. 2.9).
Gurson Tvergaard and Needleman model (GTN model)
Needleman and Tvergaard  have modified the Gurson model by comparing the prediction of the model with FE cylindrical unit cell containing a spherical void. Needleman and Tvergaard  introduce three parameters so that the approximate yield function takes the form: = eqv 0 2 + 2fq1 cosh q2 2 nn 0 1 q3f2 (2.15) where q1, q2 and q3 are parameters that adopt various forms in the literature. Needleman and Tvergaard  proposed the following constant values q1=1.5, q2=1, q3=q2 1 while other authors showed that the q-parameters are depending on triaxiality (Vadillo and Fernández-Sáez ). When q1=q2=q3=1, then Eq (2.15) takes the form of Gurson model Eq (2.13). From the associated flow rule, the macroscopic strain rate is perpendicular to the yield surface: D = _ @ @ (2.16).
Molinari and Mercier Model-2001
Molinari and Mercier  investigated the dynamic behavior of porous material with spherical voids by means of a micro-mechanical approach. From the principle of virtual work, the macroscopic stress was defined which accounts for the acceleration contribution. The explicit relationship was derived starting from the conservation of linear momentum in dynamic condition: div = (2.25). where is the mass density, is the acceleration field and is the Cauchy stress tensor. A homogenous kinematic boundary condition is applied on the external boundary of the RVE, so that v = D x (2.26).
Geometry of the RVE and formulation of the velocity field
The representative volume element (RVE) of the porous material corresponds to a hollow cylinder (circular cross section) of current internal radius a, external radius b, length 2l, see Fig. 3.1. The current porosity is defined as: f = a2 b2 (3.1).
Table of contents :
1 Résumé des travaux de thèse en Français
1.1 Contexte et objectifs de la thèse
1.2 Modèle analytique
1.2.1 Volume Elémentaire Représentatif
1.2.2 Contributions statique et dynamique du tenseur des contraintes
1.4 Conclusion et perspectives
2 Literature review
2.2 Experimental works
2.3.1 Quasi-static analysis on void growth
188.8.131.52 McClintock Model
184.108.40.206 Gurson model
220.127.116.11 Gurson Tvergaard and Needleman model (GTN model)
2.3.2 Dynamic analysis for porous materials
18.104.22.168 Carroll and Holt Model
22.214.171.124 Molinari and Mercier Model-2001
126.96.36.199 Leblond and Roy Model
188.8.131.52 Molinari et. al Model
3.1 Geometry of the RVE and formulation of the velocity field
3.1.1 Formulation of the admissible velocity field
3.2 Formulation of the micro-inertia dependent term dyn
3.2.1 Plane strain case
3.2.2 Uniaxial deformation
3.3 Quasistatic stress tensor static
4.2 Axisymmetric loading
4.3 Plane strain
4.4 Hydrostatic loading
4.5 Additional loading cases
5 Finite element modeling
5.2 Plane strain configuration
5.3 Hydrostatic loading
5.3.1 Boundary conditions inherited from the analytical model
5.3.2 Closed unit-cell
5.3.3 Validation on the reference case
5.3.4 Thin cylinder, l0=10m
5.4 Additional loading cases
5.4.1 Imposed axial strain rate (D33=constant) with combined stress imposed on the lateral surface
5.4.2 Uniaxial loading: Case 1, =1/3
5.4.3 Biaxial loading: Case 2, =2/3
5.5 Influence of the elastic properties
6 Conclusion and perspectives
A Formulation of the macroscopic dynamic stress tensor, dyn.
A.1 General formulation
A.2 Case where = 0 3
A.3 Axisymmetric case
B Analytical relationships for the quasi-static macroscopic stress, static.
C Formulation of the macroscopic dynamic stress tensor (dyn) for 2D approach.