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 A wave equation including leptons and quarks for the standard model of quantum physics in Clifford AlgebraA wave equation with mass term is studied for all particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks $u$ and $d$ with three states of color and antiquarks $\overline{u}$ and $\overline{d}$. This wave equation is form invariant under the $Cl_3^*$ group generalizing the relativistic invariance. It is gauge invariant under the $U(1)\times SU(2) \times SU(3)$ group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra $Cl_{1,5}$. All features of the standard model, charge conjugation, color, left waves, Lagrangian formalism, are linked to the geometry of this extended space-time. Development and explicit integration of a thermo-mechanical model for saturated claysThis study is devoted to the thermo-mechanical constitutive modeling for saturated stiff clays and the development of a corresponding efficient stress integration algorithm. The mechanical behavior of natural Boom Clay in isothermal conditions was first characterized. The Modified Cam Clay model (MCC) was then applied to simulate the natural Boom Clay behavior. It has been found that the MCC gives poor-quality predictions of the natural Boom Clay behavior. Thereby, an adapted Cam Clay model (ACC-2) was developed by introducing a new yield surface and a new plastic potential as well as a Two-surface plastic mechanism. This model allows satisfactory prediction of the main features of the mechanical behavior of natural Boom Clay. Moreover, the constitutive equations of this model can be formulated mathematically as in a classic elasto-plastic model. Thus, the classic stress integration algorithm can be applied. The thermal effects were considered by assessing the performance of some advanced thermo-mechanical models (Cui et al., 2000; Abuel-Naga et al., 2007; Laloui and François, 2008; 2009). It appears that all the three models can capture the main features of the thermo-mechanical behavior of saturated clays. However, each constitutive model has its own limitations or unclear points from the theoretical point of view. The stress integration algorithm of the thermo-mechanical model proposed by Cui et al. (2000) at the stress point level was also developed using a specifically designed adaptive time-stepping scheme. The computation time required to achieve a given accuracy is largely reduced with the adaptive sub-stepping considered for both mechanical and thermal loadings. A Two-surface thermo-mechanical model (TEAM model) was developed based on the Two-surface plastic mechanism. The proposed model extends the model of Cui et al. (2000) to a Two-surface formulation, considering the plastic strain coupling between the thermal and the mechanical loading paths. The simulation of drained tests shows that this model can capture the main thermo-mechanical features of natural Boom Clay along different loading paths. The TEAM model was finally extended to undrained conditions. After setting up an appropriate effective stress principle and defining a volumetric strain condition, the undrained heating process was analyzed. The validity of the thermo-hydro-mechanical constitutive equations was examined based on the data from typical tests Quantization of soluble classical constrained systemsThe derivation of the brackets among coordinates and momenta for classical constrained systems is a necessary step toward their quantization. Here we present a new approach for the determination of the classical brackets which does neither require Dirac's formalism nor the symplectic method of Faddeev and Jackiw. This approach is based on the computation of the brackets between the constants of integration of the exact solutions of the equations of motion. From them all brackets of the dynamical variables of the system can be deduced in a straightforward way.
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